Ergodicity breaking from Rydberg clusters in a driven-dissipative many-body system

It is challenging to probe ergodicity breaking trends of a quantum many-body system when dissipation inevitably damages quantum coherence originated from coherent coupling and dispersive two-body interactions. Rydberg atoms provide a test bed to detect emergent exotic many-body phases and nonergodic dynamics where the strong Rydberg atom interaction competes with and overtakes dissipative effects even at room temperature. Here, we report experimental evidence of a transition from ergodic toward ergodic breaking dynamics in driven-dissipative Rydberg atomic gases. The broken ergodicity is featured by the long-time phase oscillation, which is attributed to the formation of Rydberg excitation clusters in limit cycle phases. The broken symmetry in the limit cycle is a direct manifestation of many-body collective effects, which is verified experimentally by tuning atomic densities. The reported result reveals that Rydberg many-body systems are a promising candidate to probe ergodicity breaking dynamics, such as limit cycles, and enable the benchmark of nonequilibrium phase transition.

It is challenging to probe ergodicity breaking trends of a quantum many-body system when dissipation inevitably damages quantum coherence originated from coherent coupling and dispersive two-body interactions.Rydberg atoms provide a test bed to detect emergent exotic manybody phases and non-ergodic dynamics where the strong Rydberg atom interaction competes with and overtakes dissipative effects even at room temperature.Here we report experimental evidence of a transition from ergodic towards ergodic breaking dynamics in driven-dissipative Rydberg atomic gases.The broken ergodicity is featured by the long-time phase oscillation, which is attributed from the formation of Rydberg excitation clusters in limit cycle phases.The broken symmetry in the limit cycle is a direct manifestation of many-body interactions, which is verified by tuning atomic densities in our experiment.The reported result reveals that Rydberg many-body systems are a promising candidate to probe ergodicity breaking dynamics, such as limit cycles, and enable the benchmark of nonequilibrium phase transition.
Many-body systems typically relax to equilibrium due to ergodicity such that observable becomes invariant with time [1][2][3].Often, the equilibration is so robust such that the observable quickly seeks new fixed points in phase space after quenching control parameters, which is driven by Boltzmann's ergodic hypothesis [4].Exceptions, i.e. broken ergodicity due to symmetry breaking [5,6], have been extensively explored in integrable [7] and manybody localized systems [8,9].A recent example is the quantum many-body scars [10] of strongly interacting Rydberg atoms trapped in optical arrays [11], where nonergodic many-body dynamics take place in a constrained sub-Hilbert space [12,13].This leads to coherent revivals of the Z 2 state [14] largely due to strong Rydberg atom interactions [15][16][17][18].In the presence of dissipation it is common that quantum many-body coherence and entanglement are eliminated, leading to stationary states in long-time limit.The interplay between strong Rydberg interactions and dissipation, on the other hand, results to exotic non-equilibrium phenomena such as collective quantum jumps [19], and phase transitions [20][21][22][23][24][25][26][27][28].Optical bistability [29], and self-organized criticality [30,31] have been demonstrated experimentally.
