Quantum Langevin equations for optomechanical systems

In order to consider a more general field state let us first introduce the Weyl operators [30, 32] for the Fock A-fields, defined as

$$\begin{eqnarray*}&&{{\mathcal{W}}}_{A}(g)=\mathrm{exp}\{\displaystyle \sum _{k=1}^{2}{\displaystyle \int }_{0}^{+\infty }{g}_{k}(s){\rm{d}}{A}_{k}^{\dagger }(s)-{\rm{h}}.{\rm{c}}.\},\end{eqnarray*}$$

with g_k square integrable functions. The operator ${{\mathcal{W}}}_{A}(g)$ is unitary and the property ${A}_{k}(t){{\mathcal{W}}}_{A}(g)e(0)={\displaystyle \int }_{0}^{t}{\rm{d}}{{sg}}_{k}(s){{\mathcal{W}}}_{A}(g)e(0)$ holds, so that the action of a Weyl operator on the Fock vacuum gives a coherent state. Therefore ${{\mathcal{W}}}_{A}(g)$ is nothing but a displacement operator for the Bose fields [49]. Relying on (26), we can introduce a Weyl operator also for the B-field

$$\begin{eqnarray}&&{{\mathcal{W}}}_{T}(f)=\mathrm{exp}\{{\displaystyle \int }_{0}^{T}f(s)\;{\rm{d}}{B}_{\mathrm{th}}^{\dagger }(s)-{\rm{h}}.{\rm{c}}.\},\end{eqnarray} \tag{ 37 }$$

where f is a locally square integrable function and T denotes a suitable large time, which we will let tend to infinity in the final formulae describing the quantities of direct physical interest. Now, ${{\mathcal{W}}}_{T}(f)e(0)$ is not a coherent state for the ${B}_{\mathrm{th}}$-field, but its relevant moments can be computed by using the A-field representation (26):

$$\begin{eqnarray}\langle {{\mathcal{W}}}_{T}(f)e(0)| {B}_{\mathrm{th}}(t){{\mathcal{W}}}_{T}(f)e(0)\rangle & = & {\displaystyle \int }_{0}^{t}f(r){\rm{d}}r,\\ \langle {{\mathcal{W}}}_{T}(f)e(0)| {B}_{\mathrm{th}}(t){B}_{\mathrm{th}}(s){{\mathcal{W}}}_{T}(f)e(0)\rangle & = & {\displaystyle \int }_{0}^{t}f(u){\rm{d}}u{\displaystyle \int }_{0}^{s}f(r){\rm{d}}r,\\ \langle {{\mathcal{W}}}_{T}(f)e(0)| {B}_{\mathrm{th}}^{\dagger }(s){B}_{\mathrm{th}}(t){{\mathcal{W}}}_{T}(f)e(0)\rangle & = & N\mathrm{min}\{t,s\}+{\displaystyle \int }_{0}^{t}f(u){\rm{d}}u{\displaystyle \int }_{0}^{s}\bar{f(r)}\;{\rm{d}}r,\\ \langle {{\mathcal{W}}}_{T}(f)e(0)| {B}_{\mathrm{th}}(t){B}_{\mathrm{th}}^{\dagger }(s){{\mathcal{W}}}_{T}(f)e(0)\rangle & = & (N+1)\mathrm{min}\{t,s\}+{\displaystyle \int }_{0}^{t}f(u){\rm{d}}u{\displaystyle \int }_{0}^{s}\bar{f(r)}\;{\rm{d}}r.\end{eqnarray} \tag{ 38 }$$

A crucial step is now to consider f to be a random process and to take the state of the field characterizing the environment to be

$$\begin{eqnarray}&&{\sigma }_{\mathrm{env}}={\mathbb{E}}[{{\mathcal{W}}}_{T}(f)| e(0)\rangle \langle e(0)| {{\mathcal{W}}}_{T}{(f)}^{\dagger }].\end{eqnarray} \tag{ 39 }$$

Again, in the final formulae we will take the limit $T\to +\infty$. This is nothing but an analogue of the regular P-representation for the case of discrete modes. Indeed, in the case of a single mode the Glauber–Sudarshan P-representation of a state ρ [26 section 4.4.3] is defined by $\rho =\int {{\rm{d}}}^{2}\alpha \;P(\alpha ,\bar{\alpha })| \alpha \rangle \langle \alpha |$. If the pseudo-density P is allowed to become negative and singular, then any state can be represented in this form. When P is a true probability density, one speaks of a regular P-representation and the Glauber–Sudarshan formula describes mixtures of coherent states, including in particular thermal states [26 page 113]. In a probabilistic language, which is more suitable for generalizations to stochastic processes and fields, the fact that a state ρ has a regular P-representation can be rephrased by saying that it can be written as the expectation value $\rho ={\mathbb{E}}[| \alpha \rangle \langle \alpha | ]$, with α a complex random variable. In order to construct a thermal state it is enough to consider the case in which the distribution of α is Gaussian with vanishing mean ${\mathbb{E}}[\alpha ]=0$ and second moments ${\mathbb{E}}[{\alpha }^{2}]=0$, ${\mathbb{E}}[{| \alpha | }^{2}]={\sigma }^{2}$. Then, $\rho ={\mathbb{E}}[| \alpha \rangle \langle \alpha | ]$ turns out to be a thermal state [26 section 4.4.5].

By analogy, to construct a thermal field state with a general thermal spectrum we take f to be a Gaussian stationary stochastic process with vanishing mean, ${\mathbb{E}}[f(t)]=0$, and correlation functions

$$\begin{eqnarray}&&{\mathbb{E}}[f(t)f(s)]=0,\qquad {\mathbb{E}}[\bar{f(t)}\;f(s)]\;=:\;G(t-s).\end{eqnarray} \tag{ 40 }$$

Thanks to stationarity, the function G(t) is positive definite, so that according to Bochner's theorem [51] its Fourier transform

$$\begin{eqnarray}&&\hat{G}(\nu )={\displaystyle \int }_{-\infty }^{+\infty }{{\rm{e}}}^{-{\rm{i}}\nu t}G(t){\rm{d}}t\end{eqnarray} \tag{ 41 }$$

is a positive function, which we assume to be absolutely integrable, thus implying a finite power spectral density for the process. Since the field state ${\sigma }_{\mathrm{env}}$, defined by (39), turns out to be Gaussian, we can characterize it through the means and the correlations of the thermal field ${B}_{\mathrm{th}}$, which are immediately obtained from (38) and the properties of the process f. The only non-zero contributions are given by

$$\begin{eqnarray}{\langle {B}_{\mathrm{th}}^{\dagger }(s){B}_{\mathrm{th}}(t)\rangle }_{\mathrm{env}} & = & N\mathrm{min}\{t,s\}+{\displaystyle \int }_{0}^{t}{\rm{d}}u{\displaystyle \int }_{0}^{s}{\rm{d}}r\;G(r-u),\\ {\langle {B}_{\mathrm{th}}(t){B}_{\mathrm{th}}^{\dagger }(s)\rangle }_{\mathrm{env}} & = & (N+1)\mathrm{min}\{t,s\}+{\displaystyle \int }_{0}^{t}{\rm{d}}u{\displaystyle \int }_{0}^{s}{\rm{d}}r\;G(r-u).\end{eqnarray} \tag{ 42 }$$

To better grasp the physical content of the new state and of the formulae (42) let us introduce a set of field modes as in [49]. Using a complete orthonormal set $\{{h}_{n}\}$ in ${L}^{2}({\mathbb{R}})$, we can expand the field in terms of discrete independent temporal modes by defining them as

$$\begin{eqnarray*}&&{c}_{{h}_{n}}={\displaystyle \int }_{-\infty }^{+\infty }\bar{{h}_{n}(t)}\;{\rm{d}}{B}_{\mathrm{th}}(t).\end{eqnarray*}$$

We then obtain ${\langle {c}_{{h}_{n}}\rangle }_{\mathrm{env}}=0$, ${\langle {c}_{{h}_{n}}^{\;2}\rangle }_{\mathrm{env}}=0$, together with

$$\begin{eqnarray*}&&\underset{T\to +\infty }{\mathrm{lim}}{\langle {c}_{{h}_{n}}^{\dagger }{c}_{{h}_{n}}\rangle }_{\mathrm{env}}=N+{\mathbb{E}}[{| {\langle {h}_{n}| f\rangle }_{{L}^{2}}| }^{2}]={\displaystyle \int }_{-\infty }^{+\infty }{| {\hat{h}}_{n}(\nu )| }^{2}N(\nu ){\rm{d}}\nu ,\end{eqnarray*}$$

where we have defined the positive quantity

$$\begin{eqnarray}&&N(\nu )=N+\hat{G}(\nu )\end{eqnarray} \tag{ 43 }$$

and ${\hat{h}}_{n}(\nu )$ is the Fourier transform of h_n(t); by normalization ${\displaystyle \int }_{-\infty }^{+\infty }{| {\hat{h}}_{n}(\nu )| }^{2}{\rm{d}}\nu =1$. So, the reduced state of the single mode ${c}_{{h}_{n}}$ is exactly a thermal state expressed in the P-representation. If we take h₁ and h₂ having non-overlapping Fourier transforms we also get ${\mathrm{lim}}_{T\to +\infty }{\langle {c}_{{h}_{1}}^{\dagger }{c}_{{h}_{2}}\rangle }_{\mathrm{env}}=0$, which means that these two modes are independent. Then, $N(\nu )$ is naturally interpreted as the mean number of phonons in a given field mode c_h well peaked around the value ν of the frequency and field modes with different frequencies are independent. A value of $N(\nu )$ different from zero in a neighbourhood of ν implies that the mechanical oscillator can absorb from the bath phonons with energy around ${\hslash }\nu$. On the contrary, the approximations are such that the oscillator can emit phonons of any frequency, even when $N(\nu )=0$. The physically relevant quantity is now the combination of the two non-negative contributions N and $\hat{G}(\nu )$, rather than the values of the individual quantities. Note that the Markovian reduced dynamics of section 2 can be obtained either by considering the non-Fock representation for the thermal field, thus assuming a strictly positive $N\gt 0$ in (26) and taking $\hat{G}(\nu )\equiv 0$, or equivalently by considering a standard Fock representation and formally taking the limit of constant spectrum $\hat{G}(\nu )$ in all the physical quantities.

