Introduction

Phonon blockade1 is a quantum effect for preventing the excitation of more than one phonon in a nanomechanical resonator (NAMR), which provides us an effective way to generate single phonons. For the potential application in phononic quantum information processing2,3,4, phonon blockade has draw more and more attentions in recent years5,6,7,8,9,10,11,12,13,14,15,16,17,18. The various proposals for realizing phonon blockade so far can be classified into two types namely conventional phonon blockade (CPNB)1,5,6,7,8,9,10,11,12,13,14 and unconventional phonon blockade (UCPNB)8,15,16,17,18.

The mechanism for CPNB is attributed to the strong nonlinearity in the system1. The strong nonlinearity results in the enharmonic energy level in system, thus the second phonon cannot be excited for the large detuning. Specifically, the strong nonlinearity for mechanical mode can be induced by dispersive (far off-resonant) NAMR-qubit coupling1,5,6,7, a NAMR resonant coupled to a qubit8 or a two-Level defect9, quadratically optomechanical coupling10,11,12,13, and the coupling between nitrogen-vacancy (NV) centers and a mechanical mode14.

Different from the CPNB, UCPNB is the counter-intuitive phenomenon that strong phonon antibunching can be observed with weak nonlinearity8,15,16,17,18. Physically, the strong phonon antibunching for UCPNB is based on the destructive interference between different paths for two-phonon excitation8, that UCPNB is usually realized by coupling an auxiliary system to the mechanical mode. Recently, UCPNB was predicted in many different systems, e.g., resonant coupled NAMR-qubit system8, coupled nonlinear mechanical resonators15,16, quadratically optomechanical system17, and hybrid optomechanical system18.

In this paper, we propose to observe phonon blockade with a quadratically coupling between a NAMR and a two-level system (TLS). The quadratically coupling between a NAMR and a TLS provides us an effective way to generate two phonons at one time19,20. We note that the phonon blockade by the quadratically coupling between a NAMR and a TLS has been studied in a recent work7. However, different from the previous study7, we will focus on the crossover from the UCPNB to CPNB and discuss the phonon blockade induced by the combination of quantum interference effect and strong nonlinearity of the system, which have not been revealed in previous works.

Results

Theoretical model

In this paper, we shall investigate a system in which a nanomechanical resonator is quadratically coupled to a TLS. As shown in Fig. 1, the quadratically coupling between NAMR and TLS can be implemented in a superconducting NAMR-qubit system19 [Fig. 1(a)], or in a phononic crystal with NV centers located near the surface14,20 [Fig. 1(b)]. We assume that the NAMR is driven by a mechanical pump with amplitude εm and frequency ωb and the TLS is driven by an external field with the strength εp and frequency ωd, respectively. The Hamiltonian for the system in the rotating reframe with respect to

$$R(t)=\exp (i{\omega }_{b}{b}^{\dagger }bt+i{\omega }_{d}{\sigma }_{+}{\sigma }_{-}t)\,{\rm{is}}\,{\rm{given}}\,{\rm{by}}\,(\hslash =1)$$
$$H=2{\rm{\Delta }}{\sigma }_{+}{\sigma }_{-}+{\rm{\Delta }}{b}^{\dagger }b+J({\sigma }_{-}{b}^{\dagger 2}+{\sigma }_{+}{b}^{2})+({\varepsilon }_{m}{e}^{-i\theta }{b}^{\dagger }+{\varepsilon }_{p}{\sigma }_{+}+{\rm{H}}.\,{\rm{c}}.),$$
(1)

where b and \({b}^{\dagger }\) denote the annihilation and creation operators of the NAMR with frequency ωm; σ+ and σ− are the raising and lowering operators of TLS with the frequency splitting ω0; we assume that the frequencies satisfy the conditions, ω0 = 2ωm and ωd = 2ωb, and Δ = ωm − ωb is the detuning between NAMR and driving field. θ is the phase difference between the two external driving fields. J is the quadratically coupling strength between the NAMR and TLS. Without loss of generality, J is assumed to be real.