Here we report observation of non-ergodic many-body dynamics in a thermal gas of strongly interacting Rydberg atoms.This setting, as depicted in Fig. 1(a), is a dissipative many-body system of effective two-level atoms (ground and Rydberg states |g and |r ).Coherent laser coupling and strong long-ranged interactions in Rydberg state |r compete with dissipation (Doppler and collisional effects, electronic state decay etc).A nonequilibrium phase transition is identified by quenching the detuning ∆ provided that the Rabi frequency Ω is above a critical value, in which a bifurcation between an ergodic (E) and weakly non-ergodic (NE) phase appears.The E-phase corresponds to a weak interaction regime where the distribution of Rydberg atoms is homogeneous [Fig.1(b)].Irrelevant to initial states, excitations of all atoms end at an identical fixed point in phase space [Fig.1(c)].The NE phase features a non-trivial population revival when the laser detuning is near resonant.Long-time many-body coherence, in the order of milliseconds, is observed in the NE phase, where Rydberg population oscillations persist for a period much longer than any time scales of the relevant dissipation and laser Rabi frequency.Our analysis suggests that the broken ergodicity is induced by clustering of strongly interacting Rydberg atoms in free space [32][33][34][35], where many-body dynamics are synchronized and form the oscillatory phase, i.e. a limit cycle [Figs.1(d) and arXiv:2305.07032v2[cond-mat.quant-gas]15 May 2023 Driven Dissipation < l a t e x i t s h a 1 _ b a s e 6 4 = " M r Y l C q j G X w + y l t J Q j 5 Q d M g G 5 P J g = " > A A A B 7 3 i c b V B N S w M x E J 3 U r 1 q / q h 6 9 B I v g q e y K q M e i F 4 8 V 7 A e 0 S 8 m m 2 T Y 0 y a 5 J V i x L / 4 Q X D 4 p 4 9 e 9 4 8 9 < l a t e x i t s h a 1 _ b a s e 6 4 = " j q y l w c x B a g M h e s C Q 8  Here a 795 nm laser is split into a probe beam and an identical reference beam by a beam displacer, which are both propagating in parallel through a heated Rb cell (10 cm long).The two-photon excitation with counter-propagating beams ensures that a narrow, low velocity class of atoms are excited to Rydberg states due to velocity selections [37].The transmission signal of the probe beam is detected on a differencing photodetector as a transmission difference [29].Rydberg atoms vary for different realizations in thermal gases, such that conventional schemes based on electrically ionizing Rydberg atoms can not be used here [38,39].Through EIT, however, the transmis-sion signal (proportional to Rydberg atom populations) is measured continuously and dynamically while not demolishing atomic states; see Section I of Supplementary Materials (SM) for details.
Ergodic-breaking phase transition.We probe the Rydberg population by changing the laser detuning ∆ c linearly from red to blue side while the probe laser is on resonance.Typically ∆ c is quenched at a rate 2π × MHz/ms.By raising the probe intensity, transmission against Ω 2 p and ∆ c is recorded and shown in Fig. 2 (a).The transmission is weak if the laser is away from the resonance.Close to the resonance, we observe the transmission exhibits distinctive features depending on Ω p .
When the probe laser is weak Ω 2 p < 30.9 (2π × MHz) 2 , transmission adiabatically follows the detuning ∆ c , leading to a smooth spectrum [region I in Fig. 2 (a)].This is a dissipation dominant phase.Increasing |Ω p |, a startling difference is that the transmission spectrum oscillates with increasing detuning, marked by region II in Fig. (a).Specifically the population of Rydberg atoms bifurcates from the E-to NE-phases, which takes place when Ω 2 p > 30.9 (2π × MHz) 2 .A typical example with Ω 2 p = 35(2π × MHz) 2 is shown in Fig. 2(b), where a sudden jump is found when we increase ∆ c .After passing the jump, the spectrum oscillates and decreases gradually when further increasing ∆ c .Such nontrivial spectrum profile is unique in region II.
The non-equilibrium phase transition occurs when the population of Rydberg atoms increases from n r < n r,c (E-phase) to n r > n r,c (NE-phase), where n r,c is a critical population obtained from the experiment.We plot a phase diagram in Fig. 2(c) by recording the critical point at which the first jump appears from increasingly ∆ c .The critical point ∆ c,c is found to be where Ω p,c = 2π×5.92MHz is the critical Rabi frequency of the probe beam, b = 0.61 ± 0.04 is the fitted critical exponent.This is consistent with the prediction of the function n r [∆ c , Ω p ] − n r,c = 0, where n r [∆ c , Ω p ] is the population of the steady state calculated from the master equation.When Ω 2 p is large (region III in Fig. 2 (a)), the transmission increases sharply to the maximal value, and then changes smoothly when increasing ∆ c , shown in Fig. 2 (d).In a lattice, this corresponds to the antiferromagnetic phase [36].As our theoretical analysis shown in the SM, here atoms end at different final states, while the overall transmission is stationary.
To further understand the dynamics, we stop sweeping ∆ c near resonance and measure the evolution of the transmission.We observe a time flow of many-body state collapsing and revival periodically as given in Figs.2(e) and (f).The oscillation has a period of ∆T ≈ 0.053 ms.
More importantly, it has very long lifetimes more than 1 ms, much longer than other characteristic time scales in the thermal Rydberg atom system.
