Generation and detection of squeezed phonons in lattice dynamics by ultrafast optical excitations

The characterization of a typical pump and probe experiment involves several steps:

(1) Initial state. Before being hit by the pump laser pulse, the material under study is in thermal equilibrium: the initial phonon state, described by the density matrix ${\rho }_{\beta }$, is then a thermal one, characterized by the inverse temperature β, which in experiments frequently corresponds to room temperature. On the other hand, the laser pump pulse can be described by a coherent photon state $| \nu \rangle$ [11, 35], of high intensity $| \nu | \gg 1$. The initial total (photon and phonon) state is then given by the density matrix

$$\begin{eqnarray}&&| \nu \rangle \langle \nu | \otimes {\rho }_{\beta }.\end{eqnarray} \tag{ 1 }$$

(2) Pumping process. In view of the difference between the laser pulse's duration (≈80–100 femtoseconds) and the typical phonon evolution time scale (a few picoseconds), the pumping process can be considered instantaneous at time t = 0. Consequently, it is described by a unitary operator ${ \mathcal U }$ (discussed in detail in section 2.5, see also section 2.7), that transforms the initial state of the composite system into a new state

$$\begin{eqnarray}&&{ \mathcal U }(| \nu \rangle \langle \nu | \otimes {\rho }_{\beta }){{ \mathcal U }}^{\dagger }.\end{eqnarray} \tag{ 2 }$$

This new state contains information about both photons and phonons. Since the scattered pump pulse is discarded and we are interested only in the phonon behavior, the photon degrees of freedom can be traced out:

$$\begin{eqnarray}&&\rho ={\mathrm{Tr}}_{\mathrm{photon}}[{ \mathcal U }(| \nu \rangle \langle \nu | \otimes {\rho }_{\beta })\,{{ \mathcal U }}^{\dagger }].\end{eqnarray} \tag{ 3 }$$

The density matrix ρ thus describes the state of the phonons immediately after the excitation by the pump.

(3) Phonon dynamics. The excited phonons cannot be considered isolated from their environment, since they inevitably interact with it, albeit weakly. In this case, standard techniques describe their time-evolution by means of a master equation [1, 4, 12]

$$\begin{eqnarray}&&{\partial }_{t}\,\rho (t)={\mathbb{L}}[\rho (t)],\end{eqnarray} \tag{ 4 }$$

where the generator ${\mathbb{L}}$ takes into account the dissipative and noisy effects induced by the environment.

(4) Probing process. After a delay time t, the probing pulse acts again impulsively; in analogy with the pump process, probing can be described by a unitary operator ${ \mathcal U }$ (treated in detail in section 2.7, see also section 2.5). It acts on a total photon–phonon state of the form $| \alpha \rangle \langle \alpha | \otimes \rho (t)$, where $| \alpha \rangle$ is the corresponding coherent state of the probe laser light, with intensity $| \alpha |$ considerably smaller than the pump intensity $| \nu |$. Since we extract information about the phonons by analyzing the scattered probe light, we are only interested in the state of the photons, obtained by tracing away the phonon degrees of freedom:

$$\begin{eqnarray}&&\sigma (t)={\mathrm{Tr}}_{\mathrm{phonon}}[{ \mathcal U }(| \alpha \rangle \langle \alpha | \otimes \rho ({t})){{ \mathcal U }}^{\dagger }].\end{eqnarray} \tag{ 5 }$$

The photon state $\sigma (t)$ is completely characterized by its correlation functions

$$\begin{eqnarray}&&\langle {({\hat{a}}^{\dagger })}^{m}{\hat{a}}^{n}\rangle (t)={\mathrm{Tr}}_{\mathrm{photon}}[{({\hat{a}}^{\dagger })}^{m}{\hat{a}}^{n}\,\sigma ({t})],\end{eqnarray} \tag{ 6 }$$

where ${\hat{a}}^{\dagger }$, $\hat{a}$ are the photon creation and annihilation operators and m, n positive integers. While previous investigations have considered essentially the case $m=n=1$, in the present approach higher order photon correlation functions can be obtained. As we shall show, they allow a precise characterization of the quantum features of the phonon states.

In the following we will discuss in detail each one of these steps.

Let us consider pump and probe experiments at room temperature. Before the pump pulse hits the target, the phonons can then be assumed to be in thermal equilibrium at a given inverse temperature β; their state is thus described by the thermal density matrix

$$\begin{eqnarray}&&{\rho }_{\beta }=\displaystyle \frac{{{\rm{e}}}^{-\beta H}}{\mathrm{Tr}[{{\rm{e}}}^{-\beta H}]}.\end{eqnarray} \tag{ 7 }$$

In general, the pump laser beam can excite different vibrational modes; however, in many situations, only one mode is seen to be physically relevant due to the specific experimental setups [3, 14, 18, 29]. In such cases, the phonon hamiltonian can be taken to be $H={\rm{\Omega }}\,{\hat{b}}^{\dagger }\hat{b}$, where only one mode of energy Ω contributes, and ${\hat{b}}^{\dagger }$ and $\hat{b}$ are the corresponding phonon bosonic creation and annihilation operators satisfying the canonical commutation relation $[\hat{b},{\hat{b}}^{\dagger }]=1$. In this equilibrium state, the mean phonon number is given by

$$\begin{eqnarray}&&{n}_{\mathrm{eq}}=\mathrm{Tr}[{\rho }_{\beta }\,{\hat{b}}^{\dagger }\hat{b}]={({{\rm{e}}}^{\beta {\rm{\Omega }}}-1)}^{-1}.\end{eqnarray} \tag{ 8 }$$

We remark that the assumption of a single active phonon mode is frequently realistic. For example, crystal symmetries and laser polarization may be exploited to selectively excite a single mode [10], or the temperature dependence of some mode can be used to suppress it, even at room temperature [22]. Furthermore, the formalism we are describing is rather general and can easily be extended to account for the presence of more than one phonon excitation mode.

In pump and probe experiments, a mode-locked pulsed laser is used. Although laser pulses contains many frequencies, mode-locking allows one to consistently adopt a single-mode description, which collectively accounts for the coherent properties of the laser beam as discussed in detail in [11]. Such single-mode treatment of laser pulses interacting with matter is of course tenable only when the dynamics does not distinguish the various pulse components.

The pump laser photon state will then be described by a single mode coherent state $| \nu \rangle$, satisfying $\hat{a}| \nu \rangle =\nu | \nu \rangle$, with $\hat{a}$, ${\hat{a}}^{\dagger }$ the (collective) annihilation and creation operators, satisfying the canonical commutation relation $[\hat{a},{\hat{a}}^{\dagger }]=1$. The parameter ν is related to the laser intensity via the mean photon number $\langle \nu | {\hat{a}}^{\dagger }\hat{a}| \nu \rangle ={| \nu | }^{2}$. Since the pump laser pulse is of high intensity, we have $| \nu | \gg 1$.

