Probing nonclassicality under dissipation

Quantum physics manifests nonclassical correlations through the violation of Bell and Leggett–Garg inequalities. A renewed interest in the investigation of Leggett–Garg inequalities has gained momentum within past few years. The original motivation of the seminal work [1] by Leggett and Garg was to probe quantum coherence in macroscopic systems [2]. It is generally believed that the quantumness of a large system is destroyed by the many-body interactions with a noisy environment, which is broadly termed as decoherence. Inspired by this fact, Leggett–Garg inequalities can play as an indicator of nonclassicality for open quantum systems in a dissipative environment [3]. Two classical assumptions are made in deriving the Leggett–Garg inequalities (a₁) macrorealism: macroscopic systems are always in a definite state with well-defined pre-existing value, and (a₂) noninvasive measurability: this pre-existing value can be measured in a non-invasive way, that is, without disturbing the subsequent dynamics of the system. Violation of Leggett–Garg inequalities in the quantum regime indicate nonclassical behavior of the system due to the existence of superposition states violating assumption (a₁) and/or due to the measurement induced collapse of state that violates assumption (a₂). The first experimental violation of LGI was demonstrated in [4] after which the LGI violation was probed in a diverse range of physical systems, for example, photonic systems [5–8], nuclear magnetic resonance [9, 10], phosphorus impurities in silicon [11], nitrogen-vacancy defect in diamond [12], and most recently in superconducting flux qubit [13]. Also, quantum violation of LGI has been studied theoretically in a variety of systems like electrons in quantum dot [14–16], optomechanical system [17], using quantum nondemolition measurement applied to atomic ensemble [18], oscillating neutral kaons and neutrino oscillations [19, 20], and even in biological light-harvesting protein complex [21, 22].

Violation of LGI [23] is associated to the nonclassical dynamics and measurement correlations in the quantum system. Realistic quantum systems are open to unavoidable interaction with its surrounding environment that act as a source of decoherence and dissipation, resulting to the loss of quantumness of the system. This loss of quantumness of the open quantum system can be probed through the LGI that acts as witness of nonclassicality. The two-time correlation functions are experimentally measurable quantities and probing nonclassical signature through them is promising for open systems. In the past, the LGI violation has been discussed for closed systems [1, 2, 6–13, 18–20]. The violations of the LGI was also investigated for open systems in the Born–Markov limit [4, 5, 14–17, 21–26]. Born approximation assumes weak system-environment coupling strength to justify a perturbative approach, while Markov approximation assumes the correlation time of the reservoir to be very short, justifying a δ-correlated reservoir correlation function [3, 28, 29]. However non-Markovian effects are typically significant in many different contexts, ranging from solid state to hybrid quantum systems, quantum biology, quantum optics, and quantum chemistry [30, 31].

Earlier, we have investigated the violation of the Leggett–Garg inequality for a two level system under decoherence in a non-Markovian dephasing environment [32]. Here we extend these results to a dissipative system-environment coupling outside the Born–Markov regime. In the present paper, we consider a two-level system (qubit) spontaneously decaying under a non-Markovian bosonic environment. The non-Markovian characteristic of the model is discussed elsewhere [30, 33–42] in great detail. We briefly discuss the model and the method to calculate the non-Markovian two-time correlation functions for the two-level system and its dynamical loss of quantumness through Leggett–Garg inequalities. Our analysis is exact and valid both for weak and strong system-environment couplings, as we have not performed either the Born or the Markov approximation [28, 29]. We present our numerical results to investigate the Leggett–Garg inequalities in various system-environment parameter regime. We show that the non-Markov effect plays an important role to the dynamics of Leggett–Garg inequalities. We observe that quantum violation of LGI becomes robust against the system-environment coupling when the environment parameters are suitably tuned in the non-Markov regime. Finally, a conclusion is given at the end.

