1. Introduction

$\mathcal {PT}$-symmetric quantum mechanics initially proposed by Bender in 1998, which proved that the eigenvalues of non-Hermitian Hamiltonians with $\mathcal {PT}$ symmetry can be real, and attracted extensive attention [1–12]. These non-Hermitian Hamiltonians satisfy the $\mathcal {PT}$-symmetric condition via an appropriate balance of gain and loss [13–19]. A real-to-complex spectrum ($\mathcal {PT}$-symmetric to broken $\mathcal {PT}$-symmetric phase) transition can be observed in these non-Hermitian Hamiltonians, when the $\mathcal {PT}$-symmetry condition is broken [20–22]. The phase transition point is called the exceptional point (EP) [23–26], where both the eigenvalues and the corresponding eigenstates of the system coalesce.

Despite the successful development of $\mathcal {PT}$-symmetric quantum mechanical theory, the experimental realization of $\mathcal {PT}$ symmetry quantum systems remains challenging. Optical systems provide fertile ground for theoretical and experimental studies of $\mathcal {PT}$ symmetry since the paraxial optical diffraction equation is equivalent to the quantum-mechanical Schrödinger equation [27]. The $\mathcal {PT}$ symmetry has been experimentally demonstrated in various optical systems, such as resonators, waveguides, and lattices [28–33]. Based on the similarity of the optical and acoustic field, $\mathcal {PT}$ symmetry has also been studied theoretically and experimentally in acoustic or phonon regimes [34–36]. However, experimental realizations of acoustic or phonon $\mathcal {PT}$ symmetry have mainly focused on optomechanical systems and are rarely performed in other systems [37–39].

Superconducting quantum circuits (SQC) are ideal platforms for constructing various quantum systems by coupling with other systems [40–43]. Superconducting qubits [44] with Josephson junction are easily connected with other quantum systems, such as microwave resonators, nanomechanical resonators, and spin ensembles [45–53]. The coupling form of superconducting qubits with other quantum systems is flexible, including transverse couplings and longitudinal couplings [54–65]. Taking advantage of these properties of SQC, some quantum systems with gain and loss are constructed by SQC to study $\mathcal {PT}$ symmetry and related non-Hermitian properties [66–68]. However, the phononic $\mathcal {PT}$-symmetric quantum systems constructed by SQC longitudinal coupling are rarely studied.

In this work, we propose a hybrid quantum system composed of two superconducting qubits and two coupled nanomechanical resonators for studying $\mathcal {PT}$ symmetry and non-Hermitian properties in the phononic system. The active (passive) subsystem is achieved by the longitudinal coupling of the resonator and the fast dissipative qubit with a blue-sideband driving (red-sideband driving). By adjusting the coupling strength between two nanomechanical resonators, the transition from $\mathcal {PT}$-symmetric to broken $\mathcal {PT}$-symmetric phase can be observed. The acoustic signal absorption, amplification, and group delay can be observed in our $\mathcal {PT}$-symmetric hybrid quantum system. These findings enable new applications for controlling the transmission of acoustic signals in quantum information processes.

This paper is organized as follows. In section 2, we introduce the hybrid quantum system that we propose for the realization of $\mathcal {PT}$ symmetry. In section 3, we construct the gain and loss subsystem via different sideband drivings of qubits and derive the effective $\mathcal {PT}$-symmetric Hamiltonian for the system. In section 4, we study the phase transitions in the $\mathcal {PT}$-symmetric hybrid quantum system by controlling the coupling strength between two resonators. We discuss the acoustic signal transmission in section 5. The unique control of output fields, especially for acoustic signal amplification, absorption, and delay, is demonstrated. The summary is presented in section 6.