Figure 1
figure 1

The schematic sketch of (a) a nanomechanical resonator coupled to a superconducting qubit19,25, (b) a phononic crystal with the NV center ensembles located near the surface14,20,26.

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To quantify the statistics of the phonons in the NAMR, we consider the equal-time second-order correlation function in the steady state defined by

$${g}_{b}^{(2)}(0)\equiv \frac{\langle {b}^{\dagger }{b}^{\dagger }bb\rangle }{{n}_{b}^{2}},$$
(2)

where \({n}_{b}\equiv \langle {b}^{\dagger }b\rangle \) is the mean phonon number. The behavior of the system is described by the master equation21 for the density matrix ρ

$$\begin{array}{rcl}\frac{d\rho }{dt} & = & -i[H,\rho ]+\kappa ({n}_{\sigma ,{\rm{th}}}+1)L[{\sigma }_{-}]\rho +\kappa {n}_{\sigma ,{\rm{th}}}L[{\sigma }_{+}]\rho \\ & & +\,\gamma ({n}_{m,{\rm{th}}}+1)L[b]\rho +\gamma {n}_{m,{\rm{th}}}L[{b}^{\dagger }]\rho ,\end{array}$$
(3)

where \(L[o]\rho =o\rho {o}^{\dagger }-({o}^{\dagger }o\rho +\rho {o}^{\dagger }o)/2\) denotes a Lindbland term for an operator o, κ is damping rate of the TLS and γ is damping rate of the NAMR; nσ,th and nm,th are the mean numbers of the thermal phonons, given by the Bose-Einstein statistics nσ,th = [exp(ℏω/kBT) − 1]−1 and nm,th = [exp(ℏωm/kBT) − 1]−1 with the Boltzmann constant kB and the environmental temperature T. The second-order correlation function \({g}_{b}^{(2)}(0)\) can be calculated by solving the master equation (3) numerically within a truncated Fock space.

Numerical results

Generally, the UCPNB can be discriminated from the CPNB by the optimal conditions for phonon blockade. Based on the Hamiltonian given in equation (1), the optimal conditions for UCPNB can be obtained analytically (the derivation is given in the the section of Methods). When θ ≠ Nπ/2 (N is an integer), the optimal conditions for UCPNB are

$${{\rm{\Delta }}}_{{\rm{opt}}}=\frac{1}{4\,\tan \,2\theta }(\gamma -\frac{\kappa }{2}\pm \sqrt{{\rm{\Psi }}}),$$
(4)
$${J}_{{\rm{opt}}}=-\frac{{\varepsilon }_{m}^{2}\,\cos \,2\theta }{{\varepsilon }_{p}\gamma }(\gamma +\frac{\kappa }{2}\pm \sqrt{{\rm{\Psi }}}),$$
(5)

where

$${\rm{\Psi }}={(\frac{\kappa }{2}-\gamma )}^{2}-2\kappa \gamma {\tan }^{2}2\theta .$$
(6)

When θ = Nπ/2, the optimal conditions for UCPNB become

$${{\rm{\Delta }}}_{{\rm{opt}}}=0,$$
(7)
$${J}_{{\rm{opt}}}=-\,\frac{{\varepsilon }_{m}^{2}\kappa \,\cos \,2\theta }{{\varepsilon }_{p}\gamma }.$$
(8)

The optimal detuning for CPNB is Δopt = 0 for resonant single-phonon driving, which is the same as the optimal detuning for the UCPNB in the special case for θ = Nπ/2.

In Fig. 2(a), we show the equal-time second-order correlation function \({g}_{b}^{(2)}(0)\) as a function of the detuning Δ/γ with θ = 0.6π ≠ Nπ/2 and J ≈ 0.4γ given by equation (5). We note that the optimal phonon blockade appears at detuning Δ ≈ −0.52γ for T = 20 mK, which is in good agreement with the analytical result Δopt ≈ −0.57γ given by equation (4). As equations (4) and (5) are the optimal conditions for UCPNB and the coupling strength is weak (J < γ), the phonon blockade discovered here should be based on the quantum interference, i.e., the UCPNB. The mean phonon number nb versus the detuning Δ/γ is shown in Fig. 2(b) for different temperatures: T = (20, 30, 40) mK. One can also find that the phonon antibunching becomes weaker with the increase of the temperature as well as the thermal phonons.

