1. Introduction
The field of open quantum systems [1] presents the most fascinating challenges, not only from a fundamental perspective but also due to their relevance in practical applications and experimental implementations [2–5]. This class of problems provides a captivating interaction of coherent Hamiltonian dynamics with the incoherent dissipative dynamics resulting from the bath [6, 7]. From a fundamental perspective, dissipative processes are now viewed as a resource that enables the engineering of quantum states of matter [6–8] and the facilitation of quantum transport features [9, 10], after previously being thought of as a nuisance since they destroy coherence [11–13]. Particle losses have been a particularly important kind of dissipative process, and recent study into cold atomic systems has made them extremely controllable. For instance, this is the situation when local losses are realized in fermionic systems [14] or weakly interacting Bose gases [15–17].
The damped harmonic oscillator is a fundamental model for studying dissipative processes in quantum systems [18–20]. It describes the interplay between a system’s inherent oscillatory motion and its interaction with an environment, often represented by a bath of harmonic oscillators [1]. This coupling leads to energy dissipation and decoherence, which are central to understanding open quantum systems. The dynamics of a damped oscillator provides an insight into such phenomena as relaxation, correlation functions, and steady-state behaviors [3, 21]. The coupling of a harmonic oscillator to a non-Markovian environment [22] gives rise to complex behavior, including intricate interactions between the system and the environment, and the emergence of non-classical effects [23]. This is due to memory effects and information backflow from the environment to the system [24], which leads to non-exponential relaxation of the quantum coherence [25] and modified steady-state properties [26]. All these are essential to understand and control the applications of quantum systems, which include quantum information processing [27] and quantum thermodynamics [28].
There has been a growing interest in studying the Markovian and non-Markovian dynamics in open quantum systems due their important applications in quantum information science [24, 25, 29, 30]. Non-Markovian effects arise due to the backflow of information from the bath to the system while Markovian effects see no retrieval of information from the bath [3, 24]. In this work, we consider a quantum system represented by a bosonic mode subject to the decay via Lindblad dynamics. We call this system a damped oscillator. Such a system is a variant of the Dicke model in the large N-limit [31, 32]. Furthermore, we couple the system to a non-Markovian bath with Lorentzian spectral density. We consider both non-rotating and rotating wave approximations of the model and analyze various properties in the steady state of the system like distribution function, effective temperature, quantum Zeno (QZ) and anti-Zeno (AQZ) effects [33, 34]. The QZ effect describes a situation where the dynamics of a given system can be frozen with continuous measurements while the AQZ effect accelerates the induced dynamics. [35, 36]. Therefore, it would be interesting to look for non-Markovian effects on the steady properties of the system and find the transition from QZ to AQZ by tuning non-Markovinity in the system.
The theoretical study of dissipative phenomena poses a significant challenge, and various methods have been employed to address the interaction of the system with its environment (bath) [1–6]. A commonly used approach involves utilizing non-Hermitian Hamiltonians [37–39]. These non-Hermitian models can be subjected to a variety of field-theoretic and numerical techniques [37, 38], such as bosonization and renormalization group analysis [39, 40]. However, it is challenging to determine how accurately these treatments can capture the dynamics of the system observables because they frequently ignore or, at most, address the quantum-jump element of the Lindblad master equation in an approximate manner. Other methods assume that the bath is Markovian and address the entire Lindblad master equation [9, 41, 42]. In this work, we focus on the Schwinger–Keldysh (SK) functional technique to tackle the non-equilibrium dynamics [43–45]. Such an approach has found its applications in condensed matter systems [46–48], cosmology [49, 50] and string theory [51, 52].
This paper is organized in the following way. Section 2 introduces model calculations in a non-rotating wave approximation, followed by section 3 on the quantum Zeno effect. Section 4 deals with rotating wave approximation. Finally, we conclude in section 5 .