The effects of the laser pulses on the crystal vibrations can generically be described in terms of first and second order processes. Since the pulse duration is much shorter than any phonon characteristic time, the laser action on the target material can be considered instantaneous, and consequently may effectively be described by a time dependent interaction hamiltonian of the form

$$\begin{eqnarray}&&{ \mathcal H }(t)=(\gamma \,{\hat{a}}^{\dagger }\,\hat{b}+{\gamma }^{* }\,\hat{a}\,{\hat{b}}^{\dagger }+\eta {({\hat{a}}^{\dagger })}^{2}\,{\hat{b}}^{2}+{\eta }^{* }\,{\hat{a}}^{2}{({\hat{b}}^{\dagger })}^{2})\delta (t),\end{eqnarray} \tag{ 9 }$$

where $\delta (t)$ embodies the impulsive character of the interaction, while γ and η are suitable complex coupling constants. To apply the scheme to a specific experiment, they have to be determined, either from first principles or phenomenologically; a study of typically possible behavior can be found in section 4.

In (9), a generic photon–phonon interaction hamiltonian has been expanded up to the second order in photon and phonon operators: higher orders can be safely neglected since their coupling constants are smaller and smaller, taking into account that the scattering processes they describe are less likely to occur. Notice that the hamiltonian (9) has been chosen to preserve the number of bosons, i.e. the annihilation of a certain number of photons results in the creation of the same number of phonons and vice versa. This assumption corresponds to the request that the interaction be gauge-invariant in all bosonic operators. The linear terms in the hamiltonian involve only phonons in a single mode at null momentum ${\bf{k}}$, while the quartic ones are not limited to ${\bf{k}}=0$ and one should thus integrate over the entire optical phonon dispersion region including processes where the momentum conservation is guaranteed by the creation of optical phonons with opposite momenta. However, the effective hamiltonian comprises only a single phonon mode at k = 0 since, in essentially the great majority of experiments so far performed, the probing process is limited to the linear regime whereby phonons at ${\bf{k}}\ne 0$ do not practically affect the observed photon number fluctuations.

Moreover, since the hamiltonian is linear and bilinear in the phonon operators, the described interaction can generate not only coherent or squeezed phonons, but also both effects at the same time—a situation that is far more likely.

The above hamiltonian is time dependent and thus the evolution operator ${ \mathcal U }(t,{t}_{0})$ involves a time-ordered exponential. In this case, however, the integral can be calculated so that the ${ \mathcal U }(t,{t}_{0})$ is the same for all ${t}_{0}\lt 0$ and $t\gt 0$. Consequently, we can describe the photon–phonon interaction by a single application of the operator

$$\begin{eqnarray}&&{ \mathcal U }=\exp [-{\rm{i}}(\gamma \,{\hat{a}}^{\dagger }\,\hat{b}+{\gamma }^{* }\,\hat{a}\,{\hat{b}}^{\dagger }+\eta \,{({\hat{a}}^{\dagger })}^{2}\,{\hat{b}}^{2}+{\eta }^{* }\,{\hat{a}}^{2}{({\hat{b}}^{\dagger })}^{2})];\end{eqnarray} \tag{ 10 }$$

it represents the action of the laser pulses on any photon–phonon state.

As mentioned before, the pump process transforms the initial total state $| \nu \rangle \langle \nu | \otimes {\rho }_{\beta }$ into ${ \mathcal U }(| \nu \rangle \langle \nu | \otimes {\rho }_{\beta }){{ \mathcal U }}^{\dagger }$. Since $| \nu \rangle$ represents a high intensity coherent photon state, the use of a mean-field approximation for the photon degrees of freedom is appropriate (see appendix A for a more detailed discussion); then, one may replace $\hat{a}$ with ν and ${\hat{a}}^{\dagger }$ with ${\nu }^{* }$, so that ${ \mathcal U }$ in (10) reduces to

$$\begin{eqnarray}&&{U}_{\nu }^{\mathrm{MF}}=\exp [-{\rm{i}}(\gamma \,{\nu }^{* }\,\hat{b}+{\gamma }^{* }\,\nu \,{\hat{b}}^{\dagger }+\eta \,{({\nu }^{* })}^{2}\,{\hat{b}}^{2}+{\eta }^{* }\,{\nu }^{2}\,{({\hat{b}}^{\dagger })}^{2})],\end{eqnarray} \tag{ 11 }$$

containing only phonon operators. Therefore, immediately after the action of the pump, the state of the composed photon–phonon system is given by

$$\begin{eqnarray}&&| \nu \rangle \langle \nu | \otimes ({U}_{\nu }^{\mathrm{MF}}\,{\rho }_{\beta }\,{({U}_{\nu }^{\mathrm{MF}})}^{\dagger }).\end{eqnarray} \tag{ 12 }$$

It turns out that ${U}_{\nu }^{\mathrm{MF}}$ can be rewritten as a combination of displacement (D) and squeezing (S) operators⁶

$$\begin{eqnarray}&&{U}_{\nu }^{\mathrm{MF}}=S({\xi }_{\nu })D({z}_{\nu }),\end{eqnarray} \tag{ 13 }$$

with

$$\begin{eqnarray}&&D({z}_{\nu })=\exp ({z}_{\nu }\,{\hat{b}}^{\dagger }-{z}_{\nu }^{* }\,\hat{b}),\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&S({\xi }_{\nu })=\exp \left(\displaystyle \frac{1}{2}{\xi }_{\nu }^{* }\,{\hat{b}}^{2}-\displaystyle \frac{1}{2}{\xi }_{\nu }{({\hat{b}}^{\dagger })}^{2}\right),\end{eqnarray} \tag{ 15 }$$

where the displacement ${z}_{\nu }$ and squeezing ${\xi }_{\nu }\equiv {r}_{\nu }{{\rm{e}}}^{{\rm{i}}{\varphi }_{\nu }}$ parameters are explicitly given by

$$\begin{eqnarray}{z}_{\nu } & = & \displaystyle \frac{| \gamma | }{2| \eta | | \nu | }[(1-\cosh (2| \eta | {| \nu | }^{2})){{\rm{e}}}^{{\rm{i}}({\theta }_{\nu }+{\theta }_{\gamma }-{\theta }_{\eta })}-\sinh (2| \eta | {| \nu | }^{2}){{\rm{e}}}^{{\rm{i}}({\theta }_{\nu }-{\theta }_{\gamma }+\pi /2)}],\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{r}_{\nu }=2| \eta | {| \nu | }^{2},\qquad {\varphi }_{\nu }=2{\theta }_{\nu }-{\theta }_{\eta }+\displaystyle \frac{\pi }{2},\end{eqnarray} \tag{ 17 }$$

with $\gamma =| \gamma | {{\rm{e}}}^{{\rm{i}}{\theta }_{\gamma }}$, $\eta =| \eta | {{\rm{e}}}^{{\rm{i}}{\theta }_{\eta }}$ and $\nu =| \nu | {{\rm{e}}}^{{\rm{i}}{\theta }_{\nu }}$.

