# The quantum Zeno and anti-Zeno effects with driving fields in the weak and
strong coupling regimes

Mehwish Majeed  Adam Zaman Chaudhry<EMAIL_ADDRESS>
###### Abstract

Repeated measurements in quantum mechanics can freeze (the quantum Zeno
effect) or enhance (the quantum anti-Zeno effect) the time-evolution of a
quantum system. In this paper, we present a general treatment of the quantum
Zeno and anti-Zeno effects for arbitrary driven open quantum systems, assuming
only that the system-environment coupling is weak. In particular, we obtain a
general expression for the effective decay rate of a two-level system
subjected to arbitrary driving fields as well as periodic measurements. We
demonstrate that the driving fields change the decay rate, and hence the
quantum Zeno and anti-Zeno behavior, both qualitatively and quantitatively. We
also extend our results to systems consisting of more than one two-level
system, as well as a two-level system strongly coupled to an environment of
harmonic oscillators, to further illustrate the non-trivial effect of the
driving fields on the quantum Zeno and anti-Zeno effects.

##  1 Introduction

Unlike classical measurements, quantum measurements in general disturb the
state of the quantum system. This back-action of measurements is a peculiar
concept of quantum mechanics which gives rise to striking phenomena such as
the quantum Zeno effect (QZE). In the QZE, repeated measurements hinder the
time evolution of the quantum system[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. However, it has also been
observed that if the measurements are not rapid enough, a reverse effect,
known as a quantum anti-Zeno effect (QAZE), can occur whereby the measurements
accelerate the quantum evolution [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36]. Both the QZE and the QAZE have gained considerable interest
theoretically and experimentally due to their huge importance in the
foundations of quantum mechanics as well as possible applications in quantum
technologies. For example, the effect of repeated measurements can be used to
infer properties of the environment of a quantum system, that is, noise
sensing [37, 38, 39]. Generally speaking, so far, the main focus of the
studies performed on QZE and the QAZE have been on the population decay model
[24, 25, 26, 27, 40, 41, 42, 28, 29, 43, 44, 45] and the dephasing model [30]
(notable exceptions include Refs. [46, 47]).

What is lacking is a rigorous general study of the QZE and QAZE in the
presence of coherent driving fields. The idea of controlling the coherent
dynamics of a quantum system by an external time-dependent field has found
widespread theoretical and experimental interest in many areas of physics and
chemistry [48]. For example, driving fields are a commonly used tool to
manipulate qubits as well as to control chemical reactions by external laser
fields [49, 50, 51, 52]. In quantum optics, it has been shown that a
frequency-modulated excitation of a two-level atom significantly modifies the
time-evolution of the system [53] . In quantum tunneling systems, it has been
demonstrated that an appropriately designed coherent drive can bring the
tunneling to an (almost) complete standstill - this is known as coherent
destruction of tunneling [54, 55]. Driving fields can even effectively remove
the interaction of the quantum system with its environment, which is precisely
the idea behind dynamical decoupling [56, 57, 58, 59, 60, 61]. Driving fields
have also been recently used in noise sensing [62].

It is then clear that driving fields, repeated measurements as well as the
environment can all drastically influence the temporal evolution of a quantum
system. Consequently, in this work, we study the QZE and the QAZE when driving
fields are applied to the quantum system as well. We start by deriving a
general expression of the effective decay rate for the driven quantum system,
provided that the system-environment coupling is weak, thereby extending the
formalism of Ref. [46]. We then obtain general expressions for the decay rate
of a two-level system subjected to arbitrary driving fields. In particular, we
consider in detail both the population decay model and the pure dephasing
model in the presence of different driving fields. For example, we show that
the effective decay rate for the driven population decay model can no longer
be obtained using the usual sinc-squared function (as can be done in the
absence of any driving fields [24]). Moreover, counter-rotating terms in the
system-environment interaction Hamiltonian can become important in the
presence of the driving fields, in contrast with the undriven case. We then
extend our results to more than one two-level system by modeling the multiple
two-level systems as a single large spin [63]. We also demonstrate that our
results can be extended to the strong system-environment coupling regime via
the well-known polaron transformation [64, 65, 66, 67, 68, 69, 70] along with
perturbation theory. All in all, our results generally indicate that the
effective decay rate is very significantly influenced by the driving fields.

##  2 Results

### Effective rate of an arbitrary driven quantum system in the weak coupling
regime

We start by writing the total Hamiltonian of a quantum system, in the presence
of driving fields, interacting with its environment as

$\hat{H}(t)=\hat{H}_{S}(t)+\hat{H}_{B}+\hat{H}_{SB}.$ (1)

Here, the first term $\hat{H}_{S}(t)$ describes the central quantum system
Hamiltonian. This carries explicit time-dependence due to the application of
external driving fields on the system; consequently, we write it as the sum of
a time-independent part $\hat{H}_{S}$ and a time-dependent part
$\hat{H}_{c}(t)$ describing the effect of the external fields. The second term
$\hat{H}_{B}$ corresponds to environment, whereas the last term $\hat{H}_{SB}$
is the coupling between them, which, for later convenience, we write in the
diagonal form $\hat{H}_{SB}=\sum_{\mu}\hat{F}_{\mu}\otimes\hat{B}_{\mu}$, with
the $\hat{F}_{\mu}$ operators belonging to the system Hilbert space and the
$\hat{B}_{\mu}$ operators living in the environment Hilbert space. From now
on, we will be suppressing the ‘hats’ on the operators - the context should
make it clear whether or not we are dealing with an operator. Keeping in mind
our objective of investigating the quantum Zeno and anti-Zeno effects in such
driven open quantum systems, our primary quantity of interest is the effective
decay rate of the system when repeated projective measurements are performed
on the system with time interval $\tau$. To calculate the effective decay
rate, we assume that the system is initially prepared in the pure state
$\left|\psi\right\rangle$. We then find the system density matrix at time
$\tau$, that is, $\rho_{S}(\tau)$, and then use this to find the survival
probability $s(\tau)$ that the system is still in state
$\left|\psi\right\rangle$ at time $\tau$. Thereafter, we can find the
effective decay rate $\Gamma(\tau)$ via
$\Gamma(\tau)=-\text{ln}\,s(\tau)/\tau$. The system density matrix at time
$\tau$ is obtained via
$\rho_{S}(\tau)=\text{Tr}_{B}[U(\tau)\rho(0)U^{\dagger}(\tau)]$, where
$\rho(0)$ is the state of total system plus environment, $\text{Tr}_{B}$ is
the partial trace with respect to environment states and $U(\tau)$ is the
total unitary time-evolution operator corresponding to the total Hamiltonian
$H(t)$. Generally speaking, for the time-dependent system-environment models
considered here, it is usually impossible to calculate the time-evolution
operator exactly. However, for weakly coupled system-environment models, we
can find the time-evolution operator $U(\tau)$ using time-dependent
perturbation theory [71]. We assume that the system-environment state is
initially of a simple product form, that is,
$\rho(0)=\left|\psi\right\rangle\left\langle\psi\right|\otimes\rho_{B}$, where
$\rho_{B}=e^{-\beta H_{B}}/Z_{B}$ is the thermal equilibrium state of the
environment with $Z_{B}=\text{Tr}_{B}[e^{-\beta H_{B}}]$. Extending the
treatment of Ref. [46] to time-dependent Hamiltonians in a straightforward
manner, we find that the decay rate of the quantum state
$\left|\psi\right\rangle$, in the presence of projective measurements and for
weak system-environment coupling, is given by ‘overlap integral’ of two
functions - the generalized filter function $Q(\omega,\tau)$ and the spectral
density of the environment $J(\omega)$ (see the Methods section), that is,

$\Gamma(\tau)=\int_{0}^{\infty}\,d\omega\,Q(\omega,\tau)J(\omega).$ (2)

