Mechanical switch of photon blockade and photon-induced tunneling

Cuilu Zhai; Ran Huang; Hui Jing; Hui Jing; Le-Man Kuang; Le-Man Kuang

doi:10.1364/OE.27.027649

1. Introduction

Achieving few-photon sources is highly desirable in modern quantum technologies, including photon transistors [1, 2], quantum repeaters [3], quantum-optical Josephson interferometer [4], as well as qubit quantum phase gates [5] and quantum nonreciprocal devices [6–10]. Photon blockade (PB) [11, 12], or single-photon blockade (1PB) for more precisely, the generation of a single photon in a nonlinear cavity can diminish the probability of generating another photon in the cavity, has been reported to realize such goal. This purely quantum effect has been experimentally demonstrated in different systems, including cavity or circuit QED systems [13–16] and cavity-free devices [17]. In addition, 1PB has also been predicted in coupled arrays of cavities [18–22] and nonlinear optical fibre [23]. Recently, two-photon blockade (2PB) [24–26], i.e., the absorption of two photons suppresses the absorption of further photons, has been realized in cavity QED system [27]. In sharp contrast to PB, photon-induced tunneling (PIT), the absorption of the first photon favors also that of the second or subsequent photons, can serve as a powerful tool in the generation of specific multi-photon states, which has been demonstrated experimentally in a photonic crystal cavity [28].

In the past decade, optomechanical system (OMS) has been proposed to study PB [29–33], PIT [34], and phonon blockade (a acoustic analog of PB) [35–38]. We note that cavity optomechanics [39, 40], exploits the interactions between light and movable mirrors, has significantly extended fundamental studies and practical applications of coherent light-matter interactions, such as optomechanically-induced transparency [41–47], storage or transduction of light signals [48], coherent mechanical lasing [49–52], and ultra-sensitive motion sensing [53–55]. Due to the high controllability in OMS, mechanical-modulated optomechanical interactions become an interesting topic, and have been harnessed to explore optical synchronization [56, 57], optical transparency and absorption [58–63], enhancement of second-order sideband generation [64], and directional light amplification [65]. However, previous studies on the role of mechanical pump in an OMS have mainly focused on the classical regimes, i.e., optical response properties instead of quantum noises.

In this paper, we propose how to introduce and control PB or PIT in the OMS with a driven oscillator. We find that 1PB and 2PB can emerge by tuning the strength of the mechanical pump. Furthermore, the mechanical switches of 1PB and PIT, 2PB and PIT, or 1PB and 2PB can be achieved with different mechanical driving strength. Our work opens up a new route to achieve quantum switch devices, which are crucial elements in quantum computations or photonic communications, and offers a way to test the quantumness of massive objects [66–70].

2. Model and solution

We consider an OMS with mechanical pump, as shown in Fig. 1(a). We note that, in OMS, mechanical driving has been experimentally realized based on microtoroidal optomechanical oscillator with an integrated electrical interface [56, 57], cascaded optomechanical resonators [58], and piezoelectric optomechanical crystal [71]. In other systems, mechanical pump was used to measure the quantum state of resonator [72], to control spin-phonon coupling [73, 74], and to break time-reversal symmetry for light propagation [75]. By driving coupled mechanical oscillators, richer physical effects can be revealed such as, mechanical mode mixing [76], coherent phonon manipulation [77], virtual exceptional points [78] and nonreciprocal cooling of phonon modes [79]. In our paper, both a mechanical pump with strength G and a weak monochromatic laser field with frequency ω_L are considered to drive the cavity. The mechanical pump G can be realized by applying a direct-current voltage to an optomechanical resonator with an integrated electrical interface [56, 57], which enables an inertial force to be applied to the mechanical mode. Moving to the rotating frame with respect to the driving laser field, the Hamiltonian of the system is given by (here after $ℏ = 1$ )

H = H_{s} + H_{p},

H_{s} = Δ_{c} a^{†} a + ω_{m} b^{†} b + g_{0} a^{†} a (b^{†} + b),

H_{p} = G (b^{†} + b) + Ω (a^{†} + a),

where, a (

a^{†}

) and b (

b^{†}

) are, respectively, the annihilation (creation) operators of the optical cavity field and the mechanical mode, with respective resonant frequencies ω_c and ω_m. g₀ represents the single-photon coupling strength between the cavity field and the mechanical resonator.

Δ_{c} = ω_{c} - ω_{L}

is the detuning between the cavity mode and the driving field. H_s is the Hamiltonian of the isolated OMS.

G (b^{†} + b)

describes the interaction between the mechanical mode and the pumping field, which is used to excite phonons. We note that, in a recent experiment, the mechanical pump strength can reach the value as high as

G / ω_{m} \sim 51

[56]. Here, we choose the experimentally accessible value

G / ω_{m} = 0.34

. The amplitude of the driving field Ω (

Ω ≪ γ_{c}

) is related to the input laser power P_in and cavity decay rate γ_c by

| Ω | = \sqrt{P_{in} γ_{c} / ω_{L}}

.

Fig. 1 (a) Schematic of the OMS with a driven oscillator. A mechanical pump with strength G is applied to the mechanical resonator. The statistics of the cavity mode is inferred from the output port by using photon-counting techniques [13, 29–33]. (b) Energy-level diagrams of the OMS with (the left) and without (the right) mechanical pump for the relevant zero-photon state $| 0 〉_{a}$ , one-photon state $| 1 〉_{a}$ , and two-photon state $| 2 〉_{a}$ . Here, $ξ_{1} = η + δ + G^{2} / ω_{m}$ , $ξ_{2} = 4 η + 2 δ + G^{2} / ω_{m}$ , $η = g_{0}^{2} / ω_{m}$ , $δ = 2 g_{0} G / ω_{m}$ .

Download Full Size | PDF

We consider $| n 〉_{a}$ and $| m 〉_{b}$ ( $n, m = 0, 1, 2...$ ) as the harmonic-oscillator number states of the cavity field and the mechanical mode, respectively. The Hamiltonian H exhibits an anharmonic energy-level configuration, which is crucial to realize 1PB and 2PB. Since the driving laser field is very weak, we can study the eigen system of this Hamiltonian by applying the unitary operator $D (n) = exp [(g_{0} n + G) (b - b^{†}) / ω_{m}]$ to $\tilde{H} = H_{s} + G (b^{†} + b)$ , where $n = a^{†} a$ . Then we obtain $\tilde{H} | n 〉_{a} | \tilde{m} (n) 〉_{b} = E_{n m} | n 〉_{a} | \tilde{m} (n) 〉_{b}$ with eigenvalues

E_{n m} = n Δ_{c} + m ω_{m} - n^{2} η - (n δ + \frac{G^{2}}{ω_{m}}),

where

η = g_{0}^{2} / ω_{m}

and

δ = 2 g_{0} G / ω_{m}

,

| n 〉_{a} | \tilde{m} (n) 〉_{b} = D (n) | n 〉_{a} | m 〉_{b}

.

| \tilde{m} (n) 〉_{b}

represents the n-photon displaced number states. Especially,

| \tilde{m} (0) 〉_{b} = exp [(G / ω_{m}) (b - b^{†})] | m 〉_{b}

. From Eq. (4), we can see that the cavity OMS is inherently nonlinear and the energy levels are nonharmonic. We note that the energy frequency shift with

n^{2} η

in Eq. (4) is caused by the nonlinear optomechanical interaction, which has been studied in previous literature [29]. Here, we focus on the mechanical pump induced frequency shift

n δ + G^{2} / ω_{m}

.

