Observation of Blackbody Radiation Enhanced Superradiance in ultracold Rydberg Gases

An ensemble of excited atoms can synchronize emission of light collectively in a process known as superradiance when its characteristic size is smaller than the wavelength of emitted photons. The underlying superradiance depends strongly on electromagnetic (photon) fields surrounding the atomic ensemble. High mode densities of microwave photons from $300\,$K blackbody radiation (BBR) significantly enhance decay rates of Rydberg states to neighbouring states, enabling superradiance that is not possible with bare vacuum induced spontaneous decay. Here we report observations of the superradiance of ultracold Rydberg atoms embedded in a bath of room-temperature photons. The temporal evolution of the Rydberg $|nD\rangle$ to $|(n+1)P\rangle$ superradiant decay of Cs atoms ($n$ the principal quantum number) is measured directly in free space. Theoretical simulations confirm the BBR enhanced superradiance in large Rydberg ensembles. We demonstrate that the van der Waals interactions between Rydberg atoms change the superradiant dynamics and modify the scaling of the superradiance. In the presence of static electric fields, we find that the superradiance becomes slow, potentially due to many-body interaction induced dephasing. Our study provides insights into many-body dynamics of interacting atoms coupled to thermal BBR, and might open a route to the design of blackbody thermometry at microwave frequencies via collective, dissipative photon-atom interactions.

An ensemble of excited atoms can synchronize emission of light collectively in a process known as superradiance when its characteristic size is smaller than the wavelength of emitted photons. The underlying superradiance depends strongly on electromagnetic (photon) fields surrounding the atomic ensemble. High mode densities of microwave photons from 300 K blackbody radiation (BBR) significantly enhance decay rates of Rydberg states to neighbouring states, enabling superradiance that is not possible with bare vacuum induced spontaneous decay. Here we report observations of the superradiance of ultracold Rydberg atoms embedded in a bath of room-temperature photons. The temporal evolution of the Rydberg |nD to |(n + 1)P superradiant decay of Cs atoms (n the principal quantum number) is measured directly in free space. Theoretical simulations confirm the BBR enhanced superradiance in large Rydberg ensembles. We demonstrate that the van der Waals interactions between Rydberg atoms change the superradiant dynamics and modify the scaling of the superradiance. In the presence of static electric fields, we find that the superradiance becomes slow, potentially due to many-body interaction induced dephasing. Our study provides insights into many-body dynamics of interacting atoms coupled to thermal BBR, and might open a route to the design of blackbody thermometry at microwave frequencies via collective, dissipative photon-atom interactions.
When the surrounding photon field is modified locally by, e.g. cavities, characters of light-matter interactions change drastically, leading to unconventional phenomena such as the paradigmatic Casimir [28] and Purcell effects [29,30]. A thermal bath of blackbody photons These two authors contributed equally to this work. * zhybai@lps.ecnu.edu.cn † zhaojm@sxu.edu.cn ‡ weibin.li@nottingham.ac.uk can modify the interaction, too. This causes tiny energy shifts to groundstate atoms, and can be detected by accurate optical clocks [31,32]. In electronically highlying Rydberg states, atoms can strongly interact with blackbody radiation (BBR) [33]. At room temperature T , BBR photons of low-frequency microwave (MW) fields can provide successive energies to couple different Rydberg states, i. e. kT > ω (k, , and ω to be the Boltzmann constant, Planck constant and transition frequency). Due to high numbers of MW photons per mode in the BBR field [34], the decay of single Rydberg atoms is orders of magnitude faster than in vacuum. The increase of decay rates [33,35] and energy shifts [36] of Rydberg atoms have been measured. The large wavelength (∼mm) of MW photons moreover permit superradiance of Rydberg atoms. Superradiance of Rydberg atoms driven by vacuum fields has been reported [6]. However, Rydberg superradiance induced by thermal BBR has only been observed in the presence of cavities [37].
tensions ∼ µm of the atomic gases [1]. We measure the superradiant decay in selective |nD 5/2 → |(n + 1)P 3/2 transition, and scaling with respect to atom numbers and Rydberg states. We identify that the superradiant decay is strongly influenced by van der Waals (vdW) interactions of Rydberg atoms, confirmed by careful theoretical analysis and large-scale numerical simulations. Our study opens a window to experimentally explore the superradiant dynamics of interacting many-body systems coupled to thermal BBR, and enable to develop blackbody thermometry at microwave frequencies through collective photon-Rydberg atom interactions.