According to the definition of reduced dynamics, the time evolved state of the mechanical oscillator is still obtained by taking the partial trace with respect to the field degrees of freedom $\rho (t)={\mathrm{lim}}_{T\to +\infty }{\mathrm{Tr}}_{\mathrm{env}}\{U(t)(\rho (0)\otimes {\sigma }_{\mathrm{env}})U{(t)}^{\dagger }\}$. However, at variance with the case in which the state of the field was taken to be the A-field vacuum, by taking the time derivative of this expression no closed evolution equation is obtained unless $N(\nu )$ is constant. Not to have a closed time-homogeneous equation for the reduced dynamics is indeed a signature of the non-Markov features of such a dynamics.

In spite of the difficulty of not having a closed master equation, the study of the reduced equilibrium state, namely ${\rho }_{\mathrm{eq}}={\mathrm{lim}}_{t\to +\infty }\;\rho \;(t)$, can still be afforded and its expression enlightens the physical role of the various parameters. Indeed, thanks to the requirement ${\mathbb{E}}[f(t)f(s)]=0$, one has that equipartition in the sense of (10) still holds. Starting from the explicit forms of position and momentum in the Heisenberg picture (see (35), (18)) one can check that the equilibrium mean values of position and momentum remain equal to zero, while the variances are still of the form (19) with N replaced by the effective mean number of excitations

$$\begin{eqnarray}&&{N}_{\mathrm{eff}}=\displaystyle \frac{{\gamma }_{{\rm{m}}}}{2\pi }{\displaystyle \int }_{-\infty }^{+\infty }\displaystyle \frac{N(\nu )}{\frac{{\gamma }_{{\rm{m}}}^{\;2}}{4}+{(\nu -{\omega }_{{\rm{m}}})}^{2}}\;{\rm{d}}\nu .\end{eqnarray} \tag{ 44 }$$

Notice that if the quantity $N(\nu )$ introduced in (43) is taken to be the constant N, corresponding to the Markovian case, then ${N}_{\mathrm{eff}}=N$. This result suggests that the final Markov approximation should be valid when $\hat{G}(\nu )$ is approximately constant in a neighbourhood of ${\omega }_{{\rm{m}}}$. In fact equation (44) represents a smearing of $N(\nu )$ around the frequency of the mechanical oscillator ${\omega }_{{\rm{m}}},$ the more peaked the smaller the damping constant ${\gamma }_{{\rm{m}}}$. Non-Markovian effects can only be relevant if $\hat{G}(\nu )$ appreciably varies in a neighbourhood of width ${\gamma }_{{\rm{m}}}$ around ${\omega }_{{\rm{m}}},$ being suppressed with decreasing ${\gamma }_{{\rm{m}}}$.

Since the equilibrium state is necessarily Gaussian, by comparing (44) with (14) we get that the new equilibrium state is again a Gibbs state with respect to the same Hamiltonian ${H}_{{\rm{m}}}$, but with an effective inverse temperature ${\beta }_{\mathrm{eff}}$ defined by setting ${N}_{\mathrm{eff}}={({{\rm{e}}}^{{\beta }_{\mathrm{eff}}{\hslash }{\omega }_{{\rm{m}}}}-1)}^{-1}$.

It is important to stress that if a set of quantum Langevin equations is considered as starting point for the description of a stochastic quantum dynamics, commutations rules and symmetrized correlations of the noises cannot be given arbitrarily. In particular, independently of the considered system, if $\{{\xi }_{i}(t)\}$ is a set of operator valued noises, the quantum correlation function ${\langle {\xi }_{i}{(t)}^{\dagger }{\xi }_{j}(t^{\prime} )\rangle }_{\mathrm{env}}$ has to be positive definite [51], in the sense that

$$\begin{eqnarray}&&\displaystyle \sum _{{ij}}{\displaystyle \int }_{0}^{+\infty }{\rm{d}}t{\displaystyle \int }_{0}^{+\infty }{\rm{d}}t^{\prime} \;\bar{{h}_{i}(t)}{\langle {\xi }_{i}{(t)}^{\dagger }{\xi }_{j}(t^{\prime} )\rangle }_{\mathrm{env}}{h}_{j}(t^{\prime} )\geqslant 0,\end{eqnarray} \tag{ 51 }$$

for every choice of the 'smooth' test functions $\{{h}_{i}\}$. Since we can always write

$$\begin{eqnarray}&&{\xi }_{i}{(t)}^{\dagger }{\xi }_{j}(t^{\prime} )=\displaystyle \frac{1}{2}\{{\xi }_{i}{(t)}^{\dagger },{\xi }_{j}(t^{\prime} )\}+\displaystyle \frac{1}{2}[{\xi }_{i}{(t)}^{\dagger },{\xi }_{j}(t^{\prime} )],\end{eqnarray} \tag{ 52 }$$

the necessary positivity condition introduced above becomes a consistency condition between commutation rules and symmetrized correlations.

Relying on (48), (49), as well as the commutation relations (32) for the noises C_q and C_p, one can immediately check this fact for the model at hand. Also for the singular noise ξ constrained by (46) one can show that the expression ${\langle \xi (t)\xi (s)\rangle }_{\mathrm{env}}$ is positive definite. These results are due to the fact that the noise fields have here been explicitly constructed in terms of the quantum Bose fields, so that commutation rules and correlations are not postulated, but rather follow from the mathematical expression of the model.

For the model at hand we denote the Fourier transform of the correlation of the noise ξ by

$$\begin{eqnarray}&&\hat{R}(\nu )=\displaystyle \frac{1}{2}{\displaystyle \int }_{-\infty }^{+\infty }{\rm{d}}t\;{{\rm{e}}}^{-{\rm{i}}\nu t}{\langle \{\xi (t+s),\xi (s)\}\rangle }_{\mathrm{env}},\end{eqnarray} \tag{ 53 }$$

so that according to (43) and (50) it reads

$$\begin{eqnarray}&&\hat{R}(\nu )=\displaystyle \frac{m{\hslash }{\gamma }_{{\rm{m}}}}{2{\omega }_{{\rm{m}}}}(\displaystyle \frac{{\gamma }_{{\rm{m}}}^{\;2}}{4}+{({\omega }_{{\rm{m}}}+\nu )}^{2})(N(\nu )+\displaystyle \frac{1}{2})+(\nu \to -\nu ),\end{eqnarray} \tag{ 54 }$$

where $(\nu \to -\nu )$ means to add the same contribution with ν replaced by $-\nu$. Note in particular that $\hat{R}(\nu )$ is an even function of the frequency.

In our treatment, which gives rise to the expression (54), the interaction with the environment is described in terms of exchange of quanta with the bosonic field representing the phonons, see (25). In models in which the system of interest is coupled to other harmonic oscillators, by some approximations it is possible to arrive to a quantum stochastic Newton equation like (45), but with a different noise spectrum. A reference expression often considered in the literature [26 (3.3.9)], [37] for the quantity $\hat{R}$ is given by

$$\begin{eqnarray}&&{\hat{R}}_{{GZ}}(\nu )=\pi {\hslash }J(\nu )\mathrm{coth}\displaystyle \frac{\beta {\hslash }\nu }{2},\qquad J(\nu ):= \displaystyle \frac{m}{\pi }\;k(\nu )\nu .\end{eqnarray} \tag{ 55 }$$

The quantity $k(\nu )$ contains information on both coupling constant and density of modes of the bath in a neighbourhood of the frequency ν; $J(\nu )$ is often called spectral density. Also in the case of this choice of the noise correlations, it is possible to show that the equilibrium mean of $\dot{q}{(t)}^{2}$ diverges and therefore the identification of the momentum with $m\dot{q}(t)$ is not possible, but some regularization is needed. A typical choice in this context is $k(\nu )={\gamma }_{{\rm{m}}}$ [26 (3.1.1)], [1, 2, 25, 37]; this is equivalent to $J(\nu )\propto \nu$, which is known as Ohmic spectral density. The function $k(\nu )$ must be even due to the definition (53) and stationarity. Also the commutations rules (47) and the positivity requirement (51) still have to hold, leading to the requirement ${\hat{R}}_{{GZ}}(\nu )-m{\hslash }{\gamma }_{{\rm{m}}}| \nu | \geqslant 0$, satisfied at any temperature by taking $k(\nu )\geqslant {\gamma }_{{\rm{m}}}\gt 0$ for all ν. This requirement tells us that in order to have a consistent model satisfying (45), at least at large times, and preserving Heisenberg commutation relations one cannot consider a spectral density with gaps inside the expression (55) of ${\hat{R}}_{{GZ}}$.

