Abstract

Few-photon effects such as photon blockade and tunneling have potential applications in modern quantum technology. To enhance the few-photon effects in an optomechanical system, we introduce a coherent feedback loop to cavity mode theoretically. By studying the second-order correlation function, we show that the photon blockade effect can be improved with feedback. Under appropriate parameters, the photon blockade effect exists even when cavity decay rate is larger than the single-photon optomechanical coupling coefficient, which may reduce the difficulty of realizing single-photon source in experiments. Through further study of the third-order correlation function, we show that the tunneling effect can also be enhanced by feedback. In addition, we discuss the application of feedback on Schrödinger-cat state generation in an optomechanical system. The result shows that the fidelity of cat state generation can be improved in the presence of feedback loop.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Optomechanical systems involved the interaction between optical and mechanical modes due to radiation pressure force have obtained a substantial development in the last decade. Under strong driving, many interesting phenomena such as sideband cooling [16], normal-mode splitting [7,8], squeezing generation [912], macroscopic entanglement [13,14], etc., have so far been observed in experiments or theoretically researched. Under strong driving conditions, although the radiation pressure effect can be enhanced, the nonlinear effects of optomechanical system are vanished. When the cavity is weakly driven, some few-photon effects, such as blockade and tunneling, caused by the optomechanical nonlinearity can be demonstrated.

Photon blockade effects characterize the antibunching of cavity fields [15]. For the single-photon blockade, excitation of the cavity by a first photon prevent the excitation of a second one, which converts a poissonian stream of photons into a sub-poissonian [16]. Therefore, the photon blockade mechanism can reduce the probability of multiphoton inside and outside the cavity, so as to realize the single-photon source. The single-photon source is widely used in quantum information processing [17] and quantum computing [1820]. These application values have promoted proposals for realizing single-photon blockade effect [2134].

Physically, photon blockade can be realized either through single-photon resonance or by destructive interference, where the former is called conventional photon blockade, and the later is refer to unconventional photon blockade. For conventional photon blockade, the nonlinearity of the system is required to form an anharmonic energy-level structure. Optomechanical system with nonlinear interaction between cavity mode and mechanical resonator is a candidate for realizing conventional photon blockade. In most optomechanical experiments to date, the single-photon nonlinear coupling is smaller than the mechanical frequency and the cavity linewidth [35]. Therefore, it is still a challenge for experimentally observing photon blockade in optomechanical system. To enhance photon blockade effect, several proposals are focused on either improving the effective single-photon nonlinear coupling strength [36,37] or reducing effective mechanical frequency through cross-Kerr nonlinearity [38]. Nevertheless, conventional photon blockade fails when the optomechanical nonlinearity is weaker than the cavity linewidth.

In addition to photon blockade effects, quantum tunneling and cat state are other interesting phenomenon in few-photon optomechanics. For photon-induced tunneling, the excitation probability of subsequent photons is increased by a first excited photon [39]. Cat state, due to its application both in fundamental aspect about quantum-classical boundary [40] and in quantum technologies, such as quantum computing [41], is an important research field. The mechanism of preparing cat state is based on the conditional displacement dynamics. Therefore, the distance between two superposed coherent states in phase space depends on the magnitude of displacement, i.e. coupling strength. In the past few years, several theoretical works have been devoted to generate cat states in optomechanical systems [4244]. However, it is still a challenge to realize cat state in optomechanical system in experiments, partly, because of the difficulties for realizing coupling strength larger than the cavity decay rate experimentally [35]. Therefore, it is possible to create distinct cat state by reducing the effective cavity linewidth.

Feedback are important quantum technology in quantum optics [4555]. Recently, feedback in optomechanical system has aroused people’s special interest. Some many-photon phenomena based on feedback have shown great advantages, which include realizing normal-mode splitting in a weakly coupled optomechanical system [56], improving entanglement [57] and steering [58], enhancing ground state cooling of the vibrator [59]. To date, no literature is found on few-photon optomechanical effects with feedback to the best of our knowledge. In this paper, we mainly focused on studying the influence of coherent feedback on photon blockade. By simulating the equal-time second-order correlation function, relative distribution of photon-number from the corresponding Poissonian distribution and time-delay second-order correlation function, we show that the photon blockade effect can be enhanced by coherent feedback. By further discussing the equal-time third-order correlation function, we show that the photon tunneling effect can be enhanced as well. Moreover, we also discuss the cat state generation with coherent feedback in the absence of laser driving. The result shows that the fidelity of cat state generation can be improved in the presence of coherent feedback loop.

The remainder of this paper is organized as follows. Section 2 presents photon blockade effect in standard optomechanical system. Section 3 studies the photon blockade, as well as photon tunneling by introducing coherent feedback in optomechanical system. Section 4 discusses the cat state generation with coherent feedback. Discussions and conclusions are given in Section 5 and 6, respectively.

2. Photon blockade effect with standard optomechanical systems

We consider an optomechanical system in which the cavity (resonance frequency $\omega _c$) couples to mechanical resonator (resonance frequency $\omega _m$) by radiation pressure. The cavity is pumped by classical laser with driving strength $\Omega$ and frequency $\omega _l$. In the rotating frame at laser frequency $\omega _l$, the Hamiltonian for the system is written as

$$H_1 = \hbar \Delta a ^ { {\dagger} } a + \hbar \omega_{m} b^{{\dagger}}b-\hbar g_{0}a ^ { {\dagger} } a \left(b+b^{{\dagger}}\right)+\hbar \Omega \left (a+a^{{\dagger}}\right),$$
where $a$ ($a^{\dagger }$) and $b$ ($b^{\dagger }$) are the annihilation (creation) operator for the optical and mechanical mode, respectively; $\Delta =\omega _c-\omega _l$ is the cavity-laser detuning, $g_0$ is the nonlincar coupling strength. Due to the nonlinear item, the optomechanical system can be used to realize the anti-bunching effect, so as to produce the single-photon source. In order to enhance the anti-bunching, we introduce coherent feedback to the system, where output fields of the cavity in the right are fed back into the left input port through three highly reflective mirrors (HRM) and a controllable beam splitter (CBS), as shown in Fig. 1(a) in the sketch of the proposed experimental setup. To make the problem clear, we first give a brief discussion of photon blockade effect in the standard optomechanical system without the coherent feedback loop. The blockade effect in the presence of coherent feedback will be discussed later.

To find the conventional blockade conditions of photons (i.e., single-photon resonance conditions), one can diagonalize the Hamiltonian by applying a unitary transform $U = \exp [-g_0 a^{\dagger } a (b^{\dagger }-b)/\omega _m]$ to Eq. (1), the transformed Hamiltonian $H_2 = U H_1 U^{\dagger }$ then reads ($\hbar$=1)

$$H_2 = \Delta a^{{\dagger}} a + \omega_m b^{{\dagger}} b - \frac{g^2_0}{\omega_m}(a^{{\dagger}}a)^2 +\Omega(a e^{{-}g_0/\omega_m (b^{{\dagger}}-b)} + a^{{\dagger}} e^{g_0/\omega_m(b^{{\dagger}}-b)}).$$

Under the weak optomechanical coupling condition $g_0\ll \omega _m$, exponential factors in Eq. (2) can be approximately omitted [60], and the validity of the approximation is verified in appendix for details. Therefore, the Hamiltonian can be approximated as

$$H_3 = \Delta a^{{\dagger}} a + \omega_m b^{{\dagger}}b - \frac{g^2_0}{\omega_m} (a^{{\dagger}}a)^2 + \Omega(a^{{\dagger}}+a).$$

Solving the eigenequation of $H_3$ by omitting $\Omega$ in the weak driving condition, we obtain the eigenvalues of $H_3$ as $n \Delta +m \omega _m- \frac {g^2 _0 n^2}{\omega _m}$. Correspondingly, the eigenstates are

$$|n,\tilde{m}(n)\rangle=e^{\frac{g_0n}{\omega_m}(b-b^{{\dagger}})}|n,m\rangle.$$

 figure: Fig. 1.

Fig. 1. (a) Single-mode cavity optomechanical system with coherent feedback loop. The output of cavity fields are fed back into the input port through HRM and CBS. (b) Energy level diagram of cavity mode under eigenstate representation.

