Cooperative non-equilibrium phase transition in a dilute Rydberg ensemble
Christopher Carr, Ralf Ritter, Christopher Wade, Charles S Adams, Kevin J Weatherill
NNon-equilibrium phase transition in a dilute Rydberg ensemble
C. Carr, R. Ritter, C. G. Wade, C. S. Adams, and K. J. Weatherill
Joint Quantum Centre (JQC) Durham-Newcastle, Department of Physics,Durham University, South Road, Durham, DH1 3LE, United Kingdom (Dated: November 5, 2018)We demonstrate a non-equilibrium phase transition in a dilute thermal atomic gas. The phasetransition, between states of low and high Rydberg occupancy, is induced by resonant dipole-dipoleinteractions between Rydberg atoms. The gas can be considered as dilute as the atoms are separatedby distances much greater than the wavelength of the optical transitions used to excite them. Inthe frequency domain we observe a mean-field shift of the Rydberg state which results in intrinsicoptical bistability above a critical Rydberg number density. In the time domain we observe criticalslowing down where the recovery time to system perturbations diverges with critical exponent α = − . ± .
10. The atomic emission spectrum of the phase with high Rydberg occupancyprovides evidence for a superradiant cascade.
Non-equilibrium systems displaying phase transitions arefound throughout nature and society, for example inecosystems, financial markets and climate [1]. The steadystate of a non-equilibrium system is a dynamical equi-librium between driving and dissipative processes. Inatomic physics, one of the most studied non-equilibriumphase transitions is optical bistability where the drivingis provided by a resonant laser field and the dissipation isinherent in the atom-light interaction. In most examplesof optical bistability feedback is provided by an opticalcavity, as in the pioneering work of Gibbs [2, 3]. However,bistability can also arise in systems where many dipolesare located within a volume which is much smaller thanthe optical wavelength; in this case the feedback is dueto resonant dipole-dipole interactions [4, 5]. This lattercase is known as intrinsic optical bistability [6] and has,so far, only been observed in an up-conversion processbetween densely packed Yb ions in a solid-state crys-tal host cooled to cryogenic temperatures [7]. Intrinsicoptical bistability generally cannot be observed for sim-ple two-level systems such as atomic gases, because theresonance broadening, which is larger than the line shift[8], suppresses the bistable response [9, 10].A solution to this problem is provided by highly-excited Rydberg states, where the dipole-dipole inducedlevel shifts between neighbouring states can be muchlarger than the excitation linewidth. This property ofoptical excitation of Rydberg atoms, known as dipoleblockade [11], enables a diverse range of applications inquantum many-body physics, quantum information pro-cessing [12], non-linear optics [13] and quantum optics[14–17]. An interesting feature of Rydberg systems isthat the range of the interaction can be much largerthan the optical excitation wavelength, giving rise tonon-local interactions [18]. This also creates the possi-bility of observing intrinsic optical bistability, and hencenon-equilibrium phase transitions [19] over macroscopic,optically-resolvable length scales.In this letter, we demonstrate a non-equilibrium phasetransition in a thermal Rydberg ensemble. In contrast to FIG. 1. (color online) Theoretical model for the coopera-tive optical response of the ensemble. (a) Simplified two-levelmodel with ground state | g (cid:105) and Rydberg state | r (cid:105) . The levelsare coupled by a laser with Rabi frequency Ω and detuningfrom resonance ∆. (b) Rydberg state population ρ rr as afunction of laser detuning ∆ for increasing Rabi frequencyΩ. As a result of the cooperative excitation-dependent shift,the response exhibits intrinsic optical bistability with hystere-sis dependent upon the history of the ensemble. Theoreticalparameters: interaction strength V /
Γ = −
11 and Rabi fre-quency Ω / Γ = (0 . , . , . previous experiments, we directly observe optical bista-bility in the transmission of the probe light without therequirement for cryogenics [7] or cavity feedback [2]. Wedistinguish between the phases of low and high Rydbergoccupancy using fluorescence spectroscopy and confirmthe first-order phase transition through the observationof critical slowing down in the temporal response of theensemble. Our observation of a non-equilibrium phasetransition in a dilute atomic system provides a new plat-form to study the transition between classical mean fieldand microscopic quantum dynamics [20, 21].To illustrate the origin of the non-equilibrium phasetransition in our system, we