Dissipative phase transition in the open quantum Rabi model

We demonstrate that the open quantum Rabi model (QRM) exhibits a second-order dissipative phase transition (DPT) and propose a method to observe this transition with trapped ions. The interplay between the ultrastrong qubit-oscillator coupling and the oscillator damping brings the system into a steady state with a diverging number of excitations, in which a DPT is allowed to occur even with a finite number of system components. The universality class of the open QRM, modified from the closed QRM by a Markovian bath, is identified by finding critical exponents and scaling functions using the Keldysh functional integral approach. We propose to realize the open QRM with two trapped ions where the coherent coupling and the rate of dissipation can be individually controlled and adjusted over a wide range. Thanks to this controllability, our work opens a possibility to investigate potentially rich dynamics associated with a dissipative phase transition.

Figure 1

Analytical solutions in the limit $\eta \to \infty$. (a) The asymptotic decay rate ${\kappa }_{\mathrm{ADR}}$ (solid line) vanishes at $g=g_c$. The excitation energy $\epsilon$ (dashed line) becomes zero for $g_1\le g\le g_2$ with $g_1=1$ and $g_2=g_c^{3/2}$, leading to an overdamped dynamics. (b) The steady-state expectation values for the order parameter ${|\langle a\rangle }_s|/\eta$ (dotted line), the oscillator population ${〈a^{†}a〉}_s$ (solid line), and the maximum quadrature variance $\mathrm{\Delta }{X}_s$ (dashed line). The relevant critical exponents for each quantity are indicated in the figures.

Figure 2

Finite-$\eta$ scaling relations. (a) Numerical solutions for the oscillator population ${〈a^{†}a〉}_s$ (squares) and the diverging quadrature variance $\mathrm{\Delta }{X}_s$ (circles) of the steady state follows a power-law behavior ${\eta }^{1/2}$ (solid line), whose exponent is analytically predicted by the Keldysh functional analysis. (b) The rescaled oscillator population $|g-g_c{|}^{{\nu }_x}{〈a^{†}a〉}_s$ is plotted as a function of $\eta |g-g_c{|}^{{\nu }_x/{\zeta }_x}$ for the open QRM (triangles) and $N|g-g_c{|}^{{\nu }_x/{\zeta }_x}$ for the open Dicke model (squares). Note that for the open Dicke model, a nonuniversal prefactor $c\sim 0.507$ is multiplied to the $y$ axis. All data points collapse onto a single curve.

Figure 3

(a) Realization of the open QRM using a ${}^9\mathrm{Be}^{+}-{}^{24}\mathrm{Mg}$ ion pair in a linear trap. (b) Two lasers are applied to the ${}^9\mathrm{Be}^{+}$ in order to drive the blue- and red-sideband transitions with detuning ${\delta }_1$ and ${\delta }_2$, respectively, and a Rabi frequency ${\eta }_{\mathrm{LD}}{\mathrm{\Omega }}_d$. This creates the coherent Rabi coupling between two hyperfine states of ${}^9\mathrm{Be}^{+}$ and the center-of-mass (COM) mode. (c) The ${}^{24}\mathrm{Mg}^{+}$ ion is used to implement a tunable phonon damping rate $\kappa =2{\mathrm{\Omega }}_e^2/\mathrm{\Gamma }$ via a weak red-sideband excitation to an optically excited state $|e\rangle$.

Figure 4

The effect of dephasing noise on finite-$\eta$ scaling relations. (a) Numerical solutions for the oscillator population ${〈a^{†}a〉}_s$ of the steady state for different values of dephasing rate: ${\mathrm{\Gamma }}_d/\kappa =0$ (squares), ${\mathrm{\Gamma }}_d/\kappa =7\times {10}^{-3}$ (circles), and $7\times {10}^{-2}$ (triangles). (b) The rescaled steady-state oscillator population $|g-g_c{|}^{{\nu }_x}{〈a^{†}a〉}_s$ is plotted as a function of $\eta |g-g_c{|}^{{\nu }_x/{\zeta }_x}$. The solid line is the nonequilibrium scaling function of the open QRM without any dephasing nose. While the nonequilibrium scaling function is still intact with the dephasing rate ${\mathrm{\Gamma }}_d/\kappa =7\times {10}^{-3}$ (circles), the data no longer collapse onto a single curve for ${\mathrm{\Gamma }}_d/\kappa =7\times {10}^{-2}$ (triangles). For all data, the same set of values for $\eta$ and $g$ is used.