We consider a two-level system (qubit) spontaneously decaying under an environment having a continuum of modes. The total Hamiltonian of the system plus environment within the rotating wave approximation is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{TotH} H = H_{S} + H_{E} + H_{I} \nonumber \end{align} \tag{ 1 }$$

where $H_S=\hbar \omega_0 \sigma_{+}\sigma_{-}$ describes the two-level system. The operators $\sigma_{+} = \vert 1\rangle_S\langle 0\vert$ and $\sigma_{-} = \vert 0 \rangle_S\langle 1\vert$ with ground state $\vert 0\rangle_S$, excited state $\vert 1\rangle_S$, and transition frequency $\omega_0$. The environment Hamiltonian $H_E = \sum_k \hbar \omega_k b_k^{\dagger} b_k$, which describes a collection of harmonic oscillators with Bosonic operators $b_k^{\dagger}$ and b_k. The interaction Hamiltonian is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{IntH} H_{I} = \sum_k \hbar \left(g_k \sigma_{+} \otimes b_k + g_k^{\ast} \sigma_{-} \otimes b_k^{\dagger} \right). \nonumber \end{align} \tag{ 2 }$$

Next, we go the interaction picture with respect to $H_0 = H_S + H_E$, the time evolution of the total system-plus-environment state in the interaction picture

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{Schrod} \frac{\rm d}{{\rm d}t} \vert {\tilde \Psi} (t)\rangle = -\frac{\rm i}{\hbar} {\tilde H}_I(t) \vert {\tilde \Psi} (t)\rangle, \nonumber \end{align} \tag{ 3 }$$

where

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{IntH2} {\tilde H}_I(t) = \hbar \left[ \sigma_{+} \otimes B(t) + \sigma_{-} \otimes B^{\dagger}(t) \right] \nonumber \end{align} \tag{ 4 }$$

is the interaction picture operator ${\tilde H}_I(t) = {\rm e}^{\frac{{\rm i} H_0 t}{\hbar}} H_I {\rm e}^{-\frac{{\rm i} H_0 t}{\hbar}}$ with $B(t) = \sum_k g_k b_k {\rm e}^{{\rm i}(\omega_0 - \omega_k) t}$. We start with an initial product state $\vert \Psi(0)\rangle = \left(c_0 \vert 0\rangle_{S} + c_1(0) \vert 1\rangle_{S} \right) \otimes \vert 0\rangle_{E}$, where the environment is initially in the vacuum state $\vert 0\rangle_E$. The interaction Hamiltonian conserves the total particle number, the Schrödinger equation generated by ${\tilde H}_I(t)$ will be confined to the subspace spanned by the vectors $\vert 0\rangle_{S} \otimes \vert 0\rangle_{E}$, $\vert 1\rangle_{S} \otimes \vert 0\rangle_{E}$, and $\vert 0\rangle_{S} \otimes \vert k\rangle_{E}$. The exact time evolution of $\vert \Psi(0)\rangle$ is given by [28]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{tPsi} \vert {\tilde \Psi}(t)\rangle = c_0 \vert 0\rangle_{S} \otimes \vert 0\rangle_{E} + c_1(t) \vert 1\rangle_{S} \otimes \vert 0\rangle_{E} + \sum_k c_k (t) \vert 0\rangle_{S} \otimes \vert k\rangle_{E} \nonumber \end{align} \tag{ 5 }$$

where $\vert k\rangle_{E}=b_k^{\dagger}\vert 0\rangle_{E}$ is the state with one particle in mode k. Note that the amplitude c₀ is constant in time because ${\tilde H}_I(t) \vert 0\rangle_{S} \otimes \vert 0\rangle_{E}=0$. Substituting $\vert {\tilde \Psi}(t)\rangle$ from equation (5) into the Schrödinger equation (3), one can obtain an integrodifferential equation for $c_1(t)$ as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{amp1N} \frac{\rm d}{{\rm d}t} c_1(t) = - \int_{0}^{t} {\rm d}\tau f(t-\tau) c_1(\tau), \nonumber \end{align} \tag{ 6 }$$

where $f(t-\tau)=\langle 0\vert B(t) B^{\dagger}(\tau) \vert 0\rangle_{E}$ is the two-time correlation function of the reservoir and is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f(t-\tau) = \sum_k \vert g_k\vert ^2 {\rm e}^{{\rm i}(\omega_0-\omega_k)(t-\tau)}. \nonumber \end{align} \tag{ 7 }$$

