Towards a three-step laser excitation of rubidium Rydberg states for use in a microwave CQED single-atom detector.

B. G Catchpole

A three-step excitation scheme can be utilised in the production of 63P3/2 Rydberg states from a room-temperature Rubidium gas cell through the application of Doppler-free laser spectroscopic techniques. Atoms were excited from the ground state along an optimal excitation pathway and resultant states were observed using a non-destructive, purely-optical detection strategy, through monitoring first step absorption on a strong rubidium D2 hyperfine transition. A high degree of frequency stability was attainable which would allow for the reliable and consistent production of 63P3/2 atoms; with a measured Allan deviation of 30kHz and 45kHz for the first two transitions respectively over the course of ~ one hour. With linewidths for these transitions of the order of 10.3±0.09 MHz and ~14 MHz respectively the techniques detailed provide more than sufficient frequency stability to be incorporated into a single-atom detection experiment.

I.I INTRODUCTION AND MOTIVATION Rydberg atoms1 have played a critical role in atomic physics and optical spectroscopy for many years [1,2], with orbital radii scaling as n2, they are extremely large states with loosely bound valence electrons and a large number of closely-spaced energy levels that decrease as 1/n3, converging onto the first ionization energy in what is known as the Rydberg series. With a host of useful properties including large dipole moments, transitions in the mm-wave region and long lifetimes, they have become an integral component of many cutting-edge experimental setups, from Rydberg ‘blockades’ [3,4,6] with ultra-cold atoms to a wide range of ground-breaking microwave cavity quantum electrodynamics (CQED) experiments [5]. Rydberg pairs have achieved interaction distances of several microns due to large electric dipole moments and with the application of a small E-field they can form a ‘Rydberg blockade’. Long lifetimes combined with coherent control means a promising step towards the realisation of a quantum computer [7], an enhanced two-body interaction means Rydberg atoms could provide a route to fast, controlled quantum gates in atomic ensembles [3,6,8], or a photonic phase gate using the technique of electromagnetically induced transparency (EIT) [8,9]. Remarkable achievements have also been made with Rydberg atoms in the field of microwave CQED, exploiting the fundamental light-matter interaction between a single, isolated field mode contained within a superconducting high-Q cavity and Rydberg atoms, in a system known as a one-atom maser, or micromaser. Since its initial demonstration [10] in 1985 the experiment has elucidated many fundamental aspects of quantum mechanics; the creation of ‘mesoscopic’ cat-states and the observation of their decoherence in real-time via the Wigner function [11,12], for example, has facilitated a phenomenal advance in scientific understanding - from Schrodinger’s 1935 ‘reductio ad absurdum’ quantum cati [13] to monitoring real-time decoherence of ‘cat’ states due to their environmental interaction. Rydberg atoms have been used to verify the quantised nature of the electromagnetic (EM) field with vacuum Rabi oscillations (VROs) [14], a process that occurs in the strong coupling

1 An excited atom that has one or more electrons with a very high

principal (n) quantum number [2].

regime (like within a high-Q cavity) and involves the oscillatory exchange of energy (a photon) between the coupled atom-field system. Through analysis of the Rabi oscillation frequency it is possible to accurately determine the ‘Fock’ or number state (the number of photons) contained within the cavity [15] (where ‘Fock’ states form the quantum mechanical basis of the EM field). It is also through the VRO phenomena that CQED setups are able to function as an almost ideal ‘entanglement resource’, an imperative for many quantum information (QI) and quantum computing (QC) applications. Capable of generating atom-atom or atom-field entanglement, via resonant or dispersive interactions [16], the setup makes it possible to investigate the theory of non-locality by testing for violations of Bell inequalities, and demonstrate the ‘deep relationship’ between entanglement and complementarityii [17], as well as providing a viable approach for a scalable quantum computer through crossed atomic beam ‘cluster-state’ quantum computing [18]. Further achievements made with Rydberg atoms via the micromaser setup include the demonstration of collapse and revival [19,20] of the VRO (a purely QM phenomena with no classical analogy), techniques for the production of number-states on-demand (trapping states [21] and state reduction[15]), observation of the collapse of the wavefunction (Fock field-state collapse)[22] through quantum non-demolition (QND) measurements, a technique that has enabled the observation of the birth, life and death of a single photon[23], as well as ‘freezing’ the coherent evolution of a cavity field with the quantum Zeno effect [24] Rydbergs can also be used in a range of critical applications in addition to entanglement-generation, including sensitive detection of microwave photons [25], and acting as a single photon source [26,27], allowing photonic generation at a specific time and in a pre-determined direction - an essential prerequisite for many areas of QI such as quantum cryptography (BB84 protocol [28]) and for a range of QC applications.

Figure 1: Demonstration of the potential for using the phenomena of VRS as a sensitive SAD technique. In the plot, adapted from

[30], mean intracavity photon number is plotted against probe frequency for a range of values of N (the number of atoms within

the cavity). The dependence on N can clearly be identified – leading to a reliable SAD scheme. An equally useful application that has received less attention in the literature is the potential of using a microwave CQED setup for the purpose of single-atom detection (SAD). Promising developments were made at Caltech in an analogous setup for SAD in the optical regime through the analysis of Vacuum-Rabi splitting (VRS), initially for N two state atoms [29], and for N≤10 shortly after [30]. It was reported that the process of normal-mode splitting, a phenomena well understood from the behaviour of coupled pendula (a superposition of symmetric and anti-symmetric modes are produced) displayed sufficient sensitivity to the number of atoms in the atom-cavity system to enable SAD [84]. VRS is a process that involves the splitting of the transmission spectrum of a cavity into distinct peaks as a result of the presence of an atom; it is caused by VROs that occur in the strong coupling regime of a high-Q cavity as energy quanta is exchanged between the atom and field. Without an atom inside the cavity a single transmission peak is observed, however when the resonance of the atom and the cavity are tuned into coincidence and an atom is within the cavity the atom-field interaction causes a mixing of the states to produce new eigenfrequencies for the entangled atom-field state. This splitting of the cavity

resonance is directly equivalent to observing VROs in the frequency domain [83]. I.II HISTORICAL BACKGROUND With a plethora of applications, it is imperative that efficient and robust protocols are developed to provide precise, reliable and cost-effective excitation and detection schemes. Historically Rydberg states have been created with electron impact [31], where collimated beams of fast electrons are used to excite ground state atoms, or charge excitation [32] between a beam of ions and a population of neutral atoms [2]. The drawback inherent in both systems is that the kinetic energy of the interaction can make a contribution to the final energy of the Rydberg, resulting in a population with a broad spread of energy levels.

Optical excitation, made possible by the arrival of lasers in the 1960s [33,34], and more specifically with the demonstration of a tuneable dye laser in 1972 [35], has made it possible to exact much more control over the produced distribution of Rydberg energies. With optical excitation schemes a required final energy can be selected and rigorously enforced (within specific restricted excitation pathways) through specification of

the energy of the incident photon absorbed by the ground state Rydberg. These developments mean that a consistent, mono-energetic population of Rydberg atoms can now be created through attaining precise control of a lasers frequency.

Optical detection of Rydberg states has traditionally been problematic due to a small absorption cross-section; the radial component of the dipole matrix-element between the ground and high-n states is extremely small, consequently they tend to exhibit very weak absorption of fluorescence signals [36]. Indirect electronic detection techniques such as field ionisation have therefore been extensively used [2,5,12], whilst the detection of ionisation products is an efficient and widely adopted approach, it presents one major drawback – destruction of the Rydberg.