By a suitable choice of $N(\nu )$, it is possible to get $\hat{R}(\nu )$ very similar to ${\hat{R}}_{{GZ}}(\nu )$, apart from the low temperature limit. For instance, by taking

$$\begin{eqnarray}&&N(\nu )=| \nu | k(\nu )\displaystyle \frac{2{\omega }_{{\rm{m}}}}{({\Omega }_{{\rm{m}}}^{\;2}+{\nu }^{2}){\gamma }_{{\rm{m}}}({{\rm{e}}}^{\beta {\hslash }| \nu | }-1)},\end{eqnarray} \tag{ 56 }$$

we obtain

$$\begin{eqnarray}&&\hat{R}(\nu )={\hat{R}}_{{GZ}}(\nu )+m{\hslash }[\displaystyle \frac{{\gamma }_{{\rm{m}}}^{\;2}}{2{\omega }_{{\rm{m}}}}({\Omega }_{{\rm{m}}}^{\;2}+{\nu }^{2})-k(\nu )| \nu | ];\end{eqnarray} \tag{ 57 }$$

note that the difference is independent from the temperature. In (57) the compatibility with the commutation relations is guaranteed by the first term in the square brackets; so, in (56) there is no restriction on the choice of $k(\nu )$, apart from $k(\nu )\geqslant 0$, and even spectral gaps can be introduced.

Besides the case (56), the freedom in the choice of $N(\nu )$ in (54) allows to model quite different environments, for instance with a sub-Ohmic or super-Ohmic spectral density [22–24], or with a structured occupation spectrum. In optomechanical systems the quantity $\hat{R}(\nu )$ enters experimental quantities, as in the cases of homodyne/heterodyne detection discussed in section 4.3; so, in principle it is possible to test the form of $\hat{R}(\nu )$. However, essentially a frequency window of width ${\gamma }_{{\rm{m}}}$ around ${\omega }_{{\rm{m}}}$ is experimentally relevant. Some results at room temperature [52] seem to indicate a non-Ohmic spectral density around ${\omega }_{{\rm{m}}},$ a very interesting possibility, but not enough to discriminate between ${\hat{R}}_{{GZ}}(\nu )$ and the form (57) for $\hat{R}(\nu )$. At zero temperature, corresponding to $N(\nu )=0$ in our case (54) and to $\beta \to +\infty$ in (55), the difference is most evident, namely

$$\begin{eqnarray}&&\hat{R}(\nu )=m{\hslash }{\gamma }_{{\rm{m}}}\displaystyle \frac{{\Omega }_{{\rm{m}}}^{\;2}+{\nu }^{2}}{2{\omega }_{{\rm{m}}}}\end{eqnarray} \tag{ 58 }$$

versus

$$\begin{eqnarray}&&{\hat{R}}_{{GZ}}(\nu )=m{\hslash }k(\nu )| \nu | ,\qquad k(\nu )\geqslant {\gamma }_{{\rm{m}}}.\end{eqnarray} \tag{ 59 }$$

These expressions are very different, but to discriminate between them one has to consider very low temperatures and a ratio ${\gamma }_{{\rm{m}}}/{\omega }_{{\rm{m}}}$ that is not too small.

A relevant role in determining the properties of the system is played by the denominator $d(\nu )$ (89); indeed, the quantity ${\Omega }_{{\rm{m}}}[{\Delta }^{2}-{(\nu +{\rm{i}}{\gamma }_{{\rm{c}}}/2)}^{2}]/d(\nu )$ is sometimes interpreted as the effective mechanical susceptibility; see equation (17) of [25]. Most importantly note that the zeros of $d(\nu )$ determine the positions and the widths of the peaks of the fluctuation spectra: even though in principle they can be obtained by solving the fourth order algebraic equation $d(\nu )=0$, it is much more convenient to have simple expressions, even if approximate. An analysis of these zeros is given in appendix A.1 in the case in which $d(\nu )$ can be written in the form

$$\begin{eqnarray}&&d(\nu )=({(\nu +{\rm{i}}\displaystyle \frac{{\Gamma }_{{\rm{c}}}}{2})}^{2}-{\Delta }_{\mathrm{eff}}^{\ 2})({(\nu +{\rm{i}}\;\displaystyle \frac{{\Gamma }_{{\rm{m}}}}{2})}^{2}-{\omega }_{\mathrm{eff}}^{{\rm{m}}\ \ 2}).\end{eqnarray} \tag{ 91 }$$

The stability conditions (78), (79) guarantee the strict positivity of the effective damping constants ${\Gamma }_{{\rm{c}}}$ and ${\Gamma }_{{\rm{m}}}$. The quantity ${\omega }_{\mathrm{eff}}^{{\rm{m}}}$ is known as optical spring rigidity, while the ratio $({\Gamma }_{{\rm{m}}}-{\gamma }_{{\rm{m}}})/{\gamma }_{{\rm{m}}}$ is called co-operativity [5, 9].

An exact expression for the zeros is found when $\Delta ={\omega }_{{\rm{m}}}$, which allows us to put into evidence a crossing of the frequencies of the hybridized optical and mechanical modes [9]. If also the condition

$$\begin{eqnarray}&&4{G}^{2}\lt {({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}\end{eqnarray} \tag{ 92 }$$

holds, the result is

$$\begin{eqnarray}&&{\Gamma }_{{\rm{c}}}=\displaystyle \frac{{\gamma }_{{\rm{c}}}+{\gamma }_{{\rm{m}}}}{2}+\epsilon \sqrt{2{u}^{2}-2{\omega }_{{\rm{m}}}^{\;2}+\displaystyle \frac{{({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}}{8}},\end{eqnarray} \tag{ 93 }$$

$$\begin{eqnarray}&&{\Gamma }_{{\rm{m}}}=\displaystyle \frac{{\gamma }_{{\rm{c}}}+{\gamma }_{{\rm{m}}}}{2}-\epsilon \sqrt{2{u}^{2}-2{\omega }_{{\rm{m}}}^{\;2}+\displaystyle \frac{{({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}}{8}},\end{eqnarray} \tag{ 94 }$$

$$\begin{eqnarray}&&{\Delta }_{\mathrm{eff}}^{\ 2}={\omega }_{\mathrm{eff}}^{{\rm{m}}\ \ 2}=\displaystyle \frac{{\omega }_{{\rm{m}}}^{\;2}+{u}^{2}}{2}-\displaystyle \frac{{({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}}{32},\end{eqnarray} \tag{ 95 }$$

where

$$\begin{eqnarray}&&{u}^{2}=\sqrt{{({\omega }_{{\rm{m}}}^{\;2}+\displaystyle \frac{{({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}}{16})}^{2}-{G}^{2}{\omega }_{{\rm{m}}}^{\;2}},\qquad \epsilon =\{\displaystyle \genfrac{}{}{0em}{}{+1\mathrm{if}{\gamma }_{{\rm{c}}}\gt {\gamma }_{{\rm{m}}}}{-1\mathrm{if}{\gamma }_{{\rm{c}}}\lt {\gamma }_{{\rm{m}}}}.\end{eqnarray} \tag{ 96 }$$

Always for $\Delta ={\omega }_{{\rm{m}}}$, under the conditions

$$\begin{eqnarray}&&\displaystyle \frac{{\omega }_{{\rm{m}}}^{\;2}{({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}}{4}\lt {G}^{2}{\omega }_{{\rm{m}}}^{\;2}\lt {({\omega }_{{\rm{m}}}^{\;2}+\displaystyle \frac{{({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}}{16})}^{2},\qquad {\omega }_{{\rm{m}}}^{\;2}\gt \displaystyle \frac{{({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}}{16},\end{eqnarray} \tag{ 97 }$$

we get instead the result

$$\begin{eqnarray}&&{\Gamma }_{{\rm{c}}}={\Gamma }_{{\rm{m}}}=\displaystyle \frac{{\gamma }_{{\rm{c}}}+{\gamma }_{{\rm{m}}}}{2},\qquad {\Delta }_{\mathrm{eff}}=\sqrt{{x}_{\pm }},\qquad {\omega }_{\mathrm{eff}}^{{\rm{m}}}=\sqrt{{x}_{\mp }},\end{eqnarray} \tag{ 98 }$$

$$\begin{eqnarray}&&{x}_{\pm }={\omega }_{{\rm{m}}}^{\;2}-\displaystyle \frac{{({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}}{16}\pm {\omega }_{{\rm{m}}}\sqrt{{G}^{2}-\displaystyle \frac{{({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}}{4}}.\end{eqnarray} \tag{ 99 }$$

The two alternatives in (98) are completely equivalent; there is no reason to associate the frequency $\sqrt{{x}_{+}}$ to the cavity and $\sqrt{{x}_{-}}$ to the mechanical oscillator, or viceversa. A striking feature of the case $\Delta ={\omega }_{{\rm{m}}}$ is the change in the behaviour of the zeros at the critical point ${\gamma }_{{\rm{c}}}={\bar{\gamma }}_{{\rm{c}}}$, solution of ${G}^{2}={({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}/4$; recall that G² depends on ${\gamma }_{{\rm{c}}}$ due to (67) and (75).