2. Model calculations
The dynamics of a dissipative quantum system is usually given by Lindblad equation (ℏ = 1):
$\begin{eqnarray}\frac{{\rm{d}}\rho }{{\rm{d}}t}=-{\rm{i}}[H,\rho ]+{ \mathcal L }(\rho ).\end{eqnarray}$
In this equation, ρ represents the density matrix for the system under consideration. The term − i[H, ρ] represents the coherent evolution of the system while the term ${ \mathcal L }(\rho )$ describes the dissipation in the system due to various processes. For the Lindblad dynamics, it has the form $\begin{eqnarray}{ \mathcal L }(\rho )=\displaystyle \sum _{i}[2{L}_{i}\rho {L}_{i}^{\dagger }-\{{L}_{i}^{\dagger }{L}_{i},\rho \}],\end{eqnarray}$
with Li as the Lindblad operator. In our case, we assume the Lindblad operator $L=\sqrt{{{\rm{\Gamma }}}_{M}}a$ (a is the annihilation operator of the system) corresponding to particle loss at rate ΓM, so that we can write ${ \mathcal L }(\rho )={{\rm{\Gamma }}}_{M}\,[2a\rho {a}^{\dagger }-\{{a}^{\dagger }a,\rho \}]$. We now consider our system, the damped harmonic oscillator modeled as a bosonic mode subject to Lindblad dynamics with $L=\sqrt{{{\rm{\Gamma }}}_{M}}a$. Also, we couple this system to a non-Markovian bath. Such a system is a variant of the Dicke Model. The total Hamiltonian in the non-rotating wave approximation can be written as follows $\begin{eqnarray}\begin{array}{rcl}H & = & {H}_{A}+{H}_{B}+{H}_{AB}={\omega }_{0}{a}^{\dagger }a\\ & & +\displaystyle \sum _{k}{\omega }_{k}{b}_{k}^{\dagger }{b}_{k}+\displaystyle \sum _{k}(a+{a}^{\dagger })({g}_{k}{b}_{k}+{g}_{k}^{* }{b}_{k}^{\dagger }),\end{array}\end{eqnarray}$
HA describes the free Hamiltonian of the system with annihilation (creation) operators as a (a†) and ω0 as the characteristic frequency of the bosonic mode. The second term HB describes the bath with annihilation (creation) operators bk (${b}_{k}^{\dagger }$) for the k-th bath mode and energy ωk. The final term HAB captures the interaction of the system with bath and gk denotes the coupling strength of k-th bath mode and system. In the rotating wave approximation, the terms abk and ${a}^{\dagger }{b}_{k}^{\dagger }$ are ignored. This model can be obtained from the large N-limit of the Dicke model, where an ensemble of two level atoms are interacting with a bath [31, 32, 53]. The system part HA therefore represents the collective bosonic mode as a result of the thermodynamic limit. This model has been realized in various settings like dissipative Bose–Einstein condensates [54], multilevel atom-schemes [55, 56], etc.2.1. Keldysh field theory
In this section, we use SK field theoretic technique to investigate the dynamics and calculate various observables in our model. The idea of the SK functional technique is to evaluate the time evolution on a closed time contour with a forward and backward branch, such that the value of the fields match at time t = ∞. Let ∣Φ⟩ be the coherent state representing some general quantum system, then the SK action is defined through the partition function [43–45]16 ) vanishes, i.e. ω = 0 and it occurs at ${\alpha }_{c}=\frac{2[{{ \mathcal Q }}^{2}+1][{({ \mathcal Q })}^{2}+{{ \mathcal R }}^{2}]}{{{ \mathcal Q }}^{2}}$.
$\begin{eqnarray}Z=\int {\rm{D}}[{{\rm{\Phi }}}_{+}^{* },{{\rm{\Phi }}}_{+},{{\rm{\Phi }}}_{-}^{* },{{\rm{\Phi }}}_{-}]{{\rm{e}}}^{{\rm{i}}{S}_{{\rm{S}}{\rm{K}}}[{{\rm{\Phi }}}_{+}^{* },{{\rm{\Phi }}}_{+},{{\rm{\Phi }}}_{-}^{* },{{\rm{\Phi }}}_{-}]},\end{eqnarray}$
where the integration measure is given by $\begin{eqnarray}{\rm{D}}[{{\rm{\Phi }}}_{+}^{* },{{\rm{\Phi }}}_{+},{{\rm{\Phi }}}_{-}^{* },{{\rm{\Phi }}}_{-}]=\mathop{\mathrm{lim}}\limits_{N\to \infty }\displaystyle \prod _{n=0}^{N}\frac{{\rm{d}}{{\rm{\Phi }}}_{+}^{* }{\rm{d}}{{\rm{\Phi }}}_{+}}{\pi }\frac{{\rm{d}}{{\rm{\Phi }}}_{-}^{* }{\rm{d}}{{\rm{\Phi }}}_{-}}{\pi }.\end{eqnarray}$