Here, the generalized filter function is given by

$Q(\omega,\tau)=\dfrac{2}{\tau}\text{Re}\left(\sum_{\mu\nu}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}f_{\mu\nu}(\omega,t^{\prime})\text{Tr}_{S}\bigr{[}P_{\perp}\widetilde{F}_{\nu}(t-t^{\prime})\rho_{S}(0)\widetilde{F}_{\mu}(t)\bigr{]}\right),$
(3)

where $F_{\mu}(t)=U_{S}^{\dagger}(t)H_{S}(t)U_{S}(t)$, with $U_{S}(t)$ being
the unitary time-evolution operators corresponding to $H_{S}(t)$ only, and
$P_{\perp}$ is the projector onto the system subspace orthogonal to
$\left|\psi\right\rangle\left\langle\psi\right|$. The environment correlation
function is
$C_{\mu\nu}(t)=\text{Tr}_{B}[\rho_{B}e^{iH_{B}t}B_{\mu}e^{-iH_{B}t}B_{\nu}]$,
which can generally be simplified to the form
$C_{\mu\nu}(t)=\sum_{k}|g_{k}|^{2}f_{\mu\nu}(\omega_{k},t)$, where $g_{k}$ is
the coupling between the system and the $k^{\text{th}}$ mode of the
environment. The function $f_{\mu\nu}(\omega_{k},t)$ then contains the
remaining information about $C_{\mu\nu}(t)$. The sum over the modes is
typically converted to an integral over the environment frequencies via the
substitution
$\sum_{k}|g_{k}|^{2}(\ldots)\rightarrow\int_{0}^{\infty}\,d\omega\,J(\omega)(\ldots.)$,
thereby introducing the spectral density function $J(\omega)$ of the
environment. It should be noted that $Q(\omega,\tau)$ depends not only on the
frequency of the measurements, the way that system is coupled to its
environment, the state of system that is repeatedly prepared, and part of the
environment correlation function $f_{\mu\nu}(\omega,t)$; most importantly for
us, it also depends on the driving fields applied. Similar analytical
expressions to account for the effect of driving fields have been considered
before [49, 50, 51, 52]. However, our expression takes into account the effect
of both measurements as well as the concurrent application of driving fields
for arbitrary system-environment models, and we do not make any assumptions
regarding the driving fields such as the adiabatic approximation [72, 73]. We
also note that there are different ways to define the survival probability and
hence the decay rate, as well as different ways of identifying the Zeno and
anti-Zeno regimes. For example, one can also look at the history of
measurements [74, 75] when calculating the survival probability [34].
Similarly, we identify the Zeno and anti-Zeno regimes by looking at when the
decay rate $\Gamma(\tau)$ is an increasing function (the Zeno regime) or a
decreasing function (the anti-Zeno regime) [46]; an alternative approach is to
compare the measurement modified decay rate with the decay rate without
measurement [76].

### General expression of the decay rate for a driven two-level system

To apply our formalism to a two-level system, we first note that, without loss
of generality, we can assume the initial state to be $\left|e\right\rangle$,
where $\sigma_{z}\left|e\right\rangle=\left|e\right\rangle$, since we can
always choose our coordinate system in this manner. We then check, with time
interval $\tau$, whether or not the system is still in this state or not. The
projector onto the orthogonal subspace is $\left|g\right\rangle\left\langle
g\right|$, where $\sigma_{z}\left|g\right\rangle=-\left|g\right\rangle$. Now
comes a key insight. No matter what the external driving fields are, the
system unitary time-evolution operator corresponding to $U_{S}(t)$ can always
be written in the form

$U_{S}(t)=e^{-i\alpha(t)\sigma_{z}/2}e^{-i\beta(t)\sigma_{y}/2}e^{-i\gamma(t)\sigma_{z}/2},$
(4)

where $\alpha(t)$, $\beta(t)$, and $\gamma(t)$ are time-dependent Euler
angles. These are arbitrary functions of time with the constraint that
$U_{S}(t=0)=\mathds{1}$. The corresponding Hamiltonian $H_{S}(t)$ can be
worked out from Schrodinger’s equation. We find that

$\displaystyle H_{S}(t)$
$\displaystyle=\dfrac{\sigma_{z}}{2}\dfrac{\partial\alpha(t)}{\partial
t}+\dfrac{1}{2}\dfrac{\partial\beta(t)}{\partial
t}\bigr{(}\cos[\alpha(t)]\sigma_{y}-\sin[\alpha(t)]\sigma_{x}\bigr{)}+\dfrac{1}{2}\dfrac{\partial\gamma(t)}{\partial
t}\cos[\beta(t)]\sigma_{z}\,$
$\displaystyle+\dfrac{1}{2}\dfrac{\partial\gamma(t)}{\partial
t}\bigr{(}\sin[\beta(t)]\cos[\alpha(t)]\sigma_{x}+\sin[\beta(t)]\sin[\alpha(t)]\sigma_{y}\bigr{)}.$
(5)

In other words, by choosing the functions $\alpha(t)$, $\beta(t)$, and
$\gamma(t)$ appropriately, we can work backwards to find the corresponding
system Hamiltonian.

With this form of $U_{S}(t)$, we can work out the effective decay rate. Using
our previous general expression given in Eq. (3), we find that the filter
function $Q(\omega,\tau)$ is

$Q(\omega,\tau)=\frac{2}{\tau}\text{Re}\left(\sum_{\mu\nu}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}f_{\mu\nu}(\omega,t^{\prime})G_{\mu}(t)\bar{G}_{\nu}(t-t^{\prime})\right),$
(6)

where

$\displaystyle
G_{\mu}(t)=e^{i(\alpha(t)+\gamma(t))}\cos^{2}[\dfrac{\beta(t)}{2}]F_{\mu
eg}-e^{-i(\alpha(t)-\gamma(t))}\sin^{2}[\dfrac{\beta(t)}{2}]F_{\mu
ge}+e^{i\gamma(t)}\dfrac{\sin[\beta(t)]}{2}(F_{\mu gg}-F_{\mu ee}),$ (7)

and

$\displaystyle\bar{G}_{\nu}(t)=e^{-i(\alpha(t)+\gamma(t))}\cos^{2}[\dfrac{\beta(t)}{2}]F_{\nu
ge}-e^{i(\alpha(t)-\gamma(t))}\sin^{2}[\dfrac{\beta(t)}{2}]F_{\nu
eg}+e^{-i\gamma(t)}\dfrac{\sin[\beta(t)]}{2}(F_{\nu gg}-F_{\nu ee}),$ (8)

with $F_{\mu ij}=\left\langle i\right|F_{\mu}\left|j\right\rangle$. In the
following sections, we use this form of the filter function with various
$\alpha(t)$, $\beta(t)$, and $\gamma(t)$ to evaluate the effective decay rate
for different system-environment models.

### Application to the driven population decay model in weak coupling regime

To illustrate our formalism, we first consider a single two-level system
interacting with an environment of harmonic oscillators in the presence of
external driving fields. The total system-environment Hamiltonian is written
as (we set $\hbar=1$ throughout)

$H(t)=\frac{\varepsilon_{0}}{2}\sigma_{z}+H_{c}(t)+\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}+\sum_{k}(g_{k}^{*}b_{k}\sigma_{+}+g_{k}b_{k}^{\dagger}\sigma_{-}),$
(9)

where $H_{S}(t)=\frac{\varepsilon_{0}}{2}\sigma_{z}+H_{c}(t)$ is the system
Hamiltonian with $\varepsilon_{0}$ representing the energy spacing of the two-
level system, while $H_{c}(t)$ is a time-dependent external driving field
acting on the system. $H_{B}=\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}$ is the
environment Hamiltonian with $b_{k}$ and $b_{k}^{\dagger}$ representing the
usual annihilation and creation operators, and
$H_{SB}=\sum_{k}(g_{k}^{*}b_{k}\sigma_{+}+g_{k}b_{k}^{\dagger}\sigma_{-})$ is
the system-environment interaction Hamiltonian with $g_{k}$ denoting the
coupling strength between the central two-level system and the environment
oscillators. As usual, $\sigma_{z}$ is the standard Pauli spin-1/2 matrix, and
$\sigma_{+}(\sigma_{-})$ are the raising (lowering) operators. Note that we
have made the rotating-wave approximation (RWA) here for the system-
environment interaction, which means that we have neglected those processes
which do not conserve energy [17, 77].

Figure 1: Filter function for the driven population decay model with the RWA.
(a) Graph of $Q(\omega,\tau)$ versus $\omega$ in the presence of time-
dependent driving fields:
$\alpha(t)=\varepsilon_{0}t+(V_{0}/\Omega)\sin(\Omega t)$, $\beta(t)=0$ and
$\gamma(t)=0$ with $V_{0}=1$ and $\Omega=5$ (dashed, red curve), $V_{0}=5$ and
$\Omega=1$ (large-dashed, magenta curve), and $V_{0}=5$ and $\Omega=5$ (dot-
dashed, green curve). $\alpha(\tau)=\varepsilon_{0}\tau$, $\beta(\tau)=0$ and
$\gamma(\tau)=0$ for the solid, black curve. Throughout, we use dimensionless
units with $\hbar=1$ and we have set $\varepsilon_{0}=1$. Here we have used a
relatively small measurement interval, that is, $\tau=0.1$. (b) Same as (a),
except that now $\tau=1$.

