Observation of a Dissipation-Induced Classical to Quantum Transition

We report here the experimental observation of a dynamical quantum phase transition in a strongly interacting open photonic system. The system studied, comprising a Jaynes-Cummings dimer realized on a superconducting circuit platform, exhibits a dissipation driven localization transition. Signatures of the transition in the homodyne signal and photon number reveal this transition to be from a regime of classical oscillations into a macroscopically self-trapped state manifesting revivals, a fundamentally quantum phenomenon. This experiment also demonstrates a small-scale realization of a new class of quantum simulator, whose well controlled coherent and dissipative dynamics is suited to the study of quantum many-body phenomena out of equilibrium.

An understanding of the physics of systems far from equilibrium [1] encompasses deep issues of fundamental importance such as dissipation, decoherence, emergence of classicality from intrinsically quantum systems [2], symmetry breaking and bifurcations, and how equilibrium is itself established [3][4][5][6]. Unraveling this intricate physics is essential to making sense of the world around us, which is fundamentally nonequilibrium and yet displays complex emergent structure. Much of the important recent progress in experimental condensed matter physics has explored the equilibrium regime of strongly correlated synthetic matter (e.g. ultra-cold atoms in optical lattices [7]), but it has been a long-standing goal to understand what new phenomena may arise as these systems are pushed away from equilibrium. With the rapid technological advances in solid state quantum optics [8,9], it is now becoming possible to experimentally study strongly correlated photons, and to build model systems whose open nature gives rise to rich emergent behavior. Interaction with an environment has been argued to provide a mechanism for the emergence of classical behavior [2] from a quantum system. It is also possible, as our work explicitly demonstrates, that dissipation into an environment can qualitatively change this picture, where initially classical dynamics crosses over into one which is fundamentally quantum in nature.
In this experiment we explore a localization transition in a dissipative photonic system [22] realized in the circuit quantum electrodynamics (cQED) architecture [8,9], a solid state realization of cavity QED [23]. As a system supporting phase-coherent photonic states and controlled nonlinearity (tunable in situ on nanosecond timescales) reaching well into the strong-coupling regime even at the single photon level, it opens up the possibility of experimental condensed matter physics with strongly correlated photons. The flexibility in engineering model Hamiltonians and environmental couplings makes it an exemplary candidate for carrying out certain classes of quantum simulations [24,25] of important but difficult to study problems [26][27][28][29][30][31]. The dynamics of polaritons in driven dissipative Jaynes-Cummings chains have been studied theoretically, where a transition from classical to non-classical steady state fields, with varying interaction, tunneling and drive strengths, observable in the density-density correlation functions, have been suggested [32,33] [34].
The physics of a single qubit coupled to a superconducting microwave resonator is well described by the Jaynes-Cummings Hamiltonian (we choose units where 2π = 1) with ν c (ν a ) the bare cavity (qubit) frequency and g the qubit-cavity coupling rate,â,â † representing the photon annihilation and creation operators, andσ ± the Pauli pseudo-spin operators. The photon-qubit interaction induces an anharmonicity in the spectrum of the Jaynes-Cummings Hamiltonian which leads to an effective on-site repulsion for photons [30]. Multiple Jaynes-Cummings sites can be coupled to form a lattice with various symmetries and topologies [26,27,[35][36][37]. Here we study the smallest nontrivial chain, coupling a pair of identical Jaynes-Cummings sites through a photon hopping term (with rate J, and subscript s = L/R specifying the left and right sites) to form a dimer [22] Interaction with the environment is described through a Markovian Lindblad master equation governing the dynamics of the reduced density matrix of the dimer where the Liouvillian super-operator L[Ô] = 2ÔρÔ † −Ô †Ôρ −ρÔ †Ô describes the cavity photon and qubit relaxation rates at κ and γ respectively. Dephasing for our choice of qubits (transmons) can be made much weaker than the above two channels [38,39], and hence is ignored in our theoretical description.
We first discuss the semiclassical dynamics of the dimer in the absence of dissipation (κ = γ = 0). This can be done via the Heisenberg equations of motion and fully factorizing expectation values of spin-photon operator products, yielding a set of eight coupled differential equations for expectation values of the qubit and cavity field operators. A useful representation is in terms of real and imaginary parts of the cavity field R s = Re â s , I s = Im â s , with the angles parametrizing the spin direction n on the qubit Bloch sphere, n s = (sin(θ s ) cos(φ s ), sin(θ s ) sin(φ s ), cos(θ s )), these equations arė for the dynamics of the cavity (s denotes the cavity opposite to s), anḋ for the qubits. In writing these equations we have assumed the qubits to be resonant with the respective cavity modes they are coupled to (ν a = ν c ), and work in the rotating frame. We define the photon number on the left and right as N L/R = â † L/Râ L/R , the total photon number as their sum N = N L + N R , and the photon imbalance as Z = (N L − N R )/N . For special choices of initial conditions, the dynamics can be restricted to certain sub-manifolds of the phase space.