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In the diagonalized representation, there is no interaction between cavity mode and oscillator. Therefore, we can ignore the effects of the oscillator to photon blockade temporarily, and consider the cavity mode alone. This treatment ignores the phonon sideband effect [61,62], but can simplify the problem. For the phonon sideband effect, we will discuss it later. By employing $a=\sum _n \sqrt {n+1}|n\rangle \langle n+1|$ to Eq. (3), we can rewrite the Hamiltonian as

$$H_3 = \sum_{n}[(n \Delta - \frac{g^2 _0 n^2}{\omega_m})\left| n \right\rangle \left\langle n \right| + \Omega\sqrt{n+1}\left(\left| n \right\rangle \left\langle n+1 \right| +\left| n+1 \right\rangle \left\langle n \right|\right)].$$

By the above equation, we give the energy level diagram of the cavity mode in the absence of laser driving, i.e. $\Omega =0$, shown in Fig. 1(b). The energy gap between two adjacent energy levels $|n\rangle$ and $|n+1\rangle$ is $\Delta -g_0^2(2n+1)/\omega _m$, which is dependent of $n$. Under the weak driving condition, if $|0\rangle$ and $|1\rangle$ is resonant (i.e., $\Delta =g^2_0/\omega _m$), there will be detuning $2g_0^2/\omega _m$ between $|1\rangle$ and $|2\rangle$ (i.e., two-photon off-resonance), which makes the single-photon blockade possible. However, the existence of cavity dissipation can broaden the width of the energy level, which will destroy the realization of photon blockade effects (see Appendix for details).

To illustrate the above analysis, we simulate the equal-time second-order correlation function

$$g^{(2)} (0) = \frac {\langle a^{{\dagger}} a^{{\dagger}} a a \rangle} {{\langle a^{{\dagger}} a \rangle}^2}$$
by solving master equation
$$\begin{aligned}\dot{\rho} =& - i[H_{1},\rho] +\frac{\kappa}{2} (2a \rho a^{{\dagger}} - a^{{\dagger}} a \rho - \rho a^{{\dagger}} a)\\ & + \frac{\gamma}{2}(\bar{n}_{th}+1) (2b \rho b^{{\dagger}} - b^{{\dagger}} b \rho -\rho b^{{\dagger}} b)+ \frac{\gamma}{2}\bar{n}_{th} (2b^{{\dagger}} \rho b - b b^{{\dagger}} \rho -\rho b b^{{\dagger}}) \end{aligned}$$
in steady state, where $\kappa$ and $\gamma$ are the damping rate of cavity and mechanical oscillator, respectively. $\bar {n}_{th}$ is the thermal excited number from bath of mechanical mode.

In Fig. 2, we present $g^{(2)}(0)$ and average photon number $\langle a^{\dagger } a \rangle$ with the parameters $\omega _m/2\pi = 10$ MHz, $g_0/2\pi = 2.5$ MHz, and $\gamma /2\pi =0.01$ MHz. As shown in Fig. 2(a), the minimum $g^{(2)}(0)$ is located at the single-photon resonant condition $\Delta =\Delta _0$ with $\Delta _0=g_0^2/\omega _m$, which is not surprising since $g^{(2)} (0)= (2P_2 + 6P_3+12P_4)/(P_1+2P_2+3P_3+4P_4)^2$ in four-photon truncated subspace (see Eq. (4) in Appendix), where $P_n$ is the probability of the $n$ photon. In this case, owing to the off-resonance as shown in Fig. 1(b), the probabilities of higher photon excited states are suppressed. On the contrary, when $\Delta = 2 \Delta _0$, we can see that a peak for $g^{(2)} (0) > 1$ appears for different $\kappa$, which means that the two-photon tunneling happens due to the two-photon resonant condition $\Delta = 2 \Delta _0$. Obviously, the larger $\kappa$, the larger $g^{(2)}(0)$ at $\Delta = \Delta _0$, and smaller $g^{(2)}(0)$ at $\Delta = 2 \Delta _0$, which means large cavity decay will wreck both the blockade and tunneling effects. The destruction effect for resonant transition is especially obvious at three-photon resonant condition $\Delta = 3 \Delta _0$, where the peak disappears for large cavity decay. Moreover, the average photon number is lower at single-photon blockade condition with larger $\kappa$, as shown in Fig. 2(b), which means large cavity decay will hurt the preparation of single-photon source. In Fig. 2(c), we numerically simulated $g^{(2)} (0)$ as functions of $g_0/\omega _m$ and $\kappa /\omega _m$. We see that the cavity decay has negative effects on the phonon sideband effect. For strong single-photon coupling, the strong photon blockade effect can be observed in the deep-resolved-sideband region $\kappa /\omega _m < 0.01$ during each phonon sideband as shown in the white dotted rectangle in Fig. 2(c). By increasing coupling strength $g_0$ and reducing cavity decay $\kappa$ within a certain range, $g^{(2)} (0)$ can be smaller. Until now, several works have been focused on enhancing the blockade effect by improving the coupling strength. Figure 2(d) shows that the single-photon coupling $g_0/\omega _m$ has no influence on average photon number $\left \langle a^{\dagger } a \right \rangle$ at the single-photon resonant condition $\Delta = \Delta _0$, while $\left \langle a^{\dagger } a \right \rangle$ will decease with the increase of cavity decay rate $\kappa$. In this paper, we aim at reducing the effective cavity decay to enhance the photon blockade effect.

 figure: Fig. 2.

Fig. 2. Plot of $g^{(2)} (0)$ as a function of $\Delta /\Delta _0$ in (a), and $\left \langle a^{\dagger } a \right \rangle$ versus $\Delta /\Delta _0$ in (b) at different values of $\kappa$. Plot of $Log_{10}[g^{(2)} (0)]$ as functions of $g/\omega _m$ and $\kappa /\omega _m$ in (c), and $Log_{10}\left \langle a^{\dagger } a \right \rangle$ versus $g/\omega _m$ and $\kappa /\omega _m$ in (d) . We set $\omega _m/2\pi = 10$ MHz, $g_0=0.25\omega _m$ for (a) and (b), $\Delta = \Delta _0$ for (c) and (d). Other parameters are $\Omega =0.00125 \omega _m$, $\bar {n}_{th}=0$ and $\gamma =0.001 \omega _m$.

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3. Photon blockade effect with coherent feedback in optomechanical system

To address the problem proposed above, we add a coherent feedback loop to the optomechanical system, as shown in Fig. 1(a). The Heisenberg-Langevin equations for this model can be expressed as

$$ \dot{a} ={-}(i\Delta +\frac{\kappa_1}{2}+\frac{\kappa_2}{2})a +ig_{0} a (b+b^{{\dagger}})-i\Omega+\sqrt{\kappa_1} a_{1}^{in}+\sqrt{\kappa_2} a_{2}^{in}, $$
$$ \dot{b} ={-}(i\omega_{m}+ \frac{\gamma}{2}) b +i g_{0} a^{{\dagger}} a +\sqrt{\gamma}b_{in}, $$
where $\kappa _1$ and $\kappa _2$ are the decay of cavity in the left and right, respectively; $a_{j}^{in}$ ($j=1,2$) and $b_{in}$ are the corresponding noise operators of cavity and mechanical mode. By the standard input-output formula, we can obtain
$$a^ {out} = \sqrt{\kappa_2}a-a_{2}^{in}.$$

In the coherent feedback loop, the output cavity fields are send back to the input port, as shown in Fig. 1(a). The new input fields of cavity are superposition of the original input fields and returned output fields. Strictly speaking, since it takes time for the fields to return, the returned fields will have a time delay. For a $5-$cm cavity with a $10-$cm feedback loop, the delay time is about $10^{-10}$ s. Therefore, the delay can be safely neglected for mechanical resonator with its frequency of $10$ MHz [57]. Then, the modified input operator is

$$ a^{in}_{fb} = r e^{i \theta} a ^{out} + t a^{in}_1,$$
where $r$ and $t$ are the reflection and transmission coefficients, with $r^2 + t^2=1$; $\theta$ is the additional phase shift caused by CBS. In deriving the above equation, we do not consider the loss of light in the coherent feedback loop. Thereby, the modified motion equation for cavity mode, in the presence of coherent feedback loop, is
$$ \dot{a} ={-}(i \widetilde{\Delta} + \frac{\widetilde{\kappa}}{2}) a + i g_{0} a (b+b^{{\dagger}})-i \Omega+\sqrt{\widetilde{\kappa}} A_{in},$$
where the modified dissipation $\widetilde {\kappa }=\kappa _1 + \kappa _2-2r \sqrt {\kappa _1 \kappa _2} \cos {\theta }$ and detuning $\widetilde {\Delta } = \Delta - r \sqrt {\kappa _1 \kappa _2} \sin {\theta }$. Obviously, the dissipation $\widetilde {\kappa }$ can be smaller by adjusting $r$ and $\theta$. For convenience, we set $\theta =2n\pi$ and $\kappa _1 =\kappa _2=\kappa$, therefore we have $\widetilde {\kappa }=2 \kappa (1-r)$ and $\widetilde {\Delta }=\Delta$. The new input noise operator $A_{in}$ is
$$ A_{in} = \frac{\sqrt{\kappa_1} t a^{in}_1 + (\sqrt{\kappa_2}-\sqrt{\kappa_1} r e^{i \theta}) a^{in}_2} {\sqrt{\widetilde{\kappa}}}.$$

