Universal non-equilibrium properties of dissipative Rydberg gases
M. Marcuzzi, E. Levi, S. Diehl, J. P. Garrahan, I. Lesanovsky
UUniversal non–equilibrium properties of dissipative Rydberg gases
Matteo Marcuzzi, Emanuele Levi, Sebastian Diehl, Juan P. Garrahan, and Igor Lesanovsky School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
We investigate the out-of-equilibrium behavior of a dissipative gas of Rydberg atoms that featuresa dynamical transition between two stationary states characterized by different excitation densities.We determine the structure and properties of the phase diagram and identify the universality classof the transition, both for the statics and the dynamics. We show that the proper dynamical orderparameter is in fact not the excitation density and find evidence that the dynamical transition isin the “model A” universality class, i.e. it features a non-trivial Z symmetry and a dynamicswith non-conserved order parameter. This sheds light on some relevant and observable aspects ofdynamical transitions in Rydberg gases. In particular it permits a quantitative understanding of arecent experiment [C. Carr et al. , Phys. Rev. Lett. , 113901 (2013)] which observed bistablebehaviour as well as power-law scaling of the relaxation time. The latter emerges not due to criticalslowing down in the vicinity of a second order transition, but from the non-equilibrium dynamicsnear a so-called spinodal line. PACS numbers: 64.60.Ht, 64.60.My, 05.30.-d, 32.80.Ee
Introduction.—
The study of the emergence of collec-tive behavior in many-body systems continues to be avery active field of research. Fundamental insights, suchas the onset of universality and its consequences [1–3] arecentral for our understanding of matter in general. Inrecent years, there has been a growing interest in under-standing dynamical phase transitions [4–6] in the contextof driven open many-body quantum systems [7–19], andprogress in the manipulation of ultracold atoms [20] hasmade it possible to access and explore many-body phe-nomena under precisely controllable experimental condi-tions [21–25].In this context, a class of systems that offers a richand intricate physics is represented by so-called Ryd-berg gases [26–31], i.e., atomic clouds in which atoms arelaser-excited to high-lying energy levels. The main con-sequence of the population of such orbitals is a consider-able increase [27, 28] in the interaction strength. This isat the heart of several non-trivial dynamical phenomena,both for closed systems undergoing coherent evolutionand showing enhanced spatial (anti-)correlations [32–35],and for open ones, in which the interplay between drivingand dissipation leads instead to intermittency [36], glassybehavior [37] and bistable behavior [38].The dissipative case has been recently studied via amean-field approach [39–41], numerical calculations inone dimension [42–44] and an approximate rate equa-tion description in higher dimensions [31, 45–47]. Theseinvestigations highlighted the presence of various station-ary regimes and the existence of first and second orderphase transitions. In addition, experiments have startedto probe the static and dynamic features of these systemsrevealing a bimodal behavior of the excitation density[29] and the presence of an optical bistability [30].The aim of this work is to shed light on the bistabletransition in a dissipative Rydberg gas with particular fo- cus on its dynamics and to connect the findings to recentexperimental studies. For the stationary state, the tran-sition is related to the spontaneous breaking of a Z sym-metry and falls into the Ising universality class. The ef-fective static order parameter is an appropriately shiftedRydberg excitation density. The dynamics is found to beof “model A” type according to the standard classifica-tion of Ref. [4], i.e., it is akin to a classical Ising modelsubject to a noise-induced spin-flipping process whichdoes not preserve the total magnetization. However,within the dynamical framework it becomes clear thatthe dynamical order parameter is not formally identicalto the Rydberg excitation density and the Z symmetryidentified in the static case must be non-trivially general-ized. Linking to recent experimental studies [30], we notethat the dynamic transitions observed there take in factplace near the so-called spinodal lines of the mean-fieldphase diagram. The connection established to “model A”physics allows us, moreover, to extract a universal scalinglaw for relaxation times for which quantitative agreementwith experiment is found. We believe that this perspec-tive will be useful for analyzing and understanding thedynamical phenomena observed in other related experi-ments, such as the one presented in Ref. [29]. The model.—
