Tunable phonon blockade in weakly nonlinear coupled mechanical resonators via Coulomb interaction

Realizing quantum mechanical behavior in micro- and nanomechanical resonators has attracted continuous research effort. One of the ways for observing quantum nature of mechanical objects is via the mechanism of phonon blockade. Here, we show that phonon blockade could be achieved in a system of two weakly nonlinear mechanical resonators coupled by a Coulomb interaction. The optimal blockade arises as a result of the destructive quantum interference between paths leading to two-phonon excitation. It is observed that, in comparison to a single drive applied on one mechanical resonator, driving both the resonators can be beneficial in many aspects; such as, in terms of the temperature sensitivity of phonon blockade and also with regard to the tunability, by controlling the amplitude and the phase of the second drive externally. We also show that via a radiation pressure induced coupling in an optomechanical cavity, phonon correlations can be measured indirectly in terms of photon correlations of the cavity mode.

where, H om describes the standard linearized optomechanical interaction 22 as given below, with effective optomechanical coupling, G, and detuning, Δ a , in a frame rotating at the drive frequency ω l : a om 1 1 The Hamiltonian for the mechanical resonators is given by p q q . Here, b 1 ( † b 1 ) and b 2 ( † b 2 ) are the annihilation (creation) operators for the two mechanical resonator modes with damping rate γ. Hereafter, we will call the mode 'b 1 ' as the primary mode and the mode 'b 2 ' as the secondary mode. Here, H free is the free Hamiltonian of the two mechanical resonators, H nl is the Hamiltonian describing the Kerr nonlinearity, U, in both the mechanical resonators and H co represents the Coulomb interaction Hamiltonian of the two charged mechanical oscillators. The primary and the secondary mechanical modes are driven by pumps with frequencies ω p and ω q respectively with the corresponding pump amplitudes Ω 1 and Ω 2 and an initial phase difference φ; which is described by the term H drive . Hereafter, we will assume that ω q = ω p . In passing, it may be noted that a somewhat similar model has been discussed by some authors in the context of photon blockade in a system of coupled optical cavities 40,41 . Also, as regards the physical realization of a Kerr nonlinearity is concerned, one may obtain it by introducing an ancilla two-level system in the resonator assuming far off-resonance interactions 20 . It is worthwhile to note that the first analysis of phonon blockade in a system described by the effective Hamiltonians, given in eqs (1)(2)(3), was also given by Miranowicz et al. 20 .
In the Coulomb interaction Hamiltonian H co , k e denotes the electrostatic constant, d is the equilibrium separation of the two charged oscillators in absence of any interaction between them, and x 1 and x 2 are the small oscillations of the two mechanical oscillators from their equilibrium positions. Now, assuming that the deviations are small compared to the equilibrium separation, i.e.  x x d { , } 1 2 , one can expand Here, the first term is a constant term and the second one is a linear term which can be absorbed into the definition of the equilibrium positions. The last term consists of two parts: one part refers to the small frequency shift of the original frequencies and can be neglected by renormalising the mechanical frequencies, and the other part is the coupling term between the oscillators. Therefore, we obtain the Coulomb interaction between the mechanical oscillators as [37][38][39] The charge contained in the electrodes are given by q 1 = C 1 V 1 , and q 2 = −C 2 V 2 , where C j is the capacitance of the bias gate on the resonator M j . Therefore, H co can be obtained as In the weak-coupling regime, considering only the resonant terms, the Coulomb interaction Hamiltonian reduces to In the following, we will study the occurrence of phonon blockade in the primary resonator by analyzing the phonon statistics by means of the zero-time delay second-order correlation function given by, Phonon blockade with a single drive. First, we will consider the case when there is no