Ergodicity breaking from Rydberg atom clusters.The oscillation effect observed in Fig. 2(e) and (f) corresponds to the broken ergodicity, which is induced by inhomogeneous Rydberg excitation in the thermal gas, resulting in spatial clusters which violate translational invariance [3,35].Here we model the non-ergodic dynamics by a Lindblad master equation ρ = L(ρ), where the generator L( 29], and Hamiltonian Ĥ, where V jk = C 6 /|R j −R k | 6 (C 6 to be the state-dependent dispersion coefficient) represents the van der Waals interaction between two Rydberg atoms locating at R j and R k .Operator σx j = (|r j g j |+|g j r j |)/2 flips the atomic state and nj = |r j r j | is Rydberg density operator of the j-th atom.In the experiment a large fraction of atoms is excited.We consider up to 10 3 atoms that are randomly distributed in space such that spatial configurations are largely explored.We numerically solve the       many-body dynamics with the discrete truncated-Wigner method [40,41], which is suitable for dealing with interacting many atom systems (N 1).
We show mean values n r = j nj of the Rydberg population by sweeping the laser detuning in Fig. 3(a), which exhibit a similar profile with the optical transmission (Fig. 2(b)).The high level of similarity permits us to gain insights from the simulation into the observed dynamics.In Fig. 3(b1)-(b2), examples of the Rydberg population are shown when detuning is frozen at ∆/γ = −2.35(marked by the vertical line in Fig. 3(a)).It is found that dynamics of various atoms are largely different in an initial transient period, such that the underlying trajectories explore the phase space.At a later stage (γt > 50), some atoms reach a steady population and becomes dynamically inactive, see Fig. 3(b1).A large fraction of the atoms, on the other hand, is dynamically active, as shown in Fig. 3(b2).Surprisingly, the overall oscillations of different Rydberg atoms are synchronized at a later time, though each oscillations are at different amplitudes and frequencies.In Fig. 3(c), we plot n r corresponding to the data shown in Fig. 3(b1)-(b3).After a transient period (A in Fig. 3(c)), the mean population enters an oscillation phase.Importantly we find that oscillations of different atoms are strongly synchronized, and centered at a single frequency (inset of Fig. 3(c)).Such synchronized oscillation reproduces the essential character observed in our experiment, as shown in Fig. 2(e).We shall point out that the EIT setting allows to monitor transmission continuously, i.e. probing many-body dynamics without destroying Rydberg populations.
A closer look shows that the active atoms form spatial clusters.Here the Rydberg level is shifted by the attractive van der Waals interaction, while Rydberg excitation happens under the antiblockade condition where the red detuned laser compensates the interaction [33,42,43].Features of the Rydberg distribution can be analyzed by Hopkins statistic [44], which gives a figure of merit of cluster tendency of Rydberg atom spatial distribution.Our numerical calculations show that the Hopkins statistic is larger than 0.5 in the limit cycle phase, strongly suggesting formation of spatial clusters.More details of the analysis can be found in SM.
Stability of the ergodic breaking phase.The nonergodic dynamics is stable, against temperature and den-  4 (a), there are obvious dips in the oscillation region (marked by the yellow lines).The characteristic lines of these oscillations are gradually faded out when we decrease the atomic density, see Fig. 4(a)-Fig.4(c) for comparison.Due to the resonance condition of probe field, the photoelectric signal under lower density is much larger than higher density.This effect causes a lower criticality of phase transition at relatively lower atomic densities.
We also extract the probe transmission data marked by the double arrows shown in Fig. 4 (d-f).In Fig. 4 (d),  A, B and C indicate dips in the oscillations when sweeping the detuning.The dips are gradually disappeared when we decrease the atomic density, as seen in Fig. 4  (e-f).The reason is that average distances between Rydberg atoms are larger if atomic densities are low.This results to weaker Rydberg atom interactions, such that dissipation overrides the dynamics.If we decrease the atomic density further, the phase transition would disappear completely and the probe transmission spectrum becomes smooth.The obtained density-dependent oscillations behavior manifests strong many-body characters in the non-equilibrium dynamics.