The eigenenergy spectrum of the Hamiltonian $\tilde{H}$ limited in the zero-, one-, and two-photon cases is shown in Fig. 1(b). For the OMS without mechanical pump, no PB occurs when the input laser frequency is $ω_{10} = ω_{c} - η - δ$ , since the transition $| 0 〉_{a} \to | 1 〉_{a}$ is detuned by δ. However, by applying the mechanical pump with strength G, the transition $| 0 〉_{a} \to | 1 〉_{a}$ is resonantly driven by the same input laser, but the transition $| 1 〉_{a} \to | 2 〉_{a}$ is detuned by $2 η$ , which features the 1PB effect. Similarly, with the mechanical pump, 2PB corresponding to the transitions $| 0 〉_{a} | \tilde{0} (0) 〉_{b} \to | 2 〉_{a} | \tilde{0} (2) 〉_{b}$ and $| 0 〉_{a} | \tilde{0} (0) 〉_{b} \to | 2 〉_{a} | \tilde{2} (2) 〉_{b}$ can occur for the driving frequency $ω_{20} = ω_{c} - 2 η - δ$ and $ω_{21} = ω_{c} - 2 η - δ + ω_{m}$ , respectively, but can not emerge in the system without the mechanical pump.

To confirm this intuitive picture, we analytically calculate the second-order and the third-order correlation functions of cavity photons with zero-time delay, i.e., $g^{(μ)} (0) = 〈 a^{† μ} a^{μ} 〉 / 〈 a^{†} a 〉^{μ}$ for μ = 2 and 3. In general, $g^{(2)} (0) > 1$ corresponds to PIT with super-Poissonian light and $g^{(2)} (0) < 1$ corresponds to 1PB with sub-Poissonian light, signifying nonclassical correlation. The condition $g^{(2)} (0) \to 0$ means a complete 1PB. Considering the higher-order correlation functions, we use the conditions

g^{(μ)} (0) ≪ 1, and g^{(μ)} (0) ≫ 1,

characterize 1PB and PIT, respectively. In order to prove 2PB where two-photon bunching and three-photon antibunching, it is sufficient to fulfill a necessary criterion [27], i.e.,

\begin{array}{l} g^{(2)} (0) > 1, \\ g^{(3)} (0) < 1. \end{array}

For the sufficient small Ω, only the lower energy levels of the system are excited. By treating the weak optical driving term as a perturbation, the general state of the system in the few-photon subspace can be written as

| ψ (t) 〉 = \sum_{n = 0}^{n = 3} \sum_{m = 0}^{\infty} C_{n, m} (t) | n 〉_{a} | \tilde{m} (n) 〉_{b},

where coefficients

C_{n, m}

describe the probability amplitudes of the corresponding states respectively. We phenomenologically add an anti-Hermitian term to the Hamiltonian, given in Eq. (1) [31, 80], to describe the dissipation of the cavity mode (the time in the case of

1 / γ_{c} ≪ t ≪ 1 / γ_{m}

, γ_m represents the mechanical decay). The effective non-Hermitian Hamiltonian takes the form

H_{eff} = H - i \frac{γ_{c}}{2} a^{†} a .

In terms of Eqs. (7) and (8), and the Schrödinger equation $i d ψ (t) / d t = H_{eff} ψ (t)$ , we obtain the equations of motion for the probability amplitudes

\begin{array}{l} {\dot{C}}_{0, m} = - i E_{0, m} C_{0, m} - i Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (0, 1) C_{1, m^{'}}, \\ {\dot{C}}_{1, m} = - Γ_{1, m} C_{1, m} - i Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (1, 0) C_{0, m^{'}} - i \sqrt{2} Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (1, 2) C_{2, m^{'}}, \\ {\dot{C}}_{2, m} = - Γ_{2, m} C_{2, m} - i \sqrt{2} Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (2, 1) C_{1, m^{'}} - i \sqrt{3} Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (2, 3) C_{3, m^{'}}, \\ {\dot{C}}_{3, m} = - Γ_{3, m} C_{3, m} - i \sqrt{3} Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (3, 2) C_{2, m^{'}}, \end{array}

where

Γ_{n, m} = n γ_{c} / 2 + i E_{n, m}

and

Θ_{m_{1}, m_{2}} (n_{1}, n_{2}) =_{b} 〈 {\tilde{m}}_{1} (n_{1}) | {\tilde{m}}_{2} (n_{2}) 〉_{b}

. These transition rates can be calculated by using the relations

_{b} 〈 \tilde{l} (n^{'}) | \tilde{k} (n) 〉_{b} =_{b} 〈 l | D (n - n^{'}) | k 〉_{b}

, and

_{b} 〈 l | e^{α (b^{†} - b)} | k 〉_{b} = {\begin{array}{l} \sqrt{\frac{l!}{k!}} e^{- \frac{α^{2}}{2}} {(- α)}^{k - l} L_{l}^{k - l} (α^{2}), & k \geq l \\ \sqrt{\frac{k!}{l!}} e^{- \frac{α^{2}}{2}} {(α)}^{l - k} L_{k}^{l - k} (α^{2}), & l > k \end{array}

where

L_{r}^{s} (x)

is generalized Laguerre polynomial. In the weak-optical-driving case, we have the following approximate formulas:

C_{0, m} \sim 1

,

C_{1, m} \sim Ω / γ_{c}

,

C_{2, m} \sim Ω^{2} / γ_{c}^{2}

,

C_{3, m} \sim Ω^{3} / γ_{c}^{3}

, then Eq. (9) can be solved by neglecting terms of higher order in Ω. For an initially empty cavity, we have

C_{1, m} (0) = 0

,

C_{2, m} (0) = 0

,

C_{3, m} (0) = 0

. Assuming the membrane is initially in its ground state, then the long-time solutions of the system can be obtained (see Appendix).

The equal-time second-order correlation and third-order correlation can be respectively written as $g^{(2)} (0) = 2 P_{2} / {(P_{1} + 2 P_{2})}^{2}$ and $g^{(3)} (0) = 6 P_{3} / {(P_{1} + 2 P_{2} + 3 P_{3})}^{3}$ . Here $P_{n} = \sum_{m = 0}^{\infty} {| C_{n, m} |}^{2}$ for $n = 1, 2, 3$ are the probabilities for finding a single photon, two and three photons in the cavity, respectively. Since the optical driving is very weak, we have $P_{1}^{2} ≫ P_{2}^{2} ≫ P_{3}^{2}$ . Thus, correlation functions are reduced to $g^{(2)} (0) \approx 2 P_{2} / P_{1}^{2}$ and $g^{(3)} (0) \approx 6 P_{3} / P_{1}^{3}$ . Then we get the approximate solutions of the second-order and third-order correlation functions

g^{(2)} (0) = \frac{4 {(χ_{1} - δ)}^{2} + γ_{c}^{2}}{4 {(χ_{2} - δ)}^{2} + γ_{c}^{2}},

g^{(3)} (0) = \frac{4 {[{(χ_{1} - δ)}^{2} + γ_{c}^{2}]}^{2}}{[4 {(χ_{2} - δ)}^{2} + γ_{c}^{2}] [4 {(χ_{3} - δ)}^{2} + γ_{c}^{2}]},

where

χ_{n} = Δ_{c} - n η

. For the single-photon resonance case,

Δ_{c} = η + δ

, the second-order correlation function is given as

g_{SPR}^{(2)} (0) = γ_{c}^{2} / (4 η^{2} + γ_{c}^{2})

. With strong-coupling, i.e.,

g_{0} > γ_{c}

, we have

g_{SPR}^{(2)} (0) ≪ 1

, which means the probability of exciting the single-photon state is higher than that of preparing a two-photon state, i.e., 1PB. For the two-photon resonance case,

Δ_{c} = 2 η + δ

, the second-order correlation function is given as

g_{TPR}^{(2)} (0) = (4 η^{2} + γ_{c}^{2}) / γ_{c}^{2}

. We have

g^{(2)} (0) ≫ 1

, which indicates PIT.