2. Setup

We consider a superconducting hybrid quantum system for studying $\mathcal {PT}$ symmetry as shown in Fig. 1(a). This system is composed of two coupled nanomechanical cantilever resonators and two superconducting flux qubits that are magnetically coupled to two cantilevers via an external magnetic tip. The magnetic tip is attached to the cantilever and produces a time-varying magnetic field. The motion of the cantilever can be described by the Hamiltonian $H_{ri} = \hbar \omega _{ri}b_{i}^{\dagger }b_{i}$ $(i=1,2)$, where $b_{i}^{\dagger }(b_{i})$ is the creation (annihilation) operator for the fundamental vibrational mode with frequency $\omega _{ri}$. The qubit is a gap-tunable flux qubit that is composed of a qubit loop and a superconducting quantum interference device (SQUID) [69–71], for which the Hamiltonian is [72,73]

 

Fig. 1. (a) Schematic of a superconducting hybrid quantum system for studying $\mathcal {PT}$ symmetry. Each silicon nanomechanical cantilever resonator with dimensions $(l,w,t)$ is coupled with a flux qubit via an external magnetic tip attached to the cantilever. Two resonators separated by a distance $d$ are coupled indirectly through capacitive interactions with nearby wires. The inset shows the lumped-circuit layout of the flux qubit. (b) Circuit model that describes the coupling between two silicon nanomechanical cantilever resonators.

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The first term of $H_{qi}$ is the energy bias, which satisfies $\epsilon _i(\Phi _{zi}) = 2I_{pi}(\Phi _{zi} - \Phi _0/2)$. Here, $I_{pi}$ is the persistent current in the qubit loop, $\Phi _0=\hbar /2\pi$ is the flux quantum, and $\Phi _{zi}$ is the external flux of the qubit loop that can be produced by the flux driving. In our system, the qubit loop is inductively coupled to two identical microwave control lines symmetrically distributed on both sides of the qubit via a mutual kinetic inductance $M_i$ [74,75]. We add a magnetic flux threading through the qubit loop $\Phi _{\text {ext}}=\Phi _0$ to minimize the flux noise. Then we apply a time-dependent current $I_i(t)=\varepsilon _{pi} \text {cos}(\omega _{pi} t)$ in the microwave control line, where $\varepsilon _{pi}$ is the microwave current amplitude and $\omega _{pi}$ is the frequency of the microwave current. The first term of $H_{qi}$ can be re-expressed as $H_{qi} = \eta _i I_i(t)\bar {\sigma }_{z}$, where $\eta _i = M_iI_{pi}$ is the coupling strength between qubit loop and microwave control line [72].

The second term of $H_{qi}$ is the qubit gap $\Omega _i(\Phi _{xi})$ that dependents on the external flux $\Phi _{xi}$ threading the SQUID loop, which can be expressed as $\Phi _{xi} = \int _{Si}[B_{\text {total,i}}(x,y,z) + B_{\text {ext,i}}]dxdy \simeq \int _{Si}[\frac {\partial B_{\text {total},i,z}(x,y,z)}{\partial z}|_{z=z_{0}}(z-z_{0})+B_{\text {total},i,z}(x,y,z_{0}) + B_{\text {ext,i}}]dxdy$ [46,52]. Here, $B_{\text {total},i,z}(x,y,z)$ is the $z$ component of the total fields created by the magnetic tip of the $i$ th resonator, $B_{\text {ext,i}}$ is an applied external magnetic field, $S_i$ is the coupling area of the SQUID loop, $z_{0}$ is the average position of the SQUID loop, and $\frac {\partial B_{\text {total,i,z}}(x,y,z)}{\partial z}|_{z=z_{0}}$ is the magnetic field gradient at the position of the qubit [46,76]. The size of the magnetic tip we use here is $\sim 100 \ \text {nm}$, as reported in Ref. [77]. The magnetic gradient does not change very much in the area under the tip, which can be considered as a constant $G_{mi}$ in the effective coupling area. Assume the static part of $\Omega _i(\Phi _{xi})$ is $\hbar \omega _{qi}/2$, which is controlled by an applied external magnetic field. The qubit gap can be expressed as $\Omega _i(\Phi _{xi}) = \hbar \omega _{qi}/2 + I_{pi}G_{mi} S_{\text {eff,i}}b_{0i}(b_i^{\dagger } + b_i)\bar {\sigma }_{xi}$, where $b_{0i}(b_i^{\dagger } + b_i) = z -z_{0}$, and $b_{0i} = \sqrt {\hbar /2m\omega _{ri}}$ is the zero-point fluctuation amplitude of the mechanical resonator with mass $m_i$. We can find that the second term of $H_{qi}$ includes the coupling between the qubit and the nanomechanical cantilever, that is, $H_{\text {int}}=g_{i}(b_i^{\dagger } + b_i)\bar {\sigma }_{x}$, where the coupling strength $g_i =I_{pi}G_{mi} S_{\text {eff,i}}b_{0i}/\hbar$. In the new basis of the eigenstates of the qubit ${|e\rangle = (|+\rangle + |-\rangle )/\sqrt {2}}$ and ${|g\rangle = (|+\rangle - |-\rangle )/\sqrt {2}}$, the qubit Hamiltonian can be written as

