Driven-Dissipative Criticality within the Discrete Truncated Wigner Approximation

We present an approach to the numerical simulation of open quantum many-body systems based on the semiclassical framework of the discrete truncated Wigner approximation. We establish a quantum jump formalism to integrate the quantum master equation describing the dynamics of the system, which we find to be exact in both the noninteracting limit and the limit where the system is described by classical rate equations. We apply our method to simulation of the paradigmatic dissipative Ising model, where we are able to capture the critical fluctuations of the system beyond the level of mean-field theory.

Figure 1

Open-system dynamics within the discrete truncated Wigner approximation. (a) Bloch sphere representation for a spin-$1/2$ particle, where the spin points in the $-z$ direction. This initial state is sampled from a discrete four-point Wigner quasiprobability distribution $w_{(p,q)}$, which are $w_{(0,0)}=w_{(0,1)}=0$ and $w_{(1,0)}=w_{(1,1)}=1/2$. The probability for a spin to point along the $\pm x$, $\pm y$, and $\pm z$ directions $(p_{\pm 1}^{x,y,z})$ is given by the sum over the vertical, diagonal, and horizontal lines, respectively [43, 58]. (b) Classical trajectories corresponding to two different initial configurations (dotted and dashed lines) for single spin and $g/{\gamma }_d=5$. The averaged time evolution of $⟨S^z(t)⟩$ over ${10}^5$ trajectories is shown as a continuous line.

Figure 2

Benchmarking against exact results for a single spin. The time evolution of $⟨S^z(t)⟩$, same as in Fig. 1, is compared with the exact result $S_{\text{exact}}^z(t)$ for $g/{\gamma }_d=5$. The inset shows the numerically obtained steady state from $⟨S^z(t)⟩$ in the long-time limit and the exact steady state for varying $g/{\gamma }_d$.

Figure 3

Liquid-gas transition. The spin-up density $n_{\uparrow }$ as a function of $g/{\gamma }_d$ for $V/{\gamma }_d=5$ for both the forward (continuous line) and reverse sweep (dashed line). The inset depicts $n_{\uparrow }$ as a function of $g/{\gamma }_d$ and $V/{\gamma }_d$, where the dashed line is the location of the susceptibility peak, see the main text. All results are shown for a $10\times 10$ lattice and 3200 trajectories.

Figure 4

Driven-dissipative criticality. (a) Value ${\chi }_0$ and position $g_0$ of the susceptibility peak for varying system sizes between $10\times 10$ and $16\times 16$, derived from Gaussian fits to the susceptibility $\chi (g)=(\partial n_{\uparrow }/\partial g)$. Results were obtained using up to 12000 trajectories. (b) Universal scaling close to the critical point obtained by fitting the susceptibility data to the finite-size scaling function Eq. (8), which yields the critical exponents $\gamma =1.69\pm 0.07$ and $\nu =0.99\pm 0.04$, and the critical point $g_c/{\gamma }_d=2.94\pm 0.14$.