Fig. 2 The correlation functions $g^{(2)} (0)$ and $g^{(3)} (0)$ versus $Δ_{c} / ω_{m}$ for different strengths of mechanical pump. In (a) and (b), the solid and dashed lines correspond to the analytical and numerical solutions of $g^{(2)} (0)$ , respectively. In (c-f), the dashed and dotted lines correspond to $g^{(2)} (0)$ and $g^{(3)} (0)$ , respectively. The other parameters are taken as $g_{0} / ω_{m} = 0.5$ , $Ω / ω_{m} = 0.01$ , $γ_{c} / ω_{m} = 0.3$ , $γ_{m} / ω_{m} = 0.001$ and ${\bar{n}}_{m} = 0$ (T = 0).

Download Full Size | PDF

In order to confirm our analytical results, now we turn to the numerical solution case. The quantum fluctuations of the environmental will introduce damping to the cavity field and mechanical oscillator, as required by the fluctuation-dissipation theorem. After taking into account both optical and mechanical dissipations, the dynamical evolution of the system is described by the master equation

\begin{matrix} \dot{ρ} = i [ρ, H] + \frac{γ_{c}}{2} (2 a ρ a^{†} - a^{†} a ρ - ρ a^{†} a) + \frac{γ_{m}}{2} ({\bar{n}}_{m} + 1) (2 b ρ b^{†} - b^{†} b ρ - ρ b^{†} b) \\ + \frac{γ_{m}}{2} {\bar{n}}_{m} (2 b^{†} ρ b - b b^{†} ρ - ρ b b^{†}), \end{matrix}

where we assume that the cavity field is connected with a vacuum bath, γ_m represents the mechanical decay, and

{\bar{n}}_{m}

is the average thermal photon number related to the temperature by

{\bar{n}}_{m} = {[exp (ω_{m} / k_{B} T_{M}) - 1]}^{- 1}

, k_B is the Boltzmann constant, T_M is the temperature of the environment. By numerically solving Eq. (12), the steady state of the system and correlation functions can be obtained.

In Fig. 2(a), we plot the optical correlation function $g^{(2)} (0)$ versus optical detuning for G = 0 and $G = 0.34 ω_{m}$ respectively. Here, the solid curves are plotted using the numerical solution of Eq. (12), while the dashed curves are based on the analytical solution in Eq. (10). The analytical results, including correlation function $g^{(3)} (0)$ , agree well with the numerical one. We show that, for the OMS without mechanical pump (G = 0), $g^{(2)} (0)$ has a peak around $Δ_{c} = 0.56 ω_{m}$ , indicates that PIT occurs when the driving laser frequency is $ω_{L} = ω_{c} - 0.56 ω_{m}$ . In sharp contrast, by applying mechanical pump with strength $G = 0.34 ω_{m}$ , 1PB happens for the same driving light, i.e., $g^{(2)} (0) = 0.33$ , which can be seen more clearly in Fig. 2(b).

In addition, 2PB can also be generated with mechanical pump, as shown in Figs. 2(c) and 2(d). Without mechanical pump, we find no 2PB around $Δ_{c} = - 0.3 ω_{m}$ . However, 2PB occurs when the mechanical driving strength is $G = 0.34 ω_{m}$ since the correlation functions $g^{(2)} (0)$ and $g^{(3)} (0)$ fulfill the criteria given in Eq. (6). Moreover, by tuning mechanical driving strength, different types of PB can be generated with the fixed optical driving frequency. Figures 2(e) and 2(f) show that, 1PB occur around $Δ_{c} = 0.56 ω_{m}$ with mechanical driving strength $G = 0.34 ω_{m}$ . For the same driving laser, 2PB corresponding to the transitions $| 0 〉_{a} | \tilde{0} (0) 〉_{b} \to | 2 〉_{a} | \tilde{0} (2) 〉_{b}$ and $| 0 〉_{a} | \tilde{0} (0) 〉_{b} \to | 2 〉_{a} | \tilde{2} (2) 〉_{b}$ can emerge with $G = 0.18 ω_{m}$ and $G = 1.2 ω_{m}$ respectively. This generation of 1PB or 2PB with different mechanical pump indicates a mechanical switch of purely quantum effects.

3. Mechanical switch of PB and PIT

To study the mechanical switch between PIT and PB, we plot correlation functions $g^{(2)} (0)$ or $g^{(3)} (0)$ versus the mechanical pumping strength G for $Δ_{c} = 0.56 ω_{m}$ in Figs. 3(a)–(c). We find three types of mechanical switches can be achieved as follows:

Mechanical switch of 1PB and PIT—As shown in Fig. 3(a), we find the maximum value of $g^{(2)} (0)$ , i.e., $g^{(2)} (0) = 3.99$ at G = 0, corresponding to PIT. By increasing of G, $g^{(2)} (0)$ becomes smaller and reaches the minimum value $g^{(2)} (0) = 0.33$ at $G = 0.34 ω_{m}$ , i.e., 1PB, which indicates a switching behavior of 1PB and PIT can be achieved with different mechanical driving strength. This mechanical switch can be intuitively understood by considering the energy-level structure of the system. When $Δ_{c} = 0.56 ω_{m}$ , and the input laser with frequency $ω_{c} - 0.56 ω_{m}$ , there is a two-photon resonance with the transition $| 0 〉_{a} \to | 2 〉_{a}$ for G = 0; hence the absorption of the first photon favors also that of the second or subsequent photons, i.e., resulting in PIT. However, the same light is resonantly coupled to the transition $| 0 〉_{a} \to | 1 〉_{a}$ , but not resonantly coupled to the transition $| 1 〉_{a} \to | 2 〉_{a}$ for $G = 0.34 ω_{m}$ , which leading to 1PB. Thus, a mechanical switch of PIT and 1PB can be achieved by tuning mechanical pump.

Mechanical switch of 2PB and PIT—Fig. 3(b) shows that the third-order correlation function $g^{(3)} (0)$ also becomes smaller by applying stronger mechanical pump. At G = 0, $g^{(μ)} (0) > 1$ for $μ = 2, 3$ indicates PIT, as aforementioned. Interestingly, by increasing G, this PIT effect can be transformed into 2PB effect with $g^{(2)} (0) > 1$ and $g^{(3)} (0) < 1$ around $G = 0.18 ω_{m}$ . Also, 2PB can be converted into PIT by decreasing G. Together with the mechanical switch of 1PB and PIT, these mechanical switches of bunching light and few photons can find applications for quantum control of light in photonic communication and quantum technologies [81–83].