begin by considering thesimple two-level atom shown in Fig. 1(a) with groundstate | g (cid:105) and Rydberg state | r (cid:105) . The levels are coupledby a laser with Rabi frequency Ω and detuning from res-onance ∆. Using standard semi-classical analysis, the a r X i v : . [ phy s i c s . a t o m - ph ] A ug FIG. 2. (color online) (a) Three-photon excitation scheme toRydberg states in cesium. (b) Schematic of the experimen-tal setup. The three excitation lasers co-propagate through a2 mm vapour cell. The non-equilibrium dynamics are probedby measuring the transmission of the probe laser or analysingthe emitted fluorescence. (c) Experimental optical response∆ T as a function of Rydberg laser detuning ∆ r for RydbergRabi frequency Ω r increasing from (i) to (iii). Experimen-tal parameters: ground state density N = 4 . × cm − ,probe Rabi frequency Ω p = 2 π ×
37 MHz, coupling Rabifrequency Ω c = 2 π ×
77 MHz and Rydberg Rabi frequencyΩ r = 2 π × (14,36,74) MHz. time evolution of the system can be described using theLindblad master equation for the single-particle densitymatrix. To include the effect of dipole-dipole interactionsbetween atoms in state | r (cid:105) , we use a classical approxima-tion to the many-body quantum dynamics [20] and intro-duce a mean-field shift of the Rydberg state, proportionalto its steady state population. Coherent dynamics havebeen observed in thermal ensembles by operating on ul-trashort time scales, where the Rydberg interactions ap-pear as a dephasing of the Rabi oscillations due to thebroad distribution of inter-atomic distances [22]. How-ever, over longer time scales or in steady state as in ourexperiment, the interactions between Rydberg atoms re-sult in a mean-field shift. In this limit, the laser detuning∆ → ∆ − V ρ rr where V is the dipole-dipole interactionterm and ρ rr is the population of the Rydberg state. Theinteraction term V corresponds to the sum of the dipole-dipole interaction over the excitation volume [23].The optical Bloch equations for the two-level system can be written as˙ ρ gr = i Ω (cid:18) ρ rr − (cid:19) + i (∆ − V ρ rr ) ρ gr − Γ2 ρ gr (1a)˙ ρ rr = − Ω Im( ρ gr ) − Γ ρ rr (1b)where the off-diagonal coherence terms ρ rg = ρ ∗ gr and thediagonal population terms ρ gg = 1 − ρ rr . The steady-statesolution for the Rydberg population ρ rr as a function oflaser detuning ∆ is shown in Fig. 1(b) for increasing Rabifrequency Ω. As the cooperative shift is dependent on theRydberg population, the lineshape becomes asymmetri-cal and eventually exhibits bistability with hysteresis de-pendent on the direction in which the detuning is varied(shown by the arrows). In the bistable region, there isalso an unstable state (shown by the dashed curve) whichcannot be measured experimentally. At the critical tran-sition, there is an abrupt change in the atomic dynamics.The solution of Equation (1) can be found in the Supple-mentary Material [25].To experimentally observe a non-equilibrium phasetransition in a dilute medium we use a resonant multi-photon excitation scheme in a thermal Cs vapour, asshown in Fig. 2(a) [26]. In the simple theoretical anal-ysis above Doppler averaging is not considered; howeverby using a multi-photon scheme we excite only a narrowvelocity distribution of atoms [27] and can therefore ac-cess a regime where the mean-field shift between Rydbergstates far exceeds the Doppler width of the excitation.The optical Bloch model for the multi-photon schemeis presented in the Supplementary Material [28]. Aschematic of the experimental setup is shown in Fig. 2(b).A thermal vapour of Cs atoms is confined in a quartzcell with an optical path length of 2 mm. The atoms aredriven into the 26p / Rydberg state using three exci-tation lasers which co-propagate through the cell. Theprobe laser, with wavelength λ p = 852 . p and waist w p = 150 µ m is frequency sta-bilised to the | s / , F = 4 (cid:105) → | p / , F (cid:48) = 5 (cid:105) transition.The coupling laser, with wavelength λ c = 1469 . c and waist w c = 80 µ m is stabilisedto the | p / , F (cid:48) = 5 (cid:105) → | s / , F (cid:48)(cid:48) = 4 (cid:105) transition usingexcited state polarisation spectroscopy [29] . Finally, theRydberg laser with wavelength λ r = 790 . r and waist w r = 80 µ m, is tuned around theresonance between the excited-state 7s / and the Ryd-berg state 26p / .For a multi-photon transition to a Rydberg state, thetransmission of the probe light resonant with the opti-cal transition is increased by population shelving in theRydberg state [30] and provides a direct readout of theRydberg population. The change in probe laser trans-mission ∆ T as a function of Rydberg laser detuning ∆ r is shown in Fig. 2(c) for increasing Rydberg Rabi fre-quency Ω