For a continuous distribution of environmental modes described by $\mathcal{P}(\omega)$, the correlation function can be expressed through the spectral density $J(\omega)$ of the environment:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle f(t-\tau) = \int {\rm d}\omega J(\omega) {\rm e}^{{\rm i}(\omega_0 - \omega)(t-\tau)}, \nonumber \end{align} \tag{ 8 }$$

where $J(\omega) = \sum_k \vert g_k\vert ^2 \delta(\omega-\omega_k) = \mathcal{P}(\omega) \vert g(\omega)\vert ^2$ and $g(\omega)$ is the frequency-dependent coupling. The spectral density $J(\omega)$ contain all the information about the distribution of environmental modes and also the coupling between the system and the environment. For an initial state $\rho_{T}(0) = |\Psi(0)\rangle \langle \Psi(0)|$, the reduced density operator of the system in the interaction picture ${\tilde \rho}_{S} (t) = {\rm Tr}_E \{\vert {\tilde \Psi}(t)\rangle \langle {\tilde \Psi}(t)\vert \}$ is determined by the function $c_1(t)$. We can calculate the exact two-time correlation function $\langle \sigma_{+}(t_1) \sigma_{-}(t_2) \rangle_I$ in the interaction picture as follows

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nonumber \langle \sigma_{+}(t_1) \sigma_{-}(t_2) \rangle_I &= {\rm Tr}_{S \oplus E} \left({\tilde U}^{\dagger}(t_1) \sigma_{+} {\tilde U}(t_1) {\tilde U}^{\dagger}(t_2) \sigma_{-} {\tilde U}(t_2) \rho_T(0) \right) \nonumber \\ \nonumber &= {\rm Tr}_{S \oplus E} \left(\sigma_{+} {\tilde U}(t_1-t_2) \sigma_{-} {\tilde U}(t_2) \vert \Psi(0)\rangle \langle \Psi(0)\vert {\tilde U}^{\dagger}(t_1) \right) \nonumber \\ &= {\rm Tr}_{S \oplus E} \left(\sigma_{+} {\tilde U}(t_1-t_2) \sigma_{-} \vert {\tilde \Psi}(t_2)\rangle \langle {\tilde \Psi}(t_1)\vert \right) \label{Exact1} \nonumber \end{align} \tag{ 9 }$$

where ${\tilde U}(t)$ is the unitary time evolution operator generated by the Hamiltonian ${\tilde H}_I(t)$ and ${\tilde U}(t_1-t_2)={\tilde U}(t_1) {\tilde U}^{\dagger}(t_2)$. In equation (9), we substitute the time evolved wave functions $|{\tilde \Psi}(t_1)\rangle$ and $\vert {\tilde \Psi}(t_2)\rangle$ using equation (5) to finally obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \langle \sigma_{+}(t_1) \sigma_{-}(t_2) \rangle_I = c_1(t_2) c_1^{\ast}(t_1), \nonumber \end{align} \tag{ 10 }$$

where we have used $\sigma_{+}{\tilde U}(t_1-t_2) \sigma_{-} \vert {\tilde \Psi}(t_2)\rangle =c_1(t_2) \vert 1\rangle_{S} \otimes \vert 0\rangle_{E}$. Another two-time correlation function $\langle \sigma_{-}(t_1) \sigma_{+}(t_2) \rangle_I$ can also be obtained exactly as follows

