Quantum Sensing

C. L. Degen∗

Department of Physics, ETH Zurich,Otto Stern Weg 1, 8093 Zurich,Switzerland.

F. Reinhard†

Walter Schottky Institut and Physik-Department,Technische Universitat Munchen,Am Coulombwall 4, 85748 Garching,Germany.

P. Cappellaro‡

Research Laboratory of Electronics and Department of Nuclear Science & Engineering,Massachusetts Institute of Technology,77 Massachusetts Ave.,Cambridge MA 02139,USA.

(Dated: November 9, 2016)

“Quantum sensing” describes the use of a quantum system, quantum properties or quan-tum phenomena to perform a measurement of a physical quantity. Historical examplesof quantum sensors include magnetometers based on superconducting quantum interfer-ence devices and atomic vapors, or atomic clocks. More recently, quantum sensing hasbecome a distinct and rapidly growing branch of research within the area of quantumscience and technology, with the most common platforms being spin qubits, trapped ionsand flux qubits. The field is expected to provide new opportunities – especially withregard to high sensitivity – in applied physics and other areas of science. In this review,we provide an introduction to the basic principles, methods and concepts of quantumsensing from the viewpoint of the interested experimentalist.

CONTENTS

I. Introduction 2Content 2

II. Definitions 3A. Quantum sensing 3B. Quantum sensors 3

III. Examples of quantum sensors 4A. Neutral atoms as magnetic field sensors 4

1. Atomic vapors 42. Cold clouds 4

B. Trapped ions 5C. Rydberg atoms 6D. Atomic clocks 6E. Solid state spins – Ensemble sensors 6

1. NMR ensemble sensors 62. NV center ensembles 7

F. Solid state spins - Single spin sensors 7G. Superconducting circuits 8

1. SQUIDs 82. Superconducting qubits 8

H. Elementary particle qubits 81. Muons 8

∗ degenc@ethz.ch† friedemann@quantum-minigolf.org‡ pcappell@mit.edu

2. Neutrons 8I. Other sensors 9

1. Single electron transistors 92. Optomechanics 93. Photons 9

IV. The quantum sensing protocol 10A. Quantum sensor Hamiltonian 10

1. Internal Hamiltonian 102. Signal Hamiltonian 103. Control Hamiltonian 11

B. The protocol 11C. First example: Ramsey measurement 12D. Second example: Rabi measurement 13E. Slope and variance detection 13

1. Slope detection (linear detection) 132. Variance detection (quadratic detection) 13

V. Sensitivity 14A. Quantum projection noise 14B. Decoherence 14C. Initialization, manipulation and readout noise 14

1. Single shot readout 152. Averaged readout 153. Total readout uncertainty 16

D. Sensitivity and minimum detectable signal 161. Slope detection 162. Variance detection 16

E. Allan variance 17F. Quantum Cramer Rao Bound for parameter

estimation 17

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VI. Sensing of AC signals 18A. Time-dependent signals 18B. Ramsey sequence revisited 18C. Spin echo sequence 19D. Multipulse sensing sequences 19

1. Modulation function 192. CP and PDD sequences 193. Lock-in detection 204. Other types of multipulse sensing sequences 205. AC signals with random phase and/or random

amplitude 21E. Waveform reconstruction 21F. Correlation sequences 22

VII. Noise spectroscopy 22A. Noise processes 23B. Decoherence, dynamical decoupling and filter

functions 231. Decoherence function χ(t) 232. Filter function Y (ω) 233. Dynamical decoupling 24

C. Relaxometry 241. Basic theory of relaxometry 242. T1 relaxometry 253. T ∗2 and T2 relaxometry 264. T1ρ relaxometry 26

VIII. Dynamic range and adaptive sensing 27A. Phase estimation protocols 27

1. Quantum phase estimation 282. Adaptive phase estimation 293. Non-adaptive phase estimation 294. Comparison of phase estimation protocols 29

B. Experimental realizations 29

IX. Ensemble sensing 30A. Ensemble sensing 30B. Heisenberg limit 30C. Entangled states 30

1. GHZ an N00N states 302. Squeezing 313. Parity measurements 324. Other types of entanglement 32

X. Sensing assisted by auxiliary qubits 33A. Quantum logic clock 33B. Storage and retrieval 33C. Quantum error correction 33

XI. Outlook 34

References 35

I. INTRODUCTION

Can we find a promising real-world application ofquantum mechanics? This question has intrigued physi-cists ever since its development in the early twentiethcentury. Today, quantum computers (Deutsch, 1985; Di-Vincenzo, 2000) and quantum cryptography (Gisin et al.,2002) are widely believed to be the most promising ones.

Interestingly, however, this belief might turn out tobe incomplete. In recent years a different class of ap-plications has emerged that employs quantum mechan-ical systems as sensors for various physical quantitiesranging from magnetic and electric fields, to time and

frequency, to rotations, to temperature and pressure.“Quantum sensors” capitalize on the central weaknessof quantum systems – their strong sensitivity to externaldisturbances. This trend in quantum technology is curi-ously reminiscent of the history of semiconductors: here,too, sensors – for instance light meters based on seleniumphotocells (Weston, 1931) – have found commercial ap-plications decades before computers.

Although quantum sensing as a distinct field of re-search in quantum science and engineering is quite recent,many concepts are well-known in the physics communityand have resulted from decades of developments in high-resolution spectroscopy, especially in atomic physics andin nuclear magnetic resonance. Notable examples includeatomic clocks, atomic vapor magnetometers, and super-conducting quantum interference devices. What can beconsidered as “new” is that quantum systems are increas-ingly investigated at the “single atom” level, that entan-glement is used as a resource for increasing the sensitivity,and that quantum systems and quantum manipulationsare specifically designed for use as sensors.

The focus of this review is on the key concepts andmethods of quantum sensing, with particular attentionto practical aspects that emerge from non-ideal experi-ments. As “quantum sensors” mostly qubits will be con-sidered – quantum systems that can be described throughtwo quantum states. Although an overview over actualimplementations of qubits is given, the review will notcover any of those implementation in specific detail. Itwill also not cover related fields including atomic clocksor photon-based sensors. In addition, theory will onlybe considered up to the point necessary to introduce thekey concepts of quantum sensing. The motivation be-hind this review was to offer an introduction to studentsand researchers new to the field, and to provide a basicreference for researchers already active in the field.

Content

The review starts by suggesting some basic definitionsfor “quantum sensing” and by noting the elementary cri-teria for a quantum system to be useful as a quantumsensor (Section II). The next section provides an overviewof the most important physical implementations (SectionIII). The discussion then moves on to the core conceptsof quantum sensing, which include the basic measure-ment protocol (Section IV) and the sensitivity of a quan-tum sensor (Section V). Sections VI and VII cover theimportant area of time-dependent signals and quantumspectroscopy.

The remaining sections introduce some advanced quan-tum sensing techniques. These include adaptive meth-ods developed to greatly enhance the dynamic rangeof the sensor (Section VIII), and techniques that in-volve multiple qubits (Sections IX and X). In partic-

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ular, entanglement-enhanced sensing, quantum storageand quantum error correction schemes are discussed. Thereview then concludes with a brief outlook on possible fu-ture developments (Section XI).

There have already been several reviews that covereddifferent aspects of quantum sensing. An excellent intro-duction into the field is the review by (Budker and Roma-lis, 2007) on atomic vapor magnetometry and the paperby (Taylor et al., 2008) on magnetometry with nitrogen-vacancy centers in diamond. Entanglement-assisted sens-ing, often referred to as “quantum metrology” or “secondgeneration quantum sensors” are covered by (Bollingeret al., 1996), (Giovannetti et al., 2006) and (Giovannettiet al., 2011). In addition, many excellent reviews cov-ering different implementations of quantum sensors areavailable; these will be noted in Section III.

II. DEFINITIONS

A. Quantum sensing

“Quantum sensing” is typically used to describe one ofthe following:

I. Use of a quantum object to measure a physicalquantity (classical or quantum). The quantumobject is characterized by quantized energy lev-els. Specific examples include electronic, magneticor vibrational states of superconducting or spinqubits, neutral atoms, or trapped ions.

II. Use of quantum coherence (i.e., wave-like spatial ortemporal superposition states) to measure a phys-ical quantity.

III. Use of quantum entanglement to improve the sen-sitivity of a measurement, beyond what is possibleclassically.

Of these three definitions, the first two are rather broadand cover many physical systems. This even includessome systems that are not strictly “quantum’. An exam-ple is classical wave interference as it appears in optical ormechanical systems (Faust et al., 2013; Novotny, 2010).However, since quantum sensors according to definitions Iand II are often close to applications, we will mostly focuson these definitions and discuss them extensively in thisreview. While these types of sensors might not exploitthe full power of quantum mechanics, as for type-III sen-sors, they already can provide several advantages, mostnotably operation at nano-scales that are not accessibleto classical sensors.

The third definition is more stringent and a truly“quantum” definition. Because the definition relies onentanglement, more than one sensing qubit is requiredto form a type III quantum sensor. A well-known ex-ample is the use of maximally entangled states to reach

FIG. 1 Basic features of a two-state quantum system. |0〉is the lower energy state and |1〉 is the higher energy state.Quantum sensing exploits changes in the transition frequencyω0 or the transition rate Γ in response to an external signalV .

a Heisenberg-limited measurement. Type III quantumsensors are discussed in Section X.

B. Quantum sensors

In analogy to the diVincenzo criteria for quantum com-putation (DiVincenzo, 2000), a set of necessary attributescan be listed for a quantum system to function as a quan-tum sensor. These attributes include three original di-Vincenzo criteria plus a fourth requirement that is spe-cific to quantum sensing:

(V1) The quantum system has discrete, resolvable en-ergy levels. Specifically, we will assume it to be atwo-level system (or an ensemble of two-level sys-tems) with a lower energy state |0〉 and an upperenergy state |1〉 that are separated by a transitionenergy E = ~ω0. 1 (see Fig. 1).

(V2) It must be possible to initialize the quantum systeminto a well-known state and to read out its state.

(V3) The quantum system can be coherently manip-ulated, typically by time-dependent fields. Thiscondition is not strictly required for all proto-cols; examples that fall outside of this criterionare continuous-wave spectroscopy or relaxation ratemeasurements.

The fourth attribute is specific to quantum sensing, andrequires that:

(V4) The quantum system interacts with some relevantphysical quantity V (t), like an electric or mag-netic field. This interaction could be both a field-dependent shift of the quantum system’s energylevels such as a Stark or Zeeman shift, or the driv-ing of transitions between energy levels by a time-dependent field (see Fig. 1). The interaction is

1 Note that this review uses ~ = 1 and expresses all energies inunits of angular frequency.

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quantified by a coupling or transduction parameterof the form γ = ∂E/∂V which relates changes inthe transition energy E to changes in the externalparameter V .

Apart from the above four criteria, experimental imple-mentations of quantum sensors will have two additionalimportant characteristics. One of them is to what kind ofexternal parameter(s) the quantum sensor responds to.Charged systems, like trapped ions, will be immenselysensitive to electrical fields, while spin-based systems willmainly respond to magnetic fields. Some quantum sen-sors may respond to several physical parameters.

A second key characteristic of a quantum sensor is itsintrinsic sensitivity. The “quality” of a quantum sen-sor is determined by a compromise between maximuminteraction strength with wanted signals, and maximumsuppression of unwanted noise. Later in this review (Sec-tion V) we will see that the intrinsic sensitivity can beroughly quantified by the figure of merit

sensitivity ∝ γ√Tχ , (1)

where γ is the above coupling parameter and Tχ is arelaxation or decoherence time that reflects the immunityof the quantum sensor against noise, and where a largefigure of merit corresponds to a high sensitivity.

III. EXAMPLES OF QUANTUM SENSORS

We now give an overview of the most important exper-imental implementations of quantum sensors, followingthe summary in Table I.

A. Neutral atoms as magnetic field sensors

Alkaline atoms are suitable sensing qubits fulfilling theabove definitions (Kitching et al., 2011). Their groundstate spin - a coupled angular momentum of electron andnuclear spin - can be both prepared and read out opti-cally by the strong spin-selective optical dipole transitionbetween the electronic ground and p-type excited state.

1. Atomic vapors

In the simplest implementation, a thermal vapor ofatoms serves as a quantum sensor for magnetic fields(Budker and Romalis, 2007; Kominis et al., 2003).Trapped in a cell at or above room temperature, atomsare spin-polarized by an optical pump beam. Magneticfield sensing is based on the Zeeman effect of a smallexternal field. This induces coherent precession of theatomic spin in a classical picture or, equivalently, transi-tions between spin sublevels in a quantum picture, which

are detected by an optical probe beam. Despite their su-perficial simplicity, these sensors achieve sensitivities inthe range of 100 aT/

√Hz (Dang et al., 2010) and ap-

proach a theory limit of < 10 aT/√

Hz, placing themon par with superconducting SQUIDs (see below) as themost sensitive magnetometers to date. This is owing tothe surprising fact that relaxation and coherence timesof spins in atomic vapors can be pushed to the second tominute range (Balabas et al., 2010). These long relax-ation and coherence times are achieved by coating cellwalls to preserve the atomic spin upon collisions, and byoperating in the spin exchange relaxation-free (“SERF”)regime of high atomic density and zero magnetic fieldwhere motional averaging drastically suppresses decoher-ence by atomic collisions (Happer and Tang, 1973). Va-por cells have been miniaturized to few mm3 small vol-umes (Shah et al., 2007) and have been used to demon-strate entanglement-enhanced sensing (Fernholz et al.,2008; Wasilewski et al., 2010). The most advanced appli-cations of vapor cells are the detection of neural activity(Jensen et al., 2016; Livanov et al., 1978) which has founduse in magnetoencephalography (Xia et al., 2006). Vaporcells also promise complementary access to high-energyphysics, detecting anomalous dipole moments from cou-pling to exotic elementary particles (Budker et al., 2014).

2. Cold clouds

The advent of laser cooling in the 1980’ies spawneda revolution in atomic sensing. The reduced velocityspread of cold atoms enabled sensing with longer interro-gation times using spatially confined atoms, freely fallingalong specific trajectories in vacuum or trapped.

Freely falling atoms have enabled the developmentof atomic gravimeters (Kasevich and Chu, 1992; Peterset al., 1999) and gyrometers (Gustavson et al., 1997,2000). In these devices an atomic cloud measures ac-celeration by sensing the spatial phase shift of a laserbeam along its freely falling trajectory.

Trapped atoms have been employed to detect and im-age magnetic fields at the microscale, by replicating Lar-mor precession spectroscopy on a trapped Bose-Einsteincondensate (Vengalattore et al., 2007) and by direct driv-ing of spin-flip transitions by microwave currents (Ock-eloen et al., 2013) or thermal radiofrequency noise insamples (Fortagh et al., 2002; Jones et al., 2003). Sens-ing with cold atoms has found application in solid statephysics by elucidating current transport in microscopicconductors (Aigner et al., 2008).

Arguably the most advanced demonstrations ofentanglement-enhanced quantum sensing (“DefinitionIII”) have been implemented in trapped cold atoms. En-tanglement – in the form of spin squeezing (Winelandet al., 1992) – has been produced by optical non-destructive measurements of atomic population (Appel

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Implementation Qubit(s) Measuredquantity(ies)

Typicalfrequency

Initalization Readout Typea

Neutral atomsAtomic vapor Ground state

spinMagnetic field,Rotation,Time/Frequency

DC-GHz Optical Optical II–III

Cold clouds Electron spins Magnetic field,Acceleration,Time/Frequency

DC-GHz Optical Optical II–III

Trapped ion(s)Long-lived Time/Frequency THz Optical optical IIelectronic state Rotation Optical Optical IIVibrational mode Electric field, Force MHz Optical Optical II

Rydberg atoms Rydberg states Electric field DC,GHz Optical Optical IISolid state spins (ensembles)

NMR sensors Nuclear spins Magnetic field DC Thermal Pick-up coil IINV center ensembles Electron spins Magnetic field,

Electric field,Temperature,Pressure, Rotation

DC-10 GHz Optical Optical II

Solid state spins (single spins)P donor in Si Electron spin Magnetic field DC-10GHz Thermal ElectricalSemiconductor quantumdots

Electron spin Magnetic field,Electric field

DC-10GHz Electrical, Optical Electrical, Optical I–II

Single NV center Electron spin Magnetic field,Electric field,Temperature,Pressure, Rotation

0-10 GHz Optical Optical II

Superconducting circuitsSQUID Supercurrent Magnetic field DC-GHz Thermal Electrical I–IIFlux qubit Circulating

currentsMagnetic field DC-GHz Thermal Electrical II

Charge qubit Chargeeigenstates

Electric field DC-GHz Thermal Electrical II

Elementary particlesMuon Muonic spin Magnetic field DC Radioactive decay Radioactive decay IINeutron Nuclear spin Magnetic field,

Phonon densityDC Bragg scattering Bragg scattering II

Other sensorsSET Charge

eigenstatesElectric field DC–100MHz Thermal Electrical I

Optomechanics Phonons Force, Acceleration,Mass, Magneticfield, Voltage

kHz-GHz Thermal Optical I

Interferometer Photons, (Atoms,Molecules)

Displacement,Refractive Index

– III

TABLE I Experimental implementations of quantum sensors. aSensor type refers to the three definitions of quantum sensingon page 3.

et al., 2009; Bohnet et al., 2014; Cox et al., 2016; Hostenet al., 2016a; Leroux et al., 2010a; Louchet-Chauvetet al., 2010; Schleier-Smith et al., 2010b) and atomic in-teractions (Esteve et al., 2008; Riedel et al., 2010). It hasimproved the sensitivity of magnetometry devices beyondthe shot noise limit (Ockeloen et al., 2013).

B. Trapped ions

Ions, trapped in vacuum by electric or magnetic fields,have been explored as single-particle quantum sensors.The most advanced applications employ the quantizedmotional levels as sensing qubits for electric fields andforces. These levels are strongly coupled to the electricfield by dipole-allowed transitions and sufficiently (MHz)spaced to be prepared by Raman cooling and read out bylaser spectroscopy. The sensor has a predicted sensitiv-

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ity of 500 nV/m/√

Hz or 1 yN/√

Hz for the force actingon the ion (Biercuk et al., 2010; Maiwald et al., 2009).Trapped ions have been extensively used to study electricfield noise above surfaces. Presumably, this noise arisesfrom charge fluctuations induced by adsorbents. Electri-cal field noise is a severe source of decoherence for iontraps and superconducting quantum processors (Brown-nutt et al., 2015; Labaziewicz et al., 2008) and a keylimiting factor in ultrasensitive force microscopy (Kuehnet al., 2006; Tao and Degen, 2015).

