# Quantum Brownian Motion in the Caldeira-Leggett Model
with a Damped Environment

Lester Buxton1,2 Marc-Thomas Russo1,2 Jim Al-Khalili1 Andrea Rocco1,3
<EMAIL_ADDRESS>1School of Mathematics and Physics, 2Leverhulme Quantum
Biology Doctoral Training Centre and 3Department of Microbial Sciences,
University of Surrey, GU2 7XH, Guildford, UK

(August 28, 2024)

###### Abstract

We model a quantum system coupled to an environment of damped harmonic
oscillators by following the approach of Caldeira-Leggett and adopting the
Caldirola-Kanai Lagrangian for the bath oscillators. In deriving the master
equation of the quantum system of interest (a particle in a general
potential), we show that the potential is modified non-trivially by a new
inverted harmonic oscillator term, induced by the damping of the bath
oscillators. We analyze numerically the case of a particle in a double-well
potential, and find that this modification changes both the rate of
decoherence at short times and the well-transfer probability at longer times.
We also identify a simple rescaling condition that keeps the potential fixed
despite changes in the environmental damping. Here, the increase of
environmental damping leads to a slowing of decoherence.

Open quantum systems, Dissipation, Caldirola-Kanai Lagrangian

##  I Introduction

Dissipative dynamics cannot be derived from time-independent Hamiltonians or
Lagrangians. While classical equations of motion can be defined so as to
include phenomenological dissipation, in quantum mechanics the lack of
Hamiltonian or Lagrangian functions based on first principles makes quantum
dissipative systems hard to describe.

Historically, this problem has been regarded as a major conceptual obstacle to
the quantization of dissipative systems, and has sparked a considerable effort
to identify a dynamical derivation of dissipative features. The celebrated
work of Caldeira and Leggett Caldeira83  tackles this issue by defining a
microscopic model of dissipation based on the interaction of a system of
interest with its own environment, defined as a thermal bath of harmonic
oscillators. While the elimination of the environmental degrees of freedom by
projection leads to a dissipative system described macroscopically by the
Langevin equation, the original system, before the reduction, is fully
Hamiltonian, and as such can be quantized with standard procedures.

The approach of Caldeira83  has been extended to consider both Markovian and
non-Markovian dynamics Lally22 , different types of systems of interest,
different structured environments Gribben20 , with repercussions in the broad
fields of quantum technologies, quantum optics and quantum computing, and more
generally in all those applications where controlling the behaviour of a
quantum system coupled to its environment is necessary Schlosshauer04 .

In this paper we further extend the approach of Caldeira83  to the case where
the quantum environment is itself experiencing dissipative processes.

Our work follows in spirit the approach presented in Plyukhin19  in the
classical case, where classical bath oscillators undergo dissipation,
described by equations of motion with friction terms. However, in contrast to
Plyukhin19 , in the quantum case we need to define an appropriate
phenomenological Lagrangian formalism in order to be able to adopt the path
integral approach defined in Caldeira83 . The solution to this problem is
provided by considering the proposal by Caldirola Caldirola41  and Kanai
Kanai48 , who introduced a method to describe dissipation using time-dependent
Hamiltonians and Lagrangians, obtained by assuming the mass of the system to
increase exponentially in time Crespo90 ; Choi13 . While this approach has
been received with some degree of controversy Dekker81 ; Hensel92 , it has
nonetheless been used widely to describe dissipative systems Baskoutas94 ;
Dodonov07 ; Tokieda17 ; Tokieda21 .

Here, we use the propagator associated with a damped oscillator described by
the Caldirola-Kanai Lagrangian Kerner58 ; Um87  to implement damping on the
bath oscillators in the Caldeira-Leggett model. The damped environment so
defined can still be integrated out analytically and the corresponding
influence functional derived. This leads to a master equation for the system
of interest in which the potential experienced by the system is modified by a
new damping-dependent term which takes the form of an inverted harmonic
potential.

We study the time evolution and dynamics of a particle of mass $m$ in a
symmetric double well potential by focusing on two relevant processes: (1) the
decoherence process at short times for the particle initially in a
superposition of identical Gaussian wave functions, one in each well, and (2)
the dynamics of the particle at long times, starting as a single Gaussian
distribution on one side of the well. We then explore the time-dependence of
the subtle interplay between two parameters: the strength of the damping felt
by the environment oscillators and the relaxation rate of the system of
interest (the particle). For the case of the particle starting on one side of
the barrier, we explore transitions between the regimes of parameter space in
which well-transfer (whether via tunnelling through the barrier or classical
over-the-barrier hopping) is reduced (a quantum Zeno effect misra ) or
enhanced due to thermal excitation by the environment (an anti-Zeno effect)
Kofman00 ; Facchi01 . Furthermore, by requiring that the modified potential
felt by the system is invariant upon rescaling of the environmental damping,
we find that dissipation decreases and decoherence slows down as the
environmental damping is increased.