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nonumber \langle \sigma_{-}(t_1) \sigma_{+}(t_2) \rangle_I &= {\rm Tr}_{S \oplus E} \left({\tilde U}^{\dagger}(t_1) \sigma_{-} {\tilde U}(t_1) {\tilde U}^{\dagger}(t_2) \sigma_{+} {\tilde U}(t_2) \rho_T(0) \right) \nonumber \\ \nonumber &= {\rm Tr}_{S \oplus E} \left(\sigma_{-} {\tilde U}(t_1-t_2) \sigma_{+} {\tilde U}(t_2) \vert \Psi(0)\rangle \langle \Psi(0)\vert {\tilde U}^{\dagger}(t_1) \right) \nonumber \\ &= {\rm Tr}_{S \oplus E} \left(\sigma_{-} {\tilde U}(t_1-t_2) \sigma_{+} \vert {\tilde \Psi}(t_2)\rangle \langle {\tilde \Psi}(t_1)\vert \right). \label{Exact2} \nonumber \end{align} \tag{ 11 }$$

Again by substituting $|{\tilde \Psi}(t_1)\rangle \,{\rm and} \,\vert {\tilde \Psi}(t_2)\rangle$ explicitly in equation (11) and using the fact that $\sigma_{-} {\tilde U}(t_1-t_2) \sigma_{+} \vert {\tilde \Psi}(t_2)\rangle = c_0 c_1(t_1-t_2) \vert 0\rangle_{S} \otimes \vert 0\rangle_{E}$, one can show

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \langle \sigma_{-}(t_1) \sigma_{+}(t_2) \rangle_I = \vert c_0\vert ^2 c_1(t_1-t_2). \nonumber \end{align} \tag{ 12 }$$

Using the transformation ${\tilde U}(t)=U_0^{\dagger}(t)U(t)$ with $U_0(t)$ and $U(t)$ being the unitary time evolution operators generated by H₀ and H respectively, it is then straightforward to obtain the two-time correlation functions in the usual Heisenberg picture

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nonumber \langle \sigma_{+}(t_1) \sigma_{-}(t_2) \rangle &= \langle \sigma_{+}(t_1) \sigma_{-}(t_2) \rangle_I \exp \{-{\rm i}\omega_0 (t_2 - t_1) \} \nonumber \\ \label{ttc1} &= c_1(t_2) c_1^{\ast}(t_1) \exp \{-{\rm i}\omega_0 (t_2 - t_1) \} \nonumber \end{align} \tag{ 13 }$$

and

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \nonumber \langle \sigma_{-}(t_1) \sigma_{+}(t_2) \rangle &= \langle \sigma_{-}(t_1) \sigma_{+}(t_2) \rangle_I \exp \{{\rm i}\omega_0 (t_2 - t_1) \} \nonumber \\ \label{ttc2} &= \vert c_0\vert ^2 c_1(t_1-t_2) \exp \{{\rm i}\omega_0 (t_2 - t_1) \}. \nonumber \end{align} \tag{ 14 }$$

Leggett–Garg inequalities test the correlations of a single system measured at different times for which we need to calculate the two-time correlation functions '$\langle O(t_j) O(t_i) \rangle$' of an observable O. We can construct the simplest LGI as follows. Consider the measurement of an observable $O(t)$ of a two level system which is found to take a value $+1$ or $-1$, depending on the system being in state $\vert + \rangle$ or $\vert - \rangle$. Now perform a series of three set of experimental runs starting from identical initial condition (at time $t = 0$) such that in the first set of runs O is measured at times t₁ and $t_2 = t_1 + \tau$; in the second, at t₁ and $t_3 = t_1 + 2 \tau$; in the third at t₂ and t₃ (where $t_3 > t_2 > t_1$). The temporal correlations $\langle O(t_j) O(t_i) \rangle$ can be obtained from such measurements. Leggett and Garg [1] followed the standard classical argument (assumptions a₁ and a₂) leading to a Bell-type inequality, with times t_i and t_j playing the role of apparatus settings. According to the classical assumption a₁, for any set of runs corresponding to the same initial state, any individual $O(t)$ has a well-defined pre-existing value prior to measurement. According to assumption a₂, the value of $O(t_j)$ or $O(t_i)$ in any pair does not depend on whether any prior or subsequent measurement has been made on the system, so the joint measurements $O(t_j) O(t_i)$ are independent of the sequence in which they are measured. Hence for classical systems the combination $O(t_2) O(t_1) + O(t_3) O(t_2) - O(t_3) O(t_1)$ has an upper bound of $+1$ and lower bound of $-3$. Replacing all the individual product terms in this expression by their averages over the entire ensemble for each sets of runs, one obtains the following form of LGI