optomechanical interaction. The master equation describing the evolution of the system is given by: is the Liouvillian operator for the mode b i and n th,i = 1/ [exp(ħω m /k B T) − 1] denotes the thermal phonon number in that mode at environmental temperature T. We will consider n th,1 = n th,2 = n th for the rest of the paper. The Hamiltonian describing the mechanical resonators in a rotating frame with the mechanical drive frequency is given by where, Δ = ω m − ω p is the detuning from the mechanical pump frequency. We will calculate g (0) b (2) numerically by solving Eq. (4) in the weak-driving limit i.e. for γ (2) 1 1 1 1 ss 1 1 ss 2 , where ρ ss is the steady-state density matrix. Before solving the master equation numerically, in order to obtain the optimal parameters for unconventional phonon blockade, we develop an analytical model in the following. Firstly, we consider the case when the secondary mechanical resonator is not driven, i.e. Ω 2 = 0. At low temperature, and assuming a weak pumping condition, the low-energy levels dictated by the Hamiltonian is shown in Fig. 2(a). The counterintuitive phonon antibunching, that occurs owing to the quantum interference effect, could be understood from the sketch. There are two possible paths to reach the state |20〉:(a) the direct excitation from (solid arrow) and (b) tunnel-coupling-mediated transition (dotted arrows). The destructive interference between these two paths, under appropriate conditions, results in the phenomenon of unconventional phonon blockade. Assuming that the system is initially prepared in the |00〉 state, we consider the following ansatz:  The coefficients C ij 's can be obtained by solving the Schrödinger equation 2 2 2 is the non-Hermitian Hamiltonian that includes the damping of the mechanical oscillators. Following an iterative method prescribed by Bamba et al. in connection with photon blockade in coupled photonic molecules 30 , in the limit of weak Ω 1 , at steady-state, the optimal parameters are obtained as follows: The limit for the coupling, J, in this case is that the value of J must be larger than γ/ 2. In Fig. 2(b), we show the variation of the zero time-delay second-order correlation function g (0) b (2) by solving the master equation, i.e. Eq. (4), in a truncated Fock space. Here, is plotted as functions of the normalized coupling strength J/γ and nonlinearity U/γ for U ≤ γ, with optimal values of Δ as derived in Eq. (7). The black dashed curve shows the optimal values of U calculated in Eq. (7). It is observed that for the optimal conditions, phonon blockade can be obtained in the weakly nonlinear regime.
To demonstrate these results more clearly, in Fig. 3 is depicted as a function of Δ/γ for different values of J/γ. The value of U is considered to be U opt . For J/γ = 0.8, 0.95, and 1.5, the optimal values of Δ/γ found from the analytical calculations are ≈0.11, 0.16, and 0.24 respectively. The corresponding optimal values of U/γ are 0.98, 0.52, and 0.18 respectively. From the plots, it is evident that the numerically calculated results show complete agreement with the optimal values of the parameters calculated from the approximate analytical model. With weak coupling strengths of J/γ = 0.8 and 0.95, ≈ .
, while for a moderate value of J/γ = 1.5, is on the order of 0.01. We also demonstrate the second-order correlation function, , as a function of the normalized time delay τ/ (2π/J) in Fig. 3(b). Considering optimal parameters, when J/γ = 0.8 and 0.95, ≈ .
at τ = 0, and for increasing delay times . Similarly, for J/γ = 1.5, ≈ . g (0) 0 01 b (2) at τ = 0, and for higher delay times and finally reaches the value 1. Therefore, the plots demonstrate that the phonons are antibunched and have sub-Poissonian distribution. Now, in order to see the influence of environmental phonon population on the phonon blockade characteristics, in Fig. 3(c), we show the variation of g (0) b (2) as a function of the bath phonon number, n th . For J/γ = 0.8, g (0) b (2) reaches 1 at n th ≈ 0.001, whereas, for J/γ = 0.95 and 1.5, upto n th = 4.5 × 10 −4 and n th = 1.5 × 10 −4 respectively. Therefore, it is evident that the environmental thermal population has undesirable effect on the observation of phonon blockade.