The development of purely optical three-step excitation and detection procedure in 2009 [36] transformed the field, demonstrating frequency stabilisation to an optimal excitation pathway of Rydberg hyperfine transitions [37] through spectroscopy of the rubidium hyperfine manifold they were able to produce 63P3/2 Rydberg states on demand. Obtaining such high-resolution spectroscopic information from a room-temperature gas cell relies upon the application of Doppler-free spectroscopic techniques such as polarisation spectroscopy [38,39] and the use of electromagnetically induced transparency [40,41]. This offers a far superior scheme, which achieves the detection and excitation of Rydberg states simultaneously as part of a non-destructive procedure. Whilst the development of a monochromatic light source has undoubtedly been instrumental, the ‘purely optical detection’ [36,42,43,44,45] scheme utilised in this work also builds upon a number of important developments that have ushered in the ‘era of precision laser spectroscopy’ [44]. Early optical spectroscopy involved analysis of absorption lines on a spectrograph as the only available light sources were broadband, incoherent and non-tuneable [44], fortunately the situation improved with the introduction of the tuneable dye laser in the early 70s. With the first external-cavity diode laser in 1977 [46], achieved through the coupling of a continuous wave (cw)

GaA1As-diode laser to an external resonator it became possible to achieve a single-axial mode line-width of 32MHz that could be coarse-tuned over >10nm and fine-tuned over a 500MHz range [46]. Diode lasers have since become ubiquitous in high-resolution spectroscopy due to their low-cost, narrowband output, and ease-of-use [46], and with line-widths of <1MHz now commercially available [47] the spectral resolution of laser absorption spectroscopy is now only limited by Doppler-broadening of the signal produced by the gaseous sample. II.I THEORETICAL BACKGROUND With narrow linewidth lasers readily available the principal obstacle to performing high-resolution spectroscopy in a room temperature gas cell are restrictions introduced by the motion of the atomic vapour itself, in a process known as Doppler broadening (the broadening of spectral lines due to the Doppler effect). As the cell contains rubidium gas with a Maxwell-Boltzmann velocity distribution atoms move randomly in all directions, each component of velocity taking a distribution of values [48]. This range of different velocities give rise to a range of Doppler shifts, the cumulative effect of which is inhomogeneous line broadening of the spectral signal [49]. The broadening mechanism is inhomogeneous as each atom interacts differently with the laser beam as frequency detuning (and therefore absorption and emission) depends on the velocity of each specific atom [50]. Fortunately, in conjunction with the introduction of tuneable diode lasers, the 1970s also saw the development of powerful non-linear spectroscopic techniques [51,52,53,54] to eliminate the broadening mechanism and allow sub-Doppler resolution [44]. Early Doppler-free spectroscopy was first demonstrated in 1942 [55], well before the development of lasers, with collimated atomic beam spectroscopy. The approach requires the collimation axis to be perpendicular to the atomic beam, a thin vertical slit produces a small angular spread, giving a non-isotropic atomic sample to significantly reduce broadening effects [44,50] - the extent of the broadening is decreased as velocity components along the transmission axis are reduced [55].

Figure 2: Population densities of the two levels as a function of velocity for three different laser frequencies. Absorption of pump

and probe beams by ground state atoms (N1 ( )) depends on the Doppler shift of the laser frequency to the transition frequency through the velocity of the atoms. The three figures show the situation for a laser frequency below, equal to, and above the atomic

resonance frequency, respectively. Figure taken from [50]

Saturated absorption spectroscopy (SAS), first demonstrated in the early 1970s [51,56] was the first technique to negate Doppler broadening in a gas cell, exploiting a velocity-selective saturation of absorption to produce a Doppler-free spectral signal. A beam splitter is used to divide the laser into a strong ‘pump’ beam and a less intense ‘probe’, which are then made to counter-propagate through the gas cell. The intensity of the two beams is arranged such that (Iprobe<<Isat) and (Ipump ≥ Isat), during propagation through the cell the pump beam causes the ‘saturation’ of the transition as large

numbers of atoms in the =0 velocity group are excited. This leads to a reduction in the population difference between the levels, consequently ‘burning’ a hole into the lower level population density (see fig.2) [50]. The probe beam consequently encounters significantly less

=0 atoms during propagation, meaning a more intense probe signal is transmitted through the cell. Conversely, the absorption spectra for the probe beam contains a large dip, or hole, in what is known as a ‘Lamb dip’, at the resonant frequency [57], the technique is referred to as ‘velocity selective’ as the Lamb dip created by the pump is burnt into the distribution of ground state atoms

with υ≈0, thereby enabling a Doppler-free spectral signal to be produced [44]. Generally, the pump interacts with atoms in the

velocity class υ = (ω-ω0)/k within the cell, exciting a significant proportion to the upper level. When the laser is far from the atomic resonance the two beams interact with a completely different velocity class of atoms,

yielding a typical Doppler-broadened spectral signal [50]. However, close to resonance, the

frequency difference between the beams is Δω≈0,

meaning υpump = υprobe ≈0, both beams therefore

interact with the same (υ≈0) velocity class - as only atoms with a non-zero axial velocity component can be resonant with probe and pump simultaneously[44]. As the hole is burnt into the zero-velocity class of atoms, and only these can contribute to the signal, the ‘Lamb dip’ is free from Doppler broadening and the spectra obtained is therefore naturally Doppler-free. The earliest SAS experiment, performed at Stanford in 1971 involved the investigation of the sodium D-transition with a repetitively pulsed tuneable dye laser. With a 7MHz laser linewidth the hyperfine splitting of the 3S1/2 and 3P1/2 states were resolved for the first time using SAS techniques [51]. The same group also reported the first SAS of Hydrogen a year later with a successful elimination of the Doppler broadening

effect for the H Balmer line at 6563Å [54], this allowed the fine structure to be resolved for the first time and provided the first optical determination of the Lamb shift between the 2S1/2 and 2P1/2 states (a result that had a phenomenal impact on the field of QED!) [44,54]. The work at Stanford enabled an order of magnitude improvement in accuracy for the Rydberg constant R∞ [44] – an impressive achievement considering that the attainable resolution was limited by the large bandwidth of the pulsed lasers utilised in these pioneering experiments, cw lasers that are predominantly used in modern experiments have a much narrower bandwidth [50].

Sub-Doppler laser spectroscopy was further enhanced four years later with another paper from the Stanford group in which they demonstrated another sensitive high-resolution, sub-Doppler spectroscopic technique, based on the light-induced birefringence and dichroism (different polarisation states experience a varying absorption) of a room-temperature gas cell, in a technique known as polarisation spectroscopy [38,39]. It was already acknowledged that signal magnitude in SAS depends on the relative polarisation of the two beams [38,58], but they were the first to utilise this induced optical anisotropy in a ‘sensitive polarisation detection’ scheme [38]. Monitoring the nonlinear interaction of a pump and probe during counter-propagation through a gas cell via induced changes in polarisation [38]

they were able to resolve the Balmer-β line of atomic Hydrogen near 4860Å with a cw dye laser, and revealed the stark splitting of the fine structure components [38]. As with the SAS setup, polarisation spectroscopy exploits a single laser, split into a counter-propagating pump and probe to produce a Doppler-free spectral signal when both beams interact with the same (zero-velocity) class of atoms [38]. The key difference between the schemes is that now a weak, linearly-polarised probe is used to ‘interrogate’ the birefringence of the Rb gas cell, while the strong, counterpropagating, circularly polarised pump beam is used to induce the necessary optical anisotropy [38,39]. The linearly-polarised probe can be decomposed into two circularly polarised beams, rotating in the same, and the opposite directions as the pump (in the case of a weak pump the two components can be considered separately), consequently the optical anisotropy created by the polarising pump causes the two components to see different refractive indices and absorption coefficients [38]. As any anisotropy encountered by the probe causes a rotation of its polarisation, analysis of the relative signal intensities after a polarising beam splitter (PBS) provides an extremely sensitive spectroscopic technique. The beam is decomposed into its horizontally and vertically polarised components by a PBS and it is the difference in intensity between the two

components that produces the polarisation spectroscopy signal. [39]. Instead of adding the signals to create a typical saturation spectrum, opting to subtract the two polarisation signals offers a number of advantages over the earlier technique of SAS, particularly for the application of Doppler-free spectroscopic techniques to laser frequency stabilisation. The approach is still essentially a form of SAS, as any change to the complex refractive index is proportional to the pump intensity [39]. At low intensities, the interaction of counter-propagating circularly polarised beams with different values of angular momenta can also be seen in terms of a velocity-selective hole-burning for the different (degenerate) sublevels of the magnetic quantum

number mF [38] found within each hyperfine level. Features of the scheme that are particularly useful for laser stabilisation applications include an increased sensitivity with a greatly improved signal-to-background ratio [38] on the ‘closed’ D2 transitions (indicated in red in Fig. 3) which are locked to during this experiment and the fact that the technique naturally differentiates the saturation spectrum [45], removing the necessity to frequency modulate the laser [39,71]. The unambiguous dispersion curve that is produced (the derivative of the sub-Doppler spectral signal) means an atomic resonance can be easily locked to without modulating the laser frequency to produce an error signal [38,44,45] - as is the case with the typical approach of ‘dither locking’ to a Doppler-free spectral feature [39]. Furthermore, the produced signal is embedded within a larger dispersion curve which is the differential of the Doppler background [45], this allows a far more robust lock to be achieved as each individual dispersion curve is located at a different voltage (see Fig.11) [45]. As an alkali metal, ground state rubidium has a single valence electron in the outer 5s orbital that surrounds the completely-filled orbital shells with smaller n-values. The ground state configuration is 1s22s22p63s23p63d104s24p65s, where the values 1-5 represent the principal quantum number n; s, p and d describe values of orbital angular momentum (l) of 0, 1 and 2 respectively and the superscript indicates the number of electrons with the corresponding n and l values [48].

Figure 3: Fine and hyperfine splitting of the 5S and 5P states in 85Rb, with the D1 and D2 transitions also represented. Arrows

between hyperfine states indicate selection-rule allowed transitions (∆F=0,±1), whilst the red arrows indicate ‘closed’ transitions that are not permitted an alternative decay channel.

