Dissipative Dicke Phase Transition

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

Ferdinand Brennecke, Rafael Mottl, Kristian Baumann1, Renate Landig, Tobias Donner2, and Tilman Esslinger

Institute for Quantum Electronics, ETH Zürich, CH-8093 Zürich, Switzerland

Edited by Peter Zoller, University of Innsbruck, Innsbruck, Austria, and approved June 4, 2013 (received for review April 13, 2013) We experimentally study the influence of dissipation on the driven strong-enough coupling, this causes a spatial self-organization of Dicke quantum phase transition, realized by coupling external the atoms on a wavelength-periodic checkerboard pattern, which degrees of freedom of a Bose–Einstein condensate to the light is a realization of the driven-dissipative Dicke phase transition field of a high-finesse optical cavity. The cavity provides a natural (21, 22). The fluctuations triggering the phase transition are dissipation channel, which gives rise to vacuum-induced fluctua- atomic density fluctuations, and they are generated by long-range tions and allows us to observe density fluctuations of the gas in atom–atom interactions that are mediated by exchange of cavity real-time. We monitor the divergence of these fluctuations over photons (23). In the presence of cavity decay, vacuum fluctua- two orders of magnitude while approaching the phase transition, tions enter the cavity, interfere with the coherent pump laser fi and observe a behavior that deviates signi cantly from that field, and drive the system to a steady state with increased density fl expected for a closed system. A correlation analysis of the uctu- fluctuations (24, 25). In turn, the cavity decay offers natural ac- ations reveals the diverging time scale of the atomic dynamics and cess to the system properties via the light field leaking out of the allows us to extract a damping rate for the external degree of cavity, which allows us to measure the density fluctuations in freedom of the atoms. We find good agreement with our theoret- real-time (13). Except for the natural quantum backaction of this ical model including dissipation via both the cavity field and the atomic field. Using a dissipation channel to nondestructively gain continuous measurement process (26), the system remains un- information about a quantum many-body system provides a perturbed by our observation. PHYSICS unique path to study the physics of driven-dissipative systems. System Description

driven-dissipative phase transitions | critical behavior | Dicke model | Hamiltonian Dynamics. As described in our previous work (21, 23), fi quantum gas | cavity QED we place a BEC of N atoms inside an ultrahigh- nesse optical cavity and pump the atoms transversally with a far-detuned standing-wave laser field (Fig. 1A). The closed-system dynamics xperimental progress in the creation, manipulation, and is described by the Dicke model (21, 22) (SI Appendix), Eprobing of atomic quantum gases has made it possible to study highly controlled many-body systems and to access their Zλ ^ ^†^ ^ 2 ^ ^† ^ phase transitions. This unique approach to quantum many-body H = Zωa a + Zω0Jz + pffiffiffiffi a + a Jx; [1] physics has substantiated the notion of quantum simulation for N key models of condensed matter physics (1, 2). There has been ω ω increasing interest in generalizing such an approach to non- where denotes the detuning between pump laser frequency p and dispersively shifted cavity resonance frequency ωc, and Z is equilibrium zero-temperature or quantum phase transitions in ’ π driven-dissipative systems (3), as occurring in condensed matter Planck s constant divided by 2 . The annihilation operator of a cavity photon in a frame rotating at ω is given by a^. The systems coupled to light (4, 5) or in open electronic systems (6, p atomic dynamics is captured in a two-mode description, consist- 7). Among the most tantalizing questions is how vacuum fluctua- ing of the macroscopically populated zero-momentum mode ψ tions from the environment influence the critical behavior at a 0 of the BEC, and an excited momentum mode ψ , carrying in a phase transition via quantum backaction. Related to this question 1 symmetric superposition one photon momentum along the ±x is whether driven-dissipative phase transitions give rise to new direction and one along the ±z direction; this defines an effec- universal behavior, and under which conditions they exhibit clas- tive two-level system with energy splitting Zω = Z2k2=m, where k sical critical behavior with an effective temperature (8–12). 0 denotes the optical wave vector and m the atomic mass. The Coupling quantum gases to the field of an optical cavity is atomic ensemble of N such two-level systems can be described a particularly promising approach to realize a driven-dissipative ^ ^ ^ ^ by collective spin operators Jx, Jy,andJz. The expectation value hJxi quantum many-body system with a well-understood and con- measures the checkerboard density modulation that results from trolled dissipation channel. A further advantage of this scheme is the interference between coherent populations of the two mat- that the dissipation channel of the cavity mode can be directly ter wave modes and can be identified as order parameterpffiffiffi of used to investigate the system in a nondestructive way via the the phase transition. The coupling strength λ ∝ P between leaking cavity field (13). Combining the experimental setting of fi – atomic motion and light eld can be experimentally controlled cavity quantum electrodynamics with that of quantum gases (14 via the power P of the transverse pump field, and represents the 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 fl fl Author contributions: F.B., T.D., and T.E. designed research; F.B., R.M., K.B., and T.D. study the in uence of cavity dissipation on the uctuation performed research; F.B., R.M., and T.D. analyzed data; and F.B., R.M., K.B., R.L., T.D., spectrum at the Dicke phase transition by connecting these and T.E. wrote the paper. approaches. We nondestructively observe diverging fluctuations The authors declare no conflict of interest. of the order parameter when approaching the critical point, and This article is a PNAS Direct Submission. fi nd a distinct difference with respect to predictions for the 1Present address: Departments of Applied Physics and Physics, and E. L. Ginzton Labora- closed (i.e., nondissipative) system. tory, Stanford University, Stanford, CA 94305. In our experimental system, density wave excitations in a 2To whom correspondence should be addressed. E-mail: [email protected]. – Bose Einstein condensate (BEC) are coupled via a coherent This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. laser field to the mode of a standing-wave optical cavity. For 1073/pnas.1306993110/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1306993110 PNAS Early Edition | 1of5 Downloaded by guest on September 25, 2021 D E † 2 λ2 ^ ^ A a rate ω. Toward λcr, the variance b+b of the resulting density ﬂuctuations diverges,D whereasE the gas still does not show ^ ^† a density modulation b + b = 0 .

