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# Dynamical phase transition in the open Dicke model

Edited by Peter Zoller, University of Innsbruck, Innsbruck, Austria, and approved January 15, 2015 (received for review September 4, 2014)

## Significance

Nonequilibrium phenomena in quantum many-body systems are not well understood to date. This applies in particular for open systems, coupled to an external bath. We use a Bose–Einstein condensate in a high-finesse optical resonator with ultralow bandwidth to emulate the open Dicke model. In well-controlled sweeps across the Hepp–Lieb–Dicke phase transition, we observe hysteretic dynamics showing power-law scaling with respect to the transition time, which suggests an interpretation in terms of a Kibble–Zurek mechanism. Our observations indicate the possibility of universal behavior in the presence of dissipation.

## Abstract

The Dicke model with a weak dissipation channel is realized by coupling a Bose–Einstein condensate to an optical cavity with ultranarrow bandwidth. We explore the dynamical critical properties of the Hepp–Lieb–Dicke phase transition by performing quenches across the phase boundary. We observe hysteresis in the transition between a homogeneous phase and a self-organized collective phase with an enclosed loop area showing power-law scaling with respect to the quench time, which suggests an interpretation within a general framework introduced by Kibble and Zurek. The observed hysteretic dynamics is well reproduced by numerically solving the mean-field equation derived from a generalized Dicke Hamiltonian. Our work promotes the understanding of nonequilibrium physics in open many-body systems with infinite range interactions.

Although equilibrium phases in quantum many-body systems have been explored for a long time with great success, nonequilibrium phenomena in such systems are far less well understood (1). A paradigm for exploring nonequilibrium dynamics is the quench scenario, where a system parameter is subjected to a sudden change between two values associated with different equilibrium phases. Quantum degenerate atomic gases with their unique degree of control are particularly adapted for experimental quench studies (2, 3). For isolated quantum many-body systems a wealth of theoretical and experimental investigations of quench dynamics has appeared recently (4⇓⇓⇓⇓⇓⇓–11). A natural extension of such studies is to consider driven open systems, where dynamical equilibrium states can arise via a competition between dissipation and driving, and nonequilibrium transitions between such phases can occur as a function of some external control parameter (12⇓⇓–15). A nearly ideal experimental platform for this endeavor are quantum degenerate atomic gases subjected to optical high-finesse cavities, where the usual extensive control in cold gas systems can be combined with a precisely engineered coupling to the external bath of vacuum radiation modes (16).

Here, we study a dynamical phase transition in the open Dicke model emulated in an atom–cavity system prepared near zero temperature. The Dicke model is a paradigmatic scenario of quantum many-body physics, still subject to intensive research despite a history more than half a century long (17⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–28). It describes the interaction of *N* two-level atoms with a common mode of the electromagnetic radiation field. Hepp and Lieb already pointed out in the 1970s that upon varying the coupling strength, this model possesses a second-order equilibrium quantum phase transition between a homogeneous phase, in which each atom interacts separately with the radiation mode, and a collective phase in which all atomic dipoles align to form a macroscopic dipole moment (19, 22). It has been early suspected that the critical properties of the externally pumped open Dicke model should give rise to nonlinear hysteretic behavior in dynamical experiments (20, 21). The dynamical properties of the open Dicke model and of related many-body atom–cavity systems in the presence of dissipation are subject of extensive recent theoretical research (24, 25, 28⇓⇓–31).

With the atomic levels chosen to be momentum states of the external motion, the open Dicke model has been recently implemented experimentally by coupling a Bose–Einstein condensate (BEC) to a high-finesse resonator pumped by an external optical standing wave (32). A transition from a homogeneous phase (consisting of the condensate with no photons in the cavity) into a collective phase (with the atoms forming a density grating trapped in a stationary intracavity optical standing wave) was observed at a critical pump strength close to the expected equilibrium transition boundary. A related transition with thermal atoms has been studied in earlier work (33, 34). In the formation of the collective phase the

In the present work, the main innovation is the use of a cavity with ultranarrow bandwidth on the order of the single-photon recoil frequency (36, 37). The time scales for dissipation of the intracavity field and the coherent atomic evolution are similar. We can thus dynamically access the nonadiabatic regime, where both quantities are not in equilibrium and hence explore nonequilibrium critical properties of the Dicke model in quench experiments. For different signs of the effective detuning

## Experimental Scheme

In our experiment, outlined in Fig. 1*A*, a cigar-shaped BEC of *SI Appendix*). The frequency *SI Appendix*). The cavity possesses an extremely low dissipation rate associated with the loss of photons. The field decay rate *B*, to be resonantly coupled. For a uniform atomic sample and left circularly polarized light, the TEM_{00} resonance frequency is dispersively shifted by an amount *SI Appendix*).