With the model specified, we now move to find the effective decay rate of the
two-level system using the formalism described before. As is usually the case
in studies of the quantum Zeno and anti-Zeno effects, we initially prepare our
system in the excited state $\left|e\right\rangle$ such that
$\sigma_{z}\left|e\right\rangle=\left|e\right\rangle$, and we then repeatedly
check, with time interval $\tau$, whether or not the system is still in the
excited state. To calculate the decay rate using our formalism, we note that
$F_{1}=\sigma_{+}$, $F_{2}=\sigma_{-}$,
$C_{\mu\nu}(t)=\text{Tr}_{B}[\rho_{B}\widetilde{B}_{\mu}(t)B_{\nu}]$,
$\widetilde{B}_{\mu}(t)=e^{iH_{B}t}B_{\mu}e^{-iH_{B}t}$,
$B_{1}=\sum_{k}g_{k}^{*}b_{k}$, and $B_{2}=\sum_{k}g_{k}b_{k}^{\dagger}$. In
the limit of zero temperature, we find that $f_{12}(\omega,t)=e^{-i\omega t}$,
while $f_{11}=f_{22}=f_{21}=0$. Moreover, we find
$G_{1}(t)=e^{i(\alpha(t)+\gamma(t))}\cos^{2}[\beta(t)/2]$, and
$\bar{G}_{2}(t-t^{\prime})=e^{-i(\alpha(t-t^{\prime})+\gamma(t-t^{\prime}))}\cos^{2}[\beta(t-t^{\prime})/2]$.
Using these results, we obtain

$Q(\omega,\tau)=\frac{2}{\tau}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}\cos[\alpha(t)-\alpha(t-t^{\prime})+\gamma(t)-\gamma(t-t^{\prime})-\omega
t^{\prime}]\cos^{2}[\beta(t)/2]\cos^{2}[\beta(t-t^{\prime})/2].$ (10)

In general, this can be a very complicated function. Therefore, we first
consider the simplest case where $\alpha(t)=\varepsilon_{0}t$, while
$\beta(t)=\gamma(t)=0$. This corresponds to [see Eq. (2)]
$H=\frac{\varepsilon_{0}}{2}\sigma_{z}$, that is, the usual population decay
model with no driving field. After performing the integrals, we get
$Q(\omega,\tau)=\tau\text{sinc}^{2}[\dfrac{(\varepsilon_{0}-\omega)\tau}{2}]$,
thereby reproducing the well-known sinc-squared function for $Q(\omega,\tau)$.
Our formalism, on the other hand, allows us to go much further. The next case
that we can consider is $\alpha(t)=\varepsilon_{0}t+(V_{0}/\Omega)\sin(\Omega
t)$ with $\beta(t)=\gamma(t)=0$, which corresponds to
$H_{S}=\frac{\varepsilon_{0}}{2}\sigma_{z}$ and $H_{c}(t)=V_{0}\cos(\Omega
t)\sigma_{z}/2$, with $V_{0}$ the amplitude of the applied sinusoidal field
and $\Omega$ its frequency. Using the Jacobi-Auger identity $e^{ix\sin(\Omega
t)}=\sum_{l=-\infty}^{\infty}J_{l}(x)e^{il\Omega t}$, with $J_{l}(x)$ being
the Bessel functions of the first kind [78], we find that now

$Q(\omega,\tau)=\sum_{m,n=-\infty}^{\infty}\frac{\tau}{\varepsilon_{0}-\omega+m\Omega}A_{mn}\bigr{(}\text{sinc}^{2}[\dfrac{(m-n)\Omega\tau}{2}](m-n)\Omega+\text{sinc}^{2}[\dfrac{(\varepsilon_{0}-\omega+n\Omega)\tau}{2}](\varepsilon_{0}-\omega+n\Omega)\bigr{)},$
(11)

with $A_{mn}=J_{m}(V_{0}/\Omega)J_{n}(V_{0}/\Omega)$. It is clear that the
filter function is no longer a simple sinc-squared function - although the
average value of the sinusoidal applied field is zero, the filter function
changes in a very non-trivial manner. In particular, it is clear that the
filter function $Q(\omega,\tau)$ is no longer generally peaked at
$\omega=\epsilon_{0}$, even for changing measurement interval $\tau$. Rather,
the second term in Eq. (11) makes it particularly clear that such a simple
conclusion no longer holds in the driven case, and in fact the peak of the
filter function changes as the measurement interval changes. Carrying on, we
can also consider $\alpha(t)=\varepsilon_{0}t$, with non-zero $\beta(t)$
(while $\gamma(t)=0$). This corresponds to the driving field
$H_{c}(t)=\dfrac{1}{2}\dfrac{\partial\beta(t)}{\partial
t}\bigr{(}\cos[\varepsilon_{0}t]\sigma_{y}-\sin[\varepsilon_{0}t]\sigma_{x}\bigr{)}$.
In these cases, $Q(\omega,\tau)$ needs to be calculated numerically, but the
point is that in all such cases, the filter function, and hence the decay
rate, changes in a very non-trivial manner. Similarly, we can also study non-
zero values of $\gamma(t)$; the part of the Hamiltonian which contributes in
$H_{c}(t)$ due to $\gamma(t)$ is of form
$\dfrac{1}{2}\dfrac{\partial\gamma(t)}{\partial
t}\cos[\beta(t)]\sigma_{z}+\dfrac{1}{2}\dfrac{\partial\gamma(t)}{\partial
t}\bigr{(}\sin[\beta(t)]\cos[\alpha(t)]\sigma_{x}+\sin[\beta(t)]\sin[\alpha(t)]\sigma_{y}\bigr{)}$.

We now illustrate the change in the filter function as a result of these
driving fields. Our results are shown in Fig. 1 where we demonstrate the
behavior of the filter function $Q(\omega,\tau)$ as a function of oscillator
frequency $\omega$ for two different measurement intervals $\tau$ both with
and without driving fields. We have set $\beta(t)=\gamma(t)=0$, while
$\alpha(t)=\varepsilon_{0}t$ for the solid, black curve (the undriven case)
and $\alpha(t)=\varepsilon_{0}t+(V_{0}/\Omega)\sin(\Omega t)$ for the other
curves (the driven cases). For the small measurement interval case illustrated
in Fig. 1(a), the different filter functions practically overlap - this is
simply a manifestation of the convergence to the Zeno limit in the small
measurement interval scenario even in the presence of the driving fields.
However, for relatively large measurement interval $\tau$, the filter function
for the population decay model (given by the usual sinc-squared function) is
qualitatively different from the cases where we place the central system in a
time-dependent external field [see Fig. 1(b)]. It can be seen that for the
solid, black curve (the no driving case), the filter function $Q(\omega,\tau)$
is sharply peaked at $\varepsilon_{0}=\omega$ for $\tau=1$, and changes very
appreciably in the presence of strong driving fields (the dot-dashed green and
long-dashed magenta curves). The long-dashed magenta curve corresponds to
relatively lower frequency ($V_{0}=5$ and $\Omega=1$), and to a first
approximation, this filter function can be obtained by considering that the
peak of the usual sinc-squared filter function is now shifted to
$\varepsilon_{0}+V_{0}$. However, for strong driving fields with higher
frequencies (the dot-dashed green curve), such a naive picture is no longer
applicable. Looking at Eq. (11) and using the fact that for higher
frequencies, the Bessel functions are rapidly decaying so that only a few
terms in the sum are important, it is clear that not only is the frequency
$\omega=\varepsilon_{0}$ important in the filter function, but also other
frequencies such as $\omega=\varepsilon_{0}+\Omega$,
$\omega=\varepsilon_{0}-\Omega$, and so on. This leads to a much richer and
complicated filter function, whose peak in fact also changes as the
measurement interval $\tau$ is varied. As a result, we can expect that the
effective decay rate is non trivially modified.

Figure 2:  Filter function for the driven population decay model with the RWA.
(a) Graph of $Q(\omega,\tau)$ versus $\omega$ in the presence of time-
dependent driving fields: $\alpha(t)=\varepsilon_{0}t$, $\beta(t)=-(e^{-\chi
t}-1)/\chi$ and $\gamma(t)=0$ with $\chi=0.2$ (dashed, blue curve),
$\alpha(t)=\varepsilon_{0}t$, $\beta(t)=\upsilon t$ and $\gamma(t)=0$ with
$\upsilon=5$ (dot-dashed, orange curve), $\alpha(t)=\varepsilon_{0}t$,
$\beta(t)=0$ and $\gamma(t)=0$ (solid, black curve). We have used $\tau=1$.
(b) Here $\alpha(t)=\varepsilon_{0}t$, $\beta(t)=\upsilon t$ and
$\gamma(t)=\xi t$, with $\upsilon=5$ and $\xi=1$ (dashed, red curve),
$\upsilon=1$ and $\xi=5$ (long-dashed, magenta curve), and $\upsilon=5$ and
$\xi=5$ (dot-dashed, green curve). The solid black curve shows
$\alpha(t)=\varepsilon_{0}t$, $\beta(t)=0$ and $\gamma(t)=0$. We have again
used $\tau=1$.