Independently, the ground state spin sublevels of ionsare magnetic-field-sensitive qubits analogous to neutralatoms discussed above. Being an extremely clean sys-tem, trapped ions have demonstrated sensitivities of or-der 0.1 nT/

√Hz (Maiwald et al., 2009) and served as a

testbed for advanced sensing protocols such as dynamicaldecoupling (Biercuk et al., 2009; Kotler et al., 2011). Re-cently, trapped ions have also been proposed as rotationsensors, via matter-wave Sagnac interferometry (Camp-bell and Hamilton, 2016). Their use in practical appli-cations, however, has proven difficult. Because they op-erate on a single-ion basis, they are not competitive toensemble sensors such as atomic vapors. Although theirsmall size would be an advantage for applications in mi-croscopy, operation of ion traps in close proximity to sur-faces has proven difficult.

C. Rydberg atoms

Rydberg atoms – atoms in highly excited electronicstates – are remarkable quantum sensors for electric fieldsfor a similar reason as trapped ions: The loosely confinedelectron in a highly excited orbit is easily displaced byelectric fields in a classical picture. In a quantum picture,its motional states are coupled by strong electric dipoletransitions and experience strong Stark shifts (Herrmannet al., 1986; Osterwalder and Merkt, 1999). Preparationand readout of states is possible by laser excitation andspectroscopy.

As their most spectacular sensing application, Rydbergatoms in vacuum have been employed as single-photondetectors for microwave photons in a cryogenic cavityin a series of experiments that has been highlighted bythe Nobel prize in Physics in 2012 (Gleyzes et al., 2007;Haroche, 2013; Nogues et al., 1999).

Recently, Rydberg states have become accessible inatomic vapour cells (Kubler et al., 2010). They havebeen applied for sensing of weak electric fields, mostlyin the GHz frequency range (Fan et al., 2015; Sedlaceket al., 2012), and have been suggested as a candidate fora primary traceable standard for microwave power.

D. Atomic clocks

At first sight, atomic clocks – qubits with transitions soinsensitive that their level splitting can be regarded as ab-solute and serve as a frequency reference – do not seem toqualify as a quantum sensor since this very definition vio-lates criterion (V4). Their operation as a clock, however,employs identical protocols as the operation of quantumsensors, in order to repeatedly compare the qubit’s tran-sition to the frequency of an unstable local oscillator andsubsequently lock the latter to the former. Therefore, anatomic clock can be equally regarded a quantum sensormeasuring and stabilizing the phase drift of a local os-cillator. Vice versa, quantum sensors as discussed abovecan be regarded as clocks that operate on purpose ona bad, environment-sensitive clock transition in order tomeasure external fields.

Today’s most advanced atomic clocks employ opticaltransitions in single ions (Huntemann et al., 2016) oratomic clouds trapped in an optical lattice (Bloom et al.,2014; Hinkley et al., 2013; Takamoto et al., 2005). Inter-estingly, even entanglement-enhanced sensing has founduse in actual devices, since some advanced clocks employmulti-qubit quantum logic gates for readout of highly sta-ble but optically inactive clock ions (Rosenband et al.,2008; Schmidt et al., 2005).

E. Solid state spins – Ensemble sensors

1. NMR ensemble sensors

Some of the earliest quantum sensors have been basedon ensembles of nuclear spins. Magnetic field sensorshave been built that infer field strength from their Lar-mor precession, analogous to neutral atom magnetome-ters described above (Kitching et al., 2011; Packard andVarian, 1954; Waters and Francis, 1958). Initialization ofspins is achieved by thermalization in an externally ap-plied field, readout by induction detection. Although thesensitivity of these devices (10 pT/

√Hz) (Lenz, 1990) is

inferior to their atomic counterparts, they have foundbroad use in geology, archaeology and space missionsthanks to their simplicity and robustness. More recently,NMR sensor probes have been developed for in-situ anddynamical field mapping in clinical MRI systems (Zancheet al., 2008).

Spin ensembles have equally served as gyroscopes(Fang and Qin, 2012; Woodman et al., 1987), exploitingthe fact that Larmor precession occurs in an indepen-dent frame of reference and therefore appears frequency-shifted in a rotating laboratory frame. In the most ad-vanced implementation, nuclear spin precession is readout by an atomic magnetometer, which is equally usedfor compensation of the Zeeman shift (Kornack et al.,2005). These experiments reached a sensitivity of 5 ·

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10−7 rad/s/√

Hz, which is comparable to compact imple-mentations of atomic interferometers and optical Sagnacinterferometers.

2. NV center ensembles

Much excitement has recently been sparked by ensem-bles of nitrogen-vacancy centers (NV centers) – electronicdefect spins in diamond that can be optically initializedand read out. Densely-doped diamond crystals promiseto deliver “frozen vapor cells” of spin ensembles that com-bine the strong (electronic) magnetic moment and effi-cient optical readout of atomic vapor cells with the highspin densities achievable in the solid state. Althoughthese advantages are partially offset by a reduced coher-ence time (T2 < 1 ms at room temperature, as comparedto T2 > 1 s for vapor cells), the predicted sensitivityof diamond magnetometers (250 aT/

√Hz/cm−3/2) (Tay-

lor et al., 2008) or gyroscopes (10−5 rad/s/√

Hz/mm3/2)(Ajoy and Cappellaro, 2012; Ledbetter et al., 2012) wouldbe competitive with their atomic counterparts.

Translation of this potential into competitive devicesremains challenging, with two technical hurdles stand-ing out. First, efficient fluorescence detection of largeNV ensembles is difficult while absorptive and dispersiveschemes are not easily implemented (Clevenson et al.,2015; Jensen et al., 2014; Le Sage et al., 2012). Second,spin coherence times are reduced by 100−1000× in high-density ensembles owing to interaction of NV spins withparasitic substitutional nitrogen spins incorporated dur-ing high-density doping (Acosta et al., 2009). As a con-sequence, even the most advanced devices are currentlylimited to ∼ 1 pT/

√Hz (Wolf et al., 2015) and operate

several orders of magnitude above the theory limit. As atechnically less demanding application, NV centers in amagnetic field gradient have been employed as spectrumanalyzer for high frequency microwave signals (Chipauxet al., 2015).

While large-scale sensing of homogeneous fields re-mains a challenge, micrometer-sized ensembles of NVcenters have found application in imaging applications,serving as detector pixels for microscopic mapping ofmagnetic fields. Most prominently, this line of researchhas enabled imaging of magnetic organelles in magne-totactic bacteria (Le Sage et al., 2013) and microscopicmagnetic inclusions in meteorites (Fu et al., 2014) as wellas contrast-agent-based magnetic resonance microscopy(Steinert et al., 2013).

F. Solid state spins - Single spin sensors

Readout of single spins in the solid state – a ma-jor milestone on the road towards quantum comput-ers – has been achieved both by electrical and opti-

cal schemes. Electrical readout has been demonstratedwith phosphorus dopants in silicon (Morello et al.,2010) and electrostatically-defined semiconductor quan-tum dots (Elzerman et al., 2004). Optical readout wasshown with single organic molecules (Wrachtrup et al.,1993a,b), optically active quantum dots (Kroutvar et al.,2004; Vamivakas et al., 2010), and defect centers in crys-talline materials including diamond (Gruber et al., 1997)and silicon carbide (Christle et al., 2015; Widmann et al.,2015). In addition, mechanical detection of single para-magnetic defects in silica (Rugar et al., 2004) and real-time monitoring of few-spin fluctuations (Budakian et al.,2005) has been demonstrated.

Out of all solid state spins, NV centers in diamond havereceived by far the most attention for sensing purposes.This is in part due to the convenient room-temperatureoptical detection, and in part due to their stability in verysmall crystals and nanostructures. The latter permitsuse of NV centers as sensors in high-resolution scanningprobe microscopy (Balasubramanian et al., 2008; Cher-nobrod and Berman, 2005; Degen, 2008), as biomarkerswithin living organisms (Fu et al., 2007), or as stationaryprobes in the surface of diamond sensor chips. Quantumsensing with NV centers has been considered in severalrecent focused reviews (Rondin et al., 2014; Schirhaglet al., 2014).

Single NV centers have been employed and/or pro-posed as sensitive magnetometers (Balasubramanianet al., 2008; Maze et al., 2008; Taylor et al., 2008), elec-trometers (Dolde et al., 2011), pressure sensors (Dohertyet al., 2014) and thermometers (Hodges et al., 2013;Kucsko et al., 2013; Neumann et al., 2013; Toyli et al.,2013), using the Zeeman, Stark and temperature shiftsof their spin sublevels. The most advanced experimentsin terms of sensitivity have employed near-surface NVcenters in bulk diamond crystals. This approach has en-abled sensing of (nm)3-sized voxels of nuclear or elec-tronic spins deposited on the diamond surface (Loretzet al., 2014; Mamin et al., 2013; Shi et al., 2015; Stau-dacher et al., 2013) and of distant nuclear spin clusters(Shi et al., 2014). Other applications included the studyof ballistic transport in the Johnson noise of nanoscaleconductors (Kolkowitz et al., 2015), of the helimagneticphase of skyrmion materials (Dussaux et al., 2016), aswell as of spin waves (van der Sar et al., 2015; Wolfeet al., 2014) and relaxation in nanomagnets (Schafer-Nolte et al., 2014; Schmid-Lorch et al., 2015).

Integration of NV centers into scanning probes has en-abled imaging of magnetic fields with sub-100 nm resolu-tion, with applications to nanoscale magnetic structuresand domains (Balasubramanian et al., 2008; Maletinskyet al., 2012; Rondin et al., 2012), vortices and domainwalls (Rondin et al., 2013; Tetienne et al., 2014, 2015),superconducting vortices (Pelliccione et al., 2016; Thielet al., 2016), mapping of currents (Chang et al., 2016)and nanoscale magnetic resonance imaging (Haeberle

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et al., 2015; Luan et al., 2015; Pelliccione et al., 2014;Rugar et al., 2015; Schmid-Lorch et al., 2015).

NV centers in ∼10-nm-sized nanodiamonds have beeninserted into living cells. They have been employed forparticle tracking (McGuinness et al., 2011) and in vivotemperature measurements (Kucsko et al., 2013) andcould enable real-time monitoring of metabolic processes.

G. Superconducting circuits

1. SQUIDs

Superconducting Quantum Interference Devices(SQUIDs) are simultaneously one of the oldest and oneof the most sensitive type of magnetic sensor (Clarkeand Braginski, 2004; Fagaly, 2006; Jaklevic et al., 1965).These devices – interferometers of superconductingconductors – measure magnetic fields with a sensitivitydown to 10 aT/

√Hz (Simmonds et al., 1979). Their

sensing mechanism is based on the Aharonov-Bohmphase imprinted on the superconducting wave functionby an encircled magnetic field, which is read out by asuitable circuit of phase-sensitive Josephson junctions.

From a commercial perspective, SQUIDs can be con-sidered the most advanced type of quantum sensor, withapplications ranging from materials characterization insolid state physics to clinical magnetoencephalographysystems for measuring tiny (∼ 100 fT) stray fields of elec-tric currents in the brain. In parallel to the developmentof macroscopic (mm-cm) SQUID devices, miniaturizationhas given birth to sub-micron sized “nanoSQUIDs” withpossible applications in nanoscale magnetic, current, andthermal imaging (Halbertal et al., 2016; Vasyukov et al.,2013). Note that because SQUIDs rely on spatial ratherthan temporal coherence, they are more closely related tooptical interferometers than to the spin sensors discussedabove.

2. Superconducting qubits

Meanwhile, temporal quantum superpositions of su-percurrents or charge eigenstates have become accessiblein superconducting qubits (Clarke and Wilhelm, 2008;Martinis et al., 2002; Nakamura et al., 1999; Vion et al.,2002; Wallraff et al., 2004). They have been employedfor quantum sensing experiments, using the many es-tablished quantum sensing protocols to be discussed inSections IV–VII: Specifically, noise in these devices hasbeen thoroughly studied from the sub-Hz to the GHzrange, using Ramsey interferometry (Yan et al., 2012;Yoshihara et al., 2006), dynamical decoupling (Bylanderet al., 2011; Ithier et al., 2005; Nakamura et al., 2002;Yan et al., 2013; Yoshihara et al., 2006), and T1 relax-ometry (Astafiev et al., 2004; Yoshihara et al., 2006).These studies have been extended to discern charge from

flux noise by choosing qubits with a predominant electric(charge qubit) or magnetic (flux qubit) dipole moment,or by tuning bias parameters in situ (Bialczak et al.,2007; Yan et al., 2012). Extending these experiments tothe study of extrinsic samples appears simultaneously at-tractive and technically challenging, since superconduct-ing qubits have to be cooled to temperatures of only few10mK. Very promising sensitivities, 3.3 pT/

√Hz for op-

eration at 10 MHz were demonstrated (Bal et al., 2012).

H. Elementary particle qubits

Interestingly, elementary particles have been employedas quantum sensors long before the development ofatomic and solid state qubits. This somewhat paradox-ical fact is owing to their straightforward initializationand readout, as well as their targeted placement in rele-vant samples by irradiation with a particle beam.

1. Muons

Muons are produced from positive pions in proton-proton collisions by the reaction π+ → µ+ + νµ. Parityviolation of the weak interaction automatically initializesthe muon spin to be collinear with the particle’s momen-tum. Readout of the muon spin is straightforward bymeasuring the emission direction of positrons from thesubsequent decay µ+ → e+ + νe + νµ, which are prefer-ably emitted along the muon spin (Brewer and Crowe,1978).

Crucially, muons can be implanted into solid state sam-ples and serve as local probes of their nanoscale envi-ronment for their few microseconds long lifetime. Lar-mor precession measurements have been used to inferthe intrinsic magnetic field of materials. Despite its ex-otic nature, the technique of muon spin rotation (µSR)has become and remained a workhorse tool of solid statephysics. In particular, it is a leading technique to mea-sure the London penetration depth of superconductors(Sonier et al., 2000).

2. Neutrons

Slow beams of thermal neutrons can be spin-polarizedby Bragg reflection on a suitable magnetic crystal. Spinreadout is feasible by a spin-sensitive Bragg analyzer andsubsequent detection. Spin rotations (single qubit gates)are easily implemented by application of localized mag-netic fields along parts of the neutron’s trajectory. Asa consequence, many early demonstrations of quantumeffects, such as the direct measurement of Berry’s phase(Bitter and Dubbers, 1987), have employed neutrons.

Sensing with neutrons has been demonstrated in mul-tiple ways. Larmor precession in the magnetic field of

9

samples has been employed for three-dimensional tomog-raphy (Kardjilov et al., 2008). The most establishedtechnique, neutron spin echo, employs quantum sensingtechniques to measure small (down to neV) energy lossesof neutrons in inelastic scattering events (Mezei, 1972).Here, the phase of the neutron spin, coherently precessingin an external magnetic field, serves as a clock to measurea neutron’s time of flight. Inelastic scattering in a samplechanges a neutron’s velocity, resulting in a different timeof flight to and from a sample of interest. This differenceis imprinted in the spin phase by a suitable quantum sens-ing protocol, specifically a Hahn echo sequence whose πpulse is synchronized with passage through the sample.

I. Other sensors

In addition to the many implementations of quantumsensors already discussed, three further devices deservedspecial attention for their future potential or for theirfundamental role in developing quantum sensing method-ology.

1. Single electron transistors

Single electron transistors (SET’s) sense electric fieldsby measuring the tunneling current across a submicronconducting island sandwiched between tunneling sourceand drain contacts. In the “Coulomb blockade regime” ofsufficiently small (typically ≈ 100 nm) islands, tunnelingacross the device is only allowed if charge eigenstates ofthe island lie in the narrow energy window between theFermi level of source and drain contact. The energy ofthese eigenstates is highly sensitive to even weak exter-nal electric fields, resulting in a strongly field-dependenttunneling current (Kastner, 1992; Schoelkopf, 1998; Yooet al., 1997). SETs have been employed as scanningprobe sensors to image electric fields on the nanoscale,shedding light on a variety of solid-state-phenomena suchas the fractional quantum Hall effect or electron-holepuddles in graphene (Ilani et al., 2004; Martin et al.,2008). In a complementary approach, charge sensing bystationary SETs has enabled readout of optically inac-cessible spin qubits such as phosphorus donors in silicon(Morello et al., 2010) based on counting of electrons (By-lander et al., 2005).

2. Optomechanics

Phonons – discrete quantized energy levels of vibration– have recently become accessible at the “single-particle”level in the field of optomechanics (Aspelmeyer et al.,2014; O’Connell et al., 2010), which studies high-qualitymechanical oscillators that are strongly coupled to light.

While preparation of phonon number states and theircoherent superpositions remains difficult, the devicesbuilt to achieve these goals have shown great promise forsensing applications. This is mainly due to the fact thatmechanical degrees of freedom strongly couple to nearlyall external fields, and that strong optical coupling en-ables efficient actuation and readout of mechanical mo-tion. Specifically, optomechanical sensors have been em-ployed to detect minute forces (12 zN/

√Hz, (Moser et al.,

2013)), acceleration (100 ng/√

Hz, (Cervantes et al.,2014; Krause et al., 2012)), masses (2 yg/

√Hz, (Chaste

et al., 2012)), magnetic fields (200 pT/√

Hz, (Forstneret al., 2014)), spins (Degen et al., 2009; Rugar et al.,2004), and voltage (5 pV/

√Hz, (Bagci et al., 2014)).

While these demonstrations have remained at the level ofclassical sensing in the sense of this review, their futureextension to quantum-enhanced measurements appearsmost promising.

3. Photons

While this review will not discuss quantum sensingwith photons, due to the breadth of the subject, severalfundamental paradigms have been pioneered with opti-cal sensors including light squeezing and photonic quan-tum correlations. These constitute examples of quantum-enhanced sensing according to our “Definition III”.