In the general case, an approximate expression can be obtained under the conditions

$$\begin{eqnarray}&&\displaystyle \frac{{\gamma }_{{\rm{m}}}}{{\gamma }_{{\rm{c}}}}\ll 1,\qquad | \chi (\Delta )| \ll 1,\qquad | \chi (\Delta )| | 1-\displaystyle \frac{{\Delta }^{2}}{{\omega }_{{\rm{m}}}^{\;2}}-\displaystyle \frac{{\gamma }_{{\rm{c}}}^{\;2}}{4{\omega }_{{\rm{m}}}^{\;2}}| \ll 1,\end{eqnarray} \tag{ 100 }$$

where

$$\begin{eqnarray}&&\chi (\Delta )=\displaystyle \frac{{G}^{2}{\omega }_{{\rm{m}}}\Delta }{(\frac{{({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}}{4}+{(\Delta -{\omega }_{{\rm{m}}})}^{2})(\frac{{({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}}{4}+{(\Delta +{\omega }_{{\rm{m}}})}^{2})}.\end{eqnarray} \tag{ 101 }$$

The result for the damping constants is

$$\begin{eqnarray}&&{\Gamma }_{{\rm{m}}}\simeq {\gamma }_{{\rm{m}}}+\chi (\Delta )({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}}),\qquad {\Gamma }_{{\rm{c}}}\simeq {\gamma }_{{\rm{c}}}-\chi (\Delta )({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}}).\end{eqnarray} \tag{ 102 }$$

The expressions for ${\omega }_{\mathrm{eff}}^{{\rm{m}}\ \ 2}$ and ${\Delta }_{\mathrm{eff}}^{\ 2}$ are then obtained by inserting ${\Gamma }_{{\rm{m}}}$ and ${\Gamma }_{{\rm{c}}}$ in the equations (A.12). The compatibility conditions (A.13) have to hold. As one can see, when $\Delta \gt 0$ (red detuning), we have an increasing of the mechanical damping constant, ${\Gamma }_{{\rm{m}}}\gt {\gamma }_{{\rm{m}}}$, and a decreasing of the spring rigidity, ${\omega }_{\mathrm{eff}}^{{\rm{m}}}\lt {\omega }_{{\rm{m}}}$.

By integrating in their frequency dependence the fluctuation spectra one obtains the second moments of position and momentum in the equilibrium state:

$$\begin{eqnarray}&&{\langle {q}^{2}\rangle }_{\mathrm{eq}}-{\langle q\rangle }_{\mathrm{eq}}^{2}=\displaystyle \frac{{\hslash }}{2\pi m{\Omega }_{{\rm{m}}}}{\displaystyle \int }_{{\mathbb{R}}}{S}_{q}(\nu ){\rm{d}}\nu ,\qquad {\langle {p}^{2}\rangle }_{\mathrm{eq}}=\displaystyle \frac{m{\hslash }{\Omega }_{{\rm{m}}}}{2\pi }{\displaystyle \int }_{{\mathbb{R}}}{S}_{p}(\nu ){\rm{d}}\nu ,\end{eqnarray} \tag{ 103 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{\langle \{q,p\}\rangle }_{\mathrm{eq}}=\displaystyle \frac{{\hslash }}{2\pi }{\displaystyle \int }_{{\mathbb{R}}}{S}_{{qp}}(\nu ){\rm{d}}\nu .\end{eqnarray} \tag{ 104 }$$

All these quantities are finite since the integrands behave as ${\nu }^{-2}$ for $| \nu | \to +\infty$. Moreover, the reduced equilibrium state of the mechanical oscillator is a Gaussian state characterized by (103) and (104) together with ${\langle q\rangle }_{\mathrm{eq}}={\hslash }{g}_{0}{| \zeta | }^{2}/(m{\Omega }_{{\rm{m}}}^{\;2})$, ${\langle p\rangle }_{\mathrm{eq}}=0$.

On the contrary the integral of ${\nu }^{2}{S}_{q}(\nu )$, which would give the fluctuations at equilibrium of $\sqrt{m{\Omega }_{{\rm{m}}}/{\hslash }}\;\dot{q}$, does not exist. This fact is related to the features of the noise in the thermal part and, as already noticed right before section 3.3.1, this noticeably implies that the standard identification of $m\dot{q}$ with momentum is not possible. The expression of ${S}_{q}(\nu )$ coincides with the one given in [1, 25], where however ${\hat{R}}_{{GZ}}(\nu )$ with Ohmic spectral density appears instead of $\hat{R}(\nu )$. While in the case of [1, 25] ${S}_{q}(\nu )\;\asymp \;{\nu }^{-3}$, still ${\dot{q}}^{2}$ does not have a finite mean and also in this case the identification of momentum and velocity is not possible. Notice that the expressions for ${S}_{p}(\nu )$ and ${S}_{{qp}}(\nu )$ have not been obtained before. In particular ${S}_{{qp}}(\nu )\ne 0$ implies that the fluctuations of position and momentum are actually correlated.

The mean energy of the harmonic oscillator at equilibrium takes the form

$$\begin{eqnarray}&&{\langle {H}_{{\rm{m}}}\rangle }_{\mathrm{eq}}=\displaystyle \frac{1}{2}\;m{\Omega }_{{\rm{m}}}^{\;2}{\langle q\rangle }_{\mathrm{eq}}^{2}+{\langle H\rangle }_{\mathrm{fl}},\end{eqnarray} \tag{ 105 }$$

where the contribution due to fluctuations is given by

$$\begin{eqnarray}&&{\langle H\rangle }_{\mathrm{fl}}=\displaystyle \frac{{\hslash }}{4\pi }{\displaystyle \int }_{{\mathbb{R}}}{\rm{d}}\nu [{\Omega }_{{\rm{m}}}({S}_{q}(\nu )+{S}_{p}(\nu ))+{\gamma }_{{\rm{m}}}{S}_{{qp}}(\nu )].\end{eqnarray} \tag{ 106 }$$

It is convenient and natural to split this contribution into three distinct terms, distinguishing a radiation pressure term from the rest and further dividing the thermal contributions into two, putting into evidence a contribution which is not proportional to position fluctuations and does not have a definite sign. We thus introduce the dimensionless quantities

$$\begin{eqnarray}&&{{\mathcal{N}}}_{\mathrm{rp}}=\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{{\mathbb{R}}}\displaystyle \frac{{\Omega }_{{\rm{m}}}^{\;2}+{\nu }^{2}}{2{\omega }_{{\rm{m}}}{\Omega }_{{\rm{m}}}}\;{S}_{q}^{\mathrm{rp}}(\nu ){\rm{d}}\nu ,\qquad {{\mathcal{N}}}_{\mathrm{th}}=\displaystyle \frac{1}{2\pi }{\displaystyle \int }_{{\mathbb{R}}}\displaystyle \frac{{\omega }_{{\rm{m}}}}{{\Omega }_{{\rm{m}}}}\;{S}_{q}^{\mathrm{th}}(\nu ){\rm{d}}\nu ,\end{eqnarray} \tag{ 107 }$$

as well as

$$\begin{eqnarray}{{\mathcal{M}}}_{\mathrm{th}}(\Delta ) & = & \displaystyle \frac{1}{2\pi }{\displaystyle \int }_{{\mathbb{R}}}{\rm{d}}\nu \;\displaystyle \frac{{G}^{2}{\gamma }_{{\rm{m}}}\Delta }{2{| d(\nu )| }^{2}}\{[\displaystyle \frac{1}{2}\;{G}^{2}\Delta -\nu \;\displaystyle \frac{{\gamma }_{{\rm{c}}}{\gamma }_{{\rm{m}}}}{2}+({\omega }_{{\rm{m}}}+\nu )\\ & & \times ({\nu }^{2}-{\Delta }^{2}-\displaystyle \frac{{\gamma }_{{\rm{c}}}^{\;2}}{4})](N(\nu )+\displaystyle \frac{1}{2})+(\nu \to -\nu )\},\end{eqnarray} \tag{ 108 }$$

so that the fluctuation contribution can be written as

$$\begin{eqnarray}&&{\langle H\rangle }_{\mathrm{fl}}={\hslash }{\omega }_{{\rm{m}}}({{\mathcal{N}}}_{\mathrm{rp}}+{{\mathcal{N}}}_{\mathrm{th}}+{{\mathcal{M}}}_{\mathrm{th}}(\Delta ));\end{eqnarray} \tag{ 109 }$$

by construction we have ${{\mathcal{N}}}_{\mathrm{rp}}+{{\mathcal{N}}}_{\mathrm{th}}+{{\mathcal{M}}}_{\mathrm{th}}(\Delta )\geqslant 1/2$. As it appears, the mean energy density cannot be obtained from the knowledge of S_q alone, but extra terms are present. Moreover, the contribution proportional to ${{\mathcal{M}}}_{\mathrm{th}}(\Delta )$ can be negative. Depending on the parameter values, the extra terms can be actually quite small. It is important to stress that the given expression for the mean energy of the resonator holds for any temperature of the phonon bath, including the case of zero temperature.

