Quantum Langevin equations for optomechanical systems

We provide a fully quantum description of a mechanical oscillator in the presence of thermal environmental noise by means of a quantum Langevin formulation based on quantum stochastic calculus. The system dynamics is determined by symmetry requirements and equipartition at equilibrium, while the environment is described by quantum Bose fields in a suitable non-Fock representation which allows for the introduction of temperature. A generic spectral density of the environment can be described by introducing its state trough a suitable P-representation. Including interaction of the mechanical oscillator with a cavity mode via radiation pressure we obtain a description of a simple optomechanical system in which, besides the Langevin equations for the system, one has the exact input-output relations for the quantum noises. The whole theory is valid at arbitrarily low temperature. This allows the exact calculation of the stationary value of the mean energy of the mechanical oscillator, as well as both homodyne and heterodyne spectra. The present analysis allows in particular to study possible cooling scenarios and to obtain the exact connection between observed spectra and fluctuation spectra of the position of the mechanical oscillator.


Introduction
Optomechanical systems in the quantum regime are very important for quantum information processing and for testing fundamental issues of quantum mechanics [1][2][3][4][5][6][7][8][9][10]. Their theoretical analysis therefore calls for a first principle description. In particular since the focus is on quantum effects, the theoretical models must be fully consistent with quantum mechanics. Actually the correct quantum description of a mesoscopic mechanical oscillator and of the thermal noise affecting it is not a trivial task, and there is not a unique accepted model for them [11][12][13][14][15][16][17][18][19].
The first aim of this paper is therefore to obtain an accurate quantum mechanical description of a mechanical oscillator taken to be part of an opto-mechanical device. The oscillator cannot be considered as a Brownian particle, but rather as a mesoscopic mechanical system, say a movable mirror mounted on a vibrating structure. Dissipative effects are essentially due to the interaction with phonons. Our strategy will be to introduce reasonable physical requirements leading to a master equation in Lindblad form, valid for any temperature of the thermal bath. We then translate these results into quantum Langevin equations and we show how to obtain a suitable non-Markovian generalization at this level of description. Relying on these results we can consider the description of the simplest optomechanical system, that is a moving mirror interacting with an electromagnetic mode in a cavity via radiation pressure [1,[5][6][7]20]. Again a suitable analysis of the composite system and of the monitoring of the emitted light calls for a consistent quantum description. We shall obtain this result by the use of quantum Langevin equations, directly deducing them from a unitary dynamics, and exploiting the theory of measurements in continuous time.
The paper is organized as follows. In Section 2 we determine the dynamics of the mechanical oscillator only. The basic assumption in performing this step is the use of a Markovian master equation with a quadratic generator and having a unique equilibrium state. Its structure is further determined by suitable symmetry requirements and by physical constraints on the behaviour of the mean values of position and momentum. In Section 3 we introduce the quantum Langevin equations for the mechanical oscillator alone, starting from the Markovian case and extending it to include memory effects in Section 3.2. The whole presentation is based on the notions of quantum noise [21,22] and of input-output fields [23][24][25], as well as on the use of quantum stochastic calculus [26,27]. As a result the possibility of a nonflat noise spectrum is accounted for within a consistent description of the quantum oscillator. The non-Markovian effects are introduced by modifying the state of the fields describing the noise in the overall unitary dynamics constructed by quantum stochastic calculus. This in contrast with standard approaches in which the quantum noises and their correlation properties are introduced in a phenomenological way. The difference is of relevance especially at zero temperature. A quantum optomechanical system is studied in Section 4 by using the quantum Langevin approach. Again a fully consistent quantum mechanical description is given, valid for any temperature. In such a new framework the typical effect of laser cooling is discussed. Then, the continuous monitoring of the emitted light is introduced in Sections 4.3.1 (homodyne detection) and 4.3.2 (heterodyne detection). The treatment is well based in the theory of measurements in continuous time. Detection of the emitted light is usually assumed to give a direct measurement of the fluctuations of the position of the mechanical component. We show that this is true, but only for not too low temperatures; at very small temperatures, interference terms are important and the direct connection with such fluctuations is lost. We finally summarize and discuss our results in Section 5.

Damped mechanical oscillator: the master equation approach
As a first step towards the construction of models of optomechanical systems valid in the quantum regime at low temperatures, we consider the reduced dynamics of an open mechanical oscillator. A fully consistent quantum description of a massive nanomechanical component, kept at the simplest possible level, will be our basic building block in order to consider more complex dynamics. We therefore formulate in the first instance a Markovian description for the mechanical oscillator, which we build up relying on general physical constraints and symmetry requirements. Our strategy will be to consider non-Markovian effects at the level of the quantum Langevin equations in Section 3.2, once a well defined stochastic framework for the quantum description of an open mechanical oscillator has been settled, fully consistent at any temperature.

