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Abstract
We present a proposal to enhance the cross-Kerr coupling between the cavity and the mechanical oscillator significantly. Specifically, the periodic modulation of the mechanical spring constant induces strong mechanical parametric amplification, which leads to the cross-Kerr nonlinear enhancement. Also, we discuss its application in photon-phonon blockade and phonon-number measurement. We find that under the strong cross-Kerr coupling condition, not only the photon-phonon blockade effect is dramatically enhanced but also different phonon number is clearly distinguished. Our results offer an alternative approach to perform quantum manipulation between photon and phonon.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optomechanical systems, in which light is coupled to mechanical oscillator via the electromagnetic radiation pressure, have become a research focus in fundamental and quantum physics [1–4]. Due to the very particular light-matter interaction, the optomechanical cavity provides an excellent platform for generating squeezed light [5–7], observing optomechanically induced transparency [8–10] and normal-mode splitting [11,12], etc. Note that these achievements can be realized under the condition of strong linearized optomechanical coupling. However, the intrinsic nonlinearity of the optomechanical coupling becomes negligible.
To enhance the nonlinear coupling, Heikkilä et al. have proposed theoretically and observed experimentally the cross-Kerr effect in the Josephson-junction setup, which can boost the underlying nonlinear optomechanical coupling by several orders of magnitude [13,14]. In this case the cross-Kerr interaction gives rise to the change in the cavity refractive index that depends on the phonon number in the oscillator. Subsequently, many theoretical works focus on the cross-Kerr type of coupling between the cavity and the mechanical oscillator. For example, the cross-Kerr coupling leads to the frequency shift and the optimal cooling or heating [15], strengthens the steady-state optomechanical entanglement [16], and also affects the optical bistable behavior [17]. Furthermore, Ref. [18] showed that it can turn the optical bistability into the tristable behavior. Very recently, in Refs. [19,20] the authors proposed to enhance the cross-Kerr coupling by applying strong mechanical driving or optical parametric amplification.
Although the cross-Kerr coupling and mechanical parametric amplification [21–26] have been extensively investigated separately, there are few studies which employ mechanical parametric amplification to enhance the cross-Kerr coupling. In this paper, we consider the case in which the optical mode couples to the mechanical mode through the optomechanical and cross-Kerr interactions, and analyze the effect of the cross-Kerr interaction on the photon-phonon blockade and the phonon-number detection. We find that the cross-Kerr coupling strength can be significantly enhanced by the strong mechanical parametric amplification, which is physically different from the previous proposals [19,20]. Further, we show that the enhanced cross-Kerr coupling can induce the strong photon-phonon blockade phenomenon and distinguish different phonon number. This work would be helpful for exploring quantum manipulation [27], quantum computing [28], and optical isolation [29].
2. Model and Hamiltonian
We consider a generalized optomechanical system depicted in Fig. 1 with the Hamiltonian (setting $\hbar =1$)
Fig. 1. Schematic picture of the generalized optomechanical system, where a cavity mode is coupled to a mechanical mode through both optomechanical radiation-pressure and cross-Kerr interactions with coupling strengths $g_{0}$ and $g_{ck}$. In addition, the parametric driving of the mechanical oscillator is obtained by the periodic modulation of the spring constant $k(t)$. Two weak probe fields with amplitude $\epsilon _{pi}$ and frequency $\omega _{pi}$ $(i=1,2)$ are applied to the cavity and the mechanical oscillator, respectively.
Specifically, a parametric amplification in the mechanical oscillator is used to introduce a preferred squeezed mechanical mode $b_{s}^{\dagger }(b_{s})$, which satisfies a squeezing transformation $b=\cosh (r)b_{s}-e^{-i\Phi _{d}}\sinh (r)b_{s}^{\dagger }$, with a squeezing parameter $r=(1/4)\ln [(\Delta _{m}+\lambda )/(\Delta _{m}-\lambda )]$ [24–26]. Then, in terms of $b_{s}^{\dagger }(b_{s})$, the total Hamiltonian in Eq. (1) becomes
Fig. 2. The values (a) $g/g_{0}$ and (b) $g_{s}/g_{0}$ versus the squeezing parameter $r$ for $g_{ck}\approx 0.25g_{0}$.
