Limit cycle phase and Goldstone mode in driven dissipative systems

In this article, we theoretically investigate the first- and second-order quantum dissipative phase transitions of a three-mode cavity with a Hubbard interaction. In both types, there is a mean-field (MF) limit cycle phase where the local U(1) symmetry and the time-translational symmetry of the Liouvillian superoperator are spontaneously broken. In MF, this spontaneous symmetry breaking manifests itself through the appearance of an unconditionally and fully squeezed state at the cavity output, connected to the well-known Goldstone mode. By employing the Wigner function formalism, hence, properly including the quantum noise, we show that away from the thermodynamic limit and within the quantum regime, fluctuations notably limit the coherence time of the Goldstone mode due to the phase diffusion. Our theoretical predictions suggest that interacting multimode photonic systems are rich, versatile test beds for investigating the crossovers between the mean-field picture and quantum phase transitions, a problem that can be investigated in various platforms including superconducting circuits, semiconductor microcavities, atomic Rydberg polaritons, and cuprite excitons.

Figure 1

MF dissipative phase diagram of a three-mode harmonic cavity, i.e., $2{\omega }_2={\omega }_1+{\omega }_3$, as a function of (a) the interaction strength $V_0$ and (b) the laser detuning ${\mathrm{\Delta }}_0$. In each panel the yellow (${\mathrm{\Pi }}_1$), orange (${\mathrm{\Pi }}_2$), and red (${\mathrm{\Pi }}_3$) regions correspond to the presence of one, two (bistability), and three stable (tristability) stationary states for the pumped mode, respectively. In (a) the detuning is fixed at ${\mathrm{\Delta }}_0=-3$ and in (b) the interaction strength has the constant value $V_0=1$. The dotted vertical lines [labeled (I) and (II)] at $V_0=0.1$ and ${\mathrm{\Delta }}_0=-3$ indicate the cuts through the phase diagram studied in subsequent figures. Rates are in units of ${\gamma }_0$.

Figure 2

Population of the 1st and the 2nd mode in a harmonic cavity, i.e., ${\delta }_D=0$, as a function of the pumping rate ${\mathrm{\Omega }}_0$ calculated from MF (solid black lines for stable and dashed lines for unstable solutions) and SDE (purple diamonds). Solid red line in panels (c), (d) show the density matrix solutions for comparison. $V_0=0.1$ in panels (a), (b) and $V_0=1$ in (c), (d). In both cases ${\mathrm{\Delta }}_0=-3$ and all rates are normalized to ${\gamma }_0$.

Figure 3

Number of photons in the 1st, 3rd modes of a three-mode anharmonic cavity ($2{\omega }_2\ne {\omega }_1+{\omega }_3$), as a function of the (a) interaction strength $V_0$ and (b) anharmonicity ${\delta }_D$, determined from MF, where all rates are in units of ${\gamma }_0$. In (a) ${\delta }_D=5$ and in (b) $V_0=1$ and the laser is always resonantly pumping the 2nd mode ${\mathrm{\Delta }}_0=0$. (${\mathrm{\Pi }}_1$), (${\mathrm{\Pi }}_3$) indicate two different phases of zero and nonzero population of the first mode. The dotted vertical lines [labeled (I) and (II)] at $V_0=0.1$ and ${\delta }_D=5$ indicate the cuts through the phase diagram studied in subsequent figures.

Figure 4

Population of the 1st (3rd) and the 2nd mode, in an anharmonic cavity with ${\delta }_D=5$, as a function of the pumping rate (${\mathrm{\Omega }}_0$) calculated from MF (black dots), SDE (purple diamonds). Solid red line in panels (c), (d) shows the DM solutions for comparison. $V_0=0.1$ in panels (a), (b) and $V_0=1$ in (c), (d). In both cases ${\mathrm{\Delta }}_0=0$ and the rates are normalized to ${\gamma }_0$.

Figure 5

Temporal behavior of the mean fields ${\alpha }_j\left(t\right)$ within the MF (${\mathrm{\Pi }}_3$) phase in a three-mode (a) harmonic cavity at ${\mathrm{\Omega }}_0=1.85,{\mathrm{\Delta }}_0=-3$ and (b) anharmonic cavity at ${\mathrm{\Omega }}_0=2,{\delta }_D=5$, and $V_0=1$. In both panels the blue, green, and red lines correspond to the 1st, 2nd, and 3rd mode, respectively. The time is in units of ${\gamma }_0^{-1}$, and $T_{LC}$ indicates the limit-cycle period.

Figure 6

Output $X,P$ spectra of the modes in the (a), (b) harmonic cavity at ${\mathrm{\Delta }}_0=-3,{\mathrm{\Omega }}_0=1.85$, and (c), (d) anharmonic cavity at ${\mathrm{\Delta }}_0=0,{\mathrm{\Omega }}_0=2$, calculated from the MF-Bogoliubov. In each panel the solid blue, red, and green lines correspond to the spectrum of the pumped ($m_2$), symmetric ($m_{+}$), and antisymmetric ($m_{-}$) modes, respectively. Due to its divergence, the momentum of the antisymmetric mode is scaled down in panels (b), (d). The frequency axis is in units of ${\gamma }_0$.

Figure 7

Histograms of number state occupation probability $p_n$ and color maps of the Wigner function of the (a)–(d) 1st mode and (e)–(h) 2nd mode, in a three-mode harmonic cavity when ${\mathrm{\Delta }}_0=-3,V_0=1$ for different pumping rates ${\mathrm{\Omega }}_0$ highlighted as (A, B, C, D) in Figs. 2, 2. In each phase-space map the white dashed lines show the axes ($X=0,P=0$) in the $XP$ plane and the light blue stars or circles correspond to the predicted MF.

