## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# Real-time observation of fluctuations at the driven-dissipative Dicke phase transition

Edited by Peter Zoller, University of Innsbruck, Innsbruck, Austria, and approved June 4, 2013 (received for review April 13, 2013)

## Abstract

We experimentally study the influence of dissipation on the driven Dicke quantum phase transition, realized by coupling external degrees of freedom of a Bose–Einstein condensate to the light field of a high-finesse optical cavity. The cavity provides a natural dissipation channel, which gives rise to vacuum-induced fluctuations and allows us to observe density fluctuations of the gas in real-time. We monitor the divergence of these fluctuations over two orders of magnitude while approaching the phase transition, and observe a behavior that deviates significantly from that expected for a closed system. A correlation analysis of the fluctuations reveals the diverging time scale of the atomic dynamics and allows us to extract a damping rate for the external degree of freedom of the atoms. We find good agreement with our theoretical model including dissipation via both the cavity field and the atomic field. Using a dissipation channel to nondestructively gain information about a quantum many-body system provides a unique path to study the physics of driven-dissipative systems.

Experimental progress in the creation, manipulation, and probing of atomic quantum gases has made it possible to study highly controlled many-body systems and to access their phase transitions. This unique approach to quantum many-body physics has substantiated the notion of quantum simulation for key models of condensed matter physics (1, 2). There has been increasing interest in generalizing such an approach to nonequilibrium zero-temperature or quantum phase transitions in driven-dissipative systems (3), as occurring in condensed matter systems coupled to light (4, 5) or in open electronic systems (6, 7). Among the most tantalizing questions is how vacuum fluctuations from the environment influence the critical behavior at a phase transition via quantum backaction. Related to this question is whether driven-dissipative phase transitions give rise to new universal behavior, and under which conditions they exhibit classical critical behavior with an effective temperature (8⇓⇓⇓–12).

Coupling quantum gases to the field of an optical cavity is a particularly promising approach to realize a driven-dissipative quantum many-body system with a well-understood and controlled dissipation channel. A further advantage of this scheme is that the dissipation channel of the cavity mode can be directly used to investigate the system in a nondestructive way via the leaking cavity field (13). Combining the experimental setting of cavity quantum electrodynamics with that of quantum gases (14⇓⇓⇓–18) led to the observation of quantum backaction heating caused by cavity dissipation (19, 20), as well as to the realization of the nonequilibrium Dicke quantum phase transition (21). Here, we study the influence of cavity dissipation on the fluctuation spectrum at the Dicke phase transition by connecting these approaches. We nondestructively observe diverging fluctuations of the order parameter when approaching the critical point, and find a distinct difference with respect to predictions for the closed (i.e., nondissipative) system.

In our experimental system, density wave excitations in a Bose–Einstein condensate (BEC) are coupled via a coherent laser field to the mode of a standing-wave optical cavity. For strong-enough coupling, this causes a spatial self-organization of the atoms on a wavelength-periodic checkerboard pattern, which is a realization of the driven-dissipative Dicke phase transition (21, 22). The fluctuations triggering the phase transition are atomic density fluctuations, and they are generated by long-range atom–atom interactions that are mediated by exchange of cavity photons (23). In the presence of cavity decay, vacuum fluctuations enter the cavity, interfere with the coherent pump laser field, and drive the system to a steady state with increased density fluctuations (24, 25). In turn, the cavity decay offers natural access to the system properties via the light field leaking out of the cavity, which allows us to measure the density fluctuations in real-time (13). Except for the natural quantum backaction of this continuous measurement process (26), the system remains unperturbed by our observation.

## System Description

### Hamiltonian Dynamics.

As described in our previous work (21, 23), we place a BEC of *N* atoms inside an ultrahigh-finesse optical cavity and pump the atoms transversally with a far-detuned standing-wave laser field (Fig. 1*A*). The closed-system dynamics is described by the Dicke model (21, 22) (*SI Appendix*),where *ω* denotes the detuning between pump laser frequency and dispersively shifted cavity resonance frequency , and *ℏ* is Planck’s constant divided by 2π. The annihilation operator of a cavity photon in a frame rotating at is given by . The atomic dynamics is captured in a two-mode description, consisting of the macroscopically populated zero-momentum mode of the BEC, and an excited momentum mode , carrying in a symmetric superposition one photon momentum along the ±*x* direction and one along the ±*z* direction; this defines an effective two-level system with energy splitting , where *k* denotes the optical wave vector and *m* the atomic mass. The atomic ensemble of *N* such two-level systems can be described by collective spin operators , , and . The expectation value measures the checkerboard density modulation that results from the interference between coherent populations of the two matter wave modes and can be identified as order parameter of the phase transition. The coupling strength between atomic motion and light field can be experimentally controlled via the power *P* of the transverse pump field, and represents the collective two-photon Rabi frequency of the underlying scattering process between pump and cavity field (Fig. 1*A*). When *λ* reaches the critical coupling strength , Hamiltonian **1** gives rise to a second-order quantum phase transition (27) toward a phase characterized by a nonzero order parameter , and a coherent cavity field, . At this phase transition, a discrete symmetry is broken, resulting in the coherent cavity field oscillating either in or out of phase with the pump field (28). Below the critical point, the system is in the normal phase, , where only fluctuations of the order parameter, , give rise to an incoherent cavity field with , and the relative pump-cavity time phase is undefined.

