Protected State Enhanced Quantum Metrology with Interacting Two-Level Ensembles

Laurin Ostermann,1 Helmut Ritsch,1 and Claudiu Genes1,2

1Institut fur Theoretische Physik, Universitat Innsbruck, Technikerstrasse 25, A-6020 Innsbruck, Austria2IPCMS (UMR 7504) and ISIS (UMR 7006), Universite de Strasbourg and CNRS, 67000 Strasbourg, France

(Received 19 July 2013; published 18 September 2013)

Ramsey interferometry is routinely used in quantum metrology for the most sensitive measurements

of optical clock frequencies. Spontaneous decay to the electromagnetic vacuum ultimately limits the

interrogation time and thus sets a lower bound to the optimal frequency sensitivity. In dense ensembles

of two-level systems, the presence of collective effects such as superradiance and dipole-dipole interaction

tends to decrease the sensitivity even further. We show that by a redesign of the Ramsey-pulse sequence

to include different rotations of individual spins that effectively fold the collective state onto a state close

to the center of the Bloch sphere, partial protection from collective decoherence is possible. This allows

a significant improvement in the sensitivity limit of a clock transition detection scheme over the

conventional Ramsey method for interacting systems and even for noninteracting decaying atoms.

DOI: 10.1103/PhysRevLett.111.123601 PACS numbers: 42.50.Ar, 42.50.Lc, 42.72.�g

The precise measurement of time using suitable atomictransitions is a major achievement of quantum metrology.The Ramsey interferometry procedure plays a crucial roleas it allows an accurate locking of the microwave or opticaloscillator to the transition frequency in the atom. Typicalearly realizations were based on atomic beams or later onlaser-cooled atomic fountains [1], where the atoms wouldinteract with two consecutive Rabi pulses. With opticallattices [2,3] time measurements were expected to becomeeven more accurate due to longer interaction times and theelimination of collisions (see [4,5] for recent reviews).

To reduce quantum projection noise (scaling as 1=ffiffiffiffiN

pfor N atoms) and to speed up the measurement, setupsusually involve an as large as possible number of atoms. Ina finite volume, of course, this brings collective effects likesuperradiance and dipole-dipole shifts to the table [6].While some techniques rely on the engineering of particu-lar geometries without the need to alter the internal atomicstates [7], exploiting the uncertainty principle by employ-ing squeezed states [8–10] can be helpful as well to achieveless noise with lower atom numbers [11,12]. These tech-niques rely heavily on entanglement [13,14] among atomsand require very careful preparation and isolation of theensemble.

When, finally, interrogation times reached the lifetime ofthe excited state, spontaneous emission became a criticalfactor for the contrast of the Ramsey fringes. Interestingly,despite the use of long-lived clock states, for multipleatoms in close proximity to each other, collective sponta-neous emission can still reach a detrimental magnitude,namely, proportional to the atom number [15,16]. Whilethis is usually limited to volumes of the order of a cubicwavelength, in regular arrays, such as an optical lattice, theeffect can extend over tens of lattice sites [17].

In this Letter, we propose a strategy that works on thelevel of the Ramsey pulses and which we dub the

asymmetric Ramsey technique (ART), in contrast to theconventional symmetric Ramsey technique (SRT) thatemploys only identical �=2 pulses applied to all atoms.While the conventional SRT excites superposition states,which possess a maximum dipole moment and thus aremost sensitive to superradiance, this new approach allowsthe selection of long-lived collective states (or ‘‘darkstates’’) to improve the sensitivity of the clock signal.The procedure requires a modification of the Ramsey steps:after the initial �=2 pulse is applied, each atomic coher-ence is rotated by a distinct phase, resulting in a subradiantcollective state with a vanishing classical dipole (with alifetime which can be even longer than that of the inde-pendent atoms [18]). The phase-spread is then reversed andfollowed by detection; this leads to a significant improve-ment in the sensitivity limit over the conventional SRT.Model.—We assume N identical two-level emitters

with levels jgi and jei separated by !0 in a geometrydefined by ri for i ¼ 1; . . . ; N. For the Pauli ladderoperators ��

i and subsequently �xi ¼ �þ

i þ ��i , �y

i ¼�ið�þi � ��

i Þ, and �zi ¼ �þ

i ��i � ��

i �þi , unitary

rotations RðjÞ� ½’� ¼ expði’��

j =2Þ where � 2 fx; y; zg aredefined. The independent atom decay rates are �; thecooperative nature of decay for atom pairs i, j is reflectedby mutual decay rates �ij (in the following we use

�ii ¼ �). Via the vacuum, dipole-dipole interactions occurcharacterized by the frequency shifts �ij. Both functions

depend on rij [6,19]. The dynamics of the system can be

described via the master equation

@�

@t¼ i½�;H� þL½��; (1)

where the Hamiltonian is given by

H ¼ !