The phonon state after the pump process

$$\begin{eqnarray}&&\rho ={U}_{\nu }^{\mathrm{MF}}\,{\rho }_{\beta }\,{({U}_{\nu }^{\mathrm{MF}})}^{\dagger },\end{eqnarray} \tag{ 18 }$$

obtained from (12) by tracing out the photon degrees of freedom, is then a squeezed coherent thermal state; it represents the initial state of the phonon dynamics, whose description requires taking into account the weak interaction with the environment (see figure 1 for a phase–space representation of all the above introduced states).

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Phonons in the target material can hardly be considered as an isolated system, since they inevitably interact with the environment formed by the remaining crystal degrees of freedom, e.g. electrons or impurities. However, the interaction is in general weak and furthermore, initially, phonons and the environment can be considered uncorrelated. In such situations, the standard approach to open quantum systems allows to derive through the so-called 'weak coupling limit' techniques a Lindblad type master equation for the dynamics of the phonon state $\rho (t)$, that takes the general form [1, 4]

$$\begin{eqnarray}{\partial }_{t}\rho (t) & = & -{\rm{i}}[H,\rho (t)]\\ & & +\displaystyle \frac{1}{2}\displaystyle \sum _{k}{\lambda }_{k}(2{V}_{k}\rho (t){V}_{k}^{\dagger }-{V}_{k}^{\dagger }{V}_{k}\rho (t)-\rho (t){V}_{k}^{\dagger }{V}_{k}).\end{eqnarray} \tag{ 19 }$$

The first line alone would give rise to the usual unitary evolution generated by the phonon free hamiltonian $H={\rm{\Omega }}\,{\hat{b}}^{\dagger }\hat{b}$, while the additional contribution in the second line arises due to the presence of the environment; the quantities V_k represent suitable phonon operators, while the real and positive constants ${\lambda }_{k}$ parametrize noisy and dissipative effects. Two different, physically relevant situations [12] will be studied explicitly:

Thermal environment. Experimental evidence indicates that, after being excited by the pump pulse, phonons relax in a few picoseconds to a thermal equilibrium state, at a final temperature possibly higher than the initial one. We have thus considered the typical master equation that describes this situation:

$$\begin{eqnarray}{\partial }_{t}\rho (t) & = & -{\rm{i}}[{\rm{\Omega }}\,{\hat{b}}^{\dagger }\hat{b}\rho (t)]\\ & & +\displaystyle \frac{{\lambda }_{T}}{2}[({n}_{{\rm{f}}}+1)(2\hat{b}\rho (t){\hat{b}}^{\dagger }-{\hat{b}}^{\dagger }\hat{b}\rho (t)-\rho (t){\hat{b}}^{\dagger }\hat{b})\\ & & +{n}_{{\rm{f}}}(2{\hat{b}}^{\dagger }\rho (t)\hat{b}-\hat{b}{\hat{b}}^{\dagger }\rho (t)-\rho (t)\hat{b}{\hat{b}}^{\dagger })],\end{eqnarray} \tag{ 20 }$$

where ${n}_{{\rm{f}}}={({{\rm{e}}}^{{\beta }_{{\rm{f}}}{\rm{\Omega }}}-1)}^{-1}$ is the thermal mean phonon number at the final inverse temperature ${\beta }_{{\rm{f}}}$, while ${\lambda }_{T}\geqslant 0$ is the phenomenological constant parametrizing the strength of noise and dissipation. The dynamics generated by this equation has a unique long-time asymptotic state, the Gibbs state at inverse temperature ${\beta }_{{\rm{f}}}$.

Dephasing environment. A typical effect due to the presence of an external environment is loss of coherence, a phenomenon usually referred to as 'dephasing'. This occurs when the number of phonons is constant in time, since due to the Heisenberg uncertainty principle, the conjugate variable, the 'phase', must then be completely undetermined. A typical master equation describing dephasing effects is given by [12]

$$\begin{eqnarray}&&{\partial }_{t}\rho (t)=-{\rm{i}}[{\rm{\Omega }}\,{\hat{b}}^{\dagger }\hat{b},\rho (t)]-\displaystyle \frac{{\lambda }_{{\rm{D}}}}{2}[{\hat{b}}^{\dagger }\hat{b},[{\hat{b}}^{\dagger }\hat{b},\rho (t)]],\end{eqnarray} \tag{ 21 }$$

where ${\lambda }_{{\rm{D}}}\geqslant 0$ is the phenomenological parameter describing the damping of the system. This master equation describes the situation in which there is no energy exchange between the phonons and their surroundings; indeed, the phonon number operator ${\hat{b}}^{\dagger }\hat{b}$ is left invariant by the time evolution generated by (21), which, as a consequence, does not possess a unique asymptotic state.

Besides their physical interpretation and applicability in the description of the phonon dynamics, these two different master equations can also be solved analytically.

While the pump pulse is intense in order to excite the system out of equilibrium, the probe pulse is less intense and its use is to record the properties of the phonon state through the analysis of the properties of the scattered probe photons. By varying the time delay between pump and probe pulses, the relaxing phonon dynamics can thus be systematically examined.

As in the case of the pump, the probe laser pulses can be described using a coherent state $| \alpha \rangle$, so that the density matrix representing the state of the compound photon–phonon system just before probing can still be written in factorized form, $| \alpha \rangle \langle \alpha | \otimes \rho (t)$ (compare with (1)). However, one has $| \alpha | \ll | \nu |$, since in general the probe pulses are less intense than the pump ones. In such a case, the unitary operator ${ \mathcal U }\equiv \exp (-{\rm{i}}{G})$ in (10), with $G=\gamma \,{\hat{a}}^{\dagger }\,\hat{b}+{\gamma }^{* }\,\hat{a}\,{\hat{b}}^{\dagger }+\eta \,{({\hat{a}}^{\dagger })}^{2}\,{\hat{b}}^{2}+{\eta }^{* }\,{\hat{a}}^{2}{({\hat{b}}^{\dagger })}^{2}$, representing in general the action of the laser pulses on the target material, cannot be written as in (11), since the probe pulse can no longer be considered semiclassical, and therefore no mean-field approximation can be used. Nevertheless, when acting on $| \alpha \rangle \langle \alpha | \otimes \rho (t)$, ${ \mathcal U }$ can always be decomposed as

$$\begin{eqnarray}&&{ \mathcal U }={U}_{\alpha }+{{\rm{\Delta }}}_{\alpha },\end{eqnarray} \tag{ 22 }$$

in terms of the new unitary operator

$$\begin{eqnarray}&&{U}_{\alpha }\equiv \exp (-{{\rm{i}}{G}}_{\alpha }),{G}_{\alpha }=\gamma \,{\hat{a}}^{\dagger }\,\hat{b}+{\gamma }^{* }\,\hat{a}\,{\hat{b}}^{\dagger }+\eta \,{({\alpha }^{* })}^{2}\,{\hat{b}}^{2}+{\eta }^{* }\,{\alpha }^{2}{({\hat{b}}^{\dagger })}^{2}\end{eqnarray} \tag{ 23 }$$

and the difference ${{\rm{\Delta }}}_{\alpha }$

$$\begin{eqnarray*}{{\rm{\Delta }}}_{\alpha }\equiv { \mathcal U }-{U}_{\alpha } & = & {\displaystyle \int }_{0}^{1}{\rm{d}}{s}\,{\partial }_{s}[{U}_{\alpha }^{1-s}\ {{ \mathcal U }}^{s}]= & {\rm{i}}{\displaystyle \int }_{0}^{1}{\rm{d}}{s}\,{U}_{\alpha }^{1-s}(G-{G}_{\alpha }){{ \mathcal U }}^{s}.\end{eqnarray*}$$