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle C_3 = C_{21} + C_{32} - C_{31} \leqslant 1. \label{LGI1} \nonumber \end{align} \tag{ 15 }$$

Using similar arguments one can derive an LGI for measurements at four different times, t₁, t₂, t₃ and $t_4 = t_1 + 3\tau$ given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle C_4 = C_{21} + C_{32} + C_{43} - C_{41} \leqslant 2. \label{LGI2} \nonumber \end{align} \tag{ 16 }$$

To avoid possible time-ordering ambiguities [20], we consider the symmetric combination of the two-time correlation functions

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle C_{ji} = \langle \{O(t_j), O(t_i) \} \rangle/2. \label{Cji} \nonumber \end{align} \tag{ 17 }$$

The anticommutator $\{O(t_j), O(t_i)\} = \left(O(t_j) O(t_i) + O(t_i) O(t_j) \right)$ is Hermitian [27, 32]. We investigate the dynamics of the Leggett–Garg inequality for a two-level system under spontaneous decay, with the measurement operator $O=\sigma_x$ and the two-time correlators given by equation (17). Consequently, the two-time correlation function $C_{21}$ is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{C21A} C_{21} &= \langle \sigma_x(t_2) \sigma_x(t_1) + \sigma_x(t_1) \sigma_x(t_2) \rangle /2 \nonumber \\ \nonumber &= \frac{1}{2} \{\langle \sigma_{+}(t_1) \sigma_{-}(t_2) \rangle + \langle \sigma_{-}(t_1) \sigma_{+}(t_2) \rangle + \langle \sigma_{+}(t_2) \sigma_{-}(t_1) \rangle + \langle \sigma_{-}(t_2) \sigma_{+}(t_1) \rangle \}, \nonumber \end{align} \tag{ 18 }$$

as the two-time correlation functions $\langle \sigma_{-}(t_1) \sigma_{-}(t_2) \rangle$ and $\langle \sigma_{+}(t_1) \sigma_{+}(t_2) \rangle$ vanish for any pair of time t₁ and t₂. Then combining equations (13), (14) and (18) we have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{C21B} C_{21} = {\rm Re} \{c_1(t_2) c_1^{\ast}(t_1) {\rm e}^{{\rm -i}\omega_0(t_2-t_1)} + \vert c_0\vert ^2 c_1(t_2-t_1) {\rm e}^{{\rm -i}\omega_0(t_2-t_1)} \}, \nonumber \end{align} \tag{ 19 }$$

since $\langle \sigma_{+}(t_2) \sigma_{-}(t_1) \rangle$ and $\langle \sigma_{-}(t_2) \sigma_{+}(t_1) \rangle$ are the complex conjugates of $\langle \sigma_{+}(t_1) \sigma_{-}(t_2) \rangle$ and $\langle \sigma_{-}(t_1) \sigma_{+}(t_2) \rangle$ respectively.

For experimental investigation of the nonclassicality or quantumness through Leggett–Garg inequality, we propose to consider a two level quantum emitter (a solid state qubit) positioned close to a two-dimensional metal-dielectric interface [43–45]. The quantum emitter coupled to the metal-surface electromagnetic modes can be described by the Hamiltonian (1), and the problem can be solved exactly using Wigner–Weisskopf approach as discussed in section 2. Dynamics of the excited-state population and reversible coherent dynamics for this physical system was studied recently [43, 44], but our main motivation in this work is to find quantum signatures through the two-time correlation functions and to probe nonclassicality of the open system using Leggett–Garg inequality. The spectral density of the metal-surface electromagnetic field is strongly modified in presence of the quantum emitter. A recent research revealed [44] that with small enough separation between the quantum emitter and the metal-dielectric interface, the spectral density (which comprises information about the density of the surface electromagnetic field, and also the coupling between quantum emitter and the metal surface) can take a form of the Lorentzian spectral distribution. Taking this as physical motivation (see appendix), we consider a Lorentzian spectral density of the environment