Squeezing of light – the creation of partially-entangledstates with phase or amplitude fluctuations below thoseof a classical coherent state of the light field – has beenproposed (Caves, 1981) and achieved (Slusher et al.,1985) long before squeezing of spin ensembles (Hald et al.,1999; Wineland et al., 1992). It has meanwhile been em-ployed to boost contrast in microscopy of weakly absorb-ing objects (Brida et al., 2010) and has improved theperformance of actual devices such as gravitational wavedetectors (Ligo Collaboration”, 2011).

In addition, quantum correlations between photonshave proven to be a powerful resource for imaging.This has been noted very early in the famous Hanbury-Brown-Twiss experiment, where bunching of photons isemployed to filter atmospheric aberrations and to per-form “super-resolution” measurements of stellar diame-ters smaller than the diffraction limit of the telescopeemployed (Hanbury Brown and Twiss, 1956). Whilethis effect can still be accounted for classically, a recentclass of experiments has exploited non-classical correla-tions to push the spatial resolution of microscopes be-low the diffraction limit (Schwartz et al., 2013). Viceversa, multi-photon correlations have been proposed andemployed to create light patterns below the diffractionlimit for superresolution lithography (Boto et al., 2000;D’Angelo et al., 2001).

The most advanced demonstrations of entanglement-enhanced sensing have been performed with single pho-

10

tons or carefully assembled few-photon Fock states. Mostprominently, these include Heisenberg-limited interfer-ometers (Higgins et al., 2007; Holland and Burnett, 1993;Mitchell et al., 2004; Nagata et al., 2007; Walther et al.,2004). In these devices, entanglement between photonsor adaptive measurements are employed to push sensitiv-ity beyond the 1/

√N scaling of a classical interferometer

where N is the number of photons (see Section IX).

IV. THE QUANTUM SENSING PROTOCOL

In this Section, we describe the basic methodology forperforming measurements with quantum sensors. Ourdiscussion will focus on a generic scheme where the quan-tum sensor is initialized, interacts with a physical quan-tity for some time t, and is then read out. Parameter es-timation (Braunstein and Caves, 1994; Braunstein et al.,1996; Goldstein et al., 2010) and phase estimation (Ki-taev, 1995; Shor, 1994) can then be used to reconstructthe physical quantity from the readouts.

Experimentally, the protocol is typically implementedas an interference measurement using pump-probe spec-troscopy, although other schemes are possible. The pro-tocol can then be optimized for detecting weak signals orsmall signal changes with the highest possible sensitivityand precision.

A. Quantum sensor Hamiltonian

Before discussing the actual sensing protocol, a morespecific model of the quantum sensor is needed. In par-ticular, we will require that the quantum sensor can berepresented by a two-level quantum system and be de-scribed by the generic Hamiltonian

H(t) = H0 + HV (t) + Hcontrol(t) . (2)

We remark that the focus on two-level systems is not asevere restriction because many properties of more com-plex quantum systems can be modeled through a qubitsensor (Goldstein et al., 2010). H0 is the qubit internalHamiltonian, HV (t) is the Hamiltonian associated witha signal V (t), and Hcontrol(t) is the control Hamiltonian.We will assume that H0 is known and that Hcontrol(t)can be used to manipulate or tune the sensor in a con-trolled way. Our goal then is to estimate V (t) via theeffect HV (t) has on the qubit.

1. Internal Hamiltonian

H0 describes the internal Hamiltonian of the quantumsensor in the absence of any signal. Typically, the inter-nal Hamiltonian is static and defines the energy eigen-

states |0〉 and |1〉,

H0 = E0|0〉〈0|+ E1|1〉〈1| , (3)

where E0 and E1 are the eigenenergies and ω0 = E1−E0

is the transition energy2 between the states. Note thatthe presence of an energy splitting E1 − E0 6= 0 is nota priori necessary, but it represents the typical situationfor most implementations of quantum sensors. The qubitinternal Hamiltonian may contain additional interactionsthat are specific to a quantum sensor, such as dipolarinteractions between spin qubits. Moreover, the internalHamiltonian will contain some time-dependent stochasticinteractions or interactions with the environment that areresponsible for decoherence and relaxation.

2. Signal Hamiltonian

The signal V (t) can be measured thanks to a couplingbetween the sensor qubit and the signal, which gives riseto the signal Hamiltonian HV (t). Because the signal V (t)is small, it is usually detected as a perturbation to H0.The signal Hamiltonian can then be separated into twoqualitatively different contributions, one that commuteswith H0 and one that does not commute with H0,

HV (t) = HV||(t) + HV⊥(t) , (4)

where HV|| is the parallel (commuting, secular) and

HV⊥ the transverse (non-commuting) component, respec-tively. The two components can quite generally be cap-tured by

HV||(t) = 12γV||(t) {|1〉〈1| − |0〉〈0|} ,

HV⊥(t) = 12γ{V⊥(t)|1〉〈0|+ V †⊥(t)|0〉〈1|

}, (5)

where V||(t) and V⊥(t) are functions with units of energy,respectively. γ is the coupling or transduction parameterof the qubit to the signal V (t). Examples of couplingparameters include the Zeeman shift parameter (gyro-magnetic ratio) of spins in a magnetic field, with unitsof Hz/T, or the linear Stark shift parameter of electricdipoles in an electric field, with units of Hz/(Vm−1). Al-though the coupling is often linear, this is not required;a counterexample is the quadratic Stark effect.

The parallel and transverse components of a signalhave distinctly different effects on the quantum sensor.A commuting perturbation HV|| leads to shifts of the en-ergy levels and an associated change of the transitionfrequency ω0. This indicates, for example, a change ina static external field. A non-commuting perturbationHV⊥ , by contrast, can induce transitions between levels,

2 ~ = 1; see footnote on page 3

11

manifesting through an increased transition rate Γ. Mostoften, this requires the signal to be time-dependent (reso-nant with the transition) in order to have an appreciableeffect on the quantum sensor.

An important class of signals are vector signal ~V (t),in particular those provided by electric or magneticfields. The interaction between a vector signal ~V (t) ={Vx, Vy, Vz}(t) and a qubit can be described by the sig-nal Hamiltonian

HV (t) = γ~V (t) · ~σ , (6)

where ~σ = {σx, σy, σz} is a vector of Pauli matrices. Fora vector signal, the two signal functions V||(t) and V⊥(t)are

V||(t) = Vz(t) ,

V⊥(t) = Vx(t) + iVy(t), (7)

where the z direction is defined by the qubit’s quantiza-tion axis. The corresponding signal Hamiltonian is

HV (t) = γRe[V⊥(t)]σx + γIm[V⊥(t)]σy + γV||(t)σz . (8)

3. Control Hamiltonian

For most quantum sensing protocols it is required tomanipulate the qubit either before, during, or after thesensing process. This is achieved via a control Hamilto-nian Hcontrol(t) that allows implementing a standard setof quantum gates (Nielsen and Chuang, 2000). The mostcommon gates in quantum sensing include the Hadamardgate and the Pauli-X and Y gates, which may be ap-proximately carried out in practice by π/2 and π pulses.In particular, Hadamard gates are used to create super-position states, |0〉 ↔ (|0〉 ± |1〉)/

√2 ≡ |±〉, while the

Pauli gates are used to flip the states, |0〉 ↔ |1〉 and|±〉 ↔ |∓〉. Advanced sensing schemes employing morethan one sensor qubit furthermore rely on conditionalgates, especially controlled-NOT gates to generate en-tanglement or for state swaps in conjunction with mem-ory qubits and controlled phase shifts in quantum phaseestimation.

The control Hamiltonian can also include control fieldsfor systematically tuning the transition frequency ω0.This capability is frequently exploited in noise spec-troscopy experiments.

B. The protocol

The central step in a quantum sensing experiment isthe measurement of the overlap between an initial sensingstate |ψ0〉, prepared at time zero, and a final sensing state|ψ(t)〉 = U(0, t)|ψ0〉 at a later time t, where U(0, t) is thepropagator of the Hamiltonian H. Quite generally, for an

1. Initialize

5. Project, Readout

4. Transform

2. Transform

6. Repeat and average

“0” with probability “1” with probability

7. Estimate signal

3. Evolve for time t

FIG. 2 Basic steps of the quantum sensing process: 1. Thequantum system is initialized in a well-known state |0〉 and2. possibly transferred into a suitable initial sensing state|ψ0〉. 3. The quantum system evolves for a time t under the

Hamiltonian H(t) into the final sensing state |ψ(t)〉. This isthe actual sensing period. 4. The state is transformed intoa suitable observable |α〉 and 5. read out by projection onto|0〉 and |1〉. 6. By averaging over N readouts, the transitionprobability p is estimated. 7. The signal V (t) is inferredfrom one or several probability values, recorded with differentcontrol Hamiltonians or sensing times.

appropriately selected initial state |ψ0〉, the final sensingstate is given by

|ψ(t)〉 =1

2(1 + e−iφ(t))|ψ0〉+

1

2(1− e−iφ(t))|ψ1〉 , (9)

and the overlap is given by the coefficient

〈ψ0|ψ(t)〉 = 〈ψ0|U(0, t)|ψ0〉 =1

2(1 + e−iφ(t)) , (10)

where φ(t) represents a relative phase that was accumu-lated by the qubit over time t, and |ψ1〉 is the state or-thogonal to |ψ0〉. The coefficient 〈ψ0|ψ(t)〉, although notdirectly accessible, determines the measurable probabil-ity

p = 1− |〈ψ0|ψ(t)〉|2 (11)

that the qubit changed its state during t. This transitionprobability p represents the output quantity of the sensingprocess.

12

Although the evolution of the qubit under U(0, t) is thekey step in the sensing process, a few additional steps arerequired to complete a measurement. These steps can besummarized in the following basic protocol, which is alsosketched in Fig. 2:

1. The quantum sensor is initialized into a knownstate, for example |0〉.

2. |0〉 is transformed into the initial sensing state|ψ0〉 = Up|0〉 using a set of control pulses Up, ifneeded. In many cases, |ψ0〉 is a superpositionstate.

3. The state |ψ0〉 is allowed to evolve under the Hamil-tonian H for a time t. At the end of the sens-ing period, the sensor is in the final sensing state|ψ(t)〉 = U(0, t)|ψ0〉, where U(0, t) is the propaga-tor of H.

4. If needed, |ψ(t)〉 is transformed into a superpositionof observable states |α〉 = U ′p|ψ(t)〉 = c0|0〉+ c1|1〉,where c0 = 〈0|α〉 and c1 = 〈1|α〉. The reverse trans-formation is often given by U ′p = (Up)

†, but can alsobe judiciously altered. Assuming a properly chosenset of unitary transformations Up and U ′p, the co-efficient c0 = 〈0|α〉 ∝ 〈ψ0|ψ(t)〉 then represents theoverlap between the initial and final sensing states.

5. The final state of the quantum sensor is readout. We will assume that the readout is pro-jective, although more general positive-operator-valued-measure (POVM) measurements may bepossible (Nielsen and Chuang, 2000). Projectivestate readout yields an answer “0” with probability|〈0|α〉|2 = 1−p and an answer “1” with probability|〈1|α〉|2 = p, where p is the above transition prob-ability. The “answer” is converted into a physicalquantity x, for example a voltage, current, photoncount or polarization, that can be detected by asuitable measurement apparatus.

Steps (1-5) represent a single measurement cycle. To gaina precise estimate for p, the measurement cycle needs tobe repeated:

6. Steps (1-5) are repeated and averaged over a largenumber of cycles N . The repetition can be doneby running the protocol sequentially on the samequantum system, or in parallel by averaging overan ensemble of N identical (and non-interacting)quantum systems.

Step 6 only provides one value for the transition prob-ability p. While a single value of p may sometimes besufficient to estimate a signal V , it is in many situationsconvenient or required to record a set of values {pk}:

7. The transition probability p is measured as a func-tion of time t or of a parameter of the controlHamiltonian Hcontrol, and the desired signal V isinferred from the data record {pk} using a suitableprocedure.

More generally, a set of measurements can be optimizedto efficiently extract a desired parameter from the signalHamiltonian (see Section VIII). Most protocols presentedin the following implicitly use such a strategy for gaininginformation about the signal.

Although the above protocol is generic and simple, it issufficient to describe most sensing experiments. For ex-ample, classical continuous-wave absorption and trans-mission spectroscopy can be considered as an averagedvariety of this protocol. Also, the time evolution canbe replaced by a spatial evolution to describe a classicalinterferometer.

To illustrate the protocol, we consider two elementaryexamples, one for detecting a parallel signal V|| and onefor detecting a transverse signal V⊥. These exampleswill serve as the basis for the more refined sequences pre-sented in later Sections.

C. First example: Ramsey measurement

A first example is the measurement of the static en-ergy splitting ω0 (or equivalently, a static perturbationV||). The canonical approach for this measurement is aRamsey interferometry measurement (Lee et al., 2002;Taylor et al., 2008):

1. The quantum sensor is initialized into |0〉.2. Using a π/2 pulse (Hadamard gate), the quantum

sensor is brought into the superposition state

|ψ0〉 = |+〉 ≡ 1√2

(|0〉+ |1〉) . (12)

3. The superposition state evolves under the Hamilto-nian H0 for a time t. The superposition state picksup the relative phase φ = ω0t, and the state afterthe evolution is

|ψ(t)〉 =1√2

(|0〉+ e−iω0t|1〉) , (13)

up to an overall phase factor.

4. Using a second π/2 pulse, the state |ψ(t)〉 is con-verted back to the measurable state

|α〉 =1

2(1 + e−iω0t)|0〉+

1

2(1− e−iω0t)|1〉 . (14)

5,6. The final state is read out. The transition proba-bility is

p = 1− |〈0|α〉|2

= sin2(ω0t/2) =1

2[1− cos(ω0t)]. (15)

13

By recording p as a function of time t, an oscillatoryoutput (“Ramsey fringes”) is observed with a frequencygiven by ω0 (see Fig. 3). Thus, the Ramsey measure-ment can directly provide a measurement for the energysplitting ω0.

D. Second example: Rabi measurement

A second elementary example is the measurement ofthe transition matrix element |V⊥|:

1. The quantum sensor is initialized into |ψ0〉 = |0〉.3. In the absence of the internal Hamiltonian, H0 = 0,

the evolution is given by HV⊥ = 12γV⊥σx = ω1σx,

where ω1 is the Rabi frequency. The state afterevolution is:

|ψ(t)〉 = |α〉 =1

2(1 + e−iω1t)|0〉+ 1

2(1− e−iω1t)|1〉 . (16)

5,6. The final state is read out. The transition proba-bility is:

p = 1− |〈0|α〉|2 = sin2(ω1t/2). (17)

In a general situation where H0 6= 0, the transi-tion probability represents the solution to Rabi’s originalproblem (Sakurai and Napolitano, 2011),

p =ω2

1

ω21 + ω2

0

sin2

(√ω2

1 + ω20 t

). (18)

Hence, only time-dependent signals with frequency ω ≈ω0 affect the transition probability p, a condition knownas resonance. From this condition it is clear that a Rabimeasurement can provide information not only on themagnitude V⊥, but also on the frequency ω of a signal(Aiello et al., 2013; Fedder et al., 2011).

E. Slope and variance detection

A key task for a quantum sensor is the detection ofsmall signals. We can modify the protocols to optimizethe quantum sensor for these tasks. In particular, it isoften convenient to “bias” the measurement point so asto reach the best possible sensitivity (see Sec.V). Themeasurement output is then given by the difference δp =p−p0, where p is the probability measured in the presenceof the signal and p0 is the probability measured in theabsence of the signal. We will refer to p0 as the bias pointof the measurement.

1. Slope detection (linear detection)

The Ramsey interferometer is most sensitive to smallperturbations V|| when operated at the point of maximum

0

1

(a)

(b)

signal intensity V||

p(V||)

V V

p

p

FIG. 3 Transition probability p(V||) for a Ramsey experi-ment as a function of the signal V||. (a) Slope detection: Thequantum sensor is operated at the p0 = 0.5 bias point (redclosed dot). A change in the sensor phase by φ = γV||t leadsto a linear change in the transition probability by δp = φ/2(open circle). (b) Variance detection: The quantum sensor isoperated at the p0 = 0 or p0 = 1 bias point (closed blue dot).A change in sensor phase by φ leads to a quadratic changein the transition probability by δp = φ2/4. The experimentalreadout error σp translates into a signal error σV accordingto the slope or curvature of the Ramsey fringe.

slope where p0 = 0.5, see Fig. 3(a). This bias point isreached when ω0t = kπ/2, with k = 1, 3, 5, .... Aroundp0 = 0.5, the transition probability is linear in V|| and t,producing an output reading

δp =1

2[1− cos(ω0t+ γV||t)]−

1

2

≈ ±1

2γV||t, (19)

where the sign is determined by k.

Note that slope detection has a limited linear rangeset by γV||t � 1. This limitation can restrict the dy-namic range of the quantum sensor. Section VIII dis-cusses adaptive sensing techniques that have been devel-oped to extend the dynamic range.

2. Variance detection (quadratic detection)

If the magnitude of V|| fluctuates between measure-ments so that 〈V||〉 = 0, readout at p0 = 0.5 will yield noinformation about V||, since 〈p〉 ≈ p0 = 0.5. In this situ-ation, it is advantageous to detect the signal variance bybiasing the measurement to a point of minimum slope,ω0t = kπ, corresponding to the bias points p0 = 0 andp0 = 1. If the interferometer is tuned to p0 = 0, a signalwith variance 〈V 2

|| 〉 = V 2rms gives rise to a mean transition

probability that is quadratic in Vrms and t,

δp = p =

⟨1

2[1− cos(ω0t+ γV||t)]

⟩≈ 1

4γ2V 2

rmst2. (20)

This relation holds for small γVrmst � 1. If the fluctua-tion is Gaussian, the result can be extended to any large

14

value of γVrmst,

p =1

2

[1− exp(−γ2V 2

rmst2/2)

]. (21)

Variance detection is especially important for detectingac signals when their synchronization with the sensingprotocol is not possible (Section VI.D.5), or when thesignal represents a noise source (Section VII).

V. SENSITIVITY

One of the most important features of quantum sensorsis their ability to perform highly sensitive measurements.In this Section, we outline the key factors setting the sen-sitivity of a quantum sensor. We will see that the min-imum detectable signal per unit time is approximatelygiven by

vmin ≈√

2e

γC√Tχ

(22)

for slope detection and by

vmin ≈√

2e

γ√C 4

√T 3χ

(23)

for variance detection, under suitable assumptions thatwill be discussed later. Here, Tχ denotes the sensor’s co-herence time and C ≤ 1 a dimensionless constant quanti-fying readout efficiency. Both relations can be intuitivelymotivated through Fig. 3: Experimental detection of theprobability p will have a non-zero error σp. This errortranslates into a corresponding error σV via the slope orcurvature of the Ramsey fringe.

