# Moving mirror-field dynamics under intrinsic decoherence

Alejandro R. Urzúa1 and Héctor M. Moya-Cessa2

1Instituto de Ciencias Físicas, Cuernavaca, Morelos
2Instituto Nacional de Astrofísica, Óptica y Electrónica, Tonantzintla, Puebla

###### Abstract

We study the decaying dynamics in the mirror-field interaction by means of the
intrinsic decoherence scheme. Factorization of the mirror-field Hamiltonian
with the use of displacement operators, allows us to calculate the explicit
solution to Milburn’s equation for arbitrary initial conditions. We show
expectation values, correlations, and Husimi functions for the solutions
obtained.

##  1 Introduction

The presence of decoherence in quantum mechanical systems is an undesirable
effect coming mainly from two sources: interaction between internal degrees of
freedom, and the interaction with the environment via some coupling [1].
Intrinsic decoherence [2, 3] is a topic that has recently raised great
interest, as it has become a way to describe phase decay which is one of the
most damaging effects of quantum physical systems, making mandatory its study.
Therefore, quantifying the rate at which an isolated system loses its phase
coherence in time is needed in order to try to avoid such a harmful
phenomenon. We will apply its study here to an optomechanical system. Such
systems deal with the interaction of light and mechanical devices coupled by
means of their internal degrees of freedom in the most simple setting [4, 5,
6]. One realization of such systems is a cavity containing a quantized
electromagnetic field coupled to a mirror experiencing harmonic motion; in
which interesting phenomena can occur due to the feedback action between the
subsystems, like optomechanical entanglement [7], or the creation of real
photons in the cavity due to the dynamical Casimir effect and how the phase
damping of the intrinsic decoherence intervene in the time dynamics [8].

Since the seminal work by Milburn [2] on the possibility to model decoherence
based on a simple modification of unitary Schrödinger evolution, much work has
been done pointing in several directions, depending on how decoherence is
tried to be diminished, or toked in advance. Moreover, intrinsic decoherence
produces a Master Equation of the Lindblad type making it possible to compare
with other methods that describe open systems, like master equations and
quantum trajectories.

The last decade has been very prolific in this field we want to board, the
community has taken the task to study a vast amount of systems that can
experience intrinsic decoherence. We can see research on quantification of
non-classicality [9], qubits for quantum information processing [10], bipolar
spin systems [11], quantum dot correlations [12], Heisenberg XYZ spin chains,
quantum-memory-assisted two-qubit, and the temporal evolution of quantum
correlations [13, 14, 15], quantum-memory-assisted entropic uncertainty,
mixedness, and entanglement dynamics in two-qubit system [16], symmetric spin-
orbit model [17], two-qubit quantum Fisher information [18], two-qubit
maximally entangled Bell states [19], trapped ions [20], isolated Heisenber
and Aubry-André spin models [21], two coupled quantum dots [22], N-level
atomic system [23], two-coupled qubit two-level cavity [24], two-level atom
[25], state transfer in spin channels [26], Heisenber anisotropic interaction
[27], nonlocal advantage of quantum coherence [28], qutrit teleportation [29],
and quantum dense coding [30]. When we look at specific examples related
optomechanical systems, we find notable works on Jaynes-Cummings under
intrinsic decoherence [31, 32, 33], bimodal multiquanta Jaynes-Cummings [34],
entanglement two Tavis-Cummings (no-RWA JC) [35], and ultra-strong coupled
harmonic oscillator in cavities [36].

In this manuscript, we analyze the effects of intrinsic decoherence in a
mirror and a quantized field coupled by means of radiation pressure and
harmonic mechanical motion. By using solutions recently proposed by us to
study the one-dimensional displaced harmonic oscillator [37] and the three-
coupled one-dimensional harmonic oscillators [38], we solve the complete form
of the equation instead of the master equations Milburn obtained at first.

##  2 Moving mirror-field interaction

The standard setup for the moving mirror-field interaction is a cavity
enclosing the radiation, such as the quintessential Fabry-Pérot interferometer
[39], where two perfectly reflecting mirrors, one fixed and the other
experimenting harmonic motion described by the macroscopic canonical position
$\hat{q}(t)$; inside the cavity, there’s a coherent driven field
$\hat{a}\left(\hat{a}^{\dagger}\right)$ pumped by a tunable laser [5]. The
interaction between the mirror and the field is given by radiation pressure
coupled with the (microscopical) number of modes of the field and the
(macroscopical) position quadrature of the mechanical device. The Hamiltonian
for the moving mirror-field interaction then reads [40],

$\displaystyle\hat{H}=\omega\hat{n}+\nu\hat{N}+\chi\hat{n}\left(\hat{b}^{\dagger}+\hat{b}\right)$
(1)

where we have defined the number operators: $\hat{n}=\hat{a}^{\dagger}\hat{a}$
for the field, and $\hat{N}=\hat{b}^{\dagger}\hat{b}$ for the mirror. The
parameters $\omega$, $\nu$, and $\chi$ are the frequencies of the field, the
mirror, and the coupling strength, respectively. This Hamiltonian may be
rewritten in a diagonal form with the help of a composite _displacement
operator_ in the mirror eigenbasis $(\hat{b})$ as

