Phonon antibunching effect in coupled nonlinear micro/nanomechanical resonator at finite temperature

Figure 2 shows the numerical result of the equal-time second-order correlation function as a function of the nonlinearity U and mechanical coupling J at zero temperature. We observe that the phonon antibunching effect occurs in the parameter regime where the product JU is relatively small. Importantly, as opposed to the conventional phonon blockade effect, in this coupled-mechanical oscillator system, phonons exhibit antibunching even when the nonlinearity is negligible $(U\ll \gamma)$, although this is at the cost of a large J. In addition, in the regime of large U and J, g⁽²⁾(0) becomes extremely large, which means that phonons are superbunched.

To find the optimal condition for phonon antibunching, we analytically solve the steady-state master equation for the density matrix. Under the weak driving condition $F\ll \gamma$, the average mechanical excitations would be much lower than 1, and the Hilbert space of the total system can be truncated to $m+n=2$. In this case, we denote $| 0,0\rangle \rightarrow|1\rangle$, $| 0,1\rangle \rightarrow|2\rangle$, $| 0,2\rangle \rightarrow|3\rangle$, $| 1,0\rangle \rightarrow|4\rangle$, $| 1,1\rangle \rightarrow|5\rangle$, and $| 2,0\rangle \rightarrow|6\rangle$. Hence, the density matrix operator $\hat{\rho}$ can be written as $\hat{\rho}=\sum_{m,n=1}^{6}\rho_{mn}| m\rangle \langle n|$. In addition, weak mechanical excitations would result in $\rho _{00}\simeq 1\gg \rho_{01},\rho _{10}\gg \rho _{02},\rho _{20},\rho _{11}$. Then, the equal-time second-order correlation function can be approximately expressed as

$$\begin{equation} g^{( 2) }( 0) \simeq \frac{2\rho _{66}}{\rho _{44}^{2}}. \end{equation} \tag{ 5 }$$

Obviously, if $\rho _{66}=0$, then $g^{( 2) }( 0) =0$, which implies a low probability of having two phonons in the first mechanical mode. After performing tedious calculations using the perturbation theory, we obtained the expression for the matrix element $\rho _{66}$ in the zero-temperature case $(n_{th}=0)$,

$$\begin{equation} \rho _{66}=\frac{F^{4}| U( J^{2}+2\tilde{\Delta}^{2}) +2 \tilde{\Delta}^{3}| ^{2}}{2J^{8}| U+2\tilde{\Delta} | ^{2}}, \end{equation} \tag{ 6 }$$

where $\tilde{\Delta}=\Delta -i\gamma /2$. In the calculation, we further assume a large mechanical coupling J; otherwise, the expression for matrix elements would be more complicated. Then, we can find the following condition for perfect antibunching:

$$\begin{eqnarray} 2UJ^{2}+4( \Delta +U) \Delta ^{2}-( 3\Delta +U) \gamma ^{2} &= 0, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} 8U\Delta +12\Delta ^{2}- \gamma ^{2} &= 0 \end{eqnarray} \tag{ 8 }$$

which is mathematically the same as the results in the photon blockade case [26]. We plot the optimal condition in fig. 2 (white dashed curve), which is in good agreement with the numerical result.

In the situation where the mechanical coupling J is much greater than the mechanical damping γ, the optimal parameters required to achieve $\rho_{66}=0$ can be approximated as

$$\begin{eqnarray} \Delta _{optimal} &\simeq \frac{\gamma }{2\sqrt{3}}, \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} U_{optimal} &\simeq \frac{2\gamma ^{3}}{3\sqrt{3}J^{2}}. \end{eqnarray} \tag{ 10 }$$

Obviously, in this case, the nonlinearity that is required to observe that an optimal blockade can be made extremely small ($U_{optimal}/\gamma =2\gamma^{2}/ ( 3\sqrt{3}J^{2}) \cdot \ll 1$), contrary to the condition of conventional phonon blockade $(U/ \gamma \gg 1)$.

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We then study the phonon number probability distribution $P_{m}=\sum_{n=0}^{\infty }\rho _{mn,mn}$ of mechanical mode 1 when the phonon blockade occurs. For comparison, we also show the probability distribution of a coherent state with the same mean phonon number $| \alpha | ^{2}=\sum_{m=0}^{\infty }P( m)$, which obeys the Poisson distribution $P( n) =| \alpha | ^{2n}e^{-| \alpha | ^{2}}/n!$. As shown in fig. 3, the largest probability is occupied by the vacuum state. The one-phonon state is the next most probable. The two-phonon state is significantly less probable than the single-phonon state. Compared to the coherent state (the same mean phonon number with the blockade case), the probability for the two-phonon state is small, which indicates that the state of mechanical mode 1 has a nonclassical distribution. Recently, similar results have also been observed in optical counterparts [22].