SSK represents the SK action and the fields on a forward and backward branch of the Keldysh contour are given by Φ+, Φ− respectively. Φ* is the complex conjugate of Φ. This action can be further simplified using the Keldysh rotation by defining new fields ${{\rm{\Phi }}}_{c}=\frac{{{\rm{\Phi }}}_{+}+{{\rm{\Phi }}}_{-}}{\sqrt{2}}$ and ${{\rm{\Phi }}}_{q}=\frac{{{\rm{\Phi }}}_{+}-{{\rm{\Phi }}}_{-}}{\sqrt{2}}$. Φc, Φq are known as classical and quantum fields, respectively. This is attributed to the fact that Φc has a non-vanishing expectation value, while Φq does not have one. The same is true for conjugate fields. Next, we simplify our problem using the coherent states for the system and the bath. Let ∣ζ⟩ be the coherent state representing the system, i.e. let a∣ζ⟩ = ζ∣ζ⟩ and ∣ηk⟩ be the coherent state representing the state of bath, then we can write the SK action in Fourier space as $\begin{eqnarray}S={S}_{\zeta }+{S}_{\eta }+{S}_{\zeta \eta }.\end{eqnarray}$
Sζ is the action for the system and can be written as $\begin{eqnarray}{S}_{\zeta }=\int {\rm{d}}\omega \left(\begin{array}{c}{\zeta }_{c}^{* },{\zeta }_{q}^{* }\end{array}\right)\left(\begin{array}{cc}0 & \omega -{\omega }_{0}-{\rm{i}}{{\rm{\Gamma }}}_{M}\\ \omega -{\omega }_{0}-{\rm{i}}{{\rm{\Gamma }}}_{M} & 2{\rm{i}}{{\rm{\Gamma }}}_{M}\end{array}\right)\left(\begin{array}{c}{\zeta }_{c}\\ {\zeta }_{q}\end{array}\right),\end{eqnarray}$
where ζc/q = ζc/q(ω). For the bath, we have $\begin{eqnarray}\begin{array}{rcl}{S}_{\eta } & = & \displaystyle \sum _{k}\displaystyle \int {\rm{d}}\omega \left(\begin{array}{c}{\eta }_{kc}^{* },{\eta }_{kq}^{* }\end{array}\right)\left(\begin{array}{cc}0 & \omega -{\omega }_{k}-{\rm{i}}\epsilon \\ \omega -{\omega }_{k}-{\rm{i}}\epsilon & 2{\rm{i}}\epsilon \end{array}\right)\\ & & \times \,\left(\begin{array}{c}{\eta }_{kc}\\ {\eta }_{kq}\end{array}\right).\end{array}\end{eqnarray}$
Here, also ηkc/q = ηkc/q(ω). The ε → 0+ serves as a regularization parameter. The interaction between the system and the bath is represented through Sζη and is given by $\begin{eqnarray}\begin{array}{rcl}{S}_{\zeta \eta } & = & -\displaystyle \sum _{k}\displaystyle \int {\rm{d}}\omega \left[{g}_{k}({\zeta }_{{\rm{c}}}^{* }(\omega ){\eta }_{kq}(\omega )+{\zeta }_{q}^{* }(\omega ){\eta }_{k{\rm{c}}}(\omega )\right.\\ & & \left.+{\zeta }_{{\rm{c}}}(\omega ){\eta }_{kq}(-\omega )+{\zeta }_{q}(\omega ){\eta }_{k{\rm{c}}}(-\omega ))+{\rm{c}}.{\rm{c}}.\right],\end{array}\end{eqnarray}$
where c. c. means complex conjugate. We now integrate out the η-fields using the Gaussian integration to get the effective action for the ζ-fields, i.e. for the system: ${Z}_{{\rm{eff}}}\,=\int D[{\eta }_{c,q}^{* },{\eta }_{c,q}]{{\rm{e}}}^{{\rm{i}}{S}_{{\rm{eff}}}[{\eta }_{c}^{* },{\eta }_{q}^{* },{\eta }_{c},{\eta }_{q}]}$ where $\begin{eqnarray}{S}_{{\rm{eff}}}=\int {\rm{d}}\omega \,{\zeta }_{4}^{\dagger }(\omega )\left(\begin{array}{cc}0 & {{ \mathcal D }}^{{\rm{ad}}}(\omega )\\ {{ \mathcal D }}^{{\rm{re}}}(\omega ) & {{ \mathcal D }}^{K}(\omega )\end{array}\right){\zeta }_{4}(\omega ).\end{eqnarray}$
Here, the four component vector ${\zeta }_{4}(\omega )=[{\zeta }_{c}(\omega ),{\zeta }_{c}(-\omega ),{{\zeta }_{q}(\omega ),\zeta (-\omega )]}^{{\rm{T}}}$, T is transpose. ${{ \mathcal D }}^{{\rm{re}}},\,{{ \mathcal D }}^{{\rm{ad}}},{{ \mathcal D }}^{K}$ represent respectively, the inverse of the retarded, advanced and Keldysh Green’s functions. These Green’s functions can be explicitly written as $\begin{eqnarray}{{ \mathcal D }}^{K}=2{\rm{i}}{{\rm{\Gamma }}}_{M}\,{\rm{diag(1,\; 1)}}\end{eqnarray}$
and $\begin{eqnarray}{{ \mathcal D }}^{{\rm{re}}}=\left(\begin{array}{cc}\omega -{\omega }_{0}+{\rm{i}}{{\rm{\Gamma }}}_{M}+\sum (\omega ) & \sum (\omega )\\ {\left[\sum (-\omega )\right]}^{* } & -\omega -{\omega }_{0}-{\rm{i}}{{\rm{\Gamma }}}_{M}+{\left[\sum (-\omega )\right]}^{* }\end{array}\right).\end{eqnarray}$
Σ(ω) represents the self energy and its explicit form is given by $\begin{eqnarray}{\rm{\Sigma }}(\omega )=-\frac{1}{2}\displaystyle \sum _{k}\frac{| {g}_{k}{| }^{2}{\omega }_{k}}{{\omega }^{2}-{{\omega }_{k}}^{2}}.\end{eqnarray}$
Next, we define the bath spectral density ${ \mathcal J }(\omega )\,={\sum }_{k}| {g}_{k}{| }^{2}\,\delta (\omega -{\omega }_{k})$, so that its form can be written phenomenologically as having Lorentzian shape [57] $\begin{eqnarray}{ \mathcal J }(\omega )=\frac{1}{2\pi }\frac{\alpha {\lambda }^{2}}{{(\omega -{\omega }_{0})}^{2}+{\lambda }^{2}}.\end{eqnarray}$
