Abstract
We theoretically investigate the phonon statistics of a quadratically coupled optomechanical system, in which an effective secondorder nonlinear interaction between an optical mode and a mechanical mode is induced by a strong optical driving field on twophonon redsideband resonance. We show that strong phonon antibunching can be observed even if the strength of the effective secondorder nonlinear interaction is much weaker than the decay rates of the system, by driving the optical and mechanical modes with weak driving fields respectively. Moreover, the phonon statistics can be dynamically controlled by tuning the strengths and the phase difference of the weak driving fields. The scheme proposed here can be used to realize tunable singlephonon sources with quadratically optomechanical coupling.
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Introduction
Phonon blockade^{1}, in analogy to the Coulomb blockade^{2}, photon blockade^{3} and Rydberg blockaded^{4,5,6,7,8}, is a quantum phenomenon that only one phonon can be excited in a nonlinear mechanical oscillator when it is driven by external fields. Phonon blockade has already been studied in a mechanical resonator coupled to a superconducting qubit in the dispersive^{1,9,10,11} and resonant^{12,13} regimes. Effective phononphonon interactions can be induced by the qubit and strong phonon antibunching effect can be observed for large coupling strength and moderate detuning between the mechanical resonator and the qubit.
In the past decades, optomechanical systems have drew great attention in researches on the foundations of quantum theory and quantum information processing (for reviews, see refs^{14,15,16,17,18}.). Recently, two different groups studied phonon statistics in quadratically coupled optomechanical systems^{19,20}. Seok and Wright found that antibunched singlephonon field appears for large optomechanical cooperativity^{19}. Hong Xie et al. found that strong effective phononphonon nonlinear interaction as well as phonon blockade can be induced by a strong optical driving field in the quadratically coupled optomechanical system^{20}.
In contrast to refs^{1,9,10,11,12,19,20}, where the phonon blockade is induced by strong effective phononphonon interactions, an interferencebased phonon blockade called unconventional phonon blockade (UCPNB) was studied in ref.^{13}. UCPNB, due to the destructive interference between different paths for twophonon excitation, can be obtained with weak effective phononphonon interactions is similar to the unconventional photon blockade in a weakly nonlinear system of photonic molecule^{21,22,23,24,25,26,27,28,29,30,31,32,33}. More recently, UCPNB was studies in a coupled nonlinear mechanical system with weak nonlinearity^{34}.
In this paper, we shall theoretically investigate UCPNB in a quadratically coupled optomechanical system. An effective secondorder nonlinear interaction between an optical mode and a mechanical mode can be induced when the quadratically coupled optomechanical system is driven by a strong optical driving field on twophonon redsideband resonance. Beside the strong optical driving field, the optical and mechanical modes are also driven by a weak optical and mechanical fields respectively. Different from the previous studies^{19,20}, we will show that strong phonon antibunching can be observed even if the strength of the effective secondorder nonlinear interaction is much weaker than the decay rates of the system. Moreover, the phonon statistics can be dynamically controlled by tuning the strengths and the phase difference of the weak driving fields. The proposal provides a simple way to realize tunable singlephonon sources with quadratically optomechanical coupling.
Results
Theoretical model and analytical results
We study a quadratically coupled optomechanical system in which an optical mode is coupled to the second order of the position of a mechanical mode, as schematically shown in Fig. 1. The optical mode with frequency ω_{ c } is driven by a strong driving field with the strength \({{\rm{\Omega }}}_{L}\gg \{{\gamma }_{c},{\gamma }_{m}\}\) and frequency ω_{ L }, where γ_{ c } and γ_{ m } are the damping rates of the optical and mechanical modes and Δ_{ c } ≡ ω_{ c } − ω_{ L } is the frequency detuning between the strong driving field and the optical mode. Meanwhile, the optical mode and mechanical mode (frequency ω_{ m }) are driven by weak external fields with strengths {ε_{ p }, ε_{ m }} < {γ_{ c }, γ_{ m }} and frequencies {ω_{ p }, ω_{ d }}, with the detuning between the optical driving fields δ_{ p } = ω_{ p } − ω_{ L }. The Hamiltonian for quadratically coupled optomechanical system in the rotating reference frame with optical frequency ω_{ L } takes the form (ħ = 1)
where A and A^{†} (B and B^{†}) denote the annihilation and creation operators for the optical mode (mechanical mode), g > 0 describes the strength of the quadratic optomechanical coupling between the optical and mechanical modes, and H.c. stands for Hermitian conjugate. The quadratically optomechanical coupling can be found in the optomechanical crystals^{35}, FabryPerot cavities with membraneinthemiddle^{36,37,38,39}, and some other optomechanical systems^{40,41,42,43}.