Moreover, the parametric interaction can be also suppressed by choosing the proper parameters $\Delta _{m}$ and $\lambda$, so that the condition $\omega _{s}\gg g_{p}$ is met, as shown in Fig. 3. In this case, the fifth term of Eq. (2) becomes the oscillating term with high frequencies $\pm 2\omega _{s}$, which can be safely neglected under the rotating-wave approximation (RWA). Consequently, the above Hamiltonian can be reduced to
The validity of the approximate Hamiltonian is numerically checked in the following discussions. Note that in Fig. 3, a large squeezing parameter $r$ could be obtained and the cross-Kerr coupling strength $g_{s}$ could be significantly enhanced when $\Delta _{m}$ infinitely approaches $\lambda$, which leads the realization of the strong-coupling regime, i.e., $g_{s}>\kappa$ (the cavity decay rate $\kappa$). In our calculation, we choose the following parameters similar to [24]: $\kappa =0.1$MHz, $\gamma =10^{-2}\kappa$, $\Delta _{m}=4\times 10^{3}\kappa$, and $\delta =2\times 10^{-2}\kappa$, where the symbol $\delta$ denotes the difference between the parameters $\Delta _{m}$ and $\lambda$ as $\delta =\Delta _{m}-\lambda$. With these parameters, the cross-Kerr coupling strength could be enhanced three orders of magnitude. This cross-Kerr enhancement is because a single-phonon state in the squeezed mechanical mode corresponds to an exponentially growing number of phonons in the original mechanical oscillator.
Fig. 3. The values $g_{s}/\kappa$, $g_{p}/\omega _{s}$, and $r$ versus the amplitude $\lambda /\kappa$ and the frequency detuning $\Delta _{m}/\kappa$. The parameters are scaled by the cavity decay rate $\kappa$, i.e., $g_{ck}=10^{-2}\kappa$, (a) $\Delta _{m}=4\times 10^{3}\kappa$, and (b) $\lambda =4\times 10^{3}\kappa$.
3. Photon-phonon blockade and phonon-number measurement
One important application of the strong cross-Kerr interaction is the realization of photon-phonon blockade effect [20,34,35]. The physical mechanism for the creation of the photon-phonon blockade is the cross-Kerr-induced anharmonicity of the level spacing. To study the influence of the cross-Kerr interaction on the photon-phonon blockade, we use the equal-time second-order photon-phonon correlation function in the steady state ($t\rightarrow \infty$)
These quantities characterize directly the photon-phonon statistical properties, which has been measured by the correlation experiment [36]. The correlation function $g_{ss}^{(2)}(0)<1$ (or $g^{(2)}(0)<1$) represents nonclassical antibunching effect, and the limit $g_{ss}^{(2)}(0)\rightarrow 0$ (or $g^{(2)}(0)\rightarrow 0$) corresponds to photon-phonon blockade.