Here, α can be considered as the effective system-bath coupling, λ represents the spectral width of the bath. Therefore, Σ(ω) can be simplified to the form $\begin{eqnarray}\begin{array}{rcl}{\rm{\Sigma }}(\omega ) & = & -\frac{1}{2}\displaystyle \int {\rm{d}}{\omega }^{{\prime} }\frac{{\omega }^{{\prime} }\,{ \mathcal J }({\omega }^{{\prime} })}{{\omega }^{2}-{\omega }^{{\prime} 2}}\\ & = & \frac{\alpha \lambda {\omega }_{0}}{4}\left[\frac{{\omega }_{0}^{2}+{\lambda }^{2}-{\omega }^{2}}{{({\omega }^{2}+{\omega }_{0}^{2}+{\lambda }^{2})}^{2}-4{\omega }_{0}^{2}{\omega }^{2}}\right].\end{array}\end{eqnarray}$
Next, we obtain spectrum of the system from ${\rm{\det }}{{ \mathcal D }}^{{\rm{re}}}=0$, and we get the following dispersion relation $\begin{eqnarray}{\omega }_{\pm }=-{\rm{i}}{{\rm{\Gamma }}}_{M}\pm \sqrt{{\omega }_{0}^{2}-2{\omega }_{0}{\rm{\Sigma }}(\omega )}.\end{eqnarray}$
In order to plot these solutions, we first take note of the parameters involved. These parameters are an energy scale of the system ω0, spectral width of the bath λ, Markovian decay rate ΓM. The relaxation time scale for the system is given by ${\tau }_{S}\sim {\omega }_{0}^{-1}$, while the bath correlation time scale is τB ∼ λ−1. The competition of these scales determine the Markovian or non-Markovian behavior in the system. If τS > τB, the dynamics are Markovian while vice-versa implies non-Markovian effects. We therefore, define two new parameters ${ \mathcal Q }={\omega }_{0}/\lambda $ and ${ \mathcal R }=\lambda /{{\rm{\Gamma }}}_{M}$. Therefore, in terms of ${ \mathcal Q }$ and ${ \mathcal R }$, we identify ${ \mathcal Q }\lt \,\lt 1$, as the Markovian regime while the non-Markovian regime is ${ \mathcal Q }\gt \,\gt 1$, Additionally, a weaker condition but not necessary, can be imposed through values of ${ \mathcal R }$. For ${ \mathcal R }\gt 1$ would identify as Markovian, while ${ \mathcal R }\lt 1$ as a non-Markovian regime. In terms of these parameters, we rewrite the self energy function in the following (z → ω/λ): $\begin{eqnarray}{\rm{\Sigma }}(z)=\frac{\alpha { \mathcal Q }}{4}\left[\frac{{{ \mathcal Q }}^{2}-{z}^{2}+1}{{({z}^{2}+{{ \mathcal Q }}^{2}+1)}^{2}-4{{ \mathcal Q }}^{2}{z}^{2}}\right],\end{eqnarray}$
and the dispersion relation becomes $z=-{\rm{i}}{{ \mathcal R }}^{-1}\pm \sqrt{{{ \mathcal Q }}^{2}-2{ \mathcal Q }{\rm{\Sigma }}(z)}$. In the limiting cases where ${ \mathcal Q }\gt \,\gt 1$ and ${ \mathcal Q }\lt \,\lt 1$, we have $\begin{eqnarray}{\rm{\Sigma }}(z)=\left\{\begin{array}{ll}\frac{\alpha { \mathcal Q }}{4({{ \mathcal Q }}^{2}-{z}^{2})}\quad & { \mathcal Q }\gt \,\gt 1,\\ \frac{\alpha { \mathcal Q }}{4}[\frac{1-{z}^{2}}{{(1+{z}^{2})}^{2}}]\quad & { \mathcal Q }\lt \,\lt 1.\end{array}\right.\end{eqnarray}$
Now we see that for ${ \mathcal Q }\gt \,\gt 1$, the Σ(z) peaks around $z=\pm { \mathcal Q }$, i.e ω = ± ω0. Therefore, for a narrow bath spectral density, characteristic equation becomes $\begin{eqnarray}{z}^{4}+({{ \mathcal R }}^{-2}+2{{ \mathcal Q }}^{2}){z}^{2}-{{ \mathcal Q }}^{2}(\alpha +{{ \mathcal R }}^{-2})=0.\end{eqnarray}$
The roots of this equation are ${\omega }_{\pm }=\pm \sqrt{{z}_{\pm }}$ with ${z}_{\pm }=-0.5({{ \mathcal R }}^{-2}+2{{ \mathcal Q }}^{2})\pm 0.5\sqrt{{({{ \mathcal R }}^{-2}+2{{ \mathcal Q }}^{2})}^{2}+4{{ \mathcal Q }}^{2}(\alpha +{{ \mathcal R }}^{-2}).}$ Since the sign of the imaginary part of the eigenfrequency ω± provides the stability of a steady state solution. A negative imaginary part of ω± represents the stable solution while the positive part of ω± means an unstable solution. Therefore, there exists at least one root with a positive imaginary part, hence unstable. In case of ${ \mathcal Q }\lt \,\lt 1$, the bath has no substantial effects on the system and the spectrum remains unchanged at ω± ∼ − iΓM ± ω0. Furthermore, we define critical coupling αc to be the point where at least one of the roots of equation (In figure 1, we plot the real and imaginary parts of