Quantum correlations^[1-2] have become necessary resources in quantum information sciences and technologies^[3-5]. The presence of nonclassical correlations can distinguish quantum worlds from classical worlds^[6-8]. Bell inequalities^[9] play a fundamental role in the nature of correlations between spatially separated systems. Test of the violations of Bell or Clauser-Horne-Shimony-Holt (CHSH) inequalities can demonstrate quantum nonlocality^[10]. Spatial correlations can be quantitatively evaluated by quantum entanglement^[11-12], quantum discord^[13-14], quantum coherence^[15-17] and other characteristics^[18-19]. From the viewpoint of time sequential measurements, Leggett and Garg put forward a similar inequality whose violation can be qualitatively used to estimate temporal correlations^[20]. Leggett-Garg inequalities (LGIs) must be verified by sequential measurements acting on a single system at different times, which is distinct from Bell inequalities that concern multiple parties spatially separated from each other. The violations of LGIs are inconsistent with two assumptions of macroscopic realism and noninvasive measurability^[21]. The macroscopic realism of physical states implies that measurements on a system simply reveal the values which exist at previous times. Noninvasive measurability ensures that such measurements can be performed without disturbing the dynamical evolution of the states.

In recent years, many experimental tests of LGIs have been carried out in setups including Rydberg atoms^[22], photonic systems^[23], superconducting circuits^[24] and a quantum computer^[25]. With the rapid development of quantum engineering technologies, feasible quantum systems have extended from qubits to multilevel quantum systems^[26-28]. Among these fantastic quantum systems, electrically controllable quantum dots molecules have been extensively applied in quantum optics and condensed matter physics because of their high nonlinear optical susceptibility, large electrical dipole moments of band transition and great flexibility in designing devices^[29-30]. A unity of two or more quantum dots with closely spaced couplings can be referred to as a quantum dots molecule which is modeled as one kind of artificial multilevel atom^[31]. Electric voltages can be used to generate electron tunnels between two neighboring quantum dots^[32]. Meanwhile, level transitions can be induced by direct couplings between a cavity and a quantum dots molecule^[33]. How do electron tunnels, cavity couplings and environmental noises affect quantum correlations in such systems? This question motivates us to theoretically evaluate temporal and spatial quantum correlations in hybrid multilevel systems.

The structure of this paper is organized as follows. In Sec. 2, we present witnesses about temporal and spatial quantum correlations. The general expression of LGI based on two-time correlation functions is given by the open system approach. Temporal correlations are qualitatively estimated by violations of LGI and quantum coherence based on $ {l_1} $-norm is used to measure spatial correlations. In Sec. 3, we describe the model and Hamiltonian of an electrically controllable quantum dots molecule in a cavity. Quantum correlations with respect to level transitions are established by external voltages and cavity couplings. In Sec. 4, we provide our main analytical and numerical results about the dynamics of temporal and spatial correlations. The explicit expressions of the LGI violation and quantum coherence are analytically obtained by the quantum dynamic method. The quantum master equation for a quantum dots molecule is exploited to numerically study environmental effects on temporal correlation. We will discuss our conclusions in the last section.

As a temporal analog version of Bell inequalities, one simple form of LGI^[21] can be obtained by performing three measurements of time such that ${t_0} < {t_1} < {t_2}$. According to two-time correlation functions for a dichotomic observable $\hat \Omega (t)$, we define the ${\text{LGI}}$ as,

$ {K_3} = C({t_0},{t_1}) + C({t_1},{t_2}) - C({t_0},{t_2}) \leqslant 1\quad. $ (1)