Let us now consider more complicated driving fields such that we have non-zero
values of $\beta(t)$ and $\gamma(t)$. As an example, we consider
$\alpha(t)=\varepsilon_{0}t$ and $\beta(t)=\upsilon t$, while $\gamma(t)=0$.
This introduces oscillatory fields in the system Hamiltonian [see Eq. (2)],
and the filter function now changes as shown in Fig. 2(a). It is clear that
adding in the control fields now greatly reduces the filter function (see the
dot-dashed orange curve), and is thus expected to lead to a decrease in the
effective decay rate. We can also consider what happens if these oscillating
control fields are ‘damped’ - this is illustrated by the dashed blue curve. We
have checked as that as the fields become more damped, the filter function
starts to coincide with the undriven scenario (the solid black curve).
Proceeding along these lines, we can also work out the filter function when
$\gamma(t)$ is also non-zero, further illustrating the drastic effect of the
driving fields on the filter function [see Fig. 2(b)].

Figure 3:  Effective decay rate for the driven population decay model with the
RWA. (a) Graph of $\Gamma(\tau)$ versus $\tau$ in the presence of time-
dependent driving fields:
$\alpha(t)=\varepsilon_{0}t+(V_{0}/\Omega)\sin(\Omega t)$, $\beta(t)=0$ and
$\gamma(t)=0$ with $V_{0}=1$ and $\Omega=5$ (dashed, red curve), $V_{0}=5$ and
$\Omega=1$ (long-dashed, magenta curve), and $V_{0}=5$ and $\Omega=5$ (dot-
dashed, green curve). The solid black curve shows
$\alpha(t)=\varepsilon_{0}t$, $\beta(t)=0$ and $\gamma(t)=0$. We remind the
reader that we have used dimensionless units with $\hbar=1$ and we have set
$\varepsilon_{0}=1$. Here we have used $G=0.01$ and $\omega_{c}=10$. (b) We
now have $\alpha(t)=\varepsilon_{0}t$, $\beta(t)=-(e^{-\chi t}-1)/\chi$ and
$\gamma(t)=0$ with $\chi=0.2$ (dashed, blue curve);
$\alpha(t)=\varepsilon_{0}t$, $\beta(t)=\upsilon t$ and $\gamma(t)=0$ with
$\upsilon=5$ (dot-dashed, orange curve); $\alpha(t)=\varepsilon_{0}t$,
$\beta(t)=0$ and $\gamma(t)=0$ (solid, black curve). We now have $G=0.05$ and
$\omega_{c}=10$.

Having illustrated the effect of the driving fields on the filter function, we
now demonstrate how this translates to a change in the effective decay rate
and thereby the quantum Zeno and anti-Zeno behavior. The behavior of the
effective decay rate $\Gamma(\tau)$ versus measurement interval $\tau$ is
shown in Fig. 3 for different driving fields. Let us note how the behavior of
$\Gamma(\tau)$ helps us to identify the quantum Zeno and anti-Zeno regimes. If
the effective decay rate $\Gamma(\tau)$ decreases by shortening the
measurement interval $\tau$, we are in the quantum Zeno regime while if it
increases, then we are in the quantum anti-Zeno regime [41, 44, 30, 46].
Moreover, to actually compute the effective decay rate, we need to specify the
spectral density of the environment. Throughout this work, we will use an
Ohmic spectral density $J(\omega)=G\omega e^{-\omega/\omega_{c}}$, where $G$
stands for the dimensionless coupling strength between the system and its
environment, and $\omega_{c}$ is the cutoff frequency. As we have discussed
before, for weak system-environment coupling, the effective decay rate is the
overlap integral of the spectral density of the environment $J(\omega)$ and
the generalized filter function $Q(\omega,t)$. If the peak value of the filter
function is near the cutoff frequency of spectrum of environment then there
will be a significant overlap between $J(\omega)$ and $Q(\omega,t)$, giving an
enhanced decay rate. On the other hand, if the peak value of generalized
filter function is well beyond the $\omega_{c}$, then that minimizes the
overlap between $J(\omega)$ and $Q(\omega,t)$ , leading to a reduced decay
rate. The dynamically modified filter function in the presence of driving
fields [see Figs. 1 and 2] affects the overlap of $J(\omega)$ with
$Q(\omega,t)$, and thus can either accelerate or inhibit the decay rate as
compared to the undriven scenario. In particular, it is clear from Fig. 3(a)
that for the simple population decay case with no driving fields (the solid
black curve), there is a single local crossover between quantum Zeno and anti-
Zeno regime, meaning that, in short time regime, effective decay rate
$\Gamma(\tau)$ decreases by decreasing the measurement interval $\tau$ while
for large measurement interval, it increases by decreasing the measurement
interval. In the presence of driving fields, not only is the effective decay
rate greatly affected (see the long-dashed magenta curve), but also there are
multiple Zeno and anti-Zeno regimes (see the dashed red curve and the dot-
dashed green curve). For long-dashed magenta curve, the peak value of the
filter function is at approximately $\omega=5.6$ [see Fig. 1(b)] which is more
near to the peak of the spectrum of the environment as compared to the solid
black curve (for which the peak is at $\omega=1$). As a result, this gives
maximum overlap of $Q(\omega,\tau)$ with $J(\omega)$ for the magenta curve,
which consequently enhances the effective decay rate compared to the undriven
case. For the dot-dashed green curve, we observed previously that a large
driving field frequency means that the peak of the filter function keeps
changing as the measurement interval changes, and this leads to multiple Zeno
and anti-Zeno regimes. However, at short measurement intervals, all the curves
agree. This is expected, since, as we have seen before, with very fast
measurements, the filter function becomes the same, leading to the same decay
rate. We have also looked at what happens with $\beta(t)\neq 0$. Having seen
how the filter function is influenced by such driving fields in Fig. 2(a), we
illustrate what happens to the corresponding decay rate in Fig. 3(b) for the
same $\beta(t)$ as used in Fig. 2(a), where $\beta(t)$ is a damped function
for the dashed blue curve, and it is a linear function of $t$ for the dot-
dashed orange curve. Since a linear function of $t$ in the presence of non-
zero $\varepsilon_{0}$ leads to a reduction in the peak value of the filter
function [see Fig. 2(a)], the overlap between $Q(\omega,\tau)$ and $J(\omega)$
reduces for $\omega_{c}=10$, leading to a reduction in the effective decay
rate as compared to the solid black curve. On the other hand, if $\beta(t)$ is
a damped function, effects of the oscillating fields are suppressed, meaning
that the effective decay rate is increased for the dashed blue curve as
compared to dot-dashed orange curve. Once again, it is clear that the driving
fields greatly influence the decay rate both quantitatively and qualitatively.

We next discuss the effect of the non-rotating terms of the system-environment
interaction on the dynamics of the central system in the presence of a driving
field. The total Hamiltonian is now given by

$H(t)=\frac{\varepsilon_{0}}{2}\sigma_{z}+H_{c}(t)+\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}+\sigma_{x}\sum_{k}(g_{k}^{*}b_{k}+g_{k}b_{k}^{\dagger}).$
(12)

Notice the different form of the system-environment coupling as compared to
before - the system-environment Hamiltonian now contains the ‘non-rotating’
terms $\sigma_{+}b_{k}^{\dagger}$ and $\sigma_{-}b_{k}$. To calculate filter
function $Q(\omega,t)$ now, we first evaluate the environment correlation
function. With $F=\sigma_{x}$, we find
$G(t)=-e^{-i(\alpha(t)-\gamma(t))}\sin^{2}[\beta(t)/2]+e^{i(\alpha(t)+\gamma(t))}\cos^{2}[\beta(t)/2]$
and
$\bar{G}(t-t^{\prime})=-e^{i(\alpha(t-t^{\prime})-\gamma(t-t^{\prime}))}\sin^{2}[\beta(t-t^{\prime})/2]+e^{-i(\alpha(t-t^{\prime})+\gamma(t-t^{\prime}))}\cos^{2}[\beta(t-t^{\prime})/2]$.
Also, $f(\omega,t)=e^{-i\omega t}$ at zero temperature. Putting all this
together,

$\displaystyle
Q(\omega,\tau)=\frac{2}{\tau}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}\bigr{(}D_{1}(t,t^{\prime})+D_{2}(t,t^{\prime})+D_{3}(t,t^{\prime})+D_{4}(t,t^{\prime})\bigr{)},$
(13)

where

$\displaystyle D_{1}(t,t^{\prime})=\cos[\gamma(t)-\gamma(t-t^{\prime})-\omega
t^{\prime}]\cos[\alpha(t)]\cos[\beta(t)]\cos[\alpha(t-t^{\prime})]\cos[\beta(t-t^{\prime})],$
(14) $D_{2}(t,t^{\prime})=-\sin[\gamma(t)-\gamma(t-t^{\prime})-\omega
t^{\prime}]\cos[\alpha(t-t^{\prime})]\cos[\beta(t-t^{\prime})]\sin[\alpha(t)],$
(15) $D_{3}(t,t^{\prime})=\sin[\gamma(t)-\gamma(t-t^{\prime})-\omega
t^{\prime}]\cos[\alpha(t)]\cos[\beta(t)]\sin[\alpha(t-t^{\prime})],$ (16)
$D_{4}(t,t^{\prime})=\cos[\gamma(t)-\gamma(t-t^{\prime})-\omega
t^{\prime}]\sin[\alpha(t)]\sin[\alpha(t-t^{\prime})].$ (17)

