1 Introduction

Quantum correlations including coherence [1], entanglement [2, 3], mutual information, quantum discord [4], etc., have been widely theoretically studied. They provide profound insights into black hole physics and even cosmology [5,6,7,8]. They are often needed as the essential resources to carry out certain quantum information processing tasks, e.g., quantum teleportation, quantum computation, quantum cryptography, and quantum metrology  [9,10,11,12,13]. One of the major obstacles to realizing the quantum information technologies is the environmentally induced decoherence and dissipation effect on the quantum systems, which may give rise to quantum correlation degradation. However, an external environment can also provide indirect interactions between the subsystems through the existing correlations. A phenomenological and illuminating example is the independent atoms immersed in external quantum fields and weakly coupled to them through Unruh–DeWitt detector interaction [7, 8, 14]. The Unruh–DeWitt detector interaction has been used to explore spacetime structures and quantum information [15,16,17,18]. The atoms can be usually treated in a nonrelativistic approximation, as independent n-level systems (qubits or harmonic oscillator typically), with negligible size, while the environment can be described by a set of quantum fields in a given quantum state, typically either a thermal state or simply the vacuum state. Despite the simplified setting, this can also provide heuristic insight into the relationship between the environment and the quantum correlations. It has been shown that there are certain scenarios where the environment may create quantum correlations rather than destroy them in both flat spacetime [19,20,21,22,23] and curved spacetime backgrounds [7, 8, 24,25,26,27]. Meanwhile, other quantum information measures, such as Fisher information, have also been studied [28,29,30]. Connecting the spacetime structure and the quantum information encoded in the open system is interesting. On the one hand, we can infer the spacetime structure from the quantum information extracted from the system. On the other hand, we can understand the system’s evolution once we know the spacetime structure.

A black hole, which relates to gravity, quantum theory, and thermodynamics, is a fascinating and challenging physics subject. The Hawking radiation emerges at the horizon as a pure quantum effect [33], which the open quantum system theory can also understand. Concretely, consider a single two-level atom system near the horizon, treat the massless scalar field in the curved background as the environment, and then compute the spontaneous excitation rate of the atom. This reveals that close to the horizon, a ground-state detector in the vacuum would spontaneously excite with an excitation rate, as in the case when there is a thermal bath around the atom [33, 34]. Therefore, near-horizon geometry plays a crucial role in the character of a black hole spacetime [32]. Moreover, it was shown that around the horizon of a Kerr spacetime, the scalar field theory could be reduced to a two-dimensional effective field theory [34,35,36]. This insight makes studying the quantum correlations near the black hole convenient.

In this paper, we study the quantum correlations near the horizon of the Kerr black hole by using the dimensional reduction method. Both equilibrium and nonequilibrium scenarios are discussed. The equilibrium scenario means that two detectors couple to the field and co-rotate with the black hole at the same location near the black hole horizon. We start with a product state initially. An equilibrium steady state can be found at the final time in a certain scenario. We focus on the equilibrium steady state and study its quantum correlations. We found that the quantum correlations (including entanglement, coherence, discord, and mutual information) in a two-qubit system vary non-monotonically with the mass of the black hole and are amplified by the angular momentum. The von Neumann entropy, which measures the entanglement between the system and environment, behaves oppositely compared to the quantum correlations concerning the mass and the angular momentum. Moreover, we found that the local acceleration or local curvature can suppress the quantum correlations within the system but enhance the correlation with the environment.

For the nonequilibrium case, two qubits are fixed at a certain location near the black hole horizon in the co-rotating frame, one of two isolated from the environment. The system evolves from the maximally entangled state, and the information will scramble to the environment. At a fixed time, we found the behaviors of the quantum correlations are very similar to the equilibrium case: the quantum correlations in the two-qubit system vary non-monotonically with the mass of the black hole and are amplified by the angular momentum, while the curvature suppresses the quantum correlations within the system. We quantify the system’s entropy production rate (EPR) and found that it decreases in time. We study the lower bound of the EPR under the Born–Markov approximation. The EPR decreases at first and then increases to a constant as the black hole mass becomes larger and increases when the angular momentum becomes larger. The EPR decreases monotonically with the local curvature. We define the decay rate for the quantum correlation. The decay rates of the quantum correlations increase at first and then decrease with respect to the mass. The more significant angular momentum suppresses the decay rates of the correlations. The local curvature enhances the decay rate of the quantum correlations and EPR, but reduces the decay rate of the von Neumann entropy. We conclude that the spacetime structure can influence the quantum correlations of the system.

Another nonequilibrium scenario is also investigated. There are two types of massless scalar fields equivalent to two different independent baths coupled to two interacting qubits, respectively. The nonequilibrium is measured by the difference in the radius separation of two qubits. We investigate the quantum correlations of the nonequilibrium steady state. On the whole, the correlations can survive and be maintained at a steady value when \(\Delta r\) is large. The entanglement, discord, and mutual information behave non-monotonically in certain parameters. This shows that nonequilibrium can amplify the quantum correlations. The coherence monotonically decreases to a constant, and the von Neumann entropy decreases to a constant, which indicates that the nonequilibrium is harmful to producing the correlation between the system and the environment. The flux increases to a constant as the difference in \(\Delta r\) or nonequilibrium increases. The EPR as a measure of thermodynamics cost increases as the \(\Delta r\) increases. Not only the local acceleration or local curvature can impact the correlations, but the nonequilibrium also does.

The organization of our paper is as follows. In Sect. 2, we will describe the simplest model, which can be used for the later generalization. Then we review the basic formulations, including the master equation describing the system of the detector in the Born–Markov approximation. In Sect. 3, we introduce the certain quantum correlations we are interested in. In Sect. 4, the dimensional reduction technique is used to investigate the massless scalar field in a Kerr spacetime, and two types of vacuums are discussed. In Sect. 5, we discuss the quantum correlations of the equilibrium steady state for the two atom detectors near the Kerr black hole. Importantly, we study the nonequilibrium case. In Sect. 6, we consider the quantum correlations in the curved spacetime in a specific nonequilibrium transient scenario. In Sect. 7, we study the nonequilibrium quantum correlations in curved spacetime at the steady state. At last, we will draw a conclusion in Sect. 8

2 Master equation for an open quantum system

Our main objective in this section is to formulate the time evolution of an open quantum system and to obtain the Gorini–Kassakowski–Sudarshan–Lindblad (GSKL) master equation, which properly describes the non-unitary behaviors and can be obtained by partial trace over the environmental baths, i.e., the massless probe scalar field placed on the Kerr black hole spacetime background. Generally, the open quantum setup can be described by the following Hamiltonian

$$\begin{aligned} H_\mathrm{total}=H_{0}+H_{1}=H_\mathrm{sys}+H_\mathrm{field}+H_{I}. \end{aligned}$$
(1)

Here, \(H_\mathrm{sys}\) is the Hamiltonian of the atom or the detector. For the single two-level atom, internal dynamics will be driven by a \(2\times 2\) Hamiltonian matrix. On a given basis, it can be assumed to have the form \(\frac{\omega }{2}\sigma _{z}\), where \(\sigma _{z}\) is the Pauli matrix, while \(\omega \) represents the gap between the two energy eigenvalues. Then, the atom Hamiltonian becomes \(H_\mathrm{sys}=\frac{\omega }{2}\sigma _{z}\). We assume that the Hamiltonian describing the interaction between the atom and the scalar field can be taken in the form of the Unruh–DeWitt detector interaction as \( H_{I}=\mu (\sigma _{+}+\sigma _{-})\phi (x(\tau ))\), in which \(\mu \) is the coupling constant. Also, we set \(\sigma _{+}(\sigma _{-})\) as the atomic raising (lowering) operator, and \(\phi (x)\) corresponds to the scalar field operator in Kerr spacetime. The time evolution of the total system in the proper time \(\tau \) is governed by the von Neumann equation

$$\begin{aligned} \partial _{\tau }\rho _\mathrm{total}=-i[H_\mathrm{total},\rho _\mathrm{total}]. \end{aligned}$$
(2)

For convenience, one usually performs a unitary transformation to transform the above Liouville–von Neumann equation into the interaction picture

$$\begin{aligned} \partial _{\tau }\rho _\mathrm{total}^{I}=-i[H_{I}^{I},\rho _\mathrm{total}^{I}]. \end{aligned}$$
(3)