Relativistic effects leads to the splitting of these atomic energy levels, producing what is known as fine structure, this is due to the fact that an electron has spin angular momentum in addition to orbital angular momentum, as evidenced by the 1922 Stern-Gerlach experiment [45,73]. It is the interaction between the magnetic dipole moment (µs) of the electron and the internal magnetic field (B) of the atom, known as the spin-orbit interaction, that causes fine structure splitting (in the electron rest frame the charged orbiting nucleus generates a magnetic field). The energy correction is referred to as an orientation energy (-µs∙B) as it is determined by the relative orientation of the two magnetic vectors [48]. The notation used to describe the resultant energy levels is known as a term symbol, which takes the form (2s+1)LJ, where S and L are the sum total of the electronic spin angular (si) momentum and of the orbital angular (li) momentum respectively, J is the total electronic angular momentum, which is the sum of the total spin and orbital angular momenta (J=L+S) and has allowed values of |L-S| to L+S. The quantum numbers S, L and J are the magnitudes of S, L and J, respectively and the value of (2S+1) is known as the spin multiplicity, which for this spin 1/2 system takes a constant value of 2 (for this reason

the superfluous value will be omitted henceforth, in preference of the more appropriate notation: nLJ ) [48]. For rubidium there is only one value possible for L and S as they both sum to zero for filled orbitals, meaning there is only one valence electron remaining that can contribute. The 5s ground state configuration gives a value of L=0 and S=1/2, and consequently a value of 1/2 for J, meaning the fine structure state is described as 5S1/2. The next level (5p) is described by L=1 and S=1/2, leading to values of 1/2 and 3/2 for J, as such the two allowed fine structure states are described as 5P1/2 and 5P3/2 (illustrated in Figure 3) [48]. Due to the spin-spin interaction between the electron and the nucleus there is a set of hyperfine sublevels embedded within each fine structure level that is also attributable to orientation energy, but in this case it is due to two magnetic dipoles in different orientations. The magnetic moment of the nucleus depends on the nuclear structure and is proportional to the spin angular momentum I of the nucleus, whose magnitude is given by the quantum number I. Hyperfine energy levels are determined by the total angular momentum F

of the atom, which is the sum of I and J (F=I+J); F is the magnitude of F, and can take the allowed values of |J-I| to J+I [48]. Naturally occurring rubidium comprises of two isotopes, stable 85Rb (72.2%) and the radioactive 87Rb (27.8%) with total nuclear spins of I=5/2 and I=3/2 respectively. Differing magnetic dipole moments for the two isotopes leads to different spin-spin orientation energies and as a consequence, different energies for different F values [48]. Each hyperfine level is further split into multiple Zeeman magnetic sub-levels (mF=2F+1), with each corresponding to a different projection of the total angular momentum onto the atomic axis, meaning that in essence mF tracks the quantised precession of the nuclear spin. Calculating F=J+I for 85Rb yields the resultant splitting, two hyperfine levels (F=5/2-1/2 (=2) and F=5/2+1/2 (=3)) are found within the 5S1/2 and 5P1/2 fine structure, whereas, due to the additional J value caused by a non-zero L value, four hyperfine levels are found within the 5P3/2 fine structure (F=5/2-3/2 (=1), F=5/2-1/2 (=2), F=5/2+1/2 (=3) and F=5/2+3/2 (=4)), as illustrated in fig. 3. Therefore, for an alkali atom with nuclear spin I there are two possible values of angular momentum available for the ground state (I±1/2) and four for the excited state (I±1/2, I±3/2) on the D2 transition line [39,60]. Due to electric dipole selection rules (∆F=0,±1) only certain transitions between the hyperfine states are permitted, as indicted by the arrows in Fig.3. Of these allowed transitions there are two indicated in red (Fg=3 <--> Fe=4 and Fg=2 <--> Fe=1) that give a much larger contribution to the spectral signal as they are ‘closed’ atomic transitions [39]. Selection rules dictate that these atoms do not have a permitted ‘decay channel’ to an alternative hyperfine ground state as it would require F>±1. As the other D2 transitions have alternative decay mechanisms, the polarisation spectrum for these

closed transitions is much more pronounced due to the effects of hyperfine optical pumping (also known as velocity-selective optical pumping) [60]; this results in enhanced frequency stability if a laser is locked to the 5S1/2 (F=3) to 5P3/2 (F=4) transition [39], as is the case in the three-step excitation scheme employed in this work. Technically, it is only the Fg=3 to Fe=4 transition from the Fg=3 ground state that provides a genuine saturated absorption feature, as the other allowed transitions (Fg=3 --> Fe=2 or 3) have a permitted decay mechanism into the Fg=2 ground state [60]. Consequently, hyperfine pumping is an extremely efficient method to deplete the number of atoms available for absorption by the probe - producing a significant increase in transmission (and therefore reduced absorption). As the 5S1/2 ground state splitting is ~3GHz (see fig.8) any atoms that are optically pumped to the Fg=2 hyperfine ground state are invisible to the probe, which is tuned to the 5S1/2 (F=3) to 5P3/2 (F=4) resonance; as such the Fg=2 state is referred to as a ‘dark’ state, that is ‘transparent’ to the probe beam [60]. Atoms in this ‘dark’ ground state are completely unavailable for absorption and therefore are unable to contribute to rotation of the probe polarisation. [39,60]. Population of a ‘dark’ ground state by means of hyperfine optical pumping is far more efficient at creating reduced probe-absorption than the established technique of transition saturation in SAS [60] as the process completely removes atoms from the pump–probe system. Atoms in the ‘dark’ state remain transparent to the probe until they are transferred back to the ground state through atomic collisions. As the time-scale for collisional redistribution is much longer than the time it takes for the laser to propagate through the cell, hyperfine, velocity-selective optical pumping to the dark ground state represents an extremely efficient sink for atoms from the absorbing population [60].

Figure 4: Three different EIT configurations: (a) ladder-type, (b) -type and (c) V-type; and are used to denote the coupling

and probe laser frequencies, respectively. Figure taken from [41]

Whilst polarisation spectroscopy is a versatile and powerful spectroscopic tool, the technique is unable to detect signals from weaker transitions with longer atomic lifetimes; consequently other techniques have been developed to produce Doppler-free signals in these instances. One such technique is known as quantum amplification (QA), analogous to ‘electron shelving’ in trapped ions it allows detection of a weaker resonance signal via the response of a strong (shorter lifetime) atomic transition with which it shares a common state. The process takes advantage of the lifetime discrepancy to produce an EIT, a quantum interference effect that occurs when a probe beam propagates through a gaseous medium with reduced absorption, a feature caused by the presence of a weak-transition beam that is of interest [61]. The probe encounters reduced-absorption when the transition of interest is coupled to a common (upper or ground) state of a strong probe transition. Driving atoms to an alternative state with a much longer lifetime via a ‘coupling’ beam reduces their availability to participate in absorption-emission cycles on the strong probe transition – leading to a reduced absorption of the probe and the spectral signal of interest. Applications of EIT to atomic physics and quantum optics experiments are extensive, offering solutions to a range of fundamental problems in QC and QI processing [41]; commercial applications include lasing without population inversion [74], the ability to reduce the speed of light to only 17ms-1 [75], quantum memory [76], and optical switches [77,78]: EIT techniques in a rubidium ladder scheme have been used to create an optical switch [78] that is capable of operating at 1MHz [41].

The technique was introduced at Stanford in 1991, with the first demonstration that “an opaque atomic transition could be rendered transparent to radiation at its resonance

frequency” [40]. In what is now known as a -type EIT scheme a coherence was induced between two independent ground states (|1> and |3>), as illustrated in fig. 4(b). The two states were coupled to a common excited state |2> and it was shown that the medium became transparent to the probe “when the Rabi frequency of the coupling field between |3> and |2> exceeded the inhomogeneous width of the |1>-|2> transition.” [8, 40] In contrast to the techniques described previously, the induced transparency is not attributable to either saturation or hole-burning phenomena, instead it is can be seen as a consequence of destructive interference between two dressed states, the result of populating the independent ground state |3> is the production of an equal and opposite dipole moment to that created by the resonance of interest [40,62]. Alternatively, in the bare-state picture, the EIT and the rapidly varying RI are viewed as the result of coherence between the levels, induced by the probe and coupling beams [40,62]. Whilst the original observation of EIT in a strontium gas cell utilised circularly polarised

( co-propagating lasers in a -type scheme [40], the technique has since diversified to include ladder-type and V-type EIT schemes (see

Fig.4) [41]. -type EIT schemes undoubtedly show more promise for commercial applications, however ladder-type EIT (where a coherence is induced between a ground state and an excited state via an intermediate state) [62] has also been wxploited in many research environments [8,37,41,61].