Physical constraint and symmetry requirements
We formulate now our assumptions, starting from the existence of a well defined positive Markovian dynamics, describing damping and translationally invariant up to the harmonic potential. A weak equipartition condition and the existence of a unique stationary state in Gibbs form, as we shall see, will essentially fix the structure of the reduced dynamics.
Assumption 1 (Positive Markovian dynamics with quadratic generator). The evolution of the statistical operator of the oscillator is governed by a Markovian master equation preserving the positivity of the states. The generator of the dynamics is at most quadratic in the position and momentum operators of the mechanical oscillator.
As a first assumption we consider a time-homogeneous and linear time evolution. Such a dynamics can be expressed in the form d dt with L a suitable generator or Liouville operator, at most quadratic in the position and momentum operators of the mechanical oscillator q and p, so as to have at most a quadratic potential term and a friction effect proportional to the momentum of the mechanical oscillator. In the case of linear systems it is known that positivity and complete positivity of the dynamics are actually equivalent [11], therefore according to [28,29] the generator L must have the standard Lindblad structure. The most general quadratic Liouville operator is obtained in terms of two Lindblad operators [11] R j = 1 and a generic selfadjoint quadratic Hamiltonian for the mechanical system where all the constants are taken to be real, so that L takes the form Assumption 2 (Damping). The "kinetic energy" term is non negative and the mean values of position and momentum decay to zero with an oscillating behaviour.
Apart from the trivial requirement of a positive kinetic energy term, we further look for a dynamics describing the oscillating decay of the mean values of q and p to zero. This condition complies with the Markovian and quadratic approximations, which are expected to be good only for small damping. Denoting by X t the mean value of a quantum operator with the state ρ(t) solution of the master equation we have for position and momentum The eigenvalues of the associated dynamical matrix are − Im u|v ± κ 2 0 /4 − h p h q , so that in order to have an oscillating decaying dynamics rather than an overdamped one we need Im u|v > 0 and κ 2 0 /4 < h p h q . In particular h p and h q have the same sign and are non-vanishing. By asking the kinetic energy to be positive we get h p > 0, which also gives h q > 0. Then, we can write h p = 1/m and h q = mΩ 2 m . Finally, the vanishing of the equilibrium means imply f q = 0, f p = 0. The Hamiltonian term therefore becomes where, besides a contribution in the form of the free Hamiltonian of a harmonic oscillator with a strictly positive frequency Ω m satisfying Ω 2 m > κ 2 0 /4, one has an additional term in the form of an anticommutator. Introducing the positive coefficients the generator can be written in the form where in particular the constraints hold, which provide the necessary and sufficient conditions for the dynamics described by (5) to be in Lindblad form and therefore completely positive [11,28]. Let us note that an alternative way to get the same positivity condition is to ask the generalized Heisenberg uncertainty relation q 2 t p 2 t − ( {p, q} t /2) 2 ≥ 2 /4 to hold for any time and any initial state [30].
Assumption 3 (Translational invariance). The reduced dynamics is translation invariant up to the harmonic potential. This requirement is equivalent to the validity of the classical equations of motion for the mean values of position and momentum.
A further natural requirement is translational invariance up to the harmonic potential. It can be expressed as the invariance of the generator under the generic translation q → q + x, p → p, corresponding to homogeneity of space, so that the dissipative effects due to the interaction with the environment do not depend on the position of the oscillator. In view of the evolution equations for the mean values translational invariance is equivalently formulated asking the mean values of q and p to obey the classical equations, in which the momentum is proportional to the derivative of the position. The condition is verified provided κ 0 = γ m . The dynamical matrix giving the evolution of the mean values (7) has therefore eigenvalues −γ m /2 and −γ m /2 ± i Ω 2 m − γ 2 m /4, which naturally leads to introduce the damped frequency ω m of the mechanical oscillator in terms of its bare frequency Ω m Thus in particular the dynamical matrix giving the evolution of the mean values of the quadratic quantities (8) has eigenvalues −γ m and −γ m ± 2iω m . Let us note that according to Ω 2 m > γ 2 m /4 > 0 we have ruled out the case Ω m = 0, which corresponds to a quantum Brownian particle, that is a massive particle not bounded by a potential in a translation invariant environment [18,31,32,[34][35][36] (see [37] for a recent review).
Assumption 4 (Equipartition). At equilibrium the mean kinetic energy and the mean potential energy have to be equal.
At this stage we further have to determine the diffusion coefficients D qp , D qq and D pp appearing in (5). At variance with previous approaches aiming to determine such terms by referring to effective environmental models of bosonic oscillators [13,14], we will rely on the study of features of the equilibrium state. Indeed previous work [16,17] has shown that, while using careful approximations a positive dynamics can be obtained in this framework, the final results are valid only from medium to high temperatures of the thermal bath. A requirement often considered in the literature is that the equilibrium state should be the canonical thermal state determined by the standard Hamiltonian of a harmonic oscillator. However, it is known that this requirement is incompatible with positivity and translational invariance [11,33]. This incompatibility induced some authors to renounce to translational invariance [15], or to accept nonpositive dynamical equations and to give more relevance to obtaining time evolutions very close to the classical ones [13,14,38]. A non positive dynamics can be satisfactory when the system is near the classical regime, but this approach becomes questionable when quantum effects are searched for [39,40]. Rather than giving up positivity or translational invariance, we will weaken the request on the equilibrium state. Since the eigenvalues of the dynamical matrix associated to (8) have a positive real part, existence of a unique attractive equilibrium state is granted, and thanks to the linearity of the equations the equilibrium state is actually Gaussian and determined by the mean values at equilibrium. We are thus lead to ask the equipartition condition which gives equal weight to the mean kinetic and potential energy at equilibrium. By setting in (8) the time derivatives equal to zero, and κ 0 = γ m as follows from translational invariance, we come to In particular the equilibrium means turn out to be Assumption 5 (Gibbs state and temperature dependence). The diffusion coefficients determine an equilibrium state in Gibbs form. The transformation from the position and momentum operators to the mode annihilation and creation operators are temperature independent.
We now want to exploit the residual freedom we have in the choice of the diffusion coefficients to get a Gibbs state as equilibrium state. However, as we already noticed, it cannot be the Gibbs state generated by H 0 . Thanks to the Gaussianity of the equilibrium state we can write ρ eq ∝ exp −ca † m a m for a suitable chosen mode operator with r > 0, ∈ C and r Im = 1/(2 ). The positive constant c can always be written as βω m , with ω m the damped frequency of the system (9). We can therefore write with β a positive constant which can be interpreted as the inverse temperature of the equilibrium state of the mechanical oscillator. The case of a vanishing temperature is given by where ψ 0 is the vacuum state annihilated by a m . Now, we require the transformation (13) from the position and momentum operators q, p to the mode operators a m , a † m to be temperature independent, so that such is the effective Hamiltonian ω m a † m a m . The temperature dependence of the coefficients D qq and D pp is then determined by asking the equilibrium state to be given by (14), or (15) respectively for the case of zero effective temperature. To simplify the notation we introduce the positive quantity which gives the mean number of excitations. As shown in Appendix A, also exploiting (11), one obtains together with in which the adimensional quantity τ has modulus equal to one. Note that thanks to these definitions the positivity conditions (6) hold for all possible values of the parameters.

Master equation for the mechanical oscillator
The explicit relationships between the mode operators a m , a † m , satisfying the standard canonical commutation relations, and the position and momentum operators are In terms of the mode operators the Hamiltonian part of the generator can be written as with H 0 given by (4), and the equilibrium state becomes Let us note that this form of the equilibrium state does not come from a direct requirement, but rather it follows from all the considered assumptions. In particular we stress the fact that the operator H m is not the Hamiltonian of the isolated oscillator, but includes a term containing γ m which comes from the interaction with the bath. Combining (12) and (17) we have in particular indeed satisfying (6), so that the term related to damping gives a negative contribution to the equilibrium mean value H m eq arising from energy exchange with the bath. The Lindblad operators R j appearing in (2) now read R 1 = γ m (N + 1) a m , R 2 = √ γ m N a † m so that the Liouville operator can be finally written as Let us stress that despite the fact that the expression (23) has the form of the generator for an optical oscillator [21], the relations (19) connecting a m , a † m with q, p account for the description of a mechanical oscillator. Let us note that a master equation for a mechanical oscillator with the Liouville operator (23) and the relation (19) between mode and position/momentum operators was already proposed in [12,Sects. 6,7]. However, at odds with the present approach, the starting point of that proposal was a scheme of canonical quantization of dissipative classical systems.