In the thermodynamic limit, the fluctuations of the order parameter in the normal phase can be described with bosonic creation and annihilation operators and according to (*SI Appendix*). The interaction term in Eq. **1** then becomes , and couples the bare states under parity conservation of the total number of excitations . Here, is the number of photons stored in the cavity and is the number of excitations in the momentum mode (Fig. 1). The ground state of the closed, coupled system is a two-mode squeezed state (29, 30) with admixtures of the even parity states only . For , the cavity is almost only virtually populated, i.e., the admixture of states with is suppressed by . The quantum fluctuations of the Hamiltonian system then correspond dominantly to pairs of atoms in the excited momentum mode ; they are created and annihilated by quasi-resonant scattering of a pump photon into the cavity mode and back into the pump field at a rate . Toward , the variance of the resulting density fluctuations diverges, whereas the gas still does not show a density modulation .

### Dissipative Dynamics.

In the case of the open system, the tiny population of states with becomes important. As this decays via cavity dissipation, the ladder of states with odd parity is incoherently populated (Fig. 1). The microscopic process corresponds to the loss of a cavity photon at one of the mirrors before the coherent scattering back into the pump beam can be completed. The system will thus leave its ground state and irreversibly evolve into a nonequilibrium steady state with additional density fluctuations and a constant energy flow from the pump laser to the cavity output. The variance of the resulting incoherent cavity population has been predicted to diverge at with a critical exponent of 1.0 compared with the closed system exponent of 0.5 (24, 25). The depletion of the ground state takes place at rate , where *κ* is the decay rate of the cavity field (22). In the experiment, can be tuned and we choose , such that the rate of decay processes is almost an order of magnitude below the long-range interaction rate .

The observable in our experiment is light leaking out of the cavity. Because for the cavity field adiabatically follows the atomic motion, the cavity output field provides a sensitive tool to monitor the order parameter and its fluctuations in real time (*SI Appendix*) (13). Checkerboard density fluctuations with variance induce a finite incoherent population of the cavity field according to . Cavity decay amounts to a continuous measurement of the intracavity light field and causes, due to inherent matter-light entanglement, a quantum backaction upon the atomic system (26). The role of the photons leaking out of the cavity is thus twofold: they drive the system to a steady state of enhanced fluctuations and reveal real-time information about the total density fluctuations.

## Results

### Data Acquisition.

Using this concept, we experimentally observe density fluctuations of the atomic ensemble in the normal phase while approaching the phase transition. We prepare the system with ^{87}Rb atoms at an intermediate coupling of and at a detuning of . Then, the transverse pump-laser power is linearly increased within a data acquisition time of 0.8 s to a value slightly beyond the critical point. For our parameters, and (23), the rate at which the steady state is approached is for (22). We can therefore assume the system to be in steady state throughout the measurement. Fig. 2 *Inset* displays the data of a single experimental run, where we monitor the stream of photons leaking out of the cavity with a single-photon counting module. From the photon count rate *r* we deduce the intracavity photon number , taking into account the measured total detection efficiency of *η* = 5(1)% and the independently calibrated background count rate . We observe a progressively increasing photon count rate with increasing transverse pump laser power, until a steep rise marks the transition point to the ordered phase. The exact position of the transition depends on the total number of atoms, which fluctuates by 10% between repeated experimental runs. Therefore, we define a threshold for the count rate to detect the transition point (*SI Appendix*), which allows us to convert the time axis into linearly increasing coupling.

### Mean Intracavity Photon Number.

We average the signal of 372 experimental runs and observe the divergence of the intracavity photon number over three orders of magnitude, ending in a steep increase after passing the critical point (Fig. 2). We compared the measured intracavity photon number with the cavity field fluctuations expected from the ground state of the closed system (29). Our data clearly shows an enhanced cavity field occupation with respect to the Hamiltonian system (Fig. 2, solid black line), and this is in accordance with the presented picture that cavity decay increases fluctuations. However, the magnitude of the observed fluctuations is well below the theoretical expectation (24, 25) for a cavity decay at rate *κ* (Fig. 2, dashed black line), indicating the presence of a further dissipation channel, which damps out atomic momentum excitations.