We employ the standard description ofa Rydberg gas in terms of (fictitious) interacting spin-1 / |↓(cid:105) and |↑(cid:105) correspondto the atomic ground and Rydberg states respectively.The many-body dynamics of the system’s density matrix (cid:98) ρ is governed by the quantum master equation (QME) ∂ t (cid:98) ρ = − i [ H, (cid:98) ρ ] + ( L + L ) [ (cid:98) ρ ] with Hamiltonian H = Ω (cid:88) k (cid:98) σ xk + ∆ (cid:88) k (cid:98) n k + (cid:88) k (cid:54) = p V kp (cid:98) n k (cid:98) n p . (1)Here Ω is the Rabi frequency and ∆ the detuning of theexcitation laser with respect to the ground state – Ryd- a r X i v : . [ c ond - m a t . s t a t - m ec h ] J un berg state transition. The interaction between two atomspositioned at (cid:126)r k and (cid:126)r p is, e.g., of van der Waals type V kp = C | (cid:126)r k − (cid:126)r p | − . Moreover, we have defined theexcitation density (cid:98) n k = ( k + (cid:98) σ zk ) /
2, with { (cid:98) σ xk , (cid:98) σ yk , (cid:98) σ zk } being the usual quantum spin operators acting on the k -th site. Dissipation within our model is described by thedissipator of Lindblad form L j [( · )] = (cid:88) k (cid:20) L jk ( · ) L † jk − (cid:110) L † jk L jk , ( · ) (cid:111)(cid:21) . (2)Relating to previous experimental observations [27, 48],we account for two dissipation mechanisms: One is inde-pendent atomic decay (at rate Γ) from the Rydberg stateto the ground state, with the corresponding jump opera-tor being L k = √ Γ (cid:98) σ − k = √ Γ ( (cid:98) σ xk − i (cid:98) σ yk ). The second oneis dephasing of the Rydberg state relative to the groundstate, occurring at rate K with L k = √ K (cid:98) n k . Mean-field equations of motion.—
A mean-field treat-ment of the Rydberg gas has been already conductedto some extent in other works, see, e.g., [39]; here wejust briefly summarize the derivation of the equations ofmotion. We consider the complete set of one-atom ob-servables { k , (cid:98) σ xk , (cid:98) σ yk , (cid:98) n k } and calculate their respectiveaverages (cid:110) , (cid:126)S (cid:111) ≡ { , S x , S y , n } according to (cid:104) ( · ) (cid:105) =tr { (cid:98) ρ ( · ) } . Applying the QME, assuming spatial unifor-mity and factorising all quadratic expectations yields theclosed set of dynamical equations ˙ S x = − (∆ + V n ) S y − Γ+ K S x ˙ S y = 2Ω − n + (∆ + V n ) S x − Γ+ K S y ˙ n = Ω S y − Γ n, (3)with V = 2 (cid:80) p V kp the mean-field interaction energy. Stationary regime.—
Introducing the effective param-eters a = 2 + 14 Γ(Γ + K )Ω , b = (cid:18) V Ω (cid:19) ΓΓ +
K , c = ∆ V (4)allows us to formulate the problem in a concise way. Wecan eliminate S x and S y from the stationary solutions of(3), thus obtaining an algebraic equation for the station-ary average number of excitations n , n (cid:104) a + b ( c + n ) (cid:105) = 1 . (5)This expression is a cubic real polynomial in n and there-fore the number of real roots may vary from 1 to 3 de-pending on the specific values taken by ( a, b, c ) within thephysically allowed space { a ≥ , b ≥ } . In Fig. 1 we re-port the corresponding phase diagram in the a − b planefor different choices of c . The stable phase of the sys-tem corresponds to the parameter domain displaying onlyone acceptable solution. Complementary to this domain FIG. 1. (Color online) Phase diagram in the a − b plane [asdefined in Eq. (4)] for three different values of c . The shadedareas correspond to domains portraying three stationary realsolutions. Their boundaries identify the spinodal lines. Theblack curve represents the path threaded by the critical pointwhen varying c , corresponding to the projection of the criticalline { a c , b c , c } = (cid:8) − / (8 c ) , − / (8 c ) , c (cid:9) onto the a − b plane.The horizontal dotted line is the lower bound b min = 512 / c = − .