The upper index I represents the operator in the interaction picture, and the unitary transformation reads \(\rho _\mathrm{total}^{I}(\tau )=e^{iH_{0}\tau }\rho _\mathrm{total}e^{-iH_{0}\tau }\) and \(H_{I}^{I}(\tau )=e^{iH_{0}\tau }H_{I}e^{-iH_{0}\tau }\) for \(\rho _\mathrm{total}\) and \(H_{I}\), respectively. By integrating the above equation Eq. (3), we get

$$\begin{aligned} \rho _\mathrm{total}^{I}(\tau )=\rho _\mathrm{total}^{I}(0)-i \int _{0}^{\tau }\mathrm{d}s[H_{I}^{I}(s),\rho _\mathrm{total}^{I}(s)]. \end{aligned}$$
(4)

Inserting Eq. (4) back to Eq. (3) and tracing out the field (or environmental) degrees of freedoms, we arrive at

$$\begin{aligned} \frac{\mathrm{d}\rho _\mathrm{sys}^{I}(\tau )}{\mathrm{d}\tau }=-\int _{0}^{\tau }\mathrm{Tr}_\mathrm{field}{[H_{I}^{I}(\tau ),[H_{I}^{I}(s),\rho _\mathrm{total}^{I}(s)]]}, \end{aligned}$$
(5)

where we have taken \(\mathrm{Tr}_\mathrm{bath}[H_{I}^{I}(\tau ),\rho _\mathrm{total}^{I}(0)]=0\), meaning that the interaction does not create any dynamics in the bath initially. Eq. (5) still contains the density matrix of the total system \(\rho _\mathrm{total}^{I}(\tau )\) on its right-hand side. In order to eliminate \(\rho _\mathrm{total}^{I}(\tau )\) from the equation of motion, one can perform a first approximation, known as the Born approximation: the coupling between the system and the bath is weak such that the influence of the bath is small. Thus, one can consider the bath as almost unchanged, and then a tensor product may approximately characterize the state of the total system at time \(\tau \)

$$\begin{aligned} \rho _\mathrm{total}^{I}(\tau )=\rho _\mathrm{sys}^{I}(\tau )\bigotimes \rho _\mathrm{field}. \end{aligned}$$
(6)

Inserting the tensor product into the exact equation of motion given by Eq. (5), one obtains a closed integral–differential equation for the reduced density matrix,

$$\begin{aligned}&\frac{\mathrm{d}\rho _\mathrm{sys}^{I}(\tau )}{\mathrm{d}\tau }\nonumber \\&=-\int _{0}^{\tau }\mathrm{Tr}_\mathrm{field}{[H_{I}^{I}(\tau ),[H_{I}^{I}(s),\rho _\mathrm{sys}^{I}(s)\bigotimes \rho _\mathrm{field}]]}. \end{aligned}$$
(7)

In order to simplify the above equation further, one can perform the Markov approximation, in which the integrand \(\rho _\mathrm{sys}^{I}(s)\) is firstly replaced by \(\rho _\mathrm{sys}^{I}(\tau )\). In this way, one can obtain an equation of motion for the reduced system’s density matrix in which the time development of the system’s state at time \(\tau \) depends only on the present state.

$$\begin{aligned} \frac{\mathrm{d}\rho _\mathrm{sys}^{I}}{\mathrm{d}\tau }=-\int _{0}^{\tau }\mathrm{Tr}_\mathrm{field}{[H_{I}^{I}(\tau ),[H_{I}^{I}(s),\rho _\mathrm{sys}^{I}(\tau )\bigotimes \rho _\mathrm{field}]]}.\nonumber \\ \end{aligned}$$
(8)

The Markov approximation grouped with the Born approximation is often regarded as the Born–Markov approximation. However, under this approximation alone, the resulting master equation does not guarantee the generation of a quantum dynamical semigroup. One performs a further secular approximation which involves averaging over and discarding the rapidly oscillating terms in the master equation [37]. With the aid of all these approximations, one can go back to the Schrödinger picture where we obtain the following Markov master equation in Kossakowski–Lindblad form  [38]:

$$\begin{aligned} \frac{\mathrm{d}\rho _\mathrm{sys}(\tau )}{\mathrm{d}\tau }= & {} -i[H_\mathrm{eff},\rho _\mathrm{sys}(\tau )] +\mathscr {L}[\rho _\mathrm{sys}(\tau )]\nonumber \\= & {} -i[H_\mathrm{eff},\rho _\mathrm{sys}(\tau )]+\sum _{j=1}^{3}\Big [2L_{j}\rho _\mathrm{sys} L_{j}^{\dag }\nonumber \\&-L_{j}^{\dag }L_{j}\rho _\mathrm{sys}-\rho _\mathrm{sys}L_{j}^{\dag }L_{j}\Big ], \end{aligned}$$
(9)

where \(H_\mathrm{eff}\) and \(L_{j}\) are given as

$$\begin{aligned} \begin{aligned}&H_\mathrm{eff}=\frac{\Omega }{2}\sigma _{z}=\frac{\omega +i(\mathscr {K}(-\omega ) -\mathscr {K}(\omega ))}{2}\sigma _{z},\\&L_{1}=\sqrt{\frac{\gamma _{-}}{2}}\sigma _{-},\quad L_{2} =\sqrt{\frac{\gamma _{+}}{2}}\sigma _{+},\quad L_{3}=\sqrt{\frac{\gamma _{0}}{2}} \sigma _{z}, \end{aligned} \end{aligned}$$
(10)

where \(\gamma _{\pm }=\mu ^{2}\int _{-\infty }^{+\infty }e^{\mp i\omega s}G^{+}(s-i\epsilon )\mathrm{d}s\) and \(\gamma _{0}=0\), and \(G^{+}(s-i\epsilon )=\langle 0|\phi (x)\phi (x')|0\rangle \) is the Wightman function of the massless scalar field (\(s=\tau -\tau '\) here). And \(\mathscr {K}(\lambda )=\frac{P}{i\pi }\int \frac{\mathscr {G}(\omega )}{\omega -\lambda } \mathrm{d}\omega \) (P denotes the principal value) where \(\mathscr {G}(\omega )\) is the Fourier transformation of \(G^{+}\).

3 Measures of quantum correlations

In this section, we introduce certain important measures required for quantifying the quantum correlations. Coherence, being at the heart of interference phenomena, plays a central role in physics, as it enables applications that are impossible within classical mechanics or ray optics and can be measured as [1]

$$\begin{aligned} \mathscr {C}_{l_{1}}=\sum _{i\ne j}\mid \rho _{ij}\mid . \end{aligned}$$
(11)

Quantum entanglement has remained a major resource for accomplishing quantum information processing tasks such as teleportation [10], quantum key distribution [11], and quantum computing [2].

Concurrence has been extensively used so far in many contexts among many measures of entanglement of a two-qubit system. The concurrence of a two-qubit mixed state \(\rho \) is defined as [3]

$$\begin{aligned} \mathscr {C}=\mathrm{Max}(0,\lambda _{1}-\lambda _{2}-\lambda _{3}-\lambda _{4}), \end{aligned}$$
(12)

where \(\lambda _{i}\) represents the square root of the ith eigenvalue, in descending order of the matrix \(\rho \widetilde{\rho }\) with \(\widetilde{\rho }=(\sigma _{2}\bigotimes \sigma _{2})\rho ^{T}(\sigma _{2}\bigotimes \sigma _{2})\), while T denotes transposition.

A bipartite quantum state contains both classical and quantum correlations quantified jointly by their quantum mutual information—an information–theoretic measure of the total correlation in a bipartite quantum state. In particular, if \(\rho _{AB}\) denotes the density operator of a composite bipartite system AB and \(\rho _{A}\) (\(\rho _{B}\)) denotes the density operator of part A(B), respectively, then the quantum mutual information is defined as [4]

$$\begin{aligned} {I}(\rho _{AB})=S(\rho _{A})+S(\rho _{B})-S(\rho _{AB}), \end{aligned}$$
(13)

where \(S(\rho )=-\mathrm{tr}(\rho \log _{2}\rho )\) is the von Neumann entropy. If the whole system is a pure state at the initial, \(S(\rho _\mathrm{total})=0\) at any time due to the unitary evolution. Hence, \(S(\rho _{AB})\) measures the entanglement between the system and the environment when we trace out the degree of freedoms of the field.