The operators can be rewritten as the sum of their steadystate mean values and quantum fluctuation operators as: A → α + a and B → β + b, where α and β are the steadystate mean values, a and b are the quantum flucturation operators. The steadystate mean values α and β can be obtained approximately by setting the strength of the weak driving fields as zero, i.e. ε_{ m } = ε_{ p } = 0, then we have
After some standard procedures for operator linearization, the Hamiltonian for the quantum flucturation operators reads
For a strong optical driving field \({{\rm{\Omega }}}_{L}\gg \{{\gamma }_{c},{\gamma }_{m}\}\), we assume that the steadystate mean value α is much larger than the quantum flucturation operators a as \({\alpha }^{2}\gg \langle {a}^{\dagger }a\rangle \), so the term ga^{†}a(b^{†} + b)^{2} in the above equation can be neglected. In the rotating reference frame with respect to the unitary operator R(t) = exp(iδ_{ p }a^{†}at + iω_{ d }b^{†}bt), the effective Hamiltonian can be obtained under the rotatingwave approximation by neglecting the terms oscillating with high frequencies in equation (4), e.g. 2ω_{ d } and δ_{ p } + 2ω_{ d }, as
where the detunings δ = δ_{ p } − 2ω_{ d }, Δ_{ p } = Δ_{ c } − δ_{ p }, Δ_{ m } = ω_{ m } + 2gα^{2} + g − ω_{ d }, and we assume that the detunings satisfy the condition \(\{\delta ,{{\rm{\Delta }}}_{p},{{\rm{\Delta }}}_{m}\}\ll \{{\omega }_{m},{\omega }_{d}\}\). J = gα is the effective nonlinear coupling strength between the optical and mechanical modes. Without loss of generality, J, ε_{ p } and ε_{ m } are assumed to be real and the phase difference between the driving fields is denoted by θ. For simplicity, we set δ_{ p } = 2ω_{ d } and Δ_{ c } = 2(ω_{ m } + 2gα^{2} + g), then we have δ = 0 and Δ ≡ Δ_{ m } = Δ_{ p }/2, and the effective Hamiltonian \({H}_{{\rm{eff}}}^{\text{'}}\) become timeindependent as
To quantify the statistics of the phonons in the system, we consider the secondorder correlation functions in the steady state defined by
where n_{ b } ≡ 〈b^{†}(t)b(t)〉 is the mean phonon number. The dynamic behavior of the total open system is described by the master equation for the density matrix ρ^{44}
where we assume that the mean thermal photon number is negligible for the frequency of the optical mode is very high, and n_{th} is the mean number of the thermal phonons, given by the BoseEinstein statistics n_{th} = [exp(ħω_{ m }/k_{ B }T) − 1]^{−1} with the Boltzmann constant k_{ B } and the environmental temperature T. The secondorder correlation function \({g}_{b}^{(2)}(\tau )\) can be calculated by solving the master equation (8) numerically within a truncated Fock space.
It is instructive to find the optimal conditions for strong phonon antibunching before the numerical calculations of the secondorder correlation function of the phonons. Following the approach given in ref.^{22}, the optimal conditions for UCPNB can be derived analytically with the effective Hamiltonian H_{eff} given in equation (6), in the limit T → 0 and the weak driving condition \(\{{\varepsilon }_{p},{\varepsilon }_{m}\}\ll \{{\gamma }_{c},{\gamma }_{m}\}\). The derivation of the the optimal conditions is provided in the section of Methods. When θ = Nπ (N is an integer), the optimal conditions are simply given by
When θ ≠ Nπ, the optimal conditions become
where
In order to make sure that Δ_{opt} and J_{opt} given in equations (11) and (12) have real solutions, the phase θ should satisfy the condition
We take J_{opt} > 0 in the following numerical calculations, so that N should be an odd number. Without loss of generality, we choose N = 1.