The remainder of the article is arranged as follows. In Sec. II, the experiment setup is described and the fast Rydberg decay between |nD and |(n + 1)P transitions observed in the experiment is presented. In Sec. III, the lifetime of the Rydberg states is estimated and the superradiance dependence on the MW wavelength, atomic number, and BBR temperature is discussed theoretically. In Sec. IV, a master equation model including van der Waals interactions between Rydberg atoms is introduced and the mean field simulation on the master equation is carried out, with the theoretical result compared with the experimental one. In Sec. V, the scaling of the Rydberg superradiance is calculated, and the dependence of the superradiance on the particle number, Rydberg states, and BBR temperature are provided both experimentally and theoretically. In Sec. VI, a preliminary experimental result on the superradiance dynamics in the presence of MW and static electric fields is given, which manifests the signature of dipole-dipole interactions between atoms. Lastly, Sec. VII contains a summary of the research results obtained in this work.
In each experiment cycle, atoms are excited to the Rydberg state (n ≥ 60) in 6 µs (after turning off the trap laser). Due to blockade by the vdW interaction (blockade radius R b ), number N e of Rydberg atoms is varied between 10 3 to 10 4 by changing the laser power. After switching off the excitation laser, Rydberg atoms are allowed to evolve for a duration t, and then ionized by a state-dependent electric field. The ions are detected by a microchannel plate (MCP) detector with efficiency about 10%. The detail of our experiment can be found in Appendix A.
The state-selective ionization and detection method shows that Rydberg state |nD 5/2 decays immediately to the energetically closest |(n + 1)P 3/2 state [see example of atomic levels in Fig. 2(a)]. In Fig. 1(c), snapshots of ion signals are shown for state |60D 5/2 . Increasing time t, the population transfers to |61P 3/2 state rapidly where the peak is drifted towards a later time. The population of |61P 3/2 state is obtained in the red gate region. The tail in this gate when t = 0 indicates that the decay occurs slightly also during the Rydberg excitation.
The population dynamics of the system displays qualitatively different behaviors at later time. As shown in Fig. 1(d), populations change slowly when t < 2 µs. During 2 µs < t < 4 µs, a large portion of the population is transferred to state |61P 3/2 rapidly. The population in state |61P 3/2 reaches maximal at t ≈ 5 µs, and then decays to other states when t > 5 µs. Such fast decay, much shorter than the lifetime 2.6 ms in the underlying transition at room temperature, is rooted from superradiance of the Rydberg ensemble interacting with BBR photons. For convenience, quantum number J = 5/2 and J = 3/2 will be omitted in the notation from now on. 2π × 3.2 GHz. (c) The decay rate from |nD state to |(n + 1)P 3/2 for different temperature T (with n = 60, 63 and 70). At room temperature, the decay rate is significantly increased.

III. BBR ENHANCED RYDBERG SUPERRADIANCE
Decay of Rydberg atoms is affected by BBR and such effect has been experimentally observed. For example, the recent experiment has found that lifetimes in Rydberg nS state are determined by 300K BBR [38]. To identify the lifetime between Rydberg levels, the decay rate of spontaneous transition between nJ and n J states can be calculated by [35,39,40], where m e is the electron mass, ω n J nJ = |E nJ − E n J | is the transition frequency, with E nJ and E n J being energies of nJ and n J states, respectively. Energies E nL = −1/(2n 2 eff ) [in atomic units] of the Rydberg states are expressed through the effective quantum number n eff = n − µ J , where µ J is a quantum defect of Rydberg nJ-state, which can be found in Ref. [41].
The lifetime of a Rydberg state depends on background BBR temperature. Taking into account of photon number per mode at temperature T , the decay rate becomes, Γ n J nJ (T ) = Γ n J nJn ω (T ), where the thermal factor n ω (T ) = 1/[exp( ω n J nJ /k B T ) − 1] gives Bose-Einstein statistics of photon numbers at temperature T . The total decay rate is Γ nJ (T ) = n J Γ n J nJ (T ) [39,42]. For MW transitions, the photon energy is far smaller than the thermal energy, i.e. ω nn k B T , such that the thermal factor is far larger than 1.
