###### Abstract

In quantum physics, two prototypical model systems stand out due to their wide
range of applications. These are the two-level system (TLS) and the harmonic
oscillator. The former is often an ideal model for confined charge or spin
systems and the latter for lattice vibrations, i.e., phonons. Here, we couple
these two systems, which leads to numerous fascinating physical phenomena.
Practically, we consider different optical excitations and decay scenarios of
a TLS, focusing on the generated dynamics of a single phonon mode that couples
to the TLS. Special emphasis is placed on the entropy of the different parts
of the system, predominantly the phonons. While, without any decay, the entire
system is always in a pure state, resulting in a vanishing entropy, the
complex interplay between the single parts results in non-vanishing respective
entanglement entropies and non-trivial dynamics of them. Taking a decay of the
TLS into account leads to a non-vanishing entropy of the full system and
additional aspects in its dynamics. We demonstrate that all aspects of the
entropy’s behavior can be traced back to the purity of the states and are
illustrated by phonon Wigner functions in phase space.

###### keywords:

phonons; two-level system; entropy; Wigner functions; entanglement

22 3 286 Received: 04 February 2020; Accepted: 27 February 2020; Published: 29
February 2020 yes Entropy Dynamics of Phonon Quantum States Generated by
Optical Excitation of a Two-Level System Thilo Hahn, Daniel Wigger and Tilmann
Kuhn * Thilo Hahn, Daniel Wigger and Tilmann Kuhn Correspondence:
<EMAIL_ADDRESS>
##  1 Introduction

Entropy is one of the most fundamental concepts in physics. According to the
second law of thermodynamics, in a closed system, it never decreases, which
has far reaching consequences, from the limited efficiency of thermodynamic
machines Kondepudi and Prigogine (2014) to cosmological implications Gunzig et
al. (1987); la Cruz-Dombriz and Sáez-Gómez (2012). Under thermal equilibrium
conditions, it determines the state of a thermodynamic system: In a closed
system, the realized state is the one with maximal entropy; in a system in
thermal contact with a heat bath, the realized state results from an interplay
between energy and entropy and is governed by the minimum of the free energy
Toda et al. (1992). Under nonequilibrium conditions, the second law of
thermodynamics prohibits the decrease of the entropy of a closed system;
however, this does not hold for the entropy of a subsystem which is
interacting with other subsystems or with its surroundings Kubo et al. (2012);
Breuer and Petruccione (2002). In this case, the study of the dynamics of the
entropy of these subsystems provides valuable information on the evolution of
the nature of the system’s state Eisert et al. (2010).

From the point of view of information science, entropy is closely related to
the imperfect knowledge about a system Shannon (1948); Liang et al. (2006). As
such, it plays a key role in all fields related to information processing and
communication, and, in particular, in the highly topical fields of quantum
information and communication, where the entropy is closely related to
phenomena like purity of quantum states, entanglement, and decoherence Nielsen
and Chuang (2002).

In this paper, we study the entropy dynamics in a prototypical model of
quantum mechanics and quantum information theory. It consists of two
subsystems, a quantum-mechanical two-level system (i.e., a representation of a
qubit) which can be manipulated by an external field (e.g., a light field) and
which is coupled to a harmonic oscillator (e.g., a single phonon mode or a
nanomechanical oscillator). The generation of specific quantum states of such
a harmonic oscillator and the manipulation of these states has recently
attracted much interest Hofheinz et al. (2009); O’Connell et al. (2010);
Reiter et al. (2011); Satzinger et al. (2018). Prominent examples are coherent
states and Schrödinger cat states, i.e., superpositions of coherent states. We
analyze the entropy dynamics of the two subsystems after excitation with a
short optical pulse or a pair of such pulses. In particular, we compare the
case of a unitary evolution in the absence of damping processes, when the
coupled system remains in a pure state, with the case of a decaying two-level
system resulting in a mixed state also of the combined system. We will show
that the analysis of the time-dependent entropy provides interesting insight
into the nature of the quantum state of the two subsystems.

##  2 Theory

We consider a two-level system (TLS) which can be excited and de-excited by a
resonant optical field ${\bf E}$. Additionally, a pure dephasing coupling to a
single phonon (ph) mode is taken into account. Thus, the Hamiltonian reads
Mahan (1981)

$H=\hbar\Omega\left|x\right>\left<x\right|-\big{[}{\bf M}\cdot{\bf
E}(t)\left|x\right>\left<g\right|+{\bf M}^{\ast}\cdot{\bf
E}^{\ast}(t)\left|g\right>\left<x\right|\big{]}+\hbar\omega_{\rm
ph}\hat{b}^{\dagger}\hat{b}+\hbar
g\big{(}\hat{b}+\hat{b}^{\dagger}\big{)}\left|x\right>\left<x\right|\ .$ (1)

The states $\left|g\right>$ and $\left|x\right>$ describe ground and excited
state of the TLS with an energy splitting of $\hbar\Omega$, respectively. The
time dependent optical driving is mediated by the dipole matrix element ${\bf
M}$. Phonons with the discrete energy $\hbar\omega_{\rm ph}$ are created and
annihilated by $\hat{b}^{\dagger}$ and $\hat{b}$, respectively. For
simplicity, the coupling constant of the exciton-phonon interaction $g$ is
supposed to be real.

Such a TLS system coupled to a single bosonic mode is a prototypical model
that can be considered for the description of various solid state systems. For
the TLS, one might think of an exciton in a single semiconductor quantum dot
Zrenner et al. (2002) or excitations of defects in insulators, like diamond
Aharonovich et al. (2011) or hexagonal boron nitride Wigger et al. (2020),
while the phonon could be an optical mode Roca et al. (1994), a local mode
Gali et al. (2011), a van Hove singularity Debald et al. (2002), or the
mechanical excitation of a microresonator Munsch et al. (2017).

Phonon-induced transitions between the states $\left|g\right>$ and
$\left|x\right>$ of the TLS are negligible because of the strong energy
mismatch between the exciton energy, which is of the order of one or a few
electronvolts (eV), while phonon energies range from a few micro-electronvolts
(µeV) (for micromechanical resonators) up to a few tens of milli-electronvolts
(meV) (for optical phonon modes). The linear coupling in the phonon
displacement reflects typical electron-phonon interaction mechanisms in solids
like deformation potential coupling, piezoelectric coupling, or Fröhlich
coupling Ferry (1991). Although extensions of this model have been considered
that take a quadratic coupling to the phonons into account Munn and Silbey
(1978); Muljarov and Zimmermann (2004); Machnikowski (2006); Chenu et al.
(2019), the original independent boson model Duke and Mahan (1965) in Equation
(1) is successfully used in different contexts. It reproduces recent linear
and nonlinear spectroscopy signals Stock et al. (2011); Wigger et al. (2020),
Rabi oscillations Wigger et al. (2018), and rotations Ramsay et al. (2010) in
excellent agreement with experiments, to name just a few.

The possible states of the entire system can be separated into the phonons
forming product states with the ground state of the TLS and those forming
product states with the excited state

$\left|g\right>\otimes\left|ph_{\rm g}\right>\qquad{\rm
and}\qquad\left|x\right>\otimes\left|ph_{\rm x}\right>\ .$ (2)

From the full density matrix of the system $\rho$, we can calculate the one of
a subsystem by tracing over the respective other, i.e.,

$\rho_{\rm TLS}={\rm Tr}_{\rm ph}(\rho)\qquad{\rm and}\qquad\rho_{\rm ph}={\rm
Tr}_{\rm TLS}(\rho)\ .$ (3)

In the same way as in Reference Hahn et al. (2019), we model a decay of the
excited state with the rate $\Gamma$ via the Lindblad dissipator

$\mathcal{D}(\rho)=\Gamma\left[\left|g\right>\left<x\right|\rho\left|x\right>\left<g\right|-\frac{1}{2}\big{\\{}\left|x\right>\left<x\right|,\rho\big{\\}}\right],$
(4)

leading to the master equation for the density matrix:

$\frac{\rm d}{{\rm d}t}\rho=\frac{1}{i\hbar}[H,\rho]+\mathcal{D}(\rho)\ .$ (5)

We will not take an additional phenomenological pure dephasing of the TLS into
account. We will study different regimes of decay rates $\Gamma$ compared to
the characteristic phonon frequency $\omega_{\rm ph}$. On the one hand, when
describing, for example, optical phonons with energies in the range of tens of
meV the decay time of the TLS is typically much longer than a phonon period,
i.e., $\Gamma\ll\omega_{\rm ph}$ Wigger et al. (2020). On the other hand, when
considering typical mechanical resonators with phonon energies in the µeV
range we have $\Gamma>\omega_{\rm ph}$ Auffeves and Richard (2014).

As we have explained in Reference Hahn et al. (2019), the entire quantum state
and especially the Wigner function of the phonons can be calculated
analytically when considering a series of ultrafast laser pulses to drive the
TLS. Especially if the pulse duration is much shorter than the phonon period,
the pulses can be approximated by delta-functions as

$\frac{{\bf M}\cdot{\bf
E}(t)}{\hbar}=\sum_{j}\frac{\theta_{j}}{2}\exp\left[-i\left(\Omega-\frac{g^{2}}{\omega_{\rm
ph}}\right)t+i\phi_{j}\right]\delta(t-t_{j})\ .$ (6)

The pulses excite the TLS at times $t_{j}$ with pulse areas $\theta_{j}$ and
phases $\phi_{j}$. By this choice and the introduction of the generating
functions

$\displaystyle Y_{\alpha}(t)$
$\displaystyle=\Big{<}\left|g\right>\left<x\right|\exp(-\alpha^{\ast}\hat{b}^{\dagger})\exp(\alpha\hat{b})\Big{>}\
,$ (7a) $\displaystyle C_{\alpha}(t)$
$\displaystyle=\Big{<}\left|x\right>\left<x\right|\exp(-\alpha^{\ast}\hat{b}^{\dagger})\exp(\alpha\hat{b})\Big{>}\
,$ (7b) $\displaystyle F_{\alpha}(t)$
$\displaystyle=\Big{<}\exp(-\alpha^{\ast}\hat{b}^{\dagger})\exp(\alpha\hat{b})\Big{>}\
,$ (7c)

with $\left<\hat{A}\right>={\rm Tr}\left(\rho\hat{A}\right)$ denoting the
expectation value of an operator $\hat{A}$, a closed system of partial
differential equations for the time-evolution of the generating functions is
obtained and all phonon assisted density matrices can be calculated
analytically without approximations. Note that $F_{\alpha}$ contains the
entire information on the phonon system, while $C_{\alpha}$ describes the
phonon assisted occupation of the excited state, i.e., the phonons in
$\left|x\right>\otimes\left|ph_{\rm x}\right>$, and $Y_{\alpha}$ is the phonon
assisted coherence.