Figure 2
figure 2

(a) The equal-time second-order correlation function \({g}_{b}^{(2)}(0)\) and (b) mean phonon number nb are plotted as functions of the detuning Δ/γ for different temperatures: T = (20, 30, 40) mK. The other parameters are εm = 0.06γ, εp = 0.06γ, θ = 0.6π, κ = 10γ, ω0 = 2ωm = 2π × 8 GHz, and J ≈ 0.4γ is given by equation (5).

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In order to achieve a larger number of mean phonons and improve the robustness against the thermal phonons, we discuss the effect of the driving strengths on the phonon statistics. \({g}_{b}^{(2)}(0)\) and nb are plotted as functions of the mechanical driving strength εm/γ with optical driving strength \({\varepsilon }_{p}=10{\varepsilon }_{m}^{2}/\gamma \) in Fig. 3(a,b), or \({\varepsilon }_{p}={\varepsilon }_{m}^{2}/\gamma \) in 3(c) and 3(d). At nonzero temperature, the phonon blockade can be enhanced by properly increasing the driving strengths according to the temperature. Moreover, the mean phonon number for phonon blockade in the strong coupling regime with J ≈ 6.74γ in Fig. 3(d) is much larger than the one in the weak coupling regime with J ≈ 0.674γ in Fig. 3(b), so that the phonon blockade effect in the strong coupling regime is more robust against than the one in the weak coupling regime.

Figure 3
figure 3

(a) \({g}_{b}^{(2)}(0)\) and (b) nb are plotted as functions of the mechanical driving strength εm/γ with optical driving strength \({\varepsilon }_{p}=10{\varepsilon }_{m}^{2}/\gamma \) for different temperatures: T = (20, 30, 40) mK. (c) \({g}_{b}^{(2)}(0)\) and (d) nb are plotted as functions of the mechanical driving strength εm/γ with optical driving strength \({\varepsilon }_{p}={\varepsilon }_{m}^{2}/\gamma \) for different temperatures: T = (40, 80, 120) mK. Δ ≈ −0.574γ is obtained from equation (4), and J ≈ 0.674γ in (a) and (b), as well as J ≈ 6.74γ in (c) and (d) are obtained from equation (5). The other parameters are θ = 0.6π, κ = 10γ, and ω0 = 2ωm = 2π × 8 GHz.

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It is worth mentioning that with the enhancing of the mechanical driving strength εm as well as the coupling strength J, the optimal parameters for UCPNB fail to fit the optimal conditions for phonon blockade. \({g}_{b}^{(2)}(0)\) is plotted as a function of the coupling strength J/γ with detuning Δ given by equation (4) for different mechanical driving strengths in Fig. 4(a). When the mechanical driving εm/γ is very weak, e.g., εm/γ = 0.08, there is an optimal value of J ≈ 0.64γ, which is agree with the analytical result J ≈ 0.72γ given by equation (5) for UCPNB. According to equation (5), the optimal coupling strength J for UCPNB increases with the mechanical driving strength εm. When the mechanical driving εm/γ = 0.2, \({g}_{b}^{(2)}(0)\) decreases monotonically with the coupling strength J, which is remarkably different from the optimal coupling J ≈ 4.5γ given by equation (5) for UCPNB. Moreover, \({\mathrm{log}}_{10}[{g}_{b}^{(2)}(0)]\) is plotted as a function of the detuning Δ/γ with the coupling strength J given by equation (5) for different mechanical driving strengths in Fig. 4(b). With the increase of the mechanical driving strength, the optimal detuning for phonon blockade is shifted from Δ ≈ −0.52γ to Δ ≈ 0, which is also not in accordance with the prediction of UCPNB given by equation (4) for an invariant optimal detuning Δopt ≈ −0.57γ. In fact, when θ ≠ Nπ/2 (N is an integer), the phonon blockade in the strong coupling condition with optimal detuning Δ ≈ 0 is induced by the strong nonlinearity, i.e., CPNB. Figure 4(a,b) show the crossover from the UCPNB to the CPNB by enhancing the mechanical driving strength εm as well as the coupling strength J.