In Sec.II we introduce the Caldirola-Kanai Lagrangian into the Caldeira-
Leggett model to describe an environment of damped harmonic oscillators, and
derive the reduced master equation for the system of interest. In Sec.III we
discuss the dynamics that emerge from our model, focusing in particular on
decoherence and well-transfer processes, and in Sec.IV we present some
concluding remarks.

##  II The Caldeira-Leggett model with damped bath oscillators

In order to introduce dissipation phenomenologically in the bath oscillators,
we follow the approach of Caldirola and Kanai Caldirola41 ; Kanai48  for a
classical damped harmonic oscillator. The Lagrangian of a standard harmonic
oscillator without damping is given by

$\mathcal{L}(R,\dot{R})=\frac{1}{2}m\dot{R}^{2}-\frac{1}{2}m\omega^{2}R^{2},$
(1)

where $R$ and $\dot{R}$ are the coordinate and velocity of the oscillator,
respectively, $m$ is its mass, and $\omega$ its frequency. As in Caldirola41 ;
Kanai48 , we extend this Lagrangian to become time-dependent by assuming that
the mass of the oscillator grows exponentially in time with rate $\mu$:

$m(t)=m_{0}\textrm{e}^{\mu t}.$ (2)

This results in the Caldirola-Kanai Lagrangian Caldirola41 ; Kanai48 ,

$\mathcal{L}(R,\dot{R},t)=\textrm{e}^{\mu
t}\left(\frac{1}{2}m_{0}\dot{R}^{2}-\frac{1}{2}m_{0}\omega^{2}R^{2}\right),$
(3)

which leads to the desired equation of motion of the form

$\ddot{R}(t)+\mu\dot{R}(t)+\omega^{2}R(t)=0.$ (4)

The friction term $\mu\dot{R}$ in Eq. (4) emerges as a result of the
assumption in Eq. (2) and describes a damped harmonic oscillator.

This Lagrangian can now be used to describe the bath oscillators in the
quantum Brownian motion model of Caldeira and Leggett Caldeira83 . We begin by
defining the Lagrangian for the system of interest coupled to a heat bath of
oscillators as:

$\mathcal{L}=\mathcal{L}_{A}+\mathcal{L}_{B}+\mathcal{L}_{I}.$ (5)

Here, $\mathcal{L}_{A}$ is the Lagrangian of the system of interest,
characterised by spatial coordinate $x$ and a generic potential $V(x)$:

$\mathcal{L}_{A}=\frac{1}{2}M\dot{x}^{2}(t)-V(x).$ (6)

The term $\mathcal{L}_{B}$ in Eq. (5) represents the bath oscillators, assumed
to experience a damping process described by the Caldirola-Kanai Lagrangian

$\mathcal{L}_{B}=\sum_{k}\textrm{e}^{\mu_{k}t}\left(\frac{1}{2}m_{k}\dot{R}_{k}^{2}-\frac{1}{2}m_{k}\omega_{0_{k}}^{2}R_{k}^{2}\right).$
(7)

For the sake of simplicity we will assume from now on that $\mu_{k}\equiv\mu$
for all $k$’s. This assumption, despite being restrictive, is consistent with
the view of the bath oscillators undergoing a damping process induced on each
of them by a common secondary environment, as analyzed for instance in
Plyukhin19 .

Finally, the term $\mathcal{L}_{I}$ in Eq. (5) is the interaction Lagrangian
between system and bath, with coupling constant $C_{k}$,

$\mathcal{L}_{I}=-x\sum_{k}C_{k}R_{k}.$ (8)

By following the standard procedure Caldeira83 ; Feynman63 ; Feynman65 , we
construct the evolution equation for the reduced density matrix of the system,
$\tilde{\rho}$, namely:

$\tilde{\rho}(x,y,t)=\int
dx^{\prime}dy^{\prime}J(x,y,t;x^{\prime},y^{\prime},0)\rho_{A}(x^{\prime},y^{\prime},0).$
(9)

Here the superpropagator $J(x,y,t;x^{\prime},y^{\prime},0)$ reads

$\displaystyle J(x,y,t;x^{\prime},y^{\prime},0)=$
$\displaystyle\iint\mathcal{D}x\mathcal{D}y\mathcal{F}[x,y]\textrm{e}^{\frac{i}{\hbar}(S_{A}[x]-S_{A}[y])},$
(10)

with the influence functional $\mathcal{F}[x,y]$ given by

$\displaystyle\mathcal{F}[x,y]=\int
dR^{\prime}dQ^{\prime}dR\rho_{B}(R^{\prime},Q^{\prime},0)$
$\displaystyle\qquad\qquad\qquad\times K(x,R,R^{\prime})K^{*}(y,Q,Q^{\prime})\
.$ (11)