Our analysis of the phonon quantum states is based on their Wigner function
Schleich (2011)

$W(U,\Pi)=\frac{1}{4\pi}\int\limits_{-\infty}^{\infty}\left<U+\frac{X}{2}\right|\rho_{\rm
ph}\left|U-\frac{X}{2}\right>\exp\left(-\frac{i}{2}X\Pi\right)\,{\rm d}X\ ,$
(8)

which is a quasi-probability distribution in the phase space defined by the
quadratures $\hat{u}$ and $\hat{\pi}$ and their respective eigenstates

$\displaystyle\hat{u}$ $\displaystyle=\hat{b}+\hat{b}^{\dagger}\ ,$
$\displaystyle\hat{\pi}$
$\displaystyle=\frac{1}{i}(\hat{b}-\hat{b}^{\dagger})\ ,$ (9a)
$\displaystyle\hat{u}\left|U\right>$ $\displaystyle=U\left|U\right>\ ,$
$\displaystyle\hat{\pi}\left|\Pi\right>$ $\displaystyle=\Pi\left|\Pi\right>\
.$ (9b)

Due to their definition by the phonon annihilation and creation operators, the
quantities $U$ and $\Pi$ directly correspond to the lattice displacement and
momentum, respectively. The generating function $F_{\alpha}$, at the same
time, is a characteristic function of the Husimi Q function Gerry and Knight
(2005). From this, we can directly calculate the instructive Wigner
distribution analytically for a given pulse sequence via Schleich (2011):

$W(U,\Pi,t)=\frac{1}{4\pi^{2}}\iint\limits_{-\infty}^{\quad\infty}\exp\left(-\frac{|\alpha|^{2}}{2}\right)F_{\alpha}(t)\exp\left\\{i\left[{\rm
Re}(\alpha)\Pi+{\rm Im}(\alpha)U\right]\right\\}\,{\rm d}^{2}\alpha\ .$ (10)

In the same way, we can isolate the Wigner function $W_{\rm x}$ for the
phonons associated with the TLS being in the excited state $\left|x\right>$ by
choosing $C_{\alpha}$ instead of $F_{\alpha}$ in Equation (10). By doing the
same, but choosing $Y_{\alpha}$, we define $W_{\rm p}$ as the Wigner function
of the phonon assisted coherence. In summary, we have

$\displaystyle F_{\alpha}\to$ $\displaystyle\ W\ ,$ (11a) $\displaystyle
C_{\alpha}\to$ $\displaystyle\ W_{\rm x}\ ,$ (11b) $\displaystyle
Y_{\alpha}\to$ $\displaystyle\ W_{\rm p}\ ,$ (11c) $\displaystyle
F_{\alpha}-C_{\alpha}\to$ $\displaystyle\ W-W_{\rm x}=\ W_{\rm g}\ ,$ (11d)

where $W_{\rm g}$ is the Wigner function of the phonons associated with the
TLS being in the ground state $\left|g\right>\otimes\left|ph_{\rm g}\right>$.

Following the original definition by von Neumann Von Neumann (1996), we
investigate the time-dependent entropy of our coupled quantum system defined
by

$S=-{\rm Tr}\big{[}\rho\ln(\rho)\big{]}\ .$ (12)

In general, subadditivity states that the entropy of the full system is a
lower boundary for the sum of the entropies of the subsystems Wehrl (1978):

$S(\rho)\leq S(\rho_{\rm TLS}\otimes\rho_{\rm ph})=S(\rho_{\rm
TLS})+S(\rho_{\rm ph})\ .$ (13)

It is important to note that for every pure quantum state $\rho_{\rm pure}$,
the entropy vanishes, i.e.,

$S_{\rm pure}=-{\rm Tr}\big{[}\rho_{\rm pure}\ln(\rho_{\rm pure})\big{]}=0\ ,$
(14)

and the entropies of the subsystems coincide if the state of the full system
is pure Wehrl (1978)

$S^{\rm ph}=S(\rho_{\rm ph})=S(\rho_{\rm TLS})\ .$ (15)

To show this, following Reference Wehrl (1978) for an arbitrary composed
system, we decompose the pure state of the entire system $|\psi\rangle$ into
an orthonormal basis
$|\psi\rangle=\sum_{i,k}C_{ik}|\phi_{i}\rangle|\chi_{k}\rangle$ with the
coefficient matrix $C=(C_{ik})$. Here, $|\phi_{i}\rangle$ and
$|\chi_{k}\rangle$ are basis states of the two subsystems, respectively. From
the complete density matrix $\rho=(\rho_{ikjl})$, one obtains a reduced
density matrix by tracing over the respective other subsystem
$\rho_{1}=Tr_{2}(\rho)$:

$\displaystyle\rho_{ikjl}$
$\displaystyle=\langle\phi_{i}|\langle\chi_{k}|\rho|\chi_{l}\rangle|\phi_{j}\rangle=C_{ik}C_{jl}^{*}\
,$ (16a) $\displaystyle\Rightarrow\rho_{1,ij}$
$\displaystyle=\langle\phi_{i}|\rho_{1}|\phi_{j}\rangle=\sum_{k}C_{ik}C_{jk}^{*}=(CC^{\dagger})_{ij}\
.$ (16b)

Analogously the density matrix of the other subsystem is
$\rho_{2}=C^{\dagger}C$. Any non-vanishing eigenvalue $\lambda$ of $\rho_{1}$
with the eigenvector ${\bf y}$ is then also an eigenvalue of $\rho_{2}$ with
eigenvector ${\bf z}=C^{\dagger}{\bf y}$ because

$\displaystyle CC^{\dagger}{\bf y}$ $\displaystyle=\lambda{\bf y}\ ,$ (17a)
$\displaystyle\Rightarrow C^{\dagger}C{\bf z}$
$\displaystyle=C^{\dagger}C(C^{\dagger}{\bf y})=C^{\dagger}(CC^{\dagger}){\bf
y}=\lambda C^{\dagger}{\bf y}=\lambda{\bf z}\ ,$ (17b)

and vice versa.

In our particular system, we can choose the TLS’s states as the
$|\phi_{i}\rangle\in\\{|g\rangle,|x\rangle\\}$ and a Fock basis for the phonon
system $|\chi_{k}\rangle\in\\{|0\rangle,|1\rangle,\ldots\\}$. This means that
the coefficient matrix $C$ consists of $2\times\infty$ elements. The entropy
of the TLS, i.e., of its density matrix:

$CC^{\dagger}=\rho_{\rm TLS}=\begin{pmatrix}1-c&p\\\ p^{\ast}&c\end{pmatrix}\
,\qquad{\rm with}\quad c=\left<\left|x\right>\left<x\right|\right>\,,\
p=\left<\left|g\right>\left<x\right|\right>\ ,$ (18a) can be easily calculated
via Wehrl (1978) $S(\rho_{\rm TLS})=-\lambda_{+}{\rm
ln}(\lambda_{+})-\lambda_{-}{\rm ln}(\lambda_{-})\ ,$ (18b) where
$\lambda_{\pm}=\frac{1}{2}\pm\sqrt{\frac{1}{4}+c^{2}-c+|p|^{2}}=\frac{1}{2}\pm\frac{1}{2}|{\bf
v}|$ (18c)

are the two eigenvalues of the TLS’s density matrix with the Bloch vector
${\bf v}=\big{(}2{\rm Re}(p),2{\rm Im}(p),2c-1\big{)}$. For a pure state of
the full system, according to Equation (17), the phonon density matrix has,
despite being a quadratic infinite dimensional matrix, only two non-vanishing
eigenvalues $\lambda_{\pm}$. So, as long as the entire system is in a pure
state and we know the entropy of the TLS via Equation (18b), we can derive the
entropy of the phonon system by Equation (15).

For an arbitrary, non-pure state of the entire system, the calculation of the
entropy in a system with infinite dimensions is far from being trivial. This
is the case if, already, the initial state is a statistical mixture, e.g., at
non-vanishing temperature or when dephasing leads to a statistical mixture.
Therefore, approximations have been discussed and a reasonable version is
given by the linear entropy Wehrl (1978):

$S_{\rm lin}={\rm Tr}\big{(}\rho\big{)}-{\rm Tr}\big{(}\rho^{2}\big{)}=1-{\rm
Tr}\big{(}\rho^{2}\big{)}=1-\left<\rho\right>\ .$ (19)

To get an impression of this approximation, Figure 1a shows the function of
the full entropy $-\xi{\rm ln}(\xi)$ and the one for the linear entropy
$\xi-\xi^{2}$; similar to the presentation in Reference Wehrl (1978), note
that ${\rm Tr}(\rho)=1$. We find that the function of the approximated linear
entropy in red is always smaller than the full entropy in blue. Therefore, we
expect that the linear entropy under-estimates the full entropy.

Figure 1: (a) Functions for the full entropy in blue and the linear entropy in
red. (b) Entropies of the two-level system (TLS) with $p=0$, full entropy in
blue and linear entropy in red.

We can directly compare the linear and the full entropy of the isolated TLS
from Equation (18b), as shown in Figure 1b. There, the full entropy is plotted
in blue and the linear one in red as functions of the excited state occupation
$c$, both for $p=0$. In the limiting cases of full inversion $c=1$ and no
inversion $c=0$, the state is pure and both entropies vanish. For all other
occupations, the TLS is in a statistical mixture, and the entropy is non-zero,
and it is $S_{\rm lin}\leqslant S$. In the case of an equally distributed
mixture, i.e., $c=0.5$, the entropies are maximal and reach values of $S_{\rm
lin}=0.5$ and $S=\ln(2)\approx 0.7$.