Mechanical switch of 1PB and 2PB—In Fig. 3(c), the switching behavior of 2PB and 1PB can be achieved by tuning the mechanical driving strength between $G = 0.18 ω_{m}$ and $G = 0.34 ω_{m}$ . This mechanically controlled quantum switch is expected to play a key role in quantum engineering [84], metrology [85], and quantum information processing [86] at the single- or two-photon level.

Fig. 3 (a)-(c) Correlation functions versus the strength of mechanical pump under the conditions $Δ_{c} = 0.56 ω_{m}$ . (d) $g^{(2)} (0)$ in logarithmic scale [i.e., ${log}_{10} g^{(2)} (0)$ ] as function of $G / ω_{m}$ and $Δ_{c} / ω_{m}$ . The other parameters are the same as Fig. 2.

Download Full Size | PDF

Fig. 4 (a) Correlation function $g^{(2)} (0)$ versus thermal photon number ${\bar{n}}_{m}$ for different mechanical pump. The values of optical detuning are chosen to be related to 1PB, i.e., $Δ_{c} = 0.22 ω_{m}$ and $0.56 ω_{m}$ for G = 0 and $0.34 ω_{m}$ , respectively. (b) $g^{(3)} (0)$ versus ${\bar{n}}_{m}$ for different mechanical pump in 2PB case, i.e., $Δ_{c} = - 0.64 ω_{m}$ and $- 0.3 ω_{m}$ for G = 0 and $0.34 ω_{m}$ , respectively. The other parameters are set the same as in Fig. 2.

Download Full Size | PDF

In addition, for other optical detunings, the mechanical switch of PB and PIT can also be achieved with different mechanical pump, as shown in Fig. 3(d). We find that, with different optical detunings Δ_c, the mechanical pumping strength for achieving 1PB and PIT are respectively given by

G_{1 PB} = \frac{- 3 g_{0}^{2} + \sqrt{g_{0}^{4} + γ_{c}^{2}} + 2 Δ_{c} ω_{m}}{4 g_{0}},

G_{PIT} = \frac{- 3 g_{0}^{2} - \sqrt{g_{0}^{4} + γ_{c}^{2}} + 2 Δ_{c} ω_{m}}{4 g_{0}} .

We note that thermal phonons greatly affect the correlation $g^{(μ)} (0)$ of photons and tend to destroy PB. Thus, to show this effect, in Fig. 4(a), we plot the correlation $g^{(2)} (0)$ as a function of thermal phonon number ${\bar{n}}_{m}$ for different mechanical pump. We see that PB can be observed below the critical temperature $T_{0} \approx 19 mK$ (i.e., ${\bar{n}}_{m} = 12.6$ ) and $T_{0} \approx 24 mK$ (i.e., ${\bar{n}}_{m} = 15.1$ ) for $G = 0.34 ω_{m}$ and G = 0 with experimentally feasible parameter $ω_{m} = 200 MHz$ , respectively. As shown in Fig. 4(b), correlation $g^{(3)} (0)$ has the same behaviors by tuning the temperature.

4. Conclusion and outlook

In this paper, we analytically and numerically calculate the second-order and third-order correlation functions at zero-time delay in the OMS with mechanical pump. We find that 1PB and 2PB can occur by tuning mechanical driving strength. Moreover, the mechanical switch for (i) 1PB and PIT, (ii) 2PB and PIT, or (iii) 1PB and 2PB can be achieved with different strength of mechanical pump. These mechanical switches of quantum effects can provide more intriguing control about few-photon or single-photon emissions [87, 88].

Our work can be further extended to study mechanical engineering of more purely quantum effects, such as unconventional PB [89–93], mechanical squeezing [94, 95], collective radiance effects and mechanical-assisted entanglement [96–99]. In addition, the OMS with the mechanical pump may become an excellent candidate for exploring new applications precision metrology [100] to tunable photonics [101, 102]. More interesting than direct-current (scalar) pump, the alternating current mechanical pump and parametric mechanical pump in the OMS [103–107], including phase effects and quantum noise effect, will be studied in future work.

Appendix: The derivation of the equal-time correlation functions

The equations of motion for the probability amplitudes $C_{n, m} (n = 1, 2, 3)$ read

\begin{array}{l} {\dot{C}}_{0, m} = - i E_{0, m} C_{0, m} - i Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (0, 1) C_{1, m^{'}}, \\ {\dot{C}}_{1, m} = - Γ_{1, m} C_{1, m} - i Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (1, 0) C_{0, m^{'}} - i \sqrt{2} Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (1, 2) C_{2, m^{'}}, \\ {\dot{C}}_{2, m} = - Γ_{2, m} C_{2, m} - i \sqrt{2} Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (2, 1) C_{1, m^{'}} - i \sqrt{3} Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (2, 3) C_{3, m^{'}}, \\ {\dot{C}}_{3, m} = - Γ_{3, m} C_{3, m} - i \sqrt{3} Ω \sum_{m^{'} = 0}^{\infty} Θ_{m, m^{'}} (3, 2) C_{2, m^{'}}, \end{array}

where

\begin{matrix} Γ_{n, m} = n γ_{c} / 2 + i E_{n, m}, \\ E_{n m} = n Δ_{c} + m ω_{m} - n^{2} η - n δ - G^{2} / ω_{m}, \end{matrix}

and

Θ_{m_{1}, m_{2}} (n_{1}, n_{2}) =_{b} 〈 {\tilde{m}}_{1} (n_{1}) | {\tilde{m}}_{2} (n_{2}) 〉_{b} .

In the weak-optical-driving case, we have the following approximate formulas:

C_{0, m} \sim 1, C_{1, m} \sim Ω / γ_{c}, C_{2, m} \sim Ω^{2} / γ_{c}^{2}, C_{3, m} \sim Ω^{3} / γ_{c}^{3} .

Then we can approximately solve the equations using a perturbation method by discarding higher-order terms in each equations for lower-order variables. For an initial empty cavity, the initial condition read as: $C_{0, m} (0) = C_{0, m} (0)$ and $C_{1, m} (0) = C_{2, m} (0) = C_{3, m} (0) = 0$ . Then, the long-time solutions can be approximately obtained:

\begin{array}{l} C_{0, m} = C_{0, m} (0) e^{- i E_{0, m} t}, \\ C_{1, m} = - Ω \sum_{l = 0}^{\infty} \frac{Θ_{m, l} (1, 0)}{μ_{1, m}} C_{0, l} (0) e^{- i E_{0, l} t}, \\ C_{2, m} = \sqrt{2} Ω^{2} \sum_{n, l = 0}^{\infty} \frac{Θ_{m, n} (2, 1) Θ_{n, l} (1, 0)}{μ_{1, n} μ_{2, m}} C_{0, l} (0) e^{- i E_{0, l} t}, \\ C_{3, m} = - \sqrt{6} Ω^{3} \sum_{m^{'}, q, l = 0}^{\infty} \frac{Θ_{m, m^{'}} (3, 2) Θ_{m^{'}, q} (2, 1) Θ_{q, l} (1, 0)}{μ_{1, q} μ_{2, m^{'}} μ_{3, m}} C_{0, l} (0) e^{- i E_{0, l} t}, \end{array}

where

μ_{n, m} = - E_{0, l} - i Γ_{n, m}

.

