$\mathcal{PT}$-symmetric circuit QED

A parity-time ($\mathcal{PT}$)-symmetric system emerging from a quantum dynamics is highly desirable in order to understand the possible implications of $\mathcal{PT}$ symmetry in the next generation of quantum technologies. In this work, we address this need by proposing and studying a circuit-QED architecture that consists of two coupled resonators and two qubits (each coupled to one resonator). By means of external driving fields on the qubits, we are able to tune gains and losses in the resonators. Starting with the quantum dynamics of this system, we show the emergence of the $\mathcal{PT}$ symmetry via the selection of both driving amplitudes and frequencies. We engineer the system such that a non-number-conserving dipole-dipole interaction emerges, introducing an instability at large coupling strengths. The $\mathcal{PT}$ symmetry and its breaking, as well as the predicted instability in this circuit-QED system, can be observed in a transmission experiment.

Figure 1

A graphical illustration of the proposed circuit-QED architecture to study the physics of $\mathcal{PT}$ symmetry. Two superconductor resonators are coupled to each other with a coupling strength $J$, and two qubits with decay rates of $\gamma$, each coupled to one resonator with a coupling strength $g$, form the basic ingredients of this architecture. The inset in orange shows the structure of the qubits, whereas the inset in blue shows a possible implementation of tunable coupling between the resonators.

Figure 2

Time evolution of (a),(b) $\langle N\rangle$, (c),(d) $\langle {\sigma }_z\rangle$, and (e),(f) $\langle {\sigma }_x\rangle$ in the proposed circuit-QED architecture according to Eqs. (8) (black) and (12) (blue). For the numerical simulations, we set $\omega =1$ as the unit. The rest of the parameters used are ${\lambda }_{+}={\lambda }_{-}=2$, $\epsilon =5$, $\delta =0.1$, $g=0.05$, $G_{-}/G_{+}=0.58$, for (a), (c), (e) the initial state $\left|\psi \right(0)\rangle =|\uparrow \rangle |1\rangle$ and (b), (d), (f) the initial state $|\psi \left(0\right)\rangle =\left(\right|\uparrow \rangle +|\downarrow \rangle )|1\rangle /\sqrt{2}$.

Figure 3

Time evolution of (a), (b) $\langle N\rangle$, (c), (d) $\langle {\sigma }_z\rangle$, and (e), (f) $\langle {\sigma }_x\rangle$ in the proposed circuit-QED architecture according to Eqs. (8) (black) and (10) (blue). For the numerical simulations, we set $\omega =1$ as the unit. The rest of the parameters used are ${\lambda }_{+}=3,{\lambda }_{-}=1$, $\epsilon =5$, $\delta =0.1$, $g=0.05$, $G_{-}/G_{+}=2.1$, for (a), (c), (e) the initial state $\left|\psi \right(0)\rangle =|\uparrow \rangle |1\rangle$ and (b), (d), (f) the initial state $|\psi \left(0\right)\rangle =\left(\right|\uparrow \rangle +|\downarrow \rangle )|1\rangle /\sqrt{2}$.

Figure 4

Numerical verification of the adiabatic elimination for a single resonator coupled to a qubit with decay rates of $\gamma =1$ and $\gamma =10$. We use a Fock space of dimension $N_{\mathrm{Fock}}=300$. In (a), we plot the relative error for ${\sigma }^{+}{\sigma }^{-}$ as a function of $G_{-}/G_{+}$. In (b), we plot the relative error for $n=a^{†}a$.

Figure 5

Evolution of the eigenvalues of the proposed circuit-QED architecture as a function of the resonator-resonator coupling strength $J$. The background colors distinguish among three regions: $J<J_{c_1}$ (white), $J_{c_1}<J<J_{c_2}$ (light gray), and $J_{c_2}<J$ (light yellow). The eigenfrequency ${\omega }_{+-}$ is plotted in black, while ${\omega }_{++}$ is plotted in blue [cf. Eq. (22)]. The dashed lines correspond to the RWA (12), while the solid lines correspond to the most general case (14). In (a), we plot the real part of ${\omega }_{+\pm }$ and, in (b), the imaginary part of the eigenvalues. Here we set $\delta =1$ as the unit and the effective decay rates are ${\tilde{\gamma }}_1={\tilde{\gamma }}_2=0.1$.

Figure 6

Logarithm of the power transmitted from port $1L$ to $2R$: $ln\left(\right|T_{1L,2R}{|}^2)$, as a function of the input frequency $\omega$ and resonator-resonator coupling $J$ (in units of frequency). The parameters used in the simulation are ${\tilde{\gamma }}_1={\tilde{\gamma }}_2=0.1$ and $\kappa =0.02$. The units are the same as those in Fig. 5.

Figure 7

Eigenfrequencies for the case of unbalanced gain and loss between the resonators (B1) as a function of the resonator-resonator coupling strength $J$. The background colors distinguish among three regions: $J<J_{c_1}^{\prime }$ (white), $J_{c_1}^{\prime }<J<J_{c_2}^{\prime }$(light gray), and $J_{c_2}^{\prime }<J$ (light yellow) [cf. Eqs. (B2) and (B3)]. The eigenfrequency ${\omega }_{+-}^{\prime }$ is plotted in black, while ${\omega }_{++}^{\prime }$ is plotted in blue. The dashed lines correspond to the RWA (12), while the solid lines correspond to the most general case (14). In (a), we plot the real part of ${\omega }_{+\pm }^{\prime }$ and, in (b), the imaginary part. Here we set $\delta =1$ as the unit and the effective decay rates are ${\tilde{\gamma }}_1=0.1$, ${\tilde{\gamma }}_2=0.3$.