In Eq. (9) and Eq. (11), the total density matrix is assumed to be factorised
at time zero as $\rho_{T}(0)=\rho_{A}(0)\rho_{B}(0)$, and $K(x,R,R^{\prime})$
represents the propagator of the bath oscillators, damped as described by the
Lagrangian, Eq. (7), and driven by the system of interest through the
interaction, Eq. (8). The initial coordinates at time 0 are denoted by a
prime, so that $x(0)=x^{\prime}$, whereas the coordinates at time $t$ are not,
so that $x(t)=x$.

In order to compute the propagator $K(x,R,R^{\prime})$, we make use of the
results of Um87 , who obtained the propagator of the damped and forced quantum
harmonic oscillator

$m_{0}\ddot{R}(t)+m_{0}\mu\dot{R}(t)+m_{0}\omega^{2}R(t)=F(t),$ (12)

by using Feynman’s path integral formalism Feynman65 . To adapt the results of
Um87  to our case, it suffices to replace the external forcing term $F(t)$ in
Eq. (12) with $F(t)=x(t)\textrm{e}^{-\mu t}\sum_{k}C_{k}$. Hence, the
propagator for the bath oscillators results in

$K(x,R,R^{\prime})=A(t)\exp\\{\frac{i}{\hbar}\Big{(}a(t)R^{2}+b(t)R^{\prime
2}\\\ +c(t)RR^{\prime}+d(t)R+e(t)R^{\prime}+g(t)\Big{)}\Big{\\}},$ (13)

where the pre-factor, $A(t)$, is given by the expression

$A(t)=\left(\frac{m\omega\textrm{e}^{\mu t/2}}{2\pi i\hbar\sin\omega
t}\right)^{1/2},$ (14)

and the time-dependent coefficients in Eq. (13) are

$\displaystyle a(t)=\frac{m}{4}(-\mu+2\omega\cot\omega t)\textrm{e}^{\mu t},$
(15) $\displaystyle b(t)=\frac{m}{4}(\mu+2\omega\cot\omega t),$ (16)
$\displaystyle c(t)=-\frac{m\omega}{\sin\omega t}\textrm{e}^{\mu t/2},$ (17)
$\displaystyle d(t)=\frac{e^{\mu t/2}}{\sin(\omega
t)}\int^{t}_{0}x(t^{\prime})e^{-\mu t^{\prime}/2}\sin(\omega
t^{\prime})\textrm{d}t^{\prime},$ (18) $\displaystyle
e(t)=\frac{1}{\sin(\omega t)}\int^{t}_{0}x(t^{\prime})e^{-\mu
t^{\prime}/2}\sin(w(t-t^{\prime}))\textrm{d}t^{\prime},$ (19) $\displaystyle
g(t)=-\frac{1}{m\omega\sin(\omega
t)}\int^{t}_{0}\int^{t^{\prime}}_{0}x(t^{\prime})x(s)$ $\displaystyle\qquad
e^{-\mu(t^{\prime}+s)/2}\sin(w(t-t^{\prime}))\sin(ws)\textrm{d}s\textrm{d}t^{\prime}.$
(20)

The influence functional is therefore:

$\mathcal{F}[x,y]=\exp\\{-\frac{1}{\hbar}\bigg{[}\int^{t}_{0}f_{-}(t^{\prime})\textrm{e}^{-\frac{\mu}{2}t^{\prime}}I_{I}(t^{\prime})dt^{\prime}\\\
+\int^{t}_{0}f_{-}(t^{\prime})\textrm{e}^{-\frac{\mu}{2}t^{\prime}}I_{R}(t^{\prime})dt^{\prime}\bigg{]}\bigg{\\}},$
(21)

where

$\displaystyle
I_{I}(t^{\prime})=\int_{0}^{t^{\prime}}\alpha_{I}(t^{\prime},t^{\prime\prime})\textrm{e}^{-\frac{\mu}{2}t^{\prime\prime}}f_{+}(t^{\prime\prime})dt^{\prime\prime},$
(22) $\displaystyle
I_{R}(t^{\prime})=\int_{0}^{t^{\prime}}\alpha_{R}(t^{\prime},t^{\prime\prime})\textrm{e}^{-\frac{\mu}{2}t^{\prime\prime}}f_{-}(t^{\prime\prime})dt^{\prime\prime},$
(23)

are the convolution integrals containing the imaginary and real memory kernels
$\alpha_{I}(t^{\prime},t^{\prime\prime})$ and
$\alpha_{R}(t^{\prime},t^{\prime\prime})$