We further stress that there is not strict energy equipartition. This can be expected since the mechanical oscillator is coupled to the cavity through its position and also the counter-rotating terms contribute to the final result. In the thermal part the lack of equipartition is due to the terms proportional to Δ, which are present in ${S}_{p}^{\mathrm{th}}(\nu )$ and not in ${S}_{q}^{\mathrm{th}}(\nu )$. In the radiation pressure part the term with ${\Omega }_{{\rm{m}}}^{\;2}$ comes from the position and the one with ${\nu }^{2}$ comes from the momentum and give different contributions to the mean energy.

For a vanishing effective detuning $\Delta =0$ all the computations can be performed analytically. The second thermal contribution ${{\mathcal{M}}}_{\mathrm{th}}(\Delta )$ vanishes and the coupling constant takes the value ${G}^{2}=8{\hslash }{g}_{0}^{\;2}{E}^{2}/(m{\omega }_{{\rm{m}}}{\gamma }_{{\rm{c}}}^{\;2})$. For the spectra of the fluctuations the explicit expressions reduce to

$$\begin{eqnarray}{S}_{q}^{\mathrm{rp}}(\nu ) & = & \displaystyle \frac{{\Omega }_{{\rm{m}}}{\omega }_{{\rm{m}}}{G}^{2}{\gamma }_{{\rm{c}}}}{2({\nu }^{2}+\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4})[{(\nu -{\omega }_{{\rm{m}}})}^{2}+\frac{{\gamma }_{{\rm{m}}}^{\;2}}{4}][{(\nu +{\omega }_{{\rm{m}}})}^{2}+\frac{{\gamma }_{{\rm{m}}}^{\;2}}{4}]},\\ {S}_{q}^{\mathrm{th}}(\nu ) & = & \displaystyle \frac{{\Omega }_{{\rm{m}}}{\gamma }_{{\rm{m}}}}{2{\omega }_{{\rm{m}}}}[\displaystyle \frac{N(\nu )+\frac{1}{2}}{{(\nu -{\omega }_{{\rm{m}}})}^{2}+\frac{{\gamma }_{{\rm{m}}}^{\;2}}{4}}+(\nu \to -\nu )],\end{eqnarray} \tag{ 110 }$$

leading upon integration to

$$\begin{eqnarray*}\displaystyle \frac{1}{2}\;m{\Omega }_{{\rm{m}}}^{\;2}({\langle {q}^{2}\rangle }_{\mathrm{eq}}-{\langle q\rangle }_{\mathrm{eq}}^{\ 2}) & = & \displaystyle \frac{{\hslash }{\Omega }_{{\rm{m}}}^{\;2}}{4{\omega }_{{\rm{m}}}}(2{N}_{\mathrm{eff}}+1)+\displaystyle \frac{{\hslash }{\omega }_{{\rm{m}}}{G}^{2}(2{\gamma }_{{\rm{m}}}+{\gamma }_{{\rm{c}}})}{8{\gamma }_{{\rm{m}}}(\frac{{({\gamma }_{{\rm{m}}}+{\gamma }_{{\rm{c}}})}^{2}}{4}+{\omega }_{{\rm{m}}}^{\;2})},\\ \displaystyle \frac{1}{2m}\;{\langle {p}^{2}\rangle }_{\mathrm{eq}} & = & \displaystyle \frac{{\hslash }{\Omega }_{{\rm{m}}}^{\;2}}{4{\omega }_{{\rm{m}}}}(2{N}_{\mathrm{eff}}+1)+\displaystyle \frac{{\hslash }{\omega }_{{\rm{m}}}{G}^{2}{\gamma }_{{\rm{c}}}}{8{\gamma }_{{\rm{m}}}(\frac{{({\gamma }_{{\rm{m}}}+{\gamma }_{{\rm{c}}})}^{2}}{4}+{\omega }_{{\rm{m}}}^{\;2})},\\ \displaystyle \frac{{\gamma }_{{\rm{m}}}}{4}\;{\langle \{q,p\}\rangle }_{\mathrm{eq}} & = & -\displaystyle \frac{{\hslash }{\gamma }_{{\rm{m}}}^{\;2}}{8{\omega }_{{\rm{m}}}}(2{N}_{\mathrm{eff}}+1),\end{eqnarray*}$$

with ${N}_{\mathrm{eff}}$ as in (44). These expressions show that equipartition of the mean energy is not valid just due to the radiation pressure contributions. However equipartition approximately holds for ${\gamma }_{{\rm{c}}}\gg 2{\gamma }_{{\rm{m}}}$, which is the case typically considered in many theoretical studies and experiments. We further have for the fluctuation contributions to the mean energy

$$\begin{eqnarray*}&&{{\mathcal{N}}}_{\mathrm{rp}}=\displaystyle \frac{{G}^{2}({\gamma }_{{\rm{m}}}+{\gamma }_{{\rm{c}}})}{4{\gamma }_{{\rm{m}}}(\frac{{({\gamma }_{{\rm{m}}}+{\gamma }_{{\rm{c}}})}^{2}}{4}+{\omega }_{{\rm{m}}}^{\;2})},\qquad {{\mathcal{N}}}_{\mathrm{th}}={N}_{\mathrm{eff}}+\displaystyle \frac{1}{2},\qquad {{\mathcal{M}}}_{\mathrm{th}}(\Delta )=0.\end{eqnarray*}$$

The mean equilibrium energy of the mechanical oscillator is thus increased due to the interaction with the cavity as a consequence of the presence of the strong laser in resonance. For the values considered in figure 1 we have ${{\mathcal{N}}}_{\mathrm{rp}}\simeq 1.6\times {10}^{4}$ corresponding to a temperature of about 7.9 K.

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As discussed in many papers [1, 3, 5, 6, 53], an important effect which can be described by the quantum models of cavity optomechanics is the laser cooling of the mechanical resonator. Since, as already discussed, we cannot expect equipartition of the mean mechanical energy, we cannot speak of temperature in a strict sense. A natural way to speak about laser cooling is the comparison of the mean energy of the fluctuations of the mechanical oscillator in the presence or the absence of the stimulating laser (corresponding to $\zeta =0$). So, we have to study the value of the fluctuation contribution (109) and to compare it to its value for $\zeta =0$, which is given by ${\langle H\rangle }_{\mathrm{fl}}{| }_{\zeta =0}$ $={\langle {H}_{{\rm{m}}}\rangle }_{\mathrm{eq}}{| }_{\zeta =0}$ $={\hslash }{\omega }_{{\rm{m}}}({N}_{\mathrm{eff}}+\frac{1}{2})$.

To obtain explicit analytical formulae for the mean energy we consider the case of a constant noise spectrum, that is $N(\nu )=\mathrm{const}={{\rm{N}}}_{\mathrm{eff}}$. To actually perform the calculations we need the expressions of the zeros of $d(\nu )$; here we consider the generic case given by the Ansatz (91). By lengthy computations the integrals over ν can be exactly performed, leading to involved formulae explicitly given in appendix A.2. In order to describe cooling effects the relevant contributions can be written in the form

$$\begin{eqnarray}&&{{\mathcal{N}}}_{\mathrm{th}}=\displaystyle \frac{{\gamma }_{{\rm{m}}}}{{\Gamma }_{{\rm{m}}}}\;{\mathcal{Q}}({N}_{\mathrm{eff}}+\displaystyle \frac{1}{2}),\qquad {{\mathcal{M}}}_{\mathrm{th}}(\Delta )=\displaystyle \frac{{\gamma }_{{\rm{m}}}}{{\Gamma }_{{\rm{m}}}}\;{\mathcal{K}}({N}_{\mathrm{eff}}+\displaystyle \frac{1}{2}),\end{eqnarray} \tag{ 111 }$$

where the quantities ${\mathcal{Q}}$ and ${\mathcal{K}}$ are given in equations (A.16) and (A.17). The expression for ${{\mathcal{N}}}_{\mathrm{rp}}$ is given in (A.15). Note that, while ${\mathcal{Q}}$ is always positive, depending on the values of the parameters the quantity ${\mathcal{K}}$ can be also negative. For a large choice of the parameters ${\mathcal{Q}}$ turns out to be close to 1.

In the following figures we describe the effective cooling of the mechanical oscillator, by considering as a figure of merit the cooling factor

$$\begin{eqnarray}&&{\mathcal{C}}=\displaystyle \frac{{\gamma }_{{\rm{m}}}}{{\Gamma }_{{\rm{m}}}}({\mathcal{Q}}+{\mathcal{K}}).\end{eqnarray} \tag{ 112 }$$

We study two cases, corresponding to the parameter regions for which an exact or approximate analytic evaluation of the different contributions to the mean energy has been provided. In both cases mass and bare frequency of the mechanic oscillator are taken to be $m=2.5\times {10}^{-10}$ kg and ${\Omega }_{{\rm{m}}}=2\pi \times {10}^{7}$ Hz, while the mechanical damping factor is ${\gamma }_{{\rm{m}}}=2\pi \times {10}^{2}$ Hz. We consider a cavity of length $5\times {10}^{-4}$ m and resonance frequency ${\omega }_{{\rm{c}}}=2\pi c/(1064\times {10}^{-9})$ Hz, driven by a laser with a power of $5\times {10}^{-2}$ W. For the sake of comparison the values of the fixed parameters are taken from [25].