Quantum discord is a measure of the nonclassical correlation between two subsystems of a quantum system. It includes correlations that are due to quantum physical effects, but do not necessarily involve the concept of quantum entanglement. In fact, it is a different type of quantum correlation than the entanglement because separable mixed states (that is, with no entanglement) can have nonzero quantum discord. Sometimes, it is also identified as the measure of the quantum-ness of the correlation functions. It is defined as [4, 39]

$$\begin{aligned} \mathscr {Q}(\rho _{AB})=I (\rho _{AB})-\mathscr {C}{\mathscr {C}}(\rho _{AB}), \end{aligned}$$
(14)

where \(\mathscr {C}{\mathscr {C}}(\rho _{AB})\) is the classical correlation that depends on the projection operator, and we use the maximum in computing discord. For a general state, quantum discord is hard to compute, and only the X-type state exists in an exact expression. For any two-qubit state, the density matrix is given by the following expression:

$$\begin{aligned} \rho _{AB}= & {} \frac{1}{4}\Bigg (I_{a}\bigotimes I_{b}+\sum _{i=1}^{3}(a_{i}\sigma _{i}\bigotimes I_{b}\nonumber \\&+I_{a}\bigotimes b_{i}\sigma _{i})+\sum _{i,j=1}^{3}C_{ij}\sigma _{i}\bigotimes \sigma _{j}\Bigg ). \end{aligned}$$
(15)

For the class of “X” states, the Bloch vector is along the z axis. The above expression can be simplified as

$$\begin{aligned} \rho _{AB}= & {} \frac{1}{4}\Bigg (I_{a}\bigotimes I_{b}+(a\sigma _{z}\bigotimes I_{b}\nonumber \\&+I_{a}\bigotimes b\sigma _{z})+\sum _{i,j=1}^{3}C_{ij}\sigma _{i}\bigotimes \sigma _{j}\Bigg ). \end{aligned}$$
(16)

The quantum discord is invariant under the local unitary transformations. It has been shown that the \(\rho _{AB}\) can be further simplified as

$$\begin{aligned} \rho _{AB}= & {} \frac{1}{4}\Bigg (I_{a}\bigotimes I_{b}+(a\sigma _{z}\bigotimes I_{b}\nonumber \\&+I_{a}\bigotimes b\sigma _{z})+\sum _{i=1}^{3}C_{i}\sigma _{i}\bigotimes \sigma _{i}\Bigg ), \end{aligned}$$
(17)

with the local unitary transformations [40]. The \({\mathscr {C}}{\mathscr {C}}\) can be measured as [41]

$$\begin{aligned} \mathscr {C}{\mathscr {C}}=S(\rho _{A})-\min \left\{ S_{1},S_{2}\right\} , \end{aligned}$$
(18)

where

$$\begin{aligned} S_{1}= & {} -\frac{1+a+b+c_{3}}{4}\log _{2}\frac{1+a+b+c_{3}}{2(1+b)}\nonumber \\&\quad -\frac{1-a+b-c_{3}}{4}\log _{2}\frac{1-a+b-c_{3}}{2(1+b)}\nonumber \\&\quad -\frac{1+a-b-c_{3}}{4}\log _{2}\frac{1+a-b-c_{3}}{2(1-b)}\nonumber \\&\quad -\frac{1-a-b+c_{3}}{4}\log _{2}\frac{1-a-b+c_{3}}{2(1-b)}. \end{aligned}$$
(19)

and

$$\begin{aligned} S_2=1+f\left( \sqrt{a^{2}+C_{1}^2}\right) . \end{aligned}$$
(20)

The f(t) is defined as \(f(t)=-\frac{1-t}{2}\log _{2}(1-t)-\frac{1+t}{2}\log _{2}(1+t)\). Finally, the quantum discord is given as \(\mathscr {Q}(\rho _{AB})=I (\rho _{AB})-\mathscr {C}{\mathscr {C}}(\rho _{AB})\).

4 Massless scalar field quantized near the Kerr horizon and the two vacuums

In order to determine how the reduced density matrix evolves with proper time near the Kerr black hole, we will introduce the scalar wave equation of the Kerr black hole spacetime. The metric of Kerr spacetime in Boyer–Lindquist coordinates is given as

$$\begin{aligned} \mathrm{d}s^{2}= & {} -\frac{\Delta }{R^{2}}(\mathrm{d}t-a\sin ^{2}\theta \mathrm{d}\varphi )^2\nonumber \\&\quad +\frac{\sin ^{2}\theta }{R^{2}}\left[ (r^{2}+a^{2})\mathrm{d}\varphi -a\mathrm{d}t\right] ^{2}\nonumber \\&\quad +\frac{R^{2}}{\Delta }\mathrm{d}r^{2}+R^{2}\mathrm{d}\theta ^{2}, \end{aligned}$$
(21)

where \(\Delta =(r-r_{+})(r-r_{-})\), \(R^{2}=r^{2}+a^{2}\cos ^{2}\theta \) and \(r_{\pm }=M\pm \sqrt{M^{2}-a^{2}}\). M and a represent the mass and the angular momentum per unit mass of the black hole, respectively. The event horizon of the Kerr black hole is located at \(r=r_{+}\). Liu et al. showed that the scalar field theory in the background Eq. (21) can be reduced to a two-dimensional field theory in the near-horizon region with the dimensional reduction technique [34]. This technique was first employed for the Kerr black hole by Murata and Soda [35] and developed with a more general technique by Iso et al. [36].

First, we further write the action of the massless scalar field as

$$\begin{aligned} \begin{aligned} S[\phi ]=\frac{1}{2}\int \mathrm{d}x^{4}\sqrt{-g}g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi . \end{aligned} \end{aligned}$$
(22)

By substituting Eq. (21) into Eq. (22), and then transforming the radial coordinate r into the tortoise coordinate \(r_{*}\) defined by Eq. (23), now the action reads

$$\begin{aligned} \begin{aligned} \frac{\mathrm{d}r_{*}}{\mathrm{d}r}=\frac{1}{F(r)}=\frac{r^{2}+a^{2}}{\Delta }. \end{aligned} \end{aligned}$$
(23)
$$\begin{aligned} S[\phi ]= & {} -\frac{1}{2}\int \mathrm{d}r_{*}\mathrm{d}t\mathrm{d}\theta \mathrm{d}\varphi \sin \theta \phi \nonumber \\&\times \Bigg [-((r^{2}+a^{2})-F(r)a^{2}\sin ^{2}\theta )\partial _{t}^{2}\nonumber \\&-2a(1-F(r))\partial _{t}\partial _{\varphi }+\left( \frac{F(r)}{\sin ^{2}\theta }-\frac{a^{2}}{r^{2}+a^{2}}\right) \partial _{\varphi }^{2}\nonumber \\&+\partial _{r_{*}}(r^{2}+a^{2})\partial _{r_{*}}+\frac{F(r)}{\sin \theta }\partial _{\theta }\sin \theta \partial _{\theta }\Bigg ]\phi .\nonumber \\ \end{aligned}$$
(24)

We only consider the region near the Kerr horizon. Since \(F(r_{+})\rightarrow 0\) when \(r\rightarrow r_{+}\), we can only consider dominant terms in Eq. (24).

$$\begin{aligned} \begin{aligned} S[\phi ]=&-\frac{1}{2}\int \mathrm{d}r_{*}\mathrm{d}t\mathrm{d}\theta \mathrm{d}\varphi \sin \theta \phi \\&\times \left[ -(r^{2}+a^{2})\partial _{t}^{2}-2a\partial _{t}\partial _{\varphi }\right. \\ {}&\left. -\frac{a^{2}}{r^{2}+a^{2}}\partial _{\varphi }^{2}+\partial _{r_{*}}(r^{2}+a^{2})\partial _{r_{*}}\right] \phi . \end{aligned} \end{aligned}$$
(25)

And then we return to the r coordinate system and use a globally co-rotating coordinate system as

$$\begin{aligned} \begin{aligned} \psi&=\varphi -\frac{a}{a^{2}+r^{2}}t,\\ \xi&=t. \end{aligned} \end{aligned}$$
(26)