Note that ${U}_{\alpha }$ is akin to a mean-field approximation in only the quadratic terms, thus introducing the parameter α. The full expression for ${ \mathcal U }$ in (22) is, however, still exact. Now we introduce an approximation that goes beyond the mean-field, while still guaranteeing the tractability of the calculation. The details of its application will be discussed in section 2.8. A first order approximation of ${{\rm{\Delta }}}_{\alpha }$ suffices, so that in the probing process ${ \mathcal U }$ can be approximated by the following expression:

$$\begin{eqnarray}{{ \mathcal U }}_{\alpha } & = & {U}_{\alpha }\left\{{\mathbb{I}}-{\rm{i}}{\displaystyle \int }_{0}^{1}{\rm{d}}{s}\,{({U}_{\alpha }^{\dagger })}^{s}[\eta ({\hat{a}}^{\dagger }{}^{2}-{\alpha }^{* }{}^{2}){\hat{b}}^{2}\right.+{\eta }^{* }({\hat{a}}^{2}-{\alpha }^{2}){\hat{b}}^{\dagger }{}^{2}]{({U}_{\alpha })}^{s}\phantom{\rule[-0000]{}{3.18ex}}\}.\end{eqnarray} \tag{ 24 }$$

The state of the compound photon–phonon system after the probe pulse hits the material is then given by ${{ \mathcal U }}_{\alpha }(| \alpha \rangle \langle \alpha | \otimes \rho (t)){{ \mathcal U }}_{\alpha }^{\dagger }$. In general, the bilinear terms in the exponential dominate over the quadratic ones, so that one can safely assume $\eta \approx 0;$ however, there are situations in which, due to crystal symmetries or to specific experimental adjustments, the terms with the linear pieces in photon and phonon operators do not contribute [14], this situation is commonly considered for the sake of simplicity when the generation of a squeezed phonon state is the goal. In such cases $\gamma =0$ and only the quadratic terms survive. Both situation will be discussed in detail in the following.

Our aim is the study of the behavior of the photoexcited phonons as monitored by the probe pulses. This can be achieved by analyzing suitable scattered probe photon observables. Specifically, we shall focus on the behavior of the mean photon number $\langle {\hat{a}}^{\dagger }\hat{a}\rangle$ and on its variance ${{\rm{\Delta }}}_{{\hat{a}}^{\dagger }\hat{a}}$. While $\langle {\hat{a}}^{\dagger }\hat{a}\rangle$ is connected to the reflectivity/transmittance of the probe light and thus to the semiclassical properties of the excited phonons, higher order photon correlations, as in ${{\rm{\Delta }}}_{{\hat{a}}^{\dagger }\hat{a}}$, are needed in order to inspect its quantum aspects. The joint analysis of mean number and its variance, two experimentally accessible photon observables, provides enough information for identifying some phonon quantum features, in particular the presence of squeezing.

The mean probe photon number at time t and its corresponding variance can be suitably written as⁷ :

$$\begin{eqnarray}&&\langle {\hat{a}}^{\dagger }\hat{a}\rangle (t)=\mathrm{Tr}[({{ \mathcal U }}_{\alpha }^{\dagger }\,{\hat{a}}^{\dagger }\hat{a}\,{{ \mathcal U }}_{\alpha })\,| \alpha \rangle \langle \alpha | \otimes \rho (t)],\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{{\hat{a}}^{\dagger }\hat{a}}(t)=\langle {({\hat{a}}^{\dagger }\hat{a})}^{2}\rangle (t)-{(\langle {\hat{a}}^{\dagger }\hat{a}\rangle (t))}^{2},\end{eqnarray} \tag{ 26 }$$

with

$$\begin{eqnarray}&&\langle {({\hat{a}}^{\dagger }\hat{a})}^{2}\rangle (t)=\mathrm{Tr}[({{ \mathcal U }}_{\alpha }^{\dagger }\,{({\hat{a}}^{\dagger }\hat{a})}^{2}\,{{ \mathcal U }}_{\alpha })\,| \alpha \rangle \langle \alpha | \otimes \rho (t)],\end{eqnarray} \tag{ 27 }$$

where the trace operation is now performed on both phonon and photon degrees of freedom. These quantities can be conveniently expressed in terms of suitable phonon correlations, $\langle ({\hat{b}}^{\dagger }){}^{m}{\hat{b}}^{n}\rangle (t)={\mathrm{Tr}}_{\mathrm{phonon}}[({\hat{b}}^{\dagger }){}^{m}{\hat{b}}^{n}\rho ({\rm{t}})]$, with $m,n=0,1,2;$ their explicit forms depend on which of the two probing processes mentioned above is relevant in the examined context (details and further discussion can be found in appendix B).

First-order probing process. When the quadratic terms in the hamiltonian are negligible, i.e. $\eta \approx 0$, a situation most likely occurring in opaque materials, the impulsive unitary operator describing the probe process (24) reduces to

$$\begin{eqnarray}&&{{ \mathcal U }}_{\alpha }=\exp (-{\rm{i}}[\gamma \,{\hat{a}}^{\dagger }\,\hat{b}+{\gamma }^{* }\,\hat{a}\,{\hat{b}}^{\dagger }]).\end{eqnarray} \tag{ 28 }$$

Note that this operator does not depend on α anymore, and that the subscript merely denotes the approximation used. In this case, the mean value of the photon number can be expressed as

$$\begin{eqnarray}\langle {\hat{a}}^{\dagger }\hat{a}\rangle (t) & = & {| \alpha | }^{2}{\cos }^{2}(| \gamma | )+\langle {\hat{b}}^{\dagger }\hat{b}\rangle (t){\sin }^{2}(| \gamma | )\\ & & +{\rm{i}}\displaystyle \frac{\sin (| \gamma | )\cos (| \gamma | )}{| \gamma | }({\gamma }^{* }\,\alpha \langle {\hat{b}}^{\dagger }\rangle (t)-\gamma \,{\alpha }^{* }\,\langle \hat{b}\rangle (t)),\end{eqnarray} \tag{ 29 }$$

while the explicit expression of the variance ${{\rm{\Delta }}}_{{\hat{a}}^{\dagger }\hat{a}}(t)$ is rather more involved; it is collected in appendix B.