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J(\omega) = \frac{\gamma}{2\pi} \frac{\lambda^2}{(\omega-\omega_0)^2 + \lambda^2}, \label{spectra} \nonumber \end{align} \tag{ 20 }$$

where γ describes the coupling strength (decay rate) and λ is the spectral width of the reservoir. It is important to mention here that the Wigner–Weisskopf theory of spontaneous emission of a two-level system has been widely used recently in the context of open quantum systems where the environment is routinely characterized by a Lorentzian spectral density [30, 33–42]. With the spectral density specified by the equation (20), the exact probability amplitude $c_1(t)$ of equation (6) can be solved analytically

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle c_1(t) = c_1(0) {\rm e}^{-\frac{\lambda t}{2}}\left(\cosh \frac{\delta t}{2} + \frac{\lambda}{\delta} \sinh \frac{\delta t}{2} \right) \label{amptd} \nonumber \end{align} \tag{ 21 }$$

where $\delta=\sqrt{\lambda^2 - 2 \gamma \lambda}$. Then for this spectral density $J(\omega)$, the correlation functions $C_{21}$, $C_{32}$, $C_{43}$, $C_{31}$, and $C_{41}$ can be calculated exactly using equations (19) and (21). The parameter λ is connected to the reservoir correlation time $\tau_r$ by the relation $\tau_r \approx \lambda^{-1}$, whereas the parameter γ is related to the relaxation time scale $\tau_x$ through the relation $\tau_x \approx \gamma^{-1}$. We consider two different regime of the system-environment parameters: (a) $\lambda > 2 \gamma$, in this regime the reservoir correlation time is small compared to the relaxation time ($\tau_r < \tau_x$) of the qubit and the behavior of $c_1(t)$ shows a Markovian exponential decay (b) $\lambda < 2 \gamma$, for which the reservoir correlation time $\tau_r$ is large or of the same order as the relaxation time $\tau_x$ and non-Markovian effects become relevant [46–48]. We investigate the nonclassical transient behavior of the two-level system through Leggett–Garg inequality in the above two physically distinct regime (a) and (b). The initial environment state is considered to be in the vacuum state and the system is arbitrarily chosen to $\left\vert \Psi \right\rangle =\frac{1}{\sqrt{2}}(\left\vert + \right\rangle + \left\vert - \right\rangle )$, hence $\rho_S (0) = \vert \Psi \rangle \langle \Psi \vert$. Here $\vert +\rangle$ and $\vert -\rangle$ are the eigenstates of $\sigma_x$. In figure 1(a), we show the exact dynamics of Leggett–Garg inequality ($C_{3}$) as a function of τ for different system-environment coupling strengths γ satisfying $\lambda > 2 \gamma$. The spectral width λ is fixed at $\lambda=5\omega_0$. For simplicity, we also set $\omega_0t_1=0$. Different curves represent different coupling strengths, namely $\gamma=0.1\omega_0$ (blue), $\gamma=0.3\omega_0$ (red), $\gamma=0.5\omega_0$ (green), and $\gamma=0.8\omega_0$ (black). In this regime, the system shows nonclassical behavior (violation of LGI with $C_{3} > 1$) only when the system-reservoir coupling is relatively weak ($\gamma < 0.5\omega_0$). This violation of LGI (15) is reduced and the nonclassicality of the open system eventually vanishes for higher values of coupling γ. Next in figure 1(b), we show the dynamics of $C_{3}$ in another regime where $\lambda < 2 \gamma$. Here we fix the spectral width of the environment at $\lambda=0.1\omega_0$. Then we vary the coupling strengths as $\gamma=0.1\omega_0$ (blue), $\gamma=0.3\omega_0$ (red), $\gamma=0.5\omega_0$ (green), and $\gamma=0.8\omega_0$ (black). The two-level system now shows (see figure 1(b)) nonclassical behavior through the violation of LGI ($C_{3} > 1$) for both the weak and strong system-reservoir couplings. In this non-Markov regime ($\lambda < 2 \gamma$), the violation of LGI is much more robust with respect to the system-reservoir coupling. This indicates that the non-Markovianity actually helps us to maintain the quantumness of the system. In figure 2, we further show the exact dynamics of $C_{4}$ in two different regime (a) $\lambda > 2 \gamma$ and (b) $\lambda < 2 \gamma$. In the Markov regime (figure 2(a)), the nonclassical behavior (violation of LGI with $C_{4} > 2$) of the system is observed for weak system-reservoir couplings. In this regime ($\lambda > 2 \gamma$), the violation of LGI is very sensitive to the system-environment coupling strength γ, and nonclassicality of the two-level system completely disappears for higher values of γ. In contrast, we see a robust nonclassical behavior (violation of LGI) of the system with respect to the system-environment coupling in the non-Markovian regime (figure 2(b)). In this case, the non-Markovianity restores the quantumness of the system even if the system is strongly coupled to the environment. The LGI violation also depends on the initial time t₁ of the first measurement. We have studied numerically the effect of varying t₁ on the dynamics of Leggett–Garg inequality. It is observed that the nonclassicality of the open system will be wiped out if we allow the system to evolve under the environment for a long time before performing the measurements. It is important to mention that the nonclassical behavior (violation of LGI) of the system is significant when the measurement time interval τ is small. The small τ region is captured in the inset of individual plots.