In the new coordinates, we can rewrite Eq. (25) as

$$\begin{aligned}&S[\phi ]=\frac{1}{2}\int \mathrm{d}r_{*}\mathrm{d}t\mathrm{d}\theta \mathrm{d}\varphi \sin \theta \phi \nonumber \\&\quad \times \left[ (r^{2}+a^{2})\sin \theta \phi \left( -\frac{1}{F(r)} \partial _{\xi }^{2}+\partial _{r}F(r)\partial _{r}\right) \phi \right] \end{aligned}$$
(27)

Therefore, the angular terms disappear entirely. Using the spherical harmonics expansion \(\phi =\sum _{l,m}\phi _{lm}(\xi ,r)Y_{lm}(\theta ,\psi )\), we obtain the effective two-dimensional action

$$\begin{aligned} \begin{aligned} S[\phi ]=&\sum _{l,m}-\frac{1}{2}\int (r^{2}+a^{2})\mathrm{d}r\mathrm{d}\xi \phi _{lm}\\ {}&\times \left( -\frac{1}{F(r)}\partial _{\xi }^2+\partial _{r}F(r)\partial _{r}\right) \phi _{lm}, \end{aligned} \end{aligned}$$
(28)

where we have used the orthonormal condition for the spherical harmonics as follows:

$$\begin{aligned} \begin{aligned} \int \mathrm{d}\psi \mathrm{d}\theta \sin \theta Y_{l'm'}^{*}Y_{lm}=\delta _{ll'}\delta _{mm'}. \end{aligned} \end{aligned}$$
(29)

From the action Eq. (28), it is apparent that \(\phi \) can be considered as a (1+1)-dimensional massless scalar field in the backgrounds of the dilaton \(\Phi \). The effective two-dimensional metric near the horizon and the dilaton can be written as

$$\begin{aligned} \begin{aligned} \mathrm{d}s^{2}&=-F(r)\mathrm{d}\xi ^{2}+\frac{1}{F(r)}\mathrm{d}r^{2},\\ \Phi&=r^{2}+a^{2}. \end{aligned} \end{aligned}$$
(30)

Hence, we have reduced the four-dimensional field theory to a two-dimensional case. This is consistent with [35, 36]. From Eq. (30), we can define two types of vacuums: the Boulware vacuum and the Unruh vacuum.

According to Eq. (23), the effective two-dimensional metric Eq. (30) changes to

$$\begin{aligned} \mathrm{d}s^{2}=F(r)(-\mathrm{d}\xi ^{2}+\mathrm{d}r_{*}^{2}). \end{aligned}$$
(31)

Equation (31) is exactly conformal to Minkowski metric form. Hence, the scalar field equation reads

$$\begin{aligned} (\partial _{\xi }^{2}-\partial _{r_{*}}^{2})\phi (\xi ,r_{*})=0. \end{aligned}$$
(32)
Fig. 1
figure 1

Left panel: local acceleration \(\kappa _{r}\) vs. mass and angular momentum per mass. Right panel: partial enlarged detail. The system localizes at 1.01\(r_{+}\)

Full size image

We can derive the standard ingoing and outgoing orthonormal mode solutions of Eq. (32): \(\phi (\xi ,r_{*})\sim (e^{-i\omega (\xi +r_{*})},e^{-i\omega (\xi -r_{*})})\). The particle can be suitably defined: the modes are positive frequency modes with respect to the Killing vector field \(\frac{\partial }{\partial \xi }\) for \(\omega >0\). Only the outgoing modes \(\phi (\xi ,r_{*})=\frac{1}{\sqrt{4\pi \omega }}e^{-i\omega (\xi -r_{*})}\) are considered near the horizon. The massless scalar field near the horizon can be quantized as

$$\begin{aligned} \begin{aligned} \phi ^{B}=\sum _{\omega }\left[ a_{\omega }^{B}\phi (\xi ,r_{*})+a_{\omega }^{B\dag }\phi ^{\dag }(\xi ,r_{*})\right] , \end{aligned} \end{aligned}$$
(33)

where \(a_{\omega }^{B}\) and \(a_{\omega }^{B\dag }\) are the annihilation and creation operators acting on the Boulware vacuum state, respectively. The Fock vacuum state corresponds to \(a_{\omega }^{B}|0\rangle =0\). The Wightman function of the Boulware vacuum state can be shown as

$$\begin{aligned} \begin{aligned} G^{B+}(x,x')=-\frac{1}{4\pi ^{2}}\frac{1}{(\Delta \xi -i\epsilon )^{2}}, \end{aligned} \end{aligned}$$
(34)

with the proper \(i\epsilon \) prescription. Its Fourier transform with respect to the proper time \(\mathscr {G}^{B+}(\omega )=0\). The Boulware vacuum corresponds to our familiar notion of a vacuum state.

Now one can define the Unruh vacuum state following the method mentioned above. First, we write the Kerr spacetime line element according to Kruskal-like coordinates

$$\begin{aligned} \begin{aligned} \mathrm{d}s^{2}&=C(r)(-\mathrm{d}T^{2}+\mathrm{d}R^{2}), \end{aligned} \end{aligned}$$
(35)

where \(T=\kappa ^{-1}e^{\kappa r_{*}}\sinh \kappa \xi \), \(R=\kappa ^{-1}e^{\kappa r_{*}}\cosh \kappa \xi \), \(\kappa =\frac{r_{+}-r_{-}}{2(r_{+}^{2}+a^{2})}\), and \(C(r)=e^{-2\kappa r_{*}}F(r)\) is a finite constant near the horizon. As seen from Eq. (35), we can see that Eq. (35) is exactly conformal to Minkowski metric form. Hence, the scalar field equation reads

$$\begin{aligned} \begin{aligned} (\partial _{T}^{2}-\partial _{R}^{2})\phi (T,R)=0. \end{aligned} \end{aligned}$$
(36)

Similar to the previous proceeding, we can derive the outgoing wave equation as \(\phi (\xi ,r_{*})\sim e^{-i\omega (T-R)}\). The particle can be suitably defined: the modes are positive frequency modes with respect to the Killing vector field \(\frac{\partial }{\partial T}\) for \(\omega >0\). Near the horizon, we only consider the outgoing modes \(\phi (T,R)=\frac{1}{\sqrt{4\pi \omega }}e^{-i\omega (T-R)}\), the massless scalar field near the horizon can be quantized as

$$\begin{aligned} \begin{aligned} \phi ^{U}=\sum _{\omega }[a_{\omega }^{U}\phi (T,R)+a_{\omega }^{U\dag }\phi ^{\dag }(T,R)], \end{aligned} \end{aligned}$$
(37)

where \(a_{\omega }^{U}\) and \(a_{\omega }^{U\dag }\) are the annihilation and creation operators acting on the Unruh vacuum state, respectively. The Fock vacuum state corresponds to \(a_{\omega }^{U}|0\rangle =0\). The Wightman function of the Unruh vacuum state can be shown as

$$\begin{aligned} \begin{aligned} G^{U+}(x,x')&=-\frac{1}{4\pi ^{2}}\frac{1}{(\Delta T-i\epsilon )^{2}-\Delta R^{2}}\\&=-\frac{1}{16\pi ^{2}\kappa ^{2}\sinh ^{2}[\frac{\Delta \xi }{2\kappa }-i\epsilon ]}, \end{aligned} \end{aligned}$$
(38)

with the proper \(i\epsilon \) prescription. Its Fourier transform with respect to the proper time is given as \(\mathscr {G}^{U+}(\omega )=\frac{\omega }{2\pi }\frac{1}{1-e^{-2\pi \kappa _{r}^{-1}\omega }}\), where \(\kappa _{r}=\frac{\kappa }{\sqrt{F(r)}}\). It is found that the detector in the Unruh vacuum can spontaneously get excited with a nonvanishing probability, in the same way as the thermal radiation with Hawking temperature from a Kerr black hole. The Hawking–Unruh effect of a Kerr spacetime can be understood as a manifestation of thermalization behavior in an open quantum system [34]. We will only consider the Unruh vacuum in the following studies. The local \(\kappa _{r}\) plays an essential role when studying the characteristics of quantum correlations. This reflects the local curvature of spacetime and embodies the thermal nature of the black hole. In Fig. 1, the local acceleration \(\kappa _{r}\) decreases to a steady value as the mass increases at fixed angular momentum. The local acceleration shows non-monotonic behavior only when the mass is close to the angular momentum per mass. At fixed mass, the local acceleration keeps a steady value when the angular momentum per mass is less than the mass and only significantly decreases when the angular momentum per mass is close to the mass.