Compared with Eq. (10), we can see that the two filter functions agree for
$\beta(t)=0$. This means that in the absence of driving fields (where
$\alpha(t)=\varepsilon_{0}t$ and $\beta(t)=0$), the counter-rotating terms do
not affect the decay rate. However, in the presence of driving fields with
$\beta(t)\neq 0$, the counter-rotating terms become important, even in the
weak system-environment coupling regime we are dealing with. The influence of
the non-rotating terms is shown in Fig. 4, where the behavior of the filter
function as a function of the frequency $\omega$ is shown in Fig. 4(a) and the
behavior of the effective decay rate as a function of measurement interval is
illustrated in Fig. 4(b). Comparing Figs. 2(a) and Fig. 4(a), it is clear that
when there are no driving fields, the filter function does not change since
the solid, black curve is the same in both figures. However, as shown by the
dashed blue and dot-dashed orange curves, in the presence of driving fields
with $\beta(t)\neq 0$, the filter function does change. This correspondingly
changes the effective decay rate by modifying the overlap of $J(\omega)$ with
$Q(\omega,t)$ as can be seen by comparing Figs. 3(b) and 4(b). The counter-
rotating terms help to enhance the peak value of the filter function, leading
to an increase in the effective decay rate.

Figure 4: Filter function and effective decay rate for the driven population
decay model without the RWA. (a) Graph of $Q(\omega,\tau)$ versus $\omega$ in
the presence of time-dependent driving fields: $\alpha(t)=\varepsilon_{0}t$,
$\beta(t)=-(e^{-\chi t}-1)/\chi$ and $\gamma(t)=0$ with $\chi=0.2$ (small-
dashed, blue curve); $\alpha(t)=\varepsilon_{0}t$, $\beta(t)=\upsilon t$ and
$\gamma(t)=0$ with $\upsilon=5$ (dot-dashed-dashed, orange curve);
$\alpha(t)=\varepsilon_{0}t$, $\beta(t)=0$ and $\gamma(t)=0$ (solid, black
curve). Here we have used $\tau=1$. (b) Similar to (a), except that now we
have plotted $\Gamma(\tau)$ versus $\tau$ with $G=0.01$ and $\omega_{c}=10$.

### Application to the driven dephasing model in weak coupling regime

We now consider the pure dephasing model given by the system-environment
Hamiltonian

$H=\frac{\varepsilon_{0}}{2}\sigma_{z}+\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}+\sigma_{z}\sum_{k}(g_{k}^{*}b_{k}+g_{k}b_{k}^{\dagger}).$
(18)

Notice the different form of the system-environment interaction. With this
model, the populations of the system energy eigenstates cannot change - only
the off-diagonal coherences can change, which is why this is referred to as a
pure dephasing model[30]. The initial state usually considered in this model
is
$\left|\psi\right\rangle=\dfrac{1}{\sqrt{2}}(\left|e\right\rangle+\left|g\right\rangle)$,
with $\left\langle e\right|g\rangle=0$. However, with the formalism we have
developed, the initial state we considered was $\left|e\right\rangle$. To use
our formalism, we consequently perform a unitary operation
$U_{R}=e^{i\pi\sigma_{y}/4}$. The initial state then again becomes
$\left|e\right\rangle$, while the Hamiltonian is transformed to

$H^{(R)}=U^{(R)}HU^{(R)\dagger}=-\frac{\varepsilon_{0}}{2}\sigma_{x}+\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}-\sigma_{x}\sum_{k}(g_{k}^{*}b_{k}++g_{k}b_{k}^{\dagger}).$
(19)

To find the filter function now, we look at Eq. (2) and find that
$\alpha(t)=\pi/2$, $\beta(t)=\varepsilon_{0}t$ and $\gamma(t)=-\pi/2$ gives
the same Hamiltonian as Eq. (19). Then, using our developed formalism, we find
that $G(t)=1$ and $\bar{G}(t-t^{\prime})=1$. Consequently, assuming that the
environment is at zero temperature, we get
$Q(\omega,\tau)=\frac{2}{\tau}\frac{1-\cos(\omega\tau)}{\omega^{2}}$, which
agrees with the filter function obtained using the exact solution [46]. Next,
we add in the effect of the driving fields. To this end, we look at more
complicated time-dependent functions $\alpha(t)$, $\beta(t)$, and $\gamma(t)$.
We write the corresponding system-environment Hamiltonian as

$H(t)=-\frac{\varepsilon_{0}}{2}\sigma_{x}+H_{c}(t)+\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}-\sigma_{x}\sum_{k}(g_{k}^{*}b_{k}++g_{k}b_{k}^{\dagger}).$
(20)

To take into account the additional control fields given by $H_{c}(t)$, we
write $\alpha(t)=\pi/2+\widetilde{\alpha}(t)$ and
$\gamma(t)=-\pi/2+\widetilde{\gamma}(t)$, while $\beta(t)$ remains
$\varepsilon_{0}t$. Simple calculations then lead to the filter function

$\displaystyle
Q(\omega,\tau)=\frac{2}{\tau}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}\bigr{(}D_{1}(t,t^{\prime})+D_{2}(t,t^{\prime})+D_{3}(t,t^{\prime})+D_{4}(t,t^{\prime})\bigr{)},$
(21)

where now

$\displaystyle
D_{1}(t,t^{\prime})=\cos[\widetilde{\gamma}(t)-\widetilde{\gamma}(t-t^{\prime})-\omega
t^{\prime}]\sin[\widetilde{\alpha}(t)]\cos[\beta(t)]\sin[\widetilde{\alpha}(t-t^{\prime})]\cos[\beta(t-t^{\prime})],$
(22)
$D_{2}(t,t^{\prime})=\sin[\widetilde{\gamma}(t)-\widetilde{\gamma}(t-t^{\prime})-\omega
t^{\prime}]\sin[\widetilde{\alpha}(t-t^{\prime})]\cos[\beta(t-t^{\prime})]\cos[\widetilde{\alpha}(t)],$
(23)
$D_{3}(t,t^{\prime})=-\sin[\widetilde{\gamma}(t)-\widetilde{\gamma}(t-t^{\prime})-\omega
t^{\prime}]\sin[\widetilde{\alpha}(t)]\cos[\beta(t)]\cos[\widetilde{\alpha}(t-t^{\prime})],$
(24)
$D_{4}(t,t^{\prime})=\cos[\widetilde{\gamma}(t)-\widetilde{\gamma}(t-t^{\prime})-\omega
t^{\prime}]\cos[\widetilde{\alpha}(t)]\cos[\widetilde{\alpha}(t-t^{\prime})].$
(25)

Using these expressions, we have plotted the filter function (for $\tau=1$) in
Fig. 5(a) for different control fields. Once again, it is clear that the
driving fields greatly influence the filter function in general. For instance,
with a sinusoidal driving field
($\widetilde{\alpha}(t)=(V_{0}/\Omega)\sin(\Omega t)$,
$\beta(t)=\varepsilon_{0}t$, and $\widetilde{\gamma}(t)=0$), the filter
function is very different as compared with the undriven case (compare the
solid black curve with the dot-dashed green and long-dashed magenta curves),
with the difference becoming smaller as the driving field strength is reduced
(see the dashed red curve). Consequently, the behavior of effective decay rate
and the corresponding quantum Zeno and anti-Zeno phenomena is expected to be
greatly modified due to the different overlap of the filter function with the
spectrum of the environment. That this is indeed the case can be seen in Fig.
5(b). We see that there is a single peak in the case of the undriven pure
dephasing model. On the other hand, due to the application of external fields,
the effective decay rate increases with increasing $V_{0}$ and the effective
decay rate shows multiple Zeno and anti-Zeno regimes for fast oscillating
external fields.

Figure 5:  Filter function and effective decay rate for the driven dephasing
model. (a) Graph of $Q(\omega,\tau)$ versus $\omega$ in the presence of time-
dependent driving fields: $\widetilde{\alpha}(t)=(V_{0}/\Omega)\sin(\Omega
t)$, $\beta(t)=\varepsilon_{0}t$ and $\widetilde{\gamma}(t)=0$ with $V_{0}=1$
and $\Omega=5$ (dashed, red curve), $V_{0}=5$ and $\Omega=1$ (large-dashed,
magenta curve), and $V_{0}=5$ and $\Omega=5$ (dot-dashed, green curve). The
solid black curve is with $\widetilde{\alpha}(t)=0$,
$\beta(t)=\varepsilon_{0}t$, and $\widetilde{\gamma}(t)=0$. Again, we are
using dimensionless units with $\hbar=1$ and we have set $\varepsilon_{0}=1$.
Here we are have used $\tau=1$. (b) Similar to (a), except that now we have
plotted $\Gamma(\tau)$ versus $\tau$ with $G=0.01$ and $\omega_{c}=10$.