We start by exploring the case $\Delta ={\omega }_{{\rm{m}}}$, studied in section 4.2.1, in which the location of the poles can be evaluated exactly, provided one distinguishes two regions according to the value of the ratio ${({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}/4{G}^{2}$. No approximation is taken in the expression of the integrals giving the mean energy. If this ratio is above one, verified for a cavity damping ${\gamma }_{{\rm{c}}}\gt {\bar{\gamma }}_{{\rm{c}}}$, where the critical damping ${\bar{\gamma }}_{{\rm{c}}}$ is introduced in the comments after (99) and corresponds for the considered parameters to ${\bar{\gamma }}_{{\rm{c}}}\simeq 4.1\times {10}^{7}$, the effective damping rates ${\Gamma }_{{\rm{m}}}$ (93) and ${\Gamma }_{{\rm{c}}}$ (94) are actually distinct, while ${\Delta }_{\mathrm{eff}}={\omega }_{\mathrm{eff}}^{{\rm{m}}}$ and their expression is given by (95). By numerical computations we see that the cooling factor ${\mathcal{C}}$ is a monotonic increasing function of the cavity damping rate ${\gamma }_{{\rm{c}}},$ and around the starting point ${\bar{\gamma }}_{{\rm{c}}}$ the cooling factor takes the value $2.9\times {10}^{-5}$. In the complementary region, corresponding to ${({\gamma }_{{\rm{c}}}-{\gamma }_{{\rm{m}}})}^{2}/4{G}^{2}$ below one, the cooling factor is a decreasing function of the cavity damping rate, so that the optimal cooling is obtained for ${\gamma }_{{\rm{c}}}={\bar{\gamma }}_{{\rm{c}}}$. In this region, corresponding to ${\gamma }_{{\rm{c}}}\lt {\bar{\gamma }}_{{\rm{c}}}$, we have ${\Gamma }_{{\rm{m}}}={\Gamma }_{{\rm{c}}}$ with value given in (98), while the effective frequencies ${\Delta }_{\mathrm{eff}}$ and ${\omega }_{\mathrm{eff}}^{{\rm{m}}}$ are given by the expressions (98). To assess the relevance of the various contributions in (112) we report the values for ${\gamma }_{{\rm{c}}}={\bar{\gamma }}_{{\rm{c}}}$: we have ${\gamma }_{{\rm{m}}}/{\Gamma }_{{\rm{m}}}\simeq 3.05\times {10}^{-5}$, ${\mathcal{Q}}\simeq .997$ and ${\mathcal{K}}\simeq -4.18\times {10}^{-2}$; then, (112) gives ${\mathcal{C}}\simeq 2.91\times {10}^{-5}$, which is a very strong cooling factor.

Instead, in figure 1 we consider the case ${\gamma }_{{\rm{c}}}\gg {\omega }_{{\rm{m}}}$, that is a cavity damping much bigger than the mechanical oscillator frequency. In the exact formulae for the integrals we use the approximate expressions for ${\Gamma }_{{\rm{m}}}$ and ${\Gamma }_{{\rm{c}}}$ given in (102), relying on the conditions (100). The stationary value of the energy of the mechanical system has a marked dependence on the effective detuning Δ and the optimal cooling region, corresponding to ${\mathcal{C}}$ of the order of 10⁻³, is obtained for $\Delta \lesssim {\gamma }_{{\rm{c}}}$. In this parameter region ${{\mathcal{N}}}_{\mathrm{rp}}$ can be neglected with respect to ${\mathcal{C}}{N}_{\mathrm{eff}}$, unless the phonon bath is below 1 K, so that indeed the quantity ${\mathcal{C}}$ given in (112) properly describes the cooling effect. When the detuning Δ goes to zero the cooling factor rapidly increases in agreement with the discussion in section 4.2.3 showing the presence of heating at $\Delta =0$; in this parameter region, the cooling effect disappears also when Δ grows.

The case of a perfect coherent monochromatic local oscillator of frequency ${\omega }_{0}$ with detection of the whole emitted light [49, 55] corresponds to the continuous measurement of a field quadrature of the type

$$\begin{eqnarray}&&Q(t;\vartheta )={\mathrm{ie}}^{-{\rm{i}}\vartheta }{{\rm{e}}}^{\mathrm{iarg}\zeta }{\displaystyle \int }_{0}^{t}{{\rm{e}}}^{-{\rm{i}}{\omega }_{0}r}{\rm{d}}{B}_{\mathrm{em}}^{\ \dagger }(r)+{\rm{h}}.{\rm{c}}.;\end{eqnarray} \tag{ 113 }$$

ϑ is a free parameter which depends on the optical path and determines the observed quadrature. As a consequence of the definition we have that $[Q(t;\vartheta ),Q(s;\vartheta )]=0$. By the properties of the output fields, discussed after equation (33), the commutation rules are preserved; this gives that the output current ${Q}^{\mathrm{out}}(t;\vartheta ):= U{(t)}^{\dagger }Q(t;\vartheta )U(t)$ satisfies $[{Q}^{\mathrm{out}}(t;\vartheta ),{Q}^{\mathrm{out}}(s;\vartheta )]=0$. This is the key property expressing the fact that ${Q}^{\mathrm{out}}(t;\vartheta )$ can be measured with continuity in time. Similarly to (80) we introduce the gated Fourier transforms

$$\begin{eqnarray}&&{Q}_{T}(\nu ;\vartheta )=\displaystyle \frac{1}{\sqrt{T}}{\displaystyle \int }_{0}^{T}{{\rm{e}}}^{{\rm{i}}\nu t}{\rm{d}}Q(t;\vartheta ),\qquad {Q}_{T}^{\mathrm{out}}(\nu ;\vartheta )=\displaystyle \frac{1}{\sqrt{T}}{\displaystyle \int }_{0}^{T}{{\rm{e}}}^{{\rm{i}}\nu t}{\rm{d}}{Q}^{\mathrm{out}}(t;\vartheta ).\end{eqnarray} \tag{ 114 }$$

From the above relations we obtain the second key relation which guarantees the presence of commuting observables and therefore the consistency of the theory:

$$\begin{eqnarray}&&[{Q}_{T}^{\mathrm{out}}(\nu ;\vartheta ),{Q}_{T}^{\mathrm{out}}({\nu }^{\prime };\vartheta )]=0.\end{eqnarray} \tag{ 115 }$$

The homodyne spectrum is then given by the expression

$$\begin{eqnarray}&&S(\nu ;\vartheta )=\underset{T\to +\infty }{\mathrm{lim}}\mathrm{Tr}\{{Q}_{T}^{\mathrm{out}}(-\nu ;\vartheta ){Q}_{T}^{\mathrm{out}}(\nu ;\vartheta ){\rho }_{0}\otimes \tilde{{\sigma }_{\mathrm{env}}}\},\end{eqnarray} \tag{ 116 }$$

where the environmental state is given by (62) and ${\rho }_{0}$ is any initial state for the mechanical oscillator and the cavity mode. Note that this expression is nothing but the spectrum of the classical stochastic process representing the output, and not an ad hoc quantum definition [49 section 4]. The commutation property (115) implies that the homodyne spectrum $S(\nu ;\vartheta )$ is an even function of ν.

As shown in appendix B.1, the homodyne spectrum has both an elastic and an inelastic component

$$\begin{eqnarray}&&S(\nu ;\vartheta )={S}_{\mathrm{el}}(\nu ;\vartheta )+{S}_{\mathrm{inel}}(\nu ;\vartheta ),\end{eqnarray} \tag{ 117 }$$

which turn out to have the expressions

$$\begin{eqnarray}{S}_{\mathrm{el}}(\nu ;\vartheta ) & = & 8\pi {\gamma }_{{\rm{c}}}{| \zeta | }^{2}{(\mathrm{sin}\vartheta )}^{2}\delta (\nu ),\\ {S}_{\mathrm{inel}}(\nu ;\vartheta ) & = & {S}_{\mathrm{th}}(\nu ;\vartheta )+{S}_{\mathrm{rp}}(\nu ;\vartheta ),\end{eqnarray} \tag{ 118 }$$

with

$$\begin{eqnarray}&&{S}_{\mathrm{th}}(\nu ;\vartheta )=\displaystyle \frac{2{\gamma }_{{\rm{c}}}{\omega }_{{\rm{m}}}{G}^{2}[{(\frac{{\gamma }_{{\rm{c}}}}{2}\;\mathrm{cos}\vartheta +\Delta \mathrm{sin}\vartheta )}^{2}+{(\nu \mathrm{cos}\vartheta )}^{2}]}{{\Omega }_{{\rm{m}}}(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\Delta -\nu )}^{2})(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\Delta +\nu )}^{2})}\;{S}_{q}^{\mathrm{th}}(\nu ),\end{eqnarray} \tag{ 119 }$$