$\displaystyle\hat{H}=\hat{D}^{\dagger}\left(\frac{\chi\hat{n}}{\nu}\right)\left[\nu\hat{N}+\omega\hat{n}-\frac{\chi^{2}}{\nu^{2}}\hat{n}^{2}\right]\hat{D}\left(\frac{\chi\hat{n}}{\nu}\right),$
(2)

where
$\hat{D}^{\dagger}\left(\tfrac{\chi\hat{n}}{\nu}\right):=\exp\left(\left[\tfrac{\chi}{\nu}\hat{b}^{\dagger}-\tfrac{\chi^{*}}{\nu^{*}}\hat{b}\right]\otimes\hat{n}\right)$,
that is a tensor element of the two eigenbases, acting as a true displacement
operator on the mirror basis, and as an exponential of the number operator in
the field basis.

##### Analytical solution

There exists a straightforward solution of the Schrödinger equation associated
to Hamiltonian (2) when the initial condition is set to
$\ket{\alpha}_{f}\otimes\ket{\beta}_{m}$, giving the expression (see refs.
[40, 41] for the detailed solution via exponential disentanglement),

$\ket{\psi(t)}=e^{-|\alpha|^{2}/2}\sum\limits_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}e^{\mathrm{i}(\chi/\nu)^{2}n^{2}\left(t-\sin
t\right)}\ket{n}\ket{\phi_{n}}$ (3)

where $\ket{\phi_{n}}\equiv\ket{\beta
e^{-\mathrm{i}t}+(\chi/\nu)n(1-e^{-\mathrm{i}t})}$; with $\beta$ the amplitude
of the initial coherent state of the mirror; $\alpha$ the amplitude of the
initial coherent state of the field and $\ket{n}$ are number states of the
quantized field. This solution shows a constant number of photon modes in the
field, and periodic amplitudes in the phonon modes and position (quadrature)
in the mirror, see Figure 1. We expect that the behavior in the field is
preserved after the decoherence drive is applied and the amplitudes and
position in the mirror might experience decaying modulation depending on the
parameters involved in the intrinsic decoherence dynamic.

Figure 1: Expectation values for $(a)$ the number of phonon modes
$\braket{\hat{N}}$ and $(b)$ the position quadrature $\braket{\hat{q}}$ in the
mirror, using (3).

##### Intrinsic decoherence scheme

According to the seminal work of Milburn [2], we can observe decaying dynamics
in the Schrödinger evolution of quantum systems modifying the mathematical
structure of the governing equation. This modification takes into account the
disorder-induced stochastic phase changes in the Hamiltonian of the system at
very short times. In its closed form, the density matrix evolution turns out
to be

$\dot{\rho}_{S}\mapsto\dot{\rho}_{M}=\gamma\left(e^{-\mathrm{i}\frac{\hat{H}}{\gamma}}\hat{\rho}e^{\mathrm{i}\frac{\hat{H}}{\gamma}}-\hat{\rho}\right),$
(4)

where $\hat{H}$ is the system’s Hamiltonian, and $\gamma$ is the decay rate
parameter of the intrinsic decoherence, giving the attributes to determine the
time scale of the coherence suppression. We can, of course, expand equation
(4) in terms of $\gamma^{-1}$ to obtain the $n$-th order of approximation,
which leads to the emergence of a Lindblad master equation. But on the other
side, the equation (4) can be solved analytically using the adequate _ansatz_

$\hat{\rho}(t)=e^{-\gamma t}e^{\hat{S}t}\hat{\rho}(0),$ (5)

where we have defined the _superoperator_ [42]

$\hat{S}\hat{\rho}=\gamma
e^{-\mathrm{i}\frac{\hat{H}}{\gamma}}\hat{\rho}e^{\mathrm{i}\frac{\hat{H}}{\gamma}},$

such that the Taylor expansion of the action gives

$e^{\hat{S}t}\hat{\rho}(0)=\sum_{k=0}^{\infty}\frac{(\gamma
t)^{k}}{k!}\,\hat{\rho}_{k},$ (6)

with the $k$-th element of the density matrix $\hat{\rho}$ defined by

$\hat{\rho}_{k}:=\ket{\psi_{k}}\bra{\psi_{k}},\qquad\ket{\psi_{k}}=e^{-\frac{\mathrm{i}k}{\gamma}\hat{H}}\ket{\psi(0)}.$
(7)

Then, using (7), inserting in (6), and putting all together with (5), we
arrive at the explicit form of the solution, given by

$\hat{\rho}(t)=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\ket{\psi_{k}}\bra{\psi_{k}}.$ (8)

For the particular system we are dealing with, the $k$-th element of the
wavefunction takes the form

$\ket{\psi_{k}}=\hat{D}^{\dagger}\left(\frac{\chi\hat{n}}{\nu}\right)e^{-\frac{\mathrm{i}k}{\gamma}\left[\nu\hat{N}+\omega\hat{n}-\frac{\chi^{2}}{\nu}\hat{n}^{2}\right]}\hat{D}\left(\frac{\chi\hat{n}}{\nu}\right)\ket{\psi(0)},$
(9)

that recalls the analytical Schrödinger solution. At this point, we can apply
(9) to a direct product of initial wavefunctions for the mirror and the field:
$\ket{\psi(0)}=\ket{\phi(0)}_{m}\otimes\ket{\theta(0)}_{f}$.