Second-order probing process. On the other hand, when there is no contribution from the linear terms in phonon creation and annihilation operators ($\gamma \approx 0$), no generation of coherent states out of the pump process can be achieved. This situation describes the process in which the excited phonon state resulting from the pump process is a squeezed thermal one, a typical situation encountered in transparent materials [14, 18]; in this case, ${{ \mathcal U }}_{\alpha }$ is still given by (24), but with the simplified ${U}_{\alpha }$:

$$\begin{eqnarray}&&{U}_{\alpha }=\exp (-{\rm{i}}[\eta \,{({\alpha }^{* })}^{2}\,{\hat{b}}^{2}+{\eta }^{* }\,{\alpha }^{2}{({\hat{b}}^{\dagger })}^{2}]).\end{eqnarray} \tag{ 30 }$$

The mean value of the photon number takes now the form

$$\begin{eqnarray}\langle {\hat{a}}^{\dagger }\hat{a}\rangle (t) & = & {| \alpha | }^{2}-{\sinh }^{2}({r}_{\alpha })(1+2\langle {\hat{b}}^{\dagger }\hat{b}\rangle (t))+\sinh (2{r}_{\alpha }){\mathfrak{R}}[\langle {\hat{b}}^{2}\rangle (t){{\rm{e}}}^{-{\rm{i}}{\varphi }_{\alpha }}],\end{eqnarray} \tag{ 31 }$$

where ${r}_{\alpha }$ and ${\varphi }_{\alpha }$ are as in (17), but with ν replaced by $\alpha =| \alpha | {{\rm{e}}}^{{\rm{i}}{\theta }_{\alpha }}$, while ${\mathfrak{R}}$ signifies real part; similarly, for the variance one gets:

$$\begin{eqnarray}{{\rm{\Delta }}}_{{\hat{a}}^{\dagger }\hat{a}}(t) & = & {| \alpha | }^{2}-2(1+2\langle {\hat{b}}^{\dagger }\hat{b}\rangle (t)){\sinh }^{2}({r}_{\alpha })\\ & & -{(1+2\langle {\hat{b}}^{\dagger }\hat{b}\rangle (t))}^{2}{\sinh }^{4}({r}_{\alpha })+2\sinh (2{r}_{\alpha }){\mathfrak{R}}[\langle {\hat{b}}^{2}\rangle (t){{\rm{e}}}^{-{\rm{i}}{\varphi }_{\alpha }}]\\ & & +{\sinh }^{2}({r}_{\alpha })\sinh (2{r}_{\alpha })(1+2\langle {\hat{b}}^{\dagger }\hat{b}\rangle (t)){\mathfrak{R}}[\langle {\hat{b}}^{2}\rangle (t){{\rm{e}}}^{-{\rm{i}}{\varphi }_{\alpha }}]\\ & & -{\sinh }^{2}(2{r}_{\alpha }){({\mathfrak{R}}[\langle {\hat{b}}^{2}\rangle (t){{\rm{e}}}^{-{\rm{i}}{\varphi }_{\alpha }}])}^{2}.\end{eqnarray} \tag{ 32 }$$

In order to obtain the explicit time dependence of these two observables, $\langle {\hat{a}}^{\dagger }\hat{a}\rangle (t)$ and ${{\rm{\Delta }}}_{{\hat{a}}^{\dagger }\hat{a}}(t)$, one now has to analyze the phonon dynamics generated by the master equations (20) and (21) and compute the needed phonon correlation functions.

The thermal reservoir has a unique equilibrium state, a Gibbs state determined by the temperature of the bath. This temperature is larger than the initial equilibrium temperature of the target material (here considered to be the usual room temperature of about $300\,{\rm{K}}$), since the highly intense pump pulse is partially absorbed by the crystal. Typical temperatures reached by an absorbing opaque crystal after the pump are around $500\,{\rm{K}}$, while for transparent materials, where the absorption effects can be neglected, the increase in temperature can be estimated to be at most few degrees higher than the initial one. As discussed earlier, the pump pulse excites the phonons, producing displacement and squeezing in the initially thermal state: their subsequent dissipative relaxation back to equilibrium as driven by the presence of the thermal environment, can be studied through the analysis of the typical oscillating behavior of probe photon mean number and variance (see figure 2).

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At this stage, the difference between opaque and transparent materials needs to be taken into account; as mentioned, the final temperature is considerably higher in the case of opaque crystals, like bismuth, due to the absorption of energy from the pump pulse. Meanwhile, in transparent materials, like α-quartz, the final temperature can be expected to be only marginally higher than the initial one. In the model described here, this parameter needs to be either estimated from the crystal properties or fitted from the experimental results. The possibility of melting the target due to the absorption of energy from high intensity pump pulses establishes a natural threshold for opaque and transparent materials. For opaque materials, a limit in the intensity used in the pump pulse is naturally given by the properties of the crystal, while for non-absorbing crystal, i.e. transparent crystal targets, a broad range of intensities (typically referred to as fluence in experiments) can be used.

Concerning the probe process, there is no distinction between the two cases described above, since the probe pulse is always weak in order to avoid an additional excitation of the crystal target.

Linear probing process. The physical situation in which the probe interaction is dominated by mechanisms involving linear terms in photon and phonon operators is adequately described by the linear probing process. This regime is usually achieved by means of a probe pulse that is much weaker than the pump pulse. Some particular situations are worth discussing in detail, listed below according to characteristics of the state ρ of the excited phonons after pumping.

(i) Thermal state. If the pump pulse creates neither any displacement nor squeezing in the initial thermal equilibrium state, no oscillating terms appear in the time evolution of the mean photon number or variance. Only a thermalization process is observed, driving the phonon state to a new equilibrium state, the one supported by the thermal bath; it is characterized by the damping constant ${\lambda }_{T}$ and the bath temperature.

(ii) Coherent thermal state. Another physically likely and relevant situation is when the pump process displaces the thermal state, but does not squeeze it, i.e. $\rho =D({z}_{\nu }){\rho }_{\beta }{D}^{\dagger }({z}_{\nu })$, so that the excited phonons are described by a coherent thermal state. This particular case can be achieved by either a not so intense pump pulse or by a negligible coupling constant η. In this situation, the mean photon number and the corresponding variance both behave in the same way as a function of the time delay between pump and probe: both observables show typical damped oscillations with frequency given by the excited phonon mode Ω, as depicted in figure 2. This is also the typical behavior of the reflectivity/transmittance reported in experiments on optically excited coherent phonons.