The proper acceleration of the stationary detector near the horizon is divergent. Thus, one can define a renormalized value termed surface gravity. The surface gravity is generally the local proper acceleration multiplied by the gravitational time dilation factor (which goes to zero at the horizon). It corresponds to the Newtonian gravitational value in the nonrelativistic limit. For an asymptotic observer, we can use Newtonian gravity to obtain the surface gravity at the Schwarzschild black hole horizon \(\kappa =\frac{M}{r^2}\mid _{r=r_0}=\frac{1}{4M}\) [42]. For a \(3+1\)-dimensional asymptotically flat Kerr black hole with angular speed \(\Omega _{+}=\frac{a}{r_{+}^{2}+a^2}\), one can use it to define an effective spring constant \(k=M\Omega _{+}^{2}\). The surface gravity of the Kerr black hole can be formulated as \(\kappa _\mathrm{Kerr}=\frac{1}{4M}-k\), which decreases when M and angular speed \(\Omega _{+}\) increase [19]. One can naively consider that the reduction of the Schwarzschild black hole surface gravity compensates as a centripetal force for the detector co-rotating with the black hole. The behaviors of \(\kappa \) concerning mass and angular momentum are similar to those of \(\kappa _{r}\), except that \(\kappa \) decreases monotonically concerning mass. In our derivation steps, the effective surface gravity is changed, \(\kappa _{r}=\frac{\kappa }{\sqrt{F(r)}}\), due to dimensional reduction. Finally, it leads to nontrivial behaviors of \(\kappa _{r}\) on the mass and the angular momentum.

5 Equilibrium quantum correlations in curved spacetime

We consider that both detectors couple to the field with the same coupling. The qubit detectors, which localize at the identical location in the co-rotating frame, are initially at the product state. Generalizing Eq. (9) from one atom to two contiguous atoms, the free Hamiltonian becomes \(H_{0}=\frac{\omega }{2}\sigma _{z}^{1}+\frac{\omega }{2}\sigma _{z}^{2}\). The interaction Hamiltonian becomes \(H_{I}=\mu _{1}\sigma _{x}^{1}\Phi (x_{1})+\mu _{2}\sigma _{x}^{2}\Phi (x_{2})\) where \(\sigma _{i}^{1}=\sigma _{i}\otimes \sigma _{0}\), and \(\sigma _{i}^{2}=\sigma _{0}\otimes \sigma _{i}\). The master equation in the proper time coordinate frame reads [19, 20]

$$\begin{aligned} \begin{aligned} \frac{\mathrm{d}\rho _{sys}(\tau )}{\mathrm{d}\tau }&=-i[H_\mathrm{eff},\rho _\mathrm{sys}(\tau )]+\mathscr {L}[\rho _\mathrm{sys}(\tau )],\\ H_\mathrm{eff}&=H_{0}-\frac{i}{2}\sum _{\alpha ,\beta =1}^{2}\sum _{i,j=1}^{3}H_{ij}^{\alpha \beta }\sigma _{i}^{\alpha }\sigma _{j}^{\beta },\\ \mathscr {L}[\rho _\mathrm{sys}]&=\frac{1}{2}\sum _{\alpha ,\beta =1}^{2}\sum _{i,j=1}^{3}\mathscr {C}_{ij}^{\alpha \beta }\left[ 2\sigma _{j}^{\beta }\rho _\mathrm{sys}\sigma _{i}^{\alpha }-\left\{ \sigma _{i}^{\alpha }\sigma _{j}^{\beta },\rho _\mathrm{sys}\right\} \right] . \end{aligned} \end{aligned}$$
(39)

The GSKL matrix \(\mathscr {C}_{ij}^{\alpha \beta }\) is given by the expression

$$\begin{aligned} \begin{aligned} \mathscr {C}_{ij}^{\alpha \beta }=A^{\alpha \beta }\delta _{ij}-iB^{\alpha \beta }\epsilon _{ijk}\delta _{3k}+C^{\alpha \beta }\delta _{3i}\delta _{3j}, \end{aligned} \end{aligned}$$
(40)

where the \(A^{\alpha \beta }\) and \(B^{\alpha \beta }\) and \(C^{\alpha \beta }\) for the two atomic system are defined as

$$\begin{aligned} \begin{aligned} A^{\alpha \beta }&=\frac{\mu ^{2}}{4}(G^{\alpha \beta }(\omega )+G^{\alpha \beta }(-\omega )),\\ B^{\alpha \beta }&=\frac{\mu ^{2}}{4}(G^{\alpha \beta }(\omega )-G^{\alpha \beta }(-\omega )), \end{aligned} \end{aligned}$$
(41)

and \(C^{\alpha \beta }\) is given as \(G(0)-A^{\alpha \beta }\). Similarly, the coefficients of \(H_{ij}^{\alpha \beta }\) can be obtained by replacing \(G^{\alpha \beta }(\omega )\) with \(K^{\alpha \beta }(\omega )\) in the above equations where \(K^{\alpha \beta }(\lambda )=\frac{P}{i\pi }\int ^{\infty }_{-\infty }\frac{G^{\alpha \beta }(\omega )}{\omega -\lambda } \mathrm{d}\omega \) [19].

These results for the Hamiltonian contributions require some further comments. The \(K^{11}\) can be divided into (similar results also hold for \(K^{12}\))

$$\begin{aligned} K^{11}(\lambda )&=\frac{1}{2\pi ^{2}i}\left[ P\int _{0}^{\infty }\mathrm{d}\omega \frac{\omega }{\omega -\lambda }\right. \nonumber \\&\quad \left. +P\int _{0}^{\infty }\mathrm{d}\omega \frac{\omega }{1-e^{2\pi \kappa _{r}^{-1}\omega }}\left( \frac{1}{\omega +\lambda }-\frac{1}{\omega -\lambda }\right) \right] \nonumber \\ \end{aligned}$$
(42)

to a flat and a curvature-dependent piece. Although we do not calculate the above function concretely, the curvature-dependent second term is a finite, odd function of \(\lambda \), vanishing as \(\kappa _{r}\) becomes less. The first contribution in Eq. (42) is divergent. Despite some cancellations that occur in \(H_\mathrm{eff}\), the effective Hamiltonian turns out to be infinite in general, and its definition requires the introduction of a suitable cutoff and a renormalization procedure. The appearance of divergences comes from the nonrelativistic treatment of the two-level atoms, while any reasonable calculation of energy shifts would have required the quantum field theory approaches. In our quantum mechanical setting, the procedure needed to make \(H_\mathrm{eff}\) well defined is straightforward: perform a suitable curvature-independent subtraction so that \(H_\mathrm{eff}\) reproduces the correct quantum field theory result. However, since we are interested in analyzing the effects due to the curvature, we do not need to do this explicitly. In the following, we only consider the correlations induced by the curvature effect [19, 20, 24] by disregarding the Hamiltonian contribution in Eq. (9) and only concentrate on studying the effects induced by the dissipative part. The situation we consider is no real distance between two qubits, which means the system is in a common equilibrium environment. The presence of an equilibrium state \(\rho ^{\infty }\) can be determined by setting \(\mathscr {L}[\rho _\mathrm{sys}(\tau )]=0\). Consider a general density matrix of the two-atom system in the form of \(\rho (\tau )=\frac{1}{4}[{\textbf {1}}\otimes {\textbf {1}}+\rho _{0i}(\tau )\sigma _{0}\otimes \sigma _{i}+\rho _{i0}(\tau )\sigma _{i}\otimes \sigma _{0}+\rho _{ij}(\tau )\sigma _{i}\otimes \sigma _{j}]\), inserting it into \(\mathscr {L}[\rho _\mathrm{sys}(\tau )]=0\), and we derive the following result [19]:

$$\begin{aligned} \begin{aligned} \rho ^{\infty }_{0i}&=\rho ^{\infty }_{i0}=-\frac{R}{3+R^2}(\tau _{*}+3)\delta _{3i}\\ \rho ^{\infty }_{ij}&=\frac{1}{3+R^2}\left[ (\tau _{*}-R^2)\delta _{ij}+R^{2}(\tau _{*}+3)\delta _{3i}\delta _{3j}\right] \end{aligned} \end{aligned}$$
(43)

where \(R = B/A\), \(\tau _{*}\) is the trace of the density matrix \(\tau _{*}=\sum _{i=1}^{3}\rho _{ii}\), which is a constant of motion, and the positivity of \(\rho (0)\) requires that \(-3 \le \tau _{*} \le 1\). At the initial state, consider the direct product of two pure states: \(\rho (0)=\rho _{\vec {a}}\otimes \rho _{\vec {b}}\), where \(\rho _{\vec {a}}=\frac{1}{2}({\textbf {1}}+\vec {a}\cdot \vec {\sigma })\), \(\rho _{\vec {b}}=\frac{1}{2}({\textbf {1}}+\vec {b}\cdot \vec {\sigma })\), and \(\vec {a}\) and \(\vec {b}\) are two unit vectors. In this case, one easily finds that \(\tau _{*}=\vec {a}\cdot \vec {b}\). In this paper, we set \(\vec {a}=(0,0,1)\) and \(\vec {b}=(0,0,-1)\) such that \(\tau _{*}=-1\). The system finally reaches the steady equilibrium state since the two subsystems are coupling to a common thermal bath with the temperature depending on the curvature in a pure open system viewpoint.

