Observation of a Dissipation-Induced Classical to Quantum Transition

Here, we report 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.

Popular Summary

Our understanding of how large numbers of particles coexist in nature is based on thermodynamics and the underlying theory of statistical mechanics. The simplest case is when the particles do not interact with each other and when the system is in equilibrium, so that its macroscopically observable state does not change with time. However, the world around us is rarely in equilibrium, and most of the dynamical processes we observe occur far from equilibrium and also involve strong interactions. The study of physical systems in this regime encompasses topics of fundamental importance to science, such as dissipation, quantum decoherence, emergence of classicality from intrinsically quantum systems, symmetry breaking, and even how equilibrium is itself established. We apply tools developed for quantum computing to investigate these subjects.

We realize a system of strongly correlated photons, which, when populated with many photons, exhibits classical Josephson oscillations. A loss of photons from the system into the environment leads to a slowing down of the oscillations; at a critical number of photons, the period of the oscillations is seen to diverge, giving rise to a dynamical quantum phase transition far from equilibrium. In contrast with standard expectations, this transition is into a state displaying quantum behavior, namely, the quantum revivals of Schrödinger cat states. This experiment is the first realization of a dissipative quantum simulator built using standard solid-state fabrication technologies.

Our findings open new directions for future studies of strongly correlated many-body physics using photons, where dissipation is both central and well controlled. We expect that our results will have broad implications in fields such as condensed matter physics.

Figure 1

Numerically simulated phase diagram. Quantum dynamics (without dissipation) of the homodyne signal in left (a) and right (b) cavities, as a function of total photon number (logarithmic vertical axis, with color bar normalized by ${\xi }_1$, the homodyne signal of a coherent state with mean occupation of one photon), for the case where the system has been initialized into the $Z=1$ imbalance state, photons in a perfect coherent state, and both qubits initially in the ground state. The parameters chosen are as given in the text corresponding to the actual device used in the experiment. For photon numbers below the quantum renormalized critical number $N_c^{\mathrm{qu}}$, tunneling is dynamically suppressed and we observe collapse-revival oscillations with period scaling as $t_r=\sqrt{N}/g$, as expected for decoupled single-site Jaynes-Cummings physics [23]. At very small photon numbers, we see an “exiguous” regime where tunneling reappears. We also observe in this localized regime secondary revivals at long times (also present in the unitary evolution of the single-site Jaynes-Cummings model), which would be washed out in the presence of dissipation. The dynamics above the critical photon number displays Josephson-like oscillations with period $t_J=(2J{)}^{-1}$, becoming more linear with increasing photon number. We note that the critical number is marked by the coincidence of two time scales, $t_r=t_J$. As the initial $N$ increases, the revival period grows and ultimately matches that of the Josephson oscillation time scale, yielding the critical photon number $N_c^{\mathrm{qu}}\approx g^2/4J^2\approx 2N_c^{\mathrm{cl}}$, consistent with where the transition is observed in the simulation. Solving the dynamics at the upper limit shown necessitates the solution of the time-dependent Schrödinger equation in a Hilbert space exceeding a dimension of size $10^6$. More details regarding the difficulty of such simulations can be found in the Supplemental Material [45].

Figure 2

Time-averaged imbalance and quantum fluctuations of the imbalance. The time-averaged imbalance for varying $g/g_c^{\mathrm{cl}}$, as well as its quantum fluctuations (mean squared fluctuations), for coherent states of five different mean photon numbers (red, 20; green, 40; blue, 60; magenta, 100; black, 200). Here, the total imbalance $Z_T$ is the difference in the number of excitations on each side (photon plus qubit excitation), normalized by the total excitations $N_T$ in the system. The classical prediction puts the transition at $g/g_c^{\mathrm{cl}}=1$. The quantum fluctuations are largest in the classically delocalized regime, and lead to a renormalization of the expected quantum value of the critical coupling $g_c^{\mathrm{qu}}$ downwards (i.e., requiring smaller coupling to observe localization if the total excitation number is held fixed), and hence relocating the critical photon number $N_c^{\mathrm{qu}}$ 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 is increased, the transition gets sharper, suggesting the thermodynamic limit for this spatially finite system is given by the limit of large excitation number.

Figure 3

Device layout and initialization routine. (a) Schematic diagram of the experiment. The Jaynes-Cummings dimer is composed of two coupled transmission-line resonators each individually coupled to a transmon qubit. Intercavity coupling $J=8.7\mathrm{MHz}$ and cavity-qubit coupling $g=190\mathrm{MHz}$. The resonators are driven and monitored via coupling to external transmission lines. Initialization pulse $V_{\text{drive}}$ can be applied to the left or right cavity to generate classical oscillations in the system, while fast flux pulses $V_{L,R}$ control qubit energies at nanosecond time scales. Right cavity quadratures are monitored via a homodyne measurement. When $V_{\text{drive}}$ is applied to the right cavity, a fast microwave switch is used to block the strong reflected signal. (b) Optical micrograph of the device. (c) Initialization routine pulse waveforms. Fast flux pulses $V_{L,R}$ rapidly detune both qubits to their minimum energies to turn off photon-photon interactions. While qubits are detuned, either the left or right cavity is driven with initialization pulse $V_{\text{drive}}=\sin (2\pi {\nu }_{c}t)\sin (2\pi Jt)\mathrm{\Theta }(-t)\mathrm{\Theta }(t+m/J)$, where $m$ is an integer and $\mathrm{\Theta }$ is the Heaviside step function. Variable delay $\tau$ after the end of $V_{\text{drive}}$ allows the photon-photon interactions to be turned on at any point during the undriven linear oscillations, enabling the preparation of any desired imbalance.