Numerical results
In order to confirm the appearing of optimal UCPNB with the optimal parameters given in equations (9–13), we numerically solve the master equation (8) and calculate the secondorder correlation functions \({g}_{b}^{(2)}(\tau )\). In Fig. 2(a), the equaltime secondorder correlation functions \({g}_{b}^{(2)}(0)\) is plotted as a function of the detuning Δ/γ_{ c } with the effective coupling strength J = 0.025γ_{ c } and phase θ = π. It is clear that the optimal phonon blockade appears at the detuning Δ = 0 and this agrees well with the analytical result given in equation (9). The corresponding mean phonon number n_{ b } is plotted in Fig. 2(b). The maximal value of n_{ b } also appears at the detuning Δ = 0 for resonant driving. The dependence of \({g}_{b}^{(2)}(0)\) on the strength of the effective coupling J/γ_{ c } is shown in Fig. 2(c) for Δ = 0 and θ = π. There is a minimal value of \({g}_{b}^{(2)}(0)\) around J ≈ 0.025γ_{ c } which is in agreement with equation (10). \({g}_{b}^{(2)}(\tau )\) is plotted as a function of the normalized time delay τ/(2π/γ_{ m }) in Fig. 2(d) with Δ = 0, J = 0.025γ_{ c } and θ = π. The time duration for strong phonon antibunching is about the lifetime of the phonons.
There are two weak driving fields applied to the system with the driving strengths ε_{ p } and ε_{ m } and they can allow for dynamic control of the phonon statistics by tuning the strengths and the phase difference of driving fields. In Fig. 3 for θ = π, \({g}_{b}^{(2)}(0)\) is plotted (a) as a function of the mechanical driving strength ε_{ m }/γ_{ c } with optical driving strength ε_{ p } = 0.01γ_{ c }, (b) as a function of the optical driving strength ε_{ p }/γ_{ c } with mechanical driving strength ε_{ m } = 0.005γ_{ c }. The minimal \({g}_{b}^{(2)}(0)\) appears with mechanical driving strength ε_{ m } ≈ ±0.005γ_{ c } in Fig. 3(a) and with optical driving strength ε_{ p } ≈ 0.01γ_{ c } in Fig. 3(b). These results are consistent with the analytically expression given in equation (10). Moreover, as shown in Fig. 3(a), the phonons exhibit strong bunching as ε_{ m } = 0 but exhibit strong antibunching as ε_{ m } = 0.005γ_{ c }. These phenomena can be understand as follows: when ε_{ m } = 0, phonons only can be generated in pairs by the optical driving field, so the phonons exhibit strong bunching; when ε_{ m } ≠ 0, phonons pairs can be generated in two different ways (by optical driving field or by mechanical driving field), the strong phonon antibunching is induced by the destructive interference between the two different ways for phonon pairs generation when ε_{ m } ≈ ε_{ p }/2 = 0.005γ_{ c }. As shown in Fig. 3(b), the phonons exhibit strong antibunching as ε_{ p } = 0.01γ_{ c } but exhibit bunching as ε_{ p } > 0.02γ_{ c } or ε_{ p } < 0. So we can control the phonon statistics dynamically by tuning the strengths of driving fields.
In Fig. 4(a), we show the contour plot of \({g}_{b}^{(2)}(0)\) as a function of the phase θ/π and the detuning Δ/γ_{ c } with the effective coupling strength J given by
where
In Fig. 4(b), we show the contour plot of \({g}_{b}^{(2)}(0)\) as a function of θ/π and J/γ_{ c } with the detuning Δ given by
where
The white dashed lines refer to equation (11) in Fig. 4(a) and refer to equation equation (12) in Fig. 4(b). The white dashed lines conform very closely to optimal region (dark blue region) for phonon antibunching. Obviously, the phonon statistic properties are also dependent on the phase difference θ of the driving fields.
Different from the photon blockade in optical cavities with frequency 10^{14} Hz, where the mean thermal photon number is negligible, the effect of the thermal phonons should be considered in the investigation of phonon blockade in mechanical resonators even with microwavefrequency^{13}. In Fig. 5(a), \({g}_{b}^{(2)}(0)\) is plotted as a function of the mean thermal phonon number n_{th}. One can see that the phonon antibunching becomes weaker with the increase of the the mean thermal phonon number n_{th}. In Fig. 5(b), \({g}_{b}^{(2)}(0)\) is plotted as a function of the driving strength ε_{ m }/γ_{ c } with different mean thermal phonon number n_{th}. The optimal phonon blockade can be obtained by properly increasing the driving strengths according to the mean thermal phonon number n_{th}.