Through further derivation, we can find that the correlation function of the new noise operator $A_{in}$ satisfies: $\langle A_{in}^{\dagger }(t)A_{in}(t^{\prime })=0$ and $\langle A_{in}(t)A_{in}^{\dagger }(t^{\prime })=\delta (t-t^{\prime })$. Therefore $A_{in}$ describes vacuum noise. To sum up, by adding coherent feedback, the new Hamiltonian of the optomechanical system is modified as

$$ \widetilde{H} = \hbar \widetilde{\Delta} a^{{\dagger}} a + \hbar \omega_m b^{{\dagger}} b-\hbar g_0 a^{{\dagger}} a (b+b^{{\dagger}}) + \hbar\Omega(a + a^{{\dagger}}),$$
and the master equation of the system in the presence of coherent feedback is
$$\begin{aligned} \dot{\rho} =& - i[\widetilde{H},\rho] +\frac{\widetilde{\kappa}}{2} (2a \rho a^{{\dagger}} - a^{{\dagger}} a \rho - \rho a^{{\dagger}} a)\\ & + \frac{\gamma}{2}(\bar{n}_{th}+1) (2b \rho b^{{\dagger}} - b^{{\dagger}} b \rho -\rho b^{{\dagger}} b)+ \frac{\gamma}{2}\bar{n}_{th} (2b^{{\dagger}} \rho b - b b^{{\dagger}} \rho -\rho b b^{{\dagger}}). \end{aligned}$$

Now, we study the influence of the coherent feedback on the photon blockade effect. By numerically solving the above equation in steady state with Qutip [63], we simulate the second-order correlation function $g^{(2)}(0)$ in Fig. 3(a). We observe that $g^{(2)}(0)$ is smaller when $r=0.9$ at fixed $\kappa$ compared to $r=0$. When $\kappa >g_0$, $g^{(2)}(0)\approx 1$ without coherent feedback in the system, and $g^{(2)}(0)<1$ in the presence of coherent feedback. These phenomena mean the photon blockade effect can be enhanced when coherent feedback is introduced to the system, which can be understand since the effective dissipation $\widetilde {\kappa }=2 \kappa (1-r)$ can be reduced when we introduce the coherent feedback to the system. With the decrease of dissipation, the bandwidth of $|0\rangle$, $|1\rangle$, $|2\rangle$, $\cdots$ becomes narrower, and hence the blockade effect is better. In Fig. 3(b), average photon number $\langle a^{\dagger } a \rangle$ is presented. As $\kappa$ increases, the average photon number becomes smaller and smaller, and $\langle a^{\dagger } a \rangle$ decays rapidly without coherent feedback. The presence of coherent feedback can increase the average photon number, thus is more conducive to realize the single-photon source. In Fig. 3(c), we show the change of $g^{(2)}(0)$ and $\langle a^{\dagger } a\rangle$ with $r$ at $\kappa =0.05g_0$. It can be seen that the larger $r$ is, the better the blockade effect is, and the larger the average photon number is. Therefore, in order to improve the blockade effect, $r$ should be close to $1$.

 figure: Fig. 3.

Fig. 3. Plot of $g^{(2)} (0)$ as a function of $\kappa /g_0$ in (a), and $\left \langle a^{\dagger } a \right \rangle$ versus $\kappa /g_0$ in (b) with ($r=0.9$) and without ($r=0$) coherent feedback. (c) $g^{(2)} (0)$ and $\langle a^{\dagger }a\rangle$ (inset) as functions of $r$ with $\kappa =0.05 g_0$. (d) Relative probability population function with and without coherent feedback for $\kappa =0.01 g_0$. Other parameters are the same with Fig. 2.

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To further explore the statistical properties of photons, we use $y=Log_{10} (P_N/\mathcal {P}_N)$ to simulate a relative distribution of photon-number from the corresponding Poissonian distribution in Fig. 3(d) without and with coherent feedback, respectively, where $P_N$ is the possibility of Fock state $ \left | N \right \rangle $ and $\mathcal {P}_N = \ {{e^{ - {{\left | \alpha \right |}^2}}}{\left | \alpha \right |^{2N}}/N!\ }$ is Poissonian distribution. If $y$ is positive, it is super-Poisson distribution, otherwise it is sub-Poisson distribution. In Fig. 3(d), only the relative distribution $y$ for $N=1$ is positive, i.e. super-Poisson distribution, which implies that the single-photon blockade effect occurs. Comparing these two figures, one can see that the excitation of single-photon state $|1\rangle$ is enhanced in the presence of coherent feedback, which illustrates the enhancement of coherent feedback on blockade effect from the perspective of photon statistics.

In addition to $g^{(2)}(0)$, the time-delay second-order correlation function $g^{(2)}(\tau )$ defined by

$$ g^{(2)} (\tau) = \frac {\langle a^{{\dagger}} (t) a^{{\dagger}}(t+\tau) a(t+\tau) a (t) \rangle} {{\langle a^{{\dagger}} (t) a (t) \rangle}^2}$$
is another significant quantity to characterize the statistical properties of photons. $g^{(2)}(\tau )$ is proportional to the joint probability of detecting one photon at $t=0$ and detecting the next photon at $t=\tau$. We can observe that as time evolution, $g^{(2)}(\tau )$ approaches $1$, as shown in Fig. 4(a). Compared with $g^{(2)} (\tau )$ without coherent feedback, it is found that more time is taken for $g^{(2)}(\tau )$ evolves to $1$ in the presence of coherent feedback owing to the reducing of effective dissipation. At any moment when $\omega _m \tau$ is less than $800$, the probability of finding the second photon is lower in the presence of coherent feedback loop. This result suggests that the probability of two photons being detected continuously is reduced when coherent feedback is introduced to the system. In this paper, we consider that the mechanical frequency is in the order of $10$ MHz, thus $\omega _m \tau =800$ corresponds to $\tau \approx 10^{-4}$ s, which can guarantee the validity of omitting time delay $10^{-10}$ s in feedback loop. In Fig. 4(b), we explore the influence of coherent feedback on phonon sideband effect. Since $\widetilde {\Delta }=\Delta$, the resonant peaks between $|0,\widetilde {0}\rangle$ and $|2,\widetilde {m}\rangle$ for the cases with and without coherent feedback are both located at $g_0/\omega _m=\sqrt {m/2}$ under the conditions of single-photon blockade $\Delta =\Delta _0$, as shown in Fig. 4(b). And during each phonon sideband duration, the photon blockade effect is enhanced with coherent feedback.

 figure: Fig. 4.

Fig. 4. (a) Time evolution of time-delay second-order correlation function $g^{(2)}(\tau )$ with and without coherent feedback. (b) $g^{(2)} (0)$ versus $g_0/\omega _m$ with and without coherent feedback, where we set $\kappa =0.05 \omega _m$. Other parameters are the same with Fig. 2.