It was through the application of a ladder-type EIT scheme in 2001 that the first crucial steps were taken towards a purely optical excitation and detection scheme, when Rydberg d states up to n=8 were observed [61]. This was followed by a three-step excitation and detection scheme that traversed the 5S1/25P3/24D5/2nF pathway to produce low lying 4<n<10 Rydberg states [68]. By 2007 optical detection of highly excited Rydberg states had been reported for n=26-124 [8], the reduced-absorption created through implementation of the EIT ladder scheme produced an extremely narrow spectral signal, enabling measurements of fine structure splitting of the nd series up to n=96 [8]. The 2007 paper was therefore able to convincingly demonstrate that EIT provides an effective and non-destructive probe of high-n Rydberg energy levels, in direct contrast to the more-established indirect, destructive detection strategies that annihilate the Rydberg state so the resultant ionisation products can be analysed [2,8]. It was also demonstrated [8] that Rydberg transitions are highly insusceptible to linear and quadratic DC-Stark shifts, the electric analogue of the Zeeman effect (where spectral lines are split into multiple components due to the presence of a magnetic field) [36]. Purely optical detection of Rydberg states was also demonstrated with strontium atoms using EIT [63], using cw lasers they recorded an EIT linewidth of 5Mhz and were able to measure isotope shifts of the Rydberg states, reporting maximal EIT signals for a circularly polarised coupling laser and a linearly polarised probe beam [63]. EIT techniques were also applied to laser stabilisation in 2009 [37] when it was shown that it is possible to exploit ladder-type (cascade-type) EIT to create a signal for frequency stabilisation in the absence of a direct absorption signal [37]. Using the methods outlined above, a probe laser was locked to a rubidium D2 transition whilst a coupling laser excited atoms from the 5P3/2 state to a range of high-n states (n=19-70). As the longer-lifetime coupling transition produces a reduction in the availability of atoms to participate in the D2 transition, they produced a reduced absorption peak equivalent to the weak spectral signal in the absorption spectra of the probe laser.

An alternative approach to detecting Rydberg transitions was employed around the same time, utilising QA in a V-type EIT scheme [1]. Rydberg atoms were excited directly in a single step from the ground state with a 297nm frequency-doubled dye laser, whilst Rydberg state generation was monitored through absorption of a probe beam, resonant with a specific hyperfine transition on the D2 line. Whilst financial considerations makes it attractive to employ as few lasers as possible in any excitation and detection scheme, the lack of tuneable UV lasers necessary for direct excitation has led to the widespread application of three-step excitation schemes. Whilst the scheme outlined in [1] negates this problem, the necessity of frequency-doubling a laser brings additional complications and costs, and limits the power available from the laser. Consequently the scheme was subsequently revised, and the 297nm UV laser was replaced with diode lasers in a three-step ladder cascade excitation scheme [36]. The first three-step ladder-type scheme of this type was demonstrated as early as 1995 [7,64] and involved the excitation of lithium atoms to n~15 states by a means of the 2S2P3SnP

pathway. Lasers were locked to the 2S2P and 2P3S transitions while a third was resonant with the nP atomic transition. Through application of photo and field ionisation techniques they were able to detect Stark resonances in an electric field, evidencing the fact that low-power cw diode lasers could be applied effectively to Rydberg excitation [7]. While rubidium offers a selection of excitation pathways, due to dipole selection rules there are only two available options for visible (near-IR), single photon transitions: the 5S5P transition at 780nm and the 5S6P transition at 421nm [7]. The 780nm transition is often preferable due to the commercial availability of such lasers, as this is the wavelength used for CD-R technologies [45]. It was through the use of near-IR diode lasers at 780nm, 776nm and ~1260nm that Thoumany et al, [36] were able to follow the 5S5P5DnP pathway to high-n Rydberg states (see fig. 8) [7]. This is now a well-established route that has since been applied in the measurement of quantum defects of the nP3/2 Rydberg states in 85Rb [43] and during measurements of the absolute frequency of the 85Rb nF7/2 Rydberg states [42].

Figure 5: Level scheme for the three-step excitation pathway utilised for optimal production of 63P3/2 Rydberg states, including the

Zeeman magnetic sublevels and relative excitation probabilities. Figure adapted from [36]

Figure 6(a) 63P3/2 three-step excitation pathway, and (b): nP and nF three-step excitation pathways, both with relative excitation

probabilities. Figure adapted from [44]

(a)

(b)

Figure 7: Experimental setup used to generate 63P3/2 Rydberg states

III.I EXPERIMENTAL SET-UP AND THEORERICAL METHODOLOGY The setup depicted in Figure 7 can be used to generate 63P3/2 Rydberg states by means of the three-step laser excitation pathway illustrated in Figure 6(a). It is also possible to excite nF5/2 and nF7/2 states via alternative pathways, as illustrated in Figure 6(b), through modification of the third step polarisation and frequency. These highly-valued, high-n states can be produced through the interaction of three external-cavity tuneable diode lasers with room-temperature vapour cells, each containing the naturally occurring composition of stable 85Rb

(72.2%) and the radioactive 87Rb (27.8%). Using well-established spectroscopic techniques [38,40] Doppler-free spectral signals were produced, allowing extremely precise laser-frequency stabilisation to rubidium hyperfine transitions, an accomplishment that would enable the detection of Rydberg states by purely optical means. The grating-stabilised diode lasers are controlled with electronics that allow modification of diode temperature, injection current and fine-tuning of wavelength via a piezo-mounted diffraction grating; a feature that allows the laser to be linearly swept through a range of frequencies to produce absorption spectra (as illustrated by Figure 9).

Figure 8: Hyperfine splitting of the 5S, 5P and 5D states in 85Rb, with the D1 and D2 transitions also represented. Red arrows

between hyperfine states indicate the specific excitation pathway followed to the 5D5/2 state

The excitation pathway, detailed in Figure 6(a) and Figure 8 includes a 780.243nm transition between the 5S1/2 (F=3) and 5P3/2 (F=4) hyperfine states, a 775.979 nm transition between 5S1/2 (F=3) and 5D5/2 (F=5) and a 1256nm transition to the 63P3/2 or nF Rydberg states. From the 5D5/2 (F=5), mF =5 magnetic sublevel, electric dipole

selection rules (Δl =±1 and Δj = 0,±1) dictate that it is possible to excite nP3/2, nF5/2 or nF7/2 Rydberg states depending on the laser polarisation and frequency (Figure 6(b)). The particular three-step excitation pathway chosen for the single-atom detection scheme offers many advantages over alternative excitation strategies; direct excitation of Rydberg states from the ground state can be problematic as tuneable UV diode lasers that provide the required transition frequency are not commercially available, a limitation also inherent in two-step excitation schemes. [45] Whilst this has been negated by means of second harmonic generation, through the frequency-doubling of an IR diode laser [1], this can unnecessarily

complicate the laser setup and unduly limit the attainable power and excitation rates [7]. A three-step scheme provides enhanced control over the Rydberg states that are produced; the ability to independently determine laser polarisation for each of the three steps means optical selection can be used to address specific hyperfine and mF sub-levels, resulting in an excitation pathway with optimal coupling strengths, as indicated in Figure 5 [7,36]. Figure 6(a) depicts the three-step excitation path used to generate 63P3/2 Rydberg states, along with the angular component of the relative excitation probability for a range of transitions, such values can be obtained through calculation of the relevant 6j symbols, or sourced from the literature [36]. It is clear that through hyperfine optical pumping to the 5S1/2, mF=3 magnetic sublevel and applying circularly polarised laser light that it is possible to achieve an optimal coupling strategy through selection of the excitation pathway with the highest excitation probability, and selectively address the 63P3/2 (F=4) hyperfine level[36].

~1260 nm

Additional benefits to a three-step system include the expensive nature of UV optical fibres, compared to the low-cost and availability of 780nm lasers and fibres, the ability to use the same optics for 780nm and 776nm laser light, and a much more extensive literature base, as spectroscopy has been well documented for these transitions. [45] Laser detection was achieved with custom-built photodiodes, based on the IPL 1053DAL chip design that contain an on-chip amplifier and integrated lens. During device characterisation [45] the sensitivity was recorded as 60V/µW, with a 32V range, a noise level of 100-300µV, and a SNR of 105 for very low intensity signals [45]. The first step laser is locked to the ‘closed’ 5S1/2 (F=3) to 5P3/2 (F=4) transition of 85Rb at 780.2437 nm by means of polarisation spectroscopy [39] of the rubidium D2 transitions (illustrated in Figure 9). The technique is preferable to the more general approach of Saturated Absorption Spectroscopy (SAS) as it produces an equivalent Doppler-free spectral signal that is significantly enhanced on the required ‘closed’ 5S1/2 (F=3) to 5P3/2 (F=4) transition [39]. Furthermore, a clean error signal can be produced (Figure 11) through the simple subtraction of the two polarisation signals, eliminating the need for frequency modulation of the laser to produce a lockable signal. Early polarisation spectroscopy utilised a single detector and passed a probe beam through nearly crossed polarisers [39,81], this work employs an alternative two detector setup, with a polarisation beam splitting cube, in an approach equivalent to that employed during laser frequency stabilisation to a reflecting reference cavity [39,65]. The specific experimental arrangement used to perform polarisation spectroscopy is illustrated in the schematic below (Figure10); a portion of the first-step laser, linearly polarised by an optical (60dB) isolator is split off from the main experimental beam for frequency stabilisation,

this is then split into a pump and less-intense probe beam for use in the polarisation spectroscopy scheme.