$$\begin{eqnarray}{S}_{\mathrm{rp}}(\nu ;\vartheta ) & = & 1+\displaystyle \frac{2{\gamma }_{{\rm{c}}}{\omega }_{{\rm{m}}}{G}^{2}[{(\frac{{\gamma }_{{\rm{c}}}}{2}\;\mathrm{cos}\vartheta +\Delta \mathrm{sin}\vartheta )}^{2}+{(\nu \mathrm{cos}\vartheta )}^{2}]}{{\Omega }_{{\rm{m}}}(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\Delta -\nu )}^{2})(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\Delta +\nu )}^{2})}\;{S}_{q}^{\mathrm{rp}}(\nu )\\ & & +{\gamma }_{{\rm{c}}}{\omega }_{{\rm{m}}}{G}^{2}\;\mathrm{Re}\;[\displaystyle \frac{(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{\nu }^{2}-{\Delta }^{2})\mathrm{sin}2\vartheta -\Delta ({\gamma }_{{\rm{c}}}\mathrm{cos}2\vartheta -2{\rm{i}}\nu )}{d(\nu )(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\Delta -\nu )}^{2})(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\Delta +\nu )}^{2})}\\ & & \times (\displaystyle \frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}-{\nu }^{2}+{\Delta }^{2}-{\rm{i}}{\gamma }_{{\rm{c}}}\nu )\phantom{\displaystyle \frac{(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{\nu }^{2}-{\Delta }^{2})\mathrm{sin}2\vartheta -\Delta ({\gamma }_{{\rm{c}}}\mathrm{cos}2\vartheta -2{\rm{i}}\nu )}{d(\nu )(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\Delta -\nu )}^{2})(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\Delta +\nu )}^{2})}}].\end{eqnarray} \tag{ 120 }$$

Note that all the contributions are indeed positive as shown in appendix B.1. It is important to stress that the connection between ${S}_{q}(\nu )$ and ${S}_{\mathrm{inel}}(\nu ;\vartheta )$ is far from simple. In particular the last contribution in (120) comes from the interference of the electromagnetic part of the signal with the shot noise, as detailed in appendix B.1. This is a completely new term, in principle detectable in experiments at very low temperatures.

Let us further stress that different quadratures are incompatible and actually one can prove the general inequality [49, 55]

$$\begin{eqnarray}&&{S}_{\mathrm{inel}}(\nu ;\vartheta ){S}_{\mathrm{inel}}(\nu ;\vartheta \pm \pi /2)\geqslant 1,\end{eqnarray} \tag{ 121 }$$

which is just a form of the Heisenberg–Robertson uncertainty relations coming from the canonical commutation relations of the involved Bose fields. As a result quite different physical information can be extracted from the different quadratures.

The case $\Delta =0$. The first striking example of strong dependence on ϑ is in the case $\Delta =0$. For $\vartheta =\pi /2$ we get ${S}_{\mathrm{el}}(\nu ;\pi /2)=32\pi {E}^{2}\delta (\nu )/{\gamma }_{{\rm{c}}}$ and ${S}_{\mathrm{inel}}(\nu ;\pi /2)=1$: only the shot noise contributes to the inelastic spectrum.

On the contrary, for the quadrature with $\vartheta =0$ we get ${S}_{\mathrm{el}}(\nu ;\pi /2)=0$ and

$$\begin{eqnarray}&&{S}_{\mathrm{inel}}(\nu ;0)=1+\displaystyle \frac{2{\gamma }_{{\rm{c}}}{\omega }_{{\rm{m}}}{G}^{2}}{{\Omega }_{{\rm{m}}}(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{\nu }^{2})}\;{S}_{q}(\nu ),\end{eqnarray} \tag{ 122 }$$

where ${S}_{q}(\nu )$ is now explicitly given by (110). An important point is that in this case the interference term vanishes exactly and we have a direct connection of the homodyne spectrum with the fluctuation spectrum of the position of the mirror. This result has been found also in [37], but with the substitution $\hat{R}(\nu )\to {\hat{R}}_{{GZ}}(\nu )$ in the expression (86) for ${S}_{q}^{\mathrm{th}}(\nu ){| }_{\Delta =0}$. As a result, at least in principle, when $\Delta =0$ the homodyne observation of the quadrature with $\vartheta =\pi /2$ can give direct experimental information on the correct expression for $\hat{R}(\nu )$. At zero temperature one could experimentally discriminate between our result (58) and the standard proposal (59).

In other cases the interference term does not vanish, but it can be negligible at high temperatures. For instance, when the interference term is negligible, at least in the region where $N(\nu )\gg 1$, we recover for ${S}_{\mathrm{inel}}(\nu ;0)$ the result given in [1 section 3]. At high temperatures the inelastic homodyne spectrum allows to reconstruct the fluctuation spectrum of position, while no direct information on the fluctuation of the momentum and on the cross-correlation is obtained. Moreover, at high temperatures we have also ${S}_{q}(\nu )\simeq {S}_{q}^{\mathrm{th}}(\nu )$; by using the explicit expressions of ${S}_{q}^{\mathrm{th}}(\nu )$ (86) and $\hat{R}(\nu )$ (54) we get

$$\begin{eqnarray*}&&{S}_{\mathrm{inel}}(\nu ;0)\simeq {\gamma }_{{\rm{c}}}{\gamma }_{{\rm{m}}}{G}^{2}\;\displaystyle \frac{(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{\nu }^{2})(\frac{{\gamma }_{{\rm{m}}}^{\;2}}{4}+{({\omega }_{{\rm{m}}}+\nu )}^{2})}{{| d(\nu )| }^{2}}(N(\nu )+\displaystyle \frac{1}{2})+(\nu \to -\nu ).\end{eqnarray*}$$

This expression highlights the dependence of the homodyne spectrum on the thermal spectrum $N(\nu )$ and the characteristic polynomial $d(\nu )$ (89) of the dynamical matrix (73) of the full optomechanical system.

Squeezing. An important information about the non-classical nature of the light generated by optomechanical systems can be obtained considering the quadrature with $\vartheta =-\pi /4$. In the simple case of vanishing detuning $\Delta =0$ and vanishing temperature $N(\nu )\equiv 0$, it is possible to show from (117)–(120) that we have ${S}_{\mathrm{inel}}(0;-\pi /4)\lt 1$, at least in a certain region of the parameters. This means that in a neighbourhood of $\nu =0$ we have ${S}_{\mathrm{inel}}(\nu ;-\pi /4)\lt 1$ and the emitted light is squeezed. This result shows that such a kind of optomechanical systems can generate non-classical light [3, 7]. Note that, if light squeezing is present for certain values of the parameters, then the inequality (121) implies that the complementary quadrature is antisqueezed. Of course, experimentally it could be difficult to tune the values of the various free parameters in order to have squeezing; moreover, the elastic peak in the spectrum tends to hide the squeezing around $\nu =0$ in the inelastic spectrum.

In the case of heterodyne detection the local oscillator and the stimulating light are produced by different laser sources; now, the stimulating laser frequency ${\omega }_{0}$ and the local oscillator frequency, say μ, are in general different. Moreover, the phase difference cannot be maintained stable and this erases some interference terms. It can be shown [30, 50 section 3.5] that the balanced heterodyne detection scheme corresponds to the measurement in continuous time of the observables

$$\begin{eqnarray}&&I(\mu ;t)={\displaystyle \int }_{0}^{t}\sqrt{\varkappa }\;{{\rm{e}}}^{-\varkappa (t-s)/2}\;{{\rm{e}}}^{{\rm{i}}\mu s+{\rm{i}}\alpha }\;{\rm{d}}{B}_{\mathrm{em}}(s)+{\rm{h}}.{\rm{c}}.,\end{eqnarray} \tag{ 123 }$$

where α is a phase depending on the optical paths and $\sqrt{\varkappa }{{\rm{e}}}^{-\varkappa t/2}$, $\varkappa \gt 0$, represents the detector response function. As we shall see, the heterodyne spectrum does not depend on α. In the Heisenberg description the observables become the 'output current'

$$\begin{eqnarray*}{I}_{\mathrm{out}}(\mu ;t) & = & U{(t)}^{\dagger }I(\mu ;t)U(t)\\ & = & \sqrt{\varkappa }{\displaystyle \int }_{0}^{t}{{\rm{e}}}^{-\frac{\varkappa }{2}(t-s)+{\rm{i}}\alpha }({{\rm{e}}}^{{\rm{i}}\mu s}{\rm{d}}{B}_{\mathrm{em}}(s)+\sqrt{{\gamma }_{{\rm{c}}}}\;{{\rm{e}}}^{{\rm{i}}(\mu -{\omega }_{0})s}{a}_{{\rm{c}}}(s){\rm{d}}s)+{\rm{h}}.{\rm{c}}.\end{eqnarray*}$$

By the definition of $I(\mu ;t)$ and the properties of U(t) we get $[{I}_{\mathrm{out}}(\mu ;t),{I}_{\mathrm{out}}(\mu ;s)]=0$, which says that the output current at time t and the current at time s are compatible observables.

