Squeezed-light-induced quantum phase transition in the Jaynes-Cummings model

Quantum phase transition and quench dynamics in the Jaynes-Cummings model with squeezed light are investigated. We find a special unitary transformation that removes the nonintegrable squeezing interaction when the qubit frequency far outweighs the field frequency. The eigenenergies and the eigenstates of the normal and superradiant phases are derived analytically. We demonstrate that the system exhibits a second-order quantum phase transition at a phase-boundary-induced nonlinearly by squeezed light. This phase boundary requires neither an ultrastrong coupling regime nor multiqubits, which lessens the difficulty of the experiment. The entanglement entropy is very close to its maximum value as the squeezing strength increases in the superradiant phase. In quench dynamics, the nonlinear relation between the residual energy and the squeezing strength is obtained analytically.

Figure 1

Relative errors $E_{r_{\mathrm{np}}}$ of ground-state energy versus the ratio $\frac{\mathrm{\Omega }}{{\omega }_0}$ between the numerical solution of $H$ in Eq. (1) and the analytical solution of $H_{\mathrm{np}}$ in Eq. (7) at $g_c=\frac{1}{2}$ (blue stars) and $g_c=1$ (red circles) with different squeezing strengths: (a) $G=0.01{\omega }_0$; (b) $G=0.1{\omega }_0$; (c) $G=0.2{\omega }_0$; and (d) $G=0.4{\omega }_0$. (e) $E_{r_{\mathrm{np}}}$ versus the ratio $\frac{G}{{\omega }_0}$ between the numerical solution of $H$ in Eq. (1) and the analytical solution of $H_{\mathrm{np}}$ in Eq. (7) at $g_c=\frac{1}{2}$. $E_{r_{\mathrm{np}}}$ based on the numerical solution of $H$ in Eq. (1) is calculated as $E_{r_{\mathrm{np}}}=\left|\frac{E_{\mathrm{np}}^{\prime }-E_{\mathrm{np}}}{E_{\mathrm{np}}^{\prime }}\right|\times 100\%$, where $E_{\mathrm{np}}^{\prime }$ is the numerical eigenvalue of the ground state in $H$, which is solved directly by the matlab program's diagonalization command of $\mathrm{eig}(H$). The numerical solution has a cutoff photon number of 300.

Figure 2

Analytical solution of excitation energies $\epsilon$ (${\epsilon }_{\mathrm{np}}$ and ${\epsilon }_{\mathrm{sp}}$) versus $\frac{\lambda }{{\omega }_0}$ and $\frac{G}{{\omega }_0}$ at $\mathrm{\Omega }=20{\omega }_0$. Both ${\epsilon }_{\mathrm{np}}$ and ${\epsilon }_{\mathrm{sp}}$ vanish at a nonlinear phase boundary, locating the quantum phase transition.

Figure 3

The rescaled photon number $n_c$ as a function of $g_c$ and $\frac{G}{{\omega }_0}$: (a) numerical solution; (b) analytical solution. The rescaled ground-state energy (c) $e_G$ (i.e., $e_{\mathrm{np}}$ and $e_{\mathrm{sp}}$) and (d) its corresponding second derivative $\frac{d^2{e}_G}{d{\lambda }^2}$ as functions of $\frac{\lambda }{{\omega }_0}$ and $\frac{G}{{\omega }_0}$, where $\mathrm{\Omega }=20{\omega }_0$ and the numerical solution has a cutoff photon number of 300. $n_c$ based on the numerical solution of $H$ in Eq. (1) is calculated as $n_c=\frac{{\omega }_0}{\mathrm{\Omega }}\langle {\varphi }_{\mathrm{np}}^{\prime 0}|a^{†}a|{\varphi }_{\mathrm{np}}^{\prime 0}\rangle$, where $|{\varphi }_{\mathrm{np}}^{\prime 0}\rangle$ is the numerical eigenstate of the ground state in $H$, which is solved directly by the matlab program's diagonalization command of $\mathrm{eig}(H$).

Figure 4

The variances of (a) the position quadrature $X$ and (b) the momentum quadrature $P$ as functions of $g_c$ and $\frac{G}{{\omega }_0}$. (c) $S_q/ln\left(2\right)$ as functions of $\frac{\lambda }{{\omega }_0}$ and $\frac{G}{{\omega }_0}$ for the ground state, where $\mathrm{\Omega }=20{\omega }_0$ and the numerical solution has a cutoff photon number of 300. $S_q$ based on the numerical solution of $H$ in Eq. (1) is calculated as $S_q=-\mathrm{Tr}({\rho }_q^{\prime }ln{\rho }_q^{\prime })$, where ${\rho }_q^{\prime }$ is the reduced density matrix of the qubit by tracing out the field's freedom degree of the numerical ground state in $H$, which is solved directly by the matlab program's diagonalization command of $\mathrm{eig}(H$).

Figure 5

Residual energy $E_r$ as functions of the quench time $\tau$ for Eq. (31). Four circle-solid lines with different colors represent different final coupling constants $g_f$: 1 for blue, 0.9999 for red, 0.999 for yellow, and 0.99 for purple. Four subfigures have different squeezing coefficients: (a) $G=0.05{\omega }_0$; (b) $G=0.1{\omega }_0$; (c) $G=0.15{\omega }_0$; (d) $G=0.20{\omega }_0$. The units of the horizontal coordinate and the vertical coordinate are $\frac{1}{{\omega }_0}$ and ${\omega }_0$, respectively. The scalings $\sim {\tau }^{-1/3}$ and $\sim {\tau }^{-2}$ are denoted by black and red lines, respectively.