Here the two-time correlation function is written as $ C({t_i},{t_j}) = \langle \hat \Omega ({t_i})\hat \Omega ({t_j})\rangle $ where $\hat \Omega (t)$ denotes the time evolution of the observable $\hat \Omega $ in the Heisenberg picture. For a multilevel system, the Hamiltonian $\hat H$ has a complete set of energy levels with $\{ |\alpha \rangle ,\alpha = 1,2, \cdots ,d\} $. The dichotomic observable $\hat \Omega $ is chosen to be $ \hat \Omega = {\hat M^ + } - {\hat M^ - } $ with the corresponding values $m = \pm 1$. If the system of interest is in a specific energy state $|\alpha \rangle $ at an initial time ${t_0}$, we can define $ {\hat M^ + } = |\alpha \rangle \langle \alpha | $ and $ {\hat M^ - } = \displaystyle\sum\limits_{\beta \ne \alpha } | \beta \rangle \langle \beta | $^[26]. In this condition, the performance of the dichotomic operator $\hat \Omega $ determines whether the system is still in the state $|\alpha \rangle $ with a measurement value of $m = 1$ or whether it has undergone a transition from $|\alpha \rangle $ to other orthogonal states $|\beta \rangle $ with the other outcome. The violation of ${K_3} \leqslant 1$ is a signature of the quantum nature of the system during time evolution. We can define the violation of the inequality as ${\text{VLGI}} = \max ({K_3} - 1,0)$ characterizing the occurrence of temporal correlations. The quantity of ${\text{VLG}}{{\text{I}}_{\max }}$ represents the maximum degree to which a system can violate the Leggett-Garg inequality.

To obtain the explicit expression of time correlation functions, we need to know the time evolution process of the system. In general, a dynamical map governing the time evolution of a system from ${t_i}$ to ${t_j}$ is given by ${{\mathbf{\Phi }}_{{t_j} \leftarrow {t_i}}}$,i.e., $\rho ({t_j}) = {{\mathbf{\Phi }}_{{t_j} \leftarrow {t_i}}}\left[ {\rho ({t_i})} \right]$. In the following, we take into account a quantum dynamical semigroup which satisfies ${{\boldsymbol{\Phi }}_{{t_j} \leftarrow {t_i}}} {{\boldsymbol{\Phi }}_{{t_i} \leftarrow 0}} = {{\boldsymbol{\Phi }}_{{t_j} \leftarrow 0}}$. The map for a closed system can be expressed by the unitary operator $\rho ({t_j}) = U({t_i},{t_j})\rho ({t_i}){U^ + }({t_i},{t_j})$ where $U({t_i},{t_j}) = \hat T\exp \left[ - i\displaystyle\int_{{t_i}}^{{t_j}} {\hat H(t'){\rm{d}}t'} \right]$ is the unitary operator and $ \hat T $ is the order of time. For an open system coupled to its surroundings, the time evolution of the system can be described by the quantum Lindblad equation $ \dfrac{\partial }{{\partial t}}\rho (t) = \hat L[\rho (t)] $ where $ \hat L $ is a Lindbladian superoperator. The general form of the superoperator consists of one unitary part generated by the Hamiltonian $H$ and the other part generated by a dissipator. The two-time correlation function $C({t_i},{t_j})$ can be defined in terms of the joint probabilities, $C({t_i},{t_j}) = \displaystyle\sum\limits_{m,n = \pm } {(mn)} \cdot p(t_i^m)q(t_j^n \leftarrow t_i^m)$. Here $p(t_i^m)$ is the probability of obtaining the measurement value $m$ at a time ${t_i}$ and $q(t_j^n \leftarrow t_i^m)$ is the conditional probability of getting the value $n$ at the latter time ${t_j}$, given that the previous value m is measured at ${t_i}$. In combination with a dynamical map, the measurement probabilities are written as

$ p(t_i^m) = \;{\text{Tr}}\{ {\hat M^m} \cdot {{\mathbf{\Phi }}_{{t_i} \leftarrow 0}}\left[ {\rho (0)} \right]\} $ ()

$ q(t_j^n \leftarrow t_i^m)\; = \;{\text{Tr}}\left\{ {\hat M^n} \cdot {{\mathbf{\Phi }}_{{t_j} \leftarrow {t_i}}}\left[ {\frac{{{{\hat M}^m}\rho ({t_i}){{\hat M}^m}}}{{p(t_i^m)}}} \right]\right\} \quad. $ (2)

Joint measurability^[18] plays an important role in determining two-time correlation functions.