the roots of equation (16 ) with respect to ${ \mathcal Q }$ at large α values and ${ \mathcal R }=0.1,1,10$. The imaginary parts of the roots are shown by dotted lines while the solid lines correspond to the real part of the roots. We observe that there exists at least one root with a positive imaginary part (as shown in the inset of figure 1) signaling an instability in the system. This instability is reflected in the divergence of the number density. The system can be stabilized via tuning the values of ${ \mathcal R }$. As we progress from ${ \mathcal R }=0.1$ to ${ \mathcal R }=10$, the imaginary part of the unstable root changes sign from positive to negative as shown in the inset of figure 1(a)–(c). Therefore, in the non-rotating wave approximation, the condition ${ \mathcal R }\lt 1$ (non-Markovian) is unstable towards a transition.
Figure 1. We plot the real and imaginary parts of the roots of the characteristic equation ( |
2.2. Correlations
Next, we evaluate the steady state one point correlation function that yields the number density. The one point correlation function is related to the Keldysh Green's function in the following way16 ) (figure 1(a)–(c)). In the region ${ \mathcal R }\gt 1$, the system becomes stable for ${ \mathcal Q }\gt 1$ values and thus no region of transition. Thus, we conclude that the system can be tuned to phase transition via tuning system from Markovian to non-Markovian.
$\begin{eqnarray}C(t,{t}^{{\prime} })=\langle \{a(t),{a}^{\dagger }({t}^{{\prime} })\}\rangle ={\rm{i}}{G}^{K}(t,{t}^{{\prime} }),\end{eqnarray}$
where {A, B} = AB + BA define the anti-commutator. The Keldysh Green's function is obtained from ${{ \mathcal D }}^{K}$ using the formula ${G}^{K}=-{G}^{{\rm{re}}}{{ \mathcal D }}^{K}{G}^{{\rm{ad}}}$, with ${G}^{{\rm{re}}}(\omega )={[{{ \mathcal D }}^{{\rm{re}}}(\omega )]}^{-1}$ and ${G}^{{\rm{ad}}}(\omega )={[{G}^{{\rm{re}}}(\omega )]}^{* }$. In the steady state, the system becomes a time translation invariant and therefore the correlation function depends only on time differences $t-{t}^{{\prime} }$, which implies that for equal times, the correlation function $C(t,{t}^{{\prime} })=C(0)=\langle \{a,{a}^{\dagger }\}\rangle =2\langle n\rangle +1$; ⟨n⟩ = ⟨a†a⟩ is the average number density. Therefore, in the steady state, we compute the distribution function as follows: $\begin{eqnarray}2\langle n\rangle +1={\rm{i}}\int \frac{{\rm{d}}\omega }{2\pi }{G}_{11}^{K}(\omega ),\end{eqnarray}$
where the first diagonal entry of the Keldysh Green’s function is given by $\begin{eqnarray}{\rm{i}}{G}_{11}^{K}(\omega )=\frac{2{{\rm{\Gamma }}}_{M}[{(\omega +{\omega }_{0}-{\rm{\Sigma }})}^{2}+{{\rm{\Gamma }}}_{M}^{2}+{{\rm{\Sigma }}}^{2}]}{{({\omega }^{2}-{{\rm{\Gamma }}}_{M}^{2}-{\omega }_{0}^{2}+2{\omega }_{0}{\rm{\Sigma }})}^{2}+4{\omega }^{2}{{\rm{\Gamma }}}_{M}^{2}}.\end{eqnarray}$
In figure 2, we have plotted 2⟨n⟩ + 1 with respect to ${ \mathcal Q }$ for different ${ \mathcal R }$ values. The divergence in number density reflects the macroscopic occupation of the bosonic mode at different ${ \mathcal Q }$ values implying an instability in the system. Since the present model is the effective model obtained from the thermodynamic limit (N → ∞) of the Dicke model, this transition is well defined and the system enters into the superradiant phase with ⟨a⟩ ≠ 0 [58–60]. However, if the system is simply a single oscillator, this divergence reflects dynamic instability (not a phase transition) in the system. This instability can be attributed to the resonance effect at strong system-bath coupling [61]. Under the strong non-Markovian effects, the backflow of information leads to an uncontrolled energy dissipation causing the number density to diverge. This transition is pronounced at ${ \mathcal R }=0.1\lt 1$-value for large ${ \mathcal Q }$-values. As we increase the ${ \mathcal R }$-values, the transition point shifts to lower ${ \mathcal Q }$ and finally for ${ \mathcal R }\gt \,\gt 1$, the transition disappears. This result is consistent from the dispersion equation (