Dissipative Dynamics. In the case of the open system, the tiny population of states with na ≠ 0 becomes important. As this decays via cavity dissipation, the ladder of states with odd parity ðj ; i; j ; i; j ; i; j ; i; ...Þ B 1 0 0 1 1 2 0 3 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 ha^ aî has been predicted to diverge at λcr with a critical exponent of 1.0 compared with the closed system exponent of 0.5 (24, 25). The κ = λ2 · κ depletion of the ground state takes place at rate eff ω2 + κ2 , where κ is the decay rate of the cavity field (22). In the experiment, κeff can be tuned and we choose ω ≈ 8κ, such that the rate Fig. 1. (A) Experimental scheme: A transverse pump field (red) couples an of decay processes is almost an order of magnitude below the excited momentum mode of a BEC (blue) to a cavity mode via collective light long-range interaction rate λ2=ω. scattering at rate λ. The cavity provides a loss channel for the system through The observable in our experiment is light leaking out of the which photons can escape. Density fluctuations are inferred from the cavity. Because for κ ω0 the cavity field adiabatically follows detected cavity output field. (B) Level scheme of the system after elimination fi of the electronically excited atomic states. The ground state of the closed the atomic motion, the cavity output eld provides a sensitive tool to monitor the order parameter and its fluctuations in real coupled system is given by a coherent superposition of states jna; nbi with time (SI Appendix) (13). Checkerboard density fluctuations with even parity (black symbols). Here, na is the number of intracavity photons D E † 2 and nb is the number of momentum excitations. Decay processes drive the ^+^ fi system to a steady state, which also includes an incoherent population of variance b b induce a nite incoherent population of the À Á D E states with odd parity (gray symbols), caused by either the decay of a cavity † ω λ 2 ^ ^† 2 κ γ cavity field according to a^ a^ = 1 0 b+b . Cavity de- photon at rate 2 or the decay of a momentum excitation at rate 2 . The 4 ω λcr depicted level scheme is restricted to n ≤ 1. a cay amounts to a continuous measurement of the intracavity light field and causes, due to inherent matter-light entanglement, collective two-photon Rabi frequency of the underlying scattering a quantum backaction upon the atomic system (26). The role of process between pump and cavity fipeldffiffiffiffiffiffiffi (Fig. 1A). When λ reaches the photons leaking out of the cavity is thus twofold: they drive ωω λ = 0 1 the system to a steady state of enhanced fluctuations and reveal the critical coupling strength cr 2 ,Hamiltonian gives rise to a second-order quantum phase transition (27) toward a phase real-time information about the total density fluctuations. ^ characterized by a nonzero order parameter hJxi ≠ 0, and a coherent cavity field, haî ≠ 0. At this phase transition, a discrete Results ^ Z2 symmetry is broken, resulting in the coherent cavity field hai Data Acquisition. Using this concept, we experimentally observe oscillating either in or out of phase with the pump field (28). density fluctuations of the atomic ensemble in the normal phase Below the critical point, the system is in the normal phase, while approaching the phase transition. We prepare the system ^ 5 87 hJxi = haî = 0, where only fluctuations of the order parameter, with N = 1:6ð2Þ · 10 Rb atoms at an intermediate coupling of h^2i ≠ fi h^†î ≠ ðλ=λ Þ2 ≈ : ω = π · : ð Þ Jx 0, give rise to an incoherent cavity eld with a a 0, cr 0 55 and at a detuning of 2 10 0 5 MHz. Then, and the relative pump-cavity time phase is undefined. the transverse pump-laser power is linearly increased within In the thermodynamic limit, the fluctuations of the order a data acquisition time of 0.8 s to a value slightly beyond the parameter in the normal phase can be described with bosonic critical point. For our parameters, ω = 2π · 8:3ð2ÞkHz and ^† ^ 0 creation and annihilation operators b and b according to κ = 2π · 1:25ð5ÞMHz (23), the rate κ at which the steady state is pffiffiffiffi . eff ^ ^† π · λ = λ ^J = N b + b 2(SI Appendix). The interaction term in Eq. 1 approached is 2 1 kHz for cr (22). We can therefore as- x sume the system to be in steady state throughout the measure- † ^ ^† then becomes Zλ a^ + a^ b + b , and couples the bare states ment. Fig. 2 Inset displays the data of a single experimental run, where we monitor the stream of photons leaking out of the cavity jn ; n i under parity conservation of the total number of exci- a b with a single-photon counting module. From the photon count tations na + nb. Here, na is the number of photons stored in the rate r we deduce the intracavity photon number n = ðr − rbÞ=2κη, cavity and nb is the number of excitations in the momentum taking into account the measured total detection efficiency of η = ψ mode 1 (Fig. 1). The ground state of the closed, coupled system 5(1)% and the independently calibrated background count rate is a two-mode squeezed state (29, 30) with admixtures of the r . We observe a progressively increasing photon count rate with ðj ; i; j ; i; j ; i; ...Þ ω ω b even parity states only 0 0 1 1 0 2 . For 0, the increasing transverse pump laser power, until a steep rise marks cavity is almost only virtually populated, i.e., the admixture of the transition point to the ordered phase. The exact position of ≠ ω =ω fl states with na 0 is suppressed by 0 . The quantum uctua- the transition depends on the total number of atoms, which tions of the Hamiltonian system then correspond dominantly to fluctuates by 10% between repeated experimental runs. There- ψ pairs of atoms in the excited momentum mode 1; they are fore, we define a threshold for the count rate to detect the created and annihilated by quasi-resonant scattering of a pump transition point (SI Appendix), which allows us to convert the photon into the cavity mode and back into the pump field at time axis into linearly increasing coupling.