Summary.We have studied the non-ergodic dynamics of non-equilibrium phase transition in a strongly interacting Rydberg gases.When the system approaches to the criticality, the Rydberg population is bifurcated into E-and NE-phases.In the vicinity of the critical point, we have shown the Rydberg population is periodically oscillated for a long period of time.We have also observed the density-dependent oscillation, revealing the correspondence of a many-body effect.The ergodicity breaking observed in our experiment is explained with the formation of Rydberg clusters.Early works have revealed the importance of inhomogeneous Rydberg excitation in the study of Rydberg soft matter, such as aggregates [32][33][34]45], bistability [35], and self-organization [30,46].Our study shows that clusters of Rydberg atoms triggers non-equilibrium, non-ergodic many-body dynamics, despite the strong dissipation.Experimentally exploring broken ergodicity in a driven-dissipative Rydberg gas platform will expand the category of ergodicity of complex matter and non-equilibrium phenomenon [47,48], uncover the relation between the dissipation and the ergodicity [49], and find quantum technological applications [50,51].We have shown the dynamical evolution of the Rydberg population in Fig. 3(b1)-(b2) in the main text.Based on the dynamical behavior, the atomic system can be distinguished into two components: Most of the atoms reaches a steady population and becomes dynamically inactive (N in atoms labelled with small gray circles), and another one is non-stationary (N active atoms labelled with large red circles).We label the non-stationary solutions as the active atoms.The active atoms form spatial clusters in the Rydberg gas and give rise to the oscillatory phase into the ensemble.
For random gas, we can use the ratio N active /N to character active fraction of the system.
Here the ratio N active /N → 1 means that most of the atoms is dynamically active, and (iii) In a stationary cluster (SC) phase, H 0.9.In this regime, the number of active atom R e V A Y K 9 c c a v u D G S Z e D m p Q I 5 6 r / z V 7 c c s j V A a J q j W H c 9 N j J 9 R Z T g T O C l 1 U 4 0 J Z S M 6 w I 6 l k k a o / W x 2 8 I S c W K V P w l j Z k o b M 1 N 8 T G Y 2 0 H k e B 7 Y y o G e p F b y r + 5 3 V S E 1 7 5 G Z d J a l C y + a I w F c T E Z P o 9 6 X O F z 5 E I I F l 9 e J s 2 z a n B R P b 8 / r 9 S u 8 z i K 6 A g d o 1 M U o E t U Q 7 e o j h q I o k f 0 j F 7 R m 6 e 8 F + / d + 5 i 3 F r x 8 5 h D 9 g f f 5 A 2 I f j w Q = < / l a t e x i t > ⌦ < l a t e x i t s h a 1 _ b a s e 6 4 = " w p l L S e Z G 7 r O g H U V c P a + d C J t P 3 X o = " > A A A B 7 X i c b V D L S g N B E J y N r x h f U Y 9 e B o P g K e x K U I 9 B L x 4 j m A c k S + i d z C Z j 5 r H M z A p h y T 9 4 8 a C I 9 8 e Z k 0 z 6 r + R f X 8 7 r x S u 8 7 j K M I R H M M p + H A J N b i F O j S A g o B n e I U 3 9 I B e 0 D v 6 m L c W U D 5 z C H + A P n 8 A N F 6 Q F w = = < / l a t e x i t > x < l a t e x i t s h a 1 _ b a s e 6 4 = " F a H 4 U 5 w A

Figure 1 .
Figure 1.Ergodic and non-ergodic phases in a driven-dissipative Rydberg gas.(a) Atoms are laser excited from the ground state |g (black sphere) to Rydberg state |r (red, large sphere) with detuning ∆, Rabi frequency Ω and decay rate Γ.Due to the strong Rydberg interaction, Rydberg excitations are separated in space, where the minimal distance is determined by the blockade radius.(b) Ergodic phase.When the Rydberg atom interaction is weak, the dissipation leads to a single stationary state (fixed point).As shown in (c) any initial states on or in the Bloch sphere will decay to the fixed point (magenta dot).Blue and orange curves depict trajectories of atoms from two different initial conditions.(d) Non-ergodic phase.When the interaction is strong, closely packed Rydberg atoms form active clusters, as highlighted in the figure, leading to non-stationary dynamics.In phase space, the corresponding trajectories oscillate persistently, building up a limit cycle (e).