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In summary, we have used Leggett–Garg inequality as a nonclassicality witness for an open quantum system. We investigate the violation of LGI for a two level system (qubit) spontaneously decaying under a general non-Markovian dissipative environment. Our analysis is exact as we have calculated the two-time correlation functions exactly without using quantum regression theorem under Born–Markov approximations. We investigate the nonclassical transient behavior of the two-level system through LGI in two physically distinct regime. In the Markov regime when reservoir correlation time is small compared to the relaxation time, the LGI violation is very sensitive to the system-environment coupling strength. On the other hand, the quantum violation of LGI becomes robust against system-environment coupling when the environment parameters are suitably tuned in the non-Markov regime. The non-Markovian dynamics plays a crucial role to the dynamics of Leggett–Garg inequalities when the reservoir correlation time is large. Further experimental investigations are required to explore the nonclassicality of open quantum systems through two-time correlation functions which are experimentally measurable quantities. It will also be interesting to investigate the nonclassical dynamics of two-level open quantum system through LGI for other type of environmental spectra.

MMA acknowledges the support from the Ministry of Science and Technology of Taiwan and the Physics Division of National Center for Theoretical Sciences, Taiwan. PWC would like to acknowledge support from the Excellent Research Projects of Division of Physics, Institute of Nuclear Energy Research, Taiwan.

The Lorentzian spectral density is considered in connection to a physical system comprising of a two-level quantum emitter coupled to a two dimensional metal dielectric interface [43–45]. It has been reported [44] that with small enough separation between the quantum emitter and the metal-dielectric interface, the surface plasmon spectral density take the form of a Lorentzian distribution

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J(\omega) = \frac{3 \gamma_0 \omega_p}{8 \pi \gamma_p} \left(\frac{\omega_0}{\omega_p} \right)^3 \left(\frac{c}{\omega_e z_0} \right)^3 \frac{(\gamma_p/2)^2}{(\omega-\omega_0)^2 + (\gamma_p/2)^2} \label{SPP} \nonumber \end{align} \tag{ A.1 }$$

where $\gamma_0$ is the spontaneous decay rate of the quantum emitter in free space, $\omega_p$ is the plasma frequency, $\omega_e$ is the energy spacing of the quantum emitter, c is the speed of light. Also, $\gamma_p$ is the field-damping rate of the surface electromagnetic field, $\omega_0$ is the main peak of the Lorentzian distribution. The spectral density (A.1) has exactly the same form as (20) with $\lambda=\gamma_p/2$ and the effective coupling strength $\gamma=(3 \gamma_0 \omega_p / 4\gamma_p) (\omega_0 / \omega_p){\hspace{0pt}}^3 (c/\omega_e z_0){\hspace{0pt}}^3$.