As an example, we show that the transition frequency between |60D and |61P state in Fig. 2(a). The transition frequency ω 61P 60D ≈ 2π × 3.2 GHz is far smaller As Γ n l nl (0) ∝ (ω n l nl ) 3 , the rate becomes larger when decaying to lower states at T = 0. At T = 300K, BBR enhances decay rates corresponding to MW transitions. (c) Transition wavelength for different |nD 5/2 → |(n + a)P 3/2 transition (a = 1, 2, 3). The |nD 5/2 → |(n + 1)P 3/2 transition gives the largest wavelength (tens of mm), far larger than the spatial dimension of the trap, enabling superradiant decay. (d) Superradiance threshold parameter at T = 300K. Parameter C n l nl in the |nD → |(n + 1)P transitions is orders of magnitude larger than other transitions, due to large wavelengths and high mode densities of the MW photons.
than other transition energy [highlighted with a box]. Though MW photons of various frequencies can be emitted with higher rates, superradiance on the other hand enhances emission rates of selected transitions, depending on MW wavelengths λ n l nl . Wavelengths of the MW photon corresponding to |60D → |61P is about 92.93 mm d = 550 µm (the size of the atomic sample) or wavelengths of other MW photons. We also plot the decay rate Γ from |nD state to |(n + 1)P 3/2 as functions of temperature T in Fig. 2(c). Here Γ almost linearly increase with T . Clearly at room temperature, the decay rate is greatly enhanced byn ω (T ) 1. Superradiance in the |nD → |(n + 1)P transition is much stronger than one of the |nS → |nP transition [6], due to the low frequency MW transition.
In the experiment, frequency ω 61P ing transition [2]. The larger the parameter C n l nl is, the stronger superradiance takes place. As shown in Fig. 3(d), the threshold parameter (∼ 10 5 ) corresponding to the |60D → |61P transition is several orders of magnitude larger than that of other transitions [6].
The resulting strong superradiance exhibits sensitive dependence on numbers of the Rydberg atoms. In Fig. 4(a)-(c), net changes of the |61P population, i.e. growth of the Rydberg atom number N e when t ≥ 0, are shown. A generic feature is that populations increase rapidly and arrive at maximal values, after a slow varying stage. Increasing N e , the population dynamics become faster such that it takes less time to reach the maxima.

IV. MASTER EQUATION SIMULATION
Dipole-dipole interactions between Rydberg atoms can slow down or even destroy superradiance [43]. In our experiment, the dipole-dipole interaction plays a negligible role in the decay, where the angular average of the dipolar interactions vanishes in such large ensemble.
For vanishing dipole-dipole interaction and large single-photon detuning, the dynamics of the superradiant decay can be modeled by the quantum master equation for the many-atom density operator ρ: under a two-level approximation. The Hamiltonian in the equation is given by /|r j −r k | 6 is the vdW potential with the dispersive coefficient C D(P ) 6 ∝ n 11 . Note that influences of the vdW interaction on superradiance have not been explored so far.
In Eq. (2) the radiative decay from the upper state | ↑ to the lower state | ↓ is described by collective dissipation of Lindblad form [44] where Γ = Γ (n+1)P nD (T ) is the decay rate, whose spatial dependence can be neglected as averaging spacing between Rydberg atoms is much smaller than λ (n+1)P nD . In both (3) and (4), S j = (Ŝ j x ,Ŝ j y ,Ŝ j z ) are the Pauli matrix of the jth atom, with raising and lowering operator S k ± =Ŝ k x ± iŜ k y . For small systems (i.e., a few tens of atoms), the quantum master equation can be solved by direct diagonalization. However, the number of Rydberg excitation is large (i.e., 10 3 ∼ 10 4 ) in experiment. To efficiently simulate a large system (total particle number N t 1), one can apply the method of the discrete truncated Wigner approximation (DTWA), which is a phase space method by which the density-operator equation can be replaced by its mean-value equation with the quantum fluctuations of the system involved in random initial states [45,46].