(e)][36].The non-equilibrium dynamics is measured nondestructively through electromagnetically-induced transparency (EIT).Due to the unprecedented level of controllability, thermal Rydberg atom vapor gases provide a platform to explore and probe non-ergodicity of matter in addition to the Rydberg array simulator[17,18].The Experiment.We prepare Rubidium-85 atom gases above room temperature (typically 45 • C) with density around 9.0 × 10 10 cm −3 .The atom is excited from ground state |g to Rydberg state |r through EIT, i.e. from ground state |g = 5S 1/2 , F = 2 to intermediate state |e = 5P 1/2 , F = 3 and then to Rydberg-state |r = 51D 3/2 by a probe and coupling light field, respectively.The Rabi frequency and detuning of the probe and coupling light are denoted by Ω p and ∆ c .Both can be time dependent in our experiment.

Figure 2 .
Figure 2. Non-equilibrium phase diagram and ergodicity breaking transition.(a) The transmission spectrum as a function of Ω 2 p and ∆c with the sweep rate 2π × 6 MHz/ms.Three regions I, II and III mark the ergodic, ergodicity breaking and strongly interacting phases, respectively.(b) When sweeping the detuning, transmission grows rapidly when close to the resonance, and the oscillates around the resonance.The maximal population (marked by the circle) indicates the critical population where the many-body system changes from ergodic to non-ergodic phase.In the latter case, the transmission does not relax to equilibration.(c) The phase transition point characterized by the parameters |∆c| and Ω 2 p .The transition points are extracted from the first jump of phase revival regime of panel (a) as labeled in (b).The error bars are statistics according to the experimental fluctuations.(d) In the strongly interacting phase, the transmission increases sharply at the critical point.Before and after the rapid change, the transmission changes smoothly when sweeping the detuning.(e) The observed time flow of the many-body non-ergodic state when stopping sweep ∆c in the vicinity of the first jump, i.e., ∆c ∼ −1.5 × 2π MHz.Long time phase oscillations are found in the experiment.(f) Enlarged view of the region labelled in (e).The black curve is the fitting curve with function 1.64 + 1.60 cos (1.88 − 116t) + 1.46.The period of collapse and revival is ∆t ≈ 0.053ms.
< l a t e x i t s h a 1 _ b a s e 6 4 = " j q y l w c x B a g M h e s C Q 8V F O W E k S c m o = " > A A A B 6 H i c b V D L T g J B E O z F F + I L 9 e h l I j H x R H Y N U Y 9 E L x4 h k U c C G z I 7 9 M L I 7 O x m Z t Z I C F / g x Y P G e P W T v P k 3 D r A H B S v p p F L V n e 6 u I B F c G 9 f 9 d n J r 6 x u b W / n t w s 7 u 3 v 5 B 8 f C o q e N U M W y w W s s j j y c w C m c g w d X U I U 7 q E E D G C A 8 w y u 8 O Q / O i / P u f C x a c 0 4 2 c w x / 4 H z + A O k n j Q U = < / l a t e x i t > x < l a t e x i t s h a 1 _ b a s e 6 4 = " + u Q y N R f l h 6 Z f p B t 0 O s l + e 4 s j u B k = " > A A A B 6 H i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e