2of5 | www.pnas.org/cgi/doi/10.1073/pnas.1306993110 Brennecke et al. Downloaded by guest on September 25, 2021 measurement resolution, however, the system adiabatically fol- 2 lows the steady state, because the rate of change d=dtðλ=λcrÞ is only a few hertz (28). We attribute the damping of the oscillations in gð2ÞðτÞ 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 gð2ÞðτÞ cannot be explained by a finite admixture of the cavity field in the steady state, because ω exceeds ω0 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 α = haî, 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 co- Fig. 2. Mean intracavity photon number n (red symbols) as a function of herent and incoherent cavity field components then causes the coupling. Circles (crosses) indicate data in the normal (ordered) phase; the observed overperiod in the correlation function. error bars display the statistical error. The calculated expectations for the closed system are shown as a solid black line. Our open-system description Quantum Langevin Description. To quantitatively describe our (solid red line) includes cavity field fluctuations due to the decay of photons and momentum excitations (gray dashed-dotted line) and the finite temper- observations, we developed a theoretical model based on cou- ature of the BEC (gray dashed line), as well as a symmetry breaking coherent pled quantum Langevin equations (33) capturing the dynamics cavity field (gray dotted line). We also show the calculated fluctuations if the of the driven-dissipative system (SI Appendix). Our model ex- atomic damping rate γ would vanish (black dashed line). (Inset) The raw data plicitly takes into account the dissipation of the cavity field at of a single run (red line) is displayed together with the measured transverse rate κ, and a dissipation channel for excitations in the atomic PHYSICS pump power (gray dashed line) as a function of time. The sudden increase in momentum mode ψ . For simplicity, this dissipation channel is the photon count-rate clearly marks the transition point. 1

