Dynamical emission of phonon pairs in optomechanical systems

The multiphonon state plays an important role in quantum information processing and quantum metrology. Here we propose a scheme to realize dynamical emission of phonon pairs based on the technique of stimulated Raman adiabatic passage in a single-cavity optomechanical system, where the optical cavity is driven by two Gaussian pulse lasers. By exploring quantum trajectories of the state populations and the average phonon number, we find that the dynamical phonon-pair emission can be realized under the appropriate parameter conditions and is tunable by controlling the time interval between the consecutive pulses of pump lasers. In particular, the numerical results for the standard and generalized second-order correlation functions of the mechanical mode show that the system can behave as an antibunched phonon-pair emitter. Our proposal can be extended to achieve an antibunched $n$-phonon emitter, which has potential applications for on-chip quantum communication.

Figure 1

(a) Schematic diagram of a cavity optomechanical system consisting of a single-cavity mode coupled to a mechanical mode via radiation-pressure interaction. The optical cavity is driven by two Gaussian pulse driving fields ${\mathrm{\Omega }}_1\left(t\right)$ and ${\mathrm{\Omega }}_2\left(t\right)$. (b) The resonant transition of the effective Hamiltonian $H_{\text{eff}}^{\left(2\right)}$ at the coupling strength $g/{\omega }_m\approx 0.765$, where there is no transition between $|0,2\rangle$ and $|1,\tilde{2}\left(1\right)\rangle$. Other parameters are ${\mathrm{\Delta }}_1\equiv {\omega }_1-{\omega }_c=-g^2/{\omega }_m$ and ${\mathrm{\Delta }}_2\equiv {\omega }_2-{\omega }_c=-g^2/{\omega }_m-2{\omega }_m$.

Figure 2

(a) The ratio $g_N/{\omega }_m$ of the minimal positive value $g_N$ satisfying Eq. (13) over the mechanical frequency ${\omega }_m$ as a function of the index $N$. (b) The coefficients $|A_{m,q}^{\left(2\right)}|$ as functions of the indices $m$ and $q$ at $g/{\omega }_m\approx 0.765$. (c)–(e) The first Gaussian wave packets of two pulse lasers ${\mathrm{\Omega }}_i\left(t\right)(i=1,2)$, the zero-photon state populations $P_{|0,m\rangle }={\left|\langle 0,m|\psi \left(t\right)\rangle \right|}^2$, and the single-photon state populations $P_{|1,\tilde{m}\left(1\right)\rangle }={\left|\langle 1,\tilde{m}\left(1\right)|\psi \left(t\right)\rangle \right|}^2$ as functions of the time ${\omega }_{m}t$. Here $\left|\psi \right(t)\rangle$ is the state of the system at time $t$ in the absence of dissipation. Other parameters are ${\mathrm{\Omega }}_0/{\omega }_m=0.03$, ${\omega }_m\sigma =300$, ${\omega }_m{t}_1=1600$, ${\omega }_m{t}_2=1100$, $g/{\omega }_m\approx 0.765$, ${\mathrm{\Delta }}_1=-g^2/{\omega }_m$, and ${\mathrm{\Delta }}_2=-g^2/{\omega }_m-2{\omega }_m$.

Figure 3

(a) Two Gaussian pulse driving fields ${\mathrm{\Omega }}_i\left(t\right)(i=1,2)$ as a function of the time ${\omega }_{m}t$. (b)–(d) The state populations $P_{|0,m\rangle }\left(t\right)$ and $P_{|1,\tilde{m}\left(1\right)\rangle }\left(t\right)$ ($m=0,1,2$) as functions of the time ${\omega }_{m}t$ in the presence of dissipation. Other parameters are ${\mathrm{\Omega }}_0/{\omega }_m=0.03$, ${\omega }_m\sigma =300$, ${\omega }_m{t}_1=1600$, ${\omega }_m{t}_2=1100$, ${\omega }_{m}T=15000$, $g/{\omega }_m\approx 0.765$, $\kappa /{\omega }_m=0.002$, ${\gamma }_m/{\omega }_m=0.0004$, $T_b=n_{\text{th}}=0$, ${\mathrm{\Delta }}_1=-g^2/{\omega }_m$, and ${\mathrm{\Delta }}_2=-g^2/{\omega }_m-2{\omega }_m$.

Figure 4

(a)–(c) Quantum trajectory of the state populations $P_{|0,m\rangle }\left(t\right)$ and $P_{|1,\tilde{m}\left(1\right)\rangle }\left(t\right)$ ($m=0,1,2$). (d) Quantum trajectory of the average phonon number $\langle b^{†}b\rangle \left(t\right)$. Other parameters are ${\mathrm{\Omega }}_0/{\omega }_m=0.03$, ${\omega }_m\sigma =300$, ${\omega }_m{t}_1=1600$, ${\omega }_m{t}_2=1100$, ${\omega }_{m}T=15000$, $g/{\omega }_m\approx 0.765$, $\kappa /{\omega }_m=0.002$, ${\gamma }_m/{\omega }_m=0.0004$, $T_b=n_{\text{th}}=0$, ${\mathrm{\Delta }}_1=-g^2/{\omega }_m$, and ${\mathrm{\Delta }}_2=-g^2/{\omega }_m-2{\omega }_m$.

Figure 5

(a) One period of the equal-time second-order correlation functions $g_N^{\left(2\right)}(t,t)$ as a function of the time ${\omega }_{m}t$ with $N=1$ (the red line) and $N=2$ (the blue line) at $T_b=n_{\text{th}}=0$. The $t_{s1}$ and $t_{s2}$ correspond to the maximum value of $g_1^{\left(2\right)}(t,t)$ and the minimum value of $g_2^{\left(2\right)}(t,t)$, respectively. (b) The time-delay second-order correlation functions $g_N^{\left(2\right)}(t_{sN},t_{sN}+\tau )$ ($t_{s1}$ and $t_{s2}$ are indicated in the upper panel) with $N=1$ (the red line) and $N=2$ (the blue line) at $T_b=n_{\text{th}}=0$. (c) One period of the generalized equal-time second-order correlation function $g_2^{\left(2\right)}(t,t)$ as a function of the time ${\omega }_{m}t$ at various values $n_{\text{th}}=(0.1,0.3,0.4,0.7)$. Other parameters are ${\mathrm{\Omega }}_0/{\omega }_m=0.03$, ${\omega }_m\sigma =300$, ${\omega }_m{t}_1=1600$, ${\omega }_m{t}_2=1100$, ${\omega }_{m}T=15000$, $g/{\omega }_m\approx 0.765$, $\kappa /{\omega }_m=0.002$, ${\gamma }_m/{\omega }_m=0.0004$, ${\mathrm{\Delta }}_1=-g^2/{\omega }_m$, and ${\mathrm{\Delta }}_2=-g^2/{\omega }_m-2{\omega }_m$.