Figure 4

Low-lying dimer spectra and nonlinearities. Spectra are shown for the two flux bias points in the experiment, without (a) and with (b) photon-photon interactions. The first (orange) and second (blue) excitation manifolds are shown along with transitions (black), where transitions from the first to second excitation manifold are the same frequency as the ground to first transitions immediately below. (a) Effective photon-photon interactions can be turned off by tuning qubits to their minimum energies. While detuned, the qubits remain in their ground states and can be ignored, resulting in a simplified spectrum of two cavities in resonance. The $J$ coupling creates two linear hybridized modes with energies ${\nu }_c\pm J$, ideal for generating full linear oscillations [$Z(t)$ oscillates between $\pm 1$]. Modulating $V_{\text{drive}}$ at $J$ generates sidebands resonant with each mode and explicitly sets the phase of the resultant linear oscillations, generating an imbalanced coherent state at $t=0$. (b) Photon-photon interactions are generated by tuning both qubits into resonance with the cavities. At this bias point, the first excitation manifold has four polariton states at energies ${\nu }_c\pm g\pm J/2$. Strong single-photon nonlinearities are apparent, as no transitions to the second manifold match the energies of the first manifold. Because of the form of the Jaynes-Cummings Hamiltonian, all nonlinearities are photon number dependent and lead to linear behavior at high excitation manifolds.

Figure 5

Self-trapped regime. (a) Homodyne signal dynamics of the right cavity as a function of initial photon number and time. A 345-ns ($3/J$) initialization pulse $V_{\text{drive}}$ ending at $t=0$ generates linear oscillations between the left and right cavities while photon-photon interactions are off. The oscillations proceed for a time $\tau$ such that a perfect imbalance ($Z=-1$) is established when interactions are turned on. For $N_i<N_c$, the system is localized and the right cavity displays fast collapse and revival oscillations while the left cavity (not shown) remains empty. Inset: Fitting revival time $t_r$ as a function of drive power to the expected $\sqrt{N}$ dependence allows drive power to be mapped to initial photon number (at $t=0$), as well as the calibration of ${\xi }_1$, the homodyne signal of a coherent state with mean occupation of one photon. (b),(c) Revival time variation as a function of $\tau$ for $V_{\text{drive}}$ applied to the left (left-hand panel) and right (right-hand panel). As $\tau$ is varied, the full range of imbalance is observed. Here, the same initialization scheme as in (a) is used with $N_i=8$, which produces the cleanest collapse and revival signature. When driving the right cavity, strong reflections during $V_{\text{drive}}$ give rise to signal distortion during the dynamics (visible in the right-hand panel) that are mitigated by using a microwave switch.

Figure 6

Dissipation-driven transition and phase diagram. (a) Comparison of homodyne signal when photon-photon interactions are off (red) and on (blue), with $N_i>N_c$. The same initialization pulse $V_{\text{drive}}$ was used as in Fig. 5. With interactions off, the system undergoes oscillations and exponential decay at rate $\kappa$, which can be observed for several microseconds. Significantly, the presence of interactions causes superexponential decay of the homodyne signal as $N$ approaches $N_c$, a signature of crossover into the localized regime. (b) The time of onset of superexponential decay $t_c$ shifts with initial photon number, as shown here for initialization pulses with varying drive power lasting $11.5\mu \mathrm{s}$ ($100/J$). The use of a long initialization pulse makes it possible to drive the system to very large initial photon numbers (top to bottom: $N_i\approx 12000$, 3800, 1100, 550, 40), but introduces complications (see Supplemental Material for more details [45]). (c) Directly measuring the photon number (green) reveals that incoherent photons remain in the system after the homodyne signal (blue) has undergone superexponential decay. Oscillations in the photon number can also be observed to die out, as critical slowing-down constrains the envelope of oscillations, finally leaving only exponential decay. Here, $V_{\text{drive}}$ is $1.15\mu \mathrm{s}$ ($10/J$) and $\tau =1\mu \mathrm{s}$. Background voltages leading to distortion of the signal were removed from the photon number measurement. (d) Reconstructing the phase diagram by monitoring the homodyne signal as a function of initial photon number and time. At high powers, the dynamical transition from linear oscillations to localized behavior is marked by superexponential decay, while at low powers the collapse and revival signatures of localized behavior are observed. A 345 ns ($3/J$), an initialization pulse $V_{\text{drive}}$ ending at $t=0$ is used with $\tau =65\mathrm{ns}$, corresponding to an initial imbalance $Z\approx -0.6$. Inset: Illustration of the phase diagram showing the different dynmical regimes.

Figure 7

Simulated homodyne signal without and with dissipation. A comparison of the homodyne dynamics (shown on a logarithmic scale, normalized to ${\xi }_1$, the homodyne signal of a single-photon coherent state) without dissipation (red curve) and with (blue curve, averaged over 1000 walks in a quantum trajectory simulation of the master equation). The initial state is a coherent one with a mean of 150 photons on the left, no photons on the right, and the qubits in the ground state. The unitary dynamics (no dissipation) exhibits nonlinear oscillations due to the competition between intercavity tunneling and qubit coupling, in the delocalized regime not very far above $N_c^{\mathrm{qu}}$. 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.