C_{0, m} (0)

and

C_{0, l} (0)

are determined by the initial state of the mechanical modes. We assume that the membrane is initially in its ground state

| 0 〉_{b}

, i.e.,

C_{0, m} (0) = δ_{m, 0}

. Considering the limit case of

λ = g_{0} / ω_{m} ≪ 1

, we can expand the displacement operators up to the order 1. Thus,

〈 \tilde{m} (n + 1) | {\tilde{m}}^{'} (n) 〉_{b} = δ_{m, m^{'}} + \sqrt{m^{'} + 1} λ δ_{m, m^{'} + 1} - \sqrt{m^{'}} λ δ_{m, m^{'} - 1} .

Accordingly, with the probability amplitudes given in Eq. (19) and $P_{n} = \sum_{m = 0}^{\infty} {| C_{n, m} |}^{2}$ for $n = 1, 2, 3$ , we get the single-photon, two-photons and three-photons probabilities

\begin{array}{l} P_{1} = Ω^{2} \sum_{m = 0}^{\infty} {| \frac{U_{m}^{0}}{Q_{m}^{1}} |}^{2}, \\ P_{2} = 4 Ω^{4} \sum_{m = 0}^{\infty} {| \frac{U_{m}^{0}}{Q_{0}^{1} Q_{m}^{2}} + \frac{λ U_{m}^{1}}{Q_{1}^{1} Q_{m}^{2}} |}^{2}, \\ P_{3} = 6 Ω^{6} \sum_{m = 0}^{\infty} {| \frac{U_{m}^{0}}{Q_{0}^{1} Q_{0}^{2} Q_{m}^{3}} + \frac{λ U_{m}^{1}}{Q_{0}^{1} Q_{1}^{2} Q_{m}^{3}} + \frac{λ U_{m}^{1}}{Q_{1}^{1} Q_{1}^{2} Q_{m}^{3}} + \frac{\sqrt{2} λ^{2} U_{m}^{2}}{Q_{1}^{1} Q_{2}^{2} Q_{m}^{3}} - \frac{λ^{2} U_{m}^{0}}{Q_{1}^{1} Q_{0}^{2} Q_{m}^{3}} |}^{2}, \end{array}

where

\begin{array}{l} U_{m}^{n} = δ_{n, m} + \sqrt{n + 1} λ δ_{n + 1, m} - \sqrt{n} λ δ_{n - 1, m}, \\ Q_{m}^{n} = E_{n, m} - E_{0, 0} - i \frac{n γ_{c}}{2} . \end{array}

In the case of $g_{0} / ω_{m} ≪ 1$ , the terms with high-order can be safely neglected. Consequently the probabilities of finding single, two and three photons in the cavity are, respectively, rewritten as:

\begin{array}{l} P_{1} = \frac{Ω^{2}}{{| E_{1, 0} - E_{0, 0} - i \frac{γ_{c}}{2} |}^{2}}, \\ P_{2} = \frac{2 Ω^{4}}{{| (E_{1, 0} - E_{0, 0} - i \frac{γ_{c}}{2}) (E_{2, 0} - E_{0, 0} - i γ_{c}) |}^{2}}, \\ P_{3} = \frac{6 Ω^{6}}{{| (E_{1, 0} - E_{0, 0} - i \frac{γ_{c}}{2}) (E_{2, 0} - E_{0, 0} - i γ_{c}) (E_{3, 0} - E_{0, 0} - i \frac{3 γ_{c}}{2}) |}^{2}} . \end{array}

With $g^{(2)} (0) = 2 P_{2} / P_{1}^{2}$ and $g^{(3)} (0) = 6 P_{3} / P_{1}^{3}$ , we can obtain the equal-time second-order correlation and third-order correlation

g^{(2)} (0) = \frac{4 {(χ_{1} - δ)}^{2} + γ_{c}^{2}}{4 {(χ_{2} - δ)}^{2} + γ_{c}^{2}},

g^{(3)} (0) = \frac{4 {[{(χ_{1} - δ)}^{2} + γ_{c}^{2}]}^{2}}{[4 {(χ_{2} - δ)}^{2} + γ_{c}^{2}] [4 {(χ_{3} - δ)}^{2} + γ_{c}^{2}]},

where

χ_{n} = Δ_{c} - n η

,

η = g_{0}^{2} / ω_{m}

and

δ = 2 g_{0} G / ω_{m}

.

Funding

National Natural Science Foundation of China (NSFC) (11474087, 11774086, 11935006, 11775075, 11434011); Hunan Normal University Program for Talented Youth; Hunan Province High-level Talents Program (2017).

Acknowledgments

C. Z., R. H., and H. Jing are supported by the National Natural Science Foundation of China (NSFC) (11474087, 11774086, 11935006), HuNan Normal University Program for Talented Youth, and Hunan Province High-level Talents Program (2017). L. M. Kuang is supported by the NSFC (11775075, 11434011). The authors thank Adam Miranowicz at Adam Mickiewicz University, Jie-Qiao Liao, Huilai Zhang, Baijun Li at Hunan Normal University and Tao Liu at RIKEN for stimulating discussions.

References

1. D. E. Chang, A. S. Sørensen, E. A. Demler, and M. D. Lukin, “A single-photon transistor using nanoscale surface plasmons,” Nat. Phys. 3, 807–812 (2007). [CrossRef]

2. F.-Y. Hong and S.-J. Xiong, “Single-photon transistor using microtoroidal resonators,” Phys. Rev. A 78, 013812 (2008). [CrossRef]

3. Y. Han, B. He, K. Heshami, C.-Z. Li, and C. Simon, “Quantum repeaters based on Rydberg-blockade-coupled atomic ensembles,” Phys. Rev. A 81, 052311 (2010). [CrossRef]

4. D. Gerace, H. E. Türeci, A. Imamoǧlu, V. Giovannetti, and R. Fazio, “The quantum-optical Josephson interferometer,” Nat. Phys. 5, 281–284 (2009). [CrossRef]

5. H.-Z. Wu, Z.-B. Yang, and S.-B. Zheng, “Implementation of a multiqubit quantum phase gate in a neutral atomic ensemble via the asymmetric Rydberg blockade,” Phys. Rev. A 82, 034307 (2010). [CrossRef]

6. K. Xia, F. Nori, and M. Xiao, “Cavity-Free Optical Isolators and Circulators Using a Chiral Cross-Kerr Nonlinearity,” Phys. Rev. Lett. 121, 203602 (2018). [CrossRef] [PubMed]

7. S. Zhang, Y. Hu, G. Lin, Y. Niu, K. Xia, J. Gong, and S. Gong, “Thermal-motion-induced non-reciprocal quantum optical system,” Nat. Photonics 12, 744–748 (2018). [CrossRef]

8. R. Huang, A. Miranowicz, J.-Q. Liao, F. Nori, and H. Jing, “Nonreciprocal Photon Blockade,” Phys. Rev. Lett. 121, 153601 (2018). [CrossRef] [PubMed]