In the following, we detail the main noise sources thatdetermine σp and σV . These are, in order of fundamen-talness, (i) quantum projection noise as the final stateis measured, (ii) decoherence and relaxation of the qubitduring the sensing time, and (iii) experimental imper-fections with initialization, readout and qubit manipula-tions. We then derive the expressions for the minimumdetectable signal, Eq. (22) and (23). At the end of thesection, a formal definition of sensitivity by the quantumCramer-Rao bound (QCRB) is given.

A. Quantum projection noise

The projective readout during “Step 5” of the quantumsensing protocol (Section IV.B) does not directly producethe fractional probability value p ∈ [0...1], but a digitalvalue of “0” or “1”. The probability for obtaining answer“0” is 1− p and the probability for obtaining answer “1”is p. If one performs a series of N measurements, onemeasures, on average, N0 = (1 − p)N times a value of

“0” and N1 = pN times an answer of “1”. The estimatefor p is

p =N1

N(24)

and the uncertainty in p is given by the variance of thebinomial distribution (Itano et al., 1993),

σ2p,quantum =

1

Np(1− p) . (25)

The uncertainty in p depends on the bias point p0 of themeasurement. For slope detection, where p0 = 0.5, theuncertainty is

σ2p,quantum =

1

4N(for p0 = 0.5) (26)

Thus, the projective readout adds noise of order1/(2√N) to the probability value p. For variance detec-

tion, where ideally p0 = 0, the projection noise would inprinciple be arbitrarily low. However, the signal-to-noiseratio remains finite.

B. Decoherence

A second source of error includes decoherence and re-laxation during the sensing process. Decoherence and re-laxation cause random transitions between states or ran-dom phase pick-up during coherent evolution of the qubit(for more detail, see Section VII). The two processes leadto an apparent reduction of the measurement output δpwith increasing sensing time t.

For the present analysis of sensitivity, we assume thatdecoherence can be described through an exponential de-cay, leading to a modified output

δpobs(t) = δp(t)e−χ(t) , (27)

where δp(t) is the measurement output that would beobserved in the absence of decoherence.χ(t) is the so-called decoherence function, which is dis-

cussed in more detail in Section VII.B.1. Typical de-coherence functions found in experiments include puredephasing, χ(t) = Γt, Lorentzian noise, χ = (Γt)3, or a1/f-noise-like decay, χ(t) = (Γt)2, where Γ is a decay rate.By setting χ(Tχ) = 1, the decay rate can be associatedwith a decoherence time or relaxation time Tχ = Γ−1.The decay time Tχ is an important figure of merit forthe quantum sensor, because it represents the maximumpossible sensing time t supported by the quantum sensor.

C. Initialization, manipulation and readout noise

A third source of error is the imperfect initialization,manipulation and readout of the quantum sensor. Imper-fect initialization leads to a similar apparent reduction in

15

x0 xT x1x0 x1 x0 x1

2x

N1

N0

“ideal readout”R=0C=1

(a) (b) (c)

“single shot readout”R<1

C>0.707

“averaged readout”R>1

C<0.707

x

FIG. 4 Illustration of the sensor readout. N measurementsare performed yielding {xk}k=1...N readings on the physicalmeasurement apparatus. The readings {xk} are then binnedinto a histogram. (a) Ideal readout. All {xk} can be assignedto “0” and “1” with 100% fidelity. (b) Single shot readout.Most {xk} can be assigned, but there is an overlap betweenhistograms leading to a small error. (c) Averaged readout.{xk} cannot be assigned. The ratio between “0” and “1” isgiven by the relative position of the mean value x and theerror is determined by the histogram standard deviation σx.R and C are readout efficiency parameters that are explainedin the text.

the measurement output δp as in the case of decoherence,however, this reduction does not depend on the sensingtime t. Errors in qubit manipulations can cause many ef-fects, but will typically also lead to a reduction of δp. Inthe following, we consider the combined effect of reducedfidelity through an imperfect readout. A more generalapproach, considering, e.g., faulty initialization througha density matrix approach, will be briefly discussed in thecontext of quantum limits to sensitivity (see Sec. V.F).

For the following we assume that each readout k pro-duces a physical reading xk, where xk is, for example, avoltage, current or photon count. A series of N measure-ments then gives a set of readings {xk}k=1...N . This setcan be binned into a histogram as shown in Fig. 4. Twosituations can occur, depending on whether the readoutnoise is small or large compared to the projection noise(see Fig. 4). They can be termed the “single shot” and“averaged” readout regimes.

1. Single shot readout

In the “single shot” regime, the classical readout noiseis small. Two peaks are observed in the histogram re-flecting readouts of states “0” or “1”, respectively, as de-picted in Fig. 4(b). Measurements xk can be assigned to“0” or “1” based on a threshold value xT chosen roughlymidway between the “exact” x|0〉 and x|1〉:

N0 = number of measurements xk < xT (28)

N1 = number of measurements xk > xT (29)

The estimate for p is given through Eq. (24), p = N1/N .The choice of the threshold is not trivial; in particular,for an unbiased measurement, xT depends itself on theprobability p.

Because of the readout error, the histogram is broad-ened and some values xk will be assigned to the wrongstate due to the overlap between the peaks. The errorintroduced due to wrong assignments is

σ2p,readout =

1

N[κ0(1− κ0)p+ κ1(1− κ1)(1− p)] , (30)

where κ0 and κ1 are the fraction of measurements thatare erroneously assigned. The actual values for κ0,1 de-pend on the exact type of measurement noise and are de-termined by the cumulative distribution function of thetwo histogram peaks. Frequently, the peaks have an ap-proximately Gaussian distribution, such that

κ0 ≈1

2

[1 + erf

( |x|0〉 − xT |σx

)], (31)

and likewise for κ1, where erf(x) is the Gauss error func-tion. Moreover, if κ ≡ κ0 ≈ κ1 � 1 are small and ofsimilar magnitude,

σ2p,readout ≈

κ

N. (32)

2. Averaged readout

In the “averaged” regime, the classical readout noiseis large and only a single broad peak is seen in the his-togram, see Fig. 4c. The estimate for p and its error σpare simply given by the mean and standard error of thedistribution,

p =x− x|0〉x|1〉 − x|0〉

=1

N

N∑k=1

xk − x|0〉x|1〉 − x|0〉

, (33)

σ2p,readout =

σ2x

N(x|1〉 − x|0〉)2=R2

4N, (34)

where x = 1N

∑xk is the mean of {xk}. Here, |x|1〉−x|0〉|

is the measurement contrast and

R =2σx

|x|1〉 − x|0〉|(35)

is a parameter describing the classical noise added duringthe measurement.

As an example, we consider the optical readout of anatomic vapor magnetometer (Budker and Romalis, 2007)or of NV centers in diamond (Taylor et al., 2008). Then,x|0〉 and x|1〉 are the average numbers of photons col-lected per readout for each state. The standard error is(under suitable experimental conditions) dominated byshot noise, σx ≈

√x. The readout noise parameter is

Rshot ≈2√x

|x|1〉 − x|0〉|=

2√

1− ε/2ε√x|1〉

≈ 2

ε√x|1〉

, (36)

16

where ε = |1 − x|0〉/x|1〉| is a relative optical contrastbetween the states, 0 < ε < 1, and the last equationrepresents the approximation for ε� 1.

3. Total readout uncertainty

The two errors, quantum projection error and readouterror, can be combined into a total error

σ2p = σ2

p,quantum + σ2p,readout

≈ (1 +R2)σ2p,quantum ≈

σ2p,quantum

C2=

1

4C2N, (37)

where C = 1/√

1 +R2 ≈ 1/√

1 + 4κ is an overall read-out efficiency parameter (Taylor et al., 2008). C ≤ 1describes the reduction of the signal-to-noise ratio com-pared to an ideal readout (C = 1), see Fig. 4.

D. Sensitivity and minimum detectable signal

The three sources of signal uncertainty – quantum pro-jection noise, decoherence, and readout imperfections –can be combined to estimate the sensitivity of a quan-tum sensing measurement. In a first step, we define asignal-to-noise ratio (SNR) for the transition probability

SNR =δpobs

σp= δp(t) e−χ(t)2C

√N , (38)

given through Eqs. (27) and (37).In order to quantify the sensitivity, we define a mini-

mum detectable signal, Vmin, that yields unit SNR for atotal available measurement time T . The change in p isrelated to the change in signal δV as δp = δV q ∂

qp∂V q ∝

(γtδV )q (with q = 1 for slope detection and q = 2 forvariance detection), thus Vmin can be evaluated from (38)to be

V qmin :=eχ(t)

2C|∂qV p(t)|√N∝ eχ(t)

2Cγqtq√N

. (39)

Assuming that the duration of one measurement cycle ist + tm, where tm is the extra time needed to initialize,manipulate and readout the sensor, the number of mea-surements is N = T/(t + tm). In addition, the readoutefficiency C may be a function of tm, as a longer readouttime often allows for an improved readout. We can hencerewrite (39) as

V qmin =eχ(t)√t+ tm

2C(tm)|∂qV p(t)|√T∝ eχ(t)

√t+ tm

2C(tm)γqtq√T, (40)

We can now define the sensitivity as the minimum de-tectable signal per unit time,

vqmin = V qmin

√T =

eχ(t)√t+ tm

2C(tm)γqtq. (41)

From Eq. (41) it is clear that the sensing time t shouldbe made as long as possible. However, as t > Tχ, thedecay function χ(t) exponentially penalizes the sensingtime, imposing a sharp upper limit on t. The optimumsensitivity with respect to the sensing time t is thereforereached when t ≈ Tχ,

vqopt ≈e√Tχ + tm

2C(tm)|∂qV p(Tχ)| for t = Tχ. (42)

In addition, η can be optimized for the overhead time tmif the readout efficiency improves with available readouttime. If C does improve with readout time as C ∝ √tm,which is a typical situation when operating in the aver-aged readout regime, the optimum setting is tm ≈ t. Bycontrast, if C is independent of tm – for example, be-cause the sensor is operated in the single-shot regime orbecause readout resets the sensor – tm should be madeas short as possible.

Finally, we can evaluate Eq. (40) for a few specificsituations:

1. Slope detection

For slope detection, p0 = 0.5 and δp(t) ≈ 12γV t (Eq.

19). The minimum detectable signal per unit time is

vmin =eχ(t)√t+ tm

γC(tm)t. (43)

Assuming tm � t, we can typically find an exact opti-mum solution with respect to the acquisition time t. Forexample, in the case of a Ramsey measurement with anexponential dephasing e−χ(t) = e−t/T

∗2 , we have

vopt =

√2e

γC√T ∗2

(44)

with an optimum evolution time of t = T ∗2 /2. Othercases can be optimized by numerical minimization of Eq.(43).

2. Variance detection

For variance detection, δp ≈ 14γ

2V 2rmst

2 (Eq. 20). Theminimum detectable signal per unit time is

v2min =

2eχ(t)√t+ tm

C(tm)γ2t2≈ 2e

γ2C√T 3χ

, (45)

again in the limit of tm ≈ 0 and t ≈ Tχ. Alternatively,for the detection of a noise spectral density SV (ω), thetransition probability is δp ≈ 1

2γ2SV (ω)Tχ (see Eqs. (80)

and (94)) and

Svmin(ω) ≈ e

γ2C√Tχ

. (46)

17

Likewise, Eq. (45) can be numerically minimized for aspecific measurement and specific set of experimental pa-rameters.

E. Allan variance

In addition to the sensitivity, sensors are typically char-acterized by their stability over time. Indeed, while thesensitivity calculation suggests that one will always im-prove the minimum detectable signal by simply extend-ing the measurement time, additional variations affect-ing the sensor might make this not possible. The Al-lan variance (Allan, 1966) resp. its square root, the Al-lan deviation, quantify these effects. While the conceptis based on a classical analysis of the sensor output, itis still important for characterizing the performance ofquantum sensors. In particular, Allan deviations are typ-ically reported to evaluate the performance of quantumclocks (Hollberg et al., 2001; Leroux et al., 2010b).

Assuming that the sensor is sampled over time at con-stant intervals τ yielding the signal yj = y(tj) = y(jτ),the Allan variance is defined as

σ2Y (τ) =

1

2(M − 1)τ2

M−1∑j=1

(yj+1 − yj)2 , (47)

where M is the number of samples yj . One is typicallyinterested in knowing how σY varies with time, giventhe recorded sensor outputs. To calculate σY (t) one cangroup the data in variable-sized bins and calculate theAllan variance for each grouping. The Allan variance foreach grouping time t = mτ can then be calculated as

σ2Y (mτ) =

1

2(M − 2m)m2τ2

M−2m∑j=1

(yj+m − yj)2. (48)

The Allan deviation can also be used to reveal theperformance of a sensor beyond the standard quantumlimit (Leroux et al., 2010b), and its extension to andlimits in quantum metrology have been recently ex-plored (Chabuda et al., 2016).

F. Quantum Cramer Rao Bound for parameter estimation

The sensitivity of a quantum sensing experiment canbe more rigorously considered in the context of theCramer-Rao bound applied to parameter estimation.Quantum parameter estimation aims at measuring thevalue of a continuous parameter V that is encoded in thestate of a quantum system ρV , via, e.g., its interactionwith the external signal we want to characterize. The es-timation process consists of two steps: In a first step, thestate ρV is measured and in a second step, the estimateof V is determined by data-processing the measurementoutcomes.

In the most general case, the measurement can be de-scribed by a positive-operator-valued measure (POVM)M = {ENx } over the N copies of the quantum system.The measurement yields the outcome x with conditional

probability pN (x|V ) = Tr[E(N)x ρ⊗NV ].

With some further data processing, we arrive at theestimate v of the parameter V . The estimation uncer-tainty can be described by the probability PN (v|V ) :=∑x p

(N)est (v|x)pN (x|V ), where p

(N)est (v|x) is the probability

of estimating v from the measurement outcome x. Wecan then define the estimation uncertainty as δVN :=√∑

v[v − V ]2PN (v|V ). Assuming that the estimationprocedure is asymptotically locally unbiased, δVN obeysthe so-called Cramer-Rao bound

δVN ≥ 1/γ√FN (V ) , (49)

where

FN (V ) :=∑x

1

pN (x|V )

(∂pN (x|V )

∂V

)2

=∑x

1

Tr[E(N)x ρ⊗nV ]

(∂ Tr[E

(N)x ρ⊗nV ]

∂V

)2 (50)

is the Fisher information associated with the givenPOVM measurement.

By optimizing Eq. (49) with respect to all possiblePOVM’s, one obtains the quantum Cramer-Rao bound(QCRB) (Braunstein, 1996; Braunstein and Caves, 1994;Goldstein et al., 2010; Helstrom, 1967; Holevo, 1982;Paris, 2009)

δVN >1

γ√

maxM(N) [FN (V )]>

1

γ√NF(ρV )

, (51)

where the upper bound of maxM(N) [FN (V )] is expressedin terms of the quantum Fisher information F(ρV ), de-fined as

F(ρV ) := Tr[R−1ρV (∂V ρV )ρVR−1

ρV (∂V ρV )], (52)

with

R−1ρ (A) :=

∑j,k:λj+λk 6=0

2Ajk|j〉〈k|λj + λk

(53)

being the symmetric logarithmic derivative written in thebasis that diagonalizes the state, ρV =

∑j λj |j〉〈j|.

A simple case results when ρV is a pure state, obtainedfrom the evolution of the reference initial state |0〉 un-

der the signal Hamiltonian, |ψV 〉 = e−iHV t|0〉. Then, theQCRB is a simple uncertainty relation (Braunstein, 1996;Braunstein and Caves, 1994; Helstrom, 1967; Holevo,1982),

δVN >1

2γ√N ∆H

, (54)

18

where ∆H :=√〈H2〉 − 〈H〉2. We note that the scal-

ing of the QCRB with the number of copies, N−1/2, is aconsequence of the additivity of the quantum Fisher in-formation for tensor states ρ⊗N . This is the well-knownstandard quantum limit (SQL). To go beyond the SQL,one then needs to use entangled states (see Section IX) –in particular, simply using correlated POVMs is not suf-ficient. Thus, to reach the QCRB, local measurementsand at most adaptive estimators are sufficient, withoutthe need for entanglement.

While the quantum Fisher information (and theQCRB) provide the ultimate lower bound to the achiev-able uncertainty for optimized quantum measurments,the simpler Fisher information can be used to evaluate agiven measurement protocol, as achievable, e.g., withinexperimental constraints.

Consider for example the sensing protocols describedin Section IV. For the Ramsey protocol, the quantumsensor state after the interaction with the signal V isgiven by

ρ11(V, t) =1

2ρ12(V, t) = − i

2e−iγV te−χ(t) . (55)

Here, e−χ(t) describes decoherence and relaxation as dis-cussed with Eq. (27). If we assume to perform a projec-tive measurement in the σx basis, {|±〉} = { 1√

2(|0〉±|1〉),

giving the outcome probabilities p(x±|V ) = 〈±|ρ(V )|±〉,the Fisher information is

F =∑x

1

p(x|V )[∂V p(x|V )]

2=

t2 cos2(γV t)e−2χ

1− e−2χ sin2(γV t).

(56)The Fisher information thus oscillates between its min-imum, where γV t = (k + 1/2)π and F = 0, and itsoptimum, where γV t = kπ and F = t2e−2χ. The uncer-tainty in the estimate δV = 1/γ

√NF therefore depends

on the sensing protocol bias point. In the optimum caseF corresponds to the quantum Fisher information andwe find the QCRB

δVN =1

γ√NF

=eχ

γt√N

. (57)

Depending on the functional form of χ(t), we can furtherfind the optimal sensing time for a given total measure-ment time. Similarly, we can analyze variance detectionof random fields.

VI. SENSING OF AC SIGNALS

So far we have implicitly assumed that signals arestatic and deterministic. For many applications, how-ever, it is useful or required to detect time-varying sig-nals V (t). For example, it may be needed to determinethe amplitude, phase or frequency of a time-dependent

signal. More broadly, one might want to estimate thewaveform of a time-varying parameter or reconstruct itsfrequency spectrum. A diverse set of methods has beendeveloped for detecting time-dependent signals, includ-ing spin-echo, multipulse decoupling, continuous drivingprotocols and relaxation time measurements.