The biggest advantage of this linear entropy for our study is that it can be
directly calculated from the phonons’ Wigner functions due to the trace-
product rule Manfredi and Feix (2000) via

$S_{\rm lin}^{\rm
ph}=1-4\pi\iint\limits_{-\infty}^{\quad\infty}W(U,\Pi)^{2}\,{\rm d}U\,{\rm
d}\Pi\ .\\\ $ (20)

Note that the prefactor $4\pi$ depends on the definition of the quadratures
$U$ and $\Pi$, i.e., the scaling of the phase space.

With the separation into TLS and phonon system, the linear entropy is
calculated via

$S_{\rm lin}=1-{\rm Tr}_{\rm ph}\left[{\rm Tr}_{\rm
TLS}\big{(}\rho^{2}\big{)}\right]\ .$ (21)

For this, we again consider the density matrix of the full system as

$\displaystyle\rho$
$\displaystyle=(1-c)\left|g\right>\left<g\right|\otimes\rho_{\rm
ph,g}+c\left|x\right>\left<x\right|\otimes\rho_{\rm
ph,x}+p^{*}\left|g\right>\left<x\right|\otimes\rho_{\rm
ph,p}+p\left|x\right>\left<g\right|\otimes\rho_{\rm ph,p}^{\dagger}\ ,$ (22a)
$\displaystyle\Rightarrow{\rm Tr}_{\rm TLS}(\rho^{2})$
$\displaystyle=(1-c)^{2}\rho_{\rm ph,g}^{2}+c^{2}\rho_{\rm
ph,x}^{2}+|p|^{2}(\rho_{\rm ph,p}\rho_{\rm ph,p}^{\dagger}+\rho_{\rm
ph,p}^{\dagger}\rho_{\rm ph,p})\ ,$ (22b) leading to $\displaystyle{\rm
Tr}_{\rm TLS}(\rho)-{\rm Tr}_{\rm TLS}(\rho^{2})$
$\displaystyle=(1-c)\rho_{\rm ph,g}-(1-c)^{2}\rho_{\rm ph,g}$
$\displaystyle+c\rho_{\rm ph,x}-c^{2}\rho_{\rm ph,x}^{2}$
$\displaystyle-|p|^{2}(\rho_{\rm ph,p}\rho_{\rm ph,p}^{\dagger}+\rho_{\rm
ph,p}^{\dagger}\rho_{\rm ph,p})\ .$ (22c)

We can now use the separate parts of the Wigner function from Equation (11) to
define entropies

$\displaystyle S_{\rm lin}^{\rm i}$
$\displaystyle=\iint\limits_{-\infty}^{\quad\infty}\bigg{[}W_{\rm
i}(U,\Pi)-4\pi W_{\rm i}(U,\Pi)^{2}\bigg{]}\,{\rm d}U\,{\rm d}\Pi\ ,\qquad{\rm
with}\ {\rm i}\in\\{{\rm g,x}\\}\ ,$ (22d) $\displaystyle S_{\rm lin}^{\rm p}$
$\displaystyle=-4\pi\iint\limits_{-\infty}^{\quad\infty}|W_{\rm
p}(U,\Pi)|^{2}\,{\rm d}U\,{\rm d}\Pi\ .$ (22e)

Note that the polarization Wigner function $W_{\rm p}$ is a complex quantity.
With this and the definitions of the generating functions in Equation (7), we
can write the linear entropy in Equation (21) as

$S_{\rm lin}=S_{\rm lin}^{\rm g}+S_{\rm lin}^{\rm x}+2S_{\rm lin}^{\rm p}\ .$
(23)

To briefly summarize, for pure states of the entire system, i.e., without
dephasing or decay of the TLS, we can calculate the full entropy of the phonon
state $S_{\rm ph}$ via Equation (15). If the state is not pure, we can at
least calculate the linear entropy from the Wigner functions. We can further
distinguish between the linear entropy of the full system $S_{\rm lin}$ in
Equation (23) and the one of the phonons $S_{\rm lin}^{\rm ph}$ in Equation
(20).

##  3 Results and Discussion

###  3.1 Single Pulse Excitation

We start our study with the most basic situation, where the TLS is excited by
a single optical pulse. It is well known from previous works that a single
ultrafast excitation in general creates a statistical mixture of coherent
states in the phonon system. The excitation of the TLS means for the phonons a
shift of the equilibrium position determined by the dimensionless coupling
strength $\gamma=g/\omega_{\rm ph}$. If not stated differently, in the
following, we fix this value to $\gamma=2$ in order to separate the different
parts of the Wigner function in phase space, as will be seen later. Although
$\gamma=2$ is a rather large value for quantum dots and optical phonons, the
general physics explained in this paper will not depend on this value. Some
effects might be strengthened or weakened with a different choice of the
coupling strength, as will be highlighted later.

####  3.1.1 Phonons Generated by a Non-Decaying TLS

In the first step, we neglect the decay of the excited state by choosing
$\Gamma=0$. In this situation, the state of the full system, including the TLS
and the phonons, is pure. Therefore, the full and the linearized entropy are
zero and Equation (15) holds, meaning that the entropy of the TLS and that of
the phonons is the same. Figure 2 recapitulates the phonon dynamics for a
pulse area of $\theta=\pi/2$, i.e., an inversion of the TLS of $50\%$ or
$c=0.5$, from Reference Reiter et al. (2011). The phonon’s Wigner function
reads:

$\displaystyle W(U,\Pi,t)=$
$\displaystyle\frac{1}{4\pi}\Bigg{\\{}\exp\left[-\frac{1}{2}(U^{2}+\Pi^{2})\right]$
(24) $\displaystyle+\exp\left(-\frac{1}{2}\\{U-2\gamma[1-\cos(\omega_{\rm
ph}t)]\\}^{2}-\frac{1}{2}[\Pi-2\gamma\sin(\omega_{\rm
ph}t)]^{2}\right)\Bigg{\\}}\ ,$

and its dynamics are shown at five different times in Figure 2a. Before the
optical excitation, the phonons are in the vacuum state represented by the
Gaussian Wigner function in the center of the phase space. Half of the weight
of the phonon’s Wigner distribution is brought into the excited state subspace
by the optical pulse. This makes them move as a coherent state around the new
equilibrium position, which is shifted by $2\gamma$ in $U$-direction. This
trajectory is marked as black circle in the figure. The other half of the
phonon state remains associated with the ground state of the TLS and,
therefore, stays in the vacuum state. The full phonon state after tracing over
the TLS states is a statistical mixture of the vacuum state and a coherent
state moving around the shifted equilibrium position. After a full phonon
period at $t=t_{\rm ph}$, the Wigner function agrees with the initial
situation because the coherent state moves through the origin and overlaps
with the vacuum state. The phonon’s influence on the properties of the TLS is
shown in Figure 2b. While the occupation of the excited state stays constant
at $c=0.5$, the polarization $|p|$ starts at 0.5 directly after the optical
excitation at $t=0$ and drops rapidly to almost zero in the following. This
dephasing is inverted towards $t=t_{\rm ph}$, resulting in a full rephasing to
$|p|=0.5$. While the coherent states separate in phase space, coherence is
lost from the TLS, which already shows that the overlap of the different parts
of the Wigner function plays an important role for the properties of the
entire system.

Figure 2: (a) Dynamics of the phonon Wigner function after a single pulse
excitation of the TLS. (b) TLS dynamics with the excited state occupation $c$
in red and the polarization $|p|$ in blue. (c) Entropies of the phonon system,
$S^{\rm ph}$ in green and $S^{\rm ph}_{\rm lin}$ in red.

Finally, in Figure 2c, we show the entropy of the phonons which, as mentioned
above, agrees with the entropy of the TLS. As the initial phonon state is
pure, both entropies, $S^{\rm ph}$ in green and $S^{\rm ph}_{\rm lin}$ in red,
start at zero at $t=0$. While the different parts in phase space separate, the
entropy grows to $S^{\rm ph}_{\rm lin}\approx 0.5$ because the phonons are in
a statistical mixture that must have a non-vanishing entropy. The full entropy
follows the same dynamics as the linear one but is always larger, as
previously explained, and grows to $S^{\rm ph}\approx\ln(2)\approx 0.7$.
Reaching $t=t_{\rm ph}$, the entropies drop to zero again. The reason is the
recovered overlap of the two parts of the Wigner function. Finally, at
$t=t_{\rm ph}$, the phase space representation cannot be distinguished from
the vacuum state. Therefore, the entropy also has to agree with the one of the
pure vacuum state being zero.

####  3.1.2 Phonons Generated by a Decaying TLS

In the previous section, without any decay or additional pure dephasing of the
TLS, the quantum state of the entire system remained pure, resulting in a
vanishing entropy of the full system. It also allowed us to easily calculate
the full entropies for TLS and phonons. In this section, we consider a non-
vanishing decay rate of the occupation of the excited state into the ground
state, which naturally results in a statistical mixture in the TLS’s quantum
state that also imprints onto the phonons. Therefore, for the phonons, we can
only calculate linear entropies, according to Equation (20). In Reference Hahn
et al. (2019), we explained how the Wigner function evolves during the decay
process in the TLS. Therefore, we consider the same optical excitation with a
pulse area of $\theta=\pi$, which initially fully inverts the TLS. Without any
decay, the Wigner function would read

$W(U,\Pi,t)=\frac{1}{2\pi}\exp\left(-\frac{1}{2}\\{U-2\gamma[1-\cos(\omega_{\rm
ph}t)]\\}^{2}-\frac{1}{2}[\Pi-2\gamma\sin(\omega_{\rm ph}t)]^{2}\right)\ ,$
(25)

being a single Gaussian moving on a circle around the shifted equilibrium
position of the excited state. In Figure 3a, Wigner functions for different
decay rates $\Gamma$ are shown at $t=10t_{\rm ph}$. We find that the phonon
state gets smeared out in phase space. For rapid decays on the left, the
phonons almost completely stay in the vacuum state and look more or less like
a coherent Gaussian distribution. When looking at the corresponding linear
entropy dynamics in Figure 3b, in bright red, we see that it only increases
slightly after the optical excitation at $t=0$. When slowing down the decay
process, i.e., moving in Figure 3a more to the right, the Wigner function
smears out more and more. Accordingly it looks less and less like a coherent
state which also leads to increasing entropies in (b) when going from bright
to dark colors. Additionally, we find that the final entropy value is reached
slower because it follows the decay of the TLS. Especially for the slowest
considered decay of $\Gamma=0.1\omega_{\rm ph}$, where the full decay takes
several phonon periods, the dynamics of the linear phonon entropy develop
minima at full phonon periods $t=nt_{\rm ph}$. These are the times when the
Wigner function in Figure 3a starts overlapping itself. This is exemplarily
shown in Figure 3c for $t=t_{\rm ph}$, where the thick Gaussian part is the
oscillating coherent state. This is the first time it intersects with the
circular distribution that has already decayed into $\left|g\right>$. In
agreement with the findings in Figure 2, this leads to the temporary reduction
of the entropy in Figure 3b.