### Correlation Analysis.

Additional insight into the fluctuation dynamics and possible dissipative processes can be gained from a correlation analysis of the cavity output field. We calculate the second-order correlation function for all experimental data contributing to Fig. 2. Because the cavity field adiabatically follows the atomic dynamics, its second-order correlation function is linked to the temporal correlation function of the order parameter fluctuations . The evaluated correlations as a function of time and coupling are shown in Fig. 3, together with cuts for specific coupling values. In contrast to a purely coherent cavity output field, which would yield a flat correlation function, we observe enhanced correlations for short times, followed by damped oscillations. The frequency of these oscillations agrees with the excitation energy of the coupled system, , which softens with increasing coupling and tends toward zero at the critical point; this shows that the cavity output field indeed carries information about the incoherent fluctuations of the system, and is consistent with our previous measurement of a mode softening and a diverging response (23). A vanishing excitation frequency corresponds to a critical slowing down of the dynamics. Within our measurement resolution, however, the system adiabatically follows the steady state, because the rate of change is only a few hertz (28).

We attribute the damping of the oscillations in to the decay of atomic momentum excitations; this constitutes an additional dissipation channel caused by collisional and possibly cavity-mediated coupling of momentum excitations to Bogoliubov modes of the BEC (31, 32). The observed decay rate of cannot be explained by a finite admixture of the cavity field in the steady state, because *ω* exceeds by orders of magnitude in our system (24).

The oscillations in the second-order correlation function exhibit an overperiod, which becomes more pronounced toward the critical point; this indicates the presence of a finite coherent cavity field amplitude , which we attribute to the finite cloud size of the BEC and residual scattering of pump light at the edges of the cavity mirrors (21). Interference between the coherent and incoherent cavity field components then causes the observed overperiod in the correlation function.

### Quantum Langevin Description.

To quantitatively describe our observations, we developed a theoretical model based on coupled quantum Langevin equations (33) capturing the dynamics of the driven-dissipative system (*SI Appendix*). Our model explicitly takes into account the dissipation of the cavity field at rate *κ*, and a dissipation channel for excitations in the atomic momentum mode . For simplicity, this dissipation channel is phenomenologically modeled by a thermal Markovian bath at the BEC temperature of into which excitations in the momentum mode decay at a rate *γ* (*SI Appendix*). Due to the softening excitation frequency , the decay rate *γ* is taken as a function of the coupling rate *λ*. Our model further includes a small symmetry breaking field, which results in a coherent cavity field amplitude α already below the critical point; this is taken into account by renormalizing the order parameter with a constant offset *ζ* in Eq. **1** (21).

From the solution of the quantum Langevin equations in the thermodynamic limit we obtain the second-order correlation function of the intracavity field in the steady state (*SI Appendix*). The free parameters of our model description (*ζ* and *γ*) are extracted from fits of the model to the correlation data (Fig. 3). We obtain an order parameter offset at , which corresponds to 0.8‰ of the maximal possible order parameter *N*/2 and agrees with our earlier investigation of the symmetry breaking field (28).

The extracted damping rate *γ* is displayed in Fig. 4 as a function of coupling; it increases with increasing coupling, until it exhibits a cusp at ∼95% of the critical coupling and vanishes toward the critical point. We attribute this behavior mainly to the softening of the excitation frequency (Fig. 4, dashed line) (23), which influences the density of states into which the momentum excitations in mode can decay; at the critical point, this is expected to lead to the absence of damping of the excited momentum mode (31).

Our model describes our data very well for coupling values up to . Above this value, we observe enhanced correlations for small *τ*, which are not captured by the model, as can be seen in the uppermost subpanel of Fig. 3*C*. We believe that in this region technical fluctuations, the dynamical change in the dispersive cavity shift (34), finite-*N* effects (35), and population of higher-order momentum states start to play a role.

Using the extracted atomic damping rate and symmetry breaking field magnitude, we find very good agreement between the observed intracavity photon number and our model (Fig. 2). The inclusion of atomic damping is crucial for the quantitative description. Though cavity decay is expected to lead to a strong increase of the density fluctuations, atomic dissipation dominantly damps out these momentum excitations, such that the total fluctuations in the steady state are only moderately enhanced with respect to the ground-state fluctuations. Except for a small region close to the critical point, the dominant contribution to the observed fluctuations originates from vacuum input noise associated with dissipation via the cavity (Fig. 2, gray dashed-dotted line). Only close to the critical point, fluctuations from the thermal atomic bath are predicted to contribute because the energy of the relevant mechanical excitation vanishes toward the phase transition (23).