Langevin equations for the mechanical oscillator
So far we have obtained a quantum master equation in Lindblad form for the mechanical oscillator, only relying on general physical constraints and symmetry requirement. As for any Lindblad master equation, such a result allows us to introduce in a rigorous way a unitary dynamics involving the system of interest and suitable quantum Bose fields, which at the level of the reduced dynamics of the system exactly reproduces the master equation. That is, these quantum Bose fields effectively describe the thermal environment affecting the mechanical oscillator. A key step in constructing a unitary dilation for the system of interest is the introduction of a thermal field which does not admit a vacuum and therefore a Fock representation. A more familiar picture can be obtained by representing the thermal field as a suitable linear combination of Bose fields having a standard Fock representation with a common vacuum [41]. The standard choice for the state of the field acting as environment reproduces the Markovian dynamics for the mechanical oscillator described by the master equation. However exploiting the freedom in the choice of the state of the quantum field more general non-Markovian situations can be considered.
Let us start introducing the Hudson-Parthasarathy equation or quantum stochastic Schrödinger equation [26], which gives the evolution equation for the unitary dynamics involving the system of interest and a quantum Bose field. The proper mathematical formulation of this equation relies on the formalism of quantum stochastic calculus [27]. Within this formalism the Heisenberg equations for the system operators provide the quantum Langevin equations, while, as shown in [24], the Heisenberg equations for the Bose fields give the input-output relation of Gardiner and Collet [21,23]. We thus obtain in a unified framework all relevant physical information [25].
For the Liouville operator (23) the associated Hudson-Parthasarathy equation reads (see e.g. [24,41,42] with U (0) = 1, H m given by (20), and B th (t) a Bose thermal field satisfying the canonical commutation relations and the quantum Itô table with N the positive quantity introduced in (16). The commutation rules are better understood by introducing the formal field densities: dB th (t) = b th (t) dt. Then, these densities satisfy the standard canonical commutation relations The thermal field can be represented by means of two commuting Bose fields A 1 and A 2 in the Fock representation [24,42]. This means that such fields satisfy the canonical commutations rules [A i , A j (s)] = 0 and that there exists a common Fock vacuum, i.e. a normalized vector e(0) annihilated by all these operators: A k (t)e(0) = 0 for k = 1, 2. The field define by satisfies the canonical commutation relations (25) and the Itô table (26). Equation (24) is a quantum stochastic differential equation in Itô sense and the second line of (24) corresponds to the Itô correction. The solution U (t) of (24) is a family of unitary operators on the overall Hilbert space which represent the dynamics of the closed system corresponding to "mechanical oscillator plus field", in the interaction picture with respect to the free dynamics of the field. An heuristic, but more familiar, picture can be obtained by using the field densities introduced above. The formal expression of the unitary evolution is indeed [43] where ← − T denotes the time ordered product. The thermal field B th therefore provides the mathematical representation of the phonon field interacting with the mechanical oscillator.
Let us now consider as state of the field the A-field vacuum. In such a case taking the partial trace over the Fock space of the fields, which corresponds to take the trace over the environmental degrees of freedom in open quantum system theory, the reduced system state is given by ρ(t) = Tr env U (t) ρ(0) ⊗ |e(0) e(0)|U (t) † , with ρ(0) the initial state of the oscillator. Thanks to (24) the reduced dynamics of the mechanical oscillator can then be shown to obey exactly the master equation (23) [25]. Further, we have the important relations It is worth noticing that the thermal parameter N does not appear in the commutation rules of the field B th , but rather in the quantum correlations (30). This expresses the fact that N depends on the "state" of the field or, more precisely, N determines a non-Fock representation of the canonical commutation relations. Indeed, representations with different N are unitarily inequivalent. Note furthermore that the vacuum e(0) is not annihilated by the fields B th (t), but it plays the role of a thermal state [42,Sect. 6]; no vacuum state exists for a non-Fock Bose field.

Quantum Langevin equations and input-output relations
Relying on the previously introduced formalism we are now in the position to obtain the so-called quantum Langevin equations, providing the stochastic evolution for the system observables in Heisenberg picture. For a generic system operator X we denote as usual the Heisenberg picture as X(t) = U (t) † XU (t), with U (t) the unitary operator describing the closed dynamics of system and environment. Differentiating this expression by the rules of quantum stochastic calculus, essentially summarized by the Itô table (26), one obtains the quantum Langevin equations for the relevant system operators, namely for the mode operator By (19) we get also the equivalent equations for position and momentum in which we have introduced the Hermitian quantum noises where τ is the pase factor defined in (18). By (25) the new noises obey the commutation rules A fundamental advantage of the considered formalism is that, thanks to the unitarity of U (t), the transformation X → U (t) † XU (t) preserves all the commutation rules among system observables, in particular the Heisenberg relations between position and momentum, as can be checked also directly relying on (35). Warranting preservation of these fundamental commutation relations is indeed a basic step in providing a true quantum description of a dissipative dynamics [21,Chapts. 1,3].
We now consider the Heisenberg picture for the thermal fields and we define While B th (t) represents the field before the interaction with the oscillator, the socalled input field, B out th (t) is the field after the interaction, the so-called output field. We stress in particular that an important consequence of the Hudson-Parthasarathy which warrants the fact that the output fields obey the same commutation relations as the input fields, namely (25). Once again one has to differentiate the three contributions in U (t) † B th (t)U (t) according to the Itô table, thus coming to the input-output relation The linearity of the Heisenberg equations of motion allows for an explicit solution leading for the position and momentum Heisenberg operators to Let us stress that both the bare frequency Ω m and the damped one ω m appear in the expressions of q and p, while only ω m is involved in the expression of a m .