(iii) Squeezed thermal state. Another possibility is given by the excitation of phonons in a squeezed thermal state, $\rho =S({\xi }_{\nu }){\rho }_{\beta }\,{S}^{\dagger }({\xi }_{\nu })$. It should be noted that this case is less likely to occur since it corresponds to a physical situation in which the bilinear contributions in the effective hamiltonian describing the pumping process can be neglected. In this case the mean photon number does not present oscillations, whereas the variance oscillates with twice the phonon mode frequency Ω; furthermore, both observables contain damping factors characterized by the dissipative constant ${\lambda }_{T}$. The presence of the frequency $2{\rm{\Omega }}$ in the variance oscillations is a signal of squeezing in the excited phonon state; indeed, if the squeezing parameter ${\xi }_{\nu }$ is zero, then this contribution disappears.

(iv) Squeezed coherent thermal state. The most general situation occurs when the pumping process excites a phonon state which is both squeezed and coherent, as in (18), a likely situation in this kind of non-equilibrium optical experiments. In this case, the mean photon number oscillates with frequency Ω, with an amplitude that, besides depending on displacement, squeezing and probe intensity parameters, is damped in time by the factor ${{\rm{e}}}^{-{\lambda }_{T}t/2}$. On the other hand, the associated variance shows oscillating terms with frequency Ω, which are damped by factors ${{\rm{e}}}^{-{\lambda }_{T}t/2}$ or ${{\rm{e}}}^{-3{\lambda }_{T}t/2}$, and terms with frequency $2{\rm{\Omega }}$, damped by ${{\rm{e}}}^{-{\lambda }_{T}t}$. This is due to the different damping behavior in (42)–(44). Furthermore, one finds that the oscillating piece with frequency $2{\rm{\Omega }}$ is present only if the squeezing parameter ${\xi }_{\nu }$ is non-vanishing, i.e. only in presence of squeezed phonons. However, it is in general difficult to isolate such a contribution or to have evidence of a $2{\rm{\Omega }}$ contribution in the corresponding Fourier spectrum; indeed, at least for high intensity pump and probe pulses, commonly used in pump and probe experiments where the crystal target is non-absorbing, the mean photon number and variance show a behavior in time similar to the one shown in figure 2.

For low intensity probe pulses ${| \alpha | }^{2}\lt 1$ and pump intensity ${| \nu | }^{2}$ two orders of magnitude larger, describing experiments with absorbing materials, one can notice the occurrence of a phase shift between the oscillations of mean number and variance, which is present only if the phonon state out of the pump process has non-vanishing squeezing (see figure 3). This shift is time dependent and in general vanishes after a few oscillations (see appendix C for details). The detection of such a shift can be taken as clear signal of the presence of squeezed phonons in pump and probe experiments involving opaque materials. The visibility of the shift in actual experiments depends on the specific features of the setups: each given experimental situation fixes the magnitude of the phenomenological parameters associated with both the pump and probe processes and the dissipative dynamics and, as a consequence, also the magnitude of the shift.

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Quadratic probing process. As already mentioned before, due either to crystal symmetries or to specific experimental situations, optical excitation of phonons in some crystal targets do not involve coherent contributions: only the quartic pieces in the hamiltonian (9) are relevant [14]; in other words, $\gamma =0$. We focus below on relevant physical situations, listed according to the characteristics of the excited phonon state ρ out of the pumping process; note that, in the present situation, the displacement parameter ${z}_{\nu }$ is always vanishing.

(i) Thermal state. When the pump is not able to excite the target material, i.e. the phonon state after pumping is still a thermal state, the behavior of the mean photon number and its variance is given by exponential damping terms, signaling thermalization of the phonon state to the bath temperature.

(ii) Squeezed thermal state. In this case, while the photon mean number oscillates with frequency $2{\rm{\Omega }}$, the corresponding variance contains oscillatory contributions with frequency $2{\rm{\Omega }}$ and $4{\rm{\Omega }}$. In addition, one finds that, in both cases, these oscillations are present only when the squeezing parameter ${\xi }_{\nu }$ is non-zero: their detection would then constitute a clear signal of the presence of squeezing in the pump excited phonons.

Notice that for small values of the squeezing parameter ${\xi }_{\nu }$, the oscillating contribution with frequency $4{\rm{\Omega }}$ in the variance can be neglected; in this case, both the mean and variance oscillate with the same frequency and in addition these oscillations are perfectly in phase (the behavior is similar to the one reported in figure 2, although the oscillations now occur at frequency $2{\rm{\Omega }}$).

As discussed in section 3, the characteristic property of the dephasing bath is the conservation of the number of phonons in the material. Thus, if the pump pulse is too weak to produce excitations, leaving the phonons in thermal equilibrium, the phonon dynamics do not show any time dependence since the thermal state is a stationary state of the master equation (21). A non trivial time evolution occurs only when the pump pulse is able to generate phonon displacement and squeezing. As in the case of the thermal reservoir, also for a dephasing environment the behavior of the photon observables depends on the reading mechanism considered in the probe process.

Linear probing process. We shall first consider the situation in which the probing mechanism can be described using only the bilinear terms in the hamiltonian (9) describing photon–phonon interactions. Again we describe different cases, classified according to the phonon state ρ.

(i) Coherent thermal state. When the pump process excites coherent phonons only, the mean photon number oscillates with frequency Ω, damped by a factor ${{\rm{e}}}^{-{\lambda }_{{\rm{D}}}t/2}$. On the other hand, the variance shows oscillations both with frequency Ω and $2{\rm{\Omega }};$ however, the latter are more damped, the decaying factors being proportional to ${{\rm{e}}}^{-{\lambda }_{{\rm{D}}}t}$ and ${{\rm{e}}}^{-2{\lambda }_{{\rm{D}}}t}$, respectively. In practice, mean number and variance are seen to oscillate in time with the same frequency and in addition with the same phase. In the literature, the presence in the variance of oscillating terms with frequency $2{\rm{\Omega }}$ has been repeatedly considered to be a signal of the presence of phonon squeezing. Instead, here we see that, due to the effects of the dephasing bath, a $2{\rm{\Omega }}$ oscillating contribution can arise even in absence of squeezing, and therefore that the appearance of such oscillations in the variance cannot be taken as an unquestionable signal of squeezing.