Discussion
In summary, we have investigated the UCPNB in a quadratically coupled optomechanical system. It has been shown that strong phonon antibunching can be observed even with weak effective secondorder nonlinear interaction. The optimal conditions for UCPNB were given analytically and they well coincided with the numerical results. Moreover, the phonon statistics can be dynamically controlled by tuning the strengths and the phase difference of external driving fields. The results show that tunable singlephonon sources can be realize in the quadratically coupled optomechanical systems.
Based on the numerical results, we can estimate the experimental parameters for realizing our proposal. For instance, if we take the parameters according to the numerical simulations in ref.^{35}, ω_{ m }/2π = 225 MHz, γ_{ c }/2π = 20 MHz, g/2π = 10 kHz, and γ_{ m }/2π = 80 kHz, then the effective coupling strength J = 0.025γ_{ c } can be realized with α = 50 when the strength of the strong optical driving field is taken as Ω_{ L } ≈ 27.5 GHz. In order to reduce the negative impact of the environment on the phonon statistics, the experiments should be done under low temperature with high mechanical frequency. The mechanical resonators with frequency above 5 GHz have already be realized in many groups^{45,46}, and the mean thermal phonon number will be smaller than 10^{−4} at a temperature of 25 mK in a dilution refrigerator. So far as we know, the secondorder correlation of phonons can not be observed directly. In a recent experiment, the correlations of phonons have been observed indirectly by coupling an auxiliary optical cavity to the mechanical resonator and measuring photon correlations of the output field from the optical cavity^{47}.
Methods
In this section, we will derive the optical conditions for UCPNB analytically with the effective Hamiltonian H_{eff} given in equation (6), in the limit T → 0 and the weak driving condition \(\{{\varepsilon }_{p},{\varepsilon }_{m}\}\ll \{{\gamma }_{c},{\gamma }_{m}\}\). The wave function can be expanded on a Fock state basis as
where \(n,m\rangle \) represents the state with n photons and m phonons, and the corresponding coefficient C_{ nm }^{2} denotes the occupying probability. In the weak driving condition, i.e. \(\{{\varepsilon }_{p},{\varepsilon }_{m}\}\ll \{{\gamma }_{c},{\gamma }_{m}\}\), we will have \({C}_{00}\gg \{{C}_{10},{C}_{01},{C}_{02}\}\gg \{{C}_{11},{C}_{02},{C}_{12}\}\gg \cdots \), so the wave function can be truncated to the onephoton and twophonon states approximately. Substituting the wave function in equation (19) and the Hamiltonian in equation (6) into the Schrödinger’s equation \(id\psi \rangle /dt={H}_{{\rm{eff}}}\psi \rangle \), then the dynamical equations for the coefficients C_{ nm } are shown as
In the steady state, i.e. dC_{ nm }/dt = 0, the phonon blockade \({g}_{b}^{\mathrm{(2)}}\mathrm{(0)}\approx 0\) appears when C_{02} ≈ 0. Under the condition for phonon blockade, i.e. C_{02} ≈ 0, the coefficients C_{10}, C_{01} and C_{00} satisfy the linear equations
From equations (23) and (24), C_{10} and C_{01} are given by
Substituting C_{10} and C_{01} into equation (25), we obtain
As C_{00} ≈ 1 ≠ 0, then we get the conditions for the optimal parameters J_{opt} and Δ_{opt} as
The optimal parameters for phonon blockade given in equations (9–13) are obtained by solving the equations (29) and (30).
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Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant No.11604096.
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H.Q.S. and X.W.X. conceived the idea and carried out the calculation. N.H.L. supervised the work. All authors contributed to the interpretation of the work and the preparation of the manuscript.
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Shi, HQ., Zhou, XT., Xu, XW. et al. Tunable phonon blockade in quadratically coupled optomechanical systems. Sci Rep 8, 2212 (2018). https://doi.org/10.1038/s4159801820568x
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DOI: https://doi.org/10.1038/s4159801820568x
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