Field state and non-Markovian dynamics
In the Markovian approximation considered so far, the temperature enters the theory only through the parameter N defined in (16). This approximation can be safely described stating that the system actually sees a flat noise spectrum, or more precisely the system is only affected by the value of the bath spectrum at the frequency ω m . A more general and physically more realistic situation is to allow for a structured noise spectrum and this can be achieved without any modification of the unitary dynamics (24) and of the related Langevin equations and input-output relations. To this aim it is enough to change the state of the field by taking mixtures of coherent states [25,44]. Let us note that considering such a mixture of coherent states for the description of the state of the field is actually analogous to consider a state with a regular P -representation in the case of discrete modes (see e.g. [21]), as explained in Appendix B. As we shall see, this modification implies that the reduced dynamics of the oscillator is no more Markovian, in the sense that a closed master equation in Lindblad form for the statistical operator cannot be obtained.
3.2.1. The field state. In order to consider a more general field state let us first introduce the Weyl operators [25,27] for the Fock A-fields, defined as holds, so that the action of a Weyl operator on the Fock vacuum gives a coherent state. Therefore W A (g) is nothing but a displacement operator for the Bose fields [43]. Relying on (28), we can introduce a Weyl operator also for the B-field 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. The relevant expectation values of the thermal Bose field in a coherent state generated by the Weyl operator W T (f ) turn out to be given by A crucial step is now to consider f to be a random process so that we can construct field states with a regular P -representation and a general thermal spectrum, in analogy to the treatment detailed in Appendix B for the case of discrete modes. To this aim let in particular f be a Gaussian stationary stochastic process with vanishing mean, E[f (t)] = 0, and correlation functions Thanks to stationarity, the function G(t) is positive definite, so that according to Bochner's theorem [45] its Fourier transform is a positive function, which we assume to be absolutely integrable, thus implying a finite power spectral density for the process.
In this notation we take the state of the field (the environment) to be Again in the final formulae we will take the limit T → +∞. Since the field state σ env is Gaussian, we can characterize it through the means and the correlations of the thermal field B th , which are immediately obtained from (43) and the properties of the process f . The only non zero contributions are given by To better grasp the physical content of the new state and of the formulae (47) let us introduce a set of field modes as in [43]. Using a complete orthonormal set {h n } in L 2 (R), we can expand the field in terms of discrete independent modes by defining them as We then obtain c hn env = 0, c 2 hn env = 0, together with where we have defined the positive quantity andĥ n (ν) is the Fourier transform of h n (t); by normalization +∞ −∞ |ĥ n (ν)| 2 dν = 1. So, the reduced state of the single mode c hn is exactly the thermal state described in Appendix B. If we take h 1 and h 2 having non overlapping Fourier transforms we also get lim T →+∞ c † h1 c h2 env = 0, which means that these two modes are independent. Then, N (ν) 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 (ν) different from zero in a neighbourhood of ν implies that the mechanical oscillator can absorb from the bath phonons with energy around ν. On the contrary, the approximations are such that the oscillator can emit phonons of any frequency, even when N (ν) = 0. Note the important fact that the physically relevant quantity is now the combination of the two non negative contributions N andĜ(ν), rather than the values of the individual quantities.

The equilibrium state of the mechanical oscillator.
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 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 (ν) is constant. Not to have a closed equation for the reduced dynamics is indeed a signature of the non-Markov features of such a dynamics. However, it is at least possible to construct a master equation with a random Liouville operator, explicitly containing the stochastic process f , in such a way that the mean of the solution of such a master equation gives back the reduced state ρ(t) (see [25, pp. 226-227]).
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 > 0 in (28) and takingĜ(ν) ≡ 0, or equivalently considering a standard Fock representation and formally taking the limit of constant spectrumĜ(ν).
Equipartition. In spite of the difficulty of not having a closed master equation, the study of the reduced equilibrium state, namely ρ eq = lim t→+∞ ρ(t), can still be afforded and its expression enlightens the physical role of the various parameters. Indeed, thanks to the requirement E[f (t) f (s)] = 0, one has that equipartition in the sense of (10) still holds. Starting from the explicit solutions (40) and (41) of the quantum Langevin equations one can check that the equilibrium mean values of position and momentum still remain equal to zero, while the variances are given by where we have introduced an effective mean number of excitations through the expression Notice that if the quantity N (ν) introduced in (48)  Equilibrium state. Since the equilibrium state is necessarily Gaussian, by comparing (50) with (21) we get that the new equilibrium state is again a Gibbs state with respect to the same Hamiltonian H m , but with an effective inverse temperature β eff defined by setting N eff ≡ (e β eff ωm − 1) −1 .

Properties of the quantum noises and quantum stochastic Newton equation
Let us now come back to the quantum Langevin equations for the position and momentum operators, so as to better understand their physical meaning and the role of the noises. In order to study the properties of the noises we transform the Langevin equations (32), (33) in the form of a stochastic Newton equation.
To this aim we first have to consider the quantum noises (34) appearing in these quantum Langevin equations. The commutation relations (35) for these noises, which are state independent, guarantee the preservation of the canonical Heisenberg commutation relations. Their quantum correlations do instead reflect the physical properties of the field state σ env and can be obtained starting from the B-correlations (47). Note that Langevin equations for a mechanical oscillator of the same form and with two noises obeying the same commutations rules (35) were used also in [39]; however, at variance with the present approach, the two point correlations were simply postulated in [39], while in the present treatment they are deduced from the state of the environment.
We stress the fact that in the present formulation the momentum operator is not related to the time derivative of the position operator according to the classical relation, but rather through (32) where the quantum noise C q (t) explicitly appears. However, the connection to the classical formulation is not completely lost. In fact from (32) we can derive the relation By (34) and (47), the mean value of the r.h.s. of the equation above vanishes, while its variance is given by so that in particular also the variance goes to zero for growing t 2 − t 1 . Then the quantity v(t) = p(t)/m can actually be interpreted as the "coarse grained" velocity of the mechanical oscillator. If we use the formal field densities b th (t), b † th (t), with commutation rules (27), take as starting point the quantum Langevin equations (32) and (33) and eliminate the momentum, we can rewrite the quantum Langevin equations in the Newton form: where we have introduced the formally Hermitian quantum noise ξ(t) Most importantly the commutation relations for this noise take the singular expression While the expectation value of this noise with respect to the field state σ env is zero, its symmetrized correlation functions can be computed from the relations and have the expression Note that (51) and (53) were already introduced in [21, Sect. 3.1.2] and [40], where the commutation rules (53) were actually enforced by the requirement of preservation of the commutation rules between position and momentum. However, at variance with previous approaches, here we have provided an explicit construction of the quantum noise ξ(t) in terms of a quantum Bose field, based on a rigorous mathematical construction.
We stress the fact that the stochastic Newton equation (51) is mathematically purely formal due to the presence in (52) ofC q (t), which contains the formal derivativė b th (t) and its adjoint. Moreover, if one were to take (51), (53) and (56) as starting point for the construction of the quantum Langevin equations for position and momentum, then one should complete (51), which is an equation for q(t) only, with a suitable definition of p(t). The standard choice in this respect, considered for instance in [1,20,40], is to take p(t) = mq(t). This works out fine as far as the commutation relations of position and momentum are concerned. However, in this case the equation of motion (51) and the structure of the noise ξ(t) obeying (52) imply thatq(t) contains singular quantum fluctuations, so that it is not a well defined operator. Also p is then not a well defined operator and its variance is actually infinite. Then, one has to regularize the momentum, by subtracting the noise responsible of this divergence; this is what our construction does. The identification of the momentum is given implicitly through the first canonical equation (32), which corresponds to the coarse grained velocity, as discussed above. No divergency appears because the whole construction is based on the well defined unitary evolution (24).