$\displaystyle\alpha_{I}(t^{\prime},t^{\prime\prime})=-i\sum_{k}\frac{C_{k}^{2}}{2m\omega_{k}}\sin(\omega_{k}(t^{\prime}-t^{\prime\prime}))$
(24)

and

$\displaystyle\alpha_{R}(t^{\prime},t^{\prime\prime})$ $\displaystyle=$
$\displaystyle\sum_{k}\frac{C_{k}^{2}}{2m\omega_{k}}\coth(\frac{\hbar\omega_{k}}{2}\beta)$
(25) $\displaystyle\times$
$\displaystyle\bigg{(}\cos(\omega_{k}(t^{\prime}-t^{\prime\prime}))+\frac{\mu_{0}}{\omega_{k}}\sin(\omega_{k}(t^{\prime}+t^{\prime\prime}))$
$\displaystyle+$
$\displaystyle\frac{\mu_{0}^{2}}{\omega_{k}^{2}}\sin(\omega_{k}t^{\prime})\sin(\omega_{k}t^{\prime\prime})\bigg{)}$

where

$\omega_{k}^{2}\equiv\omega_{0_{k}}^{2}-\frac{\mu^{2}}{4}>0.$ (26)

In order to extend to a continuum of oscillators, we define the Ohmic spectral
density $J(\omega)$

$\omega J(\omega)=\left\\{\begin{array}[]{cc}2m\textrm{e}^{\mu
t}\eta\omega^{2}/\pi,&\omega<\Omega\\\ 0,&\omega>\Omega\end{array}\right.$
(27)

where we have ensured that we are consistent with the exponentially increasing
mass assumption of the Caldirola-Kanai model. Using this and the high
temperature limit $kT\gg\hbar\omega_{k}$ we obtain

$\hbar\alpha_{R}(t^{\prime},t^{\prime\prime})\approx 2\eta kT\textrm{e}^{\mu
t^{\prime}}\bigg{(}\delta(t^{\prime}-t^{\prime\prime})+\frac{\mu}{2}+\frac{\mu^{2}}{2}t^{\prime\prime}\bigg{)}$
(28)

and

$\alpha_{I}(t^{\prime},t^{\prime\prime})=i\eta\textrm{e}^{\mu
t^{\prime}}\frac{\textrm{d}}{\textrm{d}(t^{\prime}-t^{\prime\prime})}\delta(t^{\prime}-t^{\prime\prime}).$
(29)

With these, the influence functional defined in Eq. (21) can be split into the
following complex and real components,
$\mathcal{F}[x,y]=\mathcal{F}_{C}[x,y]\;\mathcal{F}_{R}[x,y]$:

$\displaystyle\mathcal{F}_{C}[x,y]=$ (30)
$\displaystyle\exp\\{\frac{iM\gamma}{\hbar}\bigg{(}\\!-\\!\int^{t}_{0}\\!\left(x(t^{\prime})-y(t^{\prime})\right)\left(\dot{x}(t^{\prime})+\dot{y}(t^{\prime})\right)dt^{\prime}$
$\displaystyle\qquad\qquad\quad\quad\;\;+\mu\int^{t}_{0}\left(x^{2}(t^{\prime})-y^{2}(t^{\prime})\right)dt^{\prime}\bigg{)}\bigg{\\}}$

and

$\displaystyle\mathcal{F}_{R}[x,y]=$ $\displaystyle\exp\\{-\frac{2M\gamma
kT}{\hbar^{2}}\bigg{(}\int^{t}_{0}\left(x(t^{\prime})-y(t^{\prime})\right)^{2}dt^{\prime}$
$\displaystyle+\int^{t}_{0}\\!\\!\int_{0}^{t^{\prime}}\\!\\!\\!\\!\\!\left(x(t^{\prime})-y(t^{\prime})\right)\textrm{e}^{\mu
t^{\prime}}\\!\left(\frac{\mu}{2}+\frac{\mu^{2}}{2}t^{\prime\prime}\\!\right)\textrm{e}^{-\frac{\mu}{2}(t^{\prime}\\!-t^{\prime\prime})}$
$\displaystyle\qquad\qquad\times(x(t^{\prime\prime})-y(t^{\prime\prime}))\textrm{e}^{-\mu
t^{\prime\prime}}\bigg{)}dt^{\prime\prime}dt^{\prime}.\bigg{)}\bigg{\\}},$
(31)

where we have introduced the relaxation rate $\gamma=\eta/2M$, and the zero
point energy divergent shift due to the interaction with the environment is
reabsorbed by defining the renormalised potential of the system of interest,
$V_{R}(x)$, as in Caldeira83 .