Before discussing the various AC sensing protocols, itis important to consider the type of information that oneintends to extract from a time-dependent signal V (t). Inthis Section VI, we will assume that the signal is com-posed of one or a few harmonic tones with a known fre-quency, and our goal will be to determine the signal’s am-plitude, phase or overall waveform. In the following Sec-tion VII, we will discuss the measurement of stochasticsignals with the intent of reconstructing the noise spec-trum or measuring the noise power in a certain band-width. The two Sections are intended to provide an in-troduction and concise overview over available AC sens-ing and spectral reconstruction techniques. Fundamen-tal limits of frequency estimation based on the quantumFisher information, which are not discussed here, havebeen considered by e.g. (Pang and Jordan, 2016).

A. Time-dependent signals

To develop a basic set of AC sensing protocols, we willfocus on a single-tone AC signal given by

V (t) = Vpk cos(2πfact+ α) . (58)

This signal has three basic parameters, including the sig-nal amplitude Vpk, the frequency fac and the relativephase α. Our aim will be to measure one or several ofthese parameters using suitable sensing protocols. Sig-nal detection can be extended to multi-tone signals bysumming over individual single-tone signals.

B. Ramsey sequence revisited

To illustrate the difference between DC and AC sens-ing, we reconsider the Ramsey measurement from SectionIV.C, with the pulse diagram given in Fig. 5(a). Thisprotocol is ideally suited for measuring static shifts of thetransition energy. But is it also capable of detecting dy-namical variations? In order to answer this question, onecan inspect the phase φ accumulated during the sensingtime t due to either a static or a time-dependent signalV (t),

φ =

∫ t

0

γV (t′) dt′ . (59)

For a static perturbation the accumulated phase is sim-ply φ = γV t. For a rapidly oscillating perturbation, bycontrast, phase accumulation is averaged over the sensing

19

t

(a)

(b)

(c)

(d)

Init Readout

time t’

FIG. 5 Pulse diagrams for DC and AC sensing sequences.Narrow blocks represent π/2 pulses (Hadamard gates) andwide blocks represent π pulses (Pauli gates), respectively. tis the total sensing time and τ is the interpulse delay. (a)Ramsey sequence. (b) Spin-echo sequence. (c) CP multipulsesequence. (d) PDD multipulse sequence.

time, and φ = γ〈V (t′)〉t ≈ 0. To answer our question, theRamsey sequence will only be sensitive to slowly varyingsignals up to some cut-off frequency ≈ t−1.

C. Spin echo sequence

Sensitivity to alternating (AC) perturbations can berestored by using time-reversal (“spin echo”) techniques(Hahn, 1950). To illustrate this, we assume that the ACsignal goes through exactly one period of oscillation dur-ing the sensing time t. The Ramsey phase φ due to thissignal is zero because the positive phase build-up duringthe first half of t is exactly canceled by the negative phasebuild-up during the second half of t. However, if the qubitis inverted at time t/2 using a π pulse (Pauli gate), thetime evolution of the second period is reversed (see Fig.5(b)). Thus, a signal oscillating in-phase with the pulsesequence produces an overall additive phase shift (Tay-lor et al., 2008), leading to a total phase accumulationφ = 2

πγVpkt 6= 0.

D. Multipulse sensing sequences

The spin echo sequence produces a maximum responsefor fields oscillating near t = Tac, where Tac = f−1

ac isone oscillation period. Thus, the optimum sensitivity isachieved only for slowly oscillating fields where Tac ≈ T2.

To extend AC sensing to higher frequency signals, se-quences with multiple π rotations can be used. Thesesequences are commonly referred to as multipulse sens-ing sequences or multipulse control sequences. We willnow show that by using multipulse sequences, it becomespossible to extract the frequency, amplitude and phase ofthe signal.

1. Modulation function

To understand the AC characteristics of multipulsesensing sequences, we consider a sequence consisting ofn π pulses separated by interpulse delay time τ (see Fig.5(c,d)). The total duration of this sequence is t = nτ .The phase accumulated from a harmonic signal V (t′) attime t is

φ =

∫ t

0

γV (t′)yn,τ (t′) dt′ , (60)

where yn,τ (t′) = ±1 is the modulation function of themulti-pulse sequence. The modulation function indicatesthe direction of time evolution (forward or backward)at any instance of time t′ due to the application of πrotations. For a general sequence of n π pulses appliedat times 0 < tj < t, with j = 1...n, we obtain the phase

φ =γVpk

2πfac[sin(α)− (−1)n sin(2πfact+ α)

+ 2

n∑j=1

(−1)j sin(2πfactj + α)]

= γVpkt×W (fac, α) (61)

This defines for any multipulse sequence a weightingfunction W (fac, α).

For composite signals consisting of several harmon-ics with different frequencies and amplitudes, the ac-cumulated phase simply represents the sum of individ-ual tones multiplied by the respective weighting functionand amplitude. Even more generally, we can expandany time-dependent signal in a Fourier series V (t′) =∑j Vpk,j cos(2πfac,jt

′ + αj).

2. CP and PDD sequences

The simplest pulse sequences used for sensing havebeen initially devised in nuclear magnetic resonance(NMR) (Slichter, 1996) and been developed further inthe context of dynamical decoupling (DD) (Viola andLloyd, 1998). They include multi-pulse sequences withn π-pulses equally spaced by τ , either Carr-Purcell (CP)pulse trains (Carr and Purcell, 1954) or periodic dynami-cal decoupling (PDD) sequences (Khodjasteh and Lidar,2005) (see Fig. 5).

For a basic CP sequence, tj = 2j−12 τ , and we obtain

the weighting function (Hirose et al., 2012; Taylor et al.,2008)

W cpn,τ =

sin(πfacnτ)

πfacnτ[1− sec(πfacτ)] cos(α+ πfacnτ).

(62)Similarly, for a PDD sequence we have tj = jτ and

W pddn,τ =

sin(πfacnτ)

πfacnτtan(πfacτ) sin(α+ πfacnτ) (63)

20

Because of the first (sinc) term, these weighting functionsresemble narrow-band filters around the center frequen-cies fac = fk = k/(2τ), with k = 1, 3, 5 an odd integer.In fact, they can be rigorously treated as filter functionsthat filter the frequency spectrum of the signal V (t).(This aspect will be considered in more detail in SectionVII). For large pulse numbers n, the sinc term becomesvery peaked and the filter bandwidth ∆f ≈ 1/(nτ) = 1/tbecomes very narrow. The narrow-band filter character-istics can be summed up as follows,

fk = k/(2τ) center frequencies (64)

∆f ≈ 1/t = 1/(2nτ) bandwidth (FWHM)(65)

W cp =2

πk(−1)

k−12 cos(α)

W pdd =2

πksin(α) peak transmission (66)

where k = 1, 3, 5, ... is the harmonic order (see Fig. 6).Thus, the interpulse delay τ can be used to tune the pass-band frequency, and the number of pulses n = t/τ can beused to tune the width of the filter. The time responseof the transition probability is

p =1

2[1− cos (WnγVpkt)] =

1

2

[1− cos

(2γVpkt cosα

kπ

)],

(67)where the last expression holds for CP sequences. Notethat the oscillation of p(t) is reduced by a factor of2 cosα/(πk) = 0.636 × cosα/k compared to the origi-nal Ramsey interferometry measurement (see Eq. 13 andFig. 7(b)). Most reported experiments utilized the k = 1resonance since it provides the strongest peak transmis-sion.

3. Lock-in detection

The phase φ acquired during a multipulse sequence de-pends on the relative phase difference α between the ACsignal and the modulation function yn,τ (t). For a signalthat is in-phase with yn,τ (t), the maximum phase accu-mulation occurs, while for an out-of-phase signal, φ = 0.

This behavior has been exploited to add further ca-pabilities to AC signal detection. (Kotler et al., 2011)have shown that both quadratures of a signal can be re-covered, allowing one to perform lock-in detection of thesignal. Furthermore, it is possible to correlate the phaseacquired during two subsequent multipulse sequences toperform high-resolution spectroscopy of AC signals (seeSection VI.F.

4. Other types of multipulse sensing sequences

Many varieties of multipulse sequences have been con-ceived with the aim of optimizing the basic CP design,

Signal V(t’)

t = n

×

=

0

Rectified signal

(a)

(b)

(c)

time t’0

Modulation function yn,(t’)+1

-1

k=1

k=3

~1/t

N=10

k=5

frequency fac [1/(2)]2 30 1 4 5 6

0.0

0.2

0.4

0.1

0.3

0.5Wn,(fac)2

FIG. 6 Modulation and weight functions of a CP multipulsesequence. (a) CP multipulse sequence according to Fig. 5(c).(b) Signal V (t′), modulation function yn,τ (t′) and “rectified”signal V (t′)× yn,τ (t′). The accumulated phase is representedby the area under the curve. (c) Weight function W 2

n,τ (fac)associated with the modulation function. k is the harmonicorder of the filter resonance.

including improved robustness against pulse errors, bet-ter decoupling performance, narrower spectral responseand sideband suppression, and avoidance of signal har-monics.

A systematic analysis of many common sequences hasbeen given by (Cywinski et al., 2008). One favorite hasbeen the XY4, XY8 and XY16 series of sequences (Gul-lion et al., 1990) owing to their high degree of pulse errorcompensation. A downside of XY type sequences aresignal harmonics (Loretz et al., 2015) and the sidebandscommon to CP sequences with equidistant pulses. Otherrecent efforts include sequences with non-equal pulsespacing (Casanova et al., 2015; Zhao et al., 2014) or se-quences composed of alternating subsequences (Albrechtand Plenio, 2015). A Floquet spectroscopy approach tomultipulse sensing has also been proposed (Lang et al.,2015).

21

5. AC signals with random phase and/or random amplitude

Often, the multipulse sequence cannot be synchronizedwith the signal or the phase α cannot be controlled.Then, incoherent signal averaging leads to cancellation,〈φ〉 = 0. In this case, it is advantageous to measure thevariance of the phase 〈φ2〉 rather than its average 〈φ〉.(Although such a signal technically represents a stochas-tic signal, which will be considered in more detail in thenext section, it is more conveniently described here.)

For a signal with fixed amplitude but random phase,the variance is

〈φ2〉 = γ2V 2rmst

2W 2n,τ (fac), (68)

where Vrms = Vpk/√

2 is the rms amplitude of the signaland W 2

n,τ is the average over α = 0...2π of the weightingfunctions,

W 2n,τ (fac) =

1

2π

∫ 2π

0

W 2n,τ (fac, α) dα (69)

For the CP and PDD sequences, the averaged functionsare given by

W 2n,τ (fac) =

sin2(πfacnτ)

2(πfacnτ)2[1− sec (πfacτ)]

2(cp),

W 2n,τ (fac) =

sin2(πfacnτ)

2(πfacnτ)2tan (πfacτ)

2(pdd), (70)

and the peak transmission at fac = k/(2τ) is W 2k =

2/(kπ)2.

The transition probability as a function of time t showsa damped oscillatory behavior (see Fig. 7(c)),

p(t) =1

2

[1− J0

(2Wn,τγVrmst

)]=

1

2

[1− J0

(2√

2γVrmst

kπ

)](71)

where J0 is the Bessel function of the first kind and wherethe second equation reflects the resonant case. If the am-plitude Vpk is not fixed, but slowly fluctuating betweendifferent measurements, the variance 〈φ2〉 must be inte-grated over the probability distribution of Vpk. A partic-ularly important situation is a Gaussian amplitude fluc-tuation with an rms amplitude Vrms. In this case, thetime response of the transition probability is

p(t) =1

2

[1− exp

(−W

2n,τγ

2V 2rmst

2

2

)I0

(W 2n,τγ

2V 2rmst

2

2

)](72)

where I0 is the modified Bessel function of the first kind(see Fig. 7(d)).

p(t)

1.0

0.5

0.00

accumulated phase = Vt2 3 4

(a)

(e)

(d)

(c)

(b)

FIG. 7 Transition probability p(t) as a function of phase ac-cumulation time t. (a) Ramsey oscillation (Eq. 15). (b) ACsignal with fixed amplitude and optimum phase (Eq. 67). (c)AC signal with fixed amplitude and random phase (Eq. 71).(d) AC signal with random amplitude and random phase (Eq.72). (e) Broadband noise with χ = Γt (Eq. 79).

E. Waveform reconstruction

The detection of AC fields can be extended to themore general task of sensing and reconstructing arbi-trary time dependent fields. As explained in the previoussection, using periodic dynamical decoupling sequencesallows one to not only optimally detect AC fields withknown frequencies, but also to estimate their frequen-cies. This could be done, e.g., by measuring the responseto pulse sequences with different pulse spacings τ .

To more systematically reconstruct the time depen-dence of an arbitrary signal, one may use a family of pulsesequences that forms a basis for the signal. A suitablebasis are Walsh dynamical decoupling sequences (Hayeset al., 2011), which apply a π pulse every time the cor-responding Walsh function (Walsh, 1923) flips its sign.Under a control sequence with m π-pulses applied at thezero-crossings of the m-th Walsh function wm(t′/t), thephase difference acquired after an acquisition period t is

φ(t) = γ

∫ t

0

V (t′)wm(t′/t)dt′ = γVmt , (73)

which is proportional to the m-th Walsh coefficient Vm ofV (t′). By measuring N Walsh coefficients (by applyingN different sequences) one can reconstruct an N -pointfunctional approximation to the field V (t′) from the N -th partial sum of the Walsh-Fourier series (Cooper et al.,2014; Magesan et al., 2013b)

VN (t′) =

N−1∑m=0

Vmwm(t′/T ), (74)

which can be shown to satisfy limN→∞ VN (t′) = V (t′). Asimilar result can be obtained using different basis func-tions, such as Haar wavelets, as long as they can be easilyimplemented experimentally.

22

multipulsesequence 1

multipulsesequence 2

V(t)

time t’

t1

ta tb

Init Readout

FIG. 8 Correlation spectroscopy. An AC sequence is inter-rupted half-way by an incremented delay time t1. Phase accu-mulation now occurs in two separated intervals (bold signal).Because the multipulse sequences are phase-sensitive, con-structive or destructive interference occurs between the twointervals with a periodicity of the AC signal. The frequencyresolution of the correlation sequence is Fourier-limited by themaximum possible t1, which can be longer than the coherencetime of the quantum sensor.

An advantage of these methods is that they provideprotection of the sensor against dephasing, while extract-ing the desired information. In addition, they can becombined with compressive sensing techniques (Candeset al., 2006; Magesan et al., 2013a; Puentes et al., 2014)to further reduce the number of acquisition needed toreconstruct the time-dependent signal. The ultimatemetrology limits in waveform reconstruction have alsobeen studied (Tsang et al., 2011).

F. Correlation sequences

According to Eq. (65), the frequency bandwidth of amultipulse sensing sequence is Fourier limited to ∆f ∼1/t. Since the maximum sensing time t is in turn lim-ited by decoherence to t . T2, the decoherence time T2

provides a lower limit to the bandwidth of the sequence.

Several schemes have been proposed and demonstratedto further narrow the bandwidth and to perform highresolution spectroscopy. All of them rely on correlation-type measurements where the outcomes of two subse-quent sensing periods are correlated. The correlationsignal can either be established between two subsequentreadouts {xk, xk+1} (Degen et al., 2007; Laraoui et al.,2010) or between two phase accumulation times ta and tbusing a single readout (Laraoui and Meriles, 2011, 2013).

The second method is illustrated in Fig. 8 in combi-nation with multipulse detection. In this example, themultipulse sequence is subdivided into two equal sens-ing periods of duration ta = tb = t/2 that are separatedby an incremented free evolution period t1. Since themultipulse sequence is phase sensitive, constructive ordestructive phase build-up occurs between the two se-quences depending on whether the free evolution periodt1 is a half multiple or full multiple of the AC signal pe-riod Tac = 1/fac. The final transition probability there-

fore oscillates with t1 as

p(t1) =1

2{1− sin[Φ cos(α)] sin[Φ cos(α+ 2πfact1)]}

≈ 1

2

{1− Φ2 cos(α) cos(α+ 2πfact1)

}(75)

where Φ = γVpkt/(kπ) is the maximum phase that canbe accumulated during either of the two multipulse se-quences. The second equation is for small signals wheresin Φ ≈ Φ. For signals with random phase α, the transi-tion probability further simplifies to

p(t1) ≈ 1

2

{1− Φ2

2cos(2πfact1)

}(76)

Since the qubit stays aligned with H0 during the freeevolution period, relaxation is no longer governed by T2,but by the typically much longer T1 relaxation time.In this way, a Fourier-limited spectral resolution set by∼ 1/T1 is possible. Moreover, if a long-lived auxiliarymemory qubit (such as a nuclear spin) is available, thestate of the sensing qubit can be stored in the mem-ory qubit (see Section X). The correlation protocol wasfurther shown to eliminate several of the shortcomingsof multipulse sequences, including signal ambiguities re-sulting from the multiple frequency windows and spectralselectivity (Boss et al., 2016).

VII. NOISE SPECTROSCOPY

In this Section, we will introduce methods for recon-structing the frequency spectrum of time-dependent sig-nals. Unlike in the previous section, we will assume thatthe signal V (t) is stochastic and composed of many differ-ent frequencies. Noise spectroscopy is an important toolin quantum sensing, as it can provide much insight intoboth external signals and the intrinsic noise of the quan-tum sensor itself. In particular, good knowledge of thenoise spectrum can help the adoption of suitable sensingprotocols (like dynamical decoupling or quantum errorcorrection schemes) to maximize the sensitivity of thequantum sensor.

The key handle to noise spectroscopy lies in the sys-tematic analysis of decoherence and relaxation. Althoughthese effects are typically considered adverse to mosttasks in quantum science and technology, they are im-portant resources in quantum sensing. In the following,we will discuss two approaches for extracting noise spec-tra. In Section VII.B we will introduce the concept offilter functions, where decoherence is systematically an-alyzed under different dynamical decoupling sequences.In Section VII.C we will discuss the alternative conceptof “relaxometry”, which has its origins in the field ofmagnetic resonance spectroscopy and is closely relatedto Fermi’s golden rule. Together, the two concepts pro-vide a comprehensive framework for analyzing stochasticsignals over a wide frequency range.