### Application to the driven large spin-boson model in weak coupling regime

We now briefly show how we can extend our formalism to more general systems in
which a large spin greater than spin-1/2 is coupled to a harmonic oscillator
environment. Such a model can describe, for instance, a collection of $N_{S}$
two-level systems coupled to a common environment[30, 63, 79]. We first define
$J_{k}=\dfrac{1}{2}\sum_{i}\sigma_{k}^{(i)}$, where $J_{k}$ ($k=x,y,z$) are
the large spin operators. As a concrete example, we consider the driven
population decay model given by the system-environment Hamiltonian

$H(t)={\varepsilon_{0}}J_{z}+H_{c}(t)+\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}+2J_{x}\sum_{k}(g_{k}^{*}b_{k}+g_{k}b_{k}^{\dagger}),$
(26)

where $\varepsilon_{0}$ is the energy level spacing for each spin-1/2
particle, and $H_{c}(t)$ is the control field Hamiltonian. Analogous to what
we did for the single spin-1/2 case, we consider the free system unitary time
evolution operator to be
$U_{S}(t)=e^{-i\alpha(t)J_{z}}e^{-i\beta(t)J_{y}}e^{-i\gamma(t)J_{z}}$. We
take the initial state to be $\left|j\right\rangle$ with
$J_{z}\left|j\right\rangle=j\left|j\right\rangle$ and $j=N_{S}/2$. Performing
the calculation for the filter function using our formalism, we find that the
filter function is exactly the same as for the single spin-1/2 case [see Eq.
(13)] except for an additional multiplicative factor of $N_{S}$ (see the
Methods section for details). That is, the effective decay rate is now
enhanced by a factor of $N_{S}$, reminiscent of the superradiance effect [80].
A similar calculation shows that if the pure dephasing model is extended to
the large spin case in an analogous manner, the effective rate is again
enhanced by a factor of $N_{S}$.

### Application to the driven spin-boson model in the strong system-
environment coupling regime

Finally, let us extend our treatment of the driven population decay model to
the strong system-environment case. We start from the system-environment
Hamiltonian $H(t)=H_{S}(t)+H_{B}+H_{SB}$ with
$H_{S}(t)=\dfrac{\varepsilon(t)}{2}\sigma_{z}+\dfrac{\Delta}{2}\sigma_{x}$,
$H_{B}=\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}$, and
$H_{SB}=\sigma_{z}\sum_{k}(g^{*}_{k}b_{k}+g_{k}b^{\dagger}_{k})$. The driving
fields are contained in $\varepsilon(t)$. Now, if the system and the
environment are interacting strongly, we cannot treat the interaction term
between system and its environment perturbatively. Instead, we can consider
performing a polaron transformation [65, 69, 66, 70, 47], which transforms our
Hamiltonian to (see the supplementary material)

$H^{(P)}(t)=e^{\chi\sigma_{z}/2}H(t)e^{-\chi\sigma_{z}/2}=\dfrac{\varepsilon(t)}{2}\sigma_{z}+\sum_{k}\omega_{k}b^{\dagger}_{k}b_{k}+\dfrac{\Delta}{2}(\sigma_{+}Y+\sigma_{-}Y^{\dagger}),$
(27)

where $P$ denotes the so-called polaron frame,
$\chi=\sum_{k}[\dfrac{2g_{k}}{\omega_{k}}b_{k}^{\dagger}-\dfrac{2g^{*}_{k}}{\omega_{k}}b_{k}]$,
and $Y=e^{\chi}$.

We see that in the polaron frame, the form of the system-environment
interaction is different. Now, if the tunneling parameter $\Delta$ is small,
we can apply time-dependent perturbation theory, treating the system-
environment coupling in the polaron frame perturbatively. As before, at the
initial time, we prepare our system in the excited state
$\left|e\right\rangle$, and then perform projective measurements on the system
with time interval $\tau$ to check whether the system is still present in
excited state $\left|e\right\rangle$ or not. It is also important to note that
initial state of system and environment in the untransformed frame cannot be
written in the simple usual product form
$\rho(0)=\left|e\right\rangle\left\langle e\right|\otimes e^{-\beta
H_{B}}/Z_{B}$, with $\rho_{B}=e^{-\beta H_{B}}/Z_{B}$ and
$Z_{B}=\text{Tr}_{B}[e^{-\beta H_{B}}]$, since the system and the environment
are strongly interacting in that frame and consequently, the initial system-
environment correlations are significant [81, 82, 83, 84, 85]. However, since
the system and its environment are effectively weakly interacting in the
polaron frame, the initial state in the polaron frame can be taken as a simple
product state. The rest of the calculation, performed in the polaron frame,
proceeds in a similar way as the weak coupling case using perturbation theory.
We eventually arrive at (see the supplementary Material)

$\Gamma(\tau)=\dfrac{\Delta^{2}}{2\tau}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}e^{-\Phi_{R}(t^{\prime})}\cos[\zeta(t)-\zeta(t-t^{\prime})-\Phi_{I}(t^{\prime})],$
(28)

where $\Phi_{I}(t)=\int_{0}^{\infty}d\omega J(\omega)\dfrac{\sin(\omega
t)}{\omega^{2}}$, $\Phi_{R}(t)=\int_{0}^{\infty}d\omega
J(\omega)\dfrac{1-\cos(\omega t)}{\omega^{2}}\coth(\dfrac{\beta\omega}{2})$,
and $\zeta(t)=\int_{0}^{t}\varepsilon(t^{\prime})\,dt^{\prime}$. Assuming, as
before, an Ohmic spectral density of the form $J(\omega)=G\omega
e^{-\omega/\omega_{c}}$, $\Phi_{I}(t)$ and $\Phi_{R}(t)$ are found to be (at
zero temperature) $\Phi_{I}(t)=G\tan^{-1}(\omega_{c}t)$ and
$\Phi_{R}(t)=\frac{G}{2}\ln(1+\omega_{c}^{2}t^{2})$.

Figure 6:  Effective decay rate for the strongly coupled driven spin-boson
model. (a) Graph of $\Gamma(\tau)$ versus $\tau$ in the presence of the time-
dependent driving fields:
$\alpha(t)=\varepsilon_{0}t+(V_{0}/\Omega)\sin(\Omega t)$, $\beta(t)=0$ and
$\gamma(t)=0$ with $V_{0}=1$ and $\Omega=5$ (dashed, red curve), $V_{0}=5$ and
$\Omega=1$ (large-dashed, magenta curve), and $V_{0}=5$ and $\Omega=5$ (dot-
dashed, green curve). The solid black curve shows
$\alpha(t)=\varepsilon_{0}t$, $\beta(t)=0$ and $\gamma(t)=0$. Here we have
used $\Delta=0.05$, $\omega_{c}=10$ and $G=1$. (b) Same as (a), except that
now $G=2$.

We now have everything we need to calculate the effective decay rate. It
should be obvious from our expressions above that for the strong system-
environment coupling regime that we are considering here, the effective decay
rate $\Gamma(\tau)$ no longer depends on an overlap integral of the
generalized filter function with the spectral density of environment. Rather,
there is now a non-linear dependence on the spectral density, leading to very
different qualitative behavior as compared with the usual weak system-
environment coupling regime. For instance, as the system-environment coupling
strength increases, $e^{-\Phi_{R}(t)}$ decreases, and thus we anticipate
$\Gamma(\tau)$ to decrease. Most importantly for us, we expect that the
driving fields have a drastic, non-trivial effect not only the value of the
effective decay rate, but also quantum Zeno and anti-Zeno behavior since the
integrand in Eq. (34) obviously depends on the driving fields. To make these
claims concrete, let us show the quantitative behavior of effective decay rate
$\Gamma(\tau)$. It is clear from Fig. 6 that not only the driving fields
affect the decay rate very significantly, but also that increasing the system-
environment coupling strength $G$ reduces the effective decay rate [compare
Figs. 6(a) and (b)], in contrast with the weak coupling regime. Here again, we
observe multiple oscillations in quantum Zeno to anti-Zeno regimes, as we
increase $\Omega$. Interestingly, Fig. 3(a) looks very similar with Fig. 6(b)
despite having coupling strength $G$ in different regimes. This similarity can
be understood noting that in both cases, the pointer states are the
eigenstates of the operator $\sigma_{z}$, and that in the strong coupling
regime, we have a population decay model in the polaron frame (see Eq. (27)).
In both cases, the driving fields modulate the energy-level splitting. These
similarities leads to the same qualitative form of Fig. 3(a) and Fig. 6(b).