Eqs.(30) and (31) generalize the standard quantum Brownian motion case
Caldeira83  for $\mu\neq 0$. However, while the second term of Eq. (30)
describes a further harmonic modification of the bare potential of the system
of interest, which as we shall see is responsible for interesting dynamical
behaviours, the $\mu$ and $\mu^{2}$-proportional terms in Eq. (31) do not
contribute to the master equation of the system. This can be seen by using the
standard procedure as in Caldeira83 , and propagating the density operator
from time $t$ to time $t+\varepsilon$ according to Eq. (9). We extend
Feynman’s infinitesimal method Feynman65  to evaluate the time integrals in
Eqs.(30) and (31) by using:

$\displaystyle S$
$\displaystyle=\int^{t+\epsilon}_{t}L(\dot{x}(t^{\prime}),x(t^{\prime}),t^{\prime})dt^{\prime}$
$\displaystyle\approx\varepsilon
L\bigg{(}\frac{x-x^{\prime}}{\varepsilon},\frac{x+x^{\prime}}{2},\frac{(t+\varepsilon)+t}{2}\bigg{)},$
(32)

where $x(t)=x^{\prime}$, $x(t+\varepsilon)=x$, $y(t)=y^{\prime}$ and
$y(t+\varepsilon)=y$. Applying Eq. (II) twice in Eq. (31), leaves a term of
order $\varepsilon^{2}$ and therefore disregarded in the derived master
equation.

Collecting the terms of order $\varepsilon$ returns the following master
equation:

$\displaystyle\partialderivative{\tilde{\rho}}{t}$ $\displaystyle=$
$\displaystyle\frac{i\hbar}{2M}\bigg{(}\partialderivative[2]{\tilde{\rho}}{x}-\partialderivative[2]{\tilde{\rho}}{y}\bigg{)}-\frac{i}{\hbar}(V_{R}(x)-V_{R}(y))\tilde{\rho}$
(33) $\displaystyle-$
$\displaystyle\gamma(x-y)\bigg{(}\partialderivative{\tilde{\rho}}{x}-\partialderivative{\tilde{\rho}}{y}\bigg{)}-\frac{2M\gamma
kT}{\hbar^{2}}(x-y)^{2}\tilde{\rho}$ $\displaystyle+$
$\displaystyle\frac{iM\gamma}{\hbar}\mu(x^{2}-y^{2})\tilde{\rho}\ .$

The first two terms in this master equation are found in the standard von
Neumann equation, and the third and fourth terms are known as relaxation and
decoherence terms, respectively Caldeira83 ; Zurek02 . However, because of the
damping of the bath oscillators, a new term
[$iM\gamma\mu(x^{2}-y^{2})\tilde{\rho}/\hbar$] now appears, which can be
subsumed into the second term in Eq. (33) as an additional potential to give a
full potential, $V_{F}$,

$V_{R}(x)\rightarrow V_{F}(x)=V_{R}(x)+V_{D}(x),$ (34)

where

$V_{D}(x)=-\frac{1}{2}M\gamma\mu x^{2},$ (35)

is an inverted harmonic potential that modifies further the (renormalized)
potential of the system of interest Tokieda21 ; Baskoutas94 . We can therefore
rewrite the master equation in the familiar form of Caldeira83 ; Zurek02 , but
with the modified potential $V_{F}$:

$\displaystyle\partialderivative{\tilde{\rho}}{t}$ $\displaystyle=$
$\displaystyle\frac{i\hbar}{2M}\bigg{(}\partialderivative[2]{\tilde{\rho}}{x}-\partialderivative[2]{\tilde{\rho}}{y}\bigg{)}-\frac{i}{\hbar}(V_{F}(x)-V_{F}(y))\tilde{\rho}$
(36) $\displaystyle-$
$\displaystyle\gamma(x-y)\bigg{(}\partialderivative{\tilde{\rho}}{x}-\partialderivative{\tilde{\rho}}{y}\bigg{)}-\frac{2M\gamma
kT}{\hbar^{2}}(x-y)^{2}\tilde{\rho}\ .$

In summary, the effect of the damping on the environmental degrees of freedom
manifests itself as a modification of the potential of the system of interest,
without directly affecting the relaxation and decoherence terms in the master
equation. However, we emphasize that this modification carries information
from the environment into the system of interest, through the combined
dependency on the product of the two parameters: the relaxation rate $\gamma$
and the damping coefficient $\mu$ in Eq. (35). As a result, the potential can
be tuned using both the environment’s damping $\mu$ and system’s relaxation
$\gamma$. Changing $\mu$ will only have the consequence of changing the shape
of the potential, whereas changing $\gamma$ will also affect the relaxation
and decoherence terms in the master equation. Moreover, the combined
dependency on $\gamma\mu$ in Eq. (35) suggests that it is also possible to
keep the full potential, Eq. (34), fixed under rescaling of $\mu$ by
adequately rescaling $\gamma$, which will have a direct impact on the
relaxational and decoherence properties of the system. In the next section we
will explore the effect of changing these variables independently of each
other, and in conjunction with each other, so as to keep the full potential
unchanged under different environmental dampings.