After interaction with λ/4 and λ/2 wave-plates respectively, the (now) circularly polarised pump and linearly polarised (45 degrees) probe beams are made to overlap during counter-propagation through a rubidium cell [45]. As discussed, this produces a pronounced dispersion curve for the ‘closed’ (F=3 --> F=4) transition that is locked during frequency stabilisation, in a process that has been detailed extensively by Pearman, et al [39] Figure 11 displays a typical polarisation spectrum and error signal produced by the experimental setup. It is clear that for the (F=3)(F=4) transition at zero MHz an unambiguous, ‘lockable’ signal is produced. As discussed, this error signal is generated by the birefringence introduced by the pump beam as it is seen differently by the two circularly polarised components of the linearly polarised probe (thereby rotating the probe polarisation), translating to a discrepancy in the relative polarisation signals recorded by the two photodiodes. Tweaking the relative intensity of the two polarisation signals, both through the signal processing electronics and via alteration to the respective waveplates allowed an optimal error signal with maximal steepness to be generated. Obtaining the largest possible gradient for the lock point, coupled with distinct separation of the dispersion curves for the various transition peaks enabled extremely stable first-step frequency locks to be achieved. Using this technique it was possible to attain Allan deviations of ≤15 kHz over ~5 min periods and ≤30 kHz over ~1hour periods (Figure 12), with an overall frequency drift of <100 KHz over 10+ min. periods and <<500 MHz over the course of an hour.

Figure 9: Doppler broadened and Doppler free spectroscopy of the D2 transitions in naturally occurring rubidium. (a) The four visible dips correspond to probe absorption by the four hyperfine 5S1/2 ground states (two for each isotope of Rb). The distinction between Doppler-broadened and Doppler-free spectroscopy can clearly be seen in both plots. Hyperfine structure is visible in the

Doppler-free spectra, with the 5S1/2 (F=3) to 5P3/2 (F=4) transition of 85Rb at 780.2437 nm indicated in (b).

(a)

(b)

Figure 11: Polarisation spectra obtained using the setup detailed in Figure 10. Both polarisation signals and the naturally generated error signal (A-B) are displayed. It is possible to identify the discrepancy between the two signals (centred at 0 MHz) for the ‘closed’

5S1/2 (F=3) to 5P3/2 (F=4) that is generated by the optical anisotropy of the gas cell (traces are offset for clarity).

Figure 10: Schematic of the experimental set-up used to perform polarisation spectroscopy. A circularly polarised pump beam (thicker line) was used to induce an optical anisotropy in

the rubidium cell that was subsequently interrogated by a linearly polarised probe. The birefringence created by the pump is seen differently by the two circularly polarised

components of the probe leading to dispersion-shaped error signal, as shown in Figure 11.

The degree of control attained through application of the described procedure is best illustrated through direct comparison with the response of an ‘unlocked’ laser; in this instance an Allan deviation of >50 kHz was recorded after 1 minute, and >75 kHz over a 5 minute period (Figure 12), whereas the overall frequency drift was measured to be >17MHz over a 10 minute period of free-running. The 5S1/2 (F=3) and 5P3/2 (F=4) transition peak was calculated to have a FWHM of 10.3±0.09 MHz (fitting error), as such a deviation of ≤30 kHz is a tolerable fluctuation for purposes of frequency stabilisation, as it means that the frequency of the laser is fluctuating by ~0.0029% of the transition linewidth. This does not mean, however that the laser linewidth is narrowed to

this extent, as it is still of the order of ~300 kHz [45]. The Allan deviation, or root Allan variance provides a good indication as to the stability of the laser lock, developed by David W. Allan as a measure of frequency stability in atomic clocks The Allan deviations plotted in Figure 12 are indicative of the typical response of the laser to stabilisation techniques and can be used to determine the optimal operating time of the frequency lock as minimal deviation periods can be identified. For example it is possible to determine minimal deviation of <15 kHz after a 3 minute period for the blue trace in Figure 12.

Figure 12: Selection of first step locks obtained with through polarisation spectroscopy. (a) Allan deviation of first-step laser locks,

(b) Comparison of Allan deviations achieved for a locked and unlocked first-step laser.

(a)

(b)

Figure 13: Schematic of the experimental set-up used to perform the co-propagating EIT technique where the response of the 780nm first-step laser (solid red line) to detuning of the second-step (dashed red line) is monitored. Linearly polarised light from

both beams is combined on a PBS cube before co-propagation through the room-temperature rubidium gas cell. The second (Glan-laser) beamsplitter after the cell only allows the vertically polarised first-step laser to propagate through for detection by the

photodiode.

The second transition is driven by a tuneable diode laser operating at 776nm; as with the 780nm stabilisation scheme a portion is split off from the main experimental beam to enable frequency stabilisation. Unlike the previous scheme, the 776 nm laser is stabilised by overlapping it with either a co- or counter-propagating portion of the 780nm first-step beam and monitoring the first step response as a function of second–step detuning, utilising a technique known as EIT. During the course of the experiment both co- and counter-propagating configurations were utilised simultaneously to verify that a first step ‘top-of-peak’ lock had been achieved, providing optimal stabilisation. Crucially, the technique provides a Doppler-free spectral signal without the need to apply a velocity-selection procedure, when the first step laser is locked to the 5S1/2 (F=3) to 5P3/2 (F=4) transition it is only resonant with the class of atoms possessing ~ zero axial velocity, consequently it is only these atoms that contribute to the EIT signal - meaning that the second step transition signal is naturally Doppler-free [45]. The second-step laser is locked to the 5P3/2 (F=4) to 5D5/2 (F=5) transition of 85Rb at 775.97880nm, as the 5D5/2 state has a much longer lifetime than the 5P3/2 level, direct spectroscopic detection is extremely challenging [36]. A technique that turns this to an advantage is known as quantum amplification (QA), analogous to ‘electron shelving’ in trapped ions, it is a process which utilises the lifetime discrepancy to produce an EIT, yielding an enhanced first step transmission

signal which is a function of the second step detuning. In essence, QA allows the detection of a weaker resonance signal via the response of a strong atomic transition, with which the weak transition shares a common state; there are a range of possible configurations that can achieve this effect, as illustrated in Figure 4. This work utilises a ‘ladder’ and a ‘lambda’ transition for the co- and counter-propagating setups respectively, although for experimental purposes the more-robust counter-propagating ‘lambda’ transition would be exclusively adopted. A ‘lambda’ transition provides a preferable approach to second step stabilisation as the first and second step transition frequencies are ~ equivalent, meaning that for a counter-propagating setup the Doppler shift of each beam almost cancel [69]. During counter-propagation, if the first step transition is detuned the second step will lock to a frequency detuned by an equal amount, in the opposite direction – keeping the sum frequency constant [45]. In contrast, any detuning to the first-step laser in the co-propagating setup will induce an unwanted frequency shift for the second-step transition. With both approaches, an excitation to the 5D level hinders multiple absorption-emission cycles for the 5S-5P transition – leading to a visibly increased, or ‘enhanced’ first-step transmission, also known as a ‘reduced absorption’ peak [36] which is a direct representation of the spectral signal for the 5P3/2 (F=4) to 5D5/2

(F=5) transition. Whilst it is not possible to realise the phenomenal amplification (values of up to 106) that have been reported with trapped ions [70] due to the gaseous nature of the sample (collisions between atoms and interaction time limitations are known to decrease the QA factor to values of ~103), it is still more than sufficient to produce an observable signal for an atomic transition which would otherwise be too weak to be detected directly [36]. As such, monitoring the 780nm first-step transmission as a function of second-step detuning provides an ideal solution to second step stabilisation [44], the lifetime of the 5D5/2 state is

=238.5ns [79], which is approximately 10 times that of the 5P3/2 state, which has been

measured to be =26.24 ns [80]. Whilst atoms are excited to the 5D state they are unable to participate in absorption of the first step, meaning

the ‘reduced absorption’ peak is (τD/τP)~10x more pronounced than the optically available spectral signal. [44,45] An alternative detection mechanism is offered by monitoring the 420nm fluorescence emitted by the 5D6P5S cascade with a photomultiplier tube (PMT) [44]. The visible fluorescence, produced when both lasers are on resonance is caused by atoms in the 6P3/2 state dropping back to the 5S ground state after being populated by atoms in the 5D level [36]. Applying a low-pass filter to the TTL pulses generated from the PMT can produce a usable SNR (above 30 when using a 1kHz low-pass filter), however the limitation that the experiment must be conducted in perfect darkness makes the approach relatively untenable [45]. In addition, a direct comparison [36] between the SNR and resolution attainable from the two approaches concluded that the enhanced transmission scheme was the superior scheme, a further consideration is that the signal obtained from a single photodiode is equivalent to measuring a full 4π solid angle of fluorescence from a vapour cell, and that the analogue signal produced by a photodiode is preferable for the purposes of frequency stabilisation [44]. The specific arrangement used to perform the EIT procedure is illustrated in Figure 13; whilst both setups were employed to optimise the first step ‘top-of peak’ lock, the counter-propagating setup approach is in many ways preferable.