i F 4 8 t 2 F p o Q 9 l s J + 3 a z S b s b o Q S + g u 8 e F D E q z / J m / / G b Z u D t j 4 Y e L w 3 w 8 y 8 I B F c G 9 f 9 d g p r 6 x u b W 8 X t 0 s 7 u 3 v 5 B + f C o r e N U M W y x W M S q E 1 C N g k t s G W 4 E d h K F N A o E P g T j 2 5 n / 8 I R K 8 1 j e m 0 m C f k S H k o e c U W O l 5 q R f r r h V d w 6 y S r y c V C B H o 1 / + 6 g 1 i l k Y o D R N U 6 6 7 n J s b P q D n e I U 3 5 9 F 5 c d 6 d j 0 V r w c l n j u E P n M 8 f 6 q u N B g = = < / l a t e x i t > y < l a t e x i t s h a 1 _ b a s e 6 4 = " V c d u h I m t G 3 1 x t I w C R H / 3 q E w y 9 5 Y = " > A A A B 6 H i c b V D L T g J B E O z F F + I L 9 e h l I j H x R H Y N U Y 9 E L x 4 h k U c C G z I 7 9 M L I 7 O x m Z t Y E C V / g x Y P G e P W T v P k 3 D r A H B S v p p F L V n e 6 u I B F c G 9 f 9 d n J r 6 x u b W / n t w s 7 u 3 v 5 B 8 f C o q e N U M W y w W s s j j y c w C m c g w d X U I U 7 q E E D G C A 8 w y u 8 O Q / O i / P u f C x a c 0 4 2 c w x / 4 H z + A O w v j Q c = < / l a t e x i t > z (a) < l a t e x i t s h a 1 _ b a s e 6 4 = " U L A / L N K R 2 N d 8 r / 8 r p k D p J 4 W t Y 5 g = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e i F 4 8 V 7 A e 0 o U y 2 m 3 b p b h J 3 N 0 I J / R N e P C j i 1 b / j z X / j t s 1 B W x 8 M P N 6 b Y W Z e k A i u j e t + O 4 W 1 9 Y 3 N r e J 2 a W d 3 b / + g f H j U 0 n G q K G v S W M S q E 6 B m g k e s a b g R r J M o h j I Q r B 2 M b 2 d + + 4 k p z e P o w U w S 5 k s c R j z k F I 2 V O r 0 h S o n E 9 M s V t + r O Q V a J l 5 M K 5 G j 0 y 1 + 9 Q U x T y S J D B W r d 9 d z E + B k q w 6 l g 0 1 I v 1 S x B O s Y h 6 1 o a o W T a z + b 3 T s m Z V Q Y k j J W t y J C 5 + n s i Q 6 n 1 R A a 2 U 6 I Z 6 W V v J v 7 n d V M T X v s Z j 5 L U s I g u F o W p I C Y m s + f J g C t G j Z h Y g l R x e y u h I 1 R I j Y 2 o Z E P w l l 9 e J a 2 L q n d Z r d 3 X K v W b P I 4 i n M A p n I M H V 1 C H O 2 h A E y g I e I Z X e H M e n R f n 3 f l Y t B a c f O Y Y / s D 5 / A G 4 9 I / G < / l a t e x i t > t < l a t e x i t s h a 1 _ b a s e 6 4 = " M p L B u T j A y R C G C q J 5 I 6 u h o i I l P 1 c = " > A A A C C 3 i c b V D L S s N A F J 3 4 r P V V d e l m a B E q S E i k q M u i I C 4 r 2 A c 0 o U y m 0 3 b o Z B J m b s Q S s n f j r 7 h x o Y h b f 8 C d f + P 0 s d D W A x c O 5 9 z L v f c E s e A a H O f b W l p e W V 1 b z 2 3 k N 7 e 2 d 3 Y L e / s N H S W K s t e x i t s h a 1 _ b a s e 6 4 = " U L A / L N K R 2 N d 8 r / 8 r p k D p J 4 W t Y 5 g = " > A A A B 7 3 i c b V B N S 8 N A E J 3 U r 1 q / q h 6 9 L B b B U 0 m k q M e i F 4 8 V 7 A e 0 o U y 2 m 3 b p b h J 3 N 0 I J / R N e P C j i 1 b / j z X / j t s 1 B W x 8 M P N 6 b Y W Z e k A i u j e t + O 4 W 1 9 Y 3 N r e J 2 a W d 3 b / + g f H j U 0 n G q K G v S W M S q E 6 B m g k e s a b g R r J M o h j I Q r B 2 M b 2 d + + 4 k p z e P o w U w S 5 k s y g I e I Z X e H M e n R f n 3 f l Y t B a c f O Y Y / s D 5 / A G 4 9 I / G < / l a t e x i t > t

Figure 3 .