##  III Analysis of Dynamics

In our analysis of the damped model, Eq. (36), we consider the effect of
$V_{D}(x)$ on the dynamics of the system of interest. We choose the
renormalised potential $V_{R}(x)$, to have a simple quartic (symmetric double
well) form in which the zero point energy divergent shift due to the
interaction of the system with the environment is incorporated into a
renormalised frequency, $\omega_{R}$,

$V_{R}(x)=\frac{M\omega_{R}^{2}}{2x_{0}^{2}}(x-x_{0})^{2}(x+x_{0})^{2}\ .$
(37)

A plot of all three potentials, $V_{R}(x)$, $V_{D}(x)$ and $V_{F}(x)$, is
shown in Fig. (1) with the following choice of parameters: $\omega_{R}=0.025$
fs-1, $M=938$ MeV/c2, $\gamma=2.5\times 10^{-4}$ fs-1, $\mu=1$ fs-1 and
$x_{0}=2.5$Å. We see that, compared to $V_{R}(x)$, both the well separation
and barrier height of $V_{F}(x)$ increase due to $V_{D}(x)$.

Figure 1: A plot of the three potentials: $V_{R}(x)$ is the renormalised
potential in the undamped master equation, $V_{D}(x)$ is the inverted harmonic
potential due to the damped environment and $V_{F}(x)$, the full potential, is
their sum.

###  III.1 Decoherence dynamics

To study the effect of a damped environment on the decoherence rate, we choose
the initial state of the system of interest to be a double-Gaussian with its
peaks centred in the wells of the potential $V_{R}$, with parameter values
defined as in the previous section. This choice gives us a potential barrier
high enough to maintain the localisation of the two peaks over the decoherence
timescale. The reduced density matrix, $\tilde{\rho}(x,y,t)$, is then evolved
in time according to Eq. (36) and the rate of decoherence measured through the
$l_{1}$-norm of coherence, $C(t)$ Baumgratz14 , by integrating over the
absolute value of the off-diagonal peak where

$C(t)=\int^{L}_{0}\int^{0}_{-L}|\tilde{\rho}(x,y,t)|\leavevmode\nobreak\
dx\leavevmode\nobreak\ dy,$ (38)

and $L$ is chosen to be large enough to include the entirety of one of the
off-diagonal peaks.

We compare the rates of decoherence for different values of the damping
parameter $\mu$, and the relaxation rate $\gamma$, relative to the undamped
case

$C_{R}(t)=\frac{C_{\textrm{damped}}(t)}{C_{\textrm{undamped}}(t)},$ (39)

where $C_{\textrm{undamped}}$ is the coherence for $\mu=0$ and corresponds to
that of the standard Caldeira-Leggett model.

We start by fixing the value of $\gamma$ and varying the value of $\mu$. The
range of $\mu$ values is chosen such that decoherence does not happen more
quickly than the damping of the bath oscillators.

In Fig. 2 we show the behaviour of $C_{R}(t)$, obtained for different values
of the damping coefficient $\mu$. Fig. (2) shows that there are two distinct
regimes. Initially, for $t\lesssim 10$ fs, decoherence in the damped case is
slower than in the undamped case, with the slowing down more pronounced the
larger the value of $\mu$. Subsequently, for $t\gtrsim 10$ fs, decoherence
appears to become faster, and the decoherence process in the damped situation
eventually becomes more efficient than in the undamped case.

We interpret these dynamics by observing that the adopted initial condition
sets the Gaussian peaks in the centres of the wells of the renormalized
potential $V_{R}$, not of $V_{F}$. As a consequence, the Gaussian peaks will
feel the repulsive force of the wider potential barrier, and, over time,
approach their equilibrium positions, within $V_{F}$.

While doing so the system will experience different rates of decoherence. In
particular the peaks are exposed initially to the gradient of $V_{F}$, which
is steeper than that of $V_{R}$ at that value of $x$. We argue that it is this
feature that is responsible for $C_{R}>1$ for $t\lesssim 10$ fs. To
demonstrate this, we examine the rate of decoherence for our double-Gaussian
while sitting on a linear potential with various constant slopes. In Fig. (3),
it is clear that as the potential slope increases, the rate of decoherence
slows down. Here, we plot $C_{g}$, the ratio of the coherence of a double-
Gaussian on a linear potential of slope $g$, against that on the reference
slope $g=1$. The reference slope has a gradient equal to that of the full
potential, $V_{F}$, at $x=2.5$ Å.