Figure 3: (a) Phonon Wigner function after a full decay of the TLS at
$t=10t_{\rm ph}$ for different decay rates $\Gamma$ as given in the picture.
(b) Linear phonon entropy as a function of time after the pulse. The decay
rate decreases from bright to dark colors. (c) Exemplary Wigner function for a
slow decay rate of $\Gamma=0.1\omega_{\rm ph}$ at $t=t_{\rm ph}$. (d) Same as
(b) but for the full linear entropy. (e) Same as (b) but for the linear
entropy of the TLS. (f) Bloch sphere of the TLS to illustrate the purity of
different states; blue shows pure and red mixed states.

Comparing the linear entropy of the phonon system in Figure 3b with the linear
entropy of the entire system $S_{\rm lin}$ in (d), we basically find the same
overall behavior. The dynamics start at zero, the final values increase for
smaller decay constants, and the final values are reached later. However, both
for small and large $\Gamma$, we find qualitative differences. Starting with
small $\Gamma$ in dark blue, especially for $\Gamma=0.1\omega_{\rm ph}$, the
curve constantly grows without developing any minima. This shows that the
reduction of phonon entropy due to the overlapping Wigner functions does not
attain to the full entropy. The reason for this is that the reduction of the
phonon entropy due to overlapping parts of the Wigner function only happens
because the state information of the TLS has been traced out. Taking the
entire coupled system into account, the overlapping phonon parts belong to
different states of the TLS and can therefore be told apart. Therefore, the
phonons do not lead to a depression of the full linear entropy. Conversely,
for large decay rates, in bright blue, especially $\Gamma=5\omega_{\rm ph}$,
we find that, on a short timescale around $t=0.1t_{\rm ph}$, a pronounced
maximum appears in $S_{\rm lin}$. This effect is not found in the phonon part
in Figure 3b and therefore stems from the TLS contribution. To understand
this, in Figure 3e, we plot the linear entropy of the TLS as green lines. The
bright and dark colors agree with the ones in Figure 3b,d. In addition, we
consider the schematic Bloch vector representation of the TLS state from
Equation (18c) in Figure 3f. The $z$ direction of the Bloch sphere depicts the
occupation of the states, where the south pole is a pure ground state
$\left|g\right>$ and the north pole a pure excited state $\left|x\right>$.
Between these points, all Bloch vectors that are on the surface of the sphere
(blue line) are superpositions
$\left|\chi\right>=N(\alpha\left|g\right>+\beta\left|x\right>)$ and are
therefore pure. In the other extreme case of the line directly connecting
north and south pole (red), the system is in a statistical mixture of
$\left|g\right>$ and $\left|x\right>$. In the center of the Bloch sphere, the
TLS is in both states with equal probability, resulting in the lowest purity.
For the entropy of the TLS in Figure 3e, this means that, directly after the
excitation into the excited state and after the full decay, the entropy is
zero. In between, the system evolved through a statistical mixture, which has
a non-vanishing entropy. The linear entropy reaches maxima with $S_{\rm
lin}^{\rm TLS}=0.5$, in agreement with the result in Figure 1b.

If we want to determine the final linear entropy $S_{\rm lin}^{\rm
ph,\infty}=S_{\rm lin}^{\rm ph}(t\to\infty)$ of the phonon state after the TLS
is fully decayed into the ground state, we can investigate the dynamics in
phase space. Note that the final entropy is only carried by the phonon part
because the TLS is in the pure ground state. As schematically shown in Figure
4a, the movement of the coherent state on the circle in the excited state
subspace and the accompanied decay into the ground state leads to a
distribution that can be seen as a continuous distribution of coherent states
with decreasing amplitude. We can parametrize the circular motion of the
Wigner function including the decay of the amplitude by

$W_{\infty}(U,\Pi)=\Gamma\int\limits_{0}^{\infty}\exp(-\Gamma
t)W_{(U_{0}(t),\Pi_{0}(t))}\,{\rm d}t\ ,$ (26)

where $W_{(U_{0}(t),\Pi_{0}(t))}$ is a Gaussian centered around
$(U,\Pi)=(U_{0}(t),\Pi_{0}(t))$. With the circular trajectory in Equation
(25), we have to consider

$U_{0}(t)=2\gamma[1-\cos(\omega_{\rm ph}t)]\qquad{\rm
and}\qquad\Pi_{0}(t)=2\gamma\sin(\omega_{\rm ph}t)\ .$ (27)

Note that, to retrieve the Wigner distribution in the ground state in Figure
3a, one has to mirror the schematic in Figure 4a. However, the final linear
entropy remains unaffected because it only depends on the general shape of the
distribution. With this, the final linear entropy reads

$\displaystyle S_{\rm lin}^{\rm ph,\infty}$
$\displaystyle=1-4\pi\iint\limits_{-\infty}^{\quad\infty}W_{\infty}^{2}(U,\Pi)\,{\rm
d}U{\rm d}\Pi$
$\displaystyle=1-4\pi\Gamma^{2}\iint\limits_{-\infty}^{\quad\infty}\iint\limits_{0}^{\quad\infty}\exp\left[-\Gamma(t+t^{\prime})\right]W_{(U_{0}(t),\Pi_{0}(t))}W_{(U_{0}(t^{\prime}),\Pi_{0}(t^{\prime}))}\,{\rm
d}t{\rm d}t^{\prime}\,{\rm d}U{\rm d}\Pi\ .$ (28)

The integral over $U$ and $\Pi$ describes the overlap of two coherent states
in phase space. In general, two coherent states with a phase space distance of
$a$ have an overlap of Schleich (2011)

$4\pi\iint\limits_{-\infty}^{\quad\infty}W_{(0,0)}W_{(a,0)}(U,\Pi)\,{\rm
d}U{\rm d}\Pi=\exp\left[-\frac{a^{2}}{4}\right]\ .$ (29)

Therefore, the entropy becomes

$S_{\rm lin}^{\rm
ph,\infty}=1-\Gamma^{2}\iint\limits_{0}^{\quad\infty}\exp\left[-\Gamma(t+t^{\prime})\right]\exp\left(-\left\\{2\gamma\sin\left[\frac{1}{2}\omega_{\rm
ph}(t-t^{\prime})\right]\right\\}^{2}\right)\,{\rm d}t{\rm d}t^{\prime}\ .$
(30)

The sine function in the exponent can be approximated by a linear function for
small frequencies or short times. Note that, although the integration is
carried out up to $t=\infty$, the exponential decay with $\Gamma$ effectively
limits the integrated time interval. Therefore, we expect this approximation
to work well for sufficiently large $\Gamma$. For the motion of the Wigner
function, this corresponds to a linear movement in phase space, as
schematically shown in Figure 4b. The corresponding linear entropy can then be
calculated to

$\displaystyle S_{\rm lin}^{\rm ph,\infty}$ $\displaystyle\approx
1-\Gamma^{2}\iint\limits_{0}^{\quad\infty}\exp\left[-\Gamma(t+t^{\prime})\right]\exp\left\\{-[\gamma\omega_{\rm
ph}(t-t^{\prime})]^{2}\right\\}\,{\rm d}t{\rm d}t^{\prime}$
$\displaystyle=1-\frac{\sqrt{\pi}\,\Gamma}{2\gamma\omega_{\rm
ph}}\exp\left[\left(\frac{\Gamma}{2\gamma\omega_{\rm
ph}}\right)^{2}\right]{\rm erfc}\left(\frac{\Gamma}{2\gamma\omega_{\rm
ph}}\right)\ ,$ (31)

where ${\rm erfc}(x)=1-{\rm erf}(x)$ and ${\rm erf}(x)$ are the error
function.

Figure 4: Final value of the entropy after the full decay of the TLS. (a)
Schematic of the phase space dynamics of the Wigner function during the decay.
(b) Approximated dynamics as a straight line. (c) Final linear entropy $S_{\rm
lin}^{\rm ph,\infty}$ as a function of the decay rate $\Gamma$, full
simulation in solid and approximation from Equation (31) in dashed lines.
Different coupling strengths are shown in blue, red, and green. The dotted
green line shows the entropy for an expansion of the squared sine-function up
to the sixth order.