Figure 8

Histograms of the number state occupation probability $p_n$ and color maps of the Wigner function of the (a)–(d) 1st mode and (e)–(h) 2nd mode in a three-mode anharmonic cavity when ${\mathrm{\Delta }}_0=0,{\delta }_D=5$ and at $V_0=1$ for different pumping rates ${\mathrm{\Omega }}_0$ highlighted as (A, B, C, D) in Figs. 4, 4. In each phase-space map the white dashed lines show the axes ($X=0,P=0$) in the $XP$ plane and the light blue stars or circles correspond to the predicted MF steady-state amplitudes.

Figure 9

SDE calculations of the (a), (c) spectral density in the laser frame (in units of ${\gamma }_0$) and (b), (d) delayed temporal correlation of $P$ quadrature of in-cavity $m_{-}$ mode in an anharmonic cavity. The upper row shows the behavior in the (${\mathrm{\Pi }}_1$) phase and the lower row shows that with the MF LC phase. The interaction is changed from $V_0=0.1$ to $V_0=0.5$ to $V_0=1.0$, yellow to red to brown, respectively. The dashed lines show the Lorentzian fit in (a), (c) and the exponential fits in (b), (d).

Figure 10

The linewidth of the $P_{-}$ quadrature (in units of ${\gamma }_0$), i.e., the Goldstone mode, within the MF LC phase as a function of the dimensionless parameter $N$. The red squares are the SDE calculation results while the solid blue line is a power-law fit to the data. The solid red line shows the average number of particles in this mode (right axis). The inset color maps show the TWF distribution of the $m_{-}$ mode at a couple of interaction strengths.

Figure 11

MF dissipative phase diagram of a single-mode cavity as a function of (a) the interaction strength $V_0$ and (b) the laser detuning ${\mathrm{\Delta }}_0$. In each panel the yellow (${\mathrm{\Pi }}_1$) and orange (${\mathrm{\Pi }}_2$) regions correspond to one and two (bistability) stationary states for the pumped mode, respectively. In (a) the detuning is fixed at ${\mathrm{\Delta }}_0=-3$ and in (b) the interaction strength has the constant value $V_0=1$. The dotted vertical lines [labeled (I) and (II)] at $V_0=0.1$ and ${\mathrm{\Delta }}_0=-3$ indicate the cuts through the phase diagram studied in subsequent figures.

Figure 12

Number of photons as a function of pumping rate (${\mathrm{\Omega }}_0$) in a single-mode cavity with two-body interaction strength of (a) $V_0=0.1$ [dotted line (I) in Fig. 11] and (b) $V_0=1$ [dotted line (II) in Fig. 11]. The black dots show the mean-field result. The solid red line is the full density matrix result, and the purple diamonds represent the SDE results. The solid green line shows the intensity fluctuation $g^{\left(2\right)}\left(0\right)$. For each interaction strength the blue histograms show the number state occupation probability $p_n$ at different pumping rates, before the bistability (first row), within the bistability (two middle rows), and after the bistability (last row). In each case, the color maps show the corresponding Wigner function distribution in the $XP$ phase space. The white dashed lines show the axes ($X=0,P=0$) in that plane. Also the light blue stars in those panels indicate the predicted MFs. We use ${\mathrm{\Delta }}_0=-3$ in all cases.

Figure 13

Inelastic output intensity spectrum in the laser reference frame, i.e., $\omega -{\omega }_L$, when the pumping rate is increased on the MF (a) lower and (b) upper branches at ${\mathrm{\Delta }}_0=-3,V_0=1$. (c) and (d) show their corresponding $X,P$ variance spectra calculated from Eq. (A11). Solid lines show $X$ spectra and dashed lines are their corresponding $P$ spectra.

Figure 14

(a) Real and (b) imaginary parts of the first three eigenvalues (EV) of the Liouvillian as a function of pumping rate ${\mathrm{\Omega }}_0$ in a single-mode cavity. The dashed and solid lines correspond to $V_0=0.1,1$, respectively, at a fixed detuning of ${\mathrm{\Delta }}_0=-3$. All rates are in terms of ${\gamma }_0$.

Figure 15

Particle number as a function of time in a single-mode cavity at different interaction strength and within the MF bistability region for (a) $V_0=0.1,{\mathrm{\Omega }}_0=6.5$, (b) $V_0=0.5,{\mathrm{\Omega }}_0=3$, and (c) $V_0=1,{\mathrm{\Omega }}_0=2.2$, determined from one trajectory of a quantum Monte Carlo simulation. In each panel the black dotted lines show the MF stationary states. The detuning is fixed at ${\mathrm{\Delta }}_0=-3$. The rates are in units of ${\gamma }_0$, and the time in units of ${\gamma }_0^{-1}$.

Figure 16

Instantaneous photon number $n\left(t\right)$ in a single-mode cavity as a function of time (in units of ${\gamma }_0^{-1}$) for $V_0=1,{\mathrm{\Delta }}_0=-3$ and at (a) ${\mathrm{\Omega }}_0=1$ and (b) ${\mathrm{\Omega }}_0=2.2$. (c), (d) show the first-order correlation of the fluctuations as a function of delay $\tau$. In each panel the solid blue line shows the density matrix (DM) calculations and the red line indicates the stochastic differential equation (SDE) results. The green lines in panels (c), (d) are the least-squares exponential fits to the correlation function tails.