To generate photons in the system, a weak probe field ($\epsilon _{p1}\ll \kappa$) is introduced to drive cavity mode. The corresponding Hamiltonian is $H_{p1}=\epsilon _{p1}(a^{\dagger }e^{-i\omega _{p1}t}+ae^{i\omega _{p1}t})$, where $\epsilon _{p1}$ and $\omega _{p1}$ are the amplitude and frequency, respectively. For observation of photon-phonon blockade, the original mechanical mode is also driven by a weak probe field with amplitude $\epsilon _{p2}$ and frequency $\omega _{p2}$, and this Hamiltonian is $H_{p2}=\epsilon _{p2}(b^{\dagger }e^{-i\omega _{p2}t}+be^{i\omega _{p2}t})$. Experimentally, the mechanical driving can be realized by the piezoelectric drive [37]. Then, in the frame rotating at the probe frequency $\omega _{pi}$ $(i=1,2)$, the Hamiltonian including two probe fields becomes
whereHere $\Delta _{c}=\omega _{cs}-\omega _{p1}$ ($\Delta _{s}=\omega _{s}-\omega _{p2}$) describes the detuning between the cavity (mechanical) mode and the probe field $\omega _{p1}$ ($\omega _{p2}$), and $\tilde {\epsilon }_{p2}=\epsilon _{p2}\cosh (r)$. Under the conditions of $\omega _{s}\thickapprox \omega _{p2}$ and $\omega _{s}\gg \epsilon _{p2}\sinh (r)$, the high-frequency oscillation terms of $H'_{p2}$ can be neglected.
Next, we solve exactly the dynamic behavior of the total system after taking into account the cavity and mechanical dissipation. The system master equation for the density matrix $\rho$ is given by
Here, $r_{e}$ and $\Phi$ are the squeezing parameter and reference phase of the squeezed reservoir [39–41], which is coupled to the mechanical oscillator. For the ideal parameter conditions ($r_{e}=r$ and $\Phi =\pm n\pi$, $n=1, 3, 5, \dots$), the thermal noise and the two-phonon correlation can be suppressed completely, that is, $N_s, M_s=0$. This can be understood from the phase matching [24]. By numerically solving the master equation (7) within a truncated Fock space, we can calculate the equal-time second-order correlation function and then evaluate the photon-phonon blockade effect in this system.
Figure 4(a) displays the correlation function $g_{ss}^{(2)}(0)$ as a function of the detuning $\Delta _{c}/\kappa$ under the condition of the mechanical resonance $\Delta _{s}=0$, where the result can be obtained by numerically solving the master equation (7) with the approximate Hamiltonian (6). Here, we assume that the cavity field is in a vacuum bath, and the mechanical oscillator is equivalently coupled to a squeezed vacuum bath with the ideal parameter matching conditions ($r_{e}=r$ and $\Phi = \pi$). It clearly shows that in the absence of the mechanical parametric amplification ($r=0$), there is no photon-phonon blockade (black dashed line), $g_{ss}^{(2)}(0)\rightarrow 1$. This phenomenon is essentially due to the weak coupling between the cavity and the mechanical oscillator, which cannot cause a sufficient anharmonicity of the energy levels. However in the insets, a proper decrease in the parameter $\delta$ can increase the squeezing parameter $r$, resulting in the enhancement of the cross-Kerr coupling strength $g_{s}$. Obviously, the antibunching effect, $g_{ss}^{(2)}(0)<1$, is gradually strengthened, and even the photon-phonon blockade, $g_{ss}^{(2)}(0)\rightarrow 0$, can be observed at the detuning $\Delta _{c}=0$. This blockade phenomenon can be explained from strong cross-Kerr-induced anharmonicity, as shown in Fig. 4(b). In the mechanical resonant case $\Delta _{s}=0$, if one photon is resonantly injected into the cavity, the $|0_{a}1_{s}\rangle \rightarrow |1_{a}1_{s}\rangle$ transition is detuned and will be suppressed for $g_{s}>\kappa$ and $g_{s}>\gamma$. Similarly, the $|1_{a}0_{s}\rangle \rightarrow |1_{a}1_{s}\rangle$ transition is also suppressed. Thus, the strong cross-Kerr coupling condition is necessary for generating photon-phonon blockade.