42 and show the excitation density n taken along the three cuts shown in panel (a): in panels(b.1) and (b.2) we show n as observed on the blue and reddashed lines, respectively, which correspond to the “thermal”and “magnetic” directions (see main text). The black dashedline which crosses the spinodal boundaries probes instead thestable-bistable transition, corresponding to the hysteresis-likeprofile in panel (b.3). is the bistable regime [36, 38, 39], with Eq. (5) featuringthree solutions, only two of which are stable. The bound-aries between stable and bimodal regimes are the spin-odal lines, where at least two solutions coincide. For anyvalue of c , the spinodal lines coalesce into a critical pointidentified by a c = − / (8 c ) and b c = − / (8 c ), whichcorresponds to having three coincident real solutions forEq. (5). This point moves along the curve b = (4 a/ shown in Fig. 1 and lies within the aforementioned phys-ical parameter space only when − / ≤ c ≤
0. Thisresults in a constraint on the physical parameters: a min-imal value b min = 512 /
27. From them, one can work outthe threshold value V min = 4(Γ + K ) below which thetransition cannot be found by just varying the laser pa-rameters Ω, ∆.We investigate now the universal features near the crit-ical point. To this end we expand Eq. (5) to leadingorder in a perturbation of the parameters around theircritical values: a = a c + δa and b = b c + δb . We thenstudy the corresponding variation of the stable solutions n st = n c + δn = − c/ δn and identify a special di-rection δb = ( − /c ) δa [in the following referred to as symmetry line , see Fig. 1(a)] along which the solutionis invariant under the transformation δn → − δn (witha more complicated one holding for S x and S y ). Thus,a Z symmetry for the stationary value of the excita-tion density n emerges, which is spontaneously broken FIG. 2. (Color online) Stylized sketch of the mean-field po-tential V , which gives rise to the dynamics governed by Eqs.(3), in the S x − n plane. The transformation from the sta-tionary basis of observables { S x , S y , n } to the dynamical one( (cid:8) S x (cid:48) , S y (cid:48) , n (cid:48) (cid:9) ), where the Z symmetry becomes manifestalso in the dynamic structure, is in general nonlinear. On theright we show stylized sections of this potential obtained bymoving along the directions n (cid:48) , S x (cid:48) and S y (cid:48) respectively. Cru-cially, the double-well structure (responsible for the ”modelA” physics) is only felt by the critical n (cid:48) , whereas alongthe other ”massive” directions the system only probes singlequadratic wells which play no role in the transition. in the bistable phase [see Fig. 1(b.1)]. When approach-ing the critical point along the symmetry line we find δn ∼ ( − δa ) / . For any other direction [e.g., the reddashed line in Fig. 1(a)] the system does not switchphases when crossing the critical point. The correspond-ing behavior, portrayed in Fig. 1(b.2), is described by | δn | ∼ | δa | / . We can thus conclude that this transitionbelongs to the (static) Ising universality class with orderparameter δn = n − n c : In fact, the magnetization m ofan Ising model, as a function of the temperature T , thecritical temperature T c and the magnetic field h is knownto obey m ( T, h = 0) ∼ | T − T c | β and m ( T c , h ) ∼ h /δ ,with mean-field exponents β = 1 / δ = 3 [3, 49]. Inanalogy, we associate the symmetry line (b.1) to the ther-mal direction and any deviation from it to the presenceof a magnetic field which explicitly breaks the Z symme-try. Finally, a generic choice of the parameters will leadto probing the spinodal behavior shown in Fig. 1(b.3),which has indeed been highlighted in previous theoreti-cal and experimental studies [30, 40]. Dynamical vs. static order parameter.—