Spatial quantum correlation can be treated quantitatively by defining the $ {l_1} $-norm as a coherence measurement^[17]. The $ {l_1} $-norm of a matrix is related to the absolute value of the elements of the matrix. Where a definite Hilbert space has a reference orthonormal basis $\left\{ {\left| j \right\rangle } \right\}$ and a $ {l_1} $-norm of coherence of a state $\rho = \displaystyle\sum_{jk} {{\rho _{_{jk}}}} \left| j \right\rangle \left\langle k \right|$ , the measure of quantum coherence for a density matrix is described as $Q(\rho ) = \displaystyle\sum\limits_{j \ne k} | {\rho _{jk}}|$. The value of spatial correlation can be given by the summation over the absolute values of all the off-diagonal elements of the density matrix. According to temporal and spatial correlations, we will investigate the quantum nature of a system in the two approaches. The quantum nature is revealed if the states of the system occur in the form of a coherent superposition of Hamiltonian eigenstates. The ${\text{LGI}}$ violation is considered a qualitative test and quantum coherence based on $ {l_1} $-norm is referred to as a quantitative estimation.

Rapid development in laser and semiconductor technologies has made it possible to fabricate hybrid systems applied to quantum information tasks^[33-36]. Among these hybrid systems, quantum dots molecules coupled to cavities have attracted much attention in recent years. It is known that level transitions between different electronic states in quantum dots molecules can be excited by external voltages and laser fields. We consider a quantum dots molecule composed of two self-assembled $({\text{In}},{\text{Ga}}){\text{As/GaAs}}$ quantum dots with a lateral quantum interaction. The interdot coupling is controlled by an electrical voltage. This system is coupled to a microscopic cavity. A scheme for a quantum dots molecule coupled to the cavity is illustrated in Figure 1(a) (color online). Some present technologies can be exploited to produce a homogeneous ensemble of molecules consisting of two dots aligned along the $[{\text{110}}]$ direction of the ${\text{GaAs}}$ substrates^[37]. A scheme for a double coupled quantum dots system is illustrated in Figure 1(b) (color online). The ground state $|0\rangle $ denotes a level where two quantum dots are not excited. The state $|2\rangle $ represents a level in which an electron is directly excited by a laser to the conduction band in the left dot. The parameter $\varDelta $ is the frequency detuning between the laser’s driving and level transition. $g$ denotes the strength of electrical dipole interactions between a molecule and cavity fields. The application of electrical voltages induces the transfer of an electron from the left dot to the conduction band of the right dot. The indirectly excited state $|1\rangle $ stands for the level transition by interdot tunnels. ${T_e}$ is the strength of the tunnel coupling which can be varied by the external voltage. The level transition ${\omega _{21}}$ is considered to be zero due to the negligible energy difference between the direct and indirect exciton.

By means of the electric dipole and rotating wave approximation, the Hamiltonian of the hybrid system in the interaction picture can be expressed as

$ \begin{split} \hat H =& \Delta |2\rangle \langle 2| + (\Delta - {\omega _{21}})|1\rangle \langle 1| + \\ &(g\hat a|2\rangle \langle 0| + {T_e}|1\rangle \langle 2| + H.c.)\quad, \end{split} $ (3)

where ${{H}}.{{c}}.$ is the Hermitian conjugate part and $\hat a$ represents the annihilation operator for the cavity mode. In order to obtain a dynamic map of the system, we can assume that the initial state of the total system $|\Psi (0)\rangle = |1\rangle |0{\rangle _c} = |10\rangle $ where $|0{\rangle _c}$ is the vacuum state for the cavity mode. The total system will evolve in the Hilbert space $\mathcal{H}$ spanned by $\{ |1\rangle |0{\rangle _c},|2\rangle |0{\rangle _c},|0\rangle |1{\rangle _c}\} $. The eigenvalues ${E_j}(j = 1,2,3)$ for the Hamiltonian can be obtained by the three real roots of the equation ${E^3} - (2\Delta - {\omega _{21}}){E^2} + ({\Delta ^2} - \Delta {\omega _{21}} - {g^2} - T_e^2)E + {g^2}(\Delta - {\omega _{21}}) = 0$. The corresponding eigenstate $|\psi \rangle $ is given by $|\psi {\rangle _j} = {c_{j1}}|10\rangle + {c_{j2}}|20\rangle + {c_{j3}}|01\rangle$ where three coefficients satisfy ${c_{j1}} = \dfrac{{{T_e}}}{{E - \Delta + {\omega _{21}}}}{c_{j2}}$, ${c_{j3}} = \dfrac{g}{E}{c_{j2}}$ and $\displaystyle\sum\limits_k | {c_{jk}}{|^2} = 1$. For the closed quantum system, the unitary operator $U({t_i},{t_j}) = U({t_j} - {t_i}) = \displaystyle\sum\limits_k {\exp } [ - i{E_k}({t_j} - {t_i})]|\psi {\rangle _k}\langle \psi |$ is related to the time interval. In a simple case of $\Delta = 0$, $ {\omega _{21}} = 0 $, we can obtain the eigenvalues and corresponding eigenstates of the hybrid system with