The results for the final entropy $S_{\rm lin}^{\rm ph,\infty}$ are shown in
Figure 4c as a function of the decay rate $\Gamma$, where the full
calculations according to the dynamics are shown as solid lines and the
approximations from Equation (31) are the dashed lines. We show the three
different coupling strengths $\gamma=2$ (blue), $\gamma=1$ (red), and
$\gamma=0.5$ (green). Comparing the different coupling strengths, we find that
a stronger coupling leads to a larger final entropy because the Wigner
function gets distributed over a larger area of phase space. As explained
before, the approximation in Equation (31) works very good for large $\Gamma$,
but we also see that the approximation works over a larger $\Gamma$ range if
the coupling strength is larger. The reason for this is that, for larger
$\gamma$, the circle of the Wigner function’s trajectory is larger, meaning
that its curvature can be better approximated by a linear motion. For the
smallest considered coupling strength $\gamma=0.5$, in green, we additionally
show the dotted line that stems from an approximation of the squared sine
function up to the sixth order, which is the next non-divergent contribution
in Equation (30). This curve is obviously a better approximation of the full
calculation, in particular, in the range $1\lesssim\Gamma/\omega_{\rm
ph}\lesssim 2$. However, in agreement with all dashed lines, it reaches
$S_{\rm lin}^{\rm ph,\infty}=1$ for $\Gamma=0$, while all full linear
entropies go to smaller values. This shows that any approximation of the sine
function will not give accurate results for very small decay rates. In fact,
it turns out that, for any finite (converging) order of the expansion of the
squared sine function in Equation (30), $S_{\rm ph}^{\rm lin,\infty}=1$ is
reached in the limit $\Gamma\to 0$. This can be understood by realizing that,
for any finite order in the expansion of the trigonometric functions in
Equation (27), the trajectory tends to infinity for $t\to\infty$, thus leading
to a delocalized Wigner function. On the other hand, the correct Wigner
function in this limit is the doughnut-shaped function similar to the
rightmost function in Figure 3a, for which the linear entropy can be
calculated analytically, yielding

$S_{\rm ph}^{\rm lin,\infty}=1-e^{-2\gamma^{2}}I_{0}\big{(}2\gamma^{2}\big{)}\
,$ (32)

with the modified Bessel function of first kind and zeroth order $I_{0}$, in
perfect agreement with the numerical results given in Figure 4c.

###  3.2 Two Pulse Excitation

The phonon quantum state gets more involved when a two-pulse excitation is
considered. As extensively studied in Reference Reiter et al. (2011); Hahn et
al. (2019), an excitation with two pulses having pulse areas of
$\theta_{1}=\theta_{2}=\pi/2$ and a delay of $t_{2}-t_{1}=t_{\rm ph}/2$ leads
to the generation of two Schrödinger cat states, each in one TLS subspace. A
Schrödinger cat state is a coherent superposition of two coherent states,
i.e., of the most classical states of a harmonic oscillator, and, as such, it
is of high interest in all areas of quantum optics Brune et al. (1992);
Ourjoumtsev et al. (2007); Deleglise et al. (2008) and, more recently, also
phononics Hofheinz et al. (2009).

####  3.2.1 Phonons Generated by a Non-Decaying TLS

We assume the same excitation scheme as just described and again disregard any
decays of the TLS. The dynamics of the Wigner function are exemplarily shown
in Figure 5a. Immediately before the second laser pulse reaches the TLS, the
phonons are in the statistical mixture previously shown in Figure 2a. The
second pulse creates a second coherent state in the excited state subspace of
the TLS, but it also makes half of the coherent state in $\left|x\right>$ go
back to the ground state. Therefore, we end up with two coherent states in
both subspaces, $\left|g\right>$ and $\left|x\right>$. As nicely seen in
Figure 5a, the corresponding Wigner function shows two of the classic dumbbell
structures of the cat state, two Gaussians and a striped structure of
alternating positive (green) and negative (orange) values between them. These
stripes indicate the interference between the two coherent states. The Wigner
function in the ground state rotates around the origin, and the one in the
excited state around the shifted equilibrium at $(U,\Pi)=(2\gamma,0)=(4,0)$ is
marked by the black circle.

Figure 5: Two pulse excitation without decay. (a) Snapshots of the Wigner
function after the second pulse. (b) Entropy dynamics after the second pulse,
solid lines are the entropies of the cat states, dashed lines the statistical
mixture from Figure 2c. The full phonon entropy is green and the linear one
red.

The entropies of the phonon state are shown in Figure 5b as solid lines. The
linear entropy in red starts at $S_{\rm lin}=0.5$ in agreement with the
entropy in Figure 2d at $t=t_{\rm ph}/2$ because at this time the second pulse
excites the TLS. In contrast to the behavior after the first pulse, here, the
entropy drops very rapidly and forms a sharp minimum after the second pulse.
In total, the entropy performs two oscillations before remaining constant at
the initial value. The same dynamics repeat themselves in an inverted form
before reaching a full period. Another striking feature is a smaller
depression around $t=t_{\rm ph}/2$. Overall, we find that, by the second
pulse, the entropy of the phonon is temporarily reduced but never increased.
The maximum entropy is here the one of a statistical mixture of two fully
separated coherent states (see Figure 2), which is obviously the same for a
statistical mixture of two cat states that are fully separated. This is the
case during the times around $t/t_{\rm ph}=1/4$ and $t/t_{\rm ph}=3/4$ (see
Figure 5a). Next, we discuss the reduced entropy around half a period in
Figure 5b. Looking at the corresponding Wigner function in Figure 5a, we find
that this is the time when one of the coherent states in the excited state
system (moving on the circle) overlaps with the vacuum state (staying at the
origin). This is the same effect as discussed in the previous sections, where
the entropy shrank when the phonon states were overlapping in phase space.
Finally, we have to understand the strong reductions of the entropy for times
around full periods. To do so, we examine the two insets in Figure 5b that
show snapshots of the Wigner function at the marked times, i.e., where the
entropy is minimal and maximal. The left one at the minimum depicts a time
where each of the two Gaussians starts to split into two, which cannot yet be
resolved in the figure because their overlap is still too large. However, the
interference terms between them also move apart. At $t=0$, their negative and
positive values are distributed in such a way that they exactly compensate
each other (see $t=0^{+}$ in Figure 5a). But, in the left inset, we see that
at this time negative and positive values add up, respectively, making for an
accurate alternating pattern. Comparing this structure of the Wigner function
with one of the cat states in Figure 5a shows a strong resemblance. So, the
reason for the strong decrease of the entropy is that, at these times, the
Wigner function can only hardly be distinguished from a single Schrödinger cat
state, which is a pure state. Likewise, we can analyze the Wigner function at
a maximum of the entropy oscillation in the right inset. In addition, here,
the Gaussians have a large overlap, but the stripes of the interferences are
aligned in such a way that the line in the center has vanishing values. This
strongly disagrees with the natural structure of a cat state interference and
makes it easily distinguishable from that pure cat state. The full phonon
entropy $S^{\rm ph}$ is shown as a green solid line. It follows the same
dynamics as the linear one but is just scaled to larger values, as discussed
before. Finally, let us remark on the additionally plotted dashed lines, which
are the respective entropy curves from Figure 2c. They exactly form envelopes
for the oscillations and therefore demonstrate that the oscillating dynamics
are again a result of the separation process of the different Wigner functions
in phase space.

The relative phase of the two laser pulses changes the phase in the cat
states, i.e., the phase of the striped structure of the Wigner function. As
long as the different parts of the phonon state are separated in phase space,
the phase has no influence on the phonon entropy. The other crucial parameter
of the phonon system is the coupling strength $\gamma$ that determines the
distance of the coherent states and the number of stripes in the interference
term, as exemplarily shown in Figure 6a,b. The influence of an increased
coupling strength is presented in Figure 6c, where the linear phonon entropy
$S^{\rm ph}_{\rm lin}$ is plotted in the same way as in Figure 5b but only for
times up to $t/t_{\rm ph}=0.25$. The red curve shows $\gamma=2$ from Figure 5b
as a reference. Looking at the larger coupling in bright red ($\gamma=4$) and
a smaller coupling in dark red ($\gamma=1.5$), we clearly see that the
oscillation of the entropy gets faster when $\gamma$ grows and more minima
evolve. The reason is that the interference terms consist of more stripes that
run through each other. At the same time, the envelope gets shorter for a
larger $\gamma$, which can be traced back to the larger spread of the Wigner
function in phase space. While the sizes of the interference terms and the
Gaussians stay the same, as shown in Figure 6a,b, the radius of the trajectory
increases. Because the angular frequency of the motion remains the same, the
two interferences separate in a shorter time, and this time determines the
envelope of the entropy in Figure 6c.

Figure 6: (a,b) Exemplary Wigner functions at $t=t_{\rm ph}/4$ for $\gamma=2$
in (a) (same as in Figure 5a) and $\gamma=4$ in (b). (c) Linear phonon entropy
$S^{\rm ph}_{\rm lin}$ as a function of time after a two-pulse excitation as
in Figure 5b. The coupling strength $\gamma$ increases from dark to bright
red.

###  3.3 Phonon Cat State Entropy Dynamics in a Decaying TLS

Next, we increase the complexity of the considered phonon state by analyzing
the influence of the decay of the TLS on the entropy dynamics of cat states.
For reasons of clarity, we consider a single Schrödinger cat state entirely in
the excited state $\left|x\right>\otimes\left|{\rm cat}\right>$ as initial
state without any optical excitation and account for a decay of the TLS into
its ground state. Although this state cannot directly be prepared by optical
pulses, in Reference Hahn et al. (2019) it is explained how it is constructed
mathematically as initial state for the simulated decay dynamics. Some
snapshots of the corresponding Wigner function dynamics are shown in Figure 7a
for a small decay rate of $\Gamma=0.1\omega_{\rm ph}$. As analyzed in
Reference Hahn et al. (2019), the combined rotation and shift of the phonon
equilibrium position due to the decay of the TLS finally leads to a Wigner
function in the shape of an eight. Note that, for a slow decay where the
coherent parts lead to a homogeneously distributed eight-shape, two
interference terms transferred into the ground state at $t/t_{\rm
ph}=(2n+1)/4$ survive the decay process.

The corresponding linear entropies are depicted in Figure 7b, where the blue
curves show the linear entropy of the full system $S_{\rm lin}$ and the red
ones the linear entropy of the phonons $S_{\rm lin}^{\rm ph}$. The decay rate
increases from dark to bright colors. We find the same dependency on the decay
rate as for a single coherent state in Figure 3b,d, the final entropy
increases for a slower decay. In addition, the behavior for very fast decays,
e.g., $\Gamma=5\omega_{\rm ph}$, is approximately the same as in Figure 3. The
entropy of the full system (light blue) forms a sharp peak due to the
evolution of the TLS through a statistical mixture, while the phonon part
(light red) basically just rises before reaching the stationary value. The
dynamics get more involved and new features appear for slow decays, e.g., for
$\Gamma=0.05\omega_{\rm ph}$. Here, the full entropy in dark blue continuously
increases to the stationary value at the end of the decay process, while the
phonon contribution in dark red is always slightly smaller and shows
additional dynamics developing multiple minima and maxima within each phonon
period. While the dynamics are rather irregular on shorter times $t<3t_{\rm
ph}$, it becomes more periodic for longer periods of time.

