Single-photon time-dependent spectra in quantum optomechanics

We consider how a single photon can probe the quantum nature of a moving mirror in the context of quantum optomechanics. In particular, we demonstrate how the single-photon time-dependent spectrum reveals the time required for the fully resolved spectrum to build up as a result of the single photon creating one phonon after the other. The time-dependent spectrum also manifests the order (in time) in which the blue and red resonances (sidebands) appear in the spectrum identifying the two different mechanisms responsible for the production of these sidebands. Our study includes the coupling of the movable mirror to a mechanical heat bath. We use quantum trajectory theory combined with the input-output formalism for all calculations, but we also show how a dressed-state picture can be used to explain positions and relative strengths of all resonances.

Figure 1

An optomechanical cavity (OMC): A Fabry–Pérot cavity with a fixed mirror on the left and a movable mirror on the right. Initially, the OMC is assumed to contain a single photon and zero phonons. The right mirror (modeled as a harmonic oscillator) is coupled to a finite-temperature heat bath with mechanical decay rate ${\gamma }_M$, while the temperature of the bath is given in terms of the average thermal phonon number $\overline{M}$. The decay rate of the optical cavity energy (due to leakage of photons through the left mirror) is denoted by $\kappa$. The output field is detected and its time-dependent spectrum is used to probe the motion of the movable mirror.

Figure 2

The single-photon time-dependent spectrum in an OMC cavity, calculated at six different times, in the strong-coupling regime within the good cavity limit. $t=3{\omega }_M^{-1},5{\omega }_M^{-1},6{\omega }_M^{-1},8{\omega }_M^{-1},10{\omega }_M^{-1}$, and $20{\omega }_M^{-1}$ correspond to thick blue dashed, red dotted, green dotted dashed, thin magenta dashed, brown thin dotted dashed, and black solid curves, respectively. The parameters are $g_M/{\omega }_M=1.2,k_M/{\omega }_M=0.2\Gamma /{\omega }_M=0.1$. Notice the presence of many sidebands and the appearance of red sidebands before the blue sidebands. We have considered the possibility of having ten phonons in the mechanical motion at most. This choice is made by noticing that, upon taking a higher number of phonons $\left(m>10\right)$, the spectrum does not change; rather only more tiny side bands appear at higher $\pm \Delta /{\omega }_M$ values which lie outside the range of the $\Delta /{\omega }_M$ axis shown.

Figure 3

An energy-level diagram explaining the appearance of dressed states as a result of coupling between bare optomechanical states. Different transitions are numbered corresponding to the different spectral peaks one obtains in the single-photon spectrum restricted to single-phonon mechanical motion (see Fig. 4).

Figure 4

The single-photon spectrum in an OMC with truncation of mechanical oscillations to a single phonon. The parameters used are $\kappa /{\omega }_M=0.25,g_M/{\omega }_M=1.25$, and $\Gamma /{\omega }_M=0.1$. Parameters are chosen to observe the fully resolved spectrum in this physically artificial situation. Also note the effect of changing the initial phonon number in the OMC from zero to one in the long-time spectrum (in particular, the interchange of peak heights with this change of initial phonon number). Red dashed and black solid curves correspond to situations when initially there is zero or one phonon in the OMC.

Figure 5

Time-dependent spectrum of a single photon emitted by an OMC in the presence of mechanical losses. $t=3{\omega }_M^{-1},5{\omega }_M^{-1},6{\omega }_M^{-1},8{\omega }_M^{-1},10{\omega }_M^{-1}$, and $40{\omega }_M^{-1}$ correspond to thick blue dashed, red dotted, green dotted dashed, thin magenta dashed, brown thin dotted dashed, and black solid curves, respectively. The parameters used are: $\kappa /{\omega }_M=0.2,{\gamma }_M/{\omega }_M=0.1,\overline{M}=0.1,g_M/{\omega }_M=1.2$, and $\Gamma /{\omega }_M=0.1$. Like Fig. 2, the spectrum is calculated in a truncated Hilbert space with the presence of at most ten phonons in the mechanical oscillations. As shown in the first three (upper) plots in the figure, OMC cavity emits a simple Lorentzian spectrum of the single photon due to phonon-leakage effect (see text). Only after $t\sim 8{\omega }_M^{-1}$ does the spectrum split and show resonances indicating the presence of phonons in the system. Note that the spectrum is plotted under the weak-damping limit $\left({\gamma }_M\overline{M}<\kappa \right)$ such that the effect of phonon input terms on the spectrum is excluded, as discussed in the text.