Figure 2. In this figure, we plot 2⟨n⟩ + 1 with respect to ${ \mathcal Q }$ for different values of ${ \mathcal R }$. As we progress through ${ \mathcal R }\lt 1$ to ${ \mathcal R }\gt \,\gt 1$, the transition point changes with no transition in steady state for ${ \mathcal R }\gt 1$. |
Next, we look at the fluctuation-dissipation relationship in the steady state. This is reflected through equation [62]23 ), we get ${ \mathcal F }(\omega )\,={\sigma }_{z}+\frac{1}{\omega }{\rm{\Sigma }}(\omega ){\sigma }_{x}$. The effective temperature is calculated as the dimensional coefficient of 1/ω in the expansion of ${ \mathcal F }(\omega )$ in powers of 1/ω. Thus, we have an effective temperature Teff = α independent of ${ \mathcal Q }$ and ${ \mathcal R }$.
$\begin{eqnarray}{G}^{K}(\omega )={G}^{{\rm{re}}}(\omega )\circ { \mathcal F }(\omega )-{ \mathcal F }(\omega )\circ {G}^{{\rm{ad}}}(\omega ),\end{eqnarray}$
where ${ \mathcal F }(\omega )$ is called as the distribution function that has the form of $2\langle n\rangle +1=\coth \frac{\beta \omega }{2}$ in equilibrium. The eigenvalues of ${ \mathcal F }(\omega )$ provide the quantitative description of the effective temperature. By solving this equation (3. Quantum Zeno effect
The phenomena of quantum Zeno and anti-Zeno effects provide an enlightening framework linking measurement with environmental couplings and back-action dynamics due to such dissipative processes [63, 64]. The QZ effect describes freezing out the dynamics while the AQZ effect represents the enhancement of dynamic evolution induced by frequent measurements [65]. Different experiments realized the QZ effects, for example systems like trapped ions [63], superconducting qubits [66], Ultracold atoms [67], semiconductor systems. QZ effects have been used to extend the lifetime of molecular states, improve the spectroscopic precision [68], etc. It has also been proposed that the Zeno effect could be used to preserve the entanglement between two atoms [69]. The goal of these studies has been to comprehend how measurements affect quantum system dynamics and how this knowledge may be used to manipulate and control quantum states.
In the steady state, we can quantify the Zeno effect through the Zeno parameter defined as follows [70, 71]:
$\begin{eqnarray}\xi =\frac{\langle n\rangle -\langle n\rangle {| }_{\alpha =0}}{\langle n\rangle },\end{eqnarray}$
where $\left\langle n\right\rangle $ is the steady state number density. In Keldysh formalism, the Zeno parameter ξ can be written as: $\begin{eqnarray}\begin{array}{rcl}\xi & = & \frac{[2\langle n\rangle +1]-[2\langle n\rangle {| }_{\alpha =0}+1]}{[2\langle n\rangle +1]-1},\\ & = & \frac{{\rm{i}}\displaystyle \int {\rm{d}}\omega \left[\right.{G}_{11}^{K}(\omega ,\alpha )-{G}_{11}^{K}(\omega ,\alpha =0)\left]\right.}{{\rm{i}}\displaystyle \int {\rm{d}}\omega {G}_{11}^{K}(\omega ,\alpha )-1}.\end{array}\end{eqnarray}$
The values of ξ determine whether the dynamics are Zeno or anti-Zeno in nature. A positive (negative) value of ξ indicates the anti-Zeno (Zeno) effect [72], i.e. there is an enhancement (suppression) in the average photon numbers associated with the bosonic mode due to its coupling α with the bath.In figure 3(a), we plot the steady state Zeno parameter ξ (non-rotating wave approximation), with respect to ${ \mathcal Q }$ for different values of ${ \mathcal R }$. From this figure we observe that for small ${ \mathcal R }=0.1$, dynamics of the system enters first into the anti-Zeno region (ξ > 0) then followed by a transition to the Zeno effect (ξ < 0). As we further increase ${ \mathcal Q }$-values, the system does not evolve to particular dynamics, i.e. ξ = 0. The inset of figure 3(a) represents the variation of ξ with ${ \mathcal Q }$ for ${ \mathcal R }=1,10$. We observe in the case of ${ \mathcal R }=1$, the system initially has Zeno dynamics (ξ < 0) while for ${ \mathcal R }=10$, the system posses anti-Zeno (ξ > 0) dynamics, and as we tune ${ \mathcal Q }$ to large values, these effects die out without showing any transition as shown for the case of ${ \mathcal R }=0.1$. This behavior can be attributed to the collective