Whilst both schemes utilise QA to detect the 5D5/2 state, and produce an equivalently intense signal, without the two-photon Doppler-shift cancelling effects, the co-propagating ladder transition displays a larger linewidth of ~18MHz [44].

In the co-propagating setup (represented in Figure 13) linearly-polarised light, split off from the first- and second-step laser is combined on a PBS before co-propagation through the cell. The second PBS, immediately after the rubidium cell is a Glan-Laser beamsplitter, which acts to dump the horizontally polarised second step beam (with an accuracy of one part in 105!!! [45] and allows the vertically polarised first step to transmit through to the photodiode unhindered. To enhance the SNR obtained from the setup, the degree of laser co- propagation was maximised and laser alignment between the two beamsplitters was optimised. Beam size and intensities were modified via adjustable ‘iris’ components and half-wave plates respectively [45]. Figure 14(a) displays an optimised EIT signal obtained through QA for the 5P3/2 (F=4) to 5D5/2 (F=5) co-propagating ‘ladder’ transition. To further enhance the signal, first and second step laser powers were diligently modified within experimentally determined parameters [44] to produce an optimised SNR. Work had already been done [44] within the group to characterise the response of the second step FWHM and peak size to variations in first and second step laser power. It was demonstrated that power broadening of the signal becomes significant for first-step powers >60µW, and that absorption of the first-step laser by the rubidium cell leads to an exponential decrease in second-step peak size for powers <60µW, this combined with saturation effects at powers >1µW means a second-step power in the region of ~10µW provides an ideal compromise between unwanted power broadening, signal saturation and over-absorption of the laser beam. The same author also demonstrated that the second step peak and FWHM are a function of the second-step laser power and deduced that a power of ~10µW allowed the best SNR and frequency resolution [44]. These guidelines were followed to produce an optimal spectral signal for locking, the power incident upon the gas cell was determined with a commercially available power meter.

Figure 14: (a) Typical 5P3/2 (F=4) to 5D5/2 (F=5) spectral feature detected using QA, as such the signal is a first-step reduced

absorption peak, obtained from the co-propagating EIT setup. (b) Typical dispersion-shaped error signal generated through FM of the spectral feature illustrated in (a).

The counter-propagating ladder-transition EIT scheme is performed in essentially the same way as for co-propagation, however due to the fact that the beams are travelling in opposite directions a Glan-laser beamsplitter is no-longer required to separate the first step EIT signal. The advantage of fewer experimental components, combined with a Doppler-shift cancellation makes the extensively-studied 5S5P5D two-photon transition a preferable approach, furthermore due to the cancellation, two-photon excitation is possible for a wider range of detunings, making the system much more robust [7].

As mentioned, the differing response of the co-propagating and counter-propagating signals to a first-step detuning provides an invaluable reference for the quality of the first-step lock. Without a frequency comb to provide high-resolution frequency measurements, this technique provides an essential diagnostic tool to ensure ‘top-of-peak’ stabilisation [44], as illustrated in Figure 15. The variation in response between the two approaches is potentially attributable to the fact that the ‘lambda’ peak sees less of a detuning effect as counter-propagation works to cancel any frequency difference, whereas the ‘ladder’ transition peak is much more susceptible - any detuning to the first step frequency will set an incorrect frequency and velocity class for the second step transition which is not compensated for by an opposite beam. The frequency splitting has also been attributed to a velocity- selection effect; the co- and counter-propagating beams experience equal and opposite second-step Doppler shifts as a result of any detuning. Away from resonance, atoms with a finite velocity along the cell axis are excited,

meaning that opposite Doppler-shifts are experienced by the opposite beams, resulting in a frequency-space separation [44]. Crucially, the two independently-derived signals display a different frequency response to a first-step detuning, as a consequence, simply ensuring that the two peaks overlap (Figure 15(b)) means there is minimal first-step detuning from the desired 5S1/2 (F=3) to 5P3/2 (F=4) transition frequency. This technique was applied before locking to the second step transition, in order to ensure maximal first step stability and overall accuracy. Frequency modulation [82] (FM) of the first-step laser produced a pronounced error signal to facilitate stabilisation to the 5P3/2 (F=4) to 5D5/2 (F=5) transition. A frequency dither is applied to the diode through the piezo and injection current to produce a dispersive lineshape that can be easily locked to (Figure 14(b)). The modulation is generated by a lock-in amplifier (LIA) which is contained in the Laselock unit, along with the proportional-integral-derivative (PID) servo electronics [44]. The probe is subjected to FM to produce side bands above and below the EIT resonance; it is through the beating of these sidebands with the resonant frequency that a detector signal at the modulation frequency is created - allowing a dispersive detection signal to be recovered through the LIA performing phase-sensitive detection [37]. The Laselock used to generate the error signal is the same versatile apparatus used to scan the laser frequency to produce the spectral features for the first two transitions.

(a)

(b)

Figure 15: The response of the co- and counter- propagating setups to a first step laser detuning. An optimal first-step lock is achieved when both reduced absorption peaks are coincident (as seen in (b)), this signifies a ~ zero Doppler-shift, caused by minimal deviation of the first step laser from the resonant transition frequency. The co-propagating peak, indicated in red can be identified as

the spectral feature with the larger linewidth. Using the techniques detailed above it was to possible to achieve an Allan deviation of ~40 kHz over <5 minute periods and ~45 kHz over 50m time periods, as illustrated in Figure 16(b). With a linewidth of the order of ~14 MHz [45] this extent of deviation represents approximately 0.003%. For an ‘unlocked’ second step laser an Allan deviation of 0.1 MHz was observed after only 1 minute and 0.25 MHz after 14 minutes (see Figure 16(a)). An overall frequency drift of 34 MHz was recorded over a 14m period for the blue unlocked laser trace in Figure 16(a), a value well over twice the of the transition linewidth. Figure 17 depicts the relative frequency stability that was achievable for the two laser beams, it can be seen that whilst the first step is routinely more stable than the second step, both are within a tolerable range of <70 kHz for the majority of instances. The first step was generally controllable to <30 kHz deviation whilst the second-step generally exhibited a 40-70 kHz deviation. IV.I SUMMARY AND CONCLUSIONS Overall the laser stabilisation achieved would be more than sufficient to act as a reliable source of 63P3/2 Rydberg atoms with the introduction of a third-step laser for the final transition. Allan deviations of ~30 kHz and ~45 kHz have been demonstrated for the two transitions respectively, in both cases this is an extremely minute fraction of the spectral linewidth. A more significant consideration to transition stability is the overall frequency drift of the laser, however, at values of <100 kHz this does not present an insurmountable problem as it still represents a very small proportion of the transition linewidth. Errors incurred during the course of the experiment are difficult to quantify, the largest

systematic error would have been the un-calibrated wavemeter used to make frequency measurements, it has been estimated that a freshly calibrated wavemeter of the same kind could produce an error in the region of 4 MHz[45] so the error incurred could be anything above 4MHz. It is unlikely that an excessively large systematic error would have been incurred without it being noticed however, as the transition wavelengths have been well established (to the extent where specific D2 transitions are used as frequency standards). Random errors in the first and second step transitions have been estimated to be ~750 kHz and ~ 1 MHz, respectively [45], along with ~3 MHz temperature shifts. V.I FUTURE WORK Following frequency stabilisation to the 780nm and 776nm transitions a portion of each laser would be used to stabilise the third step transition frequency; as such all three lasers would interact within another room-temperature rubidium gas cell. As with the second-step scheme the three beams would be separated after the gas cell by means of polarisation beams splitters thanks to their orthogonal polarisation, so that the reduced absorption of the 780nm laser could be scrutinised [36]. Circular polarisation of the lasers would be

produced by λ/4 plates for the first and second

step lasers and a λ/4 Fresnel-rhomb retarder for the 1260nm laser [44]. As with the generation of a second-step EIT signal, a first-step reduced absorption peak could be generated by sweeping the ~1260nm laser through an appropriate frequency range as it is a function of third step detuning.