23

A. Noise processes

For most of the following analysis we will assume thatthe stochastic signal V (t) is approximately Gaussian.Such noise can be described by simple noise models, like asemi-classical Gaussian noise or the Gaussian spin-bosonbath. Moreover, we assume that V (t) is largely uncorre-lated, or more specifically, that the autocorrelation func-tion

GV (t) = 〈V (t′ + t)V (t′)〉 (77)

decays on a time scale tc that is shorter than the sensingtime t. Equivalently, we require that the power spectraldensity SV (ω), (Biercuk et al., 2011),

SV (ω) =

∫ ∞−∞

e−iωtGV (t) dt , (78)

has no sharp features in the frequency region of interest.Our main aim in this Section will be measure the noisespectral density SV (ω) as a function of frequency ω.

Although the discussion will mostly focus on Gaussianand uncorrelated noise, we note that the analysis can beextended to other noise models. For correlated noise,where tc � t, the frequency and amplitude of V (t) areessentially fixed during one sensing period and the for-malism of AC sensing can be applied (see Section VI.D).A rigorous derivation for all ranges of tc, but especiallytc ≈ t has been given by (Cummings, 1962). More-over, more complex noise models beyond a first orderapproximation on the noise strength can be considered.Recently, open-loop control protocols have been intro-duced (Norris et al., 2016; Paz-Silva and Viola, 2014) tocharacterize stationary, non-Gaussian dephasing using aqubit probe.

B. Decoherence, dynamical decoupling and filter functions

There have been many proposals for examining deco-herence under different control sequences to investigatenoise spectra (Almog et al., 2011; Faoro and Viola, 2004;Young and Whaley, 2012; Yuge et al., 2011). In particu-lar, dynamical decoupling sequences based on multipulseprotocols (Section VI.D) provide a systematic means forfiltering environmental noise (Alvarez and Suter, 2011;Biercuk et al., 2011; Kotler et al., 2011). These have beenimplemented in many physical systems (Bar-Gill et al.,2012; Bylander et al., 2011; Dial et al., 2013; Kotler et al.,2013; Muhonen et al., 2014; Romach et al., 2015; Yanet al., 2012, 2013; Yoshihara et al., 2014). A brief intro-duction to the method of filter functions is presented inthe following. Filter functions are closely related to theweighting functions introduced in Section VI, Eq. (70).

1. Decoherence function χ(t)

The analysis of decoherence is simplified under the as-sumption of a Gaussian, stationary noise. Then, the lossof coherence is captured by a simple decay function ordecoherence function χ(t) with the associated transitionprobability

p(t) =1

2

(1− e−χ(t)

). (79)

Comparing with the expression for variance detection,Eq. (21), the decoherence function can be identified withthe rms phase accumulated during time t,

χ(t) =1

2φ2

rms . (80)

2. Filter function Y (ω)

The decoherence function χ(t) can be analyzed un-der the effect of different control sequences. Assumingthe control sequence has a modulation function y(t) (seeSection VI.D), the decay function is determined by thecorrelation integral (Biercuk et al., 2011)

χ(t) =1

2

∫ t

0

dt′∫ t

0

dt′′ y(t′)y(t′′)γ2GV (t′ − t′′) , (81)

where GV (t) is the autocorrelation function of V (t) (seeEq. 77). In the frequency domain the decay function canbe expressed as

χ(t) =2

π

∫ ∞0

γ2SV (ω)|Y (ω)|2dω , (82)

where |Y (ω)|2 is the so-called filter function of y(t), de-fined by the finite-time Fourier transform

Y (ω) =

∫ t

0

y(t′)eiωt′dt′ . (83)

(Note that this definition differs by a factor of ω2 fromthe one by (Biercuk et al., 2011)). Thus, the filter func-tion plays the role of a transfer function, and the decayof coherence is captured by the overlap with the noisespectrum SV (ω).

To illustrate the concept of filter functions, one canconsider the canonical example of a Ramsey sensing se-quence. Here, the filter function is

|Y (ω)|2 =sin2(ωt/2)

ω2. (84)

The decoherence function χ(t) then describes the “free-induction decay”,

χ(t) =2

π

∫ ∞0

γ2SV (ω)sin2(ωt/2)

ω2dω ≈ 1

2γ2SV (0)t ,

(85)

24

where the last equation is valid for a spectrum that isflat around ω ≈ 0 or more generally in the limit of larget. The Ramsey sequence hence acts as a simple sinc fil-ter for the noise spectrum SV (ω) with a lowpass cut-offfrequency of approximately π/t.

3. Dynamical decoupling

To perform a systematic spectral analysis of SV (ω),one can examine decoherence under various dynamicaldecoupling sequences. Specifically, we inspect the fil-ter functions of periodic modulation functions ync,τc(t),where a basic building block y1(t) of duration τc is re-peated nc times. The filter function of ync,τc(t) is givenby

Ync,τc(ω) = Y1,τc(ω)

nc−1∑k=0

eiτck

= Y1,τc(ω)e−i(nc−1)ωτc/2sin(ncωτc/2)

sin(ωτc/2), (86)

where Y1,τc(ω) is the filter function of the basic buildingblock. For large cycle numbers, Ync,τc(ω) presents sharppeaks at multiples of the inverse cycle time τ−1

c , and itcan be approximated by a series of δ functions.

Two specific examples of periodic modulation func-tions include the CP and PDD sequences considered inSection VI.D, where τc = 2τ and nc = n/2. The filterfunction for large pulse numbers n is

|Yn,τ |2 ≈∑k

2π

(kπ)2sinc[(ω − ωk)t/2]2

≈∑k

2π

(kπ)2tδ(ω − ωk) (87)

where ωk = 2π × k/(2τ) are resonances with k =1, 3, 5, .... The decay function can then be expressed bya simple sum of different spectral density components,

χ(t) =2

π

∫ ∞0

γ2SV (ω)∑k

2π

(kπ)2tδ(ω − ωk) dω

=4t

π2

∑k

γ2SV (ωk)

k2(88)

This result provides a simple strategy for reconstruct-ing the noise spectrum. By sweeping the time τ betweenpulses the spectrum can be probed at various frequen-cies. Since the filter function is dominated by the firstharmonic (k = 1) the frequency corresponding to a cer-tain τ is 1/(2τ). For a more detailed analysis the contri-butions from higher harmonics as well as the exact shapeof the filter functions has to be taken into account. Thespectrum can then be recovered by spectral decomposi-tion (Alvarez and Suter, 2011; Bar-Gill et al., 2012).

The filter analysis can be extended to more generaldynamical decoupling sequences. In particular, (Zhaoet al., 2014) consider periodic sequences with more com-plex building blocks, and (Cywinski et al., 2008) consideraperiodic sequences like the UDD sequence.

C. Relaxometry

An alternative framework for analyzing relaxation anddoherence has been developed in the context of magneticresonance spectroscopy, and is commonly referred to as“relaxometry”. The aim of relaxometry is to connectthe spectral density SV (ω) of a noise signal V (t) to therelaxation rate Γ in first-order kinetics, χ(t) = Γt. Re-laxometry is based on first-order perturbation theory andFermi’s golden rule. The basic assumptions are that thenoise process is approximately Markovian and that thenoise strength is weak, such that first-order perturbationtheory is valid. Relaxometry has found many applica-tions in magnetic resonance and other fields, especiallyfor mapping high-frequency noise based on T1 relaxationtime measurements (Kimmich and Anoardo, 2004).

1. Basic theory of relaxometry

To derive a quantitative relationship between the de-cay rate Γ and a noise signal V (t), we briefly revisit theelementary formalism of relaxometry. In a first step, V (t)can be expanded into Fourier components,

V (t) =1

2π

∫ ∞−∞

dω{V (ω)e−iωt + V †(ω)eiωt

}(89)

where V (ω) = V †(−ω). Next, we calculate the proba-bility amplitude c1 that a certain frequency componentV (ω) causes a transition between two orthogonal sensingstates |ψ0〉 and |ψ1〉 during the sensing time t. Since theperturbation is weak, perturbation theory can be applied.The probability amplitude c1 in first order perturbationtheory is

c1(t) = −i∫ t

0

dt′ 〈ψ1|HV (ω)|ψ0〉ei(ω01−ω)t′

= −i〈ψ1|HV (ω)|ψ0〉ei(ω01−ω)t − 1

i(ω01 − ω)(90)

where HV (ω) is the Hamiltonian associated with V (ω)and where ω01 is the transition energy between states|ψ0〉 and |ψ1〉. The transition probability is

|c1(t)|2 = |〈ψ1|HV (ω)|ψ0〉|2(

sin[(ω01 − ω)t/2]

(ω01 − ω)/2

)2

≈ 2π|〈ψ1|HV (ω)|ψ0〉|2tδ(ω01 − ω) (91)

25

where the second equation reflects that for large t, thesinc function approaches a δ function peaked at ω01. Theassociated transition rate is

∂|c1(t)|2∂t

≈ 2π|〈ψ1|HV (ω)|ψ0〉|2δ(ω01 − ω) . (92)

This is Fermi’s golden rule expressed for a two-level sys-tem that is coupled to a radiation field with a continuousfrequency spectrum (Sakurai and Napolitano, 2011).

The above transition rate is due to a single frequencycomponent of HV (ω). To obtain the overall transitionrate Γ, Eq. (92) must be integrated over all frequencies,

Γ =1

π

∫ ∞0

dω 2π|〈ψ1|HV (ω)|ψ0〉|2δ(ω01 − ω)

= 2|〈ψ1|HV (ω01)|ψ0〉|2

= 2γ2SV01(ω01)|〈ψ1|.|ψ0〉|2 (93)

where in the last equation, SV01is the spectral density

of the component(s) of V (t) than can drive transitionsbetween |ψ0〉 and |ψ1〉, multiplied by a transition matrixelement |〈ψ1|.|ψ0〉|2 of the operator part of HV (see Eq.5). The transition matrix element equals 1/4 for HV||

and HV⊥ ; specific examples are discussed below.The last equation (93) is an extremely simple, yet

powerful and quantitative relationship: The transitionrate equals the spectral density of the noise evaluatedat the transition frequency, multiplied by a matrix ele-ment of order unity. The expression can also be inter-preted in terms of the rms phase φrms. According to Eq.(80), φ2

rms = 2χ(t) = 2Γt, which in turn yields (setting|〈ψ1|.|ψ0〉|2 = 1

4 )

φ2rms = γ2SV01(ω01)t . (94)

The rms phase thus corresponds to the noise integratedover an equivalent noise bandwidth of 1/(2πt).

The relation between the transition rate Γ and thespectral density can be further specified for vector signals~V . In this case the transition rate represents the sum ofthe three vector components of Vj , where j = x, y, z,

Γ = 2∑

j=x,y,z

|〈ψ1|HVj (ωj)|ψ0〉|2

= 2∑

j=x,y,z

γ2SVj (ωj)|〈ψ1|σj |ψ0〉|2 (95)

where SVj(ωj) is the spectral density of Vj , ωj is a transi-

tion frequency that reflects the energy exchange requiredfor changing the state, and σj are Pauli matrices. Notethat if {|ψ0〉, |ψ1〉} are coherent superposition states, Vxand Vy represent the components of V⊥ that are in-phaseand out-of-phase with the coherence, rather than thestatic components of the vector signal ~V .

Relaxation rates can be measured between any setof sensing states {|ψ0〉, |ψ1〉}, including superposition

t

t

t

Init

(a)

(b)

(c)

Readout

Init Readout

Init Readout

FIG. 9 Common relaxometry protocols. (a) T1 relaxometry.(b) T ∗2 relaxometry. (c) T1ρ relaxometry. Narrow black boxesrepresent π/2 pulses (Hadamard gates) and the grey box in(c) represents the resonant “spin lock” Rabi field.

states. This gives rise to a great variety of possible re-laxometry measurements. For example, the method canbe used to probe different vector components Vj(t) (orcommuting and non-commuting signals V||(t) and V⊥(t),respectively) based on the selection of sensing states.Moreover, different sensing states typically have vastlydifferent transition energies, providing a means to covera wide frequency spectrum. If multiple sensing qubits areavailable, the relaxation of higher-order quantum transi-tions can be measured, which gives additional freedom toprobe different symmetries of the Hamiltonian.

An overview of the most important relaxometry pro-tocols is given in Table II and Fig. 9. They are brieflydiscussed in the following.

2. T1 relaxometry

T1 relaxometry probes the transition rate betweenstates |0〉 and |1〉. This is the canonical example of en-ergy relaxation. Experimentally, the transition rate ismeasured by initializing the sensor into |0〉 at time t′ = 0and inspecting p = |〈1|α〉|2 at time t′ = t without anyfurther manipulation of the quantum system (see Fig.9(a)). The transition rate is

(T1)−1 =1

2γ2SVx

(ω0) +1

2γ2SVy

(ω0) =1

2γ2SV⊥(ω0)

(96)

where T1 is the associated relaxation time. Thus, T1

relaxometry is only sensitive to the transverse componentof ~V .

By tuning the energy splitting ω0 between |0〉 and|1〉, for example through the application of a staticcontrol field, a frequency spectrum of SVx

(ω) can berecorded (Kimmich and Anoardo, 2004). For this reasonand because it is experimentally simple, T1 relaxometryhas found many applications. For example, single-spinprobes have been used to detect the presence of mag-netic ions (Steinert et al., 2013), spin waves in magneticfilms (van der Sar et al., 2015), or high-frequency mag-

26

Method Sensing states{|ψ0〉, |ψ1〉}

Sensitive to Vz atfrequency

Sensitive to Vx,y atfrequency

Frequency tunable via

Ramsey {|+〉, |−〉} 0 —a —Spin echo {|+〉, |−〉} 1/t —a —Dynamical decoupling {|+〉, |−〉} πk/τ , with k = 1, 3, .. —a Pulse spacing τ , resonance order kT1 relaxometry {|0〉, |1〉} — ω0 Static control fieldT1ρ relaxometry {|+〉, |−〉} ω1 —a Rabi field amplitude ω1

TABLE II Summary of noise spectroscopy methods. |±〉 = (|0〉 ± |1〉)/√

2. Vz is the longitudinal and Vx,y are the transverse

components of a vector signal ~V . aalso affected by T1 relaxation.

1 GHz1 MHz1 kHzDC

Electronic spins, superconducting qubits

1 MHz1 kHz1 HzDC

Nuclear spins, trapped ions (vibrational)

T1 relaxometry

Ramsey

Multipulse

Hahn echo

T1 relaxometry

FIG. 10 Typical spectral range of noise spectroscopy proto-cols. Scales refer to the quantum sensors discussed in SectionIII.

netic fluctuations near surfaces (Myers et al., 2014; Ro-mach et al., 2015; Rosskopf et al., 2014). T1 relaxom-etry has also been applied to perform spectroscopy ofelectronic and nuclear spins (Hall et al., 2016). More-over, considerable effort has been invested in mappingthe noise spectrum near superconducting flux qubits bycombining several relaxometry methods (Bialczak et al.,2007; Bylander et al., 2011; Lanting et al., 2009; Yanet al., 2013).

3. T ∗2 and T2 relaxometry

T ∗2 relaxometry probes the transition rate between thesuperposition states |±〉 = (|0〉 ± e−iω0t|1〉)/

√2. This

corresponds to the free induction decay observed in aRamsey experiment (Fig. 9(b)). The associated dephas-ing time T ∗2 is given by

(T ∗2 )−1 =1

2γ2SVx(ω0) +

1

2γ2SVz (0) (97)

which can be compared to Eq. (85). The out-of-phaseterm SVy in Eq. (97) involves a “bit flip” and the lon-gitudinal SVz

term involves a “phase flip”. Because aphase flip does not require energy, the spectral density

is probed at zero frequency. Since SV (ω) is often dom-inated by low-frequency noise, SVz (0) is typically muchlarger than SVx

(ω0) and the high-frequency contributioncan often be neglected.

T ∗2 relaxometry can be extended to include dephasingunder dynamical decoupling sequences. The relevant re-laxation time is then usually denoted by T2 rather thanT ∗2 . Dephasing under dynamical decoupling is more rig-orously described by using filter functions (see SectionVII.B.2).

4. T1ρ relaxometry

T1ρ relaxometry probes the transition rate between|±〉 is observed under continuous driving with a resonant(Rabi) field. This method is known as “spin locking” inmagnetic resonance (Slichter, 1996). Due to the presenceof the resonant field the degeneracy between |±〉 is liftedand the states are separated by the energy ~ω1, whereω1 � ω0 is the Rabi frequency. A phase flip therefore isno longer energy conserving. The associated relaxationtime T1ρ is given by

(T1ρ)−1 ≈ 1

2γ2SVy

(ω0) +1

2γ2SVz

(ω1) (98)

By systematically varying the Rabi frequency ω1, thespectrum SV0(ω1) can be recorded. Because Rabi fre-quencies are typically much smaller than the transitionfrequency ω0, dressed states are a useful way for extend-ing noise spectroscopy to lower frequencies (Loretz et al.,2013; Yan et al., 2013). Opposite to T1 relaxometry, T1ρ

relaxometry does not require sweeping a static controlfield for adjusting the probe frequency. This is advanta-geous because sweeping a control field, such as a magneticor electric field, can also affect the physical system understudy.

T1ρ relaxometry can be extended to include off-resonant Rabi fields. If the driving field is detunedby ∆ω from ω0, the effective Rabi frequency becomes

27

ωeff =√ω2

1 + ∆ω2, and the modified T1ρ is given by

(T1ρ)−1 ≈ 1

2

∆ω2

ω2eff

γ2SVx(ω0)

+1

2γ2SVy

(ω0) +1

2

ω21

ω2eff

γ2SVz(ωeff) . (99)

Thus, off-resonant driving can be used to extend the ac-cessible frequency range. Clearly, T1ρ → T1 as the de-tuning ∆ω becomes large.

VIII. DYNAMIC RANGE AND ADAPTIVE SENSING

“Adaptive sensing” refers to a class of technique ad-dressing the intrinsic problem of limited dynamic rangein quantum sensing: The basic quantum sensing pro-tocol cannot simultaneously achieve high sensitivity andmeasure signals over a large amplitude range. In otherwords, the dynamic range – defined by the maximum sig-nal that does not saturate the sensor divided by smallestdetectable signal – is limited.