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Now we turn to the effect of coherent feedback on the third-order correlation function $g^{(3)}(0)$ defined by

$$g^{(3)} (0) = \frac {\langle a^{{\dagger}} a^{{\dagger}} a^{{\dagger}} a a a \rangle} {{\langle a^{{\dagger}} a \rangle}^3}.$$

As shown in Fig. 5(a), $g^{(3)}(0)> g^{(2)}(0)>1$ at $\Delta =3 \Delta _0$ for both cases with and without coherent feedback, which means three-photon tunneling instead of three-photon blockade effect occurs. For $\Delta = \Delta _0$, $g^{(2)}(0)$ with coherent feedback is smaller than that without coherent feedback. For $\Delta =2 \Delta _0$ and $\Delta =3 \Delta _0$, both of $g^{(2)}(0)$ and $g^{(3)}(0)$ with coherent feedback are larger than that without coherent feedback. These results mean that the presence of coherent feedback can not only enhance the blockade effect, but also enhance the tunneling effect. The enhancement of three-photon tunneling effect with coherent feedback can be clearly seen from the relative distribution $y$ as shown in Fig. 5(b), and we will not repeat it here.

 figure: Fig. 5.

Fig. 5. (a) Second-order correlation function $g^{(2)}(0)$ (red) and third-order correlation function $g^{(3)}(0)$ (blue) with (solid) or without (dashed) coherent feedback for $\Omega =0.005g_0$. (b) Relative probability population function with and without coherent feedback under three-photon resonance condition $\Delta =3 \Delta _0$ for $\Omega =0.03g_0$. we set $\kappa =0.05g_0$. Other parameters are the same with Fig. 2.

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4. Effects of coherent feedback on Schrödinger-cat state generation

In addition to photon blockade effect and tunneling effect, the other few-photon optomechanical effects can also be enhanced by coherent feedback. In this section, we discuss the effects of coherent feedback on Schrödinger-cat state generation. Without considering the laser driving, we use a single-mode optomechanical system to generate the two-component cat state of the cavity [64]. For the original Hamiltonian $H_1$ without driving, its time evolution operator is

$$U(t) = {e^{ - i\Delta a^{{\dagger}} a t - i{\omega _m}b^{{\dagger}}b t + i{g_0}a^{{\dagger}}a(b + b^{{\dagger}}) t}}.$$

To split the above operator, we introduce a unitary transformation operator $T$ to perform on $U(t)$, where $T= e^{-(g_0/\omega _m) a^{\dagger } a(b^{\dagger }-b)}$. Using the formula that $T f(X_i) T=f(T X_i T)$, we can calculate the time evolution operator after unitary transformation as

$$T U(t) T^{{\dagger}}= e^{ - i\Delta a^{{\dagger}} a t} e^{ - i\omega_m b^{{\dagger}} b t +i k g_0 (a^{{\dagger}}a)^2 t},$$
where $k=g_0 / \omega _m$. Multiplying the above formula left by $T^{\dagger }$ and right by $T$, one can get
$$U(t) = e^{ - i\Delta a^{{\dagger}} a t} e^{i k g_0 (a^{{\dagger}}a)^2 t} e^{k a^{{\dagger}} a(b^{{\dagger}}-b)} e^{ - i\omega_m b^{{\dagger}} b t} e^{{-}k a^{{\dagger}} a(b^{{\dagger}}-b)}.$$

By the following formula

$$e^{ - i\omega_m b^{{\dagger}} b t} e^{{-}k a^{{\dagger}} a(b^{{\dagger}}-b)} e^{ i\omega_m b^{{\dagger}} b t}=e^{{-}k a^{{\dagger}} a(b^{{\dagger}} e^{{-}i\omega_m t} - b e^{i\omega_m t})},$$
we can rewrite the last two items of $U(t)$ as
$$U(t)= e^{ - i\Delta a^{{\dagger}} a t} e^{i k g_0 (a^{{\dagger}}a)^2 t} e^{k a^{{\dagger}} a(b^{{\dagger}}-b)} e^{{-}k a^{{\dagger}} a(b^{{\dagger}} e^{{-}i \omega_m t} - b e^{i\omega_m t})} e^{ - i\omega_m b^{{\dagger}} b t}.$$

Using Baker-Hausdorff formula, $U(t)$ can be further simplified to

$$U(t) = e^{ - i\Delta a^{{\dagger}} a t} e^{i k^2 (a^{{\dagger}}a)^2 (\omega_m t-\sin{\omega_m t})} e^{k a^{{\dagger}} a(\mu b^{{\dagger}} - \mu^\ast b)} e^{ - i\omega_m b^{{\dagger}} b t},$$
where $\mu = 1- e^{-i \omega _m t}$. We assume the initial state of the system as ${\left | \psi (0) \right \rangle } ={\left | \alpha \right \rangle _c } \otimes {\left | \beta \right \rangle _m }$. By employing time evolution operator $U(t)$ in Eq. (23) to the initial state ${\left | \psi (0) \right \rangle }$, we can obtain the state of the system at time $t$ as
$${\left| \psi(t) \right\rangle } = {e^{\frac{{ - {{\left| \alpha \right|}^2}}}{2}}}\sum_{n = 0}^\infty {\frac{{{\alpha ^n}}}{{\sqrt {n!} }}} {e^{i k^2 {n^2(\omega_m t-\sin{\omega_m t})}}}{\left| n \right\rangle _c} \otimes {\left| \psi_n (t) \right\rangle _m},$$
where
$${\left| \psi_n (t) \right\rangle _m} = e^{{-}i \Delta n t + iIm(\mu k n \beta^\ast e^{i\omega_m t})} {\left| \beta e^{{-}i\omega_m t} + kn\mu \right\rangle _m}.$$

By setting $\Delta =\omega _m$, we obtain $| \psi _n (t) \rangle _m=| \beta \rangle _m$ at $\omega _m t=2\pi$, which indicates that the mechanical oscillator returns to its initial state. Then the state of the cavity mode becomes

$${\left| \varsigma \right\rangle _c} = {e^{\frac{{ - {{\left| \alpha \right|}^2}}}{2}}}\sum_{n = 0}^\infty {\frac{{{\alpha ^n}}}{{\sqrt {n!} }}} {e^{i2\pi k^2 {n^2}}}{\left| n \right\rangle _c}.$$

By taking $k = 0.5$, the above state becomes

$${\left| \varsigma \right\rangle _c} = {e^{\frac{{ - {{\left| \alpha \right|}^2}}}{2}}}\sum_{n = 0}^\infty {\frac{{{\alpha ^n}}}{{\sqrt {n!} }}} {e^{\frac{{i\pi {n^2}}}{2}}}{\left| n \right\rangle _c}=({e^{i\frac{\pi }{4}}}{\left| { + \alpha } \right\rangle _c} + {e^{ - i\frac{\pi }{4}}}{\left| { - \alpha } \right\rangle _c})/\sqrt 2 .$$
Eq. (27) indicates that the two-component cat state of cavity mode can be realize by the optomechanical system.

The above an alytical expression is obtained in a closed system. The decay of cavity mode will influence the cat state preparation, and we should discuss it here. In order to intuitively observe the properties of the prepared cat state, we introduce the Wigner function

$${W_\textrm{m}}(\eta ) = \frac{2}{\pi }Tr[{D^{{\dagger}} }(\eta ){\rho _m}D(\eta ){( - 1)^{{a^{{\dagger}} }a}}],$$
where $D(\eta )=\exp (\eta a^{\dagger }-\eta ^\ast a)$. In Fig. 6(a), the Winger function in open system without coherent feedback is presented. We observe that the peaks of ${\left | { + \alpha } \right \rangle }$ and ${\left | { - \alpha } \right \rangle }$ are not obvious, which means the obtained cat state is indistinct superposition state. After introducing coherent feedback to the system, as shown in Fig. 6(b), we can see that there are two obvious peaks, thus the macroscopically distinct superposition state can be obtained by coherent feedback. The minimum value of $W_\textrm {m}(\eta )$ in Fig. 6(b) is obviously smaller than that in Fig. 6(a), which indicates that the quantum properties can be preserved by coherent feedback since a negative Wigner function is a definite indicator of nonclassical properties. In Fig. 6(c), we present the minimum value of the winger function $W_{min}$ as a function of $r$. It is apparent that the larger $r$, the smaller the minimum value of the winger function. In Fig. 6(d), we plot the fidelity between the target state $\left | \varsigma \right \rangle _c$ and obtained state by master equation as a function of cavity decay rate $\kappa$. Clearly, the fidelity decreases with the increase of $\kappa$, and fidelity can be enhanced with large $r$. Since the effective decay of the cavity mode is decreased with the increase of $r$ because of $\widetilde {\kappa }=2 \kappa (1-r)$. It is reasonable that the larger $r$, the higher the fidelity. Correspondingly, the quantum signature of cat state is maintained well with large $r$, which is consistent with the result obtained in Fig. 6(c).

 figure: Fig. 6.