2

Xi

�zi þ

Xi�j

�ij�þi �

�j ; (2)

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with ! ¼ !0 �!l (!l is the reference frequency) and theLiouvillian is

L½�� ¼ 1

2

Xi;j

�ij½2��i ��

þj � �þ

i ��j �� ��þ

i ��j �: (3)

A typical procedure in spectroscopic experiments is theRamsey method of separated oscillatory fields [20].The sequence assumes the ensemble of spins initiated inthe ground state at time ti such that hSziðtiÞ ¼ �N=2whereSz ¼ P

i�zi =2. Three stages follow: (i) a quick pulse

between ti and t ¼ 0 rotates the atoms into a collectivestate in the xy plane that exhibits maximal dipole, (ii) freeevolution for the time �, and (iii) a second quick pulse flipsthe spins up. The detected signal is a measure of populationinversion and is therefore proportional to hSziðtfÞ. An

analysis of this signal gives the sensitivity as a figure ofmerit in metrology

�! ¼ min

��Szð!; �Þ

j@!hSzið!; �Þj�; (4)

where the minimization is performed with respect to !.We follow the dynamics as described above in a

density matrix formalism. We start with �i ¼ jGihGj,transform it into �0 ¼ R1�iR

y1 , evolve it into ��

by solving Eq. (1), and finally transform it into �f ¼R2��R

y2 . The detected signal and its variance are

computed as hSzi and �Sz from �f.

As a basis of comparison, we take independent systems(�ij ¼ 0 and �ij ¼ 0 for i � j). The rotation pulses are

R1 ¼ R2 ¼ Nj R

ðjÞy ½�=2� and the resulting sensitivity is

½�!�indep ¼ min

24

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie�� � cos2ð!�ÞpffiffiffiffiN

p j� sinð!�Þj

35 ¼ e��=2

�ffiffiffiffiN

p : (5)

Further optimization with respect to the interrogationtime gives an optimal �opt ¼ 2=� and optimal sensitivity

�e=2ffiffiffiffiN

p, which shows that the main impediment of

Ramsey interferometry is the limitation in the interrogationtimes owing to the decay of the dipoles.As a principal advance of this Letter, we propose the

generalized Ramsey sequence ART (as illustrated in Fig. 1)that deviates from the typical one by a redesign of the twopulses at times t ¼ 0 and t ¼ �, intended to drive the spinsystem into states that are protected from the environmen-tal decoherence. To accomplish this, one complements thenormal �=2 pulse with a phase distribution pulse, whichfor a particular atom j is represented by a rotation around

the z direction with the angle ’ðmÞj ¼ 2�mðj� 1Þ=N,

where m ¼ 1; . . . ; ½N=2� and [N=2] is the integer beforeN=2. The first Ramsey-pulse operator is then

R1 ¼Oj

RðjÞz ½’ðmÞ

j �RðjÞy

��

2

�:

To justify the choice of the rotation angles notice that at

time t ¼ 0, for any set of ’ðmÞj , the system is in a state of

zero average collective spin: at an intuitive level this meansthat the phase-spread operation folds the system’s collec-tive state from the surface of the Bloch sphere onto a zoneclose to its center. For small atom-atom separations, col-lective states of higher symmetry are shorter lived (culmi-nating, at zero separation, with the maximally symmetricsuperradiant Dicke state [21] of rate N�). Let us then try tosketch how asymmetric states can be build by imposing the

orthogonality of a phase-spread state jc ’i ¼N

Nj¼1½jgi þ

ðei’Þðj�1Þjei�= ffiffiffi2

pto the multitude of symmetric states of

the system. While generally this is an unsolvable problem[19], we can get some insight using the symmetric state inthe single excitation subspace, the so-called W state jWi.

FIG. 1 (color online). State protective Ramsey sequence. The ensemble of N spins starts with all spins down in a collective coherentpure spin state on the surface of the collective Bloch sphere (radius N=2). Individual �=2 pulses are followed by phase encoding

operations of angles ’ðmÞj ¼ ð2�m=NÞðj� 1Þ where j ¼ 1; . . .N and m ¼ 1; . . . ; ½N=2�, which brings the total spin close to the center

of the Bloch sphere (the third-fifth steps are shown on small Bloch spheres of radius 1=2). After time �, the phase encoding operation isreversed and �=2 pulses prepare the ensemble (now in a mixed state shown on the large collective Bloch sphere) for the detection ofthe population difference signal.

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From hWjc ’i ¼P

Nj¼1ðei’Þj ¼ 0 we get the solutions

’ ¼ 2�m=N which justify the choice from above ’ðmÞj .