##  3 Discussion

In this paper, we started off by introducing a general formalism to calculate
the effective decay rate of a quantum system subjected to both periodic
projective measurements and driving fields, valid for weak system-environment
coupling strength. We then applied this formalism to derive a general
expression for the decay rate of a driven two-level system. The decay rate is
an overlap integral of the spectral density of environment and a filter
function expressed using time-dependent Euler angles. These results were
illustrated using the population decay model as well as the pure dephasing
model. In both cases, the application of the driving fields very significantly
changes the form of filter function of the central system, which then modifies
(either enhances or minimizes) the effective decay rate and, consequently, the
quantum Zeno and anti-Zeno regimes are altered. Interestingly, for the
population decay model, the driving fields can lay bare the effect of the
counter-rotating terms, even for the weak system-environment coupling regime
that we are dealing with. These results were then generalized to large spin
systems to show a possible amplification of the decay rate. Finally, we also
looked at driven two-level systems strongly coupled to an environment of
harmonic oscillators, where the effective decay rate shows a non-linear
dependence on the spectral density of the environment. We showed once again
that the effective decay rate is modified by the application of driving
fields. Our general expressions and insights should of interest in the broad
areas of quantum control and quantum state engineering, such as quantum noise
sensing, as well as in fundamental studies of the quantum Zeno and anti-Zeno
effects. For example, a quantum system can be put into the Zeno regime,
thereby protecting it from decoherence, by applying suitable control fields.
On the other hand, the decay rate can be enhanced in the anti-Zeno regime via
the applied control fields, and this can be useful for cooling the quantum
system [86].

##  4 Methods

### Effective decay rate using perturbation theory

We first discuss how to obtain effective decay rate of system under frequent
measurements, extending the treatment in Ref. [46] to time-dependent
Hamiltonians. We first write the system-environment Hamiltonian as
$H(t)=H_{F}(t)+H_{SB}$, where $H_{F}(t)=H_{S}(t)+H_{B}$ is the sum of the free
system and environment Hamiltonians and $H_{SB}$ describes the system-
environment interaction. Using the standard perturbation approach, we set
$U(t)=U_{F}(t)U_{I}(t)$, where ${U}_{F}(t)$ describes the free time evolution
of driven system and environment (this may be non-trivial and involve time-
ordering due to the possible time dependence of $H_{S}$), and ${U}_{I}(t)$ is
the leftover part that can be expanded perturbatively (assuming the system-
environment interaction to be weak). Expanding up to second order in system-
environment coupling, we can write ${U}_{I}(t)=\mathds{1}+{G}_{1}+{G}_{2}$,
where $G_{1}$ and $G_{2}$ are the first, and second order corrections
respectively. Using this expansion, the density matrix at time $\tau$ can be
written as

$\displaystyle{\rho}_{S}(\tau)\approx\text{Tr}_{B}\bigr{[}{U}_{F}(\tau)\bigr{(}{\rho}(0)+{\rho}(0){G}^{\dagger}_{1}+{G}_{1}{\rho}(0)+{\rho}(0){G}^{\dagger}_{2}+{G}_{2}{\rho}(0)+{G}_{1}{\rho}(0){G}^{\dagger}_{1}\bigr{)}{U}^{\dagger}_{F}(\tau)\bigr{]}.$

Perturbation theory tells us that
${G}_{1}=-i\int^{\tau}_{0}dt_{1}\widetilde{H}_{SB}(t_{1})$, and
${G}_{2}=-\int^{\tau}_{0}dt_{1}\int^{t_{1}}_{0}dt_{2}\widetilde{H}_{SB}(t_{1})\widetilde{H}_{SB}(t_{2})$,
where
$\widetilde{H}_{SB}(t)={U}^{\dagger}_{F}(t){{H}}_{SB}{U}_{F}(t)=\sum_{\mu}{U}_{S}^{\dagger}(t)F_{\mu}{U}_{S}(t)\otimes{{U}}^{\dagger}_{B}(t)B_{\mu}{{U}}_{B}(t)={\widetilde{F}_{\mu}}(t){\widetilde{B}}_{\mu}(t)$.
Each term can then be simplified to eventually arrive at

$\rho_{S}(\tau)=U_{S}(\tau)\biggl{(}\rho_{S}(0)+\sum_{\mu\nu}\int_{0}^{\tau}dt_{1}\int_{0}^{t_{1}}dt^{\prime}C_{\mu\nu}(t^{\prime})[\widetilde{F}_{\nu}(t_{1}-t^{\prime})\rho_{S}(0),\widetilde{F}_{\mu}(t_{1})]+\text{h.c.}\biggr{)}U_{S}^{\dagger}(\tau).$
(29)

where $C_{\mu\nu}(t^{\prime})=\text{Tr}_{B}[\frac{e^{-\beta
H_{B}}}{Z_{B}}\widetilde{B}_{\mu}(t^{\prime})\widetilde{B}_{\nu}(0)]$ are the
environment correlation functions, and h.c. denotes hermitian conjugate.

Once we have the expression for the system density matrix at time $\tau$, we
can evaluate the survival probability of the system in the initial state.
Since we are interested to investigate the system evolution due to system-
environment interaction only, we first apply the free driven system unitary
operator on both sides of Eq. (29) to remove the system evolution due to free
driven system Hamiltonian itself. Note that this may not be necessary if
$U_{S}(\tau)$ commutes with $\rho_{S}(0)$. We thereafter find the probability
that the system is still in the initial state $\left|\psi\right\rangle$ after
a projective measurement given by the projector
$\left|\psi\right\rangle\left\langle\psi\right|$ to be given by

$\displaystyle
s(\tau)=1-2\text{Re}\,\biggr{(}\sum_{\mu\nu}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}C_{\mu\nu}(t^{\prime})\text{Tr}_{S}\bigr{[}P_{\perp}\widetilde{F}_{\nu}(t-t^{\prime})\rho_{S}(0)\widetilde{F}_{\mu}(t)\bigr{]}\biggr{)},$
(30)

where $P_{\perp}$ is the projector onto the subspace orthogonal to
$\left|\psi\right\rangle\left\langle\psi\right|$. After performing a sequence
of $M$ repeated projective measurements, we find the survival probability that
the system state is still present in the initial state is
$S(M\tau)=[s(\tau)]^{M}$ if the system-environment correlations are ignored
during the evolution. We can then find the effective decay rate $\Gamma(\tau)$
by $S(M\tau)=e^{-\Gamma(\tau)M\tau}$ allowing us to write $\Gamma(\tau)=-\ln
s(\tau)/\tau$. In the weak system-environment coupling regime, we can further
write

$\displaystyle\Gamma(\tau)=\frac{2}{\tau}\text{Re}\biggr{(}\sum_{\mu\nu}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}C_{\mu\nu}(t^{\prime})\text{Tr}_{S}\bigr{[}P_{\perp}\widetilde{F}_{\nu}(t-t^{\prime})\rho_{S}(0)\widetilde{F}_{\mu}(t)\bigr{]}\biggr{)}.$

This can be cast into the form

$\Gamma(\tau)=\int_{0}^{\infty}\,d\omega\,Q(\omega,\tau)J(\omega),$

with $Q(\omega,\tau)$ given in Eq. (3).

### Finding the filter function for the driven large-spin population decay
model

In this case, using the standard angular momentum relations
$[J_{i},J_{j}]=i\varepsilon_{ijk}J_{k}$, we evaluate

$\widetilde{F}(t)=2\bigr{(}J_{x}c_{x}(t)+J_{y}c_{y}(t)+J_{z}c_{z}(t)\bigr{)},$

where

$\displaystyle c_{x}(t)$
$\displaystyle=\cos[\alpha(t)]\cos[\beta(t)\cos[\gamma(t)]-\sin[\alpha(t)]\sin[\gamma(t)],$
$\displaystyle c_{y}(t)$
$\displaystyle=\cos[\alpha(t)]\cos[\beta(t)\sin[\gamma(t)]+\sin[\alpha(t)]\cos[\gamma(t)],$
$\displaystyle c_{z}(t)$ $\displaystyle=-\cos[\alpha(t)]\sin[\beta(t)].$ (31)

For $\rho_{S}(0)=\left|j,j\right\rangle\left\langle j,j\right|$ and
$P_{\perp}=\sum_{m=1}^{j-1}\left|j,m\right\rangle\left\langle j,m\right|$ with
$J_{z}\left|j,m\right\rangle=m\left|j,m\right\rangle$, we have

$\text{Tr}_{S}\bigr{[}P_{\perp}\widetilde{F}(t-t^{\prime})\rho_{S}(0)\widetilde{F}(t)\bigr{]}=\sum_{m=1}^{j-1}\left\langle
j,m\right|\widetilde{F}(t-t^{\prime})\left|j,j\right\rangle\left\langle
j,j\right|\widetilde{F}(t)\left|j,m\right\rangle.$

We next note that $\left\langle
j,j\right|\widetilde{F}(t)\left|j,m\right\rangle=\sqrt{2j}\delta_{j-1,m}\bigr{(}c_{x}(t)-ic_{y}(t)\bigr{)}$.
This leads to the generalized filter function

$Q(\omega,t)=(2j)\frac{2}{\tau}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}\bigr{(}D_{1}(t,t^{\prime})+D_{2}(t,t^{\prime})+D_{3}(t,t^{\prime})+D_{4}(t,t^{\prime})\bigr{)},$

where expressions of $D_{1}(t,t^{\prime})$, $D_{2}(t,t^{\prime})$,
$D_{3}(t,t^{\prime})$ and $D_{4}(t,t^{\prime})$ are defined in Eqs. (14)-(17).