r . As the level of Rydberg population increases,the excitation-dependent shift first produces an asymme-try in the lineshape (ii). Eventually, when the shift is FIG. 3. (color online) Atomic emission spectra for low and high Rydberg state occupancy. The visible fluorescence spectrumis shown for (a) N = 3 . × cm − and (c) N = 4 . × cm − . In the low Rydberg occupancy phase, the spontaneousemission originates from high-lying Rydberg states as illustrated in (b). However, in the high Rydberg occupancy phase, thespontaneous emission originates from low-lying Rydberg states, as illustrated in (d), due to a superradiant cascade betweenhigh-lying Rydberg states. The ionisation limits from 6s / , 6p / and 6p / are shown by thick red vertical lines. Theblue shaded regions highlight the absence of spontaneous emission between 26p / and ionisation which would occur due toa blackbody or collisional excitation process. The thin cyan vertical lines indicate the dipole-allowed transitions. Probe Rabifrequency Ω p = 2 π ×
41 MHz, coupling Rabi frequency Ω c = 2 π ×
74 MHz and Rydberg Rabi frequency Ω r = 2 π ×
122 MHz. greater than the linewidth (iii), the lineshape exhibitsintrinsic optical bistability with hysteresis dependent onthe direction in which resonance is approached (shownby the arrows). Importantly, this bistability is measuredin steady-state and is not a transient phenomenon. Asa result, within the hysteresis window, the system canbe placed in either the low or high Rydberg occupancyphase with exactly the same experimental parameters.The change in atomic behaviour across the phase tran-sition can be analysed by measuring the spectrum of theoff-axis fluorescence. The emission spectra for the twophases of low and high occupancy are shown in Fig. 3(a)and (c), respectively. In the low phase, the dominanttransitions indicated by (i) and (ii) involve decay fromhigh-lying Rydberg states to the ground states of the s,p and d series. This behaviour, highlighted in Fig. 3(b), isconsistent with spontaneous emission where such transi-tions dominate due to the ω dependence in the EinsteinA-coefficient. This phase is characterised by the faint(green) fluorescence shown in the inset.In the high Rydberg occupancy phase the emissionspectrum is dramatically modified. The dominant spon-taneous emission transitions (i) and (ii) are no longerpresent. Instead, the spontaneous emission now orig-inates from a range of low-lying Rydberg states indi-cated by (iii) and (iv) and highlighted in Fig. 3(d). Thisphase is characterised by the strong (orange) fluorescenceshown in the inset. Importantly, the absence of emissionclose to the ionisation limit for each series, indicated by the dark (red) vertical lines, indicates that atoms are notpromoted to higher-lying Rydberg states, as would occurin a collisional or up-conversion processes [7]. We can alsoneglect the effects of thermal blackbody photons becausethe average number of photons per mode at Rydberg-Rydberg transition frequencies is much lower than theaverage number of excited atoms [31]. The transitions at455 nm and 459 nm occur in both phases and correspondto decay to 6s / from 7p / and 7p / respectively.The emission spectrum in the high occupancy phasecan be understood as a superradiant cascade to lower-lying Rydberg states [32]. Evidence for a superradi-ant cascade has also been observed in ultracold atoms[33, 34]. When the cooperativity on a particular transi-tion is high, the atoms emit collectively and in-phase withone another. The single atom lifetime of the 26p / to26s / transition is τ (cid:39) µ s but within the transitionwavelength volume V = 1 mm , we estimate the Rydbergatom number N r (cid:39) × . Consequently, we expect asuperradiant decay timescale of τ super = τ /N r (cid:39)
100 ps.As the transition wavelength and therefore the cooper-ative enhancement of the decay rate is proportional to n , the superradiant cascade eventually stops and givesrise to the observed spontaneous emission from low-lyingRydberg states as indicated in Fig. 3(iii) and (iv).By stabilising the laser frequency within the hysteresiswindow and varying the intensity of the Rydberg laser I r , it is possible to observe bistability and hysteresis inthe optical response as shown in Fig. 3(a). The system FIG. 4. (color online) Critical slowing down as the temporalsignature of a phase transition. (a) Continuous Rydberg laserintensity I r scan showing bistability and hysteresis in the opti-cal response ∆ T . (b) Discrete Rydberg laser intensity I r scanshowing the divergence of the switching time to steady-state τ around the critical transition intensity I r , crit ≈ . .The switching time diverges as ( I r − I r , crit ) α with critical ex-ponent α = − . ± .