Consistency of the quantum noises.
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 {ξ i (t)} is a set of operator valued noises, the quantum correlation function ξ i (t) † ξ j (t ) env has to be positive definite [45], in the sense that for every choice of the "smooth" test functions {h i }. Since we can always write the necessary positivity condition introduced above becomes a consistency condition between commutation rules and symmetrized correlations.
Relying on (55), (56), as well as the commutation relations (35) 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 (52) one can show that the expression ξ(t)ξ(s) 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.
so that according to (48) and (56) it readŝ where (ν → −ν) means to add the same contribution with ν replaced by −ν. Note in particular thatR(ν) is an even function of the frequency. A reference expression often considered in the literature for the quantityR is given for positive frequencies by m k(ν)ν coth β ν 2 [21, (3.3.9)], [40]. An expression of this form is derived by coupling the system of interest with other harmonic oscillators and taking a suitable continuum limit. These kinds of models are definitely different from the present approach. Indeed in our treatment the interaction with the environment is described in terms of exchange of quanta with the bosonic field representing the phonons, see (29). The quantity k(ν) contains information on both coupling constant and density of modes of the bath in a neighbourhood of the frequency ν. It is typically taken to be a constant, say γ m , in both Markovian and non-Markovian treatments [21, (3.1.1)], [1,2,20,40]. Note that the positivity requirement (57) still has to hold, leading to the requirement m γ m ν coth(β ν/2) − m γ m ν ≥ 0, satisfied at any temperature thanks to γ m ≥ 0, where the constraint actually comes from the low temperature behaviour. The expression to be compared with (60) can therefore be written with n β (ν) = (e β ν − 1) −1 the mean number of quanta of the field at temperature 1/β and frequency ν, extended by parity to the whole real axis. Also in the case of this choice, it is possible to show that the equilibrium mean ofq(t) 2 diverges and therefore the identification of the momentum with mq(t) is not possible, but some regularization is needed. At very low temperature, corresponding in the present model to N (ν) = 0, one hasR(ν) = m γ m Ω 2 m + ν 2 / (2ω m ) versusR GZ (ν) = m γ m ν. Despite this difference, it is important to recall that the relevant frequency interval is around the mechanical oscillator frequency ω m and consistency of these models requires a damping rate γ m small compared to ω m . Note that indeed at zero temperature we haveR GZ (ω m ) = m γ m ω m andR(ω m ) = 1 + γ 2 m /(2ω 2 m ) R GZ (ω m ), so that their ratio differs from 1 just by the small quantity γ 2 m /(2ω 2 m ). To extend the comparison to positive temperatures, we have to take some explicit expression for N (ν). A simple choice suggested by the low temperature behavior is to take an even expression for N (ν), with the following expression on the positive real line: In this case we obtain, for positive frequencies, so that the difference is actually temperature independent and takes the minimal value at the frequency ω m . The freedom in the choice of N (ν) allows to model quite different environments, with a structured occupation spectrum. In some situations, experimental tests [47] seem to indicate a non-Ohmic spectral density around ω m , and this can be incorporated in our setting by a simple change of the expression of N (ν). All these choices become distinguishable and relevant when γ m is not too small.

Cooling and emission spectra of an optomechanical system
As an application of the quantum description of a mechanical oscillator developed so far we consider the simplest optomechanical system [1,2,5,6,20,40], namely the mechanical oscillator is a mirror mounted on a cantilever and coupled to the light in an optical cavity by radiation pressure. The cavity is of high quality, without thermal dissipation other than the one due to the coupling between cantilever and phonons and tuned in such a way that only one electromagnetic mode is relevant. Strong laser light is injected and some light is allowed to leave the cavity so that its spectrum can be analysed.

The optomechanical model
The micro-mechanical oscillator (the mirror) is described by the operators q, p as in (19) and by the Hamiltonian H m (20). The cavity mode is described by the operators a c , a † c and by the free Hamiltonian ω c a † c a c . The free electromagnetic field is in a coherent state describing a perfectly monochromatic laser of frequency ω 0 ; however we use the equivalent description of inserting a source term for the cavity mode in the Hamiltonian and of taking the external field in the vacuum. The final Hamiltonian takes the form (63) Note the trilinear term giving the interaction between the position of the mirror and the number operator of the photons in the cavity, which represents the radiation pressure interaction; the coupling constant is usually expressed as g 0 = ω c /L, where L is the length of the cavity. The laser power is P = ω 0 E 2 /γ c , where γ c is the cavity decay rate and E the laser amplitude taken to be real.
In order to include the cavity mode interacting through radiation pressure with the mechanical oscillator, as well as the emission and absorption of the light from the free electromagnetic field, the Hudson-Parthasarathy equation (24) is modified as follows: Here B th is the thermal field with representation (28), while B em is an independent Bose field in the Fock representation, describing the electromagnetic field outside the cavity. The relevant Itô rule is dB em (t)dB † em (t) = dt, while all the other possible products vanish. Now U (t) is the unitary operator describing the dynamics of the two interacting oscillators and the fields. The latter are in a factorized state given by the tensor product of the thermal environment state (46) and the electromagnetic vacuum:σ env = σ env ⊗ |e em (0) e em (0)|.
It is convenient to eliminate the laser frequency working in the rotating frame and introducing the unitary operator V (t) = e iω0a † c act U (t), which upon differentiation obeys an equation of the form (64) albeit with H om (t) substituted by with ∆ 0 = ω c − ω 0 the nominal detuning. For a generic system operator X we define X(t) = V (t) † XV (t), so that by differentiating according to the rules of quantum stochastic calculus, as done in Section 3.1, we get the following quantum Langevin equations as well as where C q and C p are given by (34). Defining the output fields as in (36) of Section 3.1 we have besides (37) the input-output relation for the electromagnetic field dB out em (t) = dB em (t) + √ γ c e −iω0t a c (t)dt.
4.1.1. Linear approximation. In the case of a very intense laser, that is E 2 large, the dynamics can be linearized in a neighbourhood of the equilibrium mean values, determined by autoconsistency from the means of the linearized form of the quantum Langevin equations. The equilibrium mean value of the momentum is zero, while setting ζ = a c (t) eq , we find where we have introduced the effective detuning ∆, By inserting the equations (70) into (71) we obtain the autoconsistency condition this cubic equation determines ∆ as a function of the laser parameters ∆ 0 and E.
In writing and solving the linearized quantum Langevin equations it is useful to have adimensional and selfadjoint system operators. It is therefore convenient to set Then, the linearized quantum Langevin equations turn out to be where the dynamical matrix is given by The quantity G, having the dimension of a frequency, will play the role of effective coupling constant and is given by so that in particular it depends on the effective detuning ∆ through ζ given in (70). The vector of noises is given by the following field combinations: where τ is the phase factor defined in (18) and the quadratures Q 1 (t) and Q 2 (t), apart from a multiplicative factor due to the change of dimensions, coincide with the noises introduced in (34). Note the different structure of the two dynamical sub-matrices in (77). Indeed the former describes a mechanical oscillator and the latter an optical mode, corresponding to different interactions as discussed in Section 2. The same choice is taken, for instance, in [1,2,7,20,39,40], not in [5,9,10].
The linearization around the equilibrium state is meaningful provided one can ensure the existence of such a state. Its stability conditions can be obtained by applying the Routh-Hurwitz criterion to the equations for the mean values, which correspond to the system (75) with the noise term d Q(t) suppressed. The detailed results for the conditions warranting stability are given in Appendix C.