Based on the idea of the DTWA, we define mean values s k = S k for our system. Then we obtain the equations of motion of s k associated with the master equation (2), with the form In the DTWA method, we describe the initial state by a Wigner probability distribution, p k µ,aµ (µ = x, y, z; the subscript a µ denotes the index of each trajectory, k denotes the position of Rydberg atom) for certain discrete configurations of Bloch vector elements, s k µ = Ŝ k µ . Consider the eigen-expansion of the spin operators,Ŝ µ k = aµ η k µ,aµ |η k µ,aµ η k µ,aµ |, where η k µ,aµ and |η k µ,aµ denote the eigenvalues and eigen-vectors, respectively. Then, we select the "a-th" eigenvalue, λ k µ (t = 0) = η k µ,aµ /2, with probability p k µ,aµ = Tr[ρ k 0 |η k µ,aµ η k µ,aµ |]. Specifically, all the atoms initially populate in the upper state | ↑ , with initial density matrixρ k 0 = | ↑ ↑ |, which leads to fixed classical spin component along z, σ k z = −1/2, and fluctuating spin components in the orthogonal directions σ k x(y) ∈ {−1/2, 1/2}, each with 50% probability. Mean values of observable (i.e., the Rydberg population) are calculated by averaging over many trajectories. In the simulation, we consider an ensemble of Rydberg atoms separated by the blockade radius R b and with an Gaussian distribution in space. Typically we run ≥ 10 4 trajectories to obtain mean values through the ensemble average, which guarantee the convergence of DTWA results. A generalized truncated Wigner approximation (GDTWA) method is give in the Appendix B for spin-3/2 atoms when simulating dynamics involving all four levels [47].

V. SCALING OF RYDBERG SUPERRADIANCE
Without vdW interactions, the master equation can be solved analytically, yielding the solution to N ↓ [48], where t d = In(N t )/[Γ(N t + 1)] is the delay time, and the collective decay rate ∝ N t Γ. The analytical solution N ↓ predicts faster superradiant transition [dashed curves in Fig. 4(a1)-(a4)]. By taking into account of the vdW interaction, our numerical simulation agrees nicely with the experimental data [ Fig. 4(a1)-(a4)]. The slower superradiance can be understood that the vdW interaction mixes superradiant and other states. As the latter typically decay slower, such dephasing therefore increases the superradiant decay time. The number N t of Rydberg atoms used in the simulation is about 35% of the experimental value N e . This difference could attribute to the fact that only some of the Rydberg atoms in the trap are in superradiant states, as we observe atoms remain in the initial state even when t > 5 µs [ Fig. 1(d)].
We have also plotted dynamical evolution of N ↓ for state |63P and |71P in the Fig. 4(b1)-(b4) and 4(c1)-(c3). DTWA simulations capture our experimental data very well. In Fig. 4(c4), one sees that the number N t of Rydberg atoms used in the simulation is about 25 ∼ 35% of the experimental value N e for all Rydberg states (n = 60, n = 63, and n = 70). The ratio fluctuates around a constant when increasing N e for a given Rydberg state, which indicates that the experiment and corresponding simulation are consistent. The finite detection efficiency of the MCP might affect values of N e , and hence the ratio N t /N e . The time scale, however, is not affected by the detection efficiency, as the ion signal is linearly proportional to N e .
Drastically, the vdW interaction alters scaling of superradiance with respect to N e and principal quantum numbers. First, the collective decay rate Γ c changes nonlinearly with N e , which is confirmed by the numerical simulation, shown in Fig. 5(a). In contrast to the interaction-free case [see Eq. (6)], the rate Γ c ∝ N α e , where α increases from 2.24 (n = 60), to 2.96 (n = 63) and 4.07 (n = 70). Due to stronger vdW interactions (∝ n 11 ) and smaller decay rate (∝ n −3 ), the collective rate decreases in higher-lying states.
Next, we study the emission rate of MW photons, given by r(t) =Ṅ ↓ . Without vdW interactions, the emission rate can be derived from Eq. (6), which has the maximal emission rate r m = ΓN 2 t /4 at t = t d , i.e. proportional to N t quadratically.
The emission rate is obtained by fitting the experimental data [green dash-dotted curves in Fig. 4(a1)-(a4)]. In Fig. 5(b) normalized rate R(t) = 4r(t)/ΓN 2 r is shown, where N r is the largest N e among experiments for a Rydberg state |nD . For example, N r = 28300 for state |60D [see Fig. 4(a) and 5(a)]. Profiles of R(t) exhibit a single peak whose location varies with N e . The maximal value R max is 1 when N = N r and t = t d , and smaller than 1 when N e < N r [ Fig. 5(b)].