Figure 4
figure 4

(a) \({g}_{b}^{(2)}(0)\) is plotted as a function of the coupling strength J/γ with detuning Δ given by equation (4) for different mechanical driving strengths: (1) εm/γ = 0.08, (2) εm/γ = 0.1, (3) εm/γ = 0.15, (4) εm/γ = 0.2. (b) \({\mathrm{log}}_{10}[{g}_{b}^{(2)}(0)]\) is plotted as a function of the detuning Δ/γ with the coupling strength J given by equation (5) for different mechanical driving strengths: (I) εm/γ = 0.06, (II) εm/γ = 0.2, (III) εm/γ = 0.4, (IV) εm/γ = 0.6. The other parameters are εp = 0.06γ, θ = 0.6π, κ = 10γ, ω0 = 2ωm = 2π × 8 GHz, and T = 20 mK.

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Lastly, we will shown that, under the strong coupling condition for CPNB, quantum interference effect for UCPNB can also be used to enhanced the CPNB by optimizing the phase difference of the two external driving fields. In Fig. 5(a), \({\mathrm{log}}_{10}[{g}_{b}^{(2)}(0)]\) is plotted as a function of the detuning Δ/γ for different phase difference: θ/π = (0.2, 0.3, 0.5). It is clear that \({g}_{b}^{(2)}(0)\) is dependent on the phase difference θ. The minimal value of \({\mathrm{log}}_{10}[{g}_{b}^{(2)}(0)]\) by scanning the detuning Δ/γ is plotted as a function of the phase difference θ/π in Fig. 5(b). Under the strong coupling condition, the minimal value of \({\mathrm{log}}_{10}[{g}_{b}^{(2)}(0)]\) is obtained with θ = π/2. This can be understated by the fact that, when θ = π/2, the optimal detunings for UCPNB and CPNB are both Δ = 0, and the CPNB is enhanced by the quantum interference effect for UCPNB.

Figure 5
figure 5

(a) \({\mathrm{log}}_{10}[{g}_{b}^{(2)}(0)]\) is plotted as a function of the detuning Δ/γ for different phase difference: θ/π = (0.2, 0.3, 0.5). (b) The minimal value of \({\mathrm{log}}_{10}[{g}_{b}^{(2)}(0)]\) by scanning the detuning Δ/γ is plotted as a function of the phase difference θ/π. The other parameters are J = 5γ, εm = 0.2γ, εp = 0.06γ, θ = 0.6π, κ = 10γ, ω0 = 2ωm = 2π × 8 GHz, and T = 20 mK.

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Discussion

The direct detection of single phonons is still an outstanding challenge in the present experiments. In some recent experiments22,23, the phonon correlation has been measured indirectly by detecting the correlations of the emitted photons from an optical cavity optomechanically coupled to a mechanical mode, which provides us an effective way to investigate phonon statistics experimentally. In addition, indirect phonon detection has also been proposed by the interaction between the mechanical mode and a superconducting microwave resonator5 or NV centers24.

In summary, we have studied phonon blockade in a NAMR which is quadratically coupled to a TLS. We have shown that UCPNB can be observed in the weak coupling regime based on the destructive interference. In order to increase phonon number and improve the robustness against the thermal noise, we gradually enhanced the driving strengths. We have also shown the crossover from the UCPNB to the CPNB by increasing the mechanical driving strength and the coupling strength. In addition, the CPNB can be enhanced by optimizing the phase difference of the two external driving fields for the combination of quantum interference effect and strong nonlinearity of the system.