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In the previous subsection, we studied the phonon blockade effect without including the effect of environmental temperature. However, cooling a mechanical resonator down to zero temperature is not easily achieved experimentally. Therefore, it is important to study the phonon blockade at finite temperature. First, in fig. 4, we numerically plot g⁽²⁾(0) as a function of J and U at $T=0.04T_{0}$. It can be seen that the phonon antibunching effect still exists in the weak nonlinearity U, but small mechanical coupling J regime (enclosed by the white solid line in fig. 4). It is also observed that there is strong phonon antibunching in a specific regime, which is similar to that in the zero-temperature case. However, in the regime of weak nonlinearity U and large J, g⁽²⁾(0) is greater than 1, which is contrary to the zero-temperature case (see fig. 2). This behavior indicates that thermal noise destroys the phonon antibunching effect in the regime of weak nonlinearity U and large J.

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To study the effect of the environmental temperature on the phonon blockade, we present the analytical expressions of the density matrix elements $\rho _{66}$ and $\rho _{44}$ in the strong coupling regime (using the same procedure as in the zero-temperature case):

$$\begin{eqnarray} \rho _{44} &= \frac{F^{2}| \tilde{\Delta}| ^{2}}{ | J^{2}-\tilde{\Delta}^{2}| ^{2}}+n_{th}, \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} \rho _{66} &= \frac{F^{4}| U( J^{2}+2\tilde{\Delta}^{2}) +2\tilde{\Delta}^{3}| ^{2}}{2J^{8}| U+2\tilde{\Delta} | ^{2}}\nonumber\\ &\quad +\,\frac{2F^{2}| \tilde{\Delta}| ^{2}}{ J^{4}}n_{th}+n_{th}^{2}. \end{eqnarray} \tag{ 12 }$$

From eqs. (11), (12), both $\rho _{44}$ and $\rho _{66}$ contain two parts: one part determined by the temperature or the thermal phonon number, and the other part from the system itself (quantum interference). These two parts incoherently contribute to the phonon correlation. From eq. (12), it can be noted that $\rho _{66}>0$, which implies that the perfect antibunching condition no longer exists at finite temperature. However, from eq. (12) we can see that for a fixed temperature T, when the first term vanishes, the equal-time second-order correlation will take the minimal value. Therefore, the optimal condition for the strong phonon antibunching at finite temperature is the same as that at zero temperature. When the environmental temperature is small, the region for 0 < g⁽²⁾(0) < 1 still exists because the quantum interference partially contributing to phonon correlation leads to an imperfect antibunching effect. When the mechanical coupling J is small, the contribution from quantum interference is enhanced, leading to a stronger phonon antibunching, and vice versa. These conclusions are well confirmed by the numerical results in fig. 4.

Figure 5 shows g⁽²⁾(0) as a function of the environmental temperature. Based on different behaviors of the second-order correlation function, the whole regime can be divided into three parts by two boundary temperatures 0.028T₀ and 0.046T₀. In the first region (region I), g²(0) is a constant and is much smaller than 1, which means that the phononic mode has a strong antibunching effect. Therefore, we called it the quantum regime. Interestingly, the quantum correlation is hardly affected by the thermal noise in the quantum regime, which is very helpful to experimentally observe the strong phonon antibunching at low finite temperature. In the second region (region II), g⁽²⁾(0) experiences a sharp increase from $3\times 10^{-6}$ to 2 with increasing temperature. Obviously, in this regime, the quantum correlation of the phononic mode undergoes a transition from strong antibunching to bunching. We call it the crossover regime. In the third region (region III), g⁽²⁾(0) approaches another constant 2, which means that the phononic mode has the statistic property of a thermal field. This regime is called the thermal regime.

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These phenomena described in last paragraph can be explained from eqs. (5) and (12) as follows. In the quantum regime, the first term is much larger than other terms in eq. (12), i.e., quantum interference dominates, and, therefore, g²(0) mainly shows the pure quantum correlation. In the thermal regime, the third term is much larger than other terms in eq. (12), i.e., pure thermal noise dominates, which results in $g^{(2)}(0) \approx 2$. In the crossover regime, the gradually enhanced thermal noise gradually suppresses and eventually destroys the quantum correlation; hence, g²(0) monotonously increases from an almost vanishing value to 2. Two critical temperatures can be determined as follows: when the first term is equal to the second term and much greater than the third term in eq. (12), the corresponding temperature is the first boundary value. Then, we find $T_{1}=T_{0}/\ln (1+4J^{4}|U\tilde{\Delta}+2\tilde{\Delta} ^{2}| ^{2}/F^{2}|U(J^{2}+2\tilde{\Delta} ^{2}) +2\tilde{\Delta}^{3}| ^{2})$; when the third term is equal to the second term and much greater than the first term in eq. (12), we find another boundary temperature $T_{2}=T_{0}/\ln (1+J^{4}/2|F\tilde{\Delta}| ^{2})$.