In Fig. 5(c), R max as a function of N e is shown. Both the experimental data and simulation show R max ∝ N β e . Due to strong vdW interactions, the power β increases from 3.14 (n = 60), to 3.56 (n = 63) and 3.62 (n = 70). Moreover, the peak rate R max depends also on the BBR temperature. Our numerical simulations show R max ∝ T ξ , where ξ = 2.57, 3.01, 3.12 for n = 60, 63 and 70, respectively [ Fig. 5(d)]. Such dependence might enable a way to measure BBR temperatures. Furthermore, we have also applied a static electric field, which due to the induced dipole-dipole interaction. Details of the following reply. Previous studies have shown that the dipole-d many-body dynamics (see e.g. Phys. Rev. Lett. 120, 063 superradiance and dipole-dipole interaction in Rydberg gases is worth studying both experimentally and theoretically. Such of this work and will be carried out in the future. Referee B: 3. Regarding the modelling: it is well known that Rydberg gas under the given conditions is not driven by the m pair-wise interactions, which can trigger avalanche dynami the van-der-Waals term (Eq. 3) neglects the D-P interactio interaction and should strongly influence the dynamics.
Our response: We agree with Referee B that the state exc a↵ect the superradiance. Some discussions can be found in, e Dipolar interactions are absent in the experiments discusse no net electric dipoles are present when there are no extern interactions will depend on angle ✓ jk between the dipole and and k-th Rydberg atoms, i.e. V e jk / C 3 [1 3cos 2 (✓ jk )]/R 3 jk . number of Rydberg atoms, such that the average interaction place between these two states. Another feature in the presence of the MW field is that the delay time of the TOF signal is increased due to the MW field induced dipole-dipole interaction (see the previous and later replies).
Furthermore, we have also applied a static electric field, which also a↵ects the population dynamics due to the induced dipole-dipole interaction. Details of the relevant discussion are given in the following reply. Previous studies have shown that the dipole-dipole interaction leads to interesting many-body dynamics (see e.g. Phys. Rev. Lett. 120, 063601, 2018). The interplay between superradiance and dipole-dipole interaction in Rydberg gases has largely not explored so far, and is worth studying both experimentally and theoretically. Such study is certainly beyond the scope of this work and will be carried out in the future.
Referee B: 3. Regarding the modelling: it is well known that the dissipative dynamics in a dense Rydberg gas under the given conditions is not driven by the mean interactions, but by the strongest pair-wise interactions, which can trigger avalanche dynamics (see references above). Besides, the van-der-Waals term (Eq. 3) neglects the D-P interaction, which is a resonant dipole-dipole interaction and should strongly influence the dynamics.
Our response: We agree with Referee B that the state exchange dipole-dipole interaction will a↵ect the superradiance. Some discussions can be found in, e.g., Ref. [44,45]. Dipolar interactions are absent in the experiments discussed in the main text. When excited, no net electric dipoles are present when there are no external electric fields. The dipole-dipole interactions will depend on angle ✓ jk between the dipole and molecule axis that connects the jand k-th Rydberg atoms, i.e. V e jk / C 3 [1 3cos 2 (✓ jk )]/R 3 jk . In our experiment, there are a large number of Rydberg atoms, such that the average interaction for any Rydberg atoms vanishes, i.e. The MW field is applied during the superradiant decay. (b) Population of state |60D 5/2 and |61P 3/2 at t = 2µs [corresponding to Fig.1(c)]. (c) The population difference with and without MW fields. One sees that the population in state |60D 5/2 decreases, and population in state |61P 3/2 increases due to the MW field coupling. The MW field modulate the population dynamics dramatically (b). The number of Rydberg atoms is about 26300.

VI. SUPERRADIANT DYNAMICS WITH MW AND STATIC ELECTRIC FIELDS
To further identify electric field effects on the Rydberg transitions, we applied a microwave electric (MW) field with frequency 3.21573 GHz resonantly interacting with the |60D 5/2 → |61P 3/2 transition, as shown in Fig. 6(a). In the presence of the MW field, the TOF signals are changed apparently, i.e. more Rydberg atoms are transferred to the |61P state, see Fig. 6(b). It can be seen that the TOF signals in the presence of the MW field appear at the same position as the one without applying the MW field, indicating that the signal is indeed due to the |60D 5/2 → |61P 3/2 decay. This change of populations is a result of the interplay between the MW field and superradiant decay. To highlight the effect of the MW coupling, we show the population difference, i.e. subtracting the ion signal when the MW field is off (bottom) from the one (top) with the MW field. As shown in Fig. 6(c), the population in state |60D reduces (blue gate), and population in state |61P states increases (red gate), due to the MW coupling. The profile demonstrates that the superradiant decay takes place between these two states. Another feature in the presence of the MW field is that the delay time of the TOF signal is slightly increased, possibly due to the interplay between the dipolar interactions and superradiance.