Fig. 4. (a) In the mechanical resonance $\Delta _{s}=0$, the steady-state correlation function $g_{ss}^{(2)}(0)$ versus the detuning $\Delta _{c}/\kappa$ without and with the mechanical parametric amplification. The parameters are the same as in Fig. 3(a), except for $n_{th}=0$, $r_{e}=r$, $\Phi = \pi$, $\gamma =10^{-2}\kappa$, $\epsilon _{p1}=\epsilon _{p2}=10^{-2}\kappa$. (b) Anharmonic energy-level diagram of the system [see the Hamiltonian in Eq. (6)]. Here, states are labeled as $|n_{a}m_{s}\rangle$, where $|n_{a}\rangle$ and $|m_{s}\rangle$ denote the photon-number state and the phonon-number state in the squeezed mode, respectively.
To check the validity of the approximate Hamiltonian $\tilde {H}$, the transient correlation function $g^{(2)}(0)$ based on the master equation (7) is plotted in Fig. 5. It can be seen that the approximate result obtained using $\tilde {H}$ (black dashed curve) agrees well with the exact numerical result corresponding to the total Hamiltonian including the two probe fields (blue solid curve), i.e., $H\rightarrow H+H_{p1}+H_{p2}$. Compared with the case of $\tilde {H}$, these oscillation behaviors of $H$ result from the high-frequency-oscillation terms. Additionally, we note that the correlation function $g^{(2)}(0)$ approaches a steady value at time $\kappa t\approx 800$.
Fig. 5. In the resonant case $\Delta _{c}=\Delta _{s}=0$, the transient correlation function $g^{(2)}(0)$ versus the scaled time $\kappa t$. The parameters are the same as in Fig. 4(a), except for $\delta =2\times 10^{-2}\kappa$.
Another simple application of the strong cross-Kerr interaction is the quantum nondemolition measurement of the phonon number [20,42,43]. From the Hamiltonian of Eq. (6), we find that $H'_{\mathrm {app}}=(\Delta _{c}+g_{s}b_{s}^{\dagger }b_{s})a^{\dagger }a+\Delta _{s}b_{s}^{\dagger }b_{s}$ commutes with the squeezed phonon-number operator $b_{s}^{\dagger }b_{s}$ and represents the quantum nondemolition measurement of the mechanical energy [44]. In the situation, the phonon-number changes reflect a per phonon shift of $g_{s}$ in the resonance frequency of the optical cavity, which could be inferred by measuring the cavity excitation spectrum in the steady state ($t\rightarrow \infty$)
Figure 6 shows the cavity excitation spectrum $S(\Delta _{c})$ as a function of the detuning $\Delta _{c}/\kappa$ by numerically solving master equation (7) with Hamiltonian (6). It is clear that when $r=0$, the resonant peaks are corresponding to different phonon number and highly overlap. However, when $\delta =2\times 10^{-2}\kappa$, i.e., $g_{s}>\kappa$ and $g_{s}>\gamma$, the resonant peaks are located at $\Delta _{c}=-m_{s}g_{s}$ and become visible. Therefore, in the strong cross-Kerr coupling, the peaks in the excitation spectrum can also be used to distinguish different phonon number through the frequency shift $\Delta _{c}$.
Fig. 6. In the mechanical resonance $\Delta _{s}=0$, the cavity excitation spectrum $S(\Delta _{c})$ versus the detuning $\Delta _{c}/\kappa$. The parameters are the same as in Fig. 4(a), except for (a) $r=0$ and (b) $\delta =2\times 10^{-2}\kappa$.
4. Conclusion
In summary, we have proposed the realizable scheme for boosting the cross-Kerr coupling by the strong mechanical parametric amplification. In particular, we focused on the effect of the cross-Kerr interaction on the photon-phonon blockade and the phonon-number measurement. By numerically calculating the correlation function and the excitation spectrum, we found that the strong cross-Kerr coupling can strengthen the photon-phonon blockade effect and identify the phonon number, respectively. We believe these results can be meaningful in investigating the cross correlation between photon and phonon.
Funding
Shanghai Sailing Program, China (21YF1446900); Research start-up project of Shanghai Institute of Technology (YJ2021- 65); National Natural Science Foundation of China (11774089, 12034007).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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