Fröhlich condensate of phonons in optomechanical systems

We propose that the Fröhlich condensate of phonons can be realized in optomechanical systems. The system consists of a one-dimensional array of membranes coupled to the cavity field via a quadratic interaction, and the cavity is pumped by an external laser. Analytical and numerical results demonstrate that the high phonon occupancy of the lowest or highest mechanical mode is achievable depending on the detuning of the driving laser, the optomechanical strength, and the temperature. The feasibility of experimental implementation is discussed. Our results shed light on energy conversion and transfer, heat control, and multimode cooling using optomechanical systems.

Figure 1

(a) Schematic of an optomechanical system: an optical cavity interacting with a one-dimensional array of membranes. (b) Simplified model. The dissipation is due to the contact of the system with environment. The incoherent phonon pumping is not necessary for our system to get a Fröhlich condensate; it is optional.

Figure 2

Transition processes between the $l\mathrm{th}$ mode and the $j\mathrm{th}$ mode in the case of ${\omega }_l<{\omega }_j$. Here $n_c$, $n_l$, and $n_j$ represent the photon number and the phonon numbers in the $l\mathrm{th}$ mode and the $j\mathrm{th}$ mode, respectively. The red (left, $j\to l$) arrow describes the process such that one phonon in the $j\mathrm{th}$ mode is pumped to the $l\mathrm{th}$ mode by the red-detuned driving laser. The blue (right, $l\to j$) arrow describes the reverse process pumped by the blue-detuned driving laser.

Figure 3

Comparison of average phonon numbers obtained from the MCWF method and decorrelation approximation in the case of $N=5$. The solid lines represent the evolution of the average phonon numbers obtained from the MCWF method, and the dashed lines represent the steady-state phonon numbers obtained from the decorrelation approximation. The zoomed-in inset shows the steady state of the third and fourth modes. The parameters are as follows: $T=400$ mK, ${\omega }_0=100$ MHz, $k/m={\omega }_0^2/10$, $\kappa =1$ MHz, $\gamma =100$ Hz, $\mathrm{\Delta }=-k/\left(m{\omega }_0\right)$, $g_0\left|\overline{\alpha }\right|/\kappa =0.01$.

Figure 4

The steady-state ratio of phonon numbers with respect to detuning. (a) The optomechanical coupling $g_0\left|\overline{\alpha }\right|={10}^{-5}\kappa$. (b) $g_0\left|\overline{\alpha }\right|=5\times {10}^{-5}\kappa$. $N=5$. The membrane and cavity parameters we use are close to those in Refs. [36, 37]. ${\omega }_0=1$ MHz, $\kappa =100$ kHz, $\gamma =0.1$ Hz, $T=300$ mK, and the thermal phonon population ${\overline{n}}_{th}$ is on the order of ${10}^4$. The mechanical coupling is chosen to be $k/m={\omega }_0^2/10$. The upper (lower) dashed line denotes the ratio of phonon numbers in the lowest (highest) mode in thermal equilibrium.

Figure 5

(a) and (b) The steady-state ratio of phonon numbers with respect to optomechanical coupling. The temperature is fixed at $T=300$ mK. (a) Negative detuning $\mathrm{\Delta }=-k/\left(m{\omega }_0\right)$. (b) Positive detuning $\mathrm{\Delta }=k/\left(m{\omega }_0\right)$. The upper (lower) dashed line denotes the ratio of phonon numbers in the lowest (highest) mode in thermal equilibrium. (c) The steady-state ratio of phonon numbers with respect to temperature. $\mathrm{\Delta }=-k/\left(m{\omega }_0\right)$. In (a)–(c), $N=5$, the cavity and membranes parameters are the same as those used in Fig. 4.

Figure 6

The steady state of a Fröhlich condensate with $\langle {\widehat{n}}_1\rangle /\langle {\widehat{N}}_{\text{tot}}\rangle =0.99$. (a) Phonon distribution in $j\ge 2$ modes. (b) The net transition rate from the $j\mathrm{th}$ mode to the lowest mode. $N=70$, $g_0\left|\overline{\alpha }\right|=0.06\kappa$, $\mathrm{\Delta }=-k/\left(m{\omega }_0\right)$; the other cavity and membranes parameters are the same as those used in Fig. 4.

Figure 7

Critical condition of the Fröhlich condensate. (a) The minimum value of optomechanical coupling required to realize $a=0.99$ with respect to the number of membranes. The red squares represent the case of a single cavity, and the black diamonds represent the case of $N$ cavities. (b) The contour of $\langle {\widehat{n}}_1\rangle /\langle {\widehat{N}}_{\text{tot}}\rangle$ in the steady state with respect to $g_0\left|\overline{\alpha }\right|$ and temperature in the case of $N=5$. The dashed line fitted by Eq. (21) agrees well with the contour lines $\langle {\widehat{n}}_1\rangle /\langle {\widehat{N}}_{\text{tot}}\rangle =0.945$. (c) The contour of $\langle {\widehat{n}}_1\rangle /\langle {\widehat{N}}_{\text{tot}}\rangle$ in the steady state with respect to $g_0\left|\overline{\alpha }\right|$ and external phonon pumping in the case of $N=5$. The temperature $T=300$ mK is fixed. The dashed lines are fitted by Eq. (21) with ${\overline{n}}_{\text{th}}$ replaced by $\frac{{\mathrm{\Gamma }}_p}{\gamma }+{\overline{n}}_{\text{th}}$ and agree well with the contour lines $\langle {\widehat{n}}_1\rangle /\langle {\widehat{N}}_{\text{tot}}\rangle =0.966$ and $\langle {\widehat{n}}_1\rangle /\langle {\widehat{N}}_{\text{tot}}\rangle =0.98$. In (a)–(c), $\mathrm{\Delta }=-k/\left(m{\omega }_0\right)$; the cavity and membrane parameters are fixed and are the same as those used in Fig. 4.

Figure 8

Comparison of numerical results obtained from the full master equation and the reduced master equation for the case of $N=2$. (a) The steady-state phonon number in the first mode. (b) The steady-state phonon number in the second mode. $T=1$ mK, ${\omega }_0=100$ MHz, $k/m={\omega }_0^2/3$, $\kappa =1$ MHz, ${\gamma }_1={\gamma }_2=100$ Hz, $\mathrm{\Delta }=-k/\left(m{\omega }_0\right)$. The thermal phonon population is ${\overline{n}}_{1,\text{th}}=1.15,{\overline{n}}_{2,\text{th}}=0.71$. The cutoff of the Fock state is set to 10 for phonons and is set to 5 for photons in the calculation of the master equations.

Figure 9

Time evolution of the ratio of the average phonon numbers in the lowest mode to the total phonon numbers. (a) Time evolution of $\langle {\widehat{n}}_1\rangle /\langle {\widehat{N}}_{\text{tot}}\rangle$ at different optomechanical couplings. (b) The time needed for $\langle {\widehat{n}}_1\rangle /\langle {\widehat{N}}_{\text{tot}}\rangle$ to achieve 90% of the peak value with respect to optomechanical couplings. The dashed line in (a) denotes ${\overline{n}}_{1,\text{th}}/N_{\text{tot}}$ in thermal equilibrium. In (a) and (b), $N=5$; the cavity and membrane parameters are the same as those used in Fig. 4.