Fig. 6. (a) and (b) are Winger functions with ($r=0.9$) or without ($r=0$) coherent feedback, respectively. (c) is the minimum value of Winger function versus feedback coefficient $r$. (d) is the fidelity versus $\kappa /\omega _m$ under different $r$. We set $\Delta =\omega _m$ and $\kappa =0.05\omega _m$. Other parameters are the same with Fig. 2.

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5. Discussions

In this paper, we take the value of $\kappa$ less than $g_0$, which is the necessary condition for realizing conventional photon blockade effects, as has been pointed out by Ref. [65]. However, it is difficult to achieve $\kappa <g_0$ in experiments. Although the minimum $\kappa /g_0 \approx 1$ is reported experimentally in the effective optomechanical system [66]. At present, in most of the experiments in optomechanics, $\kappa /g_0$ has a value between $10^3-10^5$ [6769], thus it is necessary to discuss the influences of cavity decay to photon blockade in a much more wide range. In Fig. 7(a), we present $g^{(2)}(0)$ as a function of $\kappa$ with $\kappa$ varying from $10^2g_0$ to $10^4g_0$. We see that $g^{(2)}(0)$ is maintained at about $1$ without feedback. With feedback, $g^{(2)}(0)$ tends to $1$ from $0.15$ ($r=0.999$) and $0.0031$ ($r=0.9999$) as $\kappa /g_0$ increases. Therefore, in our scheme, it is possible to achieve photon blockade under the condition of $\kappa >g_0$ as long as $r$ is close enough to $1$. Although it is difficult to accurately manipulate $r$ close to $1$, our scheme provides a possibility to realize photon blockade effects under strong cavity dissipation.

 figure: Fig. 7.

Fig. 7. (a) and (b) are $g^{(2)}(0)$ as a function of $\kappa$ and $\bar {n}_{th}$, respectively. (c) is the fidelity for cat state generation versus $\bar {n}_{th}$. Other parameters for (a) and (b) are the same with Fig. 3 and parameters for (c) are the same with Fig. 6.

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In Fig. 7(b) and (c), we present the influences of thermal occupancy of the mechanical mode $\bar {n}_{th}$ on photon blockade and cat state generation, respectively. Figure 7(b) shows that thermal excitation of mechanical mode $\bar {n}_{th}$ has no influence on $g^{(2)}(0)$ in a certain range, which verifies the effectiveness of eliminating the mechanical mode in the diagonalized representation. Different with photon blockade, $\bar {n}_{th}$ has influences on cat state generation. Figure 7(c) shows that the fidelity for cat state generation will both decrease with the increase of $\bar {n}_{th}$ no matter what $r$ is. At $\bar {n}_{th}=10$, the fidelity for cat state generation is about $0.73$ with $r=0$ and $0.9$ with $r=0.9$, which means although thermal excitation will destroy cat state generation, the fidelity of cat state preparation can be effectively improved with coherent feedback.

6. Conclusions

In summary, we propose a scheme to preserve few-photon optomechanical effects in this paper. In our scheme the coherent feedback is introduced to the cavity. By analysing the effective system in the presence of coherent feedback, we show that the decay of cavity mode can be effectively reduced by selecting appropriate parameters. Then, we study the equal-time second-order correlation function, Poisson distribution of photons, and time-delay second-order correlation function in the presence of coherent feedback. As a result, the single-photon blockade effect can be enhanced. In addition, we explore the three-photon tunneling effect with coherent feedback in the system. By analyzing the equal-time third-order correlation function and the Poisson distribution under the condition of three-photon resonance, we prove that the three-photon tunneling effect can be enhanced with coherent feedback. Finally, we discuss the application of coherent feedback in preparing the cat state. The results show that two-component cat state of the cavity mode can be generated with high fidelity in the presence of coherent feedback. The proposed scheme can be extended to other quantum optical systems. As for the influence of coherent feedback on other quantum effects, we will further explore it in the next work.

Appendix: Analytical result of $g^{(2)}(0)$ in standard optomechanical system

In order to clarify the influence of cavity decay on the photon blockade effect, in this appendix, we approximately solve $g^{(2)}(0)$ by truncating the systems of the photon mode up to $n=4$. We can assume $|\psi \rangle = C_0 |0\rangle +C_1 |1\rangle +C_2 |2\rangle +C_3 |3\rangle +C_4 |4\rangle$ when photons are spanned to 4-photon Hilbert subspace. By phenomenally introducing cavity decay $\kappa$, Eq. (5) becomes

$$H_4 = \sum_{n}\{[n (\Delta-i\frac{\kappa}{2}) - \frac{g^2 _0 n^2}{\omega_m}]\left| n \right\rangle \left\langle n \right| + \Omega\sqrt{n+1}\left(\left| n \right\rangle \left\langle n+1 \right| +\left| n+1 \right\rangle \left\langle n \right|\right)\}.$$

By substituting $|\psi \rangle$ and Eq. (1) into Schrödinger equation, one can obtain

$$\begin{aligned}&i \dot{C_0}= \Omega C_1,\\ &i \dot{C_1}= \Omega C_0+ [(\Delta-\frac{i\kappa}{2})-\frac{g_0^2}{\omega_m}] C_1+\sqrt{2} \Omega C_2,\\ &i \dot{C_2}=\sqrt{2}\Omega C_1+[2 (\Delta-\frac{i\kappa}{2})-\frac{4g_0^2}{\omega_m}] C_2+\sqrt{3} \Omega C_3,\\ &i \dot{C_3}=\sqrt{3}\Omega C_2+[3(\Delta-\frac{i\kappa}{2})-\frac{9g_0^2}{\omega_m}] C_3+2\Omega C_4,\\ &i \dot{C_4}=2\Omega C_3+[4(\Delta-\frac{i\kappa}{2})-\frac{16g_0^2}{\omega_m}] C_4. \end{aligned}$$

By ignoring the jumping from high level to low level, the steady-state of $C_1-C_4$ can be solved using perturbation method as

$$\begin{aligned}&C_1=\frac{-\Omega}{\Delta_{\kappa}-\frac{g_0^2}{\omega_m}},\\ &C_2=\frac{\sqrt{2}\Omega^2}{(2 \Delta_{\kappa}-\frac{4g_0^2}{\omega_m})(\Delta_{\kappa}-\frac{g_0^2}{\omega_m})},\\ &C_3=\frac{-\sqrt{6}\Omega^3}{(3 \Delta_{\kappa}-\frac{9g_0^2}{\omega_m})(2 \Delta_{\kappa}-\frac{4g_0^2}{\omega_m})(\Delta_{\kappa}-\frac{g_0^2}{\omega_m})},\\ &C_4=\frac{2\sqrt{6}\Omega^4}{(4 \Delta_{\kappa}-\frac{16g_0^2}{\omega_m})(3 \Delta_{\kappa}-\frac{9g_0^2}{\omega_m})(2 \Delta_{\kappa}-\frac{4g_0^2}{\omega_m})(\Delta_{\kappa}-\frac{g_0^2}{\omega_m})}, \end{aligned}$$
where $\Delta _{\kappa }=\Delta -i\kappa /2$. Under the condition of single-photon resonance $\Delta =\Delta _0$, the second-order correlation function can be expressed as
$$g^{(2)}(0) = \frac{2|C_2|^2+6|C_3|^2+12|C_4|^2}{(|C_1|^2+2|C_2|^2+3|C_3|^2+4|C_4|^2)^2} \approx \frac{1}{1+\frac{4g_0^2}{\kappa^2 \omega_m^2}}.$$

The above equation shows that with the increase of cavity decay $\kappa$, $g^{(2)}(0)$ becomes larger, which describes the fact that cavity decay will destroy the photon blockade effect.