(a) (b) (c)

As with the second step EIT signal QA plays an important role in the procedure; due to discrepancies in the atomic lifetimes of the respective states an excitation to the 5D5/2 state would restricts absorption-emission cycles on the second-step transition, which in turn would restrict absorption-emission cycles on the first-step transition, leading to a reduced absorption peak in the first-step signal [36,42] that is representative of the spectral feature of the final transition. This transition could then be locked to using FM in the same manner as for the second step transition to generate a dispersive error signal. As previously discussed (illustrated by Figure 5), choosing specific polarisations for the respective transitions would allow for an optimal excitation path to be followed through the Zeeman magnetic sublevels, thereby allowing a maximal coupling to

be achieved. The relative excitation probabilities can be calculated via 6J symbols and clearly indicate that the most efficient excitation pathway for the production of 63P3/2 Rydberg states

involves a , , excitation for the first, second and third transitions respectively. After

application of a , excitation it would also possible to excite nF5/2 and nF7/2 Rydberg states (Figure 6(b)) and it would be possible to observe lower nF7/2 Doppler-free signals without the need to apply LIA techniques. The introduction of an optical frequency comb, as utilised in [41,42,44,45] would greatly enhance the accuracy of the results; developed by Hansch [85] the apparatus is able to relate an optical frequency to a microwave frequency with an accuracy of 10-16 [45].

Fig 16: Selection of second step locks obtained with the EIT procedure and FM. (a) Comparison of Allan deviations achieved for a

locked and unlocked second-step laser. (b) Allan deviation of second-step laser locks.

(a)

(b)

Fig 17: Comparison of Allan deviations achieved for the first- and second-step laser.

[1] P. Thoumany, T. Hansch, G. Stania, L. Urbonas and Th. Becker. Optical spectroscopy of rubidium Rydberg atoms with a 297nm frequency-doubled dye laser. Optics Letters 34, 1621 (2009) [2] T.F. Gallagher. Rydberg Atoms. Cambridge University Press (1994) [3] D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Cote and M. D. Lukin. Fast Quantum Gates for Neutral Atoms. Phys. Rev. Lett. 85, 2208 (2000) [4] E. Urban, T. A. Johnson, T. Henage, L. Isenhower, D. D. Yavuz, T. G.Walker and M. Saffman. Observation of Rydberg blockade between two atoms. Nature Physics 5, 110 (2009) [5] H. Walther, B. T. H. Varcoe, B-G. Englert and T. Becker, Cavity quantum electrodynamics. Rep. Prog. Phys. 69, 1325 (2006) [6] Lukin, M. D. et al. Dipole blockade and quantum information processing in mesoscopic atomic ensembles. Phys. Rev. Lett. 87, 037901 (2001). [7] D. P. Fahey and M. W. Noel. Excitation of Rydberg states in rubidium with near infrared diode lasers. Optics Express 19, 17002 (2011) [8] A.K. Mohapatra, T. R. Jackson and C. S. Adams. Coherent Optical Detection of Highly Excited Rydberg States Using Electromagnetically Induced Transparency. Phys. Rev. Lett 98, 113003 (2007). [9] I. Friedler, D. Petrosyan, M. Fleischhauer, and G. Kurizki. Long-range interactions and entanglement of slow single-photon pulses. Phys. Rev. A 72, 043803 (2005). [10] D. Meschede, H. Walther and G. Muller. One-Atom Maser. Phys. Rev. Lett 54, 551 (1985) [11] S. Deléglise, I. Dotsenko, C. Sayrin, J. Bernu, M. Brune, J-M. Raimond and S. Haroche Reconstruction of non-classical cavity field states with snapshots of their decoherence. Nature 455, 510 (2008) [12] J.M. Raimond Exploring the quantum world with photons trapped in cavities and Rydberg atoms. (2011) Oxford University Press. http://physinfo.fr/houches/pdf/Raimond-notes.pdf [13] E. Scrodinger. The Present Status of Quantum Mechanics. Die Naturwissenschaften. 23, 48 (1935) [14] M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J. M. Raimond, and S. Haroche. Quantum Rabi Oscillation: A Direct Test of Field Quantization in a Cavity. Phys. Rev. Lett. 76, 1800 (1996) [15] B. T. H. Varcoe, S.Brattke, M. Weidinger and H. Walther. Preparing pure photon number states of the radiation field. Nature 403, 743 (2000) [16] J. M. Raimond, M. Brune, and S. Haroche. Manipulating quantum entanglement with atoms and photons in a cavity. Rev. Mod. Phys 73, 565 (2001) [17] Haroche, S – ‘Quantum information in cavity quantum electrodynamics: logical gates, entanglement engineering and `Schrodinger-cat states’. Phil. Trans. R. Soc. Lond. A 361, 1339 (2003) [18] P. J. Blythe and B. T. H. Varcoe – ‘A cavity-QED scheme for cluster-state quantum computing using crossed atomic beams’. New Journal of Physics, 8, 231 (2006) [19] J.H. Eberly, N.B. Narozhny, and J.J. Sanchez-Mondragon "Periodic spontaneous collapse and revival in a simple quantum model". Phys. Rev. Lett. 44, 1323 (1980). [20] G. Rempe, H. Walther, and N. Klein. "Observation of quantum collapse and revival in a one-atom maser". Phys. Rev. Lett. 58, 353 (1987). [21] M. Weidinger, B. T. H. Varcoe, R. Heerlein, and H. Walther. Trapping States in the Micromaser. Phys. Rev. Lett 82, 3795 (1999) [22] Haroche, S, Raimond, J-M, et al – ‘Progressive field-state collapse and quantum non-demolition photon counting’. Nature 448, 889 (2007)

[23] S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Deleglise, U. B. Hoff, M. Brune, J-M Raimond and S. Haroche. Quantum jumps of light recording the birth and death of a photon in a cavity. Nature 446, 297 (2007) [24] J. Bernu, S. Deleglise, C. Sayrin, S. Kuhr, I. Dotsenko, M. Brune, J. M. Raimond,and S. Haroche. Freezing Coherent Field Growth in a Cavity by the Quantum Zeno Effect. Phys. Rev. Lett. 101, 180402 (2008) [25] Jones, M. L, Wilkes, G. J and Varcoe, B. T. H. – ‘Single microwave photon detection in the micromaser’. J. Phys. B: At. Mol. Opt. Phys, 42, 145501 (2009) [26] S. Brattke, B. T. H. Varcoe and H. Walther Preparing Fock states in the micromaser. Opt. Express 8, 131 (2001) [27] S. Brattke, B. T. H. Varcoe and H. Walther. Generation of Photon Number States on Demand via Cavity Quantum Electrodynamics. Phys. Rev. Lett. 86, 3534 (2001) [28] C. H. Bennett and G. Brassard, Proc. IEEE Int. Conference on Computers, Systems and Signal Processing , IEEE Press, 175 (1984). [29] Thompson, R. J, Kimble, H. J, et al – Normal-Mode Splitting and Linewidth averaging for Two-State Atoms in an Optical Cavity. Phys. Rev. Lett, 63, 240 (1989) [30] Thompson, R. J, Kimble, H. J, Rempe, G – Observation of Normal-Mode Splitting for an Atom in an Optical Cavity. Phys. Rev. Lett. 68, 1132 (1992) [31] J. Olmsted III Excitation of nitrogen triplet states by electron impact. Radiation Research 31, 191 (1967). [32] M. Haugh, T. G. Slanger, and K. D. Bayes. Electronic excitation accompanying charge exchange. Journal of Chemical Physics 44, 837 (1966) [33] Maiman, T. H. (1960). "Stimulated optical radiation in ruby". Nature 187, 493 (4736) [34] "The Nobel Prize in Physics 1964". Nobelprize.org. 17 Apr 2013 [35] T. W. Hänsch, Repetitively Pulsed Tunable Dye Laser for High Resolution Spectroscopy, Appl. Opt. 11, 895 (1972). [36] P. Thoumany, Th. Germann, T. Hansch, G. Stania, L. Urbonas and Th. Becker. Spectroscopy of rubidium Rydberg states with three diode lasers. Journal of Modern Optics 56, 2055 (2009) [37] R. P. Abel, A. K. Mohapatre, M. G. Bason, J. D. Pritchard, K. J. Weatherill, U. Raitzsch and C. S. Adams. Laser frequency stabilization to excited state transitions using electromagnetically induced transparency in a cascade system. Applied Physics Letters 94, 071107 (2009) [38] C. Wieman and T. W. Hansch. Doppler-free laser polarisation spectroscopy. Physical Review Letters, 36, 1170 (1976) [39] C. P. Pearman, C. S. Adams, S. G. Cox, P. F. Griffin, D. A. Smith and I. G. Hughes. Polarisation spectroscopy of a closed atomic transition: applications to laser frequency locking. J. Phys. B: At. Mol. Opt. Phys. 35, 5141 (2002) [40] K. J. Boller, A. Imamoglu and S. E. Harris. Observation of Electromagnetically Induced Transparency. Phys.