Energy fluctuations and laser cooling
In order to determine the equilibrium properties and the spectra of the emitted light it is convenient to consider the Fourier transform of the equations (75). We use a formulation tailored for (classical or quantum) processes starting at time zero and we define the gated Fourier transforms [9] for the Bose fields as well as for the relevant system operators Here T is a large time which we will let go to infinity in the final formulae to recover a stationary situation. From the correlations (47) and the fact that B em is a Fock field in the vacuum state we get Let us stress that, while useful, these definitions do not correspond to some continuous monitoring of position and momentum, even though S q (ν) is directly related to the observed optical spectra as we shall see in Section 4.3. Due to the fact that the crosscorrelations between the thermal and the electromagnetic field vanish, these spectra decompose in a thermal and a radiation pressure contribution according to As shown at the end of Appendix D.1, the final expression for the radiation pressure contribution reads while the thermal ones are given by where (ν → −ν) means to add the same contribution with ν replaced by −ν and the quantityR(ν) is the Fourier transform of the quantum correlations of the noise given in (60). Note that the quantities introduced in (87)-(89) are non-negative as they should be in order to have a sensible decomposition of the spectra; this property follows from their very expression and (D.7). The quantity d(ν) appearing in the denominators is related to the characteristic polynomial of the dynamical matrix A introduced in (76) and takes the form The quantity Ω m ∆ 2 − (ν + iγ c /2)

4.2.2.
The mean values at equilibrium. By integrating in their frequency dependence the fluctuation spectra one obtains the second moments of position and momentum in the equilibrium state: Note that all these quantities are finite due to the fact that all the integrands behave as ν −2 for |ν| → +∞. We have also that the reduced equilibrium state of the mechanical oscillator is a Gaussian state characterized by (92), (93) and q eq = g 0 |ζ| 2 /(mΩ 2 m ), p eq = 0.
On the contrary the integral of ν 2 S q (ν), which would give the fluctuations at equilibrium of mΩ m / 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 mq with momentum is not possible. The expression of S q (ν) coincides with the one given in [1,20], albeit witĥ R(ν) instead ofR GZ (ν). While in the latter case S q (ν) ν −3 , stillq 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 (ν) and S qp (ν) have not been obtained before. In particular the nonvanishing value of S qp (ν) implies that the fluctuations of position and momentum are actually correlated.
The mean energy of the harmonic oscillator at equilibrium can be expressed in the form where the contribution due to fluctuations is given by 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 adimensional quantities as well as so that the fluctuation contribution can be written as (98) by construction we have N rp + N th + M th (∆) ≥ 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 M th (∆) 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 ∆, which are present in S th p (ν) and not in S th q (ν). In the radiation pressure part the term with Ω 2 m comes from the position and the one with ν 2 comes from the momentum and give different contributions to the mean energy.
leading upon integration to with N eff as in (50). These expressions show that equipartition of the mean energy is not valid just due to the radiation pressure contributions. However equipartition approximately holds for γ c 2γ m , which is the case typically considered in many theoretical studies and experiments. We further have for the fluctuation contributions to the energy 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 N rp 1.6×10 4 corresponding to a temperature of about 7.9 K.