$ {E_1}\; = 0,\;{E_{2,3}} = \pm J{\text{ }} $ ()

$ |\psi {\rangle _1}\; = \frac{g}{J}|10\rangle - \frac{{{T_e}}}{J}|01\rangle $ ()

$ |\psi {\rangle _{2,3}} = \frac{1}{{\sqrt 2 }}\left( \pm \frac{{{T_e}}}{J}|10\rangle + |20\rangle \pm \frac{g}{J}|01\rangle \right)\quad, $ (4)

where the parameter $ J = \sqrt {{g^2} + T_e^2} $. The unitary generator for an interval $\tau $ can be written as

$ {\boldsymbol{U}}(\tau ) = \left( {\begin{array}{*{20}{c}} {{U_{11}}}&{{U_{12}}}&{{U_{13}}} \\ {{U_{12}}}&{{U_{22}}}&{{U_{23}}} \\ {{U_{13}}}&{{U_{23}}}&{{U_{33}}} \end{array}} \right)\quad. $ (5)

For formula 5, the elements are obtained by ${U_{11}} = \dfrac{{{g^2} + T_e^2\cos J\tau }}{{{J^2}}}$, ${U_{12}} = - \dfrac{{i{T_e}\sin J\tau }}{J}$, ${U_{13}} = - \dfrac{{g{T_e}}}{{{J^2}}}. (1 - \cos J\tau )$, ${U_{22}} = \cos J\tau $, ${U_{23}} = - \dfrac{{ig\sin J\tau }}{J}$ and ${U_{33}} = \dfrac{{T_e^2 + {g^2}\cos J\tau }}{{{J^2}}}$ . Therefore, the dynamic evolution of the system is determined by the above unitary generator.

To evaluate the temporal correlation, we chose the dichotomic operator $\hat \Omega = {\hat M^ + } - {\hat M^ - }$ where ${\hat M^ + } = |1\rangle \langle 1|$. Because the measurements satisfy the completeness $\displaystyle\sum\limits_{m = \pm } {{{\hat M}^m}} = I$, we can work out a simplified expression of $C({t_i},{t_j})$ only in terms of ${\hat M^ + }$ measurement. The general expression of $C({t_i},{t_j})$ can be simplified as,

$ \begin{split} C({t_i},{t_j}) =& 1- 2{\text{Tr}}\{ {{\hat M}^ + } \cdot {{\mathbf{\Phi }}_{{t_j} \leftarrow {t_i}}}[{{\mathbf{\Phi }}_{{t_i} \leftarrow 0}}(\rho (0))]\} -{\text{Tr}}\{ {{\mathbf{\Phi }}_{{t_j} \leftarrow {t_i}}}[{{\hat M}^ + }{{\mathbf{\Phi }}_{{t_i} \leftarrow 0}}(\rho (0))]\} - {\text{Tr}}\{ {{\mathbf{\Phi }}_{{t_j} \leftarrow {t_i}}}[{{\mathbf{\Phi }}_{{t_i} \leftarrow 0}}(\rho (0)){{\hat M}^ + }]\} +\\ &2{\text{Tr}}\{ {{\hat M}^ + }{{\mathbf{\Phi }}_{{t_j} \leftarrow {t_i}}}[{{\hat M}^ + }{{\mathbf{\Phi }}_{{t_i} \leftarrow 0}}(\rho (0))]\} + 2{\text{Tr}}\{ {{\hat M}^ + }{{\mathbf{\Phi }}_{{t_j} \leftarrow {t_i}}}[{{\mathbf{\Phi }}_{{t_i} \leftarrow 0}}(\rho (0)){{\hat M}^ + }]\} \quad. \end{split} $ (6)