role played by the non-Markovianity induced by the bath; and the competition between dissipative and coherent dynamics in the system. These effects are characterized through the parameters ${ \mathcal Q }={\omega }_{0}/\lambda $ and ${ \mathcal R }=\lambda /{{\rm{\Gamma }}}_{M}$. The parameter ${ \mathcal R }$ in the case of fixed dissipation rate ΓM, its large values reflect the broader spectral width λ − a Markovian nature. Therefore, as the system enters into Markovian dynamics, the backflow of information weakens leading to slower dynamics. This implies that the Markovian regime system does not show any QZ or AQZ effect. While in the non-Markovian case ${ \mathcal Q }\gt 1,{ \mathcal R }\lt 1$, there is a continuous backflow of information to the system leading to transitions between the AQZ and QZ regimes.
Figure 3. These plots depict the steady state Zeno parameter ξ in (a) non-rotating wave approximation (nRWA) (b) rotating wave approximation (RWA). We plot the Zeno parameter ξ with respect to ${ \mathcal Q }$ for ${ \mathcal R }=0.1,1,10$. (a) We can see for ${ \mathcal R }=0.01$, the system enters into the anti-Zeno region (ξ > 0) first, then making the transition to the Zeno region with ξ < 0. In case of nRWA, we observe that (shown in inset of (a)) for ${ \mathcal R }=1$ values, the system starts in the Zeno region, while for large ${ \mathcal R }=10$, we have an anti-Zeno effect that finally vanishes as ${ \mathcal Q }$ increases. (b) In the RWA case, ${ \mathcal R }\gt 1$, the system starts initially in the anti-Zeno region and finally transitions to the Zeno region. However, ${ \mathcal R }=0.1$, the system only shows the Zeno effect. |
4. Rotating wave approximation
In this section, we consider the RWA Hamiltonian given by
$\begin{eqnarray}H={H}_{A}+{H}_{B}+\displaystyle \sum _{k}({g}_{k}a{b}_{k}^{\dagger }+{g}_{k}^{* }{a}^{\dagger }{b}_{k}),\end{eqnarray}$
and compare the results with that of non-RWA. After tracing out the bath degrees of freedom as done earlier, we arrive at the diagonal inverse retarded Green's function ${{ \mathcal D }}_{{\rm{RWA}}}^{{\rm{re}}}$: $\begin{eqnarray}{{ \mathcal D }}_{{\rm{RWA}}}^{{\rm{re}}}=\left(\begin{array}{cc}\omega -{\omega }_{0}+{\rm{i}}{{\rm{\Gamma }}}_{M}+\tilde{{\rm{\Sigma }}}(\omega ) & 0\\ 0 & -\omega -{\omega }_{0}-{\rm{i}}{{\rm{\Gamma }}}_{M}+{[\tilde{{\rm{\Sigma }}}(-\omega )]}^{* }\end{array}\right).\end{eqnarray}$
In RWA, the self energy $\tilde{{\rm{\Sigma }}}(\omega )$ is given by $\begin{eqnarray}\tilde{{\rm{\Sigma }}}(\omega )=-\frac{1}{2}\displaystyle \sum _{k}\frac{| {g}_{k}{| }^{2}}{\omega -{\omega }_{k}}.\end{eqnarray}$
In terms of parameters ${ \mathcal Q }$ and ${ \mathcal R }$, this self energy can be simplified to the form $\begin{eqnarray}\tilde{{\rm{\Sigma }}}(z)=\frac{\alpha }{2}\frac{{ \mathcal Q }-z}{{({ \mathcal Q }-z)}^{2}+1}.\end{eqnarray}$
The dispersion relations in terms of ${ \mathcal Q }$ and ${ \mathcal R }$ are given by $\begin{eqnarray}\begin{array}{rcl}z+{\rm{i}}{{ \mathcal R }}^{-1}-{ \mathcal Q }+\tilde{{\rm{\Sigma }}}(z) & = & 0,\\ z+{\rm{i}}{{ \mathcal R }}^{-1}+{ \mathcal Q }-{[\tilde{{\rm{\Sigma }}}(-z)]}^{* } & = & 0.\end{array}\end{eqnarray}$
In figures 4(a)–(c), we plot the roots of the dispersion relation equation (30 ) as a function of ${ \mathcal Q }$ for ${ \mathcal R }=0.1,1,10$. We observe that there exists at least one root with the positive imaginary part making it unstable. Since ${ \mathcal R }\lt 1$ identifies as a non-Markovian region, from figure 4(a), the positive imaginary part vanishes at a value ${ \mathcal Q }\gt 1$ and at this value of ${ \mathcal Q }$, its real part becomes non-zero (positive). As we move away from ${ \mathcal R }\lt 1$ to ${ \mathcal R }\gt 1$, in figures 4(b)–(c), we see that the same root does not vanish for very large values of ${ \mathcal Q }$, thus making the system unstable. Therefore, in the rotating wave approximation, the non-Markovian regime ${ \mathcal Q }\gt 1,{ \mathcal R }\lt 1$ describes the stable state of the system while the Markovian dynamics implies instability in the system.
