Theoretical realization and application of parity-time-symmetric oscillators in a quantum regime

In the existing parity-time ($\mathcal{PT}$) symmetry with the balanced gain and loss, the gain is derived from semiclassical but not full quantum theories, which significantly restricts the applications of $\mathcal{PT}$ symmetry in quantum fields. In this work, we propose and analyze a theoretical scheme to realize full quantum oscillator $\mathcal{PT}$ symmetry. The quantum gain is provided by a dissipation optical cavity with a blue-detuned laser field. After adiabatically eliminating the cavity modes, we give an effective master equation, which is a complete quantum description compared with the non-Hermitian Hamiltonian, to reveal the quantum behaviors of such a gain oscillator. This kind of $\mathcal{PT}$ symmetry can eliminate the dissipation effect in the quantum regime. As an example, we apply $\mathcal{PT}$-symmetric oscillators to enhance optomechanically induced transparency.

Figure 1

(a) A typical dissipative optomechanical system and its corresponding model after linearization. This type of system can also be considered a hybrid system consisting of a transmission line resonator and a superconducting qubit. (b) Two schemes for realizing oscillator gain, corresponding to a dissipative cavity optomechanical system under the blue sideband and a gain cavity optomechanical system under the red sideband, respectively. (c) Level diagram of the blue sideband. Here, $|n,m\rangle$ denotes the state of $n$ photons and $m$ phonons in the displaced frame. (d) A gain and a dissipation mutually interacting through a phonon tunneling term of intensity $\mu$.

Figure 2

(a) Changes in the effective dissipation coefficient with varied linear coupling coefficient $G$. Comparisons of evolutions corresponding to (b) the first-order and (c) second-order mechanical quantities. Here, the blue dotted lines denote mechanical quantities of the original system, and red solid lines denote the approximate system after adiabatic elimination. The black line in the inset is the evolution without bath correction. (d) The fidelity between the oscillator states corresponds to the original and approximate systems. The inset in (d) shows the time-averaged fidelities with varied coupling coefficient $G$ (blue solid line) and the detuning $\mathrm{\Delta }$ (red dotted line). Here, the horizontal axis is $100G$ ($\mathrm{\Delta }$) for the blue (red) line. In this simulation, we set ${\omega }_m=1$ as the unit; the other dimensionless parameters are $\kappa =0.1,n_b=1000$. In (a), we set $\mathrm{\Delta }=2$ (3) for the red lines (blue lines) and $\gamma =5\times {10}^{-5}$ (${10}^{-4}$, ${10}^{-5}$) for the solid line (dashed line, line with circles). In (b), (c), and (d), we set $\gamma ={10}^{-5},G=0.04$ and $\mathrm{\Delta }=3$. The blue line in the inset in (d) corresponds to $\mathrm{\Delta }=3$, and the red line is under $G=0.04$.

Figure 3

(a) Real and (b) imaginary parts of the eigenfrequencies of supermodes as a function of coupling $\mu$. Here, blue lines correspond to ideal $\mathcal{PT}$ symmetry ($\gamma ={\gamma }^{\prime }=0.001$). Red lines denote that the system is not strictly $\mathcal{PT}$ symmetric but has an effective dissipation rate (${\gamma }^{\prime }=0.8\gamma$). (c) and (d) The dynamical behaviors of the two oscillators corresponding to $\mathcal{PT}\mathrm{SP}$ and $\mathcal{PT}\mathrm{BP}$, respectively. Here, the unit is $\omega =1$.

Figure 4

(a) ${\delta }_1$ given by the exact dynamic equation and our effective master equation (11). (b) and (c) Comparisons of the density matrices obtained by solving dynamic equations (11) (black solid lines), (16) (blue solid lines), and (17) (red dotted lines) under different bath phonon numbers. In (b), we set $n_{\mathrm{th}}=0$, and in (c) $n_{\mathrm{th}}=10$. The curves in the inset are the evolutions of $\delta$ based on exact dynamic equations (blue) and master equation (16) (red). (d) Fidelity between the system states calculated from the exact dynamic equation and effective Langevin equation (17) under $\gamma ={10}^{-4}$. Here, the blue solid line and the red dotted line correspond to $n_{\mathrm{th}}=0$ and $n_{\mathrm{th}}=10$, respectively. In the simulations, we set ${\mathrm{\Gamma }}^{\prime }=\gamma -\eta \kappa$. In (b), (c), and (d), we set $\mu =0.2\gamma$, and in (d) $G=0.1$. The other parameters and the units are the same as those in Fig. 3.

Figure 5

(a) Optical-cavity-coupled $\mathcal{PT}$-symmetric oscillators with nonlinear radiation pressure. (b) Level diagram of the OMIT.

Figure 6

(a) Real and (b) imaginary parts of $\chi$ corresponding to $\mathcal{PT}$-symmetric oscillators (blue and green lines) and dissipative oscillators (red lines). (c) and (d) Depths of transparent windows with varied single-photon coupling intensity $g$ and oscillator dissipation rate ${\gamma }_m$. The insets illustrate the changes of absorption spectra under different ${\gamma }_{\mathrm{eff}}$ [in (c)] and $g$ [in (d)]. In this simulation, the parameter unit is set as $\mathrm{\Delta }={\omega }_m=1$, and other parameters are $g_0=2.5\times {10}^{-4},E=10,\kappa =0.15$, and $\gamma =0.02$. In (a) and (b), the corresponding single-photon coupling is $2g_0$, and green lines are under $\kappa =0.2$.