9. L. Tang, J. Tang, W. Zhang, G. Lu, H. Zhang, Y. Zhang, K. Xia, and M. Xiao, “On-chip chiral single-photon interface: Isolation and unidirectional emission,” Phys. Rev. A 99, 043833 (2019). [CrossRef]

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

11. L. Tian and H. J. Carmichael, “Quantum trajectory simulations of the two-state behavior of an optical cavity containing one atom,” Phys. Rev. A 46, R6801–R6804 (1992). [CrossRef]

12. A. Imamoǧlu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly Interacting Photons in a Nonlinear Cavity,” Phys. Rev. Lett. 79, 1467–1470 (1997). [CrossRef]

13. K. M. Birnbaum, A. Boca, R. Miller, A. D. Boozer, T. E. Northup, and H. J. Kimble, “Photon blockade in an optical cavity with one trapped atom,” Nature (London) 436, 87–90 (2005). [CrossRef]

14. A. Reinhard, T. Volz, M. Winger, A. Badolato, K. J. Hennessy, E. L. Hu, and A. Imamoǧlu, “Strongly correlated photons on a chip,” Nat. Photonics 6, 93–96 (2012). [CrossRef]

15. C. Lang, D. Bozyigit, C. Eichler, L. Steffen, J. M. Fink, A. A. Abdumalikov, M. Baur, S. Filipp, M. P. da Silva, A. Blais, and A. Wallraff, “Observation of Resonant Photon Blockade at Microwave Frequencies Using Correlation Function Measurements,” Phys. Rev. Lett. 106, 243601 (2011). [CrossRef] [PubMed]

16. A. J. Hoffman, S. J. Srinivasan, S. Schmidt, L. Spietz, J. Aumentado, H. E. Türeci, and A. A. Houck, “Dispersive Photon Blockade in a Superconducting Circuit,” Phys. Rev. Lett. 107, 053602 (2011). [CrossRef] [PubMed]

17. T. Peyronel, O. Firstenberg, Q.-Y. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletić, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature (London) 488, 57–60 (2012). [CrossRef]

18. M. J. Hartmann, F. G. S. L. Brandão, and M. B. Plenio, “Strongly interacting polaritons in coupled arrays of cavities,” Nat. Phys. 2, 849–855 (2006). [CrossRef]

19. A. D. Greentree, C. Tahan, J. H. Cole, and L. C. L. Hollenberg, “Quantum phase transitions of light,” Nat. Phys. 2, 856–861 (2006). [CrossRef]

20. D. G. Angelakis, M. F. Santos, and S. Bose, “Photon-blockade-induced Mott transitions and XY spin models in coupled cavity arrays,” Phys. Rev. A 76, 031805 (2007). [CrossRef]

21. I. Carusotto, D. Gerace, H. E. Türeci, S. De Liberato, C. Ciuti, and A. Imamoǧlu, “Fermionized Photons in an Array of Driven Dissipative Nonlinear Cavities,” Phys. Rev. Lett. 103, 033601 (2009). [CrossRef] [PubMed]

22. M. J. Hartmann, “Polariton Crystallization in Driven Arrays of Lossy Nonlinear Resonators,” Phys. Rev. Lett. 104, 113601 (2010). [CrossRef] [PubMed]

23. D. E. Chang, V. Gritsev, G. Morigi, V. Vuletić, M. D. Lukin, and E. A. Demler, “Crystallization of strongly interacting photons in a nonlinear optical fibre,” Nat. Phys. 4, 884–889 (2008). [CrossRef]

24. S. S. Shamailov, A. S. Parkins, M. J. Collett, and H. J. Carmichael, “Multi-photon blockade and dressing of the dressed states,” Opt. Commun. 283, 766–772 (2010). [CrossRef]

25. A. Miranowicz, M. Paprzycka, Y.-X. Liu, J. Bajer, and F. Nori, “Two-photon and three-photon blockades in driven nonlinear systems,” Phys. Rev. A 87, 023809 (2013). [CrossRef]

26. C. J. Zhu, Y. P. Yang, and G. S. Agarwal, “Collective multiphoton blockade in cavity quantum electrodynamics,” Phys. Rev. A 95, 063842 (2017). [CrossRef]

27. C. Hamsen, K. N. Tolazzi, T. Wilk, and G. Rempe, “Two-Photon Blockade in an Atom-Driven Cavity QED System,” Phys. Rev. Lett. 118, 133604 (2017). [CrossRef] [PubMed]

28. A. Faraon, I. Fushman, D. Englund, N. Stoltz, P. Petroff, and J. Vučković, “Coherent generation of non-classical light on a chip via photon-induced tunneling and blockade,” Nat. Phys. 4, 859–863 (2008). [CrossRef]

29. P. Rabl, “Photon Blockade Effect in Optomechanical Systems,” Phys. Rev. Lett. 107, 063601 (2011). [CrossRef] [PubMed]

30. A. Nunnenkamp, K. Børkje, and S. M. Girvin, “Single-Photon Optomechanics,” Phys. Rev. Lett. 107, 063602 (2011). [CrossRef] [PubMed]

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

32. 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, 093602 (2015). [CrossRef] [PubMed]

33. 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, 053861 (2017). [CrossRef]

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

35. T. Ramos, V. Sudhir, K. Stannigel, P. Zoller, and T. J. Kippenberg, “Nonlinear Quantum Optomechanics via Individual Intrinsic Two-Level Defects,” Phys. Rev. Lett. 110, 193602 (2013). [CrossRef] [PubMed]

36. H. Seok and E. M. Wright, “Antibunching in an optomechanical oscillator,” Phys. Rev. A 95, 053844 (2017). [CrossRef]

37. H. Xie, C.-G. Liao, X. Shang, M.-Y. Ye, and X.-M. Lin, “Phonon blockade in a quadratically coupled optomechanical system,” Phys. Rev. A 96, 013861 (2017). [CrossRef]

38. L.-L. Zheng, T.-S. Yin, Q. Bin, X.-Y. Lü, and Y. Wu, “Single-photon-induced phonon blockade in a hybrid spin-optomechanical system,” Phys. Rev. A 99, 013804 (2019). [CrossRef]

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

40. M. Metcalfe, “Applications of cavity optomechanics,” Appl. Phys. Rev. 1, 031105 (2014). [CrossRef]

41. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically induced transparency,” Science 330, 1520–1523 (2010). [CrossRef] [PubMed]

42. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature (London) 472, 69–73 (2011). [CrossRef]

43. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010). [CrossRef]

44. H. Jing, S. K. Özdemir, Z. Geng, J. Zhang, X.-Y. Lü, B. Peng, L. Yang, and F. Nori, “Optomechanically-induced transparency in parity-time-symmetric microresonators,” Sci. Rep. 5, 9663 (2015). [CrossRef] [PubMed]

45. H. Lü, Y. Jiang, Y.-Z. Wang, and H. Jing, “Optomechanically induced transparency in a spinning resonator,” Photonics Res. 5, 367–371 (2017). [CrossRef]

46. K. Ullah, H. Jing, and F. Saif, “Multiple electromechanically-induced-transparency windows and Fano resonances in hybrid nano-electro-optomechanics,” Phys. Rev. A 97, 033812 (2018). [CrossRef]