To achieve the interaction between two resonators, the nanomechanical cantilevers are charged and interact capacitively with nearby wires interconnecting them [78,79]. The circuit model describes the coupling between two resonators as shown in Fig. 1(b). The effective interaction between two resonators is introduced by the variations of the resonator-wire capacitance $C_i(z_i)$ with the position of the tip $z_i = b_{0}(b_i + b_i^{\dagger })$. The phonon-phonon coupling is due to the electrostatic energy $W_{\text {el}}$ of the underlying circuit, which can be calculated by $\hbar J_{ij} = b_{0}^2(\partial ^2 W_{\text {el}}/\partial z_i \partial z_j)|_{{z_i}=0}$ [78]. We assume two resonators separated by a distance $d$ and connected by a wire of self-capacitance $C_w \approx \epsilon _0 d$, where $\epsilon _0$ is the vacuum permittivity. Then we calculate the electrostatic energy $W_{\text {el}} = -(U^2/2)(C_{\sum }C_w/C_{\sum } + C_w)$, where $C_{\sum } = C_1(z_1) + C_2(z_2)$ and $U$ is the applied voltage. The coupling Hamiltonian between two resonators is

Here, $J \equiv J_{12}\equiv J_{21}$ and $C_i(z_i) \simeq C(1 - z_i/h)$, where $h$ is the mean electrode spacing. The coupling strength $J$ between two resonators can be controlled by the applied voltage $U$.

After combing all parts, the Hamiltonian of the system can be written as (setting $\hbar =1$)

Applying a frame rotating at a frequency $\omega _{pi}$ and adopting the rotating wave approximation, the Hamiltonian of the total system becomes [80]

Here, we estimate the parameters of this Hamiltonian under experimental conditions. We consider the cantilever resonator with dimensions $(l,w,t) = (1,0.05, 0.05) \mu \text {m}$, which corresponds a fundamental frequency is approximately $\omega = 2\pi \times 50 \text {MHz}$, and the zero-point fluctuation of the phonon mode is $b_0 = 1.4 \times 10^{-13} \text {m}$ [81]. For superconducting persistent current, the range of superconducting flux qubits is $I_{p} \approx 500 \ \text {nA}$ [56]. A magnetic tip with a magnetic field gradient $G_{m} \approx 8\times 10^6 \ \text {T/m}$ is experimentally realized, which corresponds to the effective coupling area $S_{\text {eff}} \approx 80 \ \text {nm} \times 80 \ \text {nm} \sim 0.64 \times 10^{-14} \ \text {m}^2$ [77]. The coupling strength $g \approx 2\pi \times 5 \ \text {MHz} \approx 0.1 \omega$, which reaches the strong coupling regimes. The coupling of superconducting qubits and mechanical cantilevers has been experimentally achieved, and the strength of the coupling can be tuned by changing the relative positions of the magnetic tip and superconducting loop [52]. Moreover, for two resonators separated by $d = 100$ $\mu m$, $h = 100$ $\text {nm}$, and a typical electrode capacitance $C =0.1$ $\text {fF}$, we obtain the coupling strength $J = 1$ $\text {MHz}$ between two resonators with an applied voltage $U = 1$ $\text {V}$ [78]. In this way, a superconducting hybrid system with adjustable properties can be realized.