Figure 4. We plot the real and imaginary parts of the roots of the characteristic equation ( |
This instability would imply a divergence in the number density reflecting the phase transition in the system. However, if we calculate the effective temperature of the system using the same procedure as above, we get ${ \mathcal F }={\sigma }_{z}$, σz as the Pauli spin matrix. This implies that we have Teff = 0. Thus, in comparison to the non-rotating wave approximation where the steady state has a non-zero local effective temperature, the system in case of RWA thermalizes at Teff = 0.
Zeno Effect: Since the rotating wave approximation Hamiltonian has a different symmetry in comparison to the non-RWA counter part. Therefore, we look at the behavior of the Zeno effect in RWA under different parameter values. We now plot in figure 3(b), the Zeno parameter ξ as given in equation (25 ) in the rotating wave approximation as a function of quality factor ${ \mathcal Q }$ for different values of ${ \mathcal R }$. We observe in the non-Markovian regime ${ \mathcal Q }\gt 1,{ \mathcal R }\lt 1$, the system shows the quantum Zeno effect for a particular value of ${ \mathcal Q }$, while in the regions with ${ \mathcal R }=1,10$, the system starts the dynamics in the anti-Zeno regime with a transition to the Zeno regime. Compared to non-RWA, where the Zeno and anti-Zeno effect were absent in the Markovian regime, RWA exhibits an anti-Zeno effect in the Markovian regime. The transition from the AQZ to QZ regimes is observed in the RWA for all considered values of ${ \mathcal R }$.
5. Conclusions
In conclusion, we have studied the damped harmonic oscillator represented by a decaying bosonic mode coupled to a non-Markovian bath described by Lorentzian spectral density. Using the Schwinger–Keldysh formalism, an effective action for the system is obtained after integrating out the bath degrees of freedom. The steady state properties of the system were studied in rotating, as well in non-rotating wave approximations. We characterized the dynamics through two parameters ${ \mathcal Q }={\omega }_{0}/\lambda $ (quality factor) and ${ \mathcal R }=\lambda /{{\rm{\Gamma }}}_{M}$. Using these parameters, we identified ${ \mathcal Q }\gt 1,{ \mathcal R }\lt 1$ as non-Markovian while the ${ \mathcal Q }\lt 1,{ \mathcal R }\gt 1$ as Markovian regimes of the dynamics. A detailed study of the spectrum of the system in non-RWA and RWA revealed that the system undergoes an instability towards a phase transition in the former case for non-Markovian region, while in the latter case (RWA), the Markovian region is unstable. Therefore, tuning the quality factor ${ \mathcal Q }$, the system can be stabilized under both dynamics. Furthermore, we have shown that the system in non-RWA thermalizes at non zero effective temperature set by intrinsic system-bath coupling α independent of the ${ \mathcal Q }$ and ${ \mathcal R }$ while in the case of RWA, this effective temperature vanishes.
Next, we studied the steady state quantum Zeno and anti-Zeno effects characterized by Zeno parameter ξ. We have shown that in case of non-RWA, for ${ \mathcal R }\gt 1$ values the system dynamics is initially in the Zeno regime while for ${ \mathcal R }=1$ values, it is in anti-Zeno regime. As we increase the quality factor ${ \mathcal Q }$ for these ${ \mathcal R }$-values, the system shows no Zeno or anti-Zeno effects. Also, to complete the non-Markovian case ${ \mathcal Q }\gt 1,{ \mathcal R }\lt 1$, the system enters into anti-Zeno at a particular value of ${ \mathcal Q }$, then making a transition to the Zeno region. However, in the case of RWA, we observed that the dynamics are initially anti-Zeno for ${ \mathcal R }\geqslant 1$ and finally make a transition to the Zeno region. While the purely non-Markovian dynamics in the case of RWA have no appreciable anti-Zeno effect. Thus, the non-Markovinity parameters ${ \mathcal Q },{ \mathcal R }$ play an important role in tuning the system dynamics [73].