Phonon blockade with two drives. We now turn to study the phonon correlations by applying an additional drive Ω 2 on the secondary mechanical resonator. The transition paths leading to two-phonon excitation, are shown in Fig. 4. It is clear from the sketch that with the introduction of the second drive, there results in many more quantum pathways compared to the one with the single drive case, as was depicted in Fig. 2. This may be a possible reason for the endurance of phonon blockade effect upto higher number of thermal phonons in the presence of two drives, as discussed later. Analytical calculations of optimal conditions in this case gives rise to a quadratic equation in ζe −iφ :  From Eq. (9), it can be seen that for specific values of the parameters, U, J and Δ, the optimal values of ζ and φ could be obtained, and there are two optimal values of ζ and φ for a specific set of system parameters. Therefore, by applying the additional pump we can choose the optimal values of the amplitude and the phase of the second drive for different coupling strengths and detuning in the system. Figure 5(a) depicts g (0) b (2) as functions of the rescaled detuning Δ/γ and U/γ corresponding to ζ + , φ + for a weak coupling value of J = 0.5γ. In Fig. 5(b), we show the corresponding average phonon number in the primary resonator. From these plots, it is observed that for the parameter regime where g (0) b (2) is found to be on the order of 0.01, average phonon number on the order of 0.01 could be obtained. We show the variation of g (0) b (2) as a function of Δ/γ, for different values of J in Fig. 5(c), with U opt /γ = 0.5 and Δ opt /γ = 0.5, and J/γ = 0.5, 0.85 and 1. It is observed that phonon blockade could be obtained at Δ = 0.5γ, which is in agreement with Δ opt , as predicted by the analytical calculations. Figure 5 , with U opt = 0.5γ, and Δ opt = −0.5γ, and it is observed that phonon blockade could be obtained at Δ = −0.5γ.
Similarly, in Fig. 6(a,b), we show g (0) b (2) and the average phonon number in the primary resonator, as functions of the rescaled detuning Δ/γ and U/γ corresponding to ζ − , φ − for J = 0.5γ. Here also, g (0) b (2) on the order of 0.01 is obtained with average phonon number ≈0.01. In Fig. 6(c), we discuss the variation of g (0) b (2) with respect to Δ/γ for U opt = 0.5γ and Δ opt = 0.15γ and different values of J/γ = 0.5, 0.85 and 1. It is observed that phonon blockade can be obtained at Δ opt = 0.15γ. Figure 6(d) shows the variation of g (0) b (2) for U opt = 0.5γ and Δ opt = −0.15γ. In this case, phonon blockade is obtained at Δ = −0.15γ.
Next, we discuss the variation of the second-order correlation function with finite time-delay, τ g ( ) b (2) . In Fig. 7(a,c), we show τ g ( ) b (2) as a function of the normalized time delay τ/(2π/J) with different values of J for ζ + , φ + and ζ − , φ − respectively. It shows that the value of g (0) b (2) is the lowest at τ = 0 and for increasing delay times , which demonstrates that the phonons are antibunched and sub-Poissonian in nature. In Fig. 7(b,d), we discuss the effect of environmental phonon number on g (0) b (2) for different values of U/γ, with (ζ + , φ + , Δ opt /γ = 0.5, J/γ = 0.5) and (ζ − , φ − , Δ opt /γ = 0.15, J/γ = 0.5) respectively. As observed in Fig. 7(b), for optimum values of ζ + , φ + , the phonon blockade effect can be sustained upto n th ≈ 0.01 for U = 0.9γ whereas for U = 0.1γ and 0.5γ, for values of n th upto ≈0.001 and 0.006 respectively. On the other hand, for optimum values of ζ − , φ − , as shown in Fig. 7(d), the phonon blockade effect can be sustained upto n th ≈ 0.02 for U = 0.9γ. For U = 0.1γ and 0.5γ, for n th upto ≈0.001 and 0.01 respectively.