While in the homodyne scheme the spectrum of the output is analysed, in the heterodyne scheme it is usual to register only the output power as a function of the frequency μ of the local oscillator. The mean output power of the detection apparatus at large times is proportional to

$$\begin{eqnarray}&&P(\mu )=\underset{T\to +\infty }{\mathrm{lim}}\displaystyle \frac{1}{T}{\displaystyle \int }_{0}^{T}{\rm{d}}t\;\mathrm{Tr}\{{I}_{\mathrm{out}}{(\mu ;t)}^{2}{\rho }_{0}\otimes \tilde{{\sigma }_{\mathrm{env}}}\};\end{eqnarray} \tag{ 124 }$$

the limit is in the sense of the distributions in μ. As a function of μ, $P(\mu )$ is known as power spectrum. Note that to change μ means to change the frequency of the local oscillator, that is to change the measuring apparatus. In general ${I}_{\mathrm{out}}(\mu ;t)$ and ${I}_{\mathrm{out}}({\mu }^{\prime };s)$ do not commute, even for t = s. Then, there is no reason for the power spectrum to have some symmetry in μ. The heterodyne power spectrum can be decomposed in an elastic and an inelastic part

$$\begin{eqnarray}&&P(\mu )={\Sigma }_{\mathrm{el}}(\mu )+{\Sigma }_{\mathrm{inel}}(\mu ),\end{eqnarray} \tag{ 125 }$$

$$\begin{eqnarray}{\Sigma }_{\mathrm{el}}(\mu ) & = & \underset{T\to +\infty }{\mathrm{lim}}\displaystyle \frac{\varkappa {\gamma }_{{\rm{c}}}}{T}{\displaystyle \int }_{0}^{T}{\rm{d}}t{[2\;\mathrm{Re}\;(\zeta {{\rm{e}}}^{{\rm{i}}\alpha }{\displaystyle \int }_{0}^{t}{{\rm{e}}}^{-\frac{\varkappa }{2}(t-s)+{\rm{i}}(\mu -{\omega }_{0})s}{\rm{d}}s)]}^{2}\\ & = & \displaystyle \frac{\varkappa {\gamma }_{{\rm{c}}}{| \zeta | }^{2}}{\frac{{\varkappa }^{2}}{4}+{(\mu -{\omega }_{0})}^{2}}\stackrel{\varkappa \downarrow 0}{\longrightarrow }4\pi {\gamma }_{{\rm{c}}}{| \zeta | }^{2}\;\delta (\mu -{\omega }_{0}),\end{eqnarray} \tag{ 126 }$$

$$\begin{eqnarray}&&{\Sigma }_{\mathrm{inel}}(\mu )=\underset{T\to +\infty }{\mathrm{lim}}\displaystyle \frac{1}{T}{\displaystyle \int }_{0}^{T}{\rm{d}}t\;\mathrm{Tr}\{{I}_{\mathrm{inel}}{(\mu ;t)}^{2}{\rho }_{0}\otimes \tilde{{\sigma }_{\mathrm{env}}}\},\end{eqnarray} \tag{ 127 }$$

where

$$\begin{eqnarray*}{I}_{\mathrm{inel}}(\mu ;t) & = & \sqrt{\varkappa }{\displaystyle \int }_{0}^{t}{{\rm{e}}}^{-\frac{\varkappa }{2}(t-s)}({{\rm{e}}}^{{\rm{i}}\mu s+{\rm{i}}\alpha }{\rm{d}}{B}_{\mathrm{em}}(s)\\ & & +\sqrt{\displaystyle \frac{{\gamma }_{{\rm{c}}}}{2}}{{\rm{e}}}^{{\rm{i}}(\mu -{\omega }_{0})s+{\rm{i}}\vartheta }(Y(s)-{\rm{i}}X(s)){\rm{d}}s)+{\rm{h}}.{\rm{c}}.\end{eqnarray*}$$

The inelastic part of the spectrum is computed in appendix B.2. Again it is possible to identify a radiation pressure contribution and a thermal part

$$\begin{eqnarray}&&{\Sigma }_{\mathrm{inel}}(\mu )={\Sigma }_{\mathrm{rp}}(\mu )+{\Sigma }_{\mathrm{th}}(\mu ).\end{eqnarray} \tag{ 128 }$$

For simplicity we give only the expressions for $\varkappa \downarrow 0$:

$$\begin{eqnarray}&&{\Sigma }_{\mathrm{rp}}(\mu )=1+\displaystyle \frac{{\gamma }_{{\rm{c}}}{\omega }_{{\rm{m}}}{G}^{2}{S}_{q}^{\mathrm{rp}}(\mu -{\omega }_{0})}{{\Omega }_{{\rm{m}}}(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\mu -{\omega }_{0}-\Delta )}^{2})}-\;\mathrm{Im}\;\displaystyle \frac{{\gamma }_{{\rm{c}}}{\omega }_{{\rm{m}}}{G}^{2}}{d(\mu -{\omega }_{0})}\;\displaystyle \frac{\frac{{\gamma }_{{\rm{c}}}}{2}-{\rm{i}}(\mu -{\omega }_{0}+\Delta )}{\frac{{\gamma }_{{\rm{c}}}}{2}+{\rm{i}}(\mu -{\omega }_{0}-\Delta )},\end{eqnarray} \tag{ 129 }$$

$$\begin{eqnarray}&&{\Sigma }_{\mathrm{th}}(\mu )=\displaystyle \frac{{\gamma }_{{\rm{c}}}{\omega }_{{\rm{m}}}{G}^{2}{S}_{q}^{\mathrm{th}}(\mu -{\omega }_{0})}{{\Omega }_{{\rm{m}}}(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\mu -{\omega }_{0}-\Delta )}^{2})}.\end{eqnarray} \tag{ 130 }$$

Both contributions are positive as it follows from the expressions (130) and (B.7). Note the presence of the interference term in (129). By simple computations one can check that

$$\begin{eqnarray}&&{\Sigma }_{\mathrm{inel}}(\nu +{\omega }_{0})+{\Sigma }_{\mathrm{inel}}({\omega }_{0}-\nu )={S}_{\mathrm{inel}}(\nu ;\vartheta )+{S}_{\mathrm{inel}}(\nu ;\vartheta +\pi /2);\end{eqnarray} \tag{ 131 }$$

this is a fundamental relation [56 equation (9.61)] connecting heterodyne and homodyne spectra. Moreover, by inserting the definitions of the relevant quantities given in (54), (85) and (86), an explicit expression for ${\Sigma }_{\mathrm{inel}}$ can be obtained from which it is apparent that ${\Sigma }_{\mathrm{inel}}(\mu )\gt 1$: in the heterodyne detection the phase dependencies are lost and it is impossible to detect squeezing in the emitted light.

As in the homodyne case, the interference term in (129) is negligible when $N\gg 1$ and we get

$$\begin{eqnarray}&&{\Sigma }_{\mathrm{inel}}(\mu )\simeq 1+\displaystyle \frac{{\gamma }_{{\rm{c}}}{\omega }_{{\rm{m}}}{G}^{2}}{{\Omega }_{{\rm{m}}}(\frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\mu -{\omega }_{0}-\Delta )}^{2})}\;{S}_{q}(\mu -{\omega }_{0}).\end{eqnarray} \tag{ 132 }$$

When this approximation holds, the inelastic heterodyne spectrum too allows to reconstruct the asymptotic dynamics of the mirror through the position fluctuations.

To explore the behaviour of the spectrum we take $N(\nu )$ as given by (56) with a Ohmic spectral density. Then, by using the explicit expressions of ${S}_{q}^{\mathrm{rp}}$ and ${S}_{q}^{\mathrm{th}}$ and by setting $\nu =\mu -{\omega }_{0}$, we get

$$\begin{eqnarray}{\Sigma }_{\mathrm{inel}}(\nu +{\omega }_{0}) & = & 1+\displaystyle \frac{{\gamma }_{{\rm{c}}}{G}^{2}}{2{| d(\nu )| }^{2}}\left\{\phantom{{\frac{\frac{a}{a}}{a}}^{a}}{\gamma }_{{\rm{c}}}{\omega }_{{\rm{m}}}^{\;2}{G}^{2}\right.\\ & & +{\gamma }_{{\rm{m}}}(\displaystyle \frac{{\gamma }_{{\rm{c}}}^{\;2}}{4}+{(\nu +\Delta )}^{2})[\displaystyle \frac{{\gamma }_{{\rm{m}}}^{\;2}}{4}+{(\nu -{\omega }_{{\rm{m}}})}^{2}+\displaystyle \frac{4{\omega }_{{\rm{m}}}| \nu | }{{{\rm{e}}}^{\beta {\hslash }| \nu | }-1}]\}.\end{eqnarray} \tag{ 133 }$$

From this expression we see that the main features of the spectrum will be determined by the zeros of the denominator ${| d(\nu )| }^{2}$; for instance, as discussed in section 4.2.1, for $\Delta ={\omega }_{{\rm{m}}}$ we can have one or two resonance frequencies depending on the value of the cavity decay rate ${\gamma }_{{\rm{c}}}$. In figure 2 we show this phenomenon: the two distinct peaks coalesce as ${\gamma }_{{\rm{c}}}$ increases. For these values of the parameters one can check that the main contribution to the inelastic heterodyne spectrum comes from the thermal part ${\Sigma }_{\mathrm{th}}$. It can be checked that in this parameter region the behaviour of the inelastic homodyne spectrum ${S}_{\mathrm{inel}}(\nu ;0)$ is very close to the heterodyne one as depicted in figure 2. Let us notice that the behaviour shown in figure 2 does not uncover the whole rich structure of the spectrum which appears by exploring other parameter regions.

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