The above results are based on two approximations. We first examine the validity of the approximation from Eq. (2) to (3) by simulating the fidelity

$$\mathcal{F}\left(\rho_{2}, \rho_{3}\right)=\left[\operatorname{trace}\left(\sqrt{\rho_{2}} \rho_{3} \sqrt{\rho_{2}}\right)^{\frac{1}{2}}\right]^{2},$$
where $\rho _2$ and $\rho _3$ are the quantum states governed by $H_2$ and $H_3$, respectively. In Fig. 8(a), we plot the fidelity $\mathcal {F}$ with time evolution under different coupling strength $g_0$ by RK4. We see that $\mathcal {F}$ decreases with the increase of $g_0$. The fidelity maintains $1$ when $g_0=0.25 \omega _m$, which can guarantee the validity of the approximation from Eq. (2) to (3). Moreover, we present the analytical results of $g^{(2)}(0)$ and the numerical one in Fig. 8(b) by python package QuTiP [63], where dimensions of the cavity mode and mechanical mode are truncated to $8$ and $10$, respectively. The results show that the analytical results and numerical one coincide completely. Therefore, the truncation to $4$ of cavity mode is valid in the above analysis. In the main text, the dimension of the cavity mode is truncated to more than $8$ in all numerical simulations.

 figure: Fig. 8.

Fig. 8. (a) Time evolution of fidelity $\mathcal {F}$ with different coupling strength $g_0$. (b) Equal-time second-order correlation function $g^{(2)}{(0)}$ as a function of $\Delta$ with analytical solution and numerical simulation. Other parameters are the same with main text.

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Funding

Excellent young and middle-aged Talents Project in scientific research of Hubei Provincial Department of Education (Q20202503).

Acknowledgments

We would like to thank Prof. Ling Zhou and Dr. Ye-Xiong Zeng for helpful discussions. Thanks Ms. Chunhui Li for valuable suggestions on writing.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data produced by numerical simulations in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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References

  • View by:

  1. I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, “Theory of ground state cooling of a mechanical oscillator using dynamical backaction,” Phys. Rev. Lett. 99(9), 093901 (2007).
    [Crossref]
  2. F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, “Quantum theory of cavity-assisted sideband cooling of mechanical motion,” Phys. Rev. Lett. 99(9), 093902 (2007).
    [Crossref]
  3. J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
    [Crossref]
  4. J. Chan, T. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
    [Crossref]
  5. Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical resonator in strong coupling optomechanics,” Phys. Rev. Lett. 110(15), 153606 (2013).
    [Crossref]
  6. D.-G. Lai, J.-F. Huang, X.-L. Yin, B.-P. Hou, W. Li, D. Vitali, F. Nori, and J.-Q. Liao, “Nonreciprocal ground-state cooling of multiple mechanical resonators,” Phys. Rev. A 102(1), 011502 (2020).
    [Crossref]
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2021 (5)

A. de los Ríos Sommer, N. Meyer, and R. Quidant, “Strong optomechanical coupling at room temperature by coherent scattering,” Nat. Commun. 12(1), 276 (2021).
[Crossref]

X. Li, J. Xin, G. Li, X.-M. Lu, and L. F. Wei, “Quantum routings for single photons with different frequencies,” Opt. Express 29(6), 8861–8871 (2021).
[Crossref]

C. Zhao, R. Peng, Z. Yang, S. Chao, C. Li, and L. Zhou, “Atom-mediated phonon blockade and controlled-z gate in superconducting circuit system,” Ann. Phys. 533(5), 2100039 (2021).
[Crossref]

K. Wu, W.-X. Zhong, G.-L. Cheng, and A.-X. Chen, “Phase-controlled multimagnon blockade and magnon-induced tunneling in a hybrid superconducting system,” Phys. Rev. A 103(5), 052411 (2021).
[Crossref]

F.-Y. Zhang and C.-P. Yang, “Generation of generalized hybrid entanglement in cavity electro–optic systems,” Quantum Sci. Technol. 6(2), 025003 (2021).
[Crossref]

2020 (12)

A. Wallucks, I. Marinković, B. Hensen, R. Stockill, and S. Gröblacher, “A quantum memory at telecom wavelengths,” Nat. Phys. 16(7), 772–777 (2020).
[Crossref]

D. P. Lake, M. Mitchell, B. C. Sanders, and P. E. Barclay, “Two-colour interferometry and switching through optomechanical dark mode excitation,” Nat. Commun. 11(1), 2208 (2020).
[Crossref]

Y.-X. Zeng, J. Shen, M.-S. Ding, and C. Li, “Macroscopic schrödinger cat state swapping in optomechanical system,” Opt. Express 28(7), 9587–9602 (2020).
[Crossref]

F. Zou, D.-G. Lai, and J.-Q. Liao, “Enhancement of photon blockade effect via quantum interference,” Opt. Express 28(11), 16175–16190 (2020).
[Crossref]

Y. H. Zhou, X. Y. Zhang, Q. C. Wu, B. L. Ye, Z.-Q. Zhang, D. D. Zou, H. Z. Shen, and C.-P. Yang, “Conventional photon blockade with a three-wave mixing,” Phys. Rev. A 102(3), 033713 (2020).
[Crossref]

J.-S. Liu, J.-Y. Yang, H.-Y. Liu, and A.-D. Zhu, “Photon blockade by enhancing coupling via a nonlinear medium,” Opt. Express 28(12), 18397–18406 (2020).
[Crossref]

C. Zhao, X. Li, S. Chao, R. Peng, C. Li, and L. Zhou, “Simultaneous blockade of a photon, phonon, and magnon induced by a two-level atom,” Phys. Rev. A 101(6), 063838 (2020).
[Crossref]

F. Zou, X.-Y. Zhang, X.-W. Xu, J.-F. Huang, and J.-Q. Liao, “Multiphoton blockade in the two-photon jaynes-cummings model,” Phys. Rev. A 102(5), 053710 (2020).
[Crossref]

Z.-C. Duan, J.-P. Li, J. Qin, Y. Yu, Y.-H. Huo, S. Hofling, C.-Y. Lu, N.-L. Liu, K. Chen, and J.-W. Pan, “Proof-of-principle demonstration of compiled shor’s algorithm using a quantum dot single-photon source,” Opt. Express 28(13), 18917–18930 (2020).
[Crossref]

K. Zhang, J. He, and J. Wang, “Two-way single-photon-level frequency conversion between 852 nm and 1560 nm for connecting cesium d2 line with the telecom c-band,” Opt. Express 28(19), 27785–27796 (2020).
[Crossref]

D.-G. Lai, J.-F. Huang, X.-L. Yin, B.-P. Hou, W. Li, D. Vitali, F. Nori, and J.-Q. Liao, “Nonreciprocal ground-state cooling of multiple mechanical resonators,” Phys. Rev. A 102(1), 011502 (2020).
[Crossref]

B. Xiong, X. Li, S.-L. Chao, Z. Yang, W.-Z. Zhang, W. Zhang, and L. Zhou, “Strong mechanical squeezing in an optomechanical system based on lyapunov control,” Photonics Res. 8(2), 151–159 (2020).
[Crossref]

2019 (6)

B. Li, R. Huang, X. Xu, A. Miranowicz, and H. Jing, “Nonreciprocal unconventional photon blockade in a spinning optomechanical system,” Photonics Res. 7(6), 630–641 (2019).
[Crossref]

F. Zou, L.-B. Fan, J.-F. Huang, and J.-Q. Liao, “Enhancement of few-photon optomechanical effects with cross-kerr nonlinearity,” Phys. Rev. A 99(4), 043837 (2019).
[Crossref]

H. Xie, X. Shang, C.-G. Liao, Z.-H. Chen, and X.-M. Lin, “Macroscopic superposition states of a mechanical oscillator in an optomechanical system with quadratic coupling,” Phys. Rev. A 100(3), 033803 (2019).
[Crossref]

D.-Y. Wang, C.-H. Bai, S. Liu, S. Zhang, and H.-F. Wang, “Distinguishing photon blockade in a $\mathcal {PT}$PT-symmetric optomechanical system,” Phys. Rev. A 99(4), 043818 (2019).
[Crossref]

J. Guo, R. Norte, and S. Gröblacher, “Feedback cooling of a room temperature mechanical oscillator close to its motional ground state,” Phys. Rev. Lett. 123(22), 223602 (2019).
[Crossref]

S. Ghosh and T. C. H. Liew, “Dynamical blockade in a single-mode bosonic system,” Phys. Rev. Lett. 123(1), 013602 (2019).
[Crossref]

2018 (4)

M. Rossi, N. Kralj, S. Zippilli, R. Natali, A. Borrielli, G. Pandraud, E. Serra, G. Di Giuseppe, and D. Vitali, “Normal-mode splitting in a weakly coupled optomechanical system,” Phys. Rev. Lett. 120(7), 073601 (2018).
[Crossref]