Rev. Letts 66, 2593 (1991)

[41] A. J. Olson and S. K. Mayer. Electromagnetically induced transparency in rubidium. Am. J. Phys. 77, 116 (2009) [42] L. A. M. Johnson, H. O. Majeed, B. Sanguinetti, Th. Becker and B. T. H. Varcoe. Absolute frequency

measurements of 85Rb nF7/2 Rydberg states using purely optical detection. New Journal of Physics 12, 063028 (2012)

[43] B. Sanguinetti, H. O. Majeed, M. L. Jones and B. T. H. Varcoe. Precision measurements of quantum defects in the nP3/2 Rydberg states of 85Rb. J. Phys. B: At. Mol. Opt Phys. 42 (2009)) [44] Luke Johnson Thesis – Precision Laser Spectroscopy of Rubidium with a Frequency Comb. University of Leeds. (2011)

[45] Bruno Sanguinetti Thesis – Building a modern micromaser: atoms and cavities. University of Leeds (2009)

[46] Voumard, C. "External-cavity-controlled 32-MHz narrow-band cw GaA1As-diode lasers". Optics Letters 1, 61

(1977). [47]http://www.toptica.com/products/diode_lasers/research_grade_diode_lasers/tunable_diode_lasers.html [48] University of Florida - Department of Physics PHY4803L | Advanced Physics Laboratory http://www.phys.ufl.edu/courses/phy4803L/group_III/sat_absorbtion/SatAbs.pdf [49] A.E. Siegman. Lasers. (1986) [50] C. J. Foot. Atomic Physics (Oxford Master Series in Atomic, Optical and Laser Physics) (2005) [51]T. W. Hansch, I. S. Shahin and A. L. Schawlow. High-Resolution Saturation Spectroscopy of the Sodium D Lines with a Pulsed Tunable Dye Laser. Phys. Rev. Lett, 27, 707 (1971) [52] M. D. Levenson and N. Bloembergen. Observation of two-photon absorption without Doppler broadening on the 3S-5S transition in sodium vapour. Phys. Rev. Letts, 32, 645 (1974)

[53] T. W. Hansch, K. Harvey, G. Meisel and A. L. Schawlow. Two-photon spectroscopy of Na 3s-4d without Doppler broadening using a cw dye laser. Optics Communications, 11, 50 (1974) [54] T. W. Hansch, I. S. Shahin and A. L. Schawlow. Optical Resolution of the Lamb Shift in Atomic Hydrogen by

Laser Saturation Spectroscopy. Nature Physical Science 235, 63 (1972)

[55] K. W. Meissner. Applications of atomic beams in spectroscopy. Reviews of Modern Physics, 14, 68 (1942) [56] T. W. Hansch, M. D. Levenson and A. L. Schawlow. Complete hyperfine structure of a molecular iodine line.

Physical Review Letters, 26, 946 (1971)

[57] J. M. G. Duarte and S. L. Campbell. Measurement of the Hyperfine Structure of Rubidium 85 and 87. Massachusetts Institute of Technology (unpublished) (March 2009) [58] T. W. Hänsch and P. Toschek, Theory of a three-level gas laser amplifier, Z. Phys. 236, 213 (1970) [59] M. D. Levenson and S. S. Kano. Introduction to Nonlinear Laser Spectroscopy (Academic Press) (1988) [60] D. A. Smith and I. G. Hughes. The role of hyperfine pumping in multilevel systems exhibiting saturated absorption. Am. J. Phys. 72, 631 (2004) [61] J. J. Clarke and W. A. van Wijngaarden and H. Chen. Electromagnetically induced transparency using a vapor cell and a laser-cooled sample of cesium atoms. Phys. Rev. A. 64, 023818 (2001) [62] M. Xiao, Y. Li, S. Jin and J. Gea-Banacloche. Measurement of Dispersive Properties of Electromagnetically Induced Transpar3ency in Rubidium Atoms. Phys. Rev. Lett 74, 666 (1995) [63] S. Mauger, J. millen and M. P. A. Jones. Spectroscopy of strontium Rydberg states using electromagnetically induced transparency. J. Phys. B: At. Mol. Opt. Phys. 40, f319 (2007)

[64] C. H. Iu, G.D. Stevens, H. Metcalf. Instrumentation for multistep excitation of lithium atoms to Rydberg states.

Appl Opt. 34, 2640 (1995)

[65]W. Hansch and B. Couillaud. Laser frequency stabilisation by polarisation spectroscopy of a reflecting reference cavity. Opt. Commun. 35, 441 (1980) [66]A. Grabowski, R. Heidemann, R. Low, J. Stuhler, and T. Pfau. High resolution Rydberg spectroscopy of ultracold rubidium atoms. Fortschr. Phys. 54, 765 (2006)

[67]M. Viteau , J. Radogostowicz , M.G. Bason , N. Malossi , D. Ciampini , O. Morsch and E. Arimondo. Rydberg spectroscopy of a Rb MOT in the presence of applied or ion created electric fields. Opt. Express. 28, 6007 (2011) [68]J. R. Brandenberger, C. A. Regal, R. O. Jung, and M. C. Yakes. Fine-structure splittings in 2F states of rubidium via three-step laser spectroscopy. Phys. Rev. A, 65, 042510 (2002). [69] T .T. Grove et al. Two-photon two-colour diode laser spectroscopy of the Rb 5D5/2 state. Phys. Scr, 52, 271 (1995) [70] D. J. Wineland, J.C. Bergquist, W. M. Itano and R. E. Drullinger. Double-resonance and optical-pumping experiments on electromagnetically confined, laser-cooled ions. Opt. Letts. 5, 245 (1980) [71] G.P.T. Lancaster, R.S. Conroy ), M.A. Clifford, J. Arlt, K. Dholakia. A polarisation spectrometer locked diode laser for trapping cold Atoms. Optics Communications 170, 79 (1999) [72] S. Kakuma and R. Ohba. Atomic transition spectra markers for accurate frequency-modulated continuous-wave laser distance-meter. Optics Communications 239, 445 (2004) [73] Phipps, T.E.; Taylor, J.B. (1927). "The Magnetic Moment of the Hydrogen Atom". Physical Review 29, 309 (1927) [74] O. Kocharovskaya Amplification and lasing without inversion,” Phys. Rep. 219, 175 (1992) [75] L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 meters per second in an ultracold atomic gas,” Nature 397, 594 (1999) [76] M. D. Lukin, S. F. Yelin, and M. Fleiscchauer, “Entanglement of atomic ensembles by trapping correlated photon states,” Phys. Rev. Lett. 84, 4232 (200). [77] S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611 (1998). [78] J. Clarke, H. Chen, and W. A. van Wijngaarden, “Electromagnetically induced transparency and optical switching in a rubidium cascade system,” Appl. Opt. 40, 2047 (2001) [79] D. Sheng, A. Perez Galvan and L. A. Orozco. Lifetime measurements of the 5d states of rubidium. Phys. Rev A 78, 062506 (2008) [80] U. Volz and H. Schmoranzer. Precision lifetime measurements on alkali atoms and on helium by beam-gas-laser spectroscopy. Physica Scripta Volume T, 65, 48 (1996) [81] W. Demtroder. Laser Spectroscopy. Basic Concepts and Instrumentation. (2003) 3rd ed. (Springer) [82] M. Gehrtz and G. C. Bjorklund. Quantum-limited laser frequency-modulation spectroscopy. Phys. B: Lasers Opt. 32, 145 (1983) [83] R. J. Schoelkopf and S. M. Girvin. Wiring up quantum systems. Nature 451, 664 (2008) [84] R. Poldy, B. C. Buchler and J. D. Close. Single-atom detection with optical cavities. Phys. Rev A 78, 013640 (2008) [85] Th. Udem, R. Holzwarth and T. W. Hänsch. Optical frequency metrology. Nature 416, 233 (2002)

iA thought-experiment postulated to highlight the absurdities of the Copenhagen-interpretation of quantum mechanics. ii Complementarity is a fundamental principle of quantum mechanics, closely associated with Neils Bohr and Copenhagen interpretation – it asserts that objects governed by quantum mechanics, when measured, give results that depend inherently upon the type of measuring device used, and must necessarily be described in classical mechanical terms!