We also studied the second-order correlation function vs. the time delay $g^{(2)}(\tau )$ at three different environment temperatures, as shown in fig. 6. Owing to the probability oscillation between the phonon state $| 0,1\rangle$ and $| 1,0\rangle$, the second-order correlation functions oscillate with period $2\pi /J$. In the zero-temperature case (fig. 6(a)), the value of g⁽²⁾(0) becomes negligible. For $T=0.043T_{0}$ (fig. 6(b)), the amplitude of the oscillation of $g^{(2) }(\tau)$ decreases; meanwhile, the value of g⁽²⁾(0) increases to unitary. When the temperature is further increased to $T=0.1T_{0}$ (fig. 6(c)), the value of g⁽²⁾(0) is $\simeq 2$, indicating that the phonon field in the mechanical resonator 1 is a thermal field. Again, the time evolution of the second-order correlation function shows that a large thermal noise will suppress and even destroy the phonon antibunching. From fig. 6, one can see that strong phonon antibunching is demonstrated in a range about $\pi /J$ around the zero delay, which gives a limited requirement for the temporal resolution of a detector.

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We then investigate the influence of the driving force on the phonon statistics. In fig. 7, we plot g⁽²⁾(0) of the phononic mode in mechanical resonator 1 as a function of the driving force at zero and finite temperatures. First, we focus on the zero-temperature case. From eqs. (5), (11), and (12) we can obtain the equal-time second-order correlation function at zero temperature:

$$\begin{equation} g^{( 2) }( 0) =\left| \frac{( U( J^{2}+2 \tilde{\Delta}^{2}) +2\tilde{\Delta}^{3}) | J^{2}- \tilde{\Delta}^{2}| ^{2}}{( U+2\tilde{\Delta}) | \tilde{\Delta}| ^{2}J^{4}}\right| ^{2}. \end{equation} \tag{ 13 }$$

It should be noted that in the weak driving regime, g⁽²⁾(0) is independent of F, and is consistent with the numerical result shown in fig. 7. However, the analytical result clearly deviates from the numerical result in the strong driving regime. It is shown that the analytical result is valid only in the weak driving regime. The reason is that a strong driving enhances the population in the high-phonon-number states, and, therefore, decreases the antibunching effect.

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Next, we discuss the influence of the driving amplitude F on g⁽²⁾(0) in the finite-temperature case. As opposed to the zero-temperature case, it can be seen that all second-order correlation curves at finite temperature first decrease and then increase as the driving force is enhanced. Indeed, a similar phonon correlation behavior was numerically observed without physical interpretation in ref. [13]. In the weak driving limit, the analytical model well reproduces the numerical calculation. However, because the analytical model is based on the assumption that the total phonon number is small $(m+n=2)$, it fails to reproduce the increasing g⁽²⁾(0) at higher driving. These observations can be understood from eqs. (5), (11), and (12). When F is very small, the thermal noise term dominates the matrix elements $\rho _{44}$ and $\rho _{66}$, and hence, the phonon state is approximately a thermal state ($g^{(2)}(0)\approx2$). With the increase in driving strength, the contributions from the quantum-interference terms in eqs. (11) and (12) become important, leading to a decrease in g⁽²⁾(0). When the driving is sufficiently strong, the three- and higher-phonon states in resonator 1 are populated with increased probability owing to the strong coupling between resonator 1 and resonator 2. Hence, the second-order correlation function will increase again, resulting in a degradation of phonon antibunching. From the above discussions, it is clear that a very weak force is required to achieve a stronger antibunching effect for the zero-temperature case, while a suitably strong driving strength is helpful to preserve the stronger phonon antibunching effect when thermal noise exists.

Now, we discuss the experimental feasibility to observe the phonon blockade effect in the presence of unavoidable thermal noise. We assume that the two resonators in our model are silicon-based MEMS/NEMS resonators, which are doubly clamped rectangular cross-section beams with width $d=5\ \text{nm}$ and length $L=100\ \text{nm}$. According to refs. [17,32], we find that the fundamental frequency is $\omega _{0}/2\pi =d/L^{2}\sqrt{E/\rho }=4.3\ \text{GHz}$ and the nonlinear strength $U=\hbar (1.6\rho d^{4}L) ^{-1}=430\ \text{Hz}$. The decay rate of the resonator is dependent on many factors, such as the geometrical configuration, anchor loss, environmental pressure and temperature, input-output loss, and thermoelastic damping. With a suitable design and fabrication, the total decay can be controlled. For simplicity, we take a typical value of $\gamma =10\ \text{MHz}$, which corresponds to a quality factor $Q\sim 4300$. If we assume that the coupling between two mechanical resonators comes from an electrostatic force, the coupling strength J can be freely adjusted by changing the externally applied voltage. With these feasible parameters, we can then estimate the characteristic temperature $T_{0}=\hbar \omega _{0}/K_{B}\sim 196\ \text{mK}$. According to the previous discussion, for an environmental temperature below 5.3 mK, the phonon blockade is almost unaffected by the thermal noise. Such a temperature is attainable using current laboratory technology. To measure the time-delayed second-order correlation function $g^{(2)}(\tau )$, we can adapt the same method as in ref. [36]. In this reference [36], the phonon correlation can be transferred to the photon correlation, which can be measured by a typical photonic Hanbury Brown and Twiss (HBT) set-up. Here, we take coupling strength $J=20\gamma \sim 200\ \text{MHz}$, corresponding to a window width of phonon antibunching $\pi /J\sim 10\ \text{ns}$. Apparently, this time scale is much greater than a typical temporal resolution of the detector (usually $<100\ \text{ps}$) [37].