Fig. 2
figure 2

Quantum correlations at equilibrium state vs. mass or angular momentum per mass. The angular momentum per mass a is set up as 10, and the mass is changed from 10 in a, b. The mass a is set up as 10.01, and the angular momentum per mass is changed from 0.1 in c, d. The system is located at 1.01\(r_{+}\). The eigenfrequencies of the qubits are \(\omega _{1}=\omega _{2}=0.1\). The coupling \(\mu _{\alpha }=\mu _{\beta }=0.01\)

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On the above basis, we study the quantum correlations of a two-qubit system near the Kerr black hole with a global co-rotating coordinate as shown in Fig. 2. In our setting, the initial state is separable, i.e., there is no quantum correlation initially. After evolution, it has been shown that the system reaches a steady state. More remarkably, the system harvests the quantum correlations from the Unruh vacuum in Fig. 2. It is consistent with the suggestions in Refs. [21, 22, 24]. The system’s initial state is a pure state where the von Neumann entropy vanishes, and the final state is a mixed state with nonvanishing von Neumann entropy. We focus on the steady state. The quantum correlations derive from non-unitary evolution. In Fig. 2a, b, the concurrence, mutual information, discord, and coherence all decrease first and then increase to a constant as the mass of the black hole increases, while the angular momentum per mass is unchanged. On the contrary, the von Neumann entropy of the system, which measures the entanglement between the system and the environment, increases initially and then decreases to a constant. For the black hole with a larger mass, the quantum correlations are neither more sensitive to the change of the angular momentum per mass nor to the mass. The angular momentum per mass significantly affects the quantum correlations only when it is comparable to the mass. This is also demonstrated in Fig. 2c, d. The quantum correlations are boosted by the angular momentum per mass except that the von Neumann entropy decreases. The near-horizon limit suppresses the angular modes, but the angular momentum still plays a crucial role in harvesting the correlations from the field. Furthermore, the angular momentum can amplify the correlations, which is consistent with [44].

Fig. 3
figure 3

Quantum correlations at equilibrium state vs. \(\kappa _{r}\). Other parameters are the same as those in Fig. 2

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The above nontrivial result comes from the dependence of the quantum correlations on the local acceleration \(\kappa _{r}\), which is directly related to the curvature of spacetime. The quantum information between the two qubits is reduced by more significant curvature. On the contrary, the von Neumann entropy increases as shown in Fig. 3. At fixed angular momentum, the \(\kappa _{r}\) decreases to a steady value as the mass increases and only shows non-monotonous behavior when the mass is close to the angular momentum per mass. It keeps a constant value at fixed mass when the angular momentum per mass is less than mass and only significantly decreases when close to the mass. Thus, the behavior of the \(\kappa _{r}\) concerning the mass and angular momentum determines the behavior of the correlations. Moreover, \(\kappa _{r}\) appears to be inversely related to the quantum correlations in the system and similarly to the system–environment entanglement. The larger \(\kappa _{r}\) corresponds to the higher spacetime curvature and higher effective temperature, making the system more classical and weakening the quantum nature. On the contrary, the higher temperature strengthens the interaction between the system and the field. Therefore, this makes it easier for them to correlate. Here, we see the impact of the spacetime structure on the quantum correlations.

Fig. 4
figure 4

Schematic diagram

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6 Nonequilibrium transient quantum correlations in curved spacetime

To study the nonequilibrium quantum correlations in the Kerr black hole spacetime, we introduce an auxiliary system (the same two-level atom) isolated from the environment [45, 46]. The schematic diagram is shown in Fig. 4. The qubit coupled to the field is called A, while the qubit decoupled to the field is called B. The two-qubit system localizes at the same position and co-rotates with the black hole near the horizon. The field is represented by E. Initially, A and B are maximally entangled. Only the A interacts with E. As time goes by, the initial quantum correlations between A and B are expected to be transferred to the correlations between A and E. We can expand any general density matrix for the bipartite two-level atom system as follows:

$$\begin{aligned} \begin{aligned} \rho =\sum _{i,j=0}^{3}\rho _{ij}\sigma _{i}\bigotimes \sigma _{j}, \end{aligned} \end{aligned}$$
(44)

where \(\{\sigma _{i}\bigotimes \sigma _{j}|i,j \in 0,\ldots ,3\}\) forms sixteen linearly independent complete vector basis.

A good property of this basis is that the expansion coefficients \(\rho _{ij}\) are real and satisfy \(\rho ^{\dag }=\rho \) and \(\mathrm{Tr}\rho =1\). The expansion coefficients can be computed directly using \(\rho _{ij}=\mathrm{Tr}\{\rho \sigma _{i}\bigotimes \sigma _{j}\}\).

Substituting Eq. (44) into Eq. (9), we derive

$$\begin{aligned}&\sum _{i,j=0}^{3}\frac{\mathrm{d}\rho _{ij}(\tau )}{\mathrm{d}\tau }\sigma _{i}\bigotimes \sigma _{j}=-i\left[ H_\mathrm{eff},\sum _{i,j=0}^{3}\rho _{ij}(\tau )\sigma _{i}\bigotimes \sigma _{j}\right] \nonumber \\&\quad +\sum _{i.j=0}^{3}\sum _{m=1}^{3}\rho _{ij}\left[ 2L_{m}\sigma _{i}L_{m}^{\dag }-L_{m}^{\dag }L_{m}\sigma _{i}-\sigma _{i}L_{m}^{\dag }L_{m}\right] \bigotimes \sigma _{j}.\nonumber \\ \end{aligned}$$
(45)
Fig. 5
figure 5

Nonequilibrium quantum correlation evolution vs. time with the different angular momentum per mass in the unit of \(\frac{1}{\mu ^{2}}\) at \(a=0.01\), \(M=10\) (upper panel) and \(a=9.9\), \(M=10\) (lower panel). Other parameters are the same as those in Fig. 2

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We comment more about the above equation. The auxiliary atom is isolated from the environment, meaning the environment has no interaction or dissipative effect on it. During evolution, the system and the environment exchange energy and information. In the transient process, the interior of the system is unbalanced. The position of the auxiliary qubit not coupled with the field will still influence the dynamics of the system. The Lindblad operators only act on one of the two qubits, and another one is free to evolve. Although the isolated qubit does not interact with the field, it inevitably couples to the gravity field. Its energy gap will be redshifted relative to its location to the black hole. Moreover, that affects evolution. We assume that the auxiliary qubit is localized at the same point with the original qubit and that two qubits have the same gap in the following computation.

From Eq. (42), we can derive the time-dependent density matrix elements,

$$\begin{aligned} \rho _{0j}(\tau )= & {} \rho _{0j}(0),\nonumber \\ \rho _{1j}(\tau )= & {} \rho _{1j}(0)e^{-\frac{A\tau }{2}} \cos (\Omega \tau )-\rho _{2j}(0)e^{-\frac{A\tau }{2}}\sin (\Omega \tau ), \nonumber \\ \rho _{2j}(\tau )= & {} \rho _{1j}(0)e^{-\frac{A\tau }{2}}\sin (\Omega \tau ) +\rho _{2j}(0)e^{-\frac{A\tau }{2}}\cos (\Omega \tau ),\nonumber \\ \rho _{3j}(\tau )= & {} \rho _{1j}(0)e^{-A\tau }-\frac{B}{A}\rho _{0j}(0)(1-e^{-A\tau }), \end{aligned}$$
(46)

where \(A=\gamma _{+}+\gamma _{-}\) and \(B=\gamma _{+}-\gamma _{-}\). In the following, we consider that two atoms initially share a maximally entangled state, i.e., \(\rho _{00}=\rho _{11}=-\rho _{22}=\rho _{33}=\frac{1}{4}\), while the rest of \(\rho _{ij}\) vanishes.