This depicted scheme is a mathematical procedure relying on a _Poisson model_
for the short-time stochastic behavior of the system, as Milburn stated. On
the other side, selecting suitable initial conditions $\ket{\psi(0)}$ is
dependent on the particular system we have, in the case of optomechanical
devices that coupled electromagnetic fields and harmonic moving mirror, the
selection of coherent states $\ket{\alpha}$ for the field and $\ket{\beta}$
for the mirror can be viewed as obvious —in broad sense we have, of course, a
system of two-coupled harmonic oscillators.— Finally, the existence of the
limit in (8) precludes how many analytical solutions can we obtain, since for
some initial wavefunctions, the limit may not exist, nevertheless it can be
always solved numerically at some truncation term $<\infty$.

##  3 Solution and observables

###  3.1 Expectation values

If we have a suitable operator $\hat{A}$ that represents an observable $a$,
the expectation value of this operator can be obtained from the density matrix
$\hat{\rho}$ as
$\mathrm{tr}\left(\hat{\rho}\hat{A}\right)\equiv\bra{\psi}\hat{A}\ket{\psi}$,
for some wavefunction $\ket{\psi}$ in the eigenbasis of the system [43]. Now,
for the density matrix in (8) that depends on the $k$-th component
$\hat{\rho}_{k}$, the calculation of the expectation value reduces to

$\langle\hat{A}\rangle:=e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\psi_{k}}\hat{A}\ket{\psi_{k}},$ (10)

for $\ket{\psi_{k}}$ defined by (9). Thus, given the form of $\ket{\psi_{k}}$,
we need to obtain the action of the displacement operators and the exponential
maps onto the operator $\hat{A}$. At this point, we are interested in the most
descriptive features in the optomechanical system: the number of photon modes
in the field, that we predict has to be constant; the number of mechanical
modes and quadrature in the mirror, that should have a decaying dynamic
depending on the decoherence rate $\gamma$; a set of statistical estimators,
like the Hong-ou-Mandel parameters, and the covariance; finally, a
representation in phase-space of the dynamics via the Husimi
$\hat{Q}$-function.

####  3.1.1 Number of phonon modes $\braket{\hat{N}}$ in the mirror

We start calculating the expectation value of the number of phonon modes in
the mirror. The left-right action of the wavefunction $\ket{\psi_{k}}$ onto
the operator $\hat{N}$ is straightforward, since we know the expansion

$\displaystyle\bra{\psi_{k}}\hat{N}\ket{\psi_{k}}$
$\displaystyle=\bra{\psi(0)}\left(\hat{D}^{\dagger}\left(\frac{\chi\hat{n}}{\nu}\right)e^{\frac{\mathrm{i}k}{\gamma}\left[\nu\hat{N}+\omega\hat{n}-\frac{\chi^{2}}{\nu}\hat{n}^{2}\right]}\hat{D}\left(\frac{\chi\hat{n}}{\nu}\right)\right)\hat{N}\left(\hat{D}^{\dagger}\left(\frac{\chi\hat{n}}{\nu}\right)e^{-\frac{\mathrm{i}k}{\gamma}\left[\nu\hat{N}+\omega\hat{n}-\frac{\chi^{2}}{\nu}\hat{n}^{2}\right]}\hat{D}\left(\frac{\chi\hat{n}}{\nu}\right)\right)\ket{\psi(0)}$
(11)
$\displaystyle=\bra{\psi(0)}\left(\hat{D}^{\dagger}\left(\frac{\chi\hat{n}}{\nu}\right)e^{\frac{\mathrm{i}k}{\gamma}\left[\nu\hat{N}+\omega\hat{n}-\frac{\chi^{2}}{\nu}\hat{n}^{2}\right]}\right)\times\left(\hat{N}-\frac{\chi}{\nu}\hat{n}\left(\hat{b}^{\dagger}+\hat{b}\right)+\frac{\chi^{2}}{\nu^{2}}\hat{n}^{2}\right)\left(e^{-\frac{\mathrm{i}k}{\gamma}\left[\nu\hat{N}+\omega\hat{n}-\frac{\chi^{2}}{\nu}\hat{n}^{2}\right]}\hat{D}\left(\frac{\chi\hat{n}}{\nu}\right)\right)\ket{\psi(0)}$
$\displaystyle=\bra{\psi(0)}\left(\hat{N}+\frac{\chi}{\nu}\hat{n}\hat{b}^{\dagger}\left(1-e^{\frac{\mathrm{i}k}{\gamma}\nu}\right)\right.+\frac{\chi}{\nu}\hat{n}\hat{b}\left(1-e^{-\frac{\mathrm{i}k}{\gamma}\nu}\right)+\left.4\frac{\chi^{2}}{\nu^{2}}\hat{n}^{2}\sin\left(\frac{k}{2\gamma}\nu\right)^{2}\right)\ket{\psi(0)}.$