On the other hand, it is known that dipole-dipole interactions will depend on angle θ jk between the dipole and molecule axis that connects the j-and k-th Rydberg atoms, i.e. V e jk ∝ C 3 [1 − 3cos 2 (θ jk )]/R 3 jk . However without external fields, there are a large number of Rydberg atoms, such that the net interaction for any Rydberg atoms vanishes, i.e. k V jk ∝ π 0 (1 − 3 cos 2 θ) sin θdθ = 0, where we have replaced the sum by a continuous integral in the estimation.
To check this, we have carried out additional experiments by applying a static electric field to the sample. See the timing and results in Fig. 7. The electric field will align the dipoles along the direction of the field. In this case, a net dipole-dipole interaction will be induced. The presence of dipole-dipole interactions will slow down the superradiance due to many-body dephasing [48]. In our experiment, we indeed find that speed of the superradiant decay is reduced when the electric field is strong, see Fig. 7(b)-(c), where stronger electric fields give stronger dipole-dipole interactions, and hence cause slower superradiant decay.

VII. CONCLUSION AND DISCUSSION
We have observed the superradiant decay of the |nD → |(n + 1)P transition in an ensemble of lasercooled caesium Rydberg atoms in free space. The superradiance is found to be enhanced by finite temperature BBR due to high number densities of MW photons, confirmed by many-body simulations. The vdW interaction drastically modifies superradiance, leading to state dependent scaling.
Our system offers a controllable platform to investigate the interplay between strong collective dissipation and two-body Rydberg interactions. For example, superradiant dynamics will be qualitatively different at longer times when vdW interactions are strong. Dipole-dipole interactions, induced by external fields, will compete with the collective dissipation, too. Such competition can be explored experimentally in a controllable fashion. Beyond fundamental interests, our study might be useful in developing BBR thermometry in the MW domain whose sensitivities can be improved by collective lightatom interactions, with applications to improve accuracy of atomic clocks [49][50][51][52]. are Ω p = 2π × 132.05 MHz and Ω c = 2π × 6.91 MHz. The excitation region is surrounded by three pairs of fieldcompensation electrodes, which allow us to reduce stray electric fields via Stark spectroscopy, corresponding stray field less than 30 mV/cm.
To experimentally measure Rydberg population, we use a ramping electric field to ionize the Rydberg atoms.
In the experiments, the electric field is linearly increased to 256 V/cm in 3 µs, unless stated elsewhere explicitly. This field is much larger than the ionization threshold. Resultant ions are detected with a micro-channel plate (MCP) detector with a detection efficiency ∼ 10%. The detected ion signals are amplified with an amplifier and analyzed with a boxcar integrator (SR250) and then recorded with a computer.
Before measuring Rydberg atoms, we first calibrate the MCP ions detection system with two shadow images taken before and after the laser excitation. From the difference of two shadow images, we obtain the number of Rydberg excitation, N e , and therefore the gain factor, G, of the MCP ions detection system. The gain factor is defined as, where V signal is the intensity of the measured ion signal, t gate is the Boxcar gate width, and S sensitivity is the Boxcar setting sensitivity, respectively. The shadow image is usually used to detect the number of MOT atoms. From the difference of two shadow images taken before and after the Rydberg excitation, we can extract the number of Rydberg excitation, which is compared to the ionization signal of Rydberg atoms, V signal . Using Eq. (A1),  we obtain the gain factor, shown in Fig. 8. For different Rydberg excitation power and pulse duration, the averaged gain factor is G = 0.011± 0.004. Considering the detection efficiency (10%), the effective gain factor of the MCP detection system is G eff = 0.11 ± 0.04 in our experiment, which is used throughout the experiment to determine the number of Rydberg atoms.