(ii) Squeezed coherent thermal state. The most likely situation is, however, the excitation of squeezed coherent thermal phonons. In this more general case, for low intensity pulses, the dominant terms in the mean photon number and variance take the form [35]:

$$\begin{eqnarray}\langle {\hat{a}}^{\dagger }\hat{a}\rangle (t)\sim & - & | \alpha | | {z}_{\nu }| {{\rm{e}}}^{-{\lambda }_{{\rm{D}}}t/2}\sin (2| \gamma | )[{ \mathcal A }\sin ({\rm{\Omega }}t+{\phi }_{{ \mathcal A }})-{ \mathcal B }\sin ({\rm{\Omega }}t+{\phi }_{{ \mathcal B }})],\\ {{\rm{\Delta }}}_{{\hat{a}}^{\dagger }\hat{a}}(t)\sim & - & | \alpha | | {z}_{\nu }| {{\rm{e}}}^{-{\lambda }_{{\rm{D}}}t/2}\sin (2| \gamma | )[{ \mathcal C }\sin ({\rm{\Omega }}t+{\phi }_{{ \mathcal A }})-{ \mathcal D }\sin ({\rm{\Omega }}t+{\phi }_{{ \mathcal B }})],\end{eqnarray} \tag{ 49 }$$

where

$$\begin{eqnarray}&&{ \mathcal A }=\cosh ({r}_{\nu }),\qquad {\phi }_{{ \mathcal A }}={\theta }_{\alpha }-{\theta }_{\gamma }-{\theta }_{z},\end{eqnarray} \tag{ 50 }$$

$$\begin{eqnarray}&&{ \mathcal B }=\sinh ({r}_{\nu }),\qquad {\phi }_{{ \mathcal B }}={\theta }_{\alpha }-{\theta }_{\gamma }+{\theta }_{z}-{\varphi }_{\nu },\end{eqnarray} \tag{ 51 }$$

$$\begin{eqnarray}{ \mathcal C } & = & [{\cos }^{2}(| \gamma | )+(1+2{n}_{\mathrm{eq}})(2\cosh (2{r}_{\nu })-1){\sin }^{2}(| \gamma | )]\cosh ({r}_{\nu }),\end{eqnarray} \tag{ 52 }$$

$$\begin{eqnarray}{ \mathcal D } & = & [1+2({n}_{\mathrm{eq}}+(1+2{n}_{\mathrm{eq}})\cosh (2{r}_{\nu })){\sin }^{2}(| \gamma | )]\sinh ({r}_{\nu }).\end{eqnarray} \tag{ 53 }$$

Both observables oscillate with the frequency Ω; however, there is a phase-shift $\delta {\rm{\Omega }}$ between the two curves given by

$$\begin{eqnarray}&&\delta {\rm{\Omega }}=\arctan \left[\displaystyle \frac{({ \mathcal A }{ \mathcal D }+{ \mathcal B }{ \mathcal C })\sin ({\varphi }_{\nu }-2{\theta }_{z})}{{ \mathcal A }{ \mathcal C }+{ \mathcal B }{ \mathcal D }-({ \mathcal B }{ \mathcal C }+{ \mathcal A }{ \mathcal D })\cos ({\varphi }_{\nu }-2{\theta }_{z})}\right].\end{eqnarray} \tag{ 54 }$$

This is clearly visible in the plots of figure 4. Notice that this shift is constant in time; further, it vanishes as the squeezing parameter ${r}_{\nu }$ tends to zero. Therefore, in the specified physical situation, the detection of a phase shift between mean number and variance surely represents a clear signal of the presence of pump induced squeezed phonons.

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Quadratic probing process. One can similarly analyze the case in which the most relevant mechanism accounting for the photon–phonon interaction is due to the quartic terms in the proposed hamiltonian; in this case, as discussed before for the thermal bath, no displacement can be generated in the phonon states in both pumping and probing processes. The most likely situation to occur due to the action of the pump is then the excitation of a

(i) Squeezed thermal state. In this situation, the mean photon number is found to oscillate with frequency $2{\rm{\Omega }}$, while the corresponding variance has oscillating contributions with both frequencies $2{\rm{\Omega }}$ and $4{\rm{\Omega }}$. Nevertheless, when the probe pulses are of low intensity, ${| \alpha | }^{2}\lt 1$, both observables contain essentially only $2{\rm{\Omega }}$ oscillations, and in addition with exactly the same phase: the obtained behavior is qualitatively the same as the one shown in figure 2.

When the photon–phonon interaction is dominated by processes in which the interaction between the light and the lattice vibrations is such that one photon is transformed into one phonon, only linear terms in phonon creation and annihilation operators are relevant in the expression (24) of the operator ${{ \mathcal U }}_{\alpha }$ describing the probe process. In this case, since the quartic coupling η vanishes, its expression reduces to the one given in (28)

$$\begin{eqnarray}&&{{ \mathcal U }}_{\alpha }=\exp (-{\rm{i}}[\gamma \,{\hat{a}}^{\dagger }\,\hat{b}+{\gamma }^{* }\,\hat{a}\,{\hat{b}}^{\dagger }]).\end{eqnarray} \tag{ 63 }$$

In order to obtain probe photon correlations

$$\begin{eqnarray}&&\langle {({\hat{a}}^{\dagger })}^{m}{\hat{a}}^{n}\rangle ({t})=\mathrm{Tr}[{{ \mathcal U }}_{\alpha }^{\dagger }\,{({\hat{a}}^{\dagger })}^{m}{\hat{a}}^{n}\,{{ \mathcal U }}_{\alpha }\,| \alpha \rangle \langle \alpha | \otimes \rho ({t})],\end{eqnarray} \tag{ 64 }$$

one has to compute the action of this operator on monomials in photon creation and annihilation operators. This can be done conveniently by introducing the bilinear operators

$$\begin{eqnarray}{J}_{1} & = & \displaystyle \frac{{\gamma }^{* }\,\hat{a}\,{\hat{b}}^{\dagger }+\gamma \,{\hat{a}}^{\dagger }\,\hat{b}}{2| \gamma | },\\ {J}_{2} & = & {\rm{i}}\displaystyle \frac{{\gamma }^{* }\,\hat{a}\,{\hat{b}}^{\dagger }-\gamma \,{\hat{a}}^{\dagger }\,\hat{b}}{2| \gamma | },\\ {J}_{3} & = & \displaystyle \frac{{\hat{a}}^{\dagger }\hat{a}-{\hat{b}}^{\dagger }\hat{b}}{2},\end{eqnarray} \tag{ 65 }$$

that obey the su(2) algebraic commutation relations

$$\begin{eqnarray}&&[{J}_{i},{J}_{j}]={\rm{i}}{\epsilon }_{{ijk}}{J}_{k}.\end{eqnarray} \tag{ 66 }$$

For instance, since ${\hat{a}}^{\dagger }\hat{a}=N/2+{J}_{3}$, with $N={\hat{a}}^{\dagger }\hat{a}+{\hat{b}}^{\dagger }\hat{b}$ the total boson number, commuting with all J_i, one easily finds:

$$\begin{eqnarray}&&{{ \mathcal U }}_{\alpha }^{\dagger }\,{\hat{a}}^{\dagger }\hat{a}\,{{ \mathcal U }}_{\alpha }=\displaystyle \frac{N}{2}+\cos (2| \gamma | ){J}_{3}+\sin (2| \gamma | ){J}_{2}.\end{eqnarray} \tag{ 67 }$$