Figure 7: (a) Snapshots of the Wigner function during the decay into the
ground state for a decay rate of $\Gamma=0.1\omega_{\rm ph}$. (b) Linear
phonon entropies in red and full linear entropies in blue as functions of time
for different decay rates. $\Gamma$ increases from bright to dark colors. (c)
Zoom-in on one phonon period for $\Gamma=0.05\omega_{\rm ph}$. The coherent
contribution $S_{\rm lin}^{\rm coh}$ is green, the one from the interference
$S_{\rm lin}^{\rm int}$ is blue and their sum dashed blue. Next to (c) are
exemplary Wigner functions of the coherent part (bottom) and the interference
(top) for the respective minima marked by black lines.

To understand the origin of these dynamics in the phonon system, we take a
closer look at the different parts of the Wigner function. According to
Reference Reiter et al. (2011), the Wigner function of a cat state can be
separated into

$W=W_{\rm coh}+W_{\rm int}\ ,$ (33)

where $W_{\rm coh}$ describes the two coherent states that have been studied
previously, and $W_{\rm int}$ is the interference showing up as striped
structure in phase space. Under the assumption that the phonon coupling
strength, $\gamma$ is large enough such that the different parts of the Wigner
function do not significantly overlap in phase space; the linear entropy can
also be separated into two contributions, $S_{\rm lin}^{\rm coh}$ and $S_{\rm
lin}^{\rm int}$, calculated from the respective contributions of the Wigner
function, and we obtain

$S_{\rm lin}^{\rm ph}\approx S_{\rm lin}^{\rm coh}+S_{\rm lin}^{\rm int}\ .$
(34)

In Figure 7c, we show the different entropies for a short time window
$4\leqslant t/t_{\rm ph}\leqslant 5$. This already clarifies the picture a
bit. First of all, we find that $S_{\rm lin}^{\rm coh}$ (green) and $S_{\rm
lin}^{\rm int}$ (blue) are approximately of the same size and the sum of the
two parts, shown in dashed blue, agrees perfectly with the full linear phonon
entropy (red line). The coherent part $S_{\rm lin}^{\rm coh}$ has reduced
values at times $t/t_{\rm ph}=n/2$, and $S_{\rm lin}^{\rm int}$ develops
minima exactly between those times, i.e., at $t/t_{\rm ph}=(2n+1)/4$. Because
the shapes of the minima in the two contributions are not the same, the sum
appears quite involved.

After identifying the different dynamics, we have to understand their origin.
Therefore, next to Figure 7c, we plot the Wigner functions $W_{\rm coh}$ and
$W_{\rm int}$ for the respective minima, as marked by the black lines.
Starting with the coherent part at the bottom, we recognize that the situation
is equivalent to the one in Figure 3b,c. The entropy is always reduced when
the rotating Wigner function of the coherent states in the excited state
subspace (marked by black circles) overlaps with parts of the Wigner function
in the ground state subspace. In the right example, at $t=4.75t_{\rm ph}$, the
two Gaussians are clearly separated from the decayed part in the ground state.
In the left one, at $t=4.5t_{\rm ph}$, one of the coherent states overlaps
with the touching point of the two circles that are in the ground state
$\left|g\right>$. Because we start with two coherent states in
$\left|x\right>$, the periodicity of the minima is half the phonon period.
Moving on to the Wigner function of the interference contribution at the top,
we only see the expected striped patterns. The times where $S_{\rm lin}^{\rm
int}$ is reduced agree with the times $t/t_{\rm ph}=(2n+1)/4$, where the
interference terms that survive the decay process and remain also after the
full decay are transferred into the ground state system (see discussion in
Reference Hahn et al. (2019)). As seen in the depicted Wigner function on the
right, at these times (e.g., $t=4.75t_{\rm ph}$) in Figure 7c, one of the
interference terms that were already transferred into $\left|g\right>$
perfectly overlaps with the single interference that is still in the excited
state, resulting in the two separated structures in phase space. For all other
times, three contributions appear, two in $\left|g\right>$ and one in
$\left|x\right>$, as exemplarily shown on the left for $t=4.5t_{\rm ph}$.
Thus, the fundamental reason for the reduction of the entropy is that a
mixture of two cat states is more pure than a mixture of three. The perfectly
overlapping interferences on the right make the corresponding Wigner function
look more like a mixture of two states than of three.

####  3.3.1 Phonons Generated by a Decaying TLS

To conclude the discussion, we now take a look at the two pulse excitation
discussed in Section 3.2.1 and consider a non-vanishing decay rate of the TLS.
In Reference Hahn et al. (2019), it was shown that the final phonon state is
in good agreement with the eight-shaped Wigner function of the decayed single
cat state previously analyzed. However, now the phonon generation leads to a
statistical mixture of two cat states that are additionally smeared out in
phase space. This is exemplarily shown by the Wigner functions in Figure 8a.
Although the quantum state of the system is more involved, the linear entropy
of the phonons depicted as red line in Figure 8b evolves in a well-structured
manner. Especially after the excitation with the second pulse, marked by the
dashed black line, the dynamics resemble the ones in Figure 5b with an
additional increase of the curve according to the decay process. We find the
same broad depressions for half periods and stronger oscillating ones for full
periods. The small additional entropy reductions discussed in Figure 7 are
also found here, as shown by the zoom-in in Figure 5c. However, compared to
the effects of the two overlapping cat states, as previously mentioned, they
almost disappear. The linear entropy of the full system shown as a blue line
in Figure 5b is always smaller than the phonon part and grows smoothly. This
is in agreement with the situation without any decay, where the entropy of the
full system was always zero while the phonon part was non-vanishing.

Figure 8: (a) Snapshots of the Wigner function’s dynamics after the second
pulse for $\Gamma=0.05\omega_{\rm ph}$. (b) Dynamics of the linear phonon
entropy in red and the linear entropy of the full system in blue. (c) Zoom-in
on the marked short time window from (b).

##  4 Conclusions

In summary, we analyzed the entropy dynamics of a single phonon mode coupled
to an optically-driven TLS. We presented a theoretical framework that allowed
us to calculate entropies of the different parts of the system when the
quantum state of the entire system is pure and linear entropies when it is not
pure. Additionally, the concept of Wigner functions for the representation of
phonon quantum states was used. We started our discussion with the most basic
optical excitation, i.e., a single ultrafast pulse, that generated a mixture
of two coherent states in the phonon system and assumed a non-decaying TLS.
From this, we further increased the complexity of the generated phonon state
by including non-vanishing decay rates and two-pulse excitations of the TLS.
This led to Wigner functions that smeared out in phase space and the
generation of Schrödinger cat states, respectively. While the decay of the
TLS, in general, led to an increase of the system’s entropy, the complex
dynamics of the phonon states resulted in temporally significant reductions of
the phonon entropy. All these effects could be traced back to the purity of
the quantum states and the entanglement between phonons and TLS. This
extensive study on the phonon’s entropy led to a thorough understanding of the
fundamental interplay between the dynamics of the two separate parts and their
combined influence on the quantum state purity.

conceptualization, T.H., D.W. and T.K.; methodology, T.H.; software, T.H.;
validation, T.H., D.W. and T.K.; formal analysis, T.H.; investigation, T.H.;
data curation, T.H.; writing–original draft preparation, D.W.; writing–review
and editing, T.H., D.W. and T.K.; visualization, T.H. and D.W.; supervision,
D.W. and T.K.; project administration, T.K. All authors have read and agreed
to the published version of the manuscript.

This research received no external funding.

The authors declare no conflict of interest.

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# Not so presto? Outcomes of
sluggish prestin in outer hair cells

Kuni H. Iwasa 111 NIDCD, National Institutes of Health Bldg. 35A Rm 1F121A,
Bethesda, Maryland 20892 USA, email<EMAIL_ADDRESS>
###### Abstract

Prestin (SLC26A5), a protein essential for the sensitivity of the mammalian
ear, was so named from _presto_. The assumption was that this membrane protein
supports fast movement of outer hair cells (OHCs) that matches the mammalian
hearing range, up to 20 kHz in general and beyond, depending on the species.
_In vitro_ data from isolated OHCs appeared to be consistent with such
frequencies. However, some recent reports cast doubts on this assumption,
suggesting that the intrinsic transition rates of this protein are much lower,
about 3 kHz for guinea pigs, not covering the auditory frequency range of the
animal. Recent _in vivo_ data also show that the amplitude of OHC motion rolls
off well below the best frequency of the location. The present report examines
whether or not these resent observations are compatible with the physiological
function of OHCs by using simple piezoelectric models.

##  1 Introduction

The significance of prestin for the sensitivity and frequency selectivity of
the mammalian ear has been well established [1]. However, the detailed
mechanism, with which this piezoelectric membrane protein plays its
physiological role, remains not as clear. The frequency range that this
protein is capable of responding is an issue, which appears critical in
addressing this problem.

Earlier _in vitro_ studies on isolated OHCs confirmed that their fast motile
response is based on piezoelectricity [2, 3, 4]. Force generation under quasi-
isometric condition was shown to have flat frequency dependence up to 60 kHz
[5]. The characteristic frequency of the power spectrum of membrane current
due to prestin was about 40 kHz [6]. Those frequencies were considered to be
lower bounds imposed by experimental conditions.

However, recent reports on OHCs in _in vitro_ [7, 8] appear to contradict
these earlier observations. In addition, _in vivo_ data obtained with optical
coherence tomography (OCT) [9] have been interpreted as evidence against fast
motile response of OHCs.

The present report examines the implications of these two kinds of reports by
asking two questions. First, can OHCs counteract local viscous drag incurred
by the movement of the organ of Corti in the cochlea if gating rates of
prestin are as low as these recent reports suggest? Second, should the
movement of OHCs large at their best frequencies _in vivo_? The first issue is
addressed by evaluating power generation by OHCs driven by prestin with finite
transition rates. The second issue is analyzed using a system of coupled
oscillators as a model. In addition, the compatibility of these two kinds of
reports is examined.