Figure 2: Time evolution of relative coherence (damped versus undamped) for a
range of damping parameters, $\mu$, and a fixed dissipation parameter,
$\gamma=2.5\times 10^{-4}$ fs-1. Data points joined for clarity.

However, after $t\gtrsim 10$ fs, the Gaussians, having been pushed apart, will
be far enough apart for the decoherence term in Eq. (33) to dominate. In this
regime, the decoherence in the full potential is faster due to the larger
effect of the $(x-y)^{2}$ factor in the decoherence terms on the off-diagonal
peaks.

Figure 3: Time evolution of $C_{g}$ in the Caldeira-Leggett model (undamped
environment) under the influence of a range of linear potentials of gradient
$g$. Figure 4: Time evolution of relative coherence (damped versus undamped)
for a range of values of $\gamma$, and a fixed damping parameter, $\mu=1$
fs-1. Data points joined for clarity.

Next we fix the value of the damping parameter, $\mu$, and examine the
relative coherence (for damped versus undamped environments) for various
values of $\gamma$. The results are shown in Fig. (4). Here, we again see the
behaviour in the two regimes: at short timescales the decoherence is slower in
the damped environment, but becomes faster once the peaks have separated and
the decoherence term in the master equation begins to dominate. Of interest
here is that this switch over from slower decoherence to faster decoherence
relative to the undamped case happens at progressively earlier times as the
coupling between system and environment becomes stronger, suggesting that for
stronger coupling, the $\gamma$-dependent decoherence term quickly dominates
over the effect of the damping-induced modified barrier that delays
decoherence.

###  III.2 Well-transfer dynamics

We now consider the effects of a damped environment on the dynamics of a
particle initially on the right side of the potential barrier within the full
potential, $V_{F}(x)$, but which can transfer across to the left well either
by quantum tunnelling or classical over-the-barrier hopping, which together we
refer to as well-transfer. We consider an initial state that is a single
Gaussian centred at the minimum of the right hand well. We choose a value of
$\omega_{R}=0.005$ fs-1 to give a much smaller barrier, Eq. (37), than we had
in the previous section in order to allow for a significant transfer rate to
the other well. The well-transfer probability is determined by the probability
of finding the system in the left well,

$P(t)=\int^{0}_{-L}\tilde{\rho}(x,x,t)\leavevmode\nobreak\ dx.$ (40)

It is difficult to predict intuitively in advance what the effect of the
environment will be on the dynamics of the system – the particle transfer
probability in this case. On the one hand, continuous ‘measurement’ by an
environment should continuously collapse the system back to its initial state:
a quantum Zeno effect, as outlined by Misra and Sudarshan misra . However, the
environment can also act to excite (thermally activate) the system, coupling
it to higher energy eigenstates of the potential so that it is more likely to
transfer across Godbeer14 ; Godbeer15 ; Slocombe21 .

We investigate the interplay between these two opposing effects here. Fig. (5)
shows that over a time scale of 1000 fs the well-transfer probability in the
absence of an environment (the von Neumann term only in the master equation in
Eq. (33)) hovers around $P_{L}=0.05$. By adding an undamped environment
(Caldeira-Leggett model) we see a dramatic anti-Zeno effect as the system
interacts with the bath oscillators, which thermally activate the particle
through or over the barrier and the well-transfer probability is increased. If
we then add damping to the bath oscillators, the well-transfer probability is
suppressed increasingly (a Zeno effect) as the value of the damping parameter,
$\mu$, is increased until there is hardly any particle-transfer at all at
$\mu=0.5$ fs-1.

Figure 5: Time evolution of well-transfer for a range of damping parameters,
$\mu$, and a fixed dissipation parameter, $\gamma=2.5\times 10^{-4}$ fs-1.
Data points joined for clarity.

In contrast, changing the relaxation rate, $\gamma$, affects the damped model
in drastically different ways depending on the value of $\mu$. In the weak
damping case, Fig. (6), we choose $\mu=0.01$ fs-1, which is close to the
Cadeira-Leggett limit of no damping at all, and note that increasing $\gamma$
increases the well-transfer probability because its effect in the master
equation Eq. (33) is now more strongly felt in the dissipation (3rd) term and
the decoherence (4th) term.

However, for stronger damping ($\mu=1$ fs-1, Fig. (7)), we note the opposite
effect. As $\gamma$ is increased, the well-transfer probability drops
dramatically. Now, it is the final (damping) term in Eq. (33) that dominates.
This can be understood in terms of how it affects the modifying potential,
$V_{D}$, Eq. (35), which changes the shape and size of the barrier in $V_{F}$
in Eq. (36). The increasing barrier height and well separation with increasing
$\gamma$ diminishes the probability of well-transfer (Zeno effect) more
effectively than the anti-Zeno effect of the stronger coupling to the
environment we saw earlier. For intermediate environment damping strengths
($0.01<\mu<1$), there is a transition between Zeno and anti-Zeno as $\gamma$
is increased.