The discus-sion so far is incomplete as it does not include the dy-namical aspects of the system. As a first step, we performan analysis of the stability of the stationary points. Intheir neighborhood, we expand the r.h.s. of Eqs. (3) tolinear order in the deviations (e.g., δn = n − n st ), whichobey the differential equation δ ˙ (cid:126)S = M δ (cid:126)S ; the eigenvaluesof the stability matrix M constitute the rates of approachor escape from the stationary point. Whenever the solu-tion is unique, it is stable as well. When three solutionsare present, the two extremal ones are stable, while theone in the middle is unstable, cf. [39].Here, however, we focus on the universal properties that emerge near the critical point and the spinodal lines,where null eigenvalues appear. The latter are relatedto (leading) algebraic decays δn ∼ t − /ζ towards sta-tionarity. The corresponding exponent is ζ = 1 on thespinodal lines and ζ = 2 at the critical point. In thelatter case, an algebraic law of the form t − β/ ( νz ) is ex-pected on the basis of scaling arguments, with z beingthe dynamical critical exponent. The determination ofthe static universality class (Ising) provides us with themean-field exponents β = ν = 1 /
2. Thus, we concludethat z = 2, describing the dynamics of a diffusive sys-tem. In addition to the null direction, the stability ma-trix displays two non-vanishing (“massive”) eigenvalues,which identify directions that are not involved in thecritical physics (see qualitative sketch in Fig. 2). Thus,the effective order parameter for the low frequency (i.e.,long time) dynamics has only one component, and is de-scribed by the real variable δn (cid:48) . A discrete Z invari-ance of the equations of motion under the transformation δn (cid:48) → − δn (cid:48) emerges in a specific direction (i.e., the sym-metry line in Fig. 1) emanating from the critical point.This can be verified beyond the linear analysis: Applyinga quadratic transformation δ Σ i = R ij δS j + Q ijk δS j δS k ,we find that along the previously found null direction,the quadratic term vanishes as well. The absence of anyapparent conservation law strongly suggests that the dy-namics of the system at hand belongs to the (one compo-nent) “model A” universality class. We remark that thisis similar to the critical point in the driven open Dickemodel [10–13], which however constitutes a zero dimen-sional model where the mean-field exponents are exact.Consistently with this picture, along the symmetry linethe equation of motion reads δ ˙ n (cid:48) ∝ ( δn (cid:48) ) at leadingorder. In contrast, along the spinodal lines where thissymmetry is not present, the leading-order equation isof the form δ ˙ n (cid:48) ∝ ( δn (cid:48) ) and, consequently, gives riseto an exponent ζ = 1. We remark that the emergentsymmetry introduced above lies not among those identi-fied in Ref. [39] (i.e., { ∆ , V, S x } → {− ∆ , − V, − S x } and { Ω , S x , S y } → {− Ω , − S x , − S y } ), which are unbroken inboth phases. Metastable dynamics and connection to experiments.—
We now connect these findings to recent experiments[29, 30] that have investigated the dynamics of dissipa-tive Rydberg gases. The work presented in Ref. [29]has explored the phase diagram in the Ω − ∆ plane,shown in Fig. 3(a). Another experiment [30] has demon-strated a bistable behavior similar to the one presentedin Fig. 1(b.3). Moreover, a power-law behavior of the re-laxation time close to a “critical value” of the excitationlaser strength was reported.In order to gain some intuitive insight on the origin ofthese phenomena, we exploit our knowledge of the univer-sality class and introduce a phenomenological mean-fieldpotential V ( n (cid:48) ) = αn (cid:48) − β ( n (cid:48) ) + γ ( n (cid:48) ) which reflectsthe profile reported in the topmost panel on the r.h.s. of FIG. 3. (Color online) Emergence of metastable regimes in the vicinity of the spinodal lines. In panel (a) we show thephase diagram in the ∆ / Γ − Ω / Γ plane. The solid curves are the spinodal lines, whereas the dashed blue one represents thesymmetry line [see Fig 1(a)]. We furthermore display the qualitative structure of the mean-field potential V ( n (cid:48) ) for parameterscorresponding to inside ( (cid:70) ), on ( (cid:4) ) and outside ( (cid:78) ) the spinodal lines (see text). In panel (b) we show an example for therelaxation of the excitation density n (from an initial value n ) towards the stationary value n st , for parameters near one ofthese boundaries. Here we observe a metastable plateau whose lifetime τ is determined by the first crossing time of the midpoint¯ n = ( n st + n ) /
2. Panel (c, d) display a power-law divergence of τ as a function of the reduced Rabi frequency ω = (Ω − Ω c ) / Ω c varied along the red and purple curves in panel (a) [see for comparison the experimental data shown in Fig. 4 of Ref. [30]].The power changes from θ = 1 / θ = 2 / Fig. 2. The corresponding mean-field dynamics is givenby ˙ n (cid:48) = − ∂ n (cid:48) V ( n (cid:48) ). For fixed β > γ >