47. H. Lü, C. Wang, L. Yang, and H. Jing, “Optomechanically Induced Transparency at Exceptional Points,” Phys. Rev. Appl. 10, 014006 (2018). [CrossRef]

48. V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, “Storing Optical Information as a Mechanical Excitation in a Silica Optomechanical Resonator,” Phys. Rev. Lett. 107, 133601 (2011). [CrossRef] [PubMed]

49. I. S. Grudinin, H. Lee, O. Painter, and K. J. Vahala, “Phonon Laser Action in a Tunable Two-Level System,” Phys. Rev. Lett. 104, 083901 (2010). [CrossRef] [PubMed]

50. J. Zhang, B. Peng, S. K. Özdemir, K. Pichler, D. O. Krimer, G. Zhao, F. Nori, Y.-X. Liu, S. Rotter, and L. Yang, “A phonon laser operating at an exceptional point,” Nat. Photonics 12, 479–484 (2018). [CrossRef]

51. H. Jing, S. K. Özdemir, X.-Y. Lü, J. Zhang, L. Yang, and F. Nori, “ $P T$ -Symmetric Phonon Laser,” Phys. Rev. Lett. 113, 053604 (2014). [CrossRef]

52. H. Lü, S. K. Özdemir, L.-M. Kuang, F. Nori, and H. Jing, “Exceptional Points in Random-Defect Phonon Lasers,” Phys. Rev. Appl. 8, 044020 (2017). [CrossRef]

53. A. G. Krause, M. Winger, T. D. Blasius, Q. Lin, and O. Painter, “A high-resolution microchip optomechanical accelerometer,” Nat. Photonics 6, 768–772 (2012). [CrossRef]

54. E. Gavartin, P. Verlot, and T. J. Kippenberg, “A hybrid on-chip optomechanical transducer for ultrasensitive force measurements,” Nat. Nanotechnol. 7, 509–514 (2012). [CrossRef] [PubMed]

55. Z.-P. Liu, J. Zhang, S. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, “Metrology with $P T$ -Symmetric Cavities: Enhanced Sensitivity near the $P T$ -Phase Transition,” Phys. Rev. Lett. 117, 110802 (2016). [CrossRef]

56. C. Bekker, R. Kalra, C. Baker, and W. P. Bowen, “Injection locking of an electro-optomechanical device,” Optica 4, 1196–1204 (2017). [CrossRef]

57. C. G. Baker, C. Bekker, D. L. Mcauslan, E. Sheridan, and W. P. Bowen, “High bandwidth on-chip capacitive tuning of microtoroid resonators,” Opt. Express 24, 20400–20412 (2016). [CrossRef] [PubMed]

58. L. Fan, K. Y. Fong, M. Poot, and H. X. Tang, “Cascaded optical transparency in multimode-cavity optomechanical systems,” Nat. Commun. 6, 5850 (2015). [CrossRef] [PubMed]

59. W. Z. Jia, L. F. Wei, Y. Li, and Y.-X. Liu, “Phase-dependent optical response properties in an optomechanical system by coherently driving the mechanical resonator,” Phys. Rev. A 91, 043843 (2015). [CrossRef]

60. X.-W. Xu and Y. Li, “Controllable optical output fields from an optomechanical system with mechanical driving,” Phys. Rev. A 92, 023855 (2015). [CrossRef]

61. J. Ma, C. You, L.-G. Si, H. Xiong, J. Li, X. Yang, and Y. Wu, “Optomechanically induced transparency in the presence of an external time-harmonic-driving force,” Sci. Rep. 5, 11278 (2015). [CrossRef] [PubMed]

62. C. Jiang, Y. Cui, X. Bian, F. Zuo, H. Yu, and G. Chen, “Phase-dependent multiple optomechanically induced absorption in multimode optomechanical systems with mechanical driving,” Phys. Rev. A 94, 023837 (2016). [CrossRef]

63. C. Jiang, Y. Cui, Z. Zhai, H. Yu, X. Li, and G. Chen, “Tunable slow and fast light in parity-time-symmetric optomechanical systems with phonon pump,” Opt. Express 22, 28834–28847 (2018). [CrossRef]

64. H. Suzuki, E. Brown, and R. Sterling, “Nonlinear dynamics of an optomechanical system with a coherent mechanical pump: Second-order sideband generation,” Phys. Rev. A 92, 033823 (2015). [CrossRef]

65. Y. Li, Y. Y. Huang, X. Z. Zhang, and L. Tian, “Optical directional amplification in a three-mode optomechanical system,” Opt. Express 25, 18907–18916 (2017). [CrossRef] [PubMed]

66. Y.-X. Liu, A. Miranowicz, Y. B. Gao, J. Bajer, C. P. Sun, and F. Nori, “Qubit-induced phonon blockade as a signature of quantum behavior in nanomechanical resonators,” Phys. Rev. A 82, 032101 (2010). [CrossRef]

67. N. Didier, S. Pugnetti, Y. M. Blanter, and R. Fazio, “Detecting phonon blockade with photons,” Phys. Rev. B 84, 054503 (2011). [CrossRef]

68. A. Miranowicz, J. Bajer, N. Lambert, Y.-X. Liu, and F. Nori, “Tunable multiphonon blockade in coupled nanomechanical resonators,” Phys. Rev. A 93, 013808 (2016). [CrossRef]

69. X. Wang, A. Miranowicz, H.-R. Li, and F. Nori, “Method for observing robust and tunable phonon blockade in a nanomechanical resonator coupled to a charge qubit,” Phys. Rev. A 93, 063861 (2016). [CrossRef]

70. M. Wang, X.-Y. Lü, A. Miranowicz, T.-S. Yin, Y. Wu, and F. Nori, “Unconventional phonon blockade via atom-photon-phonon interaction in hybrid optomechanical systems,” arXiv:1806.03754 (2018).

71. J. Bochmann, A. Vainsencher, D. D. Awschalom, and A. N. Cleland, “Nanomechanical coupling between microwave and optical photons,” Nat. Phys. 9, 712–716 (2013). [CrossRef]

72. A. D. ÓConnell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, “Quantum ground state and single-phonon control of a mechanical resonator,” Nature (London) 464, 697–703 (2010). [CrossRef]

73. I. Yeo, P.-L. de Assis, A. Gloppe, E. Dupont-Ferrier, P. Verlot, N. S. Malik, E. Dupuy, J. Claudon, J.-M. Gérard, A. Auffèves, G. Nogues, S. Seidelin, J.-P. Poizat, O. Arcizet, and M. Richard, “Strain-mediated coupling in a quantum dot-mechanical oscillator hybrid system,” Nat. Nanotechnol. 9, 106–110 (2014). [CrossRef]

74. A. Barfuss, J. Teissier, E. Neu, A. Nunnenkamp, and P. Maletinsky, “Strong mechanical driving of a single electron spin,” Nat. Phys. 11, 820–824 (2015). [CrossRef]

75. D. B. Sohn, S. Kim, and G. Bahl, “Time-reversal symmetry breaking with acoustic pumping of nanophotonic circuits,” Nat. Photonics 12, 91–97 (2018). [CrossRef]

76. Q. Lin, J. Rosenberg, D. Chang, R. Camacho, M. Eichen, K. J. Vahala, and O. Painter, “Coherent mixing of mechanical excitations in nano-optomechanical structures,” Nat. Photonics. 4, 236–242 (2010). [CrossRef]