Figure 3. Rydberg clusters and synchronized oscillations.(a) Rydberg population nr as a function of ∆.At t = 0, ∆(0) = −10γ and atoms are in the ground state.When ∆ approaches to the resonance, Rydberg population grows and starts to oscillate.The vertical dashed line indicates the first jump point, which is observed experimentally.In a different scenario, the detuning is frozen at the first jump.It is found that most of atoms become dynamically inactive, where the population relaxes to equilibration rapidly (b1).Clusters of atoms, on the other hand, are active persistently (b2).In panel (b3) distributions of atoms in the ensemble are shown.The active and inactive atoms are denoted with gray and red dots.(c) Dynamical oscillations of the active atoms are synchronized after a transient period A. When the dynamics is synchronized (see region B for a finite duration), oscillations of atoms are centered at a single frequency (inset).

Figure 4 .
Figure 4. Density-dependent phase diagram.(a-c) The measured color-map of the probe transmission against temperatures from T = 45.2 • C, 44.3 • C, 43.4 • C, corresponding to the atomic densities of 9.52, 8.78, 8.11 × 10 10 cm −3 .The yellow lines marked in (a) show obvious character of oscillations.(d-f) The decreased oscillated probe transmission spectra under lowering temperature.The data in (d-f) are the correspondence in (a-c) marked by the double arrows.In these cases, the sweep rate is set as 2π × 11.4 MHz/ms.

FIG. S3 .
FIG. S3.Excitation of different phases.(a1) Rydberg populations n j r (j = A, B) as a function t with ∆ = −1 and Ω = 0.2.The system is occupied by UNI phase.The corresponding phase space trajectories are plotted in panel (a2).The system starts from red dot and decay to the fixed point (blue dot).(a3) shows population difference sz = s A z − s B z and the average Rydberg population n r .For UNI phase, sz = 0. (b1) Evolution of n j r with ∆ = −1 and Ω = 0.9.The system is in the OSC phase.Their phase space trajectories begin to oscillate persistently, and give rise to limit cycles [see panel (b2)].(b3) sz and n r are also non-stationary.(c1) The system is in the AF phase when increasing Ω to 2.8.(c2) The phase space trajectories decay to two different fixed points (blue dot).(c3) The order parameter sz is non-zero in AF phase.The panel (a), (b) and (c) correspond to points A, B, and C in Fig. S2(b).

Fig. S3 shows
Fig. S3 shows the dynamics of the Rydberg populations with different Ω.When Ω is small, the steady state is the UNI phase [Fig.S3(a1)] where excitation of atoms are identical, then changes to the OSC phase [Fig.S3(b1)] and then to the AF phase by increasing Ω [Fig.S3(c1)].For UNI phase, atoms decay to a single fixed point (blue dot) [Fig.S3(a2)].These unstable nonuniform fixed points lead to limit cycles [Fig.S3(b2)], in which the Rydberg population oscillates periodically in time [Fig.S3(b3)].Further increasing Ω, the FIG. S5.Dynamics of the atom ensemble.(a1) the mean values of the Rydberg population n r as a function t with Ω = 0.6 and ∆ = −4.The system is in the E phase.We plot fixed points of corresponding inactive atoms in panel (a2).All fixed points are almost identical.(b1) n j r varies with t for Ω = 2 and ∆ = −2.4.The system is in the NE regime.In panel (b2), we can find that most of inactive atoms are blockaded, i.e. with very small Rydberg population.(c1)-(c2) The system is in the SC phase when increasing Ω to 4. At later time, the average Rydberg population is stationary.Different atoms occupy very different Rydberg populations.Here a large fraction of the fixed points decay to two different values (highlighted by two red arrows).The panel (a), (b) and (c) correspond to E-, NE-, and SC-phases in Fig. S4(c).
decreases, but most of inactive atoms decay to different fixed points.It results in multistable stationary phases [S1, S2].We present three examples of the dynamical evolution in Fig. S5, covering the E-, NE-, and SC-phases.In E phase, the mean values of the Rydberg population n r is weak [Fig.S5(a1)].All atoms evolve into uniform fixed points with weak fluctuations [Fig.S5(a2)].In NE phase, a large fraction of the atom is active dynamically.At the same time, due to the strong interaction, most of inactive atoms are blockaded and play a minor role in contributing to the Rydberg population, Here n r oscillates periodically in time [Fig.S5(b1)-(b2)].In the SC phase, the number of active atoms decreases.Although clusters are found in this