Cavity-assisted dynamical quantum phase transition at bifurcation points

Coupling a quantum many-body system to a cavity can create bifurcation points in its phase diagram, where the ground state makes sudden switchings between different phases. Here we study the dynamical quantum phase transition of a transverse field Ising model coupled to a cavity. We show that an infinitesimal quench of the cavity driving at the bifurcation points induces gradual evolution of the Ising model to pass across the quantum critical point and excites quasiparticles. Meanwhile, when the driving is slowly ramped through the bifurcation points, the adiabaticity of the evolution and the number of quasiparticle excitations are strongly affected by cavity-induced nonlinearity. Introducing and manipulating cavity-induced nonlinearity hence provide an effective approach to control the dynamics and the adiabaticity of adiabatic quantum processes. This model can be implemented with superconducting quantum circuits.

Figure 1

(a) The stationary cavity displacement $x_a^{\left(s\right)}$ vs driving amplitude $\epsilon$ and detuning ${\mathrm{\Delta }}_c$. Solid (dashed) line: the bifurcation point ${\epsilon }_2$ $\left({\epsilon }_1\right)$ vs ${\mathrm{\Delta }}_c$. (b) The derivative $d\epsilon /x_a^{\left(s\right)}$ vs $x_a^{\left(s\right)}$ at detuning ${\mathrm{\Delta }}_c=-0.05$. Inset: $x_a^{\left(s\right)}$ vs $\epsilon$. Other parameters are $J_0=1,B_x=1.95,g=0.02$, and $\kappa =0.07$, all in arbitrary units; and the TFIM in our calculation has $N=120$ sites.

Figure 2

(a) The effective transverse field ${\tilde{B}}_x$ and (b) the ground-state probability $P_g$ vs time $t$ after a sudden quench of the driving amplitude. Solid lines: results for the cavity-coupled TFIM; dashed lines: results for the linearly ramped TFIM; thin solid line in (b): results using the Landau-Zener formula for the linearly ramped TFIM. The inset of (a): $d{\tilde{B}}_x/dt$ vs time $t$. Red dots: the time $t_c$ for ${\tilde{B}}_x=J_0$. The parameters are the same as those in Fig. 1.

Figure 3

(a) The effective transverse field ${\tilde{B}}_x$ vs $\epsilon$ under slow ramping of the driving amplitude. Solid (dashed) lines: results for ramping up (down) at $T=400,1200,2000$ from top to bottom (bottom to top); thin solid line: stationary solution. (b) $N_{ex}$ at $t=1.2T$ vs ramping time $T$. Blue circles and solid (dashed) line: results for cavity-coupled TFIM, ramping up (down); red stars: results using the Landau-Zener formula for linearly ramped TFIM. (c, d) $|{\lambda }_c|$ vs $T$ and $1/\sqrt{T}$. Blue circles: results from numerical simulation; solid (dashed) lines: results using the formula $|{\lambda }_c|=20|{\lambda }_c(T=400)|/\sqrt{T}$ for ramping up (down). The parameters are the same as those in Fig. 1.

Figure 4

The stationary-state average $X^{\left(s\right)}$ for the operator ${\sum }_i{\widehat{\sigma }}_{xi}$ vs driving amplitude $\epsilon$ and cavity detuning ${\mathrm{\Delta }}_c$. Solid (dashed) line: the bifurcation point ${\epsilon }_2$ $\left({\epsilon }_1\right)$ vs ${\mathrm{\Delta }}_c$. The parameters are the same as those in Fig. 1.

Figure 5

(a) The bifurcation points ${\epsilon }_{1,2}$ vs transverse field $B_x$ at a given detuning. (b) The bifurcation points ${\epsilon }_{1,2}$ vs detuning ${\mathrm{\Delta }}_c$ at a given transverse field. Solid lines: ${\epsilon }_2$; dashed lines: ${\epsilon }_1$. The parameters are the same as those in Fig. 1.

Figure 6

(a, b) The probability $P_g\left(T\right)$ vs $B_x$ and vs ${\mathrm{\Delta }}_c$ at the final time $T=400$, respectively. Blue circles: numerical results for the cavity-coupled TFIM; red squares: numerical results for the linearly ramped TFIM; red stars: results from the Landau-Zener formula for the linearly ramped TFIM. The parameters are the same as those in Fig. 1.