The origin of this problem is illustrated for a Ramseyexperiment in Fig. 11. The assignment of a frequencyω to a measured transition probability p is ambiguousdue to the periodic nature of Ramsey fringes, where ameasured value of p can correspond to multiple valuesω0, ω1, . . . , ωN where N is in principle infinite. A uniqueassignment hence requires a priori knowledge – that ωlies within ±π/(2t), or within half a Ramsey fringe, ofthe expected frequency, where t is the sensing time.

On the other hand, the sensitivity – which is propor-tional to the slope of the Ramsey fringe – improves witht and reaches its optimum when t ≈ T ∗2 . In the ab-sence of readout noise, the smallest detectable signal isgiven by ωmin = 2π/

√T ∗2 T , where T is the total mea-

surement time. The maximum signal change that canbe accommodated under this condition is ωmax = π/T ∗2 .For a measurement at the standard quantum limit thedynamic range is thus limited to

DR =ωmax

ωmin=

1

2

√T

T ∗2∝√T . (100)

Hence, at the standard quantum limit of sensitivity themeasurement can be applied only to small changes of aquantity around a fixed known value, frequently zero. Itdoes not apply to the problem of determining the valueof a large and a priori unknown quantity. Moreover, thedynamic range improves only with the square root of thetotal measurement time.

A. Phase estimation protocols

Interestingly, a family of advanced sensing techniquescan solve this latter problem more efficiently than DR ∝

/tt

init readout

t’

t/2

t/4

p(t’)

least significantdigit (LSD)

mostsignificantdigit (MSD)

.

.

.

...

maximum signal maxminimum signal min

FIG. 11 High dynamic range sensing. A series of measure-ments with different interrogation times t is combined to esti-mate a quantity of interest. The shortest measurement (low-est line) has the highest maximum signal ωmax and providesa coarse estimate of the quantity, which is subsequently re-fined by longer and more sensitive measurements. Frequently,interrogation times are scaled exponentially t = 2N t0, wheret0 is the basic time element, so that each measurement effec-tively provides one digit in a binary expansion of the quantity.

√T . The key idea is to combine measurements with dif-

ferent interrogation times t such that the least sensitivemeasurement with the highest ωmax yields a coarse esti-mate of the quantity of interest, which is subsequentlyrefined by more sensitive measurements with a smallerωmax (Fig. 11). This scheme can achieve a DR ∝ T scal-ing and is sometimes referred to as the Heisenberg limit,because it can be regarded as a 1/T scaling of sensitivityat fixed ωmax.

In all of the following we will discuss protocols that em-ploy exponentially growing interrogation times tn = t02n,where n = 0, 1, . . . , N and t0 is the smallest time element.Although other scheduling is possible, this choice allowsfor an intuitive interpretation: subsequent measurementsmeasure subsequent digits of a binary expansion of thequantity of interest. Clearly, the maximum allowed sig-nal is now ωmax = π/t0. On the other hand, the smallestdetectable signal is ωmin ≈ 2π/

√tNT . Because T ∝ tN

due to the exponentially growing interrogation times, thedynamic range of the improved protocol therefore scalesas

DR ∝√tNT

t0∝ T . (101)

This scaling is obvious from an order-of-magnitude esti-mate: adding an additional measurement step increasesboth precision and measurement duration t by a factor of2, such that precision scales linearly with total acquisitiontime T . We will now discuss three specific implementa-tions of this idea.

28

QFT-1

UU2U4

MSD

LSD

UU2U4X

X

XR(/4)

R(/2)

R(/2) MSD

LSD

UU2U4

X

X

XR()

Bayesianestimator

R()

R()

MSD

XR()

LSD

XR()

XR()'

'

'

0H

0H

0H

0H

0H

0H

0H

0H

0H

0H

0H

0H

(a) Quantum phase estimation

(b) Adaptive

(c) Non-adaptive

FIG. 12 Phase estimation algorithms. (a) Quantum phaseestimation by the inverse Quantum Fourier transform, as itis employed in prime factorization algorithms (Kitaev, 1995;Shor, 1994). (b) Adaptive phase estimation. Here, the quan-tum Fourier transform is replaced by measurement and clas-sical feedback. Bits are measured in ascending order, sub-stracting the lower digits from measurements of higher digitsby phase gates that are controlled by previous measurementresults. (C) Non-adaptive phase estimation. Measurementsof all digits are fed into a Bayesian estimation algorithm toestimate the most likely value of the phase. H represents aHadamard gate, R(Φ) a Z-rotation by the angle Φ, and U thepropagator for one time element t0. The box labeled by “x”represents a readout.

1. Quantum phase estimation

All three protocols can be considered variations of thephase estimation scheme depicted in Fig. 12(a). Thescheme has originally been put forward by (Shor, 1994)in the seminal proposal of a quantum algorithm for primefactorization and has been interpreted by (Kitaev, 1995)as a phase estimation algorithm.

The original formulation applies to the problem of find-ing the phase φ of the eigenvalue e2πiφ of a unitary opera-tor U , given a corresponding eigenvector |ψ〉. This prob-

lem can be generalized to estimating the phase shift φimparted by passage through an interferometer or expo-sure to an external field. The algorithm employs a regis-ter of N auxiliary qubits (N = 3 in Fig. 12) and preparesthem into a digital representation |φ〉 = |φ1〉 |φ2〉 . . . |φN 〉of a binary expansion of φ =

∑Nj=1 φj2

−j by a sequenceof three processing steps:

1. State preparation: All qubits are prepared in a co-herent superposition state |+〉 = (|0〉+ |1〉)/

√2 by

an initial Hadamard gate. The resulting state ofthe full register can then be written as

1√2N

2N−1∑n=0

|n〉N (102)

where |n〉N denotes the register state in binary ex-pansion |0〉N = |00 . . . 0〉, |1〉N = |00 . . . 1〉, |2〉N =|00 . . . 10〉, etc.

2. Phase encoding: The phase of each qubit is taggedwith a multiple of the unknown phase φ. Specifi-cally, qubit j is placed in state (|0〉+e2πi2jφ |1〉)/

√2.

Technically, this can be implemented by exploitingthe back-action of a controlled-U2j

-gate that is act-ing on the eigenvector ψ conditional on the state ofqubit j. Since ψ is an eigenvector of Un for ar-bitrary n, this action transforms the joint qubit-eigenvector state as

1√2(|0〉+ |1〉)⊗ |ψ〉

→ 1√2(|0〉+ e2πi2jφ |1〉)⊗ |ψ〉 (103)

Here, the back-action on the control qubit j createsthe required phase tag. The state of the full registerevolves to

1√2N

2N−1∑n=0

e2πiφn/2N |n〉 (104)

In quantum sensing, phase tagging by back-actionis replaced by the exposure of each qubit to an ex-ternal field for a time 2jt0 (or passage through aninterferometer of length 2j l0).

3. Quantum Fourier Transform: In a last step, an in-verse quantum Fourier transform (QFT) (Nielsenand Chuang, 2000) is applied to the qubits. Thisalgorithm can be implemented with polynomial ef-fort (i.e., using O(N2) control gates). The QFTrecovers the phase φ from the Fourier series (104)and places the register in state

|φ〉 = |φ1〉 |φ2〉 . . . |φN 〉 . (105)

A measurement of the register directly yields a dig-ital representation of the phase φ. To provide an

29

estimate of φ with 2−N accuracy, 2N applicationsof the phase shift U are required. Hence, the al-gorithm scales linearly with the number of applica-tions of U which in turn is proportional to the totalmeasurement time T .

Quantum phase estimation is the core component ofShor’s algorithm, where it is used to compute discretelogarithms with polynomial time effort (Shor, 1994).

2. Adaptive phase estimation

While quantum phase estimation based on the QFTcan be performed with polynomial time effort, the algo-rithm requires two-qubit gates, which are difficult to im-plement experimentally. This limitation can be circum-vented by an adaptive measurement scheme that readsthe qubits sequentially, feeding back the classical mea-surement result into the quantum circuit (Griffiths andNiu, 1996). The scheme is illustrated in Fig. 12(b).

The key idea of adaptive phase estimation is to firstmeasure the least significant bit of φ, represented by thelowest qubit in Fig. 12(b). In the measurement of thenext significant bit, this value is subtracted from the ap-plied phase. The subtraction can be implemented byclassical unitary rotations conditioned on the measure-ment result, for example by controlled R(π/n) gates asshown in Fig. 12(b). This procedure is then repeated inascending order of the bits. The QFT is thus replaced bymeasurement and classical feedback, which can be per-formed using a single qubit sensor.

In a practical implementations (Higgins et al., 2007),the measurement of each digit is repeated multiple timesor performed on multiple parallel qubits. This is possiblebecause the controlled-U gate does not change the eigen-vector ψ, so that it can be re-used as often as required.The Heisenberg limit can only be reached if the numberof resources (qubits or repetitions) spent on each bit arecarefully optimized (Berry et al., 2009; Cappellaro, 2012;Said et al., 2011). Clearly, most resources should be al-located to the most significant bit, because errors at thisstage are most detrimental to sensitivity. The implemen-tation by (Bonato, C. et al., 2016), for example, scaledthe number of resources Mj linearly according to

Mj = G+ F (N − 1− j). (106)

with typical values of G = 5 and F = 2.

3. Non-adaptive phase estimation

Efficient quantum phase estimation can also be imple-mented without adaptive feedback, with the advantageof technical simplicity (Higgins et al., 2009). Instead, aset of measurements σkxk=1...K (where K > N) is used

to separately determine each unitary phase 2jφ using aset of fixed, classical phase shifts before each readout.This set of measurements still contains all the informa-tion required to extract φ, which can be motivated by thefollowing arguments: Given a redundant set of phases, apost-processing algorithm can mimic the adaptive algo-rithm by postselecting those results that have been mea-sured using the phase most closely resembling the cor-rect adaptive choice. From a spectroscopic point of view,measurements with different phases correspond to Ram-sey fringes with different quadratures. Hence, at leastone qubit of every digit will perform its measurementon the slope of a Ramsey fringe, allowing for a precisemeasurement of 2jφ regardless of its value.

The phase φ can be recovered by Bayesian estimation.Every measurement σkx = ±1 provides information aboutφ, which is described by the a posteriori probability

p(φ|σkx), (107)

the probability that the observed outcome σkx stems froma phase φ. This probability is related to the inverse con-ditional probability p(σix|φ) – the excitation probabilitydescribing Ramsey fringes – by Bayes’ theorem. The jointprobability distribution of all measurements is obtainedfrom the product

p(φ) ∝∏i

p(φ|σix) (108)

from which the most likely value of φ is picked as thefinal result (Nusran et al., 2012; Waldherr et al., 2012).Here, too, acquisition time scales with the significance ofthe bit measured to achieve the Heisenberg limit.

4. Comparison of phase estimation protocols

All of the above variants of phase estimation achievea DR ∝ T scaling of the dynamic range. They differ,however, by a constant offset. Adaptive estimation isslower than quantum phase estimation by the QFT sinceit trades spatial resources (entanglement) into temporalresources (measurement time). Bayesian estimation inturn is slower than adaptive estimation due to additionalredundant measurements.

B. Experimental realizations

The proposals of Shor (Shor, 1994), Kitaev (Kitaev,1995) and Griffiths (Griffiths and Niu, 1996) were fol-lowed by a decade where research towards Heisenberg-limited measurements has focused mostly on the use ofentangled states, such as the N00N state (see SectionIX). These states promise Heisenberg scaling in the spa-tial dimension (number of qubits) rather than time (Gio-vannetti et al., 2004, 2006; Lee et al., 2002) and have

30

been studied extensively for both spin qubits (Bollingeret al., 1996; Jones et al., 2009; Leibfried et al., 2004, 2005)and photons (Edamatsu et al., 2002; Fonseca et al., 1999;Mitchell et al., 2004; Nagata et al., 2007; Walther et al.,2004; Xiang, G. Y. et al., 2011).

Heisenberg scaling in the temporal dimension hasshifted into focus with an experiment published in 2007,where adaptive phase estimation was employed in asingle-photon interferometer (Higgins et al., 2007). Theexperiment has subsequently been extended to a non-adaptive version (Higgins et al., 2009). Shortly after,both variants have been translated into protocols forspin-based quantum sensing (Said et al., 2011). Mean-while, high-dynamic-range protocols have been demon-strated on NV centers in diamond using both non-adaptive implementations (Nusran et al., 2012; Waldherret al., 2012) and an adaptive protocol based on quantumfeedback (Bonato, C. et al., 2016).

IX. ENSEMBLE SENSING

Up to this point, we have mainly focused on singlequbit sensors. In the following two sections, quantumsensors consisting of more than one qubit will be dis-cussed. The use of multiple qubits opens up many ad-ditional possibilities that cannot be implemented on asingle qubit sensor.

This section considers ensemble sensors where many(usually identical) qubits are operated in parallel. Apartfrom an obvious gain in readout sensitivity, multiplequbits allow for the implementation of second-generationquantum techniques, including entanglement and statesqueezing, which provide a true “quantum” advan-tage that cannot be realized with classical sensors.Entanglement-enhanced sensing has been pioneered withatomic systems, especially atomic clocks (Giovannettiet al., 2004; Leibfried et al., 2004). In parallel, statesqueezing is routinely applied in optical systems, such asoptical interferometers (Ligo Collaboration”, 2011).

A. Ensemble sensing

Before discussing entanglement-enhanced sensing tech-niques, we briefly consider the simple parallel operationof M identical single-qubit quantum sensors. This imple-mentation is used, for example, in atomic vapor magne-tometers (Budker and Romalis, 2007) or solid-state spinensembles. The use of M qubits accelerates the mea-surement by a factor of M , because the basic quantumsensing cycle (Steps 1-5 of the protocol, Fig. 2) can nowbe operated in parallel rather than sequentially. Equiv-alently, M parallel qubits can improve the sensitivity by√M per unit time.

This scaling is equivalent to the situation where M

classical sensors are operated in parallel. The scalingcan be seen as arising from the projection noise asso-ciated with measuring the quantum system, where it isoften called the Standard Quantum Limit (SQL) (Bra-ginskii and Vorontsov, 1975; Giovannetti et al., 2004) orshot noise limit. In practice, it is sometimes difficult toachieve a

√M scaling because instrumental stability is

more critical for ensemble sensors (Wolf et al., 2015).For ensemble sensors such as atomic vapor magnetome-

ters or spin arrays, the quantity of interest is more likelythe number density of qubits, rather than the absolutenumber of qubits M . That is, how many qubits can bepacked into a certain volume without deteriorating thesensitivity of each qubit. The maximum density of qubitsis limited by internal interactions between the qubits.Optimal densities have been calculated both for atomicvapor magnetometers (Budker and Romalis, 2007) andensembles of NV centers (Taylor et al., 2008; Wolf et al.,2015).

B. Heisenberg limit

The standard quantum limit can be overcome by us-ing quantum-enhanced sensing strategies to reach a morefundamental limit where the uncertainty σp (Eq. ( 26))scales as 1/M . This limit is also known as the Heisen-berg limit. Achieving the Heisenberg limit requires reduc-ing the variance of a chosen quantum observable at theexpenses of the uncertainty of a conjugated observable.This in turns requires preparing the quantum sensors inan entangled state. In particular, squeezed states (Caves,1981; Kitagawa and Ueda, 1993; Wineland et al., 1992)have been proposed early on to achieve the Heisenberglimit and thanks to experimental advances have recentlyshown exceptional sensitivity (Hosten et al., 2016a).

The fundamental limits of sensitivity (quantummetrology) and strategies to achieve them have been dis-cussed in many reviews (Giovannetti et al., 2004, 2006,2011; Paris, 2009; Wiseman and Milburn, 2009) and theywill not be the subject of our review. In the following,we will focus on the most important states and methodsthat have been used for entanglement-enhanced sensing.

C. Entangled states

1. GHZ an N00N states

To understand the benefits that an entangled state canbring to quantum sensing, the simplest example is givenby Greenberger-Horne-Zeilinger (GHZ) states. The sens-ing scheme is similar to a Ramsey protocol, however, if Mqubit probes are available, the preparation and readoutpulses are replaced by entangling operations (Fig. 13).

We can thus replace the procedure in Sec. IV.C withthe following:

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|0 H H

|0 H H

|0 H H

... ...

|0

|0

|0... ...

free

evol

utio

n

free

evol

utio

n

enta

nglin

g

enta

nglin

g

FIG. 13 Left: Ramsey scheme. Right: entangled Ramseyscheme for Heisenberg-limited sensitivity

1. The quantum sensors are initialized into |0〉⊗ |0〉⊗...⊗ |0〉 ⊗ |0〉 ≡ |00 . . . 0〉.

2. Using entangling gates, the quantum sensors arebrought into the GHZ state |ψ0〉 = (|00 . . . 0〉 +|11 . . . 1〉)/

√2.

3. The superposition state evolves under the Hamilto-nian H0 for a time t. The superposition state picksup an enhanced phase φ = Mω0t, and the stateafter the evolution is

|ψ(t)〉 =1√2

(|00 . . . 0〉+ eiMω0t |11 . . . 1〉) , (109)

4. Using the inverse entangling gates, the state |ψ(t)〉is converted back to an observable state, e.g. |α〉 =[ 12 (eiMω0t + 1) |01〉+ 1

2 (eiMω0t− 1) |11〉] |0 . . . 0〉2,M .

5,6. The final state is read out (only the first quantumprobe needs to be measured in the case above). Thetransition probability is

p = 1− |〈0|α〉|2

=1

2[1− cos(Mω0t)] = sin2(Mω0t/2). (110)

Comparing this result with what obtained in Sec. IV.C,we see that the oscillation frequency of the signal is en-hanced by a factor M by preparing a GHZ state. Thisallows using an M -times shorter interrogation time orachieving an improvement of the sensitivity (calculatedfrom the QCRB) by a factor

√M . While for M uncor-

related quantum probes the QCRB of Eq. (57) becomes

δVN,M =1

γ√NF

=eχ

γt√M N

, (111)

for the GHZ state, the Fisher information reflects thestate entanglement to give

δVN,GHZ =1

γ√NFGHZ

=eχ

γMt√N

(112)

Heisenberg-limited sensitivity with a GHZ state wasdemonstrated using three entangled Be ions (Leibfriedet al., 2004).