3. Realization of $\mathcal {PT}$ symmetric system by different sideband drives

Parity-time symmetric systems consist of non-Hermitian subsystems with balanced gain and loss configurations. In this section, we show how to construct the gain and loss subsystem based on the above setup. Performing the polariton transformation [82] to the total Hamiltonian in Eq. (6)

To build the gain and loss subsystems, we need different sideband drives for the qubits. The energy diagrams of two qubits with different sideband drives are shown in Fig. 2. For the subsystem that realizes the gain effect, the qubit drive satisfies the blue-sideband drive condition $\omega _{p1} =\omega _{q1} +\omega _r$. For the subsystem that implements the loss effect, the qubit drive satisfies the red-sideband drive condition $\omega _{p2} =\omega _{q2} -\omega _r$. After two qubits are driven with different sidebands driving, the Eq. (9) becomes

For convenience, we denote the coupling strength of qubit and resonator as $G_1 = 2\eta _1 \epsilon _1 \beta _1$ and $G_2 = 2\eta _2 \epsilon _2 \beta _2$, respectively. In order to focus more clearly on the gain and loss effects of the two resonators, the qubit’s freedom needs to be removed.

 

Fig. 2. Energy diagram of blue-sideband driving qubit $1$ with frequency $\omega _{p1} =\omega _{q1} +\omega _r$ (a) and red-sideband driving qubit $2$ with frequency $\omega _{p2} =\omega _{q2} -\omega _r$(b).

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It is well known that mechanical modes decay to the thermal environment with finite temperature $T$, and the thermal phonon number at frequency $\omega _r$ can be calculated by $n_{th}=(e^{\hbar \omega _r/k_B T} -1)^{-1} \simeq k_B T/\hbar \omega _r$. Defining the Lindblad operator $D[A,\Omega ]\hat {\rho }=\frac {\Omega }{2}(A\hat {\rho }A^{\dagger }-A^{\dagger }A\hat {\rho })+\text {H.c}$, the system density matrix $\hat {\rho }(t)$ evolution follows the master equation

In the above equation, $\Gamma _i$ represents the energy relaxing rate of the qubit, and $\xi _i =\omega _r/Q_i$ is the dissipative rate for the mechanical resonator. As demonstrated in Appendix A, the excited states of the qubit for the gain subsystem can be adiabatically eliminated under the condition $2G_1^{2}/\Gamma _1 \ll \Gamma _1$, and the excited states of the qubit of the loss subsystem can be adiabatically eliminated under the condition $2G_2^{2}/\Gamma _2 \ll \Gamma _2$. Therefore, we obtain the evolution equation for the reduced density matrix $\mu _i$ of the two mechanical cantilevers as follows [46,83]

Let $\bar {n}_i(t)= \text {Tr}\{\mu _i(t)b_i^{\dagger }b_i\}$ denotes the mean phonon number of each mechanical cantilever resonator, the classical equation which describes the evolution for $\bar {n}_i$ can be obtained [84]

It is easy to find that the photon number of resonator 1 will increase with $2G_1^2/\Gamma _1-\xi _1$, so there exists a gain effect. The phonon number of resonator 2 will decrease with $2G_2^2/\Gamma _2+\xi _2$, so there exists a loss effect. The equation of phonon number in the mechanical cantilevers versus time is calculated as:

Here we assume $2G_i^2/\Gamma _i \gg \xi _i$, that is, the coupling rate between the qubit and the resonator is much greater than the loss of the resonator. It should be noted that there exists a gain limit for resonator 1 and a loss limit for resonator 2. When the whole system reaches its steady state, which requires $d\bar {n}_1/{dt}=0$ and $d\bar {n}_2/{dt} =0$, so we have $\bar {n}_1 = -(n_{th}\xi _1\Gamma _1)/2G_1^2 -1$ and $\bar {n}_2=(n_{th}\xi _2\Gamma _2)/2G_2^2$. In the regime, $\Gamma _i \gg 2 G_i^2/\Gamma _i$, with the very large dissipative rate of qubits, the coherent transfer of phonons between the cantilever and the qubit will be destroyed due to the qubit energy decay process with the environment. The gain rate $2G_1^2/\Gamma _1$ and the loss rate $2G_2^2/\Gamma _2$ will decrease with $\Gamma _1$ and $\Gamma _2$. This means that the very strong energy dissipative rate of the qubit will destroy the gaining or losing process.

We numerically simulate the phonon number of two mechanical cantilevers by using the original Hamiltonian in Eq. (6) and compare it with the analytical results in Eq. (14). The mean phonon number of two nanomechanical reasoners versus time $gt/2\pi$ is plotted in Fig. 3(a) or 3(b), respectively. As the evolution time increases, the average phonon number of resonator 1 increases gradually, and the average phonon number of resonator 2 decreases gradually. It also can be found that as the evolution time increases, the numerical solution and the analytical solution do not fit well. This is because many high-frequency oscillatory terms are removed in solving the effective Hamiltonian. We then give the analytical and numerical solution error analysis of the gain system and the dissipative system, as shown in Fig. (3c) or (3d), respectively. When the coupling strength $g_i$ and the driving field strength $\epsilon _i$ are increased simultaneously, the numerical and analytical solutions’ errors become more considerable. The red dashed line indicates that the error between the analytical and numerical solutions of the gain or loss subsystem is $0.2$. To keep the error below $0.2$, the values of $g_i$ and $\epsilon _i$ should be in the region of the left half of the red dashed line in Fig. (3c) or (3d). In our following analysis, we all keep the error of our hybrid system below $0.2$. This error control is relatively easy to achieve in our hybrid quantum system due to the longitudinal coupling. The longitudinal coupling between the qubit and the resonant cavity will not produce a Lamb shift to the qubit, and there is no dispersive coupling between them, so the lifetime of the qubit does not depend on the resonator. In our system, we only need to consider the dissipation rate of the qubit, not the effect of the resonator on the qubit. After adiabatically eliminating the qubit’s degree of freedom, the system can be treated as two coupled harmonic resonators with the same frequency $\omega _r$. The effective non-Hermitian Hamiltonian of this hybrid quantum system can be written as

 

Fig. 3. The mean phonon number of the nanomechanical cantilever resonator as a function of time $gt/2\pi$ in the gain subsystem (a) and loss subsystem (b). The analytical and numerical solution error analysis of the gain (c) and dissipation subsystem (d). Here, we choose $\omega _r = 10$ MHz, $\Gamma _i = 4$ MHz, and $G_i= 0.05$ MHz (corresponding to $\eta _i = 2$, $\epsilon _i = 0.5$ MHz, and $g_i = 0.5$ MHz) (i=1,2).

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4. Controllable phase transition

It is well known that a $\mathcal {PT}$-symmetric Hamiltonian may exhibit a real spectrum for some certain parameter regimes. The (phase) transition from a real spectrum ($\mathcal {PT}$-symmetric phase) to a complex-valued spectrum (broken $\mathcal {PT}$-symmetric phase) occurs at the phase transition point. The eigenvalues and corresponding eigenvectors of the Hamiltonian merge simultaneously at the phase transition point, and many interesting and counterintuitive phenomena can be observed. In the following, we discuss how phase transition can take place in our hybrid quantum system.

According to Eq. (15), the coupling between two resonators leads two supermodes $B_+ = (b_1+b_2)/\sqrt {2}$ and $B_{-} = (b_1-b_2)/\sqrt {2}$, with the eigenfrequencies

The real and imaginary parts of $\omega _{\pm }$ quantify the frequencies and linewidth of these two supermodes, and the square root in the expression measures the competition between the coupling strength and the loss and gain of the resonators.