One possible choice, I 1 = R 2 = 0 and φ 1 = π/2, φ 2 = 0, leads to a set of of four coupled equations, which preserve this choice. This sub-manifold contains the dynamics corresponding to an initial condition with perfect imbalance (e.g. Z = 1 for R 1 = √ N and I 2 = 0 at t = 0).
In the absence of qubit-cavity interaction (g = 0), the reduced set of equations can be solved exactly, giving rise to harmonic coherent Josephson oscillations of the imbalance at frequency ν J = 2J. With increasing coupling g, the oscillations become anharmonic. Solving the system of differential equations numerically (subject to the initial condition with Z = 1) shows that at a classical critical coupling [40] the oscillation period diverges, exhibiting critical slowing down, and resulting in a sharp crossover between two qualitatively different regimes of classical dynamical behavior [22], signaling a dynamical phase transition. For couplings beyond the critical value, the system localizes, with the initial photons trapped nearly entirely on a single site, spontaneously breaking the left/right symmetry. As the parameters g and J are fixed for a particular device, it is helpful to recast the problem in terms of a corresponding classical critical photon number N cl c ≈ 0.13 (g/J) 2 for a given g/J. In the classical analysis, a dimer initialized with a photon number N < N cl c is expected to remain in the localized regime (noting also that the numerical prefactor determining the critical photon number is itself somewhat sensitive to the initial state.) We now discuss the full quantum dynamics of the dimer in the absence of dissipation (γ = κ = 0). High quality microwave generators acting as classical coherent sources prepare coherent states having nonzero homodyne voltages, making it possible to monitor the system by observing the homodyne quadraturesÎ = (1/2)(â +â † ) andQ = (i/2)(â † −â) (throughout this paper we define the homodyne signal as ξ = Î 2 + Q 2 , whereas the photon number is arrived at by averaging after squaring the individual quadratures, i.e. Î 2 +Q 2 . Note that the variables appearing in the classical equations of motion (4) and (5) are the expectation values of the quadrature operators). In the the limit g → 0 with finite J, initializing the system with a coherent state leads to oscillations of coherent states between the two cavities with a fixed phase difference of π/2. The oscillations here closely match the expected classical behavior of two coupled oscillators. Keeping g finite and taking J → 0, the two Jaynes-Cummings sites decouple, leading to the well-known resonant collapse and revival phenomenon for a coherent state interacting with a single qubit [23]. From the point of view of the cavity, collapse and revival is a manifestation of the formation of a Schrödinger cat state, as each component of the cat state accumulates a different phase due to the interaction with the qubit [41,42]. The use of coherent states emphasizes the stark contrast between the two dynamical regimes -one characterized by classical oscillations and a second by the spontaneous formation of the quintessential macroscopic quantum mechanical state, the Schrödinger cat, displaying collapse and quantum revivals. These two regimes are demarcated by a dynamical quantum phase transition, with the localization a manifestation of macroscopic quantum self-trapping [16]. We use here the term dynamical quantum phase transition to describe a situation where a qualitative change occurs in the properties of the excited states as a function of a Hamiltonian parameter (here g/J), instead of the ground state as in generic quantum phase transitions. The consequence of such a structural change in excited many-body states is reflected in the dynamics of appropriate observables after a quantum quench.
Inclusion of quantum fluctuations results in a renormalization of the critical coupling to its quantum value g qu c (and likewise for the critical number to N qu c ). In figure (1), we show the numerically calculated quantum dynamics of the homodyne signal ξ for initialization of the left cavity with a coherent state of the photon field of varying initial photon numbers (the qubits start out in the ground state and the right cavity in the vacuum state). We note that for the homodyne signal, while the delocalized regime is characterized by harmonic Josephson oscillations at frequency ν J = 2J as for the imbalance Z, the localized regime is marked by fast collapse-revival oscillations the period of which scales as t r = √ N /g. In the localized regime, the tunneling is dynamically suppressed and the dimer behaves like two uncoupled Jaynes-Cummings sites. The transition region around N qu c displays multi-scale oscillations. At very small photon numbers, we find two further regimes characterized by the reappearance of tunneling and secondary revivals. The richness of the quantum dynamics in the lower part of the figure is due to the finite nature of the system, namely small N and isolation from the environment. fluctuations as a function of g, subject to the initialization described above. With increasing N , the transition becomes sharper and appears to asymptote at a g qu c that is smaller than the classical value g cl c . The precise value of the renormalization of the critical coupling, g qu c /g cl c , depends on the initial quantum state. The crossover region is dominated by large quantum fluctuations and hence is not amenable to a simple mean field description. A natural question to ask is what asymptotic limit gives the semiclassical result described by a sharp transition at g cl c . Our simulations with larger qubit spin S (not shown here) indicate that the appropriate semiclassical limit is (S, N ) → ∞.