0, this equa-tion portrays a stable (one minimum) to bistable (twominima) transition at a threshold value α c = (2 β/ / γ .The experiments mentioned above are performed suchthat initially no excited atoms are present and subse-quently the excitation laser is switched on at given val-ues of ∆ and Ω. Within our qualitative framework, in thebistable phase ( α < α c ) this may lead to a fast relaxationtowards the nearest local minimum of V ( n (cid:48) ), which is notnecessarily the global one [see ( (cid:70) ) in Fig. 3(a)]. Account-ing for fluctuations (beyond mean-field) may in generalintroduce an additional time scale beyond which this pic-ture is no longer valid and a different physics emerges.However, the agreement between our predictions and ex-perimental observations highlighted below suggests thatthese features are quite robust in three dimensions andthat current experiments indeed probe this “short timephysics”.When α = α c , i.e., on a spinodal line, one of the min-ima becomes an inflection point [see ( (cid:4) ) in Fig. 3(a)].For α (cid:38) α c [see ( (cid:78) ) in Fig. 3(a)], in the proximity of thedisappeared minimum one can identify a region of vanish-ing slope of the potential, which leads to a characteristicslow dynamics in a flat landscape. This is reflected in theevolution of observables by the appearance of long-livedplateaus (see, e.g., Fig. 3(b) and Fig. (4) in Ref. [30])whose lifetime τ diverges when approaching α c . By an-alytically solving the phenomenological equations of mo-tion one obtains τ ∼ ( α − α c ) − θ with θ = 1 /
2, whichagrees with the experimental estimate θ = 0 . ± . β = 0 in V ( n (cid:48) )), a different expo-nent θ = 2 / n (cid:48) the standard experimental ob- servable is the excitation density n . Nonetheless, criticalscaling behaviors characterize the latter as well: in fact,although n is a (non-linear) combination of n (cid:48) and thetwo “massive” variables, the latter decay exponentiallyfast (cfr. Fig. 2). Hence, on long time scales the dynam-ics is determined by the n (cid:48) component. This and the re-sults of the phenomenological model are confirmed by thenumerical solution of the full dynamical equations (3):Mimicking the experimental procedure [50], i.e., comput-ing τ for different values of Ω in the proximity of the spin-odal lines while keeping V, ∆ , Γ , K fixed [see Fig. 3(a)]indeed yields algebraic divergences τ ∼ (Ω − Ω c ) − θ [seeFigs. 3(c) and (d)] with exponents θ ≈ . θ ≈ . Conclusions and outlook.—
We have found strong evi-dence for the non-equilibrium dynamics of the dissipativeRydberg gas being governed by the ”model A” universal-ity class. This is an extensively studied class [4, 6, 51–54]and in particular it is known that the lower critical di-mension is two. This would exclude the presence of aphase transition in dimension one — a question that wasraised by the authors of Ref. [40] who have found nu-merical evidence for long-ranged correlations.We moreover observed the emergence of a metastableregime in close proximity to the spinodal lines, whose life-time τ diverges algebraically, in agreement with recentexperimental results. Surprisingly enough, our mean-field approach is quantitatively accurate in determin-ing the exponent of this power-law. The accord foundbetween experimental, mean-field and phenomenologicalapproaches suggests that this feature, even far from thecritical point, is universal, i.e., it is dictated by the large-scale features of the system, rather than by its specificmicroscopic description. This hints at the presence of amore subtle transition taking place at the spinodal lines.The investigation of this phenomenon — in particularin different dimensions — constitutes a matter of futureexperimental and theoretical investigation. Acknowledgements – The research leading to these re-sults has received funding from the European ResearchCouncil under the European Union’s Seventh Frame-work Programme (FP/2007-2013) / ERC Grant Agree-ment No. 335266 (ESCQUMA), the EU-FET GrantNo. 512862 (HAIRS), the Austrian Science Fund (FWF)through the START grant Y 581-N16 and SFB FOQUS,Project No. F4006-N16. We also acknowledge financialsupport from EPSRC Grant no. EP/J009776/1. [1] J. Zinn-Justin,
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