Laser cooling.
As discussed in many papers [1,3,5,6,46], an important effect which can be described by this kind of models 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 ζ = 0). So, we have to study the value of the fluctuation contribution (98) and to compare it to its value for ζ = 0, which is given by H fl ζ=0 = H m eq ζ=0 = ω m N eff + 1 2 . To obtain explicit analytical formulae for the mean energy we consider the case of a constant noise spectrum, that is N (ν) = const = N eff . To actually perform the calculations we need to evaluate, at least approximately, the zeros of the denominator d(ν) given by (91). To this aim we introduce the Ansatz As discussed in Appendix D.2, d(ν) can be written in this way only under the compatibility conditions (D.12). By lengthy computations the integrals over ν can be exactly performed, leading to involved formulae explicitly given in Appendix D.3. In order to describe cooling effects the relevant contributions can be written in the form where the quantities Q and K are given in equations (D.27) and (D.28). The expression for N rp is given in (D. 26). Note that, while Q is always positive, depending on the values of the parameters the quantity K can be also negative. For a large choice of the parameters 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 quantity 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 Figure 1. Plot of the cooling factor C for the case in which the cavity damping is much bigger than the mechanical oscillator frequency. We explore the dependence of the cooling factor on both the effective detuning ∆ and the cavity damping rate γc, both expressed in Hz. It appears that the best cooling factor is of the order 10 −3 and corresponds to ∆ γc.
taken to be m = 2.5 × 10 −10 kg and Ω m = 2π × 10 7 Hz, while the mechanical damping factor is γ m = 2π × 10 2 Hz. We consider a cavity of length 5 × 10 −4 m and resonance frequency ω c = 2πc/(1064 × 10 −9 ) Hz, driven by a laser with a power of 5 × 10 −2 W. For the sake of comparison the values of the fixed parameters are taken from [20]. In Figure 1 we consider the case γ c ω 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 Γ m and Γ c given in (D.15), relying on the conditions (D.13). The stationary value of the energy of the mechanical system has a marked dependence on the effective detuning ∆ and the optimal cooling region, corresponding to C of the order of 10 −3 , is obtained for ∆ γ c . In this parameter region N rp can be neglected with respect to CN eff , unless the phonon bath is below 1 K, so that indeed the quantity C given in (102) properly describes the cooling effect. When the detuning ∆ goes to zero the cooling factor rapidly increases in agreement with the discussion in Section 4.2.3 showing the presence of heating at ∆ = 0; for these values of the parameters, the cooling effect disappears also for growing ∆. In Figure 2  ∆ = ω m , thus fixing the detuning to be equal to the effective mechanical frequency. In this case the location of the poles can be evaluated exactly, provided one distinguishes two regions according to the value of the ratio (γ c − γ m ) 2 /4G 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 γ c >γ c , corresponding for the considered parameters toγ c 4.1 × 10 7 , the effective damping Γ m (D.17) and Γ c (D.18) are actually distinct, while the effective frequencies ∆ eff and ω m eff do coincide and are given by the expression (D.19). The cooling factor is a monotonic increasing function of the cavity damping rate γ c , and around the starting pointγ c the cooling factor takes the value 2.9 × 10 −5 . In the complementary region, corresponding to (γ c − γ m ) 2 /4G 2 below one, the cooling factor is a decreasing function of the cavity damping rate, so that the optimal cooling is obtained for γ c =γ c . In this region, corresponding to γ c <γ c , the two effective dampings Γ m and Γ c both coincide with the average of optical and mechanical damping rates (D.21), while the effective frequencies ∆ eff and ω m eff are given by the expressions (D.22). To assess the relevance of the various contributions in (102) we report the values for γ c =γ c : we have γ m /Γ m 3.05 × 10 −5 , Q .997 and K −4.18 × 10 −2 . The radiation pressure contribution N rp can still be neglected, unless the phonon bath temperature and therefore N eff is very small. For γ c =γ c we have in particular N rp 0.73, so that its contribution to the mean equilibrium energy is only relevant for temperatures below 10 K.

Optical spectra
We consider now the monitoring of the emitted light by balanced homodyne and heterodyne detection [50,Sect. 7.2]. The aim is to see which kind of information on the mechanical oscillator can be obtained by detection of the emitted light.
4.3.1. Homodyne spectrum. The case of a perfect coherent monochromatic local oscillator of frequency ω 0 with detection of the whole emitted light [43,49] corresponds to the continuous measurement of a field quadrature of the type ϑ is a free parameter which depends on the optical path and determines the observed quadrature. For instance the field quadratures for ϑ = π/2 and ϑ = 0 are proportional to the noises defined in Eqs.
The homodyne spectrum is then given by the expression where the environmental state is given by (65) and ρ 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 [43,Sect. 4]. From the above relations we obtain the second key relation which guarantees the presence of the commuting observables and therefore the consistency of the theory: this implies also that the homodyne spectrum S(ν; ϑ) is an even function of ν. As shown in Appendix D.4, the homodyne spectrum has both an elastic and an inelastic component which turn out to have the expressions S el (ν; ϑ) = 8πγ c |ζ| 2 (sin ϑ) 2 δ(ν), with where S q (ν) is now explicitly given by (99). This result has been found also in [40], but with the substitutionR(ν) →R GZ (ν) in the expression (88) for S th q (ν) ∆=0 . As a result, at least in principle, when ∆ = 0 the homodyne observation of the quadrature with ϑ = π/2 can give direct information on the correct expression forR(ν).
The temperature dependence of S inel (ν; 0) is entirely contained in the thermal contribution to S q (ν) through N (ν). As a result, at high temperatures the interference term in (114) is negligible (also for ∆ = 0) at least in the region where N (ν) 1; in this case we get which is the result given in [1,Sect. 3]. Therefore, 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 (ν) S th q (ν); by using this further approximation in (116) and by using the explicit expressions of S th q (ν) (88) andR(ν) (60) we get This expression highlights the dependence of the homodyne spectrum on the thermal spectrum N (ν) and the characteristic polynomial d(ν) (91) of the dynamical matrix (76) 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 ϑ = −π/4. Considering the simple case of vanishing detuning ∆ = 0 and vanishing temperature N (ν) ≡ 0 we obtain for the inelastic contribution in ν = 0 If the parameters are such that 4ω 2 m G 2 +γ m γ c Ω 2 m < 2γ c Ω 2 m ω m , we get S inel (0; −π/4) < 1. This means that in a neighbourhood of ν = 0 we have S inel (ν; −π/4) < 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 any one of the inequalities (112) implies that the complementary quadrature is anti-squeezed. 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 ν = 0 in the inelastic spectrum.

Heterodyne 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 ω 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 [44], [25,Sect. 3.5] that the balanced heterodyne detection scheme corresponds to the measurement in continuous time of the observables whereα is a phase depending on the optical paths and √ κ e −κt/2 , κ > 0, represents the detector response function. As we shall see, the heterodyne spectrum does not depend onα. In the Heisenberg description the observables become the "output current" By the definition of I(µ; t) and the properties of U (t) we get [I out (µ; t), I out (µ; 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 the limit is in the sense of the distributions in µ. As a function of µ, P (µ) is known as power spectrum. Note that to change µ means to change local oscillator, that is to change the measuring apparatus. In general I out (µ; t) and I out (µ ; 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 with The inelastic part of the spectrum is computed in Appendix D.5. Again it is possible to identify a radiation pressure contribution and a thermal part For simplicity we give only the expressions for κ ↓ 0: (124) Both contributions are positive as it follows from the expressions (124) and (D.36).
Note the presence of the interference term in (123). By simple computations one can check that Σ inel (ν + ω 0 ) + Σ inel (ω 0 − ν) = S inel (ν; ϑ) + S inel (ν; ϑ + π/2); (125) this is a fundamental relation [50, Eq. (9.61)] connecting heterodyne and homodyne spectra. Moreover, by inserting the definitions of the relevant quantities given in (60), (87), (88), an explicit expression for Σ inel can be obtained from which it is apparent that Σ inel (µ) > 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 (123) is negligible when N 1 and we get 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 (ν) as given by (62). Then, by using the explicit expressions of S rp q and S th q and by setting ν = µ − ω 0 , we get From this expression we see that the main features of the spectrum will be determined by the zeros of the denominator |d(ν)| 2 ; for instance, as discussed in Appendix D.2, for ∆ = ω m we can have one or two resonance frequencies depending on the value of the cavity decay rate γ c . In Figure 3 we show this phenomenon: the two distinct peaks coalesce as γ 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 Σ th . In this parameter region the behaviour of the inelastic homodyne spectrum S inel (ν; 0) given by (114) is very close to the heterodyne one as depicted in Figure 3. Let us notice that the behaviour shown in Figure 3 does not uncover the whole rich structure of the spectrum which appears by exploring other parameter regions.