For an initial wavefunction as the product of coherent states in the field and
mirror, $\ket{\psi(0)}=\ket{\alpha}_{f}\otimes\ket{\beta}_{m}$, we have

$\displaystyle{}_{m}\bra{\beta}_{f}\bra{\alpha}\left(\hat{N}+\frac{\chi}{\nu}\hat{n}\hat{b}^{\dagger}\left(1-e^{\frac{\mathrm{i}k}{\gamma}\nu}\right)+\frac{\chi}{\nu}\hat{n}\hat{b}\left(1-e^{-\frac{\mathrm{i}k}{\gamma}\nu}\right)\right.\left.4\frac{\chi^{2}}{\nu^{2}}\hat{n}^{2}\sin\left(\frac{k}{2\gamma}\nu\right)^{2}\right)\ket{\alpha}_{f}\ket{\beta}_{m}=$
(12)
$\displaystyle|\beta|^{2}+\frac{\chi}{\nu}|\alpha|^{2}\beta^{*}\left(1-e^{\frac{\mathrm{i}k\nu}{\gamma}}\right)+\frac{\chi}{\nu}|\alpha|^{2}\beta\left(1-e^{\frac{-\mathrm{i}k\nu}{\gamma}}\right)+4\frac{\chi^{2}}{\nu^{2}}|\alpha|^{2}\left(1+|\alpha|^{2}\right)\sin\left(\frac{1}{2}\frac{k\nu}{\gamma}\right)^{2}=\bra{\psi_{k}}\hat{N}\ket{\psi_{k}}$

Therefore, the expectation value for the modes in the mirror is then

$\displaystyle\braket{\hat{N}}$ $\displaystyle=e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\psi_{k}}\hat{N}\ket{\psi_{k}}$ (13)
$\displaystyle=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\left(|\beta|^{2}+\frac{\chi}{\nu}|\alpha|^{2}\beta^{*}\left(1-e^{\frac{\mathrm{i}k\nu}{\gamma}}\right)\right.+\left.\frac{\chi}{\nu}|\alpha|^{2}\beta\left(1-e^{\frac{-\mathrm{i}k\nu}{\gamma}}\right)+4\frac{\chi^{2}}{\nu^{2}}|\alpha|^{2}\left(1+|\alpha|^{2}\right)\sin\left(\frac{1}{2}\frac{k\nu}{\gamma}\right)^{2}\right)$
$\displaystyle=e^{-\gamma t}\left[|\beta|^{2}e^{\gamma
t}+\frac{\chi}{\nu}|\alpha|^{2}\beta^{*}\left(e^{\gamma t}-e^{\gamma
te^{\frac{\mathrm{i}\nu}{\gamma}}}\right)+\frac{\chi}{\nu}|\alpha|^{2}\beta\left(e^{\gamma
t}-e^{\gamma
te^{\frac{-\mathrm{i}\nu}{\gamma}}}\right)+\frac{\chi^{2}}{\nu^{2}}|\alpha|^{2}\left(1+|\alpha|^{2}\right)\right.$
$\displaystyle\qquad\times\left.\left(2e^{\gamma t}-\left(e^{\gamma
te^{\frac{\mathrm{i}\nu}{\gamma}}}+e^{\gamma
te^{-\frac{\mathrm{i}\nu}{\gamma}}}\right)\right)\right.,$

that returns the initial value at $t=0$, $\braket{\hat{N}(t=0)}=|\beta|^{2}$.
For an initial condition $\ket{\psi(0)}=\ket{2\mathrm{i},1+2\mathrm{i}}$, in
Figure 2 we show how the number of phonon modes variates, modulates, and
decays over time. The parameters that change are the decaying rate $\gamma$
(above), the coupling strength between mirror and field $\chi$ (middle), and
the frequency of the mirror $\nu$ (below). The number of phonon modes is
strictly positive, so we are seeing how different set of parameters involved
leads to decaying dynamics in a more fast or slow fashion. There’s a relevant
feature of the dynamics when we observe the evolution over the set of
decoherence parameters $\gamma$. Despite each figure showing the creation of
phonons from the initial value of $\bar{\hat{N}}(0)=5$, when we look at the
figure above, it is clear that there’s a central value around
$\bar{\hat{N}}(t)\approx 22$ where other curves deviate when we increase or
decrease the value of $\gamma$.

Figure 2: Number of modes $\braket{\hat{N}}$ in the mirror. We variate the
parameters involved in the decoherence evolution. a) Decaying parameter
$\gamma$ between $1$ and $9$, maintaining $\chi=0.5$ and $\nu=0.9$. b)
Coupling parameter $\chi$ between $0.1$ and $0.9$, maintaining $\gamma=5$ and
$\nu=0.9$. c) Mirror oscillation frequency $\nu$ between $0.1$ and $0.9$,
maintaining $\gamma=5$ and $\chi=0.5$. The initial conditions in the
wavefunctions are $\ket{\alpha,\beta}=\ket{2\mathrm{i},1+2\mathrm{i}}$.