We first consider that the gain and loss are strictly balanced with $\gamma = \kappa$, as shown in the black-solid and red-dashed curves in Figs. 4(a) and 4(b). When $J > \kappa /2$, the coupling of two mechanical resonators produces two separate supermodes with different frequencies $\omega _r \pm \sqrt {J^2 - \kappa /8}$). However, these two supermodes have the same linewidth $(\gamma -\kappa )/4 = 0$. This regime corresponds to the $\mathcal {PT}$-symmetric regime, which is the blue area in Figs. 4(a) and 4(b). In the $\mathcal {PT}$-symmetric regime, since the coupling strength is large enough, the energy in the gain resonator can be transferred to the dissipated resonator in time, and the system can be in equilibrium. When $J < \kappa /2$, these two supermodes are degenerate with the same frequency $\omega _r$ but with different linewidths $\pm \sqrt {\kappa ^2/4-J^2}$. This regime corresponds to the broken $\mathcal {PT}$-symmetric regime, which is the pink area in Figs. 4(a) and 4(b). In broken $\mathcal {PT}$-symmetric regime, the energy in the gain resonator increases exponentially, while the energy in the dissipative resonator decreases exponentially. However, when the coupling strength is small, the gain is not sufficient to quickly compensate for the dissipation. At the critical point $J = \kappa /2$, the resonance frequencies and the linewidths of these two supermodes become degenerate. This critical point is the EP of our hybrid system, also known as the $\mathcal {PT}$ phase transition point. As shown in Figs. 4(a) and 4(b), by increasing the effective coupling strength $J$, the transition from the broken to the unbroken $\mathcal {PT}$-symmetric phase can be achieved. Unlike the $\mathcal {PT}$-symmetric system with gain and loss, when two mechanical modes have the same loss $\kappa$, the phase transition point does not exist, as shown in the blue-solid and the green-dashed curves in Figs. 4(a) and 4(b). The frequencies of these two supermodes $\omega _r \pm \sqrt {J^2}$ are different and the frequency separation between these two supermodes is proportional to the coupling strength $J$. These two supermodes always have identical linewidth $\kappa /2$.

 

Fig. 4. $\mathcal {PT}$ phase transition in our hybrid quantum system. The real (a,c) and imaginary (b,d) parts of the eigenfrequencies $\omega _{\pm }$ plotted as a function of coupling strength $J$. The results for the case of balanced gain and loss are shown in (a) and (b), and for the case of the unbalanced gain-to-loss are plotted in (c) and (d). The black-solid and red-dashed curves in (a) and (b) represent the system gain and loss balance, while the blue-solid and green-dashed curves represent the system without mechanical gain. The black-solid and red-dashed curves in (c) and (d) are calculated with $\gamma = \kappa /2$, and the blue-solid and green-dashed are calculated with $\gamma = 2\kappa$. Broken and unbroken $\mathcal {PT}$-symmetric regimes are distinguished by different background colors.

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We next consider more general cases in that gain and loss are not strictly balanced. Figures 4(c) and 4(d) show two examples with $\gamma = \kappa /2$ and $\gamma = 2\kappa$. It can be found that the gain and loss ratios are different, and the critical point of the $\mathcal {PT}$ phase transition of the system is different. The smaller the ratio of gain to loss, the greater coupling strength $J$ that needs to induce the $\mathcal {PT}$ phase change. This can be understood as when the dissipation is large, the small coupling strength cannot transfer the energy in the gain system to the dissipation system in time, so the system cannot be in a balanced state. Therefore, the greater the dissipation $\kappa$, the greater coupling strength $J$ is required to reach a balanced state.

5. Controllable acoustic output field in $\mathcal {PT}$-symmetric hybrid quantum system

Controllable acoustic signal transmission and detection play an important role in quantum information processing [8,85,86]. The $\mathcal {PT}$-symmetric system consisting non-Hermitian subsystems with balanced gain and loss configurations, can be used for signal detection and manipulation. In this section, we will study the output field of our $\mathcal {PT}$-symmetric hybrid quantum system. In particular, we will show that the $\mathcal {PT}$ symmetry parameters affect the transmission properties for both balanced gain-to-loss and unbalanced gain-to-loss cases.