The above arguments apply however to the conservative case for which the dimer is isolated and the dynamics conserves the total excitation numberN T = s=L,Rσ + sσ − s +â † sâ s . We describe below a dynamical phase transition that is of a different nature and is particular to the dimer connected to transmission lines, as studied in our experimental setup. The dynamics of such an open Jaynes-Cummings dimer described by the Master equation (3) does not conserve the total excitation number. As a consequence of this, the photon number decays exponentially and a system initially prepared in the delocalized regime with N i ≡ N (t = 0) > N qu c will at a finite time cross the phase boundary and localize, breaking the left/right symmetry, as predicted in [22]. We note that this is distinct from the scenario described above where the transition occurs as a function of parameters g/J in a system that conserves the number of excitations. This transition also differs from nonequilibrium dynamical transitions in the steady state, e.g. when a drive parameter is varied [43][44][45][46][47]. Interestingly, dissipation drives the system from classical behavior to quantum behavior, contrary to the standard intuition that dissipation always renders systems more classical (for previous work on a quantum to classical transition in a circuit QED realization of single site Jaynes-Cummings physics in the presence of an effective temperature, see [48]). The transition demonstrated in this work stands in sharp contrast to atomic and polaritonic BECs, for which the low-density dynamics is linear [16,17,49], and where dissipation drives the system into a delocalized classical state [21] [50].
Our experimental cQED realization of the Jaynes-Cummings dimer is presented in figures (3a,b). Each resonator of frequency ν c = 6.34 GHz and linewidth κ = 225 KHz is individually coupled to a transmon qubit [38,39] with strength g = 190 MHz, providing a strong effective photon-photon interaction. A coupling capacitor allows photon hopping at rate J = 8.7 MHz.
These parameters place the classical critical photon number at N cl c ≈ 62, and enable the observation of many periods of Josephson oscillations (J κ). Crucially, at fixed mean initial photon number in the localized phase, there exists an upper bound for κ beyond which the averaged revival signal is lost, and the control afforded over dissipation in this architecture allowed us to place κ well below this bound, allowing for a good resolution of the quantum revival oscillations [23].
The device is operated in both the linear and nonlinear regimes, tuned via external flux lines V L,R . To initialize the system (figure (3c)), flux bias pulses shift both qubits far out of resonance, removing photon-photon interactions and allowing efficient population of the linear dimer modes when driven by a coherent microwave tone V drive (t) at frequency ν c modulated by a sinusoid of frequency J. Once initialization is complete, and after a variable time delay τ , the nonlinearity is reintroduced by flux biasing the qubits into resonance (this point is our origin of time t = 0). The delay allows arranging any desired imbalance (and hence oscillation phase) at the beginning of the experiment. Here the imbalance oscillations cover the full range [20,21].
Calibrating the flux pulses requires locating bias points leading to minimal photon-photon interactions (this corresponds to the minimum of the qubit energy which gives the smallest resonator Lamb shift) for preparation, as well as resonance, where nonlinearity is largest. The low-lying spectra at these bias points is presented in figure (4), together with the associated single-photon nonlinearities. We developed a characterization technique useful for systems with low dissipation that relies on the Jaynes-Cummings nonlinearity, which leads to bistability with a sharp transition to a bright state as an applied continuous microwave tone is swept in power [51][52][53]. The threshold for this transition (above which the bright state behaves linearly) is sensitive to the frequency difference between the uncoupled mode being monitored and the nearest low energy polariton mode, a useful proxy for the strength of the induced nonlinearity.
Such a mapping of the two-dimensional qubit flux space identified the double minimum and resonance points (see supplementary material).
Dynamics was observed by monitoring photons escaping one of the cavities. After amplification, the signal was mixed down with a local oscillator at ν c to produce theÎ andQ quadratures, which were each sampled at 1 Gs/s. Ensemble averaging over many trials (typically 10 8 ) pro-duced the homodyne signal and photon number (defined previously in terms of the individual quadratures).
If initialized with N i < N c (N c is taken to be the critical photon number observed in the experiment) the system localizes as soon as interactions are introduced, clearly demonstrated by strong collapse and revival of the homodyne signal [54,55]. which play a more significant role with the qubits in resonance, and satisfies the condition for strong single photon nonlinearity g κ , a regime not accessible to current exciton-polariton BEC's [21]. The exponential decay of the oscillations later gives way to a super-exponential drop in homodyne signal, a signature of the crossover from delocalized to localized behavior: photon escape is a stochastic process, and for a given trial the photon number falls below N c at a random time, with an average time dependent on the initial photon number. When approaching this point, the Josephson oscillations become nonlinear, exhibiting a critical slowing down [22].  upwards. This is reflected in the buildup of a finite imbalance in the region where the classical analysis predicts no net imbalance, which is therefore a quantum localized regime. We also observe that as the number of excitations in the system are increased, the transition gets sharper, suggesting the thermodynamic limit for this spatially finite system is given by the limit of large excitation number.  When driving the right cavity, strong reflections during V drive give rise to signal distortion during the dynamics (visible in the right panel) which are mitigated by using a microwave switch.  As the tunneling is less sensitive to dissipative effects, the tendency is for dissipation to linearize the oscillations.