Figure 6: Time evolution of well-transfer for a range of dissipation
parameters, $\gamma$, and a fixed damping parameter, $\mu=0.01$ fs-1. Data
points joined for clarity. Figure 7: Time evolution of well-transfer for a
range of dissipation parameters, $\gamma$, and a fixed damping parameter,
$\mu=1$ fs-1. Data points joined for clarity.

###  III.3 Decoherence under combined $\gamma\mu$ dependency

In the study of open quantum systems, the reduced dynamics of the system of
interest can be split into the von Neumann contribution (the unitary dynamics
in the limit of an isolated system) and the additional influence of the
environment due to the dissipation and decoherence terms Zurek02 .

Thus far, we have varied each of the parameters, $\mu$ and $\gamma$,
independently. In so doing, we have been changing the full potential ,$V_{F}$,
which depends on the product $\gamma\mu$ in $V_{D}$. This inevitably alters
the unitary dynamics of the system. However, in order to explore the effect of
damping on the non-unitary terms alone, we must ensure that the potential
experienced by the particle remains fixed.

For simplicity, we consider here the free particle case ($V_{R}=0$, but still
with the presence of $V_{D}$ due to environment damping) and introduce a
rescaling of our parameters. We set $\gamma=\gamma_{0}/a$ and $\mu=\mu_{0}a$
so that $\gamma\mu=\gamma_{0}\mu_{0}$ is constant.

Figure 8: Time evolution of coherence for a range of rescaling factors, $a$,
for $\gamma_{0}=2.5\times 10^{-4}$ fs-1 and $\mu_{0}=1$ fs-1. Data points
joined for clarity.

Fig. (8) shows the time evolution of the $l_{1}$-norm of coherence $C(t)$,
defined in Eq. (38), for the case of a free particle surrounded by a bath of
damped oscillators and for which the product $\gamma\mu$ is kept fixed. As the
scaling parameter $a$ is increased, this turns up the strength of the damping
of the bath oscillators, but simultaneously reduces the relaxation rate, which
in turn results in slower decoherence. Thus, for a particle feeling a constant
external potential, stronger damping of the environment results in a slower
relaxation rate and weaker coupling between system and environment.

This also tells us that taking the limit of $\gamma\rightarrow 0$, when there
is no coupling between system and environment allowing the system to evolve
unitarily, is equivalent to taking the limit of $\mu\rightarrow\infty$ in
which the the environment oscillators are damped completely and, with no
energy, can no longer interact with the system.

##  IV Conclusions

We have presented a model of quantum Brownian motion and explored the dynamics
of the reduced density matrix in the presence of a bath of damped harmonic
oscillators. We describe the environmental damping by adopting the
phenomenological Caldirola-Kanai model of dissipation and find that this
results in an additional inverted harmonic potential that effects the unitary
evolution of the system of interest.

We considered the case of a particle in a double well potential and studied
the time dependence of both the decoherence rate and the well-transfer
probability (at appropriate and very different timescales) by varying both the
strength of the damping through the damping coefficient, $\mu$, and the
relaxation rate, $\gamma$. All our findings can be interpreted as a
consequence of the modification of the potential experienced by the system of
interest because of the environmental damping. Initially, the superposition in
the damped model experiences a steeper potential before the increase in
separation of the peaks becomes significant. After an initial transient of
$C_{R}>1$, decoherence appears faster in the damped model because the initial
coherences travel further away from the diagonal towards the more distant
wells of the full potential. Similarly, well-transfer is depressed for
progressively larger damping coefficients, because of increasingly deeper and
more distant wells of the full potential.

It is interesting to note that in the framework introduced here, both the
relaxation rate, $\gamma$, and the damping coefficient, $\mu$, enter
explicitly in the unitary dynamics term of the quantum master equation,
producing a coupling of dependencies between the system of interest and its
environment, which is not present in the Caldeira-Leggett model. This coupling
allows for a control of the decoherence process by tuning the environmental
damping. Imposing the constraint that the system dynamics are not altered upon
increasing the environmental damping leads to a weakening of the coupling
between the system and environment due to a reduced relaxation rate, and hence
slower decoherence.

## Acknowledgements

Lester Buxton and Marc-Thomas Russo were supported by the Leverhulme Doctoral
Training Centre. Andrea Rocco and Jim Al-Khalili acknowledge support from the
John Templeton Foundation, through Grant 62210. The opinions expressed in this
publication are those of the author(s) and do not necessarily reflect the
views of the John Templeton Foundation.

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