Observation of a Dissipative Phase Transition in a One-Dimensional Circuit QED Lattice

Condensed matter physics has been driven forward by significant experimental and theoretical progress in the study and understanding of equilibrium phase transitions based on symmetry and topology. However, nonequilibrium phase transitions have remained a challenge, in part due to their complexity in theoretical descriptions and the additional experimental difficulties in systematically controlling systems out of equilibrium. Here, we study a one-dimensional chain of 72 microwave cavities, each coupled to a superconducting qubit, and coherently drive the system into a nonequilibrium steady state. We find experimental evidence for a dissipative phase transition in the system in which the steady state changes dramatically as the mean photon number is increased. Near the boundary between the two observed phases, the system demonstrates bistability, with characteristic switching times as long as 60 ms—far longer than any of the intrinsic rates known for the system. This experiment demonstrates the power of circuit QED systems for studying nonequilibrium condensed matter physics and paves the way for future experiments exploring nonequilbrium physics with many-body quantum optics.

Popular Summary

Over the years, the study of phase transitions has steadily advanced the field of condensed matter physics. A phase transition denotes a sudden change in the physical properties of a system as a function of an external parameter, such as liquid freezing into solid as temperature decreases below the freezing point. Much of our understanding of phase transitions is limited to systems at equilibrium, with no flux of energy or particles over time. This assumption is rarely true, but finding phase transitions in nonequilibrium systems has remained a challenge because such systems are difficult to control. Interacting photons (particles of light) have emerged as an excellent candidate for studying nonequilibrium physics because of their innate property to leak out of any system and the ease with which they can be added back in. Here, we present experimental evidence for a nonequilibrium phase transition in which the steady state of a system changes dramatically as the mean photon number is increased.

Photons do not interact with each other ordinarily, but strong coupling between atoms and a cavity can mediate effective photon-photon interactions. In circuit quantum electrodynamics (circuit QED), we can engineer these interactions using superconducting microwave cavities and quantum bits (qubits). We study a circuit QED lattice made of a one-dimensional chain of 72 cavities, each coupled to a qubit. Phase transitions are characterized by transmitting a microwave signal of varying frequency and power through the circuit and measuring the power of the output signal.

This experiment demonstrates the power of circuit QED systems for studying nonequilibrium condensed matter physics and paves the way for future experiments on exotic states of many interacting photons.

Figure 1

72-site circuit QED lattice. (a) Coplanar waveguide resonators, each with a bare cavity frequency ${\omega }_c/2\pi =7.6(2)\mathrm{GHz}$, are capacitively coupled to form a linear chain on a $35\times 35{\mathrm{mm}}^2$ chip. Each resonator is coupled to its neighboring resonators with a hopping rate $t/2\pi =80(10)\mathrm{MHz}$ and each low-power eigenmode as a measured average photon loss rate of $\kappa /2\pi =1.6(5)\mathrm{MHz}$. At three intermediate chain sites, three-way coupling capacitors provide ports for secondary input and output lines (arrows on sides), which allow further characterization of the device. (b),(c) A transmon qubit is capacitively coupled to the center pin near the edge of each resonator in the lattice, ensuring strong coupling to the fundamental mode of each resonator. The coupled resonator-qubit system forms the fundamental unit cell of the lattice. (d) The circuit can be modeled as a linear chain of coupled oscillators, each dispersively coupled to a weakly anharmonic multilevel system. Transmission ($S_{21}$) as a function of external magnetic field ($B$) as measured experimentally (e) and predicted theoretically (f). Numerical simulations reveal qualitative agreement between theory and experiment for experimentally relevant parameters, ${\omega }_c/2\pi =7.6\mathrm{GHz}$, tunable qubit frequency, $\mathrm{\Omega }/2\pi \in [12.7,13.9]\mathrm{GHz}$, single-junction qubit frequency, ${\mathrm{\Omega }}_{\mathrm{SJ}}/2\pi =9.5\mathrm{GHz}$, $g/2\pi =410\mathrm{MHz}$ (on resonance with ${\omega }_c$), transmon charging energy, $E_c/h=350\mathrm{MHz}$, and $t/2\pi =90\mathrm{MHz}$. By changing the strength of the external magnetic field, the tunable qubit frequencies are continuously changed and produce nonperiodic dispersive shifts due to the random areas of the transmon SQUID loops. (g) The blue rectangles indicate the frequency range of the photonic modes, with the dark blue rectangle indicating the range of frequencies shown in (e) and (f) and the green line indicating the bare cavity frequency. The gray shaded region reflects the predicted uncertainty range of the qubit frequencies (including potential single-junction qubits) and curves represent the qubit frequencies as a function of external magnetic field.