Here, we assume that the detection field is applied to the dissipative subsystem, the system Hamiltonian can be expressed as

Solving Eq. (19) in the steady-state, we obtain the intracavity fields $B_1$ and $B_2$ as

Using this transmission coefficient, we discuss below how to achieve controllable acoustic signal amplification, absorption, and group delay in the $\mathcal {PT}$-symmetric hybrid quantum system.

5.1 Balanced gain-to-loss case

We first consider the case where the gain and loss of the system are balanced. The relationship between the transmission coefficient $T$ and the detuning $\Delta$ with the same coupling strength $J = 0.425\gamma$ is calculated and shown in Fig. 5(a). For the loss-loss system (represented by the black-dotted line), an Autler-Townes-splitting-like spectrum [87–94]is observed. However, for our $\mathcal {PT}$-symmetric system with gain and loss (represented by the red-solid line), signal amplification can be realized between two Autler-Townes absorption dips, where the peak is located at $\Delta = 0$. The dispersion behavior for both cases is calculated and shown in Fig. 5(b). Around the parameter regime $\Delta = 0$, both the gain-loss and loss-loss system exhibit a normal dispersion. Our $\mathcal {PT}$-symmetric gain-loss system has a larger dispersion change at $\Delta = 0$.

 

Fig. 5. Transmission coefficient $T$ and the dispersion Im($t_p$) versus detuning $\Delta$ with $J/\gamma = 0.425$ are shown in (a) and (b), respectively. The transmission coefficient $T$ as a function of detuning $\Delta$ for different coupling strengths $J$ of the gain-loss system is plotted in (c) and (d), and the loss-loss system is plotted in (e) and (f). The other system parameters are $\gamma = 0.4 \text {MHz}$, $\kappa = 0.4 \text {MHz}$, and $\kappa ^{\prime } = 0.2 \text {MHz}$.

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We next investigate how the transmission coefficient $T$ is affected by the coupling strength $J$ in the gain-loss system and the loss-loss system, respectively. The transmission coefficient $T$ versus detuning $\Delta$ for different coupling strength $J$ is calculated and shown in Figs. 5(c) and 5(d). In the broken $\mathcal {PT}$-symmetry scheme, as shown in Fig. 5(c), the transmission spectrum varies with the coupling strength $J$. When $J$ is small, the transmission spectrum appears as an absorption valley at $\Delta =0$. With the increase of the coupling strength $J$, the transmission coefficient $T$ is greater than $1$ at $\Delta =0$. When the coupling strength $J$ is close to the phase transition point, the transmission coefficient $T$ is much larger than $1$, and the amplification of the acoustic signal is realized. When the coupling strength $J$ exceeds the phase transition point, the transmission coefficient $T$ appears as two amplified peaks, as shown in Fig. 5(d). This is because beyond the phase transition point, the $\mathcal {PT}$-symmetric hybrid quantum system behaves as two supermodes, and there are two amplified peaks in the transmission spectrum.

In contrast to the $\mathcal {PT}$-symmetric system, the transmission coefficient $T$ is always small than $1$ in the loss-loss system without gain, as shown in Figs. 5(e) and 5(f). It can also be found that the transmission spectrum appears as an absorption valley when the coupling strength is weak. As the coupling strength $J$ increases, the transmission spectrum appears as two absorption valleys. According to the previous discussion, there is no phase transition point in a system without gain. Therefore, by changing the coupling strength $J$, the transmission spectrum should maintain the mode splitting. However, when the coupling strength is small, the energy exchange between the two dissipative systems is small, so the transmission spectrum appears as an absorption valley. Mode splitting becomes apparent when the coupling strength is increased. To explore the influence of coupling strength $J$ on signal absorption and amplification, we take the logarithm of the transmission coefficient $T$ at $\Delta =0$. According to Eq. (21), the transmission coefficient $T$ at $\Delta =0$ is given by

 

Fig. 6. The logarithm of the transmission coefficient $T$ versus coupling strength $J$ for detuning $\Delta = 0$. The black solid line represents the case in which gain and loss are balanced. The red solid line indicates the case where there is no gain in the system.