M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, “Measurement-based quantum control of mechanical motion,” Nature 563(7729), 53–58 (2018).
[Crossref]

R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal photon blockade,” Phys. Rev. Lett. 121(15), 153601 (2018).
[Crossref]

B. Xiong, X. Li, S.-L. Chao, and L. Zhou, “Optomechanical quadrature squeezing in the non-markovian regime,” Opt. Lett. 43(24), 6053–6056 (2018).
[Crossref]

2017 (5)

H. Flayac and V. Savona, “Unconventional photon blockade,” Phys. Rev. A 96(5), 053810 (2017).
[Crossref]

B. Sarma and A. K. Sarma, “Quantum-interference-assisted photon blockade in a cavity via parametric interactions,” Phys. Rev. A 96(5), 053827 (2017).
[Crossref]

J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, “Enhanced entanglement of two different mechanical resonators via coherent feedback,” Phys. Rev. A 95(4), 043819 (2017).
[Crossref]

M. Rossi, N. Kralj, S. Zippilli, R. Natali, A. Borrielli, G. Pandraud, E. Serra, G. Di Giuseppe, and D. Vitali, “Enhancing sideband cooling by feedback-controlled light,” Phys. Rev. Lett. 119(12), 123603 (2017).
[Crossref]

T.-S. Yin, X.-Y. Lü, L.-L. Zheng, M. Wang, S. Li, and Y. Wu, “Nonlinear effects in modulated quantum optomechanics,” Phys. Rev. A 95(5), 053861 (2017).
[Crossref]

2016 (6)

J.-Q. Liao and L. Tian, “Macroscopic quantum superposition in cavity optomechanics,” Phys. Rev. Lett. 116(16), 163602 (2016).
[Crossref]

M. Hirose and P. Cappellaro, “Coherent feedback control of a single qubit in diamond,” Nature 532(7597), 77–80 (2016).
[Crossref]

N. Német and S. Parkins, “Enhanced optical squeezing from a degenerate parametric amplifier via time-delayed coherent feedback,” Phys. Rev. A 94(2), 023809 (2016).
[Crossref]

M. Asjad, P. Tombesi, and D. Vitali, “Feedback control of two-mode output entanglement and steering in cavity optomechanics,” Phys. Rev. A 94(5), 052312 (2016).
[Crossref]

X. Li, W.-Z. Zhang, B. Xiong, and L. Zhou, “Single-photon multi-ports router based on the coupled cavity optomechanical system,” Sci. Rep. 6(1), 39343 (2016).
[Crossref]

C. U. Lei, A. J. Weinstein, J. Suh, E. E. Wollman, A. Kronwald, F. Marquardt, A. A. Clerk, and K. C. Schwab, “Quantum nondemolition measurement of a quantum squeezed state beyond the 3 db limit,” Phys. Rev. Lett. 117(10), 100801 (2016).
[Crossref]

2015 (5)

J.-M. Pirkkalainen, E. Damskägg, M. Brandt, F. Massel, and M. A. Sillanpää, “Squeezing of quantum noise of motion in a micromechanical resonator,” Phys. Rev. Lett. 115(24), 243601 (2015).
[Crossref]

H. Z. Shen, Y. H. Zhou, H. D. Liu, G. C. Wang, and X. X. Yi, “Exact optimal control of photon blockade with weakly nonlinear coupled cavities,” Opt. Express 23(25), 32835–32858 (2015).
[Crossref]

N. Yang, J. Zhang, H. Wang, Y.-X. Liu, R.-B. Wu, L.-Q. Liu, C.-W. Li, and F. Nori, “Noise suppression of on-chip mechanical resonators by chaotic coherent feedback,” Phys. Rev. A 92(3), 033812 (2015).
[Crossref]

S. M. Hein, F. Schulze, A. Carmele, and A. Knorr, “Entanglement control in quantum networks by quantum-coherent time-delayed feedback,” Phys. Rev. A 91(5), 052321 (2015).
[Crossref]

X.-Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, “Squeezed optomechanics with phase-matched amplification and dissipation,” Phys. Rev. Lett. 114(9), 093602 (2015).
[Crossref]

2014 (2)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86(4), 1391–1452 (2014).
[Crossref]

N. Yamamoto, “Coherent versus measurement feedback: Linear systems theory for quantum information,” Phys. Rev. X 4, 041029 (2014).
[Crossref]

2013 (10)

B. Chen, K. Zhang, C. Bian, C. Qiu, C.-H. Yuan, L. Q. Chen, Z. Y. Ou, and W. Zhang, “Efficient raman frequency conversion by coherent feedback at low light intensity,” Opt. Express 21(9), 10490–10495 (2013).
[Crossref]

J.-Q. Liao and C. K. Law, “Correlated two-photon scattering in cavity optomechanics,” Phys. Rev. A 87(4), 043809 (2013).
[Crossref]

J. Johansson, P. Nation, and F. Nori, “Qutip 2: A python framework for the dynamics of open quantum systems,” Comput. Phys. Commun. 184(4), 1234–1240 (2013).
[Crossref]

X.-W. Xu, Y.-J. Li, and Y.-X. Liu, “Photon-induced tunneling in optomechanical systems,” Phys. Rev. A 87(2), 025803 (2013).
[Crossref]

Z.-P. Liu, H. Wang, J. Zhang, Y.-X. Liu, R.-B. Wu, C.-W. Li, and F. Nori, “Feedback-induced nonlinearity and superconducting on-chip quantum optics,” Phys. Rev. A 88(6), 063851 (2013).
[Crossref]

P. Kómár, S. D. Bennett, K. Stannigel, S. J. M. Habraken, P. Rabl, P. Zoller, and M. D. Lukin, “Single-photon nonlinearities in two-mode optomechanics,” Phys. Rev. A 87(1), 013839 (2013).
[Crossref]

J.-Q. Liao and F. Nori, “Photon blockade in quadratically coupled optomechanical systems,” Phys. Rev. A 88(2), 023853 (2013).
[Crossref]

Y.-D. Wang and A. A. Clerk, “Reservoir-engineered entanglement in optomechanical systems,” Phys. Rev. Lett. 110(25), 253601 (2013).
[Crossref]

T. Palomaki, J. Teufel, R. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342(6159), 710–713 (2013).
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Y.-C. Liu, Y.-F. Xiao, X. Luan, and C. W. Wong, “Dynamic dissipative cooling of a mechanical resonator in strong coupling optomechanics,” Phys. Rev. Lett. 110(15), 153606 (2013).
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2011 (3)

J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, “Sideband cooling of micromechanical motion to the quantum ground state,” Nature 475(7356), 359–363 (2011).
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J. Chan, T. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478(7367), 89–92 (2011).
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P. Rabl, “Photon blockade effect in optomechanical systems,” Phys. Rev. Lett. 107(6), 063601 (2011).
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2010 (1)

S. C. Hou, X. L. Huang, and X. X. Yi, “Suppressing decoherence and improving entanglement by quantum-jump-based feedback control in two-level systems,” Phys. Rev. A 82(1), 012336 (2010).
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2009 (1)

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2008 (2)

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2007 (2)

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Data availability

Data produced by numerical simulations in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (8)