10 (standard deviation error) shownby the dashed line of best fit. Ground state density N =4 . × cm − , probe Rabi frequency Ω p = 2 π ×
57 MHz,coupling Rabi frequency Ω c = 2 π ×
116 MHz and Rydbergdetuning ∆ r = 2 π × −
220 MHz. The error bars represent thestandard deviation error on the determination of the laserintensity and switching time. switches between the low occupancy phase with probetransmission level T and the high occupancy phase withprobe transmission level T . In this case, the phase tran-sition from low to high Rydberg population occurs atcritical intensity I r , crit ≈ .The first-order phase transition between the lowand high Rydberg occupancy phases can be confirmedthrough the observation of critical slowing down. Thistemporal signature of a phase transition occurs as thesystem approaches a critical point and becomes increas-ingly slow at recovering from perturbations [35, 36]. Thetemporal response of the ensemble is measured by dis-cretely varying the Rydberg laser intensity I r and mea-suring the time τ to reach steady-state, as illustratedin the inset of Fig. 3(b). At the critical transition,the switching time diverges according to the power law τ ∝ ( I r − I r , crit ) α shown by the fitted dashed line. Thecritical exponent α = − . ± .
10 (standard devia-tion error) is consistent with previous work on first-orderphase transitions and optical bistability [37, 38].We also note that the geometry of the excitation region plays an important role in our observation of many-bodydynamics [39]. The optical path length of 2 mm providedby the vapour cell is comparable to the interaction wave-length. If the medium was much shorter, the cooperativeshift would not result in intrinsic optical bistability. Fur-thermore, if the medium was much longer, the dipoleswould not evolve with the same phase. A more completestudy of the length dependence of the effect will form thefocus of future work.In summary, we have demonstrated a cooperative non-equilibrium phase transition in a dilute thermal atomicgas. The observations that have been discussed raiseinteresting possibilities for future non-local propagationexperiments which utilise the long range cooperative in-teraction [18]. Furthermore, this work could be used toperform precision sensing [40] around the critical pointand to study resonant energy transfer [41] on optically-resolvable length scales. In addition, studies of the flu-orescence in the vicinity of the phase transition couldprovide further insight into the dynamics of strongly-interacting dissipative quantum systems [20, 21].We would like to thank S A Gardiner and U M Krohnfor stimulating discussions, R Sharples for the loan ofequipment and M P A Jones and I G Hughes for proof-reading the manuscript. CSA and KJW acknowledge fi-nancial support from EPSRC and Durham University.CSA and RR acknowledge funding through the EU MarieCurie ITN COHERENCE Network. [1] H. Haken, Naturwissenschaften , 815 (1980).[2] H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan,Phys. Rev. Lett. , 1135 (1976).[3] H. M. Gibbs, Optical Bistability: Controlling light withlight (Academic Press Inc., 1985).[4] H. J. Carmichael, and D. F. Walls, J. Phys. B: At. Mol.Phys. , 685 (1977).[5] D. F. Walls, P. D. Drummond, S. S. Hassain and H. J.Carmicheal, Sup. Prog. Theo. Phys. , 307 (19778).[6] C. M. Bowden and C. C. Sung, Phys. Rev. A , 2392(1979).[7] M. P. Hehlen, H. U. G¨udel, Q. Shu, J. Rai, S. Rai and S.C. Rand, Phys. Rev. Lett. , 1103 (1994).[8] J. Keaveney, A. Sargsyan, U. Krohn, I. G. Hughes, D.Sarkisyan, and C. S. Adams, Phys. Rev. Lett. ,173601 (2012).[9] F. A. Hopf, C. M. Bowden, and W. H. Louisell, Phys.Rev. A , 2591 (1984).[10] R. Friedberg, S. R. Hartmann, and J. T. Manassah, Phys.Rev. A , 3444 (1989).[11] M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D.Jaksch, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. ,037901 (2001).[12] M. Saffman, T. G. Walker, and K. Mølmer, Rev. Mod.Phys. , 2313 (2010).[13] J. D. Pritchard, K. J. Weatherill, and C. S. Adams, in Annual Review of Cold Atoms and Molecules (World Sci- entific, 2013).[14] Y. O. Dudin and A. Kuzmich, Science , 887 (2012).[15] T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth,A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuleti´c,Nature , 57 (2012).