Mean Intracavity Photon Number. We average the signal of 372 experimental runs and observe the divergence of the intracavity photon number n over three orders of magnitude, ending in A 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, B 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 fol- C lows the atomic dynamics, its second-order correlation function ð Þ † † g 2 ðτÞ ∝ ha^ ðτÞa^ ð0Þa^ð0Þa^ðτÞi is linked to the temporalD correla-E fl ^2ðτÞ^2ð Þ tion function of the order parameter uctuations Jx Jx 0 . 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 agreesqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi with the excitation energy of 2 the coupled system, Zωs = Zω0 1 − ðλ=λcrÞ , which softens with Fig. 3. Temporal correlations of the cavity output. (A) Color plot of the increasing coupling and tends toward zero at the critical point; measured second-order correlation function g2ðτÞ as a function of time τ and ðλ=λ Þ2 fi coupling cr . The correlation time increases with increasing coupling in this shows that the cavity output eld indeed carries information ω fl agreement with the timescale related to the lowest excitation energy s of about the incoherent uctuations of the system, and is consistent the coupled system (solid black line). (B) Correlations g2ðτÞ calculated from with our previous measurement of a mode softening and a di- the full theoretical model with parameters ζ and γ adjusted to match the 2 verging response (23). A vanishing excitation frequency corre- data. The horizontal dashed lines indicate values of ðλ=λcrÞ , along which the sponds to a critical slowing down of the dynamics. Within our data are shown (points) on C, Bottom, together with the full theory.