####  3.1.2 Position quadrature $\braket{\hat{b}^{\dagger}+\hat{b}}$ in the
mirror

Turning the attention to the position variation of the mirror, we calculate
the expectation value of twice the quadrature operator, taking the action of
the wavefunction $\ket{\psi_{k}}$ over the operator we have

$\displaystyle\bra{\psi_{k}}\hat{b}^{\dagger}$
$\displaystyle+\hat{b}\ket{\psi_{k}}=\bra{\psi(0)}\left[\hat{b}^{\dagger}e^{\frac{\mathrm{i}k\nu}{\gamma}}+\hat{b}e^{-\frac{\mathrm{i}k\nu}{\gamma}}-4\frac{\chi}{\nu}\hat{n}\sin\left(\frac{k\nu}{2\gamma}\right)^{2}\right]\ket{\psi(0)},$
(14)

that for an initial wavefunction as the product of coherent states in the
field and mirror, $\ket{\psi(0)}=\ket{\alpha}_{f}\otimes\ket{\beta}_{m}$, the
expression reduces to

$\displaystyle\bra{\psi_{k}}\hat{b}^{\dagger}+\hat{b}\ket{\psi_{k}}$
$\displaystyle=_{m}\bra{\beta}_{f}\bra{\alpha}\left[\hat{b}^{\dagger}e^{\frac{\mathrm{i}k\nu}{\gamma}}+\hat{b}e^{-\frac{\mathrm{i}k\nu}{\gamma}}-4\frac{\chi}{\nu}\hat{n}\sin\left(\frac{k\nu}{2\gamma}\right)^{2}\right]\ket{\alpha}_{f}\ket{\beta}_{m}$
(15) $\displaystyle=\beta^{*}e^{\frac{\mathrm{i}k\nu}{\gamma}}+\beta
e^{-\frac{\mathrm{i}k\nu}{\gamma}}-4\frac{\chi}{\nu}\left|\alpha\right|^{2}\sin\left(\frac{k\nu}{2\gamma}\right)^{2}$

This renders the expectation value for the position quadrature of the mirror
as

$\displaystyle\braket{\hat{b}^{\dagger}+\hat{b}}$ $\displaystyle=e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\psi_{k}}\hat{b}^{\dagger}+\hat{b}\ket{\psi_{k}}$ (16)
$\displaystyle=e^{-\gamma t}\left(\beta^{*}e^{\gamma
te^{\frac{\mathrm{i}\nu}{\gamma}}}+\beta e^{\gamma
te^{-\frac{\mathrm{i}\nu}{\gamma}}}\right.-\left.\frac{\chi}{\nu}\left|\alpha\right|^{2}\left(2e^{\gamma
t}-\left[e^{\gamma te^{\frac{\mathrm{i}\nu}{\gamma}}}+e^{\gamma
te^{-\frac{\mathrm{i}\nu}{\gamma}}}\right]\right)\right),$

where the initial value at $t=0$ is
$\braket{\left(\hat{b}^{\dagger}+\hat{b}\right)}=\beta^{*}+\beta\equiv
2\mathrm{Re}\left(\beta\right)$. In Figure 3 we show how the position
quadrature of the mirror variates, modulates, and decays over time. The
parameters that change are the decaying rate $\gamma$ (above), the coupling
strength between mirror and field $\chi$ (middle), and the frequency of the
mirror $\nu$ (below). If we look at the figure above, where the decaying rate
variates, we see again a central value $\approx 5$ where the curves change
their amplitude according to the increase or decreasing value of $\gamma$.

Figure 3: Position quadrature $\braket{\hat{b}^{\dagger}+\hat{b}}$ in the
mirror. We variate the parameters involved in the decoherence evolution. a)
Decaying parameter $\gamma$ between $1$ and $9$, maintaining $\chi=0.5$ and
$\nu=0.9$. b) Coupling parameter $\chi$ between $0.1$ and $0.9$, maintaining
$\gamma=5$ and $\nu=0.9$. c) Mirror oscillation frequency $\nu$ between $0.1$
and $0.9$, maintaining $\gamma=5$ and $\chi=0.5$. The initial conditions in
the wavefunctions are $\ket{\alpha,\beta}=\ket{2\mathrm{i},1+2\mathrm{i}}$.

####  3.1.3 Number of photons $\braket{\hat{n}}$ in the field

As we expect from the behavior of the original system (3), the expectation
value for the number of photons in the field is a constant with a value of

$\braket{\hat{n}}=\left|\alpha\right|^{2},$ (17)

because every displacement operator and their exponential map involved
commutes with the number operator $\hat{n}$.

###  3.2 Hong-ou-Mandel parameter and covariance

We now take place to quantify and analyze a couple of statistical descriptors
of the system, the Hong-ou-Mandel parameter, and the covariance between the
photon and phonon modes of the field and the mirror, respectively.