In Fig. 5, we plot various quantum correlations vs. time in the unit of \(\frac{1}{\mu ^{2}}\). In Fig. 5, all quantum correlations between two qubits decrease to zero due to the dissipative effect of the environment, but the von Neumann entropy increases to a constant in time. Comparing Fig. 5a–d, for a black hole with a larger mass relative to angular momentum, the quantum correlations decrease faster and reach a larger entropy. We study the dependence of the quantum correlations on the mass in Fig. 6a, b and on the angular momentum per mass in Fig. 6c, d. The behaviors of the quantum correlations are very similar to the equilibrium case: concurrence, mutual information, discord, and coherence all decrease first and then increase to a constant as the mass of the black hole increases, while the angular momentum per mass is unchanged. On the contrary, the entropy increases initially and decreases to a constant. The angular momentum can amplify quantum correlations, especially when the angular momentum becomes larger. Also, the quantum correlations vs. the local curvature \(\kappa _{r}\) are plotted in Fig. 7a, b. All the quantum correlations between the two atoms decrease by more significant curvature. On the contrary, the von Neumann entropy increases. This implies that the quantum correlations in this nonequilibrium model are also determined by the local curvature or local acceleration \(\kappa _{r}\). The larger \(\kappa _{r}\) makes the system more classical, weakening the quantum nature. On the contrary, the larger \(\kappa _{r}\) strengthens the interaction between the system and the field, making them more easily correlate.

Fig. 6
figure 6

a, b Quantum correlations at nonequilibrium transient state vs. mass when \(t=\frac{100}{\mu ^{2}}\) and \(a=10\). c, d Quantum correlations at nonequilibrium transient state vs. the angular momentum per mass when \(t=\frac{100}{\mu ^{2}}\) and \(m=10\). Other parameters are the same as those in Fig. 2

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Fig. 7
figure 7

Quantum correlations at nonequilibrium transient state vs. \(\kappa _{r}\) when \(t=\frac{100}{\mu ^{2}}\). Other parameters are the same as those in Fig. 2

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We quantify the nonequilibrium by considering entropy production (EP) and entropy production rate (EPR) for our setting. The initial state of the correlated system AB is denoted by \(\rho _{AB}^{i}\), and the initial state of the field is denoted by \(\rho _{E}^{i}\). In our setup, we assume there is no correlation between the system AB and E initially, so the initial state of the total system reads \(\rho _{ABE}^{i}=\rho _{AB}^{i}\bigotimes \rho _{E}^{i}\). The whole system is isolated and follows unitary evolution. At the final state of ABE, the density matrix is given by

$$\begin{aligned} \begin{aligned} \rho _{ABE}^{f}=U_{ABE}\rho _{AB}^{i}\bigotimes \rho _{E}^{i}U_{ABE}^{\dag }, \end{aligned} \end{aligned}$$
(47)

and the evolution of AE is given as

$$\begin{aligned} \begin{aligned} \rho _{AE}^{f}=U_{AE}\rho _{AE}U_{AE}^{\dag }, \end{aligned} \end{aligned}$$
(48)

where \(U_{AE}\) is associated with the unitary transformation. The entropy production of the system A w.r.t. the evolution \(U_{AE}\) can be given as [47, 48]

$$\begin{aligned} \begin{aligned} \Sigma _{A}(t_{i}:t_{f}):=I _{A:E}(t_{f})+S (\rho _{E}^{f}\mid \mid \rho _{E}^{i}), \end{aligned} \end{aligned}$$
(49)

where \(S (\rho \mid \mid \sigma )=Tr\rho \ln \rho -\rho \ln \sigma \) is the relative entropy.

The entropy production of the system AB about the evolution \(U_{AE}\) is

$$\begin{aligned} \begin{aligned} \Sigma _{AB}(t_{i}:t_{f})=I _{AB:E}(t_{f})+S (\rho _{E}^{f}\mid \mid \rho _{E}^{i}). \end{aligned} \end{aligned}$$
(50)

The entropy production can be rewritten as \(\Sigma _{AB}(t_{i}:t_{f})\)

$$\begin{aligned} \begin{aligned} \Sigma _{AB}(t_{i}:t_{f}):=\Delta I _{A:B}(t_{i}:t_{f})+\Sigma _{A}(t_{i}:t_{f}), \end{aligned} \end{aligned}$$
(51)

where \(\Delta I _{A:B}(t_{i}:t_{f}):=I _{A:B}(t_{i})-I _{A:B}(t_{f})\), \(I _{A:B}\) is mutual information between A and B. For the detailed derivation, see Appendix A. In our early setup, we have assumed a Born approximation: the coupling between the system and the bath is weak such that the influence of the bath is small. Thus, we can consider the bath is almost unchanged. Then the state of the total system at time \(\tau \) may be approximately characterized by a tensor product as Eq. (6), so that \(U_{ABE}\rho _{AB}(\tau ^{i})\bigotimes \rho _{E}U_{ABE}^{\dag }\approx \rho _{AB}(\tau ^{f})\bigotimes \rho _{E}\). Hence, \(\Sigma _{AB}(t_{i}:t_{f})\ge \Delta I _{A:B}(t_{i}:t_{f})\ge 0\) are due to the positivity of \(I _{A:B}(t_{i}:t_{f})\) and the quantum relative entropy. Considering the Born approximation, we further derive a lower bound of the entropy production:

$$\begin{aligned} \begin{aligned} \Delta I _{A:B}(t_{i}:t_{f})&\ge I _{A:E}(t_{f})\\&=S _{A}^{f}+S _{E}^{f}-S _{AE}^{f}\approx 0\\ S (\rho _{E}^{f}\mid \mid \rho _{E}^{i})\approx 0. \end{aligned} \end{aligned}$$
(52)

The first greater than or equal sign that comes from the scrambling information between A and B is greater than the increased information between A and E. The latter approximately equal sign is due to the Born approximation.

Fig. 8
figure 8

a EPR at nonequilibrium state varies with time t when angular momentum \(a=9.9\) and the mass \(M=10\). b EPR vs. mass when \(a=10\) and \(t=\frac{100}{\mu ^{2}}\). c EPR vs. the angular momentum per mass when \(m=10\) and \(t=\frac{100}{\mu ^{2}}\). d EPR vs. \(\kappa _{r}\) when \(t=\frac{100}{\mu ^{2}}\). Other parameters are the same as those in Fig. 2

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For a nonequilibrium state, the EPR (EP defined above in unit time) quantifying the dissipative cost is always larger than zero. The Born–Markov approximation is assumed such that the two positive parts are omitted. Hence, the EPR in our paper is actually a lower bound. The EPR of the system is plotted in Fig. 8. The EPR decreases in time and finally vanishes. At fixed time, the EPR varies non-monotonically with respect to the mass. The increase in angular momentum can amplify EPR. The nontrivial behavior of EPR on the mass and the angular momentum also comes from the fact that the mass and the angular momentum are directly related to the local curvature or acceleration \(\kappa _{r}\). The EPR varying with \(\kappa _{r}\) is plotted in Fig. 8d. The EPR decreases monotonically with the local curvature.