### Density Fluctuations.

We infer the variance of the density fluctuations in the normal phase by rescaling the intracavity photon number after subtracting the coherent fraction , which was deduced from the correlation analysis (Fig. 5). Due to the uncertainties in the symmetry breaking field *ζ*, this procedure results in systematical uncertainties of the deduced density fluctuations, which are reflected in the presented error bars. Our data, displayed on a log-log scale, deviates clearly in both magnitude and scaling from the expectations for the closed system. A linear fit (blue line) to the data results in an exponent of 0.9 (± 0.1). The exponent’s error is dominated by the uncertainty of the indirectly determined symmetry breaking field. Direct experimental access to the symmetry breaking field would be provided by a heterodyne detection of the cavity output field (28). Furthermore, this technique allows for the observation of phase fluctuations of the cavity field and possibly the detection of matter-light entanglement (33).

## Discussion

A scaling of fluctuations with exponent 1.0 was predicted from open-system calculations in which only cavity dissipation is taken into account (24, 25). The influence of the additional atomic dissipation rate *γ* on the scaling of the atomic density fluctuations depends on the precise scaling of this damping rate when approaching the critical point, which goes beyond the scope of this publication.

From a more general perspective, driven systems, coupled via a dissipation channel to a zero-temperature Markovian bath, are expected to resemble classical critical behavior and can then be characterized in steady state by an effective temperature that depends on the considered observable (3, 8, 9, 10, 36). In our system, the zero-temperature bath is provided by the optical vacuum modes outside the cavity. Verifying the fluctuation-dissipation theorem for the order parameter in our system would allow us to determine its effective temperature. The theoretical expectation of a critical exponent of 1.0 (24, 25) is a further indication that systems undergoing a driven-dissipative phase transition can be described to be effectively thermalized. However, answering the question whether cavity dissipation completely destroys the quantum character of the system, e.g., the entanglement between atomic and light fields, remains a challenge for future experiments (24, 33).

## Conclusion and Outlook

We have demonstrated the direct observation of diverging density fluctuations in a quantum gas undergoing the driven-dissipative Dicke phase transition. This experiment opens a route to study quantum phase transitions in open systems under well-controlled conditions. Our method directly uses the cavity dissipation channel to obtain real-time information on the fluctuations of the order parameter. In a similar way, intriguing quantum many-body states with long-range atom–atom interactions and the influence of dissipation on them can be investigated by, e.g., using multimode cavities, which allow to realize glassy and frustrated states of matter (36, 37). Adding classical optical lattices to the system would let the energy scale of contact interactions enter the dynamics and should allow the exploration of rich phase diagrams (38, 39).

## Acknowledgments

We acknowledge insightful discussions with I. Carusotto, S. Diehl, P. Domokos, S. Gopalakrishnan, S. Huber, A. Imamoglu, M. Paternostro, C. Rama, H. Ritsch, G. Szirmai, and H. Türeci. This work was supported by the European Research Council advanced grant Synthetic Quantum Many-Body Systems; the European Union, Future and Emerging Technologies (FET-Open) grant Nanodesigning of Atomic and Molecular Quantum Matter; the National Centre of Competence in Research/Quantum Science and Technology; and the European Science Foundation program Common Perspectives for Cold Atoms, Semiconductor Polaritons and Nanoscience.

## Footnotes

↵

^{1}Present address: Departments of Applied Physics and Physics, and E. L. Ginzton Laboratory, Stanford University, Stanford, CA 94305.- ↵
^{2}To whom correspondence should be addressed. E-mail: donner{at}phys.ethz.ch.

Author contributions: F.B., T.D., and T.E. designed research; F.B., R.M., K.B., and T.D. performed research; F.B., R.M., and T.D. analyzed data; and F.B., R.M., K.B., R.L., T.D., and T.E. 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.1306993110/-/DCSupplemental.

## References

- ↵
- ↵
- Lewenstein M,
- Sanpera A,
- Ahufinger V

- ↵
- ↵
- Roumpos G,
- et al.

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Torre EGD,
- Diehl S,
- Lukin MD,
- Sachdev S,
- Strack P

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Wolke M,
- Klinner J,
- Keßler H,
- Hemmerich A

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Mottl R,
- et al.

- ↵
- ↵
- ↵
- Thorne KS

- Braginsky VB,
- Khalili FY

- ↵
- ↵
- ↵
- ↵
- Walls DF,
- Milburn GJ

*Quantum Optics*(Springer, Berlin). - ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵

## Citation Manager Formats

### More Articles of This Classification

### Physical Sciences

### Related Content

- No related articles found.

### Cited by...

- No citing articles found.