Methods

Assume that the NAMR has been cooled to its ground state, we shall derive the optimal conditions for UCPNB approximately under the weak driving condition {εp, εm} < min{κ, γ}. The wave function can be expanded on a Fock state basis as

$$|\psi \rangle ={C}_{g0}|g,0\rangle +{C}_{e0}|e,0\rangle +{C}_{g1}|g,1\rangle +{C}_{g2}|g,2\rangle +\cdots ,$$
(9)

where g and e denote the ground and excited states of the TLS, and m represents the Fock state with m phonons in the NAMR, and the coefficient |Cgm|2 (|Cem|2) is the occupying probability corresponding to the state |g, m〉 (|e, m〉). Under the weak driving condition, i.e. {εp, εm} < min{κ, γ}, we will have |Cg0| ≫ {|Ce0|, |Cg1|, |Cg2|} ≫ {|Ce1|, |Cg3|} ≫ …, so the wave function can be truncated to the two-phonon states approximately.

Substituting the wave function in equation (9) and the Hamiltonian in equation (1) into the Schrödinger’s equation id|ψ〉/dt = H|ψ〉, then the dynamical equations for the coefficients Cgm and Cem are shown as

$$\frac{d}{dt}{C}_{e0}=-\,(\frac{\kappa }{2}+i2{\rm{\Delta }}){C}_{e0}-i{\varepsilon }_{p}{C}_{g0}-i\sqrt{2}J{C}_{g2},$$
(10)
$$\frac{d}{dt}{C}_{g1}=-\,(\frac{\gamma }{2}+i{\rm{\Delta }}){C}_{g1}-i{\varepsilon }_{m}{e}^{-i\theta }{C}_{g0}-i\sqrt{2}{\varepsilon }_{m}{e}^{i\theta }{C}_{g2},$$
(11)
$$\frac{d}{dt}{C}_{g2}=-\,(\gamma +i2{\rm{\Delta }}){C}_{g2}-i\sqrt{2}{\varepsilon }_{m}{e}^{-i\theta }{C}_{g1}-i\sqrt{2}J{C}_{e0}.$$
(12)

In the steady state, i.e. dCgm/dt = dCem/dt = 0, and under the condition for phonon blockade, i.e. Cg2 ≈ 0, we obtain the linear equations for the coefficients Ce0, Cg1 and Cg0 as

$$0=-\,(\frac{\kappa }{2}+i2{\rm{\Delta }}){C}_{e0}-i{\varepsilon }_{p}{C}_{g0},$$
(13)
$$0=-\,(\frac{\gamma }{2}+i{\rm{\Delta }}){C}_{g1}-i{\varepsilon }_{m}{e}^{-i\theta }{C}_{g0},$$
(14)
$$0=-\,i\sqrt{2}{\varepsilon }_{m}{e}^{-i\theta }{C}_{g1}-i\sqrt{2}J{C}_{e0}.$$
(15)

From equations (13 and 14), Ce0 and Cg1 are given by

$${C}_{e0}=\frac{-i2{\varepsilon }_{p}}{\kappa +i4{\rm{\Delta }}}{C}_{g0},$$
(16)
$${C}_{g1}=\frac{-i2{\varepsilon }_{m}{e}^{-i\theta }}{\gamma +i2{\rm{\Delta }}}{C}_{g0}.$$
(17)

Substituting Ce0 and Cg1 into equation (15), we obtain

$$0=(\frac{{\varepsilon }_{m}^{2}{e}^{-i2\theta }}{\gamma +i2{\rm{\Delta }}}+\frac{J{\varepsilon }_{p}}{\kappa +i4{\rm{\Delta }}}){C}_{g0}.$$
(18)

As |Cg0| ≈ 1 ≠ 0, then we get the conditions for the optimal parameters Jopt and Δopt as

$${\varepsilon }_{m}^{2}(\kappa \,\cos \,2\theta +4{\rm{\Delta }}\,\sin \,2\theta )+J{\varepsilon }_{p}\gamma =0,$$
(19)
$${\varepsilon }_{m}^{2}(4{\rm{\Delta }}\,\cos \,2\theta -\kappa \,\sin \,2\theta )+2J{\varepsilon }_{p}{\rm{\Delta }}=0.$$
(20)

The optimal parameters for UCPNB given in equations (4–8) are obtained by solving the equations (19 and 20).