Measurement of phonon blockade via photon correlations. Now, we study the phonon statistics in
presence of the optomechanical interaction. Here, we will show that phonon blockade in the primary mechanical resonator can be detected by studying photon statistics of the optical mode in the cavity. Considering the cavity to be at the red sideband, the Langevin equation for the cavity mode fluctuation is given by 2 so that the phonon correlation can be studied by evaluating the second-order correlation function for photons. We calculate g (0) b (2) and also the zero time-delay second-order correlation function for photon, g (0) a (2) , by solving the following master equation: where the total Hamiltonian of the system, in a frame rotating at the mechanical pump frequency ω p is given by In Fig. 8(a), we discuss the phonon correlations for only one drive applied on the primary resonator. We consider G = 0.1κ, that lies in the weak coupling regime and κ = 10γ for typical optomechanical systems. The black solid line shows g (0) b (2) in absence of the optomechanical coupling and the red dashed line shows the one in presence of the optomechanical coupling. It is observed that both the values agree well with each other in this parameter regime. Therefore, there is not any modification in the phonon blockade characteristics due to the additional coupling term induced by the optomechanical interaction in the adiabatic regime. In Fig. 8(b), we compare the phonon and photon correlations calculated by solving the master equation for the total Hamiltonian in presence of the optomechanical coupling. We observe that both the correlation functions show evidence of blockade at the same detuning value. Therefore, the photon blockade characteristics for the cavity mode can serve as an evidence of phonon blockade in the primary mechanical resonator. Further, in Fig. 8(c,d), we show g (0) b (2) for additional driving of the secondary mode i.e for Ω 2 ≠ 0, which also show similar features as the single-driving case. .

Conclusion
In conclusion, we have proposed schemes for the realization of phonon blockade in a weakly nonlinear mechanical end-mirror in an optomechanical cavity, coupled by Coulomb interaction to another weakly nonlinear mechanical resonator. Phonon correlations are characterised in terms of the second-order correlation function. Firstly, we studied the phonon blockade characteristics without considering the optomechanical interaction. By applying a single drive on the primary mechanical resonator, strong phonon blockade could be obtained with optimum values of the mechanical drive detuning and Kerr-nonlinearity. However, the phonon blockade effect is very fragile towards environmental thermal phonon number. Next, we discussed the scenario where both the mechanical resonators were driven simultaneously. In this case, the optimum values could be obtained in terms of the amplitude and the phase of the second mechanical drive, which allows more controllability of phonon blockade. Also, the phonon blockade effect could be sustained upto higher number of thermal phonons. Finally, we discussed the blockade characteristics to be observed when the optomechanical interaction was switched on. It was demonstrated that when the cavity optical field follows the resonator dynamics adiabatically, for both the single and the double mechanical drives, phonon blockade could be detected in terms of the photon correlations of the cavity mode.

Methods
The optimal conditions for phonon blockade can be determined by solving the equations for the coefficients obtained from Schrödinger equation:    In the limit of weak Ω 1 and Ω 2 , the probability of phonon excitation to higher levels becomes subsequently lower i.e. }   00  10  01  20  11 02 . The optimal condition for the complete phonon blockade in the primary resonator corresponds to the case when the probability of a phonon in state |20〉 equals zero. Under these assumptions, solving Eqs (14) and (15), the values of C 10 and C 01 at the steady-state are obtained as Now, sustituting Eqs (19) and (20) into Eqs (16)(17)(18), we obtain the following matrix equation where, the matrix elements are given by , and (b) effect of environmental temperature on g (0) b (2) for ζ + , φ + . (c) Second-order correlation function with finite time-delay τ g ( ) b (2) , and (d) effect of environmental temperature on g (0) b (2) for ζ − , φ − . Other parameters are considered to be same as in Fig. 6(c,d). The dashed black lines correspond to  b (2) in absence of the optomechanical coupling (black solid line) and in presence of optomechanical coupling (red dashed line) for additional driving of the secondary mode. (d) The phonon and photon correlations with two drives in presence of the optomechanical coupling. Other parameters are κ = 10γ, G = 0.1κ. The dashed black lines correspond to g (2) (0) = 1.