The covariance of two well-behaved operators $\hat{A}$ and $\hat{B}$ is
defined as

$\mathrm{cov}\left(\hat{A},\hat{B}\right):=\braket{\hat{A}\hat{B}}-\braket{\hat{A}}\braket{\hat{B}},$
(18)

where $\braket{\cdot}$ is the standard expected value of an operator on a
wavefunction. Note that the variance is just the autocovariance in the sense
$\mathrm{var}\left(\hat{A}\right)\equiv\mathrm{cov}\left(\hat{A},\hat{A}\right)=\braket{\hat{A}^{2}}-\braket{\hat{A}}^{2}$.
With this, we can define the Hong-ou-Mandel parameter as

$\hat{O}_{\mathrm{HMp}}:=\frac{\mathrm{var}\left(\hat{O}\right)}{\braket{\hat{O}}}.$
(19)

For our particular case, the expectation values are defined in terms of the
action of the wavefunctions $\ket{\psi_{k}}$ on the designed operator
$\hat{A}$ and $\hat{B}$ as

$\braket{\hat{A}\hat{B}}:=e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\;\frac{\left(\gamma
t\right)^{k}}{k!}\braket{\psi_{k}}{\hat{A}\hat{B}}{\psi_{k}},\qquad\braket{\hat{A}}^{2}:=\left(e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\;\frac{\left(\gamma
t\right)^{k}}{k!}\braket{\psi_{k}}{\hat{A}}{\psi_{k}}\right)^{2},$ (20)

##### Variance of $\hat{n}$.

As stated before, the expectation value of the number of photons in the field
is constant in time, then the variance evolution is effectively zero. Meaning
that the Hong-ou-Mandel parameter has nothing to say about the correlation
between the field itself.

##### Variance of $\hat{N}$.

The value of the variance for the phonon modes of the mirror is defined by
$\braket{N^{2}}-\braket{N}^{2}$, with

$\displaystyle\braket{\hat{N}^{2}}$ $\displaystyle=e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\left[|\beta|^{2}+\frac{\chi}{\nu}|\alpha|^{2}\beta^{*}\left(1-e^{\frac{\mathrm{i}k\nu}{\gamma}}\right)+\frac{\chi}{\nu}|\alpha|^{2}\beta\left(1-e^{\frac{-\mathrm{i}k\nu}{\gamma}}\right)+4\frac{\chi^{2}}{\nu^{2}}|\alpha|^{2}\left(1+|\alpha|^{2}\right)\sin\left(\frac{1}{2}\frac{k\nu}{\gamma}\right)^{2}\right]^{2}$
(21) $\displaystyle\braket{\hat{N}}^{2}$ $\displaystyle=\left[e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\left(|\beta|^{2}+\frac{\chi}{\nu}|\alpha|^{2}\beta^{*}\left(1-e^{\frac{\mathrm{i}k\nu}{\gamma}}\right)+\frac{\chi}{\nu}|\alpha|^{2}\beta\left(1-e^{\frac{-\mathrm{i}k\nu}{\gamma}}\right)+4\frac{\chi^{2}}{\nu^{2}}|\alpha|^{2}\left(1+|\alpha|^{2}\right)\sin\left(\frac{1}{2}\frac{k\nu}{\gamma}\right)^{2}\right)\right]^{2},$

where the sums in (21) can be done analytically, giving cumbersome and lengthy
expressions. Having these equations, we can calculate the Hong-ou-Mandel
parameter for the number of phonons evolution in the mirror

$\hat{N}_{\mathrm{HMp}}=\frac{\braket{\hat{N}^{2}}-\braket{\hat{N}}^{2}}{\braket{\hat{N}}},$
(22)

as shown in the Figure 4. We can see that the transition from classical to
non-classical behavior, and vice versa, is directly proportional to the
amplitude of the decoherence rate $\gamma$.

Figure 4: Hong-ou-Mandel parameter of the phonon mode number $\hat{N}$, eq.
(4). According to the definition, when the parameter goes above the unitary
amplitude, we expect classical behavior in the evolution of the system. It is
clear that this feature is directly proportional to the value of the
decoherence rate $\gamma$. We set the values $\nu=0.5$, $\chi=0.9$,
$\ket{\psi_{0}}=\ket{2\mathrm{i},1+2\mathrm{i}}$.

##### Covariance between the photon and phonon modes.

We can quantify how the evolution of the phonon modes of the mirror is
correlated to the (constant) evolution of the photon modes of the field. For
this, we take the covariance

$\mathrm{cov}\left(\hat{n},\hat{N}\right)=\braket{\hat{n}\hat{N}}-\braket{\hat{n}}\braket{\hat{N}},$
(23)

where the joint expect value $\braket{\hat{n}\hat{N}}$ is

$\displaystyle\braket{\hat{n}\hat{N}}$ $\displaystyle=e^{-\gamma
t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\bra{\psi_{k}}\hat{n}\hat{N}\ket{\psi_{k}}$ (24)
$\displaystyle=|\alpha|^{2}|\beta|^{2}+\frac{\chi}{\nu}|\alpha|^{2}\left(1+|\alpha|^{2}\right)\beta^{*}\left(1-e^{\frac{\mathrm{i}k\nu}{\gamma}}\right)+\frac{\chi}{\nu}|\alpha|^{2}\left(1+|\alpha|^{2}\right)\beta\left(1-e^{\frac{-\mathrm{i}k\nu}{\gamma}}\right)$
$\displaystyle+4\frac{\chi^{2}}{\nu^{2}}|\alpha|^{2}\left(1+3|\alpha|^{2}+|\alpha|^{2}\right)\sin\left(\frac{1}{2}\frac{k\nu}{\gamma}\right)^{2},$

that differs from $\braket{\hat{n}}\braket{\hat{N}}$ due the emergence of a
$\hat{n}^{3}$ term. Despite this, it seems that there are no new functions
involving time, just a scale in the amplitude given by the photon modes
$\hat{n}^{k}$.