The closed system evolves in accordance with the unitary generator of Eq. (5). For an initial state $|\Psi (0)\rangle = |1\rangle |0{\rangle _c}$, the reduced density matrix of the quantum dots molecule after 6 measurements can be given by,

$ \begin{split} &{{\mathbf{\Phi }}_{{t_j} \leftarrow {t_i}}}[{{\mathbf{\Phi }}_{{t_i} \leftarrow 0}}(\rho (0))] = {\text{T}}{{\text{r}}_c}[U({t_j} - {t_i})U({t_i})\rho (0){U^{\text{ + }}}({t_i}){U^{\text{ + }}}({t_j} - {t_i})] {{\mathbf{\Phi }}_{{t_j} \leftarrow {t_i}}}[{{\hat M}^ + }{{\mathbf{\Phi }}_{{t_i} \leftarrow 0}}(\rho (0))] ={\text{T}}{{\text{r}}_c}[U({t_j} - {t_i})\\ &{{\hat M}^ + }U({t_i})\rho (0){U^{\text{ + }}}({t_i}){U^{\text{ + }}}({t_j} - {t_i})] {{\mathbf{\Phi }}_{{t_j} \leftarrow {t_i}}}[{{\mathbf{\Phi }}_{{t_i} \leftarrow 0}}(\rho (0)){{\hat M}^ + }] ={\text{T}}{{\text{r}}_c}[U({t_j} - {t_i})U({t_i})\rho (0){U^{\text{ + }}}({t_i}){{\hat M}^ + }{U^{\text{ + }}}({t_j} - {t_i})]\quad, \end{split} $ (7)

where ${\text{T}}{{\text{r}}_c}$ denotes the partial tracing over the degrees of freedom of the cavity.

Assuming that ${t_2} - {t_1} = {t_1} - {t_0} = \tau $ and ${t_0} = 0$, the two-time correlation functions for the simplest case, $\Delta = 0,{\omega _{21}} = 0$ can be obtained analytically as

$ \begin{split} &C(0,\tau ) = 2U_{11}^2 - 1,\;\;C(0,2\tau ) = 2U_{11}^{'2} - 1, \\ &C(\tau ,2\tau ) = 1 - 2U_{11}^{'2} + 2U_{11}^4 - 4U_{11}^2|{U_{12}}{|^2} -\\ &\quad\quad\quad\quad\;2{U_{11}}\sum\limits_{j = 2,3} {{U_{jj}}} |{U_{1j}}{|^2}\quad, \end{split} $ (8)

where the coefficient $U_{11}' = \dfrac{{{g^2} + T_e^2\cos 2J\tau }}{{{J^2}}}$. Based on the above correlation functions, the dynamic behavior of the violation of ${\text{LGI}}$ can be shown in Figure 2. It is seen that the violation of ${\text{LGI}}$ can be enhanced by the tunnel coupling ${T_e}$. The fluctuations of the ${\text{LGI}}$ violation occur with time. At a given time, the optimal value of an electron tunnel can give rise to the maximal violation. The effect of the frequency detuning $\Delta $ on the ${\text{LGI}}$ violation is demonstrated in Figure 3. It is shown that the maximal violation of ${\text{LGI}}$ increases with a variation in the detuning. In a sense, the tunnel coupling and cavity frequency detuning contribute to the generation of the coherent superposition of the excited state $|1\rangle $ and other level states.