Summary and outlook
In this article we have shown how to give a fully quantum description of a dissipative mechanical oscillator. The combined use of master equations and quantum Langevin equations allows for the construction of a dissipative dynamics respecting symmetries and physical constraints, such as the energy equipartition at equilibrium, and subject to dissipation with an arbitrary noise spectrum. A crucial feature allowing for these results is that for a mechanical oscillator the definition of the creation and annihilation operators a m and a † m in terms of position and momentum is not the usual one, but as discussed in Section 2.2, rather depends on the damping constant γ m : the standard result is only recovered for a vanishing damping constant as can be seen from Eqs. (18) and (19). Moreover, the quantum Langevin equations for the system, and the input-output relations for the noises, for both the mechanical oscillator and for the optomechanical system, given in Section 3.1 and Section 4.1 respectively, need not be postulated: they are nothing but the Heisenberg equations of motion determined by the Hudson-Parthasarathy unitary evolutions (24) and (64). In this framework it appears that, in order to preserve the Heisenberg uncertainty relations, the momentum operator can be interpreted as the time derivative of the position operator only in a "coarse grained" picture. An help in comparing our approach to others and in discussing the structure of the noises comes from the quantum Langevin equations in Newton form (Section 3.3), which, despite the fact that they need the introduction of singular noises, do not contain the momentum operator. Indeed in the quantum case important constraints on the correlation functions of the operator noises come from the fact that they need to be positive definite and compatible with the commutation rules of such noises. In this formalism, we are further able to introduce a field analog of the P -representation for the state of the environment and this opens the possibility of treating an arbitrary noise spectrum as done in Section 3.2.
Our description of the mechanical oscillator is not very different from other proposals at medium and high temperatures of the phonon bath. Differences become relevant for very small temperatures. Indeed the dynamics we have constructed is fully "quantum" at all temperatures and this opens the possibility of constructing models of optomechanical systems which are reliable also in a deep quantum regime. As an example we have studied a prototypical system: a mechanical resonator interacting via radiation pressure with a single optical mode in a cavity. For this case we have given explicit general formulae for the fluctuation spectra of position and momentum of the mechanical resonator and for the mean mechanical energy at equilibrium. By using detection theory in continuous time, we have obtained the full expressions of the homodyne and heterodyne spectra of the emitted light. For not too low temperatures, usual results are recovered, such as laser cooling and connection between the light spectra and the fluctuations of position of the mechanical component. However, our description is valid also at very low temperatures, when semi-classical reasoning is not valid and the observation of the spectra of the emitted light is not giving a direct measurement of the mechanical fluctuations.
Many generalizations are possible [51][52][53][54], which could benefit of a systematic and consistent treatment. The simplest generalization is to include imperfections in the detection scheme and noise in the stimulating laser light. But also direct detection can be included [25] or the entanglement between resonator and optical mode can be studied. Moreover, the whole theory has in some sense "modular" properties and can be applied to more complicated systems, say when more mechanical resonators and more optical modes are involved. Let us consider now a generic temperature. By inserting (12), (13), (18) into a 2 m eq = 0, one gets D pp = m 2 Ω 2 m D qq . Finally, by the definition of N (16) and the previous results, we get equations (17).

Appendix B. The P -representation
In Section 3.2.1 we have introduced the state describing the environment by means of a suitable average over random coherent vectors for the quantum fields. Here we show that such a construction of a field state is a natural generalization of the standard P -representation for the thermal state of a harmonic oscillator.
Let us consider a harmonic oscillator of frequency ω, lowering operator a and Hamiltonian ωa † a; the coherent states are denoted by |α . The Glauber-Sudarshan P -representation of a state ρ [21,Section 4.4.3] is defined by the formula ρ = d 2 α P (α, α * )|α α|. (B.1) If the pseudo-density P is allowed to become negative and singular, then any state can be represented in this form; for instance the P -representation of a number state contains also derivatives of the Dirac delta [21, pp. 113-114]. When P is a true probability density (eventually with δ-contributions in order to include discrete distributions), one speaks of a regular P -representation and the formula (B.1) describes mixtures of coherent states, including in particular thermal states [21, p. 113], which are of special interest for the present treatment.
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 an expectation value so that it can be written in the form ρ = 1 − e −β ω e −β ωa † a upon identifying σ 2 = 1/(e β ω − 1). then, by dividing by √ T , for large T we obtain the algebraic system (A + iν1) F (T ; ν) Q(T ; ν).
By setting d(ν) = det(A + iν1) (91) and by computing the inverse of A + iν1 we obtain By inserting the expressions of the Q i (T ; ν) in terms of the fields we obtain When one of these conditions does not hold, at least two zeros of d(ν) are purely imaginary. We do not study this case.
To compute approximately Γ m we follow a suggestion given in [5,20] and based on an approximation of the mechanical susceptibility. In the expression of d(ν m ) taken from (91) we make the approximation ν m + ∆ + i γc (D. 16) because χ(∆) has the same sign as ∆, these conditions are true restrictions only for ∆ > 0.
Once we have Γ m and Γ c , we can compute ω m 2 eff and ∆ 2 eff from the equations (D.11), which do not contain approximations. When ∆ > 0 (red detuning), we have an increasing of the mechanical damping constant, Γ m > γ m , and a decreasing of the spring rigidity, ω m < ω m eff . The ratio (Γ m − γ m )/γ m is called co-operativity. In the considered approximation, the compatibility conditions become (D. 16), from which we see that such conditions do not hold for ∆ positive and small. In this situation the cavity is overdamped and the decomposition of d(ν) takes the form d(ν) = ν − ω m eff + i Γm 2 ν + ω m eff + i Γm 2 ν + i Γ1 2 ν + i Γ2 2 ; we do not study this case.
The case ∆ = ω m . An exact expression for Γ m and Γ c can be found when ∆ = ω m . We study only the case of d(ν) of the form (D.9) with four distinct zeros.
In the case Γ c = Γ m we set x = Γ c − Γ m and insert (D.11) and Γ c + Γ m = γ c + γ m into the last equation of the system (D.10); in such a way we get Let us set Then, by using the solution of the equation for which becomes (123) by expanding the absolute value and using (99).