##  2 Transition rates

First, let us derive the equation of motion for a cell, which is driven by a
motile molecule with finite intrinsic transition rates. An earlier version has
been published as Appendix to Santos-Sacchi et al. [8].

###  2.1 Rate equation

Consider a membrane molecule with two discrete conformational states C0 and C1
and let the transition rates $k_{+}$ and $k_{-}$ between them, with $k_{+}$
from C0 and C1, and $k_{-}$ the opposite, i.e.

$\displaystyle\;k_{+}$ $\displaystyle C_{0}\quad$
$\displaystyle\rightleftharpoons\quad C_{1}$ (1) $\displaystyle\;k_{-}$

Let $P_{1}$ be the probability that the molecule in state C1, which elongate
the cell. Suppose charge transfer $q(>0)$ is associated with a change $a(>0)$
in the length of the cell. Then $P_{1}$ satisfies

$\displaystyle\frac{P_{1}}{1-P_{1}}=\frac{k_{+}}{k_{-}}=\exp[\beta[q(V-V_{0})+aF]],$
(2)

where $F$ is the axial force applied on the cell. The transition rates $k_{+}$
and $k_{-}$ can be expressed as

$\displaystyle k_{+}$
$\displaystyle=\exp\left[\alpha\beta[q(V-V_{0})+aF]\right],$ (3a)
$\displaystyle k_{-}$
$\displaystyle=\exp\left[(-1+\alpha)\beta[q(V-V_{0})+aF]\right],$ (3b)

by introducing a parameter $\alpha$. For the rest of the present paper, the
dependence on the value of the parameter $\alpha$ does not appear except for
$\omega_{g}(=k_{+}+k_{-})$.

The time dependence of $P_{1}$ can be expressed by the rate equation

$\displaystyle\frac{d}{dt}P_{1}=k_{+}-(k_{+}+k_{-})P_{1}.$ (4)

Now assume that the voltage $V$ consists of a constant term $\overline{V}$ and
a small sinusoidal component with amplitude $v$ , i.e.
$V=\overline{V}+v\exp[i\omega t],$ where $\omega$ is the angular frequency and
$i=\sqrt{-1}$. Then the transition rates are time dependent due to the voltage
dependence, satisfying

$\displaystyle\frac{k_{+}}{k_{-}}=\frac{\overline{k}_{+}}{\overline{k}_{-}}(1-\beta
qv\exp[i\omega t]),$ (5)

where $\overline{k}_{+}$ and $\overline{k}_{-}$ are time independent, and
small amplitude $v$ implies $\beta qv\ll 1$. A set of $k_{+}$ and $k_{-}$ that
satisfies Eq. 5 can be expressed

$\displaystyle k_{+}$ $\displaystyle=\overline{k}_{+}(1-\alpha\beta
qv\exp[i\omega t]),$ (6a) $\displaystyle k_{-}$
$\displaystyle=\overline{k}_{-}\\{1-(1-\alpha)\beta qv\exp[i\omega t]\\}.$
(6b)

If we express $P_{1}=\overline{P}_{1}+p_{1}\exp[i\omega t]$, for the 0th and
1st order terms we have, respectively [10]

$\displaystyle\overline{P}_{1}$
$\displaystyle=\frac{\overline{k}_{+}}{\overline{k}_{+}+\overline{k}_{-}},$
(7a) $\displaystyle p_{1}$
$\displaystyle=-\frac{\overline{k}_{+}\overline{k}_{-}}{\overline{k}_{+}+\overline{k}_{-}}\cdot\frac{\beta
qv}{i\omega+\overline{k}_{+}+\overline{k}_{-}}.$ (7b)

Notice that $p_{1}$ does not depend on the factor $\alpha$.

By using the relationship
$\overline{P}_{1}=\overline{k}_{+}/(\overline{k}_{+}+\overline{k}_{-})$, Eq.
7b turns into

$\displaystyle\quad p_{1}=\frac{\beta
q\overline{P}_{\pm}}{1+i\omega/\omega_{g}}\cdot v,$ (8)

with $\overline{P}_{\pm}=\overline{P}_{1}(1-\overline{P})$ and
$\omega_{g}=\overline{k}_{+}+\overline{k}_{-}$. Here, the average values and
the amplitudes of sinusoidal components are represented by notations similar
to the voltage. Notice that Eq. 8 satisfies Boltzmann distribution. That means
the frequency dependence is determined only by the transition rates and
unaffected by mechanical load or by mechanical relaxation.

Figure 1:  A schematic representation of a cell with viscous drag. The
stiffness of the cell is $k$, and drag coefficient is $\eta$. The motile
element is labeled $P$. The mechanical displacement of the cell is represented
by $X$. The properties of OHC cannot easily be rendered as a simple
combination of elastic- and displacement elements, even though such attempts
have been made earlier [11, 12].

###  2.2 Equation of motion

Consider a system, where an OHC is associated with mass $m$, drag coefficient
$\eta$, and external elastic load $K$ (Fig. 1). Assume that the material
stiffness of the OHC is $k$ 222The actual axial stiffness is reduced by
conformational transitions of the motile elements [13]. This effect is
analogous to “gating compliance” [14]. The sum of inertial force and drag is
balanced with elastic force $F(t)$. The displacement $X(t)$ satisfies the
equation

$\displaystyle\left(m\frac{d^{2}}{dt^{2}}+\eta\frac{d}{dt}\right)X(t)=F(t).$
(9)

If we are interested in the response to the stimulus with an angular frequency
$\omega$, the equation of motion Eq. 9 can be expressed as

$\displaystyle(-\omega^{2}m+i\eta\omega)x=f(\omega),$ (10)

where $x$ and $f$ are, respectively, the amplitudes of displacement $X$ and
force $F$ with the frequency $\omega$.

If the force $F$ in Eq. 3 is due to the external load alone, we have $F=-KX$
and $f=-Kx$, which are the familiar equations of motion. In such a case, the
equilibrium condition is $X=0$ and $x=0$. In our system, however, movement is
driven by a deviation from Boltzmann distribution. If the displacement can
respond instantaneously to voltage changes, the system goes from one
equilibrium to another. In such a case, the relationship between $v$ and $p$
is expressed by

$\displaystyle
p_{\infty}=\beta\overline{P}_{\pm}\cdot\frac{qv+a^{2}N\tilde{K}p}{1+i\omega/\omega_{g}}.$
(11)

with $\tilde{K}=kK/(k+K)$. This value, $p_{\infty}$, is the goal of the change
for the variable $p$, while the current condition is represented by $p$. Note
that the quantity $p_{\infty}$ is similar to $p_{1}$ in Eq. 14 in that it
satisfies Boltzmann distribution. However, it depends both $v$ and $p$ because
the energy term has both electrical and mechanical terms as expressed by Eqs.
3.

In the presence of external mechanical load, the mechanical displacement, that
responds to a voltage change, is slowed down due to mechanical load. If the
difference between $p_{\infty}$ and $p$ is small, the driving force can be
proportional to the difference $p_{\infty}-p$. Thus, the driving force can be
expressed

$\displaystyle f=k\cdot aN(p_{\infty}-p)$ (12)

in the manner similar to the case of fast gating [11]. The presence of
external elastic load $K$ makes the displacement $x$ expressed by $x=aNp\cdot
k/(k+K)$, where $n$ is the number of motile elements. By choosing $p$ as the
variable of the equation, we have

$\displaystyle\left[-\omega^{2}m+i\eta\omega\right]p$
$\displaystyle=(k+K)(p_{\infty}-p),$ (13)

which has a familiar form for the equation of motion.

By introducing the explicit form of $p_{1}$, this equation turns into

$\displaystyle[-(\omega/\omega_{r})^{2}+i\omega/\omega_{\eta}+1+\delta^{2}]p=\frac{\beta\overline{P}_{\pm}}{1+i\omega/\omega_{g}}\cdot
qv,$ (14)

where $\omega_{r}^{2}=m/(k+K)$, $\omega_{\eta}=\eta/(k+K)$, and
$\delta^{2}=\beta\overline{P}_{\pm}na^{2}\tilde{K}/(1+i\omega/\omega_{g})$.
The quantity $\delta^{2}$ has only a minor effect on $p$. Notice if we let
$\omega_{g}\rightarrow\infty$, factor $1/(1+i\omega/\omega_{g})$ turns into
unity, we obtain the equation [12] for infinitely fast gating.

In a special case of $m=K=0$, this equation turns into

$\displaystyle(1+i\omega/\omega_{\eta})(1+i\omega/\omega_{g})p=\beta\overline{P}_{\pm}\cdot
qv,$ (15)

showing low-pass behavior with two time constants $1/\omega_{g}$ and
$1/\omega_{\eta}$.

###  2.3 Power output

Power output $W(\omega)$ by OHCs can be evaluated by calculating the work done
against viscous drag because the work against elastic load is recovered after
a cycle. The mean power output $\langle W(\omega)\rangle$ is given by $\langle
W(\omega)\rangle=(\eta/2)|\omega aNp/(k+K)|^{2}$.

Figure 2: Limiting frequency $\omega_{L}$ for prestin with gating frequency
$\omega_{g}$ against that of $\omega_{\ell}$, for infinitely fast gating.
Here, $\omega_{L}/\omega_{g}$, the limiting frequency normalized with respect
to the gating frequency is plotted against $\omega_{\ell}/\omega_{g}$

.

Eqs. 14, with the observation that $\delta^{2}$ has a relatively minor effect
on $p$, indicates that the mean power generation $\langle
W_{\infty}(\omega)\rangle$ for the special case of infinitely fast gating
differs from $\langle W(\omega)\rangle$ only in the absence of the attenuation
factor $1/[1+(\omega/\omega_{g})^{2}]$ if $v$ remains the same. However, the
factor $1/(1+i\omega/\omega_{g})$ reduces $v$ because it reduces movement of
the motile element. For this reason, we may put

$\displaystyle\langle W(\omega)\rangle<\langle
W_{\infty}(\omega)\rangle/[1+(\omega/\omega_{g})^{2}].$ (16)

Recall how an optimal limiting frequency was determined for the case, where
gating is infinitely fast. In the presence of inertia, power generation has a
peak value $P_{\infty}^{\mathrm{max}}$ due to piezoelectric resonance. A
limiting frequency $\omega_{\ell}$ for that case is obtained from
$P_{\infty}^{\mathrm{max}}=\mu\omega_{\ell}^{2},$ equating the maximal power
production with viscous loss. Here $\mu$ is proportional to viscous
coefficient $\eta$.