###  3.3 Husimi $\hat{Q}$ function

The Husimi $\hat{Q}$ function is defined as the expectation value of the
density matrix on the coherent basis as

$\mathrm{Q}(\epsilon)=\frac{1}{\pi}\bra{\epsilon}\hat{\rho}\ket{\epsilon},$
(25)

where $\ket{\epsilon}$ is the coherent state ket on a certain suitable basis.
When we are dealing with multiple subsystems, the Husimi function can be taken
as the pseudo-distribution on their respective phase space of the subsystem.
If we have a system decomposed as $\hat{H}=\hat{H}_{f}\otimes\hat{H}_{m}$, the
Husimi function is then

$\mathrm{Q}(\epsilon,\zeta)=\frac{1}{\pi}~{}_{m}\bra{\zeta}_{f}\bra{\epsilon}\hat{\rho}\ket{\epsilon}_{f}\ket{\zeta}_{m},$
(26)

where $f$ and $m$ are the subscripts for _field_ and _mirror_ subsystem,
respectively.

We know from (7) and (9) that the density matrix of the field-mirror system is
defined as

$\hat{\rho}=e^{-\gamma t}\sum\limits_{k=0}^{\infty}\frac{\left(\gamma
t\right)^{k}}{k!}\ket{\psi_{k}}\bra{\psi_{k}},$ (27)

where the $k$-th component of the wavefunction $\psi_{k}$ is

$\ket{\psi_{k}}=\hat{D}^{\dagger}\left(\frac{\chi}{\nu}\hat{n}\right)e^{-\frac{\mathrm{i}k}{\gamma}\left(\omega\hat{n}+\nu\hat{N}-\frac{\chi^{2}}{\nu^{2}}\hat{n}^{2}\right)}\hat{D}^{\dagger}\left(\frac{\chi}{\nu}\hat{n}\right)\ket{\psi(0)},$
(28)

for some initial wavefunction $\ket{\psi(0)}$. When
$\ket{\psi(0)}=\ket{\alpha}_{f}\ket{\beta}_{m}$, the Husimi function (26) can
be obtained in the coherent basis $\ket{a}_{f}\ket{b}_{m}$ as

$\mathrm{Q}\left(a,\alpha;b,\beta\right):=\frac{1}{\pi}\bra{b}\bra{a}\hat{\rho}\ket{a}\ket{b},$
(29)

that renders explicitly the terms

$\bra{b}\bra{a}\hat{\rho}\ket{a}\ket{b}\equiv\bra{b}\bra{a}\psi_{k}\rangle\langle\psi_{k}\ket{a}\ket{b}.$
(30)

The explicit calculation shows that

$\displaystyle\bra{b}\bra{a}\psi_{k}\rangle$
$\displaystyle=\bra{b}\bra{a}\hat{D}^{\dagger}\left(\frac{\chi}{\nu}\hat{n}\right)e^{-\frac{\mathrm{i}k}{\gamma}\left(\omega\hat{n}+\nu\hat{N}-\frac{\chi^{2}}{\nu^{2}}\hat{n}^{2}\right)}\hat{D}\left(\frac{\chi}{\nu}\hat{n}\right)\ket{\alpha}\ket{\beta}$
(31)
$\displaystyle=\bra{b}\hat{D}^{\dagger}\left(\frac{\chi}{\nu}|a|^{2}\right)e^{-\frac{\mathrm{i}k}{\gamma}\left(\omega
a^{*}\alpha+\nu\hat{N}-\frac{\chi^{2}}{\nu^{2}}(a^{*})^{2}(\alpha)^{2}\right)}\hat{D}\left(\frac{\chi}{\nu}|\alpha|^{2}\right)\ket{\beta}$
$\displaystyle=\bra{b+\tfrac{\chi}{\nu}|a|^{2}}e^{-\frac{\mathrm{i}k}{\gamma}\left(\omega
a^{*}\alpha+\nu\hat{N}-\frac{\chi^{2}}{\nu^{2}}(a^{*})^{2}(\alpha)^{2}\right)}\ket{\beta+\tfrac{\chi}{\nu}|\alpha|^{2}}$
$\displaystyle=e^{-\frac{\mathrm{i}k}{\gamma}\left[\omega
a^{*}\alpha+\nu\left(b+\tfrac{\chi}{\nu}|a|^{2}\right)^{*}\left(\beta+\tfrac{\chi}{\nu}|\alpha|^{2}\right)-\frac{\chi^{2}}{\nu^{2}}(a^{*})^{2}(\alpha)^{2}\right]}\braket{a}{\alpha}\braket{b+\tfrac{\chi}{\nu}}{a}{{}^{2}|\beta+\tfrac{\chi}{\nu}|\alpha|^{2}},$

for $a,\alpha,b,\beta\in\mathbb{C}$. From (31) we can obtain the dagger
$\langle\psi_{k}\ket{a}\ket{b}$, that finally gives expression