To verify our experimental signals is the field ionization of Rydberg atoms instead of the free ion signal, we have made experimental tests, with the result illustrated in Fig. 9. It is seen that the Rydberg population can be detected when the ionization field is larger than the ionization threshold of respective Rydberg states. When the electric field is lower than the threshold field, ion signals vanish (except fluctuations coming from background noise). This makes sure that the signal we measured is due to decay of Rydberg states, i.e. excluding autoionization of Rydberg atoms [53,54]. The two-photon excitation is used to pump the groundstate atoms to the initial state |nD 5/2 in the experiment. In the experiment, four states (two lower and two Rydberg states) are involved. Here, we denote |1 = |6S , |2 = |6P , |3 = |nD , and |4 = |(n + 1)P . The system can be described by the HamiltonianĤ =Ĥ a +Ĥ int in the interaction picture and rotating-wave approximation ( = 1), whereσ j αβ = |α β| j (α, β = 1, 2, 3, 4) is the transition operator of the jth atom. The dissipation effect is described by the Lindblad operator D 1 (ρ) and D 2 (ρ), where D 1 denotes the collective radiation between the Rydberg states and D 2 describes the single-body decay between state |2 (|3 ) to state |1 (|4 ) with rate Γ 12 (Γ 34 ).
For a few particles, the master equation can be solved numerically. To capture the build up of superradiant emission in a large ensemble, we employ a generalized discrete truncated Wigner approximation (GDTWA) based on a Monte Carlo sampling in phase space, where GDTWA method can effectively capture complex quantum dynamics in high spin systems [47]. The generic density matrix for a discrete system with D states takes the formρ i = D α=1,β=1 c αβ |α β|. For D = 4 (equivalent to a spin-3/2 atom), the states |α with α = 1, 2, 3, 4 associates to the spin states m s = −3/2, −1/2, 1/2, 3/2 of the spin-3/2 atom. Since (ρ i ) † =ρ i and Tr(ρ i ) = 1 and total (D 2 −1) real numbers are needed to describe an arbitrary state, which can be expressed as average values of (D 2 − 1) orthogonal observable: whereΛ [i],R/I α,β<α correspond to the real ("R") and imaginary ("I") parts of the off-diagonal parts of c αβ andΛ [i],D α to linear combinations of the real diagonal elements c αα . Note that for D = 2, the matrices are standard Pauli matrices for spin 1/2 system (see the DTWA method in main text). For D > 2, the matrices reduce to a generalized Gell-Mann matrices (GGMs) and corresponds to SU(D) group for spin-(D − 1)/2 system.
In the GDTWA method, we describe the initial state by a probability "Wigner" distribution, p [i] µ,aµ with a µ denoting the index of each trajectory [47]. The discrete set of initial configurations, {λ where η µ,aµ and |η [i] µ,aµ denote the eigenvalues and eigenvectors, respectively. Then, we select the initial condition λ To show dynamics starting from the laser excitation, we have made simulations with the following parameters: Γ = 389.9 Hz (n = 60), the number for groundstate atom N = 6000, Γ 12 = 2π × 5.2 MHz, Ω p = 2π × 132.05 MHz, Ω c = 2π × 6.91 MHz, and ∆ = 360 MHz. Here N is the number of groundstate atoms (not the number of Rydberg state atoms). Here superradiance takes place on a much longer time scale, as the number of atoms can be excited to Rydberg states is small. To mimic the experiment, we have increased the single-body decay rate by a factor of 3, in order to illustrate the |3 → |4 decay. As shown in Fig. 10(a), about 22% atoms are excited to state |3 during the laser excitation. In the mean time state |4 is populated weakly, which is seen in the experiment. Once the laser is turned off, superradiance is , |3 and |4 . All atoms initially populate at the groundstate Π N j=1 |1 j , and the probe and control fields are switched off at t = 6 µs. Before the excitation laser is turned off, the population in state |4 is very small (inset). After the laser is turned off, a fast population transfer from state |3 to state |4 is found. (b) The time evolution of the real and imaginary part of coherence ρ 34 between Rydberg states for one trajectory. Im(ρ 34 ) has a sech form due to superradiance. triggered. We see rapid population transfer |3 → |4 when t > 6 µs. When looking at the coherence ρ 34 , we find that its profile shows a hyperbolic function form, due to the emergence of superradiance.
It is not possible for us to simulate system sizes close to the experiment with the 4-level model, even using the GDTWA. In typical experiments hundreds of thousand atoms interact with laser fields. Among them, tens of thousand atoms are excited to Rydberg states. It is numerically challenging to simulate such large systems. As we focus on dynamics after the laser is switched off, this allows us to apply the two-level approximation. In this way, we can efficiently simulate dynamics of large system sizes by excluding groundstate atoms from the model.