Fig. 1.
Fig. 1. (a) Single-mode cavity optomechanical system with coherent feedback loop. The output of cavity fields are fed back into the input port through HRM and CBS. (b) Energy level diagram of cavity mode under eigenstate representation.
Fig. 2.
Fig. 2. Plot of $g^{(2)} (0)$ as a function of $\Delta /\Delta _0$ in (a), and $\left \langle a^{\dagger } a \right \rangle$ versus $\Delta /\Delta _0$ in (b) at different values of $\kappa$. Plot of $Log_{10}[g^{(2)} (0)]$ as functions of $g/\omega _m$ and $\kappa /\omega _m$ in (c), and $Log_{10}\left \langle a^{\dagger } a \right \rangle$ versus $g/\omega _m$ and $\kappa /\omega _m$ in (d) . We set $\omega _m/2\pi = 10$ MHz, $g_0=0.25\omega _m$ for (a) and (b), $\Delta = \Delta _0$ for (c) and (d). Other parameters are $\Omega =0.00125 \omega _m$, $\bar {n}_{th}=0$ and $\gamma =0.001 \omega _m$.
Fig. 3.
Fig. 3. Plot of $g^{(2)} (0)$ as a function of $\kappa /g_0$ in (a), and $\left \langle a^{\dagger } a \right \rangle$ versus $\kappa /g_0$ in (b) with ($r=0.9$) and without ($r=0$) coherent feedback. (c) $g^{(2)} (0)$ and $\langle a^{\dagger }a\rangle$ (inset) as functions of $r$ with $\kappa =0.05 g_0$. (d) Relative probability population function with and without coherent feedback for $\kappa =0.01 g_0$. Other parameters are the same with Fig. 2.
Fig. 4.
Fig. 4. (a) Time evolution of time-delay second-order correlation function $g^{(2)}(\tau )$ with and without coherent feedback. (b) $g^{(2)} (0)$ versus $g_0/\omega _m$ with and without coherent feedback, where we set $\kappa =0.05 \omega _m$. Other parameters are the same with Fig. 2.
Fig. 5.
Fig. 5. (a) Second-order correlation function $g^{(2)}(0)$ (red) and third-order correlation function $g^{(3)}(0)$ (blue) with (solid) or without (dashed) coherent feedback for $\Omega =0.005g_0$. (b) Relative probability population function with and without coherent feedback under three-photon resonance condition $\Delta =3 \Delta _0$ for $\Omega =0.03g_0$. we set $\kappa =0.05g_0$. Other parameters are the same with Fig. 2.
Fig. 6.
Fig. 6. (a) and (b) are Winger functions with ($r=0.9$) or without ($r=0$) coherent feedback, respectively. (c) is the minimum value of Winger function versus feedback coefficient $r$. (d) is the fidelity versus $\kappa /\omega _m$ under different $r$. We set $\Delta =\omega _m$ and $\kappa =0.05\omega _m$. Other parameters are the same with Fig. 2.
Fig. 7.
Fig. 7. (a) and (b) are $g^{(2)}(0)$ as a function of $\kappa$ and $\bar {n}_{th}$, respectively. (c) is the fidelity for cat state generation versus $\bar {n}_{th}$. Other parameters for (a) and (b) are the same with Fig. 3 and parameters for (c) are the same with Fig. 6.
Fig. 8.
Fig. 8. (a) Time evolution of fidelity $\mathcal {F}$ with different coupling strength $g_0$. (b) Equal-time second-order correlation function $g^{(2)}{(0)}$ as a function of $\Delta$ with analytical solution and numerical simulation. Other parameters are the same with main text.

Equations (33)

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H 1 = Δ a a + ω m b b g 0 a a ( b + b ) + Ω ( a + a ) ,
H 2 = Δ a a + ω m b b g 0 2 ω m ( a a ) 2 + Ω ( a e g 0 / ω m ( b b ) + a e g 0 / ω m ( b b ) ) .
H 3 = Δ a a + ω m b b g 0 2 ω m ( a a ) 2 + Ω ( a + a ) .
| n , m ~ ( n ) = e g 0 n ω m ( b b ) | n , m .
H 3 = n [ ( n Δ g 0 2 n 2 ω m ) | n n | + Ω n + 1 ( | n n + 1 | + | n + 1 n | ) ] .
g ( 2 ) ( 0 ) = a a a a a a 2
ρ ˙ = i [ H 1 , ρ ] + κ 2 ( 2 a ρ a a a ρ ρ a a ) + γ 2 ( n ¯ t h + 1 ) ( 2 b ρ b b b ρ ρ b b ) + γ 2 n ¯ t h ( 2 b ρ b b b ρ ρ b b )
a ˙ = ( i Δ + κ 1 2 + κ 2 2 ) a + i g 0 a ( b + b ) i Ω + κ 1 a 1 i n + κ 2 a 2 i n ,
b ˙ = ( i ω m + γ 2 ) b + i g 0 a a + γ b i n ,
a o u t = κ 2 a a 2 i n .
a f b i n = r e i θ a o u t + t a 1 i n ,
a ˙ = ( i Δ ~ + κ ~ 2 ) a + i g 0 a ( b + b ) i Ω + κ ~ A i n ,
A i n = κ 1 t a 1 i n + ( κ 2 κ 1 r e i θ ) a 2 i n κ ~ .
H ~ = Δ ~ a a + ω m b b g 0 a a ( b + b ) + Ω ( a + a ) ,
ρ ˙ = i [ H ~ , ρ ] + κ ~ 2 ( 2 a ρ a a a ρ ρ a a ) + γ 2 ( n ¯ t h + 1 ) ( 2 b ρ b b b ρ ρ b b ) + γ 2 n ¯ t h ( 2 b ρ b b b ρ ρ b b ) .
g ( 2 ) ( τ ) = a ( t ) a ( t + τ ) a ( t + τ ) a ( t ) a ( t ) a ( t ) 2
g ( 3 ) ( 0 ) = a a a a a a a a 3 .
U ( t ) = e i Δ a a t i ω m b b t + i g 0 a a ( b + b ) t .
T U ( t ) T = e i Δ a a t e i ω m b b t + i k g 0 ( a a ) 2 t ,
U ( t ) = e i Δ a a t e i k g 0 ( a a ) 2 t e k a a ( b b ) e i ω m b b t e k a a ( b b ) .
e i ω m b b t e k a a ( b b ) e i ω m b b t = e k a a ( b e i ω m t b e i ω m t ) ,
U ( t ) = e i Δ a a t e i k g 0 ( a a ) 2 t e k a a ( b b ) e k a a ( b e i ω m t b e i ω m t ) e i ω m b b t .
U ( t ) = e i Δ a a t e i k 2 ( a a ) 2 ( ω m t sin ω m t ) e k a a ( μ b μ b ) e i ω m b b t ,
| ψ ( t ) = e | α | 2 2 n = 0 α n n ! e i k 2 n 2 ( ω m t sin ω m t ) | n c | ψ n ( t ) m ,
| ψ n ( t ) m = e i Δ n t + i I m ( μ k n β e i ω m t ) | β e i ω m t + k n μ m .
| ς c = e | α | 2 2 n = 0 α n n ! e i 2 π k 2 n 2 | n c .
| ς c = e | α | 2 2 n = 0 α n n ! e i π n 2 2 | n c = ( e i π 4 | + α c + e i π 4 | α c ) / 2 .
W m ( η ) = 2 π T r [ D ( η ) ρ m D ( η ) ( 1 ) a a ] ,
H 4 = n { [ n ( Δ i κ 2 ) g 0 2 n 2 ω m ] | n n | + Ω n + 1 ( | n n + 1 | + | n + 1 n | ) } .
i C 0 ˙ = Ω C 1 , i C 1 ˙ = Ω C 0 + [ ( Δ i κ 2 ) g 0 2 ω m ] C 1 + 2 Ω C 2 , i C 2 ˙ = 2 Ω C 1 + [ 2 ( Δ i κ 2 ) 4 g 0 2 ω m ] C 2 + 3 Ω C 3 , i C 3 ˙ = 3 Ω C 2 + [ 3 ( Δ i κ 2 ) 9 g 0 2 ω m ] C 3 + 2 Ω C 4 , i C 4 ˙ = 2 Ω C 3 + [ 4 ( Δ i κ 2 ) 16 g 0 2 ω m ] C 4 .
C 1 = Ω Δ κ g 0 2 ω m , C 2 = 2 Ω 2 ( 2 Δ κ 4 g 0 2 ω m ) ( Δ κ g 0 2 ω m ) , C 3 = 6 Ω 3 ( 3 Δ κ 9 g 0 2 ω m ) ( 2 Δ κ 4 g 0 2 ω m ) ( Δ κ g 0 2 ω m ) , C 4 = 2 6 Ω 4 ( 4 Δ κ 16 g 0 2 ω m ) ( 3 Δ κ 9 g 0 2 ω m ) ( 2 Δ κ 4 g 0 2 ω m ) ( Δ κ g 0 2 ω m ) ,
g ( 2 ) ( 0 ) = 2 | C 2 | 2 + 6 | C 3 | 2 + 12 | C 4 | 2 ( | C 1 | 2 + 2 | C 2 | 2 + 3 | C 3 | 2 + 4 | C 4 | 2 ) 2 1 1 + 4 g 0 2 κ 2 ω m 2 .
F ( ρ 2 , ρ 3 ) = [ trace ( ρ 2 ρ 3 ρ 2 ) 1 2 ] 2 ,