The early overshoot in the homodyne signal results from transient behavior in the initial dynamics of the system.

SUPPLEMENTARY MATERIALS
Characterizing the Jaynes-Cummings Dimer Characterizing the system involves locating the two qubit flux bias points required for (1) efficient initialization and (2) strong photonphoton interactions during dynamics. Since both qubits are individually tunable the system is characterized by two independent voltages V L and V R . The conventional method for mapping out circuit-QED systems is to look at the frequency response in transmission at low drive power as a function of flux bias voltages. This works because the low-lying polariton mode frequencies are sensitive to qubit detunings, and allow direct measurement of the spectra.
Our early physical realizations of dimers with relatively high dissipation rates (κ ≈ 10 MHz) were successfully characterized in this manner. Figure  The bistability due to the Jaynes-Cummings nonlinearity can be used to probe the spectrum quickly and with large SNR. The technique is similar to the high fidelity qubit state readout technique of Reed et al. [53] (see also [51,52]). In their experiment, a difference in cavity frequency when the qubit is in the ground or excited state (ν ≈ ν c ± g 2 ∆ σ z , where ∆ is the detuning of the qubit from the cavity) creates a qubit state dependent threshold power, which is the drive power required to enter the bright state when the cavity is driven by a pulse at the bare resonator frequency. Setting a pulse power between the two threshold powers leads to high distinguishability between the ground and excited state of the qubit.
For the dimer, we take advantage of the fact that the threshold behavior is sensitively dependent on the proximity of low-lying polariton modes to the linear dimer modes (where qubits are decoupled but the resonator coupling leads to a symmetric (ν S ) and an antisymmetric (ν A ) mode, where ν S,A = ν c ± J, and the lower energy mode is the symmetric one). Transmission through the dimer is monitored while a continuous microwave drive (ν d = ν S,A ) is swept in power. Very In the actual experiment, as the region of the cross-over is approached in a single trial, the oscillations exhibit a critical slowing down, which appears as a randomization of the relative oscillation phase across trials, and hence an averaging (across all trials in an ensemble) to zero of the oscillations. Additionally, when an individual trial has localized, the quantum collapse and revival leads to a rapid disappearance of the homodyne signal. When further account is taken, within the toy model, of the critical slowing down of the oscillation frequency, the phase boundary is shifted toward a higher critical photon number.

Driven Dynamics
In nearly all experiments shown in this paper the initialization procedure involves a high-power drive applied when the qubits are far detuned. This ensures that the system is well above the Jaynes-Cummings threshold for linear system behavior during initialization. It is also important to note that the length of the drive pulse (typically 3/J) is also short compared to the time it takes for the system to reach a steady state where drive and dissipation balance.
Keeping the drive short ensures the initial photon number is a linear function of drive power, which is necessary for the calibration using the localized revival time behavior to be applicable at higher drive powers (see the discussion of figure (3a) in the main text). Unfortunately such short drive pulses limited the maximum initial photon number to about 500. To linearly delay the dissipation-induced transition it is necessary to exponentially increase N i , which was accomplished using a much longer initialization procedure in figure (4b) in the main text (11.5µs or 100/J).
Figure (S6) shows the same data as in figure (4b) but includes the driven dynamics observed during the initialization period before photon-photon interactions were turned on by tuning the qubits into resonance. For drive powers above the Jaynes-Cummings threshold (-5 dBm) the system behaves as expected: super-exponential decay sets in at a time logarithmically dependent on the initial photon number. For lower powers, however, the driven dynamics show some interesting features due to the finite nonlinearity of the dispersive spectrum at low photon numbers. Josephson oscillations that build up over the first few microseconds of drive disappear, followed by a reappearance of the homodyne signal microseconds later. The full account of the driven-dissipative dynamics is beyond the scope of this work, and will require further study.

Simulation of Jaynes-Cummings Lattices on Classical Computers
We briefly comment here on the difficulty of simulating the physics of Hamiltonians involving Hilbert spaces built as tensor products of many subsystems. We recall here the dimer Hamiltonian from the main body of the paperĤ with single site Jaynes-Cummings Hamiltonian beinĝ The dimension of the Hilbert space H necessary for simulating the Hamiltonian (7)