Recalling (64) above, from this result one immediately gets the expression of the mean photon number $\langle {\hat{a}}^{\dagger }\hat{a}\rangle (t)$ reported in (29). Using similar techniques, one can compute the corresponding variance [35]

$$\begin{eqnarray}{{\rm{\Delta }}}_{{\hat{a}}^{\dagger }\hat{a}}(t) & = & {| \alpha | }^{2}{\cos }^{2}(| \gamma | )+(\langle {({\hat{b}}^{\dagger }\hat{b})}^{2}\rangle (t)-\langle {\hat{b}}^{\dagger }\hat{b}{\rangle }^{2}(t)){\sin }^{4}(| \gamma | )\\ & & +\langle {\hat{b}}^{\dagger }\hat{b}\rangle (t)(2{| \alpha | }^{2}+1){\cos }^{2}(| \gamma | ){\sin }^{2}(| \gamma | )\\ & & -\displaystyle \frac{{\cos }^{2}(| \gamma | ){\sin }^{2}(| \gamma | )}{{| \gamma | }^{2}}[{({\gamma }^{* })}^{2}{\alpha }^{2}(\langle ({\hat{b}}^{\dagger }){}^{2}\rangle (t)-\langle {\hat{b}}^{\dagger }{\rangle }^{2}(t))\\ & & +{\gamma }^{2}{({\alpha }^{* })}^{2}(\langle {\hat{b}}^{2}\rangle (t)-\langle \hat{b}{\rangle }^{2}(t))+2{| \gamma | }^{2}{| \alpha | }^{2}\langle {\hat{b}}^{\dagger }\rangle \langle \hat{b}\rangle (t)]\\ & & +{\rm{i}}\displaystyle \frac{\cos (| \gamma | )\sin (| \gamma | )}{| \gamma | }({\gamma }^{* }\,\alpha \,\langle {\hat{b}}^{\dagger }\rangle (t)-\gamma \,{\alpha }^{* }\,\langle \hat{b}\rangle (t))\\ & & +2{\rm{i}}\displaystyle \frac{\cos (| \gamma | ){\sin }^{3}(| \gamma | )}{| \gamma | }[{\gamma }^{* }\,\alpha (\langle {\hat{b}}^{\dagger }{}^{2}\hat{b}\rangle (t)-\langle {\hat{b}}^{\dagger }\hat{b}\rangle \langle {\hat{b}}^{\dagger }\rangle (t))\\ & & -\gamma \,{\alpha }^{* }(\langle {\hat{b}}^{\dagger }{\hat{b}}^{2}\rangle (t)-\langle {\hat{b}}^{\dagger }\hat{b}\rangle \langle \hat{b}\rangle (t))],\end{eqnarray} \tag{ 68 }$$

where $\langle {({\hat{b}}^{\dagger })}^{m}{\hat{b}}^{n}\rangle (t)={\mathrm{Tr}}_{\mathrm{phonon}}[{({\hat{b}}^{\dagger })}^{m}{\hat{b}}^{n}\rho ({\rm{t}})]$, with $m,n\in \{0,1,2\}$, represent phonon correlations.

When processes in which one photon is absorbed and one phonon is produced, and vice versa, are not contributing to the photon–phonon interaction, i.e. the coupling constant γ is zero, the operator ${ \mathcal U }$ implementing the probing process in (24) can be approximated by:

$$\begin{eqnarray}&&{{ \mathcal U }}_{\alpha }={U}_{\alpha }\left[{\mathbb{I}}-{\rm{i}}{\int }_{0}^{1}{\rm{d}}{s}{({U}_{\alpha }^{\dagger })}^{s}(\eta (({\hat{a}}^{\dagger }){}^{2}-{({\alpha }^{* })}^{2}){\hat{b}}^{2}+{\eta }^{* }({\hat{a}}^{2}-{\alpha }^{2})({\hat{b}}^{\dagger }){}^{2}){({U}_{\alpha })}^{s}\right],\end{eqnarray} \tag{ 69 }$$

with now

$$\begin{eqnarray}&&{U}_{\alpha }=\exp [-{\rm{i}}(\eta \,{({\alpha }^{* })}^{2}\,{\hat{b}}^{2}+{\eta }^{* }\,{\alpha }^{2}\,{({\hat{b}}^{\dagger })}^{2})].\end{eqnarray} \tag{ 70 }$$

Since ${U}_{\alpha }$ involves no photon operators but only phonon operators, the photon correlations in (64) can be expressed in terms of phonon correlations only. For instance, in the case of the mean number, one finds:

$$\begin{eqnarray}&&\langle {\hat{a}}^{\dagger }\hat{a}\rangle ({t})={| \alpha | }^{2}-2\,{\mathfrak{R}}\left\{2{\rm{i}}\eta {({\alpha }^{* })}^{2}{\int }_{0}^{1}{\rm{d}}{s}\mathrm{Tr}[{({{U}}_{\alpha }^{\dagger })}^{s}\,{\hat{b}}^{2}\,{{U}}_{\alpha }^{s}\ \rho ({t})]\right\}.\end{eqnarray} \tag{ 71 }$$

Further, notice that ${U}_{\alpha }$ acts on phonon states as a squeezing operator (see (15)), with squeezing parameter ${\xi }_{\alpha }={r}_{\alpha }\,{{\rm{e}}}^{{\rm{i}}{\varphi }_{\alpha }}$ explicitly given by:

$$\begin{eqnarray*}&&{r}_{\alpha }=2| \eta | {| \alpha | }^{2},\quad {\varphi }_{\alpha }=2{\theta }_{\alpha }+\displaystyle \frac{\pi }{2}-{\theta }_{\eta },\end{eqnarray*}$$

where $\alpha =| \alpha | \,{{\rm{e}}}^{i{\theta }_{\alpha }}$ and $\eta =| \eta | {{\rm{e}}}^{{\rm{i}}{\theta }_{\eta }}$. Then, one sees that

$$\begin{eqnarray}{({U}_{\alpha }^{\dagger })}^{s}\,{\hat{b}}^{2}\,{U}_{\alpha }^{s} & = & {\hat{b}}^{2}{\cosh }^{2}({{sr}}_{\alpha })+({\hat{b}}^{\dagger }){}^{2}{{\rm{e}}}^{2{\rm{i}}{\varphi }_{\alpha }}{\sinh }^{2}({{sr}}_{\alpha })\\ & & -(1+2\,{\hat{b}}^{\dagger }\hat{b}){{\rm{e}}}^{{\rm{i}}{\varphi }_{\alpha }}\cosh ({{sr}}_{\alpha })\sinh ({{sr}}_{\alpha }).\end{eqnarray} \tag{ 72 }$$

Inserting this result in (71), one finally obtains for the mean photon number the expression reported in (31). The expression (32) for the corresponding variance ${{\rm{\Delta }}}_{{\hat{a}}^{\dagger }\hat{a}}(t)$ can be similarly computed [35].