## Hepp–Lieb–Dicke Transition

For negative detuning *A*, the observed intracavity power is plotted versus ^{−1} at fixed values of *B* this is shown for the position in the phase diagram in Fig. 2*A* marked by the white cross. Within a narrow channel around *κ*, the intracavity field is driven in phase with the pump field. Hence, interference of the two fields yields a square lattice potential with minima arranged on a Bravais lattice spanned by the primitive vectors *y*- and *z* directions, respectively. The density grating formed by trapping atoms in these minima (corresponding to the intensity maxima in Fig. 1*A*) satisfies the Bragg condition for scattering photons from the pump field into the cavity. When *κ*, the relative phase between the intracavity field and the pump field approaches

The basic structure of the observations in Fig. 2*A* can be understood as follows. At low pump powers the dynamics of the system may be described by the Heisenberg equations for the matter and light variables associated with the Dicke Hamiltonian with an additional term describing dissipation of the cavity light field at a rate *κ* (for details see *SI Appendix*). These equations possess a stationary solution describing the homogeneous phase, when all atoms populate the condensate mode at zero intracavity intensity. Linearization around this solution yields a stability matrix, whose eigenvalues are readily calculated. Their real parts denote the excitation spectrum whereas their imaginary parts denote the corresponding exponential excitation rates. Hence, if one of the eigenvalues attains a positive imaginary part, the homogeneous phase becomes unstable. The maximum of the imaginary parts of all eigenvalues, denoted by *C* versus *D* and *E* shows the excitation frequencies (blue solid lines) and the corresponding excitation rates (red dashed lines) along vertical sections in Fig. 2*C* with fixed detunings *D* (descending solid blue line starting at *C*, *A*.

## Observation of Hysteresis

As is seen in Fig. 2*A*, the phase transition for increasing *C* before the system can leave the homogeneous phase in a given time. A more complete picture is provided in Fig. 3, where the transition through the phase boundary is studied for negative detuning *A* *A* and *B*), a series of consecutively numbered momentum spectra is shown, recorded at different instances of time during the *A*. As the intracavity intensity assumes finite values, a coherent optical lattice is formed (arrow 2), as is seen from the occurrence of higher-order Bragg peaks. As the lattice depth grows (arrows 3 and 4), tunneling amplitudes decrease, and the relatively increased collisional interaction acts to reduce particle number fluctuations resulting in a loss of coherence. When ramping back to small values of

In Fig. 3*B* a mean-field calculation (based upon a Dicke Hamiltonian, *SI Appendix*) is shown for a homogeneous, infinite system without collisional interaction, which shows the same signatures as observed in Fig. 3*A* including dynamical details as the oscillation of the red trace around *A* and *B*; however, the system always follows the blue curve, when this point is passed with increasing

## Power-Law Scaling

The dependence of the threshold values *A* and *B* upon the quench time *C*–*E*. These quantities are determined as those values of *C* the values of *B*, are plotted versus *SI Appendix*). In Fig. 3 *D* and *E* we plot the experimentally observed values of *C* with *D* and *E*. Whereas in Fig. 3*D* the data nicely agree with the power-law behavior, in Fig. 3*E* this is only the case for the first half of the plot. At later times the data points assume an exponential rather than a power-law decay, which is in accordance with the observation that for long ramp times at the end of the descending ramp notable particle loss sets in. Our observations of power-law behavior of *SI Appendix*). A deeper understanding of these values would require a comprehensive extension of the concept of universality to the case of driven open systems (15).

## Matter Wave Superradiance

In the *A*, matter wave superradiance prevails (41, 42). Short superradiant pulses with a duration on the order of the intracavity photon life time are emitted by the cavity, if *A*. The atoms, initially populating the condensate mode at zero momentum, are thereby scattered into superpositions of higher momentum states. This excitation is irreversible and cannot be removed by ramping *C* and *E* shows, *C*, replotted from Fig. 2*A*). The value of this constant increases with increasing speed of the applied

Fig. 4 shows the intracavity intensity for fixed positive detuning *B*, associated with *B* are significantly detuned, and hence do not contribute. Accordingly, at the threshold value *y* direction. Reflection of the higher-momentum components at the anharmonic trap edges and collisions yields a rapid broadening of the momentum distribution (arrow 4).

## Conclusions

We have studied quench dynamics in the open Dicke model emulated by strongly coupling an atomic BEC to an optical cavity providing an extremely narrow bandwidth. Our experiment exhibits a uniquely controlled paradigm of nonequilibrium many-body dynamics in the presence of dissipation, which appears ideal for quantitative confrontations with theory also beyond mean-field approximations. We hope that this work will stimulate new theoretical efforts to better understand the connection between nonlinear dynamics and statistical mechanics in open many-body systems.

## Acknowledgments

We are grateful to Michael Thorwart, Reza Bakhtiari, Duncan O’Dell, and Helmut Ritsch for useful discussions. This work was partially supported by Deutsche Forschungsgemeinschaft (DFG) under Contracts DFG-SFB 925 and DFG-GrK 1355.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: hemmerich{at}physnet.uni-hamburg.de.

Author contributions: A.H. designed research; J.K., H.K., and M.W. performed research; L.M. and A.H. contributed new reagents/analytic tools; J.K., H.K., M.W., and A.H. analyzed data; and A.H. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1417132112/-/DCSupplemental.

Freely available online through the PNAS open access option.

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