At time � the phase spread is reversed and a �=2 pulsefollows

R2 ¼Oj

RðjÞy

��

2

�RðjÞ

z ½�’ðmÞj �:

Two-atom case.—Let us use a simple system to elucidatethe differences between SRT and ART. We consider atoms1 and 2 separated by a distance rwith � ¼ �12ðrÞ and� ¼�12ðrÞ [their dependence on r is shown in Fig. 2(a)]. Thediagonalization of the Hamiltonian is performed by a trans-formation from the bare basis fjggi; jgei; jegi; jeeig to the

collective basis fjGi; jSi; jAi; jEig with jGi ¼ jggi, jSi ¼ðjegi þ jgeiÞ= ffiffiffi

2p

, jAi ¼ ðjegi � jgeiÞ= ffiffiffi2

pand jEi ¼ jeei.

This transformation diagonalizes the dissipative dynamicsas well, and leads to two independent decay channels withdamping rates �S ¼ �þ � and �A ¼ �� � as illustratedin Fig. 2(b).

We follow the evolution of �i ¼ jGihGj in the collectivebasis and compute the detected signal and its variance from

��. For SRT one obtains hSziS ¼ 2ffiffiffi2

pReð�ES

� þ �SG� Þ

which can be calculated by solving the evolution between0 and � from the following set of coupled equations:

_�ES ¼�� 2�þ �S

2� ið!��Þ

��ES; (6a)

_�SG ¼���S

2� ið!þ�Þ

��SG þ �S�

ES: (6b)

The computation of the signal variance requires thederivation of hðSzÞ2iS ¼ 2½1þ �SS

� � �AA� þ 2Reð�EG

� Þ�,thus solving _�EE ¼ �2��EE, _�SS ¼ ��Sð�SS � �EEÞ,_�AA ¼ ��Að�AA � �EEÞ, and _�EG ¼ �ð�þ 2i!Þ�EG. In

contrast, for ARTwe get hSziA ¼ 2ffiffiffi2

pReð�EA

� � �AG� Þ and

hðSzÞ2iA ¼ 2½1þ �AA� � �SS

� � 2Reð�EG� Þ�, where the

extra coherences can be derived from the solutions of

_�EA ¼�� 2�þ �A

2� ið!þ�Þ

��EA; (7a)

_�AG ¼���A

2� ið!��Þ

��AG þ �A�

EA: (7b)

The minimum sensitivities depending on � after optimiza-tion with respect to ! can be very well approximated by

½�!�S ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1þ aSe

�2�� þ bSe��S� � cSe

��A�Þq

�e��S�=2ðe���A�S þAþ

S Þ; (8a)

½�!�A ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð1þ aAe

�2�� þ bAe��A� � cAe

��S�Þq

�e��A�=2ðe���AþA þA�

A Þ; (8b)

where a, b, c, andA� are given by the system’s geometry[19]. Assuming a separation of timescales, for example,when �A � �, �S, the sensitivity ½�!�A scales similarly tothe independent sensitivity of Eq. (5) with � replaced by�A. This holds approximately even in the intermediateregime shown in Fig. 2(c) where �A ’ 0:59�, as transpir-ing from the scaling of the blue (squares) line. For closelyspaced atoms, the result is easy to interpret and extremelyencouraging since it allows for large interrogation timesand direct improvement of the minimum sensitivity. In thegeneral case of varying the distance between atoms, forexample, to the second region of Fig. 2(a), the symmetricstate becomes subradiant instead and the symmetric pro-cedure is the optimal one, however, providing only aminimal gain over the independent atom case. This isrelevant for the case of linear atom chains separated by amagic wavelength [22], where SRT is optimal.Numerical results.—Let us now extend our model to

more general configurations of a few two-level systemsin various geometries. In principle, the configuration canbe generalized to a two-dimensional or three-dimensionallattice but one ends up with large Hilbert spaces ratherquickly that render simple numerical methods unfeasible.To illustrate the effectiveness of ART we particularize tothe two situations depicted in Fig. 3, i.e., square and lineargeometries. The results are presented in Figs. 3(a) and 3(b)for all possible phase-spread angle sets, i.e., varying the

index m of ’ðmÞj from 1 to [N=2] (N ¼ 4 for square and 5

for the chain) and for a lattice constant a=� ¼ 0:3. Toelucidate the effects of applying ART with different mindices, we consider a particular nontrivial model of fiveemitters (a=� ¼ 0:2) with uniform mutual couplings, i.e.,�ij ¼ � and �ij ¼ � for every i � j. Simultaneous diag-

onalization of the Hamiltonian and Liouvillian is then

FIG. 2 (color online). Two-atommetrology. (a) Normalizedmu-tual decay rate and dipole-dipole frequency shift for a pair of atomsas a function of r=�. For positive (negative)�, the asymmetric stateis subradiant (superradiant) (indicated by dashed/non-dashed re-gions). (b) Level scheme in the dressed basis showing the twoindependent decay channels with modified rates �S and �A.(c) Optimal sensitivity as a function of ��. The atom separation isr=� ¼ 0:3 corresponding to� � 0:41� and� � 0:29�. Themini-mum for the asymmetric addressing is reached around � ’ 2=�A.