Brennecke et al. PNAS Early Edition | 3of5 Downloaded by guest on September 25, 2021 from the thermal atomic bath are predicted to contribute because the energy of the relevant mechanical excitation vanishes toward the phase transition (23). fl Density Fluctuations.† We infer the variance of the density uc- ^ ^ 2 tuations hðb+b Þ i in the normal phase by rescaling the intracavity photon number n after subtracting the coherent fraction jαj2, 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 γ ðλ=λ Þ2 Fig. 4. Damping rate (symbols) as a function of coupling cr deduced both magnitude and scaling from the expectations for the closed from the cavity output correlation data. Error bars indicate the statistical system. A linear fit (blue line) to the data results in an exponent error derived from the fit. Open circles indicate the region above 2 of 0.9 (± 0.1). The exponent’s error is dominated by the un- ðλ=λcrÞ = 0:97, where our theoretical model deviates significantly from the γ certainty of the indirectly determined symmetry breaking field. data and values for might be inaccurate. The solid line shows the phe- fi nomenological function used to model the data in Fig. 2. The dashed line Direct experimental access to the symmetry breaking eld would fi shows the vanishing excitation frequency ωs=2π of the system. be provided by a heterodyne detection of the cavity output eld (28). Furthermore, this technique allows for the observation of phase fluctuations of the cavity field and possibly the detection of phenomenologically modeled by a thermal Markovian bath at matter-light entanglement (33). the BEC temperature of 100ð20Þ nK into which excitations ψ γ in the momentum mode 1 decay at a rate (SI Appendix). Due Discussion to the softening excitation frequency ωs, the decay rate γ is taken A scaling of fluctuations with exponent 1.0 was predicted from as a function of the coupling rate λ. Our model further includes open-system calculations in which only cavity dissipation is taken a small symmetry breaking field, which results in a coherent into account (24, 25). The influence of the additional atomic cavity field amplitude α already below the critical point; this is dissipation rate γ on the scaling of the atomic density fluctuations taken into account by renormalizing the order parameter with depends on the precise scaling of this damping rate when a constant offset ζ in Eq. 1 (21). approaching the critical point, which goes beyond the scope of From the solution of the quantum Langevin equations in the this publication. thermodynamic limit we obtain the second-order correlation From a more general perspective, driven systems, coupled via function of the intracavity field in the steady state (SI Appendix). a dissipation channel to a zero-temperature Markovian bath, are The free parameters of our model description (ζ and γ) are expected to resemble classical critical behavior and can then be extracted from fits of the model to the correlation data (Fig. 3). characterized in steady state by an effective temperature that We obtain an order parameter offset ζ = 60ð7Þ at λ = 0, which depends on the considered observable (3, 8, 9, 10, 36). In our corresponds to 0.8‰ of the maximal possible order parameter system, the zero-temperature bath is provided by the optical N/2 and agrees with our earlier investigation of the symmetry vacuum modes outside the cavity. Verifying the fluctuation-dis- breaking field (28). sipation theorem for the order parameter in our system would The extracted damping rate γ is displayed in Fig. 4 as a func- allow us to determine its effective temperature. The theoretical tion 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 ωs (Fig. 4, dashed line) (23), which influences the density of states into which the mo- ψ mentum excitations in mode 1 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 2 to ðλ=λcrÞ ’ 0:97. 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. 3C. 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 domi- Fig. 5. Variance of the checkerboard density fluctuations of the BEC, de- nantly damps out these momentum excitations, such that the duced from the intracavity photon number after subtracting the coherent total fluctuations in the steady state are only moderately en- contribution. For comparison, we show the theory of the closed system (black line), which diverges with a critical exponent of 0.5, and a linear fit hanced with respect to the ground-state fluctuations. Except for 2 (blue line) to the data for ðλ=λcrÞ ≥ 0:9, which results in an exponent of a small region close to the critical point, the dominant contri- 0:9 ± 0:1. We also plot the expected fluctuations for a BEC without coupling bution to the observed fluctuations originates from vacuum input to the cavity field (black dashed line). The horizontal error bars indicate the noise associated with dissipation via the cavity (Fig. 2, gray statistical error, and the vertical error bars result from the uncertainty in the dashed-dotted line). Only close to the critical point, fluctuations subtracted coherent field component (SI Appendix).

4of5 | www.pnas.org/cgi/doi/10.1073/pnas.1306993110 Brennecke et al. Downloaded by guest on September 25, 2021 expectation of a critical exponent of 1.0 (24, 25) is a further in- order parameter. In a similar way, intriguing quantum many-body dication that systems undergoing a driven-dissipative phase states with long-range atom–atom interactions and the inﬂuence transition can be described to be effectively thermalized. How- of dissipation on them can be investigated by, e.g., using multimode ever, answering the question whether cavity dissipation com- cavities, which allow to realize glassy and frustrated states of matter pletely destroys the quantum character of the system, e.g., the (36, 37). Adding classical optical lattices to the system would let the entanglement between atomic and light ﬁelds, remains a chal- energy scale of contact interactions enter the dynamics and should lenge for future experiments (24, 33). allow the exploration of rich phase diagrams (38, 39).

Conclusion and Outlook ACKNOWLEDGMENTS. We acknowledge insightful discussions with I. Carusotto, S. Diehl, P. Domokos, S. Gopalakrishnan, S. Huber, A. Imamoglu, M. Paternostro, We have demonstrated the direct observation of diverging density C. Rama, H. Ritsch, G. Szirmai, and H. Türeci. This work was supported by the ﬂuctuations in a quantum gas undergoing the driven-dissipative European Research Council advanced grant Synthetic Quantum Many-Body Dicke phase transition. This experiment opens a route to study Systems; the European Union, Future and Emerging Technologies (FET-Open) quantum phase transitions in open systems under well-controlled grant Nanodesigning of Atomic and Molecular Quantum Matter; the National Centre of Competence in Research/Quantum Science and Technology; and the conditions. Our method directly uses the cavity dissipation chan- European Science Foundation program Common Perspectives for Cold Atoms, nel to obtain real-time information on the ﬂuctuations of the Semiconductor Polaritons and Nanoscience.

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