If we can assume that the resonance peak is sharp, the corresponding limiting
frequency $\omega_{L}$ for a finite gating frequency $\omega_{g}$ can be
approximated by
$P_{\infty}^{\mathrm{max}}/[1+(\omega_{L}/\omega_{g})^{2}]<\mu\omega_{L}^{2}.$

The combination of these two equations leads to

$\displaystyle(\omega_{L}/\omega_{g})^{2}<\frac{1}{2}\left(\sqrt{1+4(\omega_{\ell}/\omega_{g})^{2}}-1\right).$
(17)

For $\omega_{\ell}=2\pi\times 10$ kHz [12] and $\omega_{g}=2\pi\times 3$ kHz
[8], $\omega_{L}/\omega_{g}<1.1$ (See Fig. 2). The limiting frequency is 3.3
kHz, not much higher than the gating frequency.

The assumption that led to the inequality (17) is not always satisfied because
the resonance peak disappears as the gating frequency decreases. In such
cases, however, the inequality still holds even though the real frequency
limit is lower than indicated because power production is lower without
resonance peak.

##  3 Coupled oscillators

The evaluation of maximum power output as mentioned above is based on the
assumption that OHCs operate at near resonance frequencies. That assumption in
turn requires multiple modes of motion in the organ of Corti because of the
mismatch in stiffness between OHCs and the basilar membrane [12]. In the
following, a coupled oscillator is used as a model illustrate this issue.

Here we assume an OHC is associated with an oscillator, which is weakly
coupled with another oscillator. It would be useful to examine a system of
interacting coupled harmonic oscillators explicitly due to its simplicity,
even though the cochlea is not a linear system. The following analysis can be
justified only for low input level.

###  3.1 Coupled harmonic oscillators

A textbook example of coupled harmonic oscillators are connected by an elastic
element. Typically described by

$\displaystyle\left(M_{1}\frac{d^{2}}{dt^{2}}+\eta_{1}\frac{d}{dt}+K_{1}\right)X_{1}$
$\displaystyle=K_{c}(X_{2}-X_{1})$ (18a)
$\displaystyle\left(M_{2}\frac{d^{2}}{dt^{2}}+\eta_{2}\frac{d}{dt}+K_{2}\right)X_{2}$
$\displaystyle=K_{c}(X_{1}-X_{2})+F_{2}(t),$ (18b)

where $M_{j}$ is the mass, $\eta_{j}$ drag coefficient, and $K_{j}$ the spring
of each oscillator (for j=1,2). $K_{c}$ is the stiffness of the spring that
connects two oscillators. $F_{2}(t)$ is the force that stimulates the system.
The classical analyses of such coupled oscillators are available [15, 16].
Despite the simplicity of the idea, the frequency dependence of the energy
transfer between the two oscillators is rather complex [16].

###  3.2 Amplification with negative drag

The simplest introduction of active or amplifying mechanism to the system of
interacting harmonic oscillator would be to flip the sign of the drag term in
one of the oscillators.

Let the first one with displacement $x$ is an amplifying oscillator (AO), i.e.
$\eta_{1}<0$, and the second one with displacement $y$ is a dissipating
oscillator (DO) and stimulus is applied to the DO. Let the stimulus has a
sinusoidal waveform with at an angular frequency $\omega$. Then
$F_{2}=f_{2}\exp[i\omega t]$ and the amplitudes of AO and DO are,
respectively, $x$ and $y$.

The equation of motion can be expressed in the form

$\displaystyle[-(\overline{\omega}/\overline{\omega}_{1})^{2}-i\overline{\omega}/\overline{\omega}_{a}+1+cs]x-csy$
$\displaystyle=0,$ (19a) $\displaystyle-
cx+[-\overline{\omega}^{2}+i\overline{\omega}/\overline{\omega}_{\eta}+1+c]y$
$\displaystyle=f,$ (19b)

where frequencies are normalized with respect to the resonance frequency
$\omega_{2}$ (where $\omega_{2}^{2}=M_{2}/K_{2}$) of the DO, i.e.
$\overline{\omega}=\omega/\omega_{2}$,
$\overline{\omega}_{\eta}=\omega_{\eta}/\omega_{2}$, and
$\overline{\omega}_{a}=\omega_{a}/\omega_{2}(>0)$. The negative sign on the
term involving $\overline{\omega}_{a}$ indicates negative drag that amplifies
the oscillation of the system.

The parameters introduced are $f=f_{2}/K_{2}$, $c(=\\!K_{c}/K_{2})<1$ and
$s(=\\!K_{2}/K_{1})<1$, because it is reasonable to assume the stiffness of
the basilar membrane is larger than the elastic load of OHCs.

###  3.3 OHC as amplifier

Let us start with the motile activity. The OHC equation, which has voltage
stimulation but not drag can be expressed [12]

$\displaystyle[-(\overline{\omega}/\overline{\omega}_{1})^{2}+1]p=\beta\overline{P}_{\pm}\cdot
qv.$ (20)

The electric circuit involved generates the receptor potential $v$, following

$\displaystyle v=\frac{-i_{0}\hat{r}+i\omega Nqp}{\sigma+i\omega C_{0}},$ (21)

Here $i_{0}=(e_{ec}-e_{K})/(\overline{R}_{a}+R_{m})$, where $e_{ec}$ is the
endocochlear potential, $e_{K}$ the resting potential of the OHC,
$\overline{R}_{a}$ the resting level of hair bundle resistance, $R_{m}$ the
basolateral resistance, and $C_{0}$ the regular capacitance.

Here we assume $\hat{r}$ is generated by hair bundle current by displacement
$x$, and let $\hat{r}=bx$. For higher frequencies, where capacitive current
dominates, we have

$\displaystyle v\approx\left[i\cdot\frac{i_{0}b}{\omega
C_{0}}+\frac{q}{aC_{0}}\cdot\frac{k+K}{k}\right]\cdot x$ (22)

where the letter form is for auditory frequencies, which satisfy
$\sigma\ll\omega C_{0}$.

Then the equations of motions for the coupled oscillators are

$\displaystyle\left[-(\overline{\omega}/\overline{\omega}_{1})^{2}-i\cdot
A/\overline{\omega}+1+cs-B\right]x-csy$ $\displaystyle=0,$ (23a)
$\displaystyle-
cx+[-\overline{\omega}^{2}+i\cdot\overline{\omega}/\overline{\omega}_{\eta}+1+c]y$
$\displaystyle=f,$ (23b)

where $A=\beta abNqi_{0}P_{\pm}k/[\omega_{2}C_{0}(k+K)]$ and $B=\beta
Nq^{2}\overline{P}_{\pm}/C_{0}$.

Figure 3: The amplitudes of amplifying oscillator (AO, in orange) and
dissipating oscillator (DO, in blue) plotted in logarithmic scale against
frequency $\overline{\omega}$ normalized with respect to the resonance
frequency of DO and the resonance frequency ratio $\omega_{1}/\omega_{2}$.
Parameter values used: $c=0.1,s=0.01,A=0.02,B=0.001,\omega_{\eta}=15$.

Notice that the imaginary part of the coefficient of $x$ in Eq. 23a is
negative. This means this oscillator functions as amplifier as expected. If
the gating rates of the motile molecule are finite, the RHS of Eq. 20 is
multiplied by factor $1/(1+i\omega/\omega_{g})$. For $\omega\gg\omega_{g}$,
the imaginary part of the coefficient of $x$ in Eq. 23a turns into
$B/\omega_{g}$, which is positive, implying that this oscillator no longer
functions as an amplifier.

The amplitude ratio $|x/y|$ is expressed by

$\displaystyle\left|\frac{x}{y}\right|^{2}=\frac{(cs)^{2}}{[1+cs-B-(\overline{\omega}/\overline{\omega}_{1})^{2}]^{2}+(A/\overline{\omega})^{2}},$
(24)

which indicates small values for $c$ and $s$, which should apply, make $x$
much smaller than $y$. In addition, AO can functions as amplifier for
realistic values of A and B (Fig. 3). These features are consistent with the
OCT observation. However, $x$ is smaller for frequencies lower than near
resonance. Thus the present model cannot explain the roll-off of OHC amplitude
that was experimentally observed.

##  4 Conclusions

The present analysis is based on simple models, which likely over-simplifies
possible modes of motion in the cochlea. Indeed, the organ of Corti, where
OHCs are located, is a anisotropic, heterogeneous-three dimensional body. In
addition, they do not include the nonlinearity, which is characteristic to the
cochlea, limiting the validity to movements at small amplitudes. With these
reservations, the following conclusions can be drawn.

The intrinsic transitions of prestin must be fast enough to exceed the upper
bound of the auditory range for electromotility to counteract local drag. Even
though energy can flow laterally along the cochlea, a large gap between the
intrinsic gating frequency and the best frequency of the location makes it
unlikely for OHCs function as the cochlear amplifier.

A simple local coupled oscillator model shows that OHCs can function as
cochlear amplifier, at frequencies lower than the gating frequency of prestin,
even if the amplitude of OHC length changes is much less than the amplitude of
basilar membrane movement, consistent with OCT data [9].

For these reasons, low frequency gating of prestin in recently reports [7, 8],
which contradicts older reports [5, 6], is not compatible the OCT data [9]
because low frequency gating is incapable with amplifying effect of OHCs at
best frequencies.

The present analysis, however, cannot explain larger amplitude of OHC movement
at low frequency as observed with the OCT [9]. Given the observed dependence
of the mode of motion of the organ of Corti [17, 18], one can argue
nonetheless that the modes of motion at low frequencies the elastic load on
OHCs could be less, making the amplitude larger. Verifying this expectation
requires a detailed analysis of modes of motion _in vivo_ and their dependence
on the frequency.

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