77. H. Okamoto, A. Gourgout, C.-Y. Chang, K. Onomitsu, I. Mahboob, E. Y. Chang, and H. Yamaguchi, “Coherent phonon manipulation in coupled mechanical resonators,” Nat. Phys. 9, 480–484 (2013). [CrossRef]

78. P. Renault, H. Yamaguchi, and I. Mahboob, “Virtual Exceptional Points in an Electromechanical System,” Phys. Rev. Appl. 11, 024007 (2019). [CrossRef]

79. H. Xu, L. Jiang, A. A. Clerk, and J. G. E. Harris, “Nonreciprocal control and cooling of phonon modes in an optomechanical system,” Nature (London) 568, 65–69 (2019). [CrossRef]

80. M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70, 101–144 (1998). [CrossRef]

81. K. Y. Bliokh, D. Smirnova, and F. Nori, “Quantum spin Hall effect of light,” Science 348, 1448–1451 (2015). [CrossRef] [PubMed]

82. K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1–38 (2015). [CrossRef]

83. P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler, and P. Zoller, “Chiral quantum optics,” Nature (London) 541, 473–480 (2017). [CrossRef]

84. M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009). [CrossRef]

85. V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011). [CrossRef]

86. A. Blais, J. Gambetta, A. Wallraff, D. I. Schuster, S. M. Girvin, M. H. Devoret, and R. J. Schoelkopf, “Quantum-information processing with circuit quantum electrodynamics,” Phys. Rev. A 75, 032329 (2007). [CrossRef]

87. T.-K. Hsiao, A. Rubino, Y. Chung, S.-K. Son, H. Hou, J. Pedrós, A. Nasir, G. Éthier-Majcher, M. J. Stanley, R. T. Phillips, T. A. Mitchell, J. P. Griffiths, I. Farrer, D. A. Ritchie, and C. J. B. Ford, “Single-photon Emission from an Acoustically-driven Lateral Light-emitting Diode,” arXiv:1901.03464 (2019).

88. M. Abdi and M. B. Plenio, “Quantum Effects in a Mechanically Modulated Single-Photon Emitter,” Phys. Rev. Lett. 122, 023602 (2019). [CrossRef] [PubMed]

89. T. C. H. Liew and V. Savona, “Single Photons from Coupled Quantum Modes,” Phys. Rev. Lett. 104, 183601 (2010). [CrossRef] [PubMed]

90. M. Bamba, A. Imamoǧlu, I. Carusotto, and C. Ciuti, “Origin of strong photon antibunching in weakly nonlinear photonic molecules,” Phys. Rev. A 83, 021802 (2011). [CrossRef]

91. X.-W. Xu and Y.-J. Li, “Antibunching photons in a cavity coupled to an optomechanical system,” J. Phys. B: At. Mol. Opt. Phys. 46, 035502 (2013). [CrossRef]

92. H. J. Snijders, J. A. Frey, J. Norman, H. Flayac, V. Savona, A. C. Gossard, J. E. Bowers, M. P. van Exter, D. Bouwmeester, and W. Löffler, “Observation of the Unconventional Photon Blockade,” Phys. Rev. Lett. 121, 043601 (2018). [CrossRef] [PubMed]

93. C. Vaneph, A. Morvan, G. Aiello, M. Féchant, M. Aprili, J. Gabelli, and J. Estève, “Observation of the Unconventional Photon Blockade in the Microwave Domain,” Phys. Rev. Lett. 121, 043602 (2018). [CrossRef] [PubMed]

94. H. Tan, G. Li, and P. Meystre, “Dissipation-driven two-mode mechanical squeezed states in optomechanical systems,” Phys. Rev. A 87, 033829 (2013). [CrossRef]

95. X.-Y. Lü, J.-Q. Liao, L. Tian, and F. Nori, “Steady-state mechanical squeezing in an optomechanical system via Duffing nonlinearity,” Phys. Rev. A 91, 013834 (2015). [CrossRef]

96. Q. Bin, X.-Y. Lü, T.-S. Yin, Y. Li, and Y. Wu, “Collective radiance effects in the ultrastrong-coupling regime,” Phys. Rev. A 99, 033809 (2019). [CrossRef]

97. C. Joshi, J. Larson, M. Jonson, E. Andersson, and P. Öhberg, “Entanglement of distant optomechanical systems,” Phys. Rev. A 85, 033805 (2012). [CrossRef]

98. M. Wang, X.-Y. Lü, Y.-D. Wang, J. Q. You, and Y. Wu, “Macroscopic quantum entanglement in modulated optomechanics,” Phys. Rev A. 94053807 (2016). [CrossRef]

99. J. Li, S.-Y. Zhu, and G. S. Agarwal, “Magnon-Photon-Phonon Entanglement in Cavity Magnomechanics,” Phys. Rev. Lett. 121, 203601 (2018). [CrossRef] [PubMed]

100. T. D. Stowe, K. Yasumura, T. W. Kenny, D. Botkin, K. Wago, and D. Rugar, “Attonewton force detection using ultrathin silicon cantilevers,” Appl. Phys. Lett. 71, 288–290 (1997). [CrossRef]

101. M. L. Povinelli, M. Lončar, M. Ibanescu, E. J. Smythe, S. G. Johnson, F. Capasso, and J. D. Joannopoulos, “Evanescent-wave bonding between optical waveguides,” Opt. Lett. 30, 3042–3044 (2005). [CrossRef] [PubMed]

102. M. Notomi, H. Taniyama, S. Mitsugi, and E. Kuramochi, “Optomechanical Wavelength and Energy Conversion in High-Q Double-Layer Cavities of Photonic Crystal Slabs,” Phys. Rev. Lett. 97, 023903 (2006). [CrossRef]

103. A. Eichler, J. Chaste, J. Moser, and A. Bachtold, “Parametric Amplification and Self-Oscillation in a Nanotube Mechanical Resonator,” Nano Lett. 11, 2699–2703 (2011). [CrossRef] [PubMed]

104. A. Szorkovszky, A. C. Doherty, G. I. Harris, and W. P. Bowen, “Mechanical Squeezing via Parametric Amplification and Weak Measurement,” Phys. Rev. Lett. 107, 213603 (2011). [CrossRef] [PubMed]

105. M.-A. Lemonde, N. Didier, and A. A. Clerk, “Enhanced nonlinear interactions in quantum optomechanics via mechanical amplification,” Nat. Commu. 7, 11338 (2016). [CrossRef]

106. H.-K. Lau and A. A. Clerk, “Ground state cooling and high-fidelity quantum transduction via parametrically-driven bad-cavity optomechanics,” arXiv: 1904.12984 (2019).

107. A. Motazedifard, A. Dalafi, F. Bemani, and M. H. Naderi, “Force sensing in hybrid Bose-Einstein-condensate optomechanics based on parametric amplification,” Phys. Rev. A 100, 023815 (2019). [CrossRef]

Mechanical switch of photon blockade and photon-induced tunneling

Abstract

1. Introduction

2. Model and solution

3. Mechanical switch of PB and PIT

4. Conclusion and outlook

Appendix: The derivation of the equal-time correlation functions

Funding

Acknowledgments

References

Cited By

Figures (4)

Equations (25)

Optics Express