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possible and it leads to 2N states jc ji ordered with increas-ing associated decay rates �j, as shown in Fig. 3(c). When

SRT is applied, the top part of the histogram shows that thepopulation is preferentially distributed among superradiantstates. Applying ART with m ¼ 1, the lower part of thehistogram shows that the population is distributed amongstates with lower decay rates. The corresponding gain ininterrogation times will then be roughly obtained by amean over the ��1

j ’s of the target states weighted by their

corresponding populations. ART with other m will, aswell, induce a similar state distribution populatingmostly subradiant states. The choice of the most suitableindex m then requires scanning over all the m indicesand finding the largest allowed interrogation time[as shown in Figs. 3(a) and 3(b)]. The procedure canconceivably be simplified by employing powerful numeri-cal techniques to identify the optimal m sequence.

Experimental investigations of the mechanism describedabove must mainly address the question of individualphase writing on distinguishable emitters. As one particu-lar realization, a chain of atoms excited by a laser tilted bysome angle opens up the possibility of imprinting avarying phase ’j ¼ k0ðj� 1Þa= cosðÞ for the jth atom.

Note that interestingly for a strontium magic wavelengthlattice, excitation at about 90� automatically exciteslong-lived exciton states close to the optimum. In atwo-dimensional lattice this still is fulfilled quite well byexcitation from the third direction perpendicular to theplane. For a cube the situation is more tricky and requirescareful angle optimization for which preliminary calcula-tions are promising and will be fully investigated in afuture publication.

A second possible realization of the model consideredhere and suitable for testing the validity of ART is aparticular case of an engineered bath, where the commoninteraction of atoms with a decaying optical cavity field[23], combined with the elimination of the cavity mode canlead to equal mutual coupling between any pair of atoms

and an effective N-particle interaction described byEqs. (2) and (3). The atoms can then be individuallyaddressed and, for example, by applying a magnetic fieldgradient along the atom chain, a progressive phase asneeded for the implementation of ART can be realized.A further possibility for an engineered bath is that ofseveral superconducting qubits coupled to coplanar wave-guide transmission lines and resonators [24–26]. Here, onthe one hand the distance of the particles is much smallerthan a wavelength so the effects are very large, but on theother hand the individual transition frequencies, Rabiamplitudes and phases can be controlled very well.Let us finally remark on multipartite entanglement. First,

our technique hints towards the possibility of preparingmultipartite entangled states via dissipative techniques.The phase-spreading technique prepares the initial stateas a separable state with a large contribution from aquasi-non-decaying state. After considerable evolutiontime �, the correlated environment filters out all othercontributions except for the decoherence-free state whichnecessarily presents quantum correlations (as a basis ofcomparison consider the two-atom case where jAi is maxi-mally entangled). Second, the state protection operationcan as well be tested for the protection of entanglementstored in collective states. This can, for example, beemployed in schemes where spin squeezed states areused for sensitive phase detection to minimize their deg-radation during pure dissipative evolution periods.Concluding remarks.—We have described a state protec-

tive mechanism applied to a collection of vacuum-coupledtwo-level systems that can be employed in quantummetrology applications for the enhanced detection of tran-sition frequencies. The generality of the mechanism opensthe way for investigations into more complex engineeredreservoirs (atoms in mode-structuring cavities, supercon-ducting qubits coupled to coplanar waveguide transmissionlines) and different noise models such as phase-correlatednoise (as treated in [27]). Applications involving multi-partite entanglement are also envisioned, such as the

FIG. 3 (color online). Numerical investigations. (a) and (b) Numerical results for the square and a five atom chain. (c) Results ofdiagonalization for an ideal system of five equally mutually coupled emitters. The states are ordered with an increasing effective decayrate. The occupancy is shown in the histograms for SRT vs ART sequences.

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protection of spin squeezed states during dissipative evolu-tion and the design of dissipation-induced entanglingschemes.

We are grateful to H. Zoubi, M. Skotiniotis,W. Niedenzu, and M. Holland for useful comments onthe manuscript and to S. Kramer for assistance withnumerics. We acknowledge the use of the QuTiP open-source software [28] to generate Fig. 1. Support has beenreceived from DARPA through the QUASAR project(L. O. and H. R.) and from the Austrian Science Fund(FWF) via project P24968-N27 (C.G.).

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