Unfortunately, the√M advantage in sensitivity is usu-

ally compensated by the GHZ state’s increased decoher-ence rate (Huelga et al., 1997), which is issue common to

most entangled states. Assuming, for example, that eachprobe is subjected to uncorrelated dephasing noise, therate of decoherence of the GHZ state is M time fasterthan for a product state. Then, the interrogation timealso needs to be reduced by a factor M and no net advan-tage in sensitivity can be obtained. This has lead to thequest for different entangled states that could be moreresilient to decoherence.

Similar to GHZ states, N00N states have been con-ceived to improve interferometry (Lee et al., 2002). Theywere first introduced in (Yurke, 1986) in the contextof neutron Mach-Zender interferometry as the fermionic“response” to the squeezed states proposed by (Caves,1981) for Heisenberg metrology. Using an M -particle in-terferometer, one can prepare an entangled Fock state,

|ψN00N 〉 = (|M〉a |0〉b + |0〉a |M〉b)/√

2 , (113)

where |N〉a indicates the N-particle Fock state in spatialmode a. Already for small M , it is possible to showsensitivity beyond the standard quantum limit (Kuzmichand Mandel, 1998). If the phase is applied only to modea of the interferometer, the phase accumulated is then

|ψN00N 〉= (eiMφa |M〉 a |0〉 b+ |0〉 a |M〉 b)/√

2 , (114)

that is, M times larger than for a one-photon state. Ex-perimental progress has allowed to reach “high N00N”(with M > 2) states (Mitchell et al., 2004; Monz et al.,2011; Walther et al., 2004) by using strong nonlineari-ties or measurement and feed-forward. They have beenused not only for sensing but also for enhanced lithog-raphy (Boto et al., 2000). Still, “N00N” states are veryfragile (Bohmann et al., 2015) and they are afflicted bya small dynamic range.

2. Squeezing

Squeezed states are promising for quantum-limitedsensing as they can reach sensitivity beyond the stan-dard quantum limit. Squeezed states of light have beenintroduced by (Caves, 1981) as a mean to reduce noise ininterferometry experiments. One of the most impressiveapplication of squeezed states of light (Schnabel et al.,2010; Walls, 1983) has been the sensitivity enhancementof the LIGO gravitational wave detector (Collaboration,2013; Ligo Collaboration”, 2011), obtained by injecting avacuum squeezed state in one arm of the interferometer.

Squeezing has also been extended to fermionic degreesof freedom (spin squeezing (Kitagawa and Ueda, 1993))to reduce the uncertainty in spectroscopy measurementsof ensemble of qubit probes. The Heisenberg uncertaintyprinciple bounds the minimum error achievable in themeasurement of two conjugate variables. While for typ-ical states the uncertainty in the two observables is onthe same order, it is possible to redistribute the fluctua-tions in the two conjugate observables. Squeezed states

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are then characterized by a reduced uncertainty in oneobservable at the expense of another observable. Thus,these states can help improving the sensitivity of quan-tum interferometry, as demonstrated in (Wineland et al.,1994, 1992). Similar to GHZ and N00N states, a key in-gredient to this sensitivity enhancement is entanglement(Sørensen and Mølmer, 2001; Wang and Sanders, 2003).However, the description of squeezed states is simplifiedby the use of a single collective angular-momentum vari-able.

The degree of spin squeezing can be measured byseveral parameters. For example, from the commuta-tion relationship for the collective angular momentum,∆Jα∆Jβ ≥ |〈Jγ〉|, one can naturally define a squeezingparameter

ξ = ∆Jα/√|〈Jγ〉/2 , (115)

with ξ < 1 for squeezed states. To quantify the advan-tage of squeezed states in sensing, it is advantageous todirectly relate squeezing to the improved sensitivity. Thismay be done by considering the ratio of the uncertaintieson the acquired phase for the squeezed state ∆φsq andfor the uncorrelated state ∆φ0 in, e.g., a Ramsey mea-surement. The metrology squeezing parameter, proposedby (Wineland et al., 1992), is then

ξR =

∣∣∣∣ |∆φ|sq|∆φ|0

∣∣∣∣ =

√N∆Jy(0)

|〉Jz(0)〈| . (116)

Early demonstrations of spin squeezing were obtainedby entangling trapped ions via their shared motionalmodes (Meyer et al., 2001), using repulsive interac-tions in a Bose-Einstein condensate (Esteve et al., 2008),or partial projection by measurement (Appel et al.,2009). More recently, atom-light interactions in high-quality cavities have enabled squeezing of large ensemblesatoms (Bohnet et al., 2014; Cox et al., 2016; Gross et al.,2010; Hosten et al., 2016a; Leroux et al., 2010a; Louchet-Chauvet et al., 2010; Schleier-Smith et al., 2010a) thatcan perform as atomic clocks beyond the standard quan-tum limit. Spin squeezing can be also implemented inqubit systems (Auccaise et al., 2015; Bennett et al., 2013;Cappellaro and Lukin, 2009; Sinha et al., 2003) followingthe original proposal by (Kitagawa and Ueda, 1993).

3. Parity measurements

A challenge in achieving the full potential of multi-qubit enhanced metrology is the widespread inefficiencyof quantum state readout. Metrology schemes often re-quire single qubit readout or the measurement of com-plex, many-body observables. In both cases, coupling ofthe quantum system to the detection apparatus is inef-ficient, often because strong coupling would destroy thevery quantum state used in the metrology task.

To reveal the properties of entangled states and to takeadvantage of their enhanced sensitivities, an efficient ob-servable is the parity of the state. The parity observablewas first introduced in the context of ion qubits (Bollingeret al., 1996; Leibfried et al., 2004) and it referred to theexcited or ground state populations of the ions. The par-ity has become widely adopted for the readout of N00Nstates, where the parity measures the even/odd numberof photons in a state (Gerry and Mimih, 2010). Photonparity measurements are as well used in quantum metrol-ogy with squeezed states. While the simplest methodfor parity detection would be via single photon count-ing, and recent advances in superconducting single pho-ton detectors approach the required efficiency (Natara-jan et al., 2012), photon numbers could also be mea-sured with single-photon resolution using quantum non-demolition (QND) techniques (Imoto et al., 1985) thatexploit nonlinear optical interactions. Until recently,parity detection for atomic ensembles containing morethan a few particles was out of reach. However, recentbreakthroughs in spatially resolved (Bakr et al., 2009)and cavity-based atom detection (Hosten et al., 2016b;Schleier-Smith et al., 2010b) enabled atom counting inmesoscopic ensembles containing N & 100 atoms.

4. Other types of entanglement

The key difficulty with using entangled states for sens-ing is that they are less robust against noise. Thus,the advantage in sensitivity is compensated by a con-current reduction in coherence time. In particular, it hasbeen demonstrated that for frequency estimation, anynon-zero value of dephasing cancels any advantage of themaximally entangled state over a classically correlatedstate(Huelga et al., 1997). An analogous result can beproven for magnetometry (Auzinsh et al., 2004).

Despite of this, non-maximally entangled states canprovide an advantage over product states (Shaji andCaves, 2007; Ulam-Orgikh and Kitagawa, 2001). Opti-mal states for quantum interferometry in the presenceof photon loss can, for example, be found by numericalsearches (Huver et al., 2008; Lee et al., 2009).

Single-mode states have also been considered as a morerobust alternative to two-mode states. Examples includepure Gaussian states in the presence of phase diffusion(Genoni et al., 2011), mixed Gaussian states in the pres-ence of loss (Aspachs et al., 2009) and single-mode vari-ants of two-mode states (Maccone and Cillis, 2009).

Other strategies include the creation of states that aremore robust to the particular noise the system is sub-jected to (Goldstein et al., 2011) or the use of entan-gled ancillary qubits that are not quantum sensors them-selves (Demkowicz-Dobrzanski and Maccone, 2014; Duret al., 2014; Huang et al., 2016; Kessler et al., 2014).These are considered in the next section (Section X).

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X. SENSING ASSISTED BY AUXILIARY QUBITS

In the previous section we considered potential im-provements in sensitivity derived from the availability ofmultiple quantum systems operated in parallel. A dif-ferent scenario arises when only a small number of addi-tional quantum systems is available, or when the addi-tional quantum systems do not directly interact with thesignal to be measured. Even in this situation, however,“auxiliary qubits” can aid in the sensing task. Althoughauxiliary qubits – or more generally, additional quan-tum degrees of freedom – cannot improve the sensitivitybeyond the quantum metrology limits, they can aid inreaching these limits, for example when it is experimen-tally difficult to optimally initalize or readout the quan-tum state. Auxiliary qubits may be used to increase theeffective coherence or memory time of a quantum sensor,either by operation as a quantum memory or by enablingquantum error correction.

In the following we present some of the schemesthat have been proposed or implemented with auxiliaryqubits.

A. Quantum logic clock

Clocks based on optical transitions of an ion kept in ahigh-frequency trap exhibit significantly improved accu-racy over more common atomic clocks. Single-ion atomicclocks currently detain the record for the most accurateoptical clocks, with uncertainties of 2.1× 1018 for a 87Srensemble clock (Nicholson et al., 2015) and 3.2×1018 fora single a single trapped 171Yb (Huntemann et al., 2016).

The remaining limitations on optical clocks are re-lated to their long-term stability and isolation from ex-ternal perturbations such as electromagnetic interference.These limitations are even more critical when such clocksare based on a string of ions in a trap, because of the as-sociated unavoidable electric field gradients. Only someion species, with no quadrupolar moment, can then beused, but not all of them present a suitable transition forlaser cooling and state detection beside the desired, sta-ble clock transition. To overcome this dilemma, quantumlogic spectroscopy has been introduced (Schmidt et al.,2005). The key idea is to employ two ion species: a clockion that presents a stable clock transition (and representsthe quantum sensor), and a logic ion (acting as auxil-iary qubit) that is used to prepare, via a cooling tran-sition, and readout the clock ion. The resulting “quan-tum logic” ion clock can thus take advantage of the moststable ion clock transitions, even when the ion cannotbe efficiently read out, thus achieving impressive clockperformance (Hume et al., 2007; Rosenband et al., 2008,2007). Advanced quantum logic clocks may incorporatemulti-ion logic (Tan et al., 2015) and use quantum algo-rithms for more efficient readout (Schulte et al., 2016).

B. Storage and retrieval

The quantum state |ψ〉 can be stored and retrieved inthe auxiliary qubit. Storage can be achieved by a SWAPgate (or more simply two consecutive c-NOT gates) onthe sensing and auxiliary qubits, respectively. Retrievaluses the same two c-NOT gates in reverse order. For theexample of an electron-nuclear qubit pair, c-NOT gateshave been implemented both by selective pulses (Pfenderet al., 2016; Rosskopf et al., 2016) and using coherentrotations (Zaiser et al., 2016).

Several useful applications of storage and retrieval havebeen demonstrated. A first example includes correlationspectroscopy, where two sensing periods are interruptedby a waiting time t1 (Laraoui and Meriles, 2013; Rosskopfet al., 2016; Zaiser et al., 2016). A second example in-cludes a repetitive (quantum non-demolition) readout ofthe final qubit state, which can be used to reduce theclassical readout noise (Jiang et al., 2009).

C. Quantum error correction

Quantum error correction (Nielsen and Chuang, 2000;Shor, 1995) aims at counteracting errors during quantumcomputation by encoding the qubit information into re-dundant degrees of freedom. The logical qubit is thus en-coded in a subspace of the total Hilbert space (the code)such that each error considered maps the code to an or-thogonal subspace, allowing detection and correction ofthe error. Compared to dynamical decoupling schemes,qubit protection covers the entire noise spectrum and isnot limited to low-frequency noise. On the other hand,qubit protection is typically only against one type error(phase flip or bit flip) which correspond to orthogonalcomponents of the signal (V|| or V⊥). For vector fields,quantum error correction can be used to protect againstnoise in one spatial direction while leaving the sensor re-sponsive to signals in the orthogonal spatial direction.Thus, quantum error correction suppresses noise accord-ing to spatial symmetry, and not according to frequency.

The simplest code is the 3-qubit repetition code, whichcorrects against one-axis noise with depth one (that is, itcan correct up to one error acting on one qubit). For ex-ample, the code |0〉L = |000〉 and |1〉L = |111〉 can correctagainst a single qubit flip error. Note that (Dur et al.,2014; Ozeri, 2013) equal superpositions of these two log-ical basis states are also optimal to achieve Heisenberg-limited sensitivity in estimating a global phase (Bollingeret al., 1996; Leibfried et al., 2004). While this seems toindicate that QEC codes could be extremely useful formetrology, the method is hampered by the fact that QECoften cannot discriminate between signal and noise. Inparticular, if the signal to be detected couples to the sen-sor in a similar way as the noise, the QEC code also elim-inates the effect of the signal. This compromise between

34

error suppression and preservation of signal sensitivity iscommon to other error correction methods. For example,in dynamical decoupling schemes, a separation in the fre-quency of noise and signal is required. Since QEC worksindependently of noise frequency, distinct operators forthe signal and noise interactions are required. This im-poses an additional condition on a QEC code: the quan-tum Fisher information (Giovannetti et al., 2011) in thecode subspace must be non-zero.

Several situations for QEC-enhanced sensing have beenconsidered. One possible scenario is to protect the quan-tum sensor against a certain type of noise (e.g., singlequbit, bit-flip or transverse noise), while measuring theinteraction between qubits (Dur et al., 2014; Herrera-Martı et al., 2015). More generally, one can measurea many-body Hamiltonian term with a strength propor-tional to the signal to be estimated (Herrera-Martı et al.,2015). Since this can typically only be achieved in a per-turbative way, this scheme still leads to a compromisebetween noise suppression and effective signal strength.

The simplest scheme for QEC is to use a singlegood qubit (unaffected by noise) to protect the sensorqubit (Arrad et al., 2014; Hirose and Cappellaro, 2016;Kessler et al., 2014; Ticozzi and Viola, 2006). In thisscheme, which has recently been implemented with NVcenters (Unden et al., 2016), the qubit sensor detectsa signal along one axis (e.g., a phase) while being pro-tected against noise along a different axis (e.g., againstbit flip). Because the “good” ancillary qubit can onlyprotect against one error event (or, equivalently, suppressthe error probability for continuous error), the signal ac-quisition must be periodically interrupted to perform acorrective step. Since the noise strength is typically muchweaker than the noise fluctuation rate, the correctionsteps can be performed at a much slower rate comparedto dynamical decoupling. Beyond single qubits, QEC hasalso been applied to N00N states (Bergmann and vanLoock, 2016). These recent results hint at the potentialof QEC for sensing which has just about begun to beingexplored.

XI. OUTLOOK

Despite its rich history in atomic spectroscopy andclassical interferometry, quantum sensing is an excit-ingly new and refreshing development advancing rapidlyalong the sidelines of mainstream quantum informationresearch. Like no other field, quantum sensing hasbeen uniting diverse efforts in science and technologyto create fundamental new opportunities and applica-tions in metrology. Inputs have been coming from tra-ditional high-resolution optical and magnetic resonancespectroscopy, to the mathematical concepts of parameterestimation, to quantum manipulation and entanglementtechniques borrowed from quantum information science.

Over the last decade, and especially in the last few years,a comprehensive toolset has been established that canbe applied to any type of quantum sensor. In particu-lar, these allow operation of the sensor over a wide signalfrequency range, can be adjusted to maximize sensitivityand dynamic range, and allow discrimination of differ-ent types of signals by symmetry or vector orientation.While many experiments so far made use of single qubitsensors, strategies to implement entangled multi-qubitsensors with enhanced capabilities and higher sensitivityare just beginning to be explored.

One of the biggest attractions of quantum sensors istheir immediate potential for practical applications. Thispotential is partially due to the immense range of con-ceived sensor implementations, starting with atomic andsolid-state spin systems and continuing to electronic andvibrational degrees of freedom from the atomic to themacroscale. In fact, quantum sensors based on SQUIDmagnetometers and atomic vapors are already in every-day use, and have installed themselves as the most sen-sitive magnetic field detectors currently available. Like-wise, atomic clocks have become the ultimate standardin time keeping and frequency generation. Many otherand more recent implementations of quantum sensors arejust starting to make their appearance in many differentniches. Notably, NV centers in diamond have startedconquering many applications in nanoscale imaging dueto their small size.

What lies ahead in quantum sensing? On the one hand,the range of applications will continue to expand as newtypes and more mature sensor implementations becomeavailable. Taking the impact quantum magnetometersand atomic clocks had in their particular discipline, it canbe expected that quantum sensors will penetrate much ofthe 21st century technology and find their way into bothhigh-end and consumer devices. Advances with quan-tum sensors will be strongly driven by the availabilityof “better” materials and more precise control, allowingtheir operation with longer coherence times, more effi-cient readout, and thus higher sensitivity.

In parallel, quantum sensing will profit from effortsin quantum technology, especially quantum computing,where many of the fundamental concepts have been de-veloped, such as dynamical decoupling protocols, quan-tum storage and quantum error correction, as well asquantum phase estimation. Vice versa, quantum sensinghas become an important resource for quantum technolo-gies as it provides much insight into the “environment”of qubits, especially through decoherence spectroscopy.A better understanding of decoherence in a particularimplementation of a quantum system can help the adop-tion of strategies to protect the qubit, and guide the en-gineering and materials development. The border regionbetween quantum sensing and quantum simulation, inaddition, is becoming a fertile playground for emulat-ing and detecting many-body physics phenomena. Over-

35

all, quantum sensing has the potential to fundamentallytransform our measurement capabilities, enabling highersensitivity and precision, new measurement types, andcovering atomic up to macroscopic length scales.

ACKNOWLEDGMENTS

The authors thank Jens Boss, Dmitry Budker, KevinChang, Kristian Cujia, Lukasz Cywinski, Simon Gus-tavsson, Sebastian Hofferberth, Dominik Irber, FedorJelezko, Renbao Liu, Luca Lorenzelli, Tobias Rosskopf,Jorg Wrachtrup and Jonathan Zopes for helpful com-ments and discussions. CLD acknowledges funding fromthe DIADEMS program 611143 of the European Com-mission, the Swiss NSF Project Grant 200021 137520,the Swiss NSF NCCR QSIT, and ETH Research GrantETH-03 16-1. FR acknowledges funding from theDeutsche Forschungsgemeinschaft via Emmy Noethergrant RE 3606/1-1. PC acknowledges funding from theU.S. Army Research Office through MURI grants No.W911NF-11-1-0400 and W911NF-15-1-0548 and by theNSF PHY0551153 and PHY1415345.

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