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## Acknowledgements

A. Z. C. is grateful for support from HEC under grant No
5917/Punjab/NRPU/R&D/HEC/2016.

## Author contributions statement

A. Z. C. came up with the basic idea behind this work. M. M. carried out the
calculations and plotted the graphs. Both authors contributed towards the
writing of the manuscript.

## Additional information

Competing interests: The authors declare no competing interests.

## Supplemental Material for ‘The quantum Zeno and anti-Zeno effects with
driving fields in the weak and strong coupling regimes’

In this supplemental Material, we use same symbols as introduced in our main
text. For completeness, we present the analytical expressions for the
Hamiltonian in the polaron frame. We also outline the calculation of the decay
rate $\Gamma(\tau)$ for a single two-level system interacting strongly with an
environment of harmonic oscillators.

### Spin-boson Hamiltonian in polaron frame

To transform spin-boson Hamiltonian to polaron frame, we need to find

$H^{(P)}(t)=e^{\chi\sigma_{z}/2}H(t)e^{-\chi\sigma_{z}/2}.$

We use the Hadamard lemma

$e^{A}Oe^{-A}=O+[A,O]+\dfrac{1}{2!}[A,[A,O]]+....$ (32)

where $A=\chi\sigma_{z}/2$, with
$\chi=\sum_{k}[\dfrac{2g_{k}}{\omega_{k}}b_{k}^{\dagger}-\dfrac{2g^{*}_{k}}{\omega_{k}}b_{k}]$
and
$O=H(t)=\dfrac{\varepsilon(t)}{2}\sigma_{z}+\dfrac{\Delta}{2}\sigma_{x}+\sum_{k}\omega_{k}b_{k}^{\dagger}b_{k}+\sigma_{z}\sum_{k}(g^{*}_{k}b_{k}+g_{k}b^{\dagger})$.
We find that

$e^{\chi\sigma_{z}/2}\sigma_{z}e^{-\chi\sigma_{z}/2}=\sigma_{z},$

and

$\displaystyle
e^{\chi\sigma_{z}/2}\sigma_{x}e^{-\chi\sigma_{z}/2}=\sigma_{+}e^{\chi}+\sigma_{-}e^{-\chi}.$

$\sigma_{-}$ and $\sigma_{+}$ are the standard spin-1/2 lowering and raising
operators. Carrying on further, we find

$e^{\chi\sigma_{z}/2}(\sum_{k}\omega_{k}b^{\dagger}_{k}b_{k})e^{-\chi\sigma_{z}/2}=\sum_{k}\omega_{k}b^{\dagger}_{k}b_{k}-\sigma_{z}\sum_{k}(g^{*}_{k}b_{k}+g_{k}b^{\dagger})+\sum_{k}\dfrac{|g_{k}|^{2}}{\omega_{k}}.$

Similarly

$e^{\chi\sigma_{z}/2}\biggr{(}\sigma_{z}\sum_{k}(g^{*}_{k}b_{k}+g_{k}b^{\dagger})\biggr{)}e^{-\chi\sigma_{z}/2}=\sigma_{z}\sum_{k}(g^{*}_{k}b_{k}+g_{k}b^{\dagger})-2\sum_{k}\dfrac{|g_{k}|^{2}}{\omega_{k}},$

The third term of Eq. (32) in the above expression is a constant number, so
higher order commutators are zero. Now putting all these terms back together,
the required Hamiltonian in polaron frame takes the following form

$H^{(P)}(t)=\dfrac{\varepsilon(t)}{2}\sigma_{z}+\sum_{k}\omega_{k}b^{\dagger}_{k}b_{k}+\dfrac{\Delta}{2}(\sigma_{+}Y+\sigma_{-}Y^{\dagger})-\sum_{k}\dfrac{|g_{k}|^{2}}{\omega_{k}},$

with $Y=e^{\chi}$, and $\sum_{k}\dfrac{|g_{k}|^{2}}{\omega_{k}}$ is a constant
number term that gives a constant shift in transformed Hamiltonian, and can
thus be dropped.

### Effective decay rate of spin-boson model in polaron frame

Since system-environment coupling is weak in the polaron frame, we can use
expression [see the main text]

$\Gamma(\tau)=\frac{2}{\tau}\text{Re}\biggr{(}\sum_{\mu\nu}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}C_{\mu\nu}(t^{\prime})\text{Tr}_{S}\bigr{[}P_{\perp}\widetilde{F}_{\nu}(t-t^{\prime})\rho_{S}(0)\widetilde{F}_{\mu}(t)\bigr{]}\biggr{)},$

to calculate effective decay rate . Here
$\rho_{S}(0)=\left|e\right\rangle\left\langle e\right|$ and
$P_{\perp}=\left|g\right\rangle\left\langle g\right|$. We identity
$F_{1}=\dfrac{\Delta}{2}\sigma_{+}$, $F_{2}=\dfrac{\Delta}{2}\sigma_{-}$,
$B_{1}=Y$, $B_{2}=Y^{\dagger}$,
$\widetilde{F}_{1}(t)=\dfrac{\Delta}{2}\sigma_{+}e^{i\zeta(t)}$ and
$\widetilde{F}_{2}(t)=\dfrac{\Delta}{2}\sigma_{-}e^{-i\zeta(t)}$ with
$\zeta(t)=\int_{0}^{t}dt^{\prime}\varepsilon(t^{\prime})$ leading us to

$\displaystyle\Gamma(\tau)=\frac{2}{\tau}\text{Re}\biggr{(}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}C_{12}(t^{\prime})e^{i(\zeta(t)-\zeta(t-t^{\prime}))}\biggr{)}.$
(33)

To get expression of effective decay rate, the environment correlation
function $C_{12}(t)$ needs to be worked out. We now show the details how to
find $C_{12}$. As we know
$C_{12}(t)=\text{Tr}_{B}[\rho_{B}\widetilde{B}_{1}(t)B_{2}]$, with $B1=Y$,
$B_{2}=Y^{\dagger}$,
$\widetilde{B}_{1}(t)=e^{iH_{B}^{(P)}t}Ye^{-iH_{B}^{(P)t}}$,
$H^{(P)}_{B}=\sum_{k}\omega_{k}b^{\dagger}_{k}b_{k}$, $Y=e^{\chi}$ and
$\chi=\sum_{k}[\dfrac{2g_{k}}{\omega_{k}}b_{k}^{\dagger}-\dfrac{2g^{*}_{k}}{\omega_{k}}b_{k}]$.
Next, we calculate

$\widetilde{B}_{1}(t)=e^{\sum_{k}\biggr{(}\frac{g_{k}}{\omega_{k}}b^{\dagger}_{k}e^{i\omega_{k}t}-\frac{g^{*}_{k}}{\omega_{k}}b_{k}e^{-i\omega_{k}t}\biggr{)}},$

using the fact $U^{\dagger}(t)e^{A}U(t)=e^{U^{\dagger}(t)AU(t)}$, and then
find

$\widetilde{B}_{1}(t)B_{2}=e^{-i\sum_{k}\dfrac{|g_{k}|^{2}}{\omega_{k}^{2}}\sin(\omega_{k}t)}e^{\sum_{k}\biggr{(}\frac{g_{k}}{\omega_{k}}b^{\dagger}_{k}(e^{i\omega_{k}t}-1)+\frac{g^{*}_{k}}{\omega_{k}}b_{k}(e^{-i\omega_{k}t}-1)\biggr{)}}.$

In order to convert double exponential in a single exponential to use useful
fact $\text{Tr}_{B}[\rho_{B}e^{C}]=e^{\langle C^{2}\rangle/2}$, where operator
$C$ is a linear combination of annihilation and creation operators; we use the
identity $e^{X}e^{Y}=e^{X+Y+\frac{1}{2}[X,Y]}$. Fortunately in this case,
first commutator is a constant number, so higher order commutators are zero.
Finally we have

$C_{12}(t)=e^{-i\Phi_{I}(t)}e^{-\Phi_{R}(t)}.$

where $\Phi_{I}(t)$ and $\Phi_{R}(t)$ have been defined in the main text.
Carrying on further,we have

$\Gamma(\tau)=\dfrac{\Delta^{2}}{2\tau}\int_{0}^{\tau}dt\int_{0}^{t}dt^{\prime}e^{-\Phi_{R}(t^{\prime})}\cos[\zeta(t)-\zeta(t-t^{\prime})-\Phi_{I}(t^{\prime})].$
(34)