$\bra{b}\bra{a}\psi_{k}\rangle\langle\psi_{k}\ket{a}\ket{b}=\frac{1}{\pi}e^{-\frac{\mathrm{i}k}{\gamma}\left[\omega
f(a,\alpha)+\nu
g(a,\alpha;b,\beta)-\frac{\chi^{2}}{\nu^{2}}h(a,\alpha)\right]}e^{-|a-\alpha|^{2}}e^{-\left|\left(b-\beta\right)+\tfrac{\chi}{\nu}\left(|a|^{2}-|\alpha|^{2}\right)\right|^{2}},$
(32)

where the functions inside the first exponential are defined as

$\displaystyle f(a,\alpha)=a^{*}\alpha-a\alpha^{*}$ (33) $\displaystyle
g(a,\alpha;b,\beta)=\left(b+\tfrac{\chi}{\nu}|a|^{2}\right)^{*}\left(\beta+\tfrac{\chi}{\nu}|\alpha|^{2}\right)-\left(b+\tfrac{\chi}{\nu}|a|^{2}\right)\left(\beta+\tfrac{\chi}{\nu}|\alpha|^{2}\right)^{*}$
$\displaystyle h(a,\alpha)=(a^{*})^{2}(\alpha)^{2}-(a)^{2}(\alpha^{*})^{2}.$

Finally, with the equations (8) and (32), the Husimi function is explicitly
given by

$\mathrm{Q}\left(a,\alpha;b,\beta\right)=\frac{1}{\pi}e^{-\gamma t}e^{\gamma
te^{-\frac{\mathrm{i}}{\gamma}\left[\omega f(a,\alpha)+\nu
g(a,\alpha;b,\beta)-\frac{\chi^{2}}{\nu^{2}}h(a,\alpha)\right]}}e^{-|a-\alpha|^{2}}e^{-\left|\left(b-\beta\right)+\tfrac{\chi}{\nu}\left(|a|^{2}-|\alpha|^{2}\right)\right|^{2}},$
(34)

that resembles Gaussians in phase space displaced by $\alpha$ and $\beta$, in
which amplitude is modulated by the first terms that are time-dependent.

Figure 5: Husimi $\hat{Q}$ function for the field. We set
$\ket{\psi_{0}}=\ket{2\mathrm{i},1+2\mathrm{i}}$, $\nu=0.5$, $\chi=0.9$ and
$\gamma=20$ Figure 6: Husimi $\hat{Q}$ function for the mirror. We set
$\ket{\psi_{0}}=\ket{2\mathrm{i},1+2\mathrm{i}}$, $\nu=0.5$, $\chi=0.9$ and
$\gamma=20$

In Figures 6 and 5 we show the evolution of the initial states of the field
and the mirror, respectively. We set
$\ket{\psi_{0}}=\ket{2\mathrm{i},1+2\mathrm{i}}$, $\nu=0.5$, $\chi=0.9$ and
$\gamma=20$. According to the expected value for the number of photons in the
field, we see that the $\hat{Q}$-function evolves as a harmonic oscillator; on
the other side, the $\hat{Q}$-function for the mirror evolves according from
what the Hong-ou-Mandel parameters suggest: at some point, the initial
coherent state transit from a non-classical to classical description, when
$t>4$ we see that the decoherence stall the movement around phase-space.

##  4 Analysis and conclusions

We solve the mirror-field interaction using the intrinsic decoherence scheme
from the complete Milburn equation (8). For this particular driven system, a
_decoherence parameter_ $\gamma$ defines the strength in the decaying
dynamics. We obtain the significant expected values for the number of
mechanical phonon modes, and the position quadrature; also we observe the
constant nature of the number of photon modes inside the cavity. Although the
radiation pressure is parametrically coupled to the harmonic motion of the
mirror, the number of quanta remains the same. For a better understanding of
the feedback between the quantized field and the mechanical mirror, we
calculate the covariance between the number of photons and phonons, giving
just an amplitude scaling proportional to $\hat{n}^{k}$, but no new time-
dependent dynamics. The Hong-ou-Mandel parameter is obtained for the photon
and phonon modes. The photon modes have nothing new to say, but the phonon
modes give insights into how the decoherence made the initial coherent state
in the mirror transit from a non-classical to a classical description. This
last can be seen also in the Husimi $\hat{Q}$-functions we show for the field
and the mirror, where the behavior on phase-space enables us to talk about the
quantized nature of the photon modes and the macroscopic nature of the moving
mirror.

## Declarations

### Funding

Consejo Nacional de Ciencia y Tecnología (Postdoctoral Grant 2021 and 2022)

### Acknowledgements

A.R. Urzúa thanks Dr. Francisco Récamier (ICF UNAM) for the help in
understanding the Hong-ou-Mandel parameter. A.R. also thanks ICF UNAM for the
logistical support during the postdoctoral stay.

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