Figure 2

Microwave transmission ($S_{21}$) as a function of power, exhibiting an abrupt transition to a suppressed transmission regime and a region of bistability. (a) Dispersively shifted transmission peaks show nonlinear splitting at increased power and give rise to a region of strongly suppressed transmission without resonance peaks. Data are acquired using frequency sweeps at constant power. (b) Corresponding mean-field results for transmission through a 72-site lattice, showing features qualitatively consistent with the experiment, where $\epsilon$ is the drive power and $\kappa$ is the dissipation rate (qubit and cavity) (c) Zoom into one lobe of the experimental data, showing the sharp transition to a state of suppressed transmission as the drive power is swept from low to high power using constant-frequency, linear power sweep over a 31.95-ms period. (d) Same region as in (c), but sweeping power in the opposite direction (high to low) over the same time period as (c). The transition now occurs at lower power. (e) The difference $\mathrm{\Delta }$ in the data shown in (c) and (d) uncovers the large region of hysteresis in between two distinct states ① and ②.

Figure 3

Asymptotic decay rate in the transition region. (a) Single-shot time trace of the homodyne phase in the hysteretic region for constant drive amplitude. Data show stochastic switching between two distinct metastable states ① and ② on time scales vastly exceeding those intrinsic to the system. (b) Asymptotic decay rate obtained from the sum of the characteristic switching times ${\gamma }_{1\to 2}+{\gamma }_{2\to 1}$ as a function of drive frequency and power. (c) ADR for a drive frequency of 7.6059 GHz is plotted as a function of power. $\kappa$ and $\gamma$ are included for reference to indicate that ADR can be as large as 5 orders of magnitude slower than relevant time scales of the device. When either ${\gamma }_{1\to 2}$ or ${\gamma }_{2\to 1}$ are slower than the duration of the measurement pulse ${\tau }_m$, we cannot reliably extract a characteristic switch rate. In these cases, we select the smallest extracted switching time, which is larger than $1/{\tau }_m$. Upward- (downward-)pointing triangles indicate when ${\gamma }_{1\to 2}$ (${\gamma }_{2\to 1}$) are less than $1/{\tau }_m$, circles indicate when both rates are used.

Figure 4

State characterization. To probe properties of the states within the region of bistability, two pulse sequences are used to initialize the system: (a) an “up pulse” for initialization in the high-power state and (b) a “down pulse” for initialization in the low-power state. Both (a) and (b) return to a constant amplitude $\xi$. In the region of bistability where the ADR is small, the system can be deterministically initialized in ① or ② or 193 after which, during the “hold and measure” segment of the pulse sequence, properties of the emitted light can be extracted. The power spectrum in (c) and (d) exhibits that ① and ② have drastically different emission properties and that the two states are maintained by a constant, coherent tone, even in the region of bistability [where (c) and (d) show different emission]. Similarly, the second-order correlation function $g^{(2)}(0)$ in (e) can be obtained for each state independently. Results in (c)–(e) indicate that ① can be characterized by a single, coherent [$g^{(2)}(0)=1$] tone at the drive frequency, 7.606 GHz, and that ② can be characterized by broadband emission and bunching [$g^{(2)}(0)>1$] at the drive frequency. (f) Mean-field results for the second-order correlation function for comparison. Note that the numerical values on the $x$ axes cannot be compared directly for (e) and (f).