The reduced density matrix for the quantum dots molecule is given by ${\rho _Q}(\tau ) ={\text{T}}{{\text{r}}_c}[U(\tau ) |\Psi (0)\rangle \langle \Psi (0)|{U^ + }(\tau )]$ . The quantum coherence as a measurement of spatial correlation is obtained as

$ Q(\rho ) = \frac{2}{{{J^3}}}|{T_e}\sin J\tau ({g^2} + T_e^2\cos J\tau )|\quad. $ (9)

Figure 4 (color online) illustrates the fluctuating behavior of temporal and spatial quantum correlations for a quantum dots molecule. There is a relationship between temporal and spatial correlations. In the time domain where the maximal ${\text{LGI}}$ violations happen, the values of quantum coherence remain high. When no spatial correlation is detected, there is also no violation of the Leggett-Garg inequality. However, the system of interest can maintain spatial correlations at a high level even when the ${\text{LGI}}$ is not violated. It is demonstrated that there is a distinction between temporal and spatial correlation due to different criteria.

When there is spontaneous, decay and photon leakage, the evolution of the hybrid system is subjected to environmental noise. The Markovian dynamics of the total system can be described by the quantum master equation

$ \frac{\partial }{{\partial t}}\rho (t) = - i[\hat H,\rho (t)] - \sum\limits_{k = 1,2} {\frac{\gamma }{2}} {\hat {\mathcal{D}}_k}[\rho (t)] - \frac{\kappa }{2}{\hat {\mathcal{D}}_a}[\rho (t)]\quad, $ (10)

where the dissipators ${\hat {\mathcal{D}}_k}[\rho (t)] = {\hat \sigma _{kk}}\rho - 2{\hat \sigma _{0k}}\rho {\hat \sigma _{k0}} + \rho {\hat \sigma _{kk}}$ and ${\hat{ \mathcal{D}}_a}[\rho (t)] = {\hat a^{ + }}\hat a\rho - 2\hat a\rho {\hat a^{ + }} + \rho {\hat a^{ + }}\hat a$ . Here, ${\hat \sigma _{jk}} = |j\rangle \langle k|$ denotes the transition operator from the $|k\rangle $ to $|j\rangle $ state. The parameters $\gamma $ and $\kappa $ stand for spontaneous decay from the quantum dots molecule and photon leakage of the cavity, correspondingly. According to the results^[22], a stationarity condition holds given that a quantum system evolves under Markovian dynamics. With stationarity, the evolution of the system from ${t_0}$ to ${t_1}$ is governed by the same stochastic differential equation as the evolution from ${t_1}$ to ${t_2}$. To numerically obtain the temporal correlation, one kind of Leggett-Garg-type inequality is defined as $K_3' = 2C({t_0},{t_1}) - C({t_0},{t_2}) \leqslant 1$ where the time interval is equal to $\tau $. The evolution of the violation is numerically calculated and shown in Figure 5 (color online). It is seen that the violations gradually decrease with time before finally disappearing in Figure 5(a). The suppressed behavior of the violation is similar to the decoherence of spatial quantum correlation. The spontaneous decay from a quantum dots molecule can decline the maximal degree of the violation. Similarly, it is demonstrated that the tunneling coupling can also improve the temporal correlation under environmental noise.

We have tested the violation of the Leggett-Garg inequalities to evaluate temporal quantum correlations in a quantum dots molecule coupled to a cavity. The coherent superposition between a tunneling excited state and other level states was studied by means of the ${\text{LGI}}$ violation and ${l_1}$-norm quantum coherence. When there is no spatial correlation with zero quantum coherence, no violation of the Leggett-Garg inequalities happens during the evolution. Moreover, the maximal violations exist in the time intervals where the system has high values of spatial correlations. However, there is also some discrimination between both kinds of quantum correlations. The quantum coherence can persist in time domains where there is no ${\text{LGI}}$ violation. This fact shows that the ${\text{LGI}}$ violation and ${l_1}$-norm coherence are independent measure criteria of quantum correlation. We have taken into account the impacts of spontaneous decay from quantum dots and photon leakage from a cavity on the dynamics of temporary quantum correlations. And we have proved that these elements can result in the attenuation of the quantum coherence. The decoherence of the LGI violation can lead to macroscopic realism where coherent superposition of the level states will disappear. We have found that the evolution of the hybrid system is subjected to spontaneous decay and photon leakage. The parameters with respect to environmental noise accelerate the decline of temporal quantum correlation. We have demonstrated that LGI violation can be enhanced by electron tunneling interactions. With the development of quantum information processing technology, these results can be applied to hybrid multilevel systems.