[16] D. Maxwell, D. J. Szwer, D. Paredes-Barato, H. Busche,J. D. Pritchard, A. Gauguet, K. J. Weatherill, M. P. A.Jones, and C. S. Adams, Phys. Rev. Lett. , 103001(2013).[17] P. Schauss, M. Cheneau, M. Endres, T. Fukuhara, S.Hild, A. Omran, T. Pohl, C. Gross, S. Kuhr, and I. Bloch,Nature, , 87 (2012)[18] S. Sevin¸cli, N. Henkel, C. Ates, and T. Pohl, Phys. Rev.Lett. , 153001 (2011).[19] T. E. Lee, H. H¨affner, and M. C. Cross, Phys. Rev. A , 031402 (2011).[20] T. E. Lee, H. H¨affner, and M. C. Cross, Phys. Rev. Lett. , 023602 (2012).[21] C. Ates, B. Olmos, J. P. Garrahan, and I. Lesanovsky,Phys. Rev. A , 043620 (2012).[22] T. Baluktsian, B. Huber, R. L¨ow and T. Pfau, Phys. Rev.Lett. , 123001 (2013)[23] R. Friedberg, S. R. Hartmann and J. T. Manassah, Phys.Rep. , 101 (1973)[24] N. Henkel, R. Nath, and T. Pohl, Phys. Rev. Lett. ,195302 (2010)[25] See Supplementary Material for the solution of Eq. (1).[26] C. Carr, M. Tanasittikosol, A. Sargsyan, D. Sarkisyan,C. S. Adams, and K. J. Weatherill, Opt. Lett. , 3858(2012).[27] M. Tanasittikosol, C. Carr, C.S. Adams and K.J. Weath- erill, Phys. Rev. A. , 033830 (2012).[28] See Supplementary Material for details of the four-leveloptical Bloch model.[29] C. Carr, C. S. Adams, and K. J. Weatherill, Opt. Lett. , 118 (2012).[30] P. Thoumany, T. Germann, T. H¨ansch, G. Stania, L.Urbonas and Th. Becker, J. Mod. Opt. , 2055 (2009).[31] J. M. Raimond, P. Goy, M. Cross, C. Fabre, and S.Haroche, Phys. Rev. Lett. , 201002 (2008).[35] R. Bonifacio and P. Meystre, Opt. Commun. , 131(1979).[36] M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin,S. R. Carpenter, V. Dakos, H. Held, E. H. van Nes, M.Rietkerk, and G. Sugihara, Nature
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NON-EQUILIBRIUM PHASE TRANSITION IN A DILUTE RYDBERG ENSEMBLE
SUPPLEMENTARY MATERIAL
In the main text, an analytic two level model is usedto give an intuitive understanding of the system. In thissupplementary material, we present:1. Details of the analytic two level model.2. A numerical solution of the full four level system.We justify the simplified 2-level system as an analogyfor the 4-level system by presenting qualitatively similarpredictions from both models.
ANALYTIC 2-LEVEL MODEL
The caesium vapor is described using optical Blochequations. To model the nonlinear behavior we adjustthe energy of the Rydberg state proportional to its pop-ulation (Equation 2). This is a mean field approximation.∆ → ∆ − V ρ rr (2) For a 2-level system, we start from the well knownsteady state solution of the 2-level optical bloch equations( see eg. Foot Atomic Physics (Oxford University Press,2005 ), and substitute the expression given in Eq. 2. ρ rr = Ω / − V ρ rr ) + Ω / / ρ rr (Eq. 4).For each laser detuning, we find either one (monostable)or two (bistable) stable real valued solutions. In thebistable region, we also find an intermediate, unstablesolution which we do not expect to realise experimen-tally. V ρ − V ∆ ρ + (∆ + Ω / / ρ rr − Ω / et al . [20] veryclosely. The predicted bistable response of 2-level atomicvapor is shown in Fig. 1 of the main text and closelyresembles Fig. 1 of this document FIG. 5. Normalised transmission of the probe laser with Ryd-berg laser detuning ∆ r calculated using the four level opticalBloch model. NUMERICAL 4-LEVEL MODEL
In the four level system with the probe and couplinglasers on resonance, the complete Hamiltonian, H , isgiven in a rotating frame as H = (cid:126) p p c
00 Ω c r r − r − V ρ rr ) (5)To find steady state solutions of the optical bloch equa-tions, each data point is calculated by evolving the Liou- ville Equation for the density matrix, ρ , (Eq. 6) in timeuntil a steady state is reached. The matrix γ implementsphenomenological spontaneous decay between the atomicstates. ˙ ρ = i (cid:126) (cid:2) ρ, H (cid:3) − γ (6)In the bistable region where there are two steadystates, both solutions are attained by using starting con-ditions close to one or other of the steady states. Tosimulate the intrinsic optical bistability reported in thisletter, parameters as follows were used.Ω p = 2 π ×
110 MHzΩ c = 2 π ×
200 MHzΩ r = 2 π ×
30 MHz V = 2 π × ( − .
6) MHz (7)Following the velocity class vv