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5.2 Unbalanced gain-to-loss case

Here we focus on the case where the gain and loss are not equal. The transmission coefficient $T$ versus detuning $\Delta$ is calculated and plotted in Fig. 7(a), where the coupling strength $J = 11\gamma$. For our $\mathcal {PT}$-symmetric hybrid quantum system (the blue-solid-curve), a remarkable acoustic signal amplification is observed around $\Delta = 0$. However, for the system without gain (red-dot-dashed curve), one can obtain an acoustically induced transparency spectrum [49,95–97]. The inset of Fig. 7(a) shows the center peak. The dispersion behavior for a $\mathcal {PT}$-symmetric system with gain and loss and a system without gain are shown in Fig. 7(b), where we find that both of them exhibit anomalous dispersion. However, we noticed an abrupt variation of the dispersion in $\mathcal {PT}$-symmetric system with gain and loss (see the blue solid curve in Fig. 7(b)), which is due to the enhancement of the acoustic transmission. The center dispersion variation is plotted in the inset of Fig. 7(b).

 

Fig. 7. The transmission coefficient $T$ and dispersion Im($t_p$) versus the detuning $\Delta$ with $J = 11\gamma$ are plotted in (a) and (b), respectively. The blue-solid curve (red-dot-solid curve) represents the gain-loss system (loss-loss system). Here, we set $\gamma = 0.1\text {MHz}$, $\kappa = 400 \gamma$, and $\kappa ^{\prime } = 1/2 \kappa$.

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It is well known that the group delay $\tau$ is caused by the extremely abrupt change of the refractive index. The group delay $\tau$ of our transmission field is defined as [98]

 

Fig. 8. Phase $\theta$ and group delay $\tau$ for the $\mathcal {PT}$-symmetric system with gain and loss (see the green-solid curve) and the system without gain (see the orange-dashed curve). The parameters are the same as Fig. 7.

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In addition, the acoustic signal absorption and amplification can also be achieved by changing the coupling strength $J$ in our unbalanced gain-to-loss case. The logarithm of the transmission coefficient $T$ varies with the coupling strength $J$ in the unbalanced gain-to-loss case is the same as in the balanced gain-to-loss case (shown in Fig. 6). When the gain and loss are unbalanced, the signal is also amplified most at the phase transition point in the PT-symmetric system with gain and loss, and no signal is amplified in the system without gain. It means that in our $\mathcal {PT}$ symmetric hybrid quantum system, even if gain and loss are not equal, the absorption and amplification of the acoustic signal can be realized by changing the coupling strength $J$.

6. Conclusion

In conclusion, we have proposed a hybrid quantum system consisting of two superconducting qubits and two coupled nanomechanical cantilever resonators. We show that a $\mathcal {PT}$-symmetric phononic system can be constructed by active and passive modes. Effective gain or loss of the phonon mode is introduced by longitudinal coupling of the resonator and the fast dissipative qubit with a blue-sideband or red-sideband driving, respectively. A $\mathcal {PT}$-symmetric to broken $\mathcal {PT}$-symmetric phase transition can be achieved by changing the coupling strength $J$ between resonators for both balanced and unbalanced gain-to-loss cases. When a resonant weak probe field is added to the dissipative resonator, the transition of the acoustic signal from absorption to amplification can be achieved in the balanced gain-loss case. For the case of unbalanced gain and loss, both acoustically induced transparency and anomalous dispersion can be observed around $\Delta = 0$, which results in an ultralong group delay. This hybrid quantum system for constructing a $\mathcal {PT}$-symmetric phononic system can be experimentally realized, which provides a good candidate for the controllable transmission of acoustic signals.