Fig. 9
figure 9

The decay rates of the quantum correlations with respect to the mass when \(a=10\), \(t=\frac{100}{\mu ^{2}}\) or w.r.t. the angular momentum per mass when \(m=10\), \(t=\frac{100}{\mu ^{2}}\). Other parameters are the same as those in Fig. 2

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Fig. 10
figure 10

The decay rates of the quantum correlations with respect to the mass when \(a=10\), \(t=\frac{100}{\mu ^{2}}\), or w.r.t. the angular momentum per mass when \(m=10\), \(t=\frac{100}{\mu ^{2}}\). Other parameters are the same as those in Fig. 2

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Fig. 11
figure 11

The decay rates of the quantum correlations with respect to the local acceleration or curvature \(\kappa _{r}\) when \(t=\frac{100}{\mu ^{2}}\). Other parameters are the same as those in Fig. 2

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The decay rates are defined as \(\frac{QC(t_{0})-QC(t)}{t-t_{0}}\), where QC is quantum correlations. The decay rates of correlations concerning the mass or the angular momentum per mass are plotted in Figs. 9 and 10. The decay rates of the quantum correlations behave similarly for the mass or the angular momentum per mass except for the von Neumann entropy. The decay rates of the quantum correlations increase at first and then decrease with the mass. Moreover, the decay rates of the quantum correlations decrease with the angular momentum per mass. However, the decay rate of the von Neumann entropy, which is negative growth, decreases at first and then increases with the mass and increases with respect to the angular momentum per mass. The dependence of the decay rates on mass and angular momentum can be understood through the dependence of the decay rate on \(\kappa _{r}\) and the dependence of \(\kappa _{r}\) on mass and angular momentum. The decay rates of correlations vs. the \(\kappa _{r}\) are plotted in Fig. 11. The decay rates of the entanglement, coherence, discord, mutual information, and EPR increase as \(\kappa _{r}\), but the decay rate of the von Neumann entropy is negative and decreases. This shows that the thermal effect of the \(\kappa _{r}\): the higher the effective temperature is, the more significant the dissipative effect of the environment. Once again, we see the influence of the spacetime structure on the quantum correlations of the system.

7 Nonequilibrium steady quantum correlations

To generalize the above case to the intrinsic nonequilibrium case where the detailed balanced is not preserved, we introduce another massless scalar field \(\psi \), and assume no interaction between the two fields. The motivations for introducing another field are as follows. When there are two qubits at different locations, things are more subtle. There does not appear to exist a global proper time. The proper time is radius-dependent. To derive an equation of motion for the system, we must choose an observer’s proper time as the global time to obtain a master equation. Considering that the observer is fixed at a specific location, the whole world is synchronous with the observer’s proper time. Then the observer launches the detectors to examine the temperatures of the other places and then sends the data back to him or herself. The observer must admit that a nonequilibrium bath is present where the effective temperature (\(T_\mathrm{eff}\propto \kappa _{r}\) [33, 34]) is radius-dependent. Now the field becomes nonuniform, and its temperature is location-dependent in the observer’s viewpoint, which is very difficult to tackle, especially for the two-point Wightman function between the two separable qubits. Thus, for simplicity, we introduce another type of field. Due to no direct interaction between two types of fields, the two qubits are localized at different baths in the observer’s viewpoint. The evolution dynamics of the system only is determined by the respective correlation functions of the two baths. Therefore, we can bypass the complex issues and focus on the simplified nonequilibrium scenario. We can derive the analytical steady state of the nonequilibrium scenario we are interested in. The flux and the EPR are well defined in the meantime. Furthermore, we introduce the interaction between the two qubits. The Hamiltonian is generalized as

$$\begin{aligned} \begin{aligned} H_\mathrm{total}&=H_{0}+H_{1}=H_\mathrm{sys}+H_\mathrm{field}+H_{I},\\ H_\mathrm{sys}&=\frac{\omega _{1}}{2}\sigma _{z1}+\frac{\omega _{2}}{2}\sigma _{z2}+K(\sigma _{1+}\sigma _{2-}+\sigma _{1-}\sigma _{2+}),\\ H_{I}&=\mu _{1}(\sigma _{1+}+\sigma _{1-})\phi (x_{1}(\tau _{1}))+\mu _{2}(\sigma _{2+}\\ {}&\quad +\sigma _{2-})\psi (x_{2}(\tau _{1})), \end{aligned} \end{aligned}$$
(53)

where K is the coupling of the inter-qubits, and \(\mu _{1}=\mu _{2}=0.01\). The eigenenergy and the eigenstates of two qubits system are \(E_{1}=-\frac{\omega _{1}+\omega _{2}}{2}\), \(|\lambda _{1}\rangle =|0,0\rangle \); \(E_{2}=\frac{\omega _{1}+\omega _{2}}{2}\), \(|\lambda _{2}\rangle =|1,1\rangle \); \(E_{3}=\kappa \), \(|\lambda _{3}\rangle =\cos (\theta /2)|1,0\rangle +\sin (\theta /2)|0,1\rangle \); \(E_{4}=-\kappa \), \(|\lambda _{4}\rangle =-\sin (\theta /2)|1,0\rangle +\cos (\theta /2)|0,1\rangle \). We define \(\kappa =\sqrt{K^{2}+(\omega _{1}-\omega _{2})^{2}/4}\) and \(\theta =\arctan (2K/(\omega _{1}-\omega _{2}))\). In this paper, we only consider the symmetric case, which means \(\omega _{1}=\omega _{2}\). On the eigenenergy basis, we can define two groups of transitions operators:

$$\begin{aligned} \begin{aligned} V_{1,1}&=\cos (\theta /2)(|\lambda _{1}\rangle \langle \lambda _{1}|+|\lambda _{4}\rangle \langle \lambda _{2}|),\\ V_{1,2}&=\sin (\theta /2)(|\lambda _{3}\rangle \langle \lambda _{2}|-|\lambda _{1}\rangle \langle \lambda _{4}|),\\ V_{2,1}&=\sin (\theta /2)(|\lambda _{1}\rangle \langle \lambda _{3}|-|\lambda _{4}\rangle \langle \lambda _{2}|),\\ V_{2,2}&=\cos (\theta /2)(|\lambda _{3}\rangle \langle \lambda _{2}|+|\lambda _{1}\rangle \langle \lambda _{4}|), \end{aligned} \end{aligned}$$
(54)

with transition frequency \(\Omega _{1}=E_{2}-E_{3}\) and \(\Omega _{2}=E_{2}+E_{3}\).

A stationary detector will experience a thermal bath with an effective temperature \(\frac{k_{r}}{2\pi }\) near the horizon. In his viewpoint, any two different points have different temperatures. Now, if we separate the two-qubit system with a finite distance along the radius (\(r_{*}\)), the system is equivalently the case connected to two independent baths. Therefore, one can derive a nonequilibrium master equation for two separated atoms in the observer’s frame. By using one of the two detector’s proper time, the master equation can be derived (neglecting the contribution of the principal value which can only modify the energy level)  [49,50,51]

$$\begin{aligned} \frac{\mathrm{d}\rho _\mathrm{sys}(\tau )}{\mathrm{d}\tau }&=-i[H_\mathrm{sys},\rho _\mathrm{sys}(\tau )]+\mathscr {L}_{1}[\rho _\mathrm{sys}]+\mathscr {L}_{2}[\rho _\mathrm{sys}],\nonumber \\ \mathscr {L}_{j}[\rho _\mathrm{sys}]&=\sum _{\mu =1}^{2}G^{i}(-\omega _{\mu })\left( 2V_{j,\mu }\rho _\mathrm{sys}V_{j,\mu }^{\dagger }-{\rho _\mathrm{sys},V_{j,\mu }^{\dagger }V_{j,\mu }}\right) \nonumber \\&\quad +G^{i}(\omega _{\mu })\left( 2V_{j,\mu }^{\dagger }\rho _\mathrm{sys}V_{j,\mu }-{\rho _\mathrm{sys},V_{j,\mu }V_{j,\mu }^{\dagger }}\right) ,\nonumber \\ \end{aligned}$$
(55)

\(G^{i}\) corresponds to the Fourier transform of the Green function for different fields. From Eq. (55), it is sufficient to obtain the steady state, and the concrete expression is in Appendix B. The frequency gap of the two identical atoms is different when the two atoms keep different separation distances from a stationary detector’s viewpoint due to the redshift effect. However, we are not interested in the redshift effect here and always set the frequency gap of two atoms to be the same.

Fig. 12
figure 12

We consider a nonequilibrium scenario: one of the two atoms is stationary at \(1.006r_{+}\), while another is at \((1.006+\Delta r)r_{+}\). Quantum correlations at nonequilibrium steady state vary with the separation distance for \(a=10\) or \(M=10\)

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Fig. 13
figure 13

We consider a nonequilibrium scenario: one of the two atoms is stationary at \(1.006r_{+}\), while another is at \((1.006+\Delta r)r_{+}\). Quantum correlations at nonequilibrium steady state vary with separation distance for \(a=10\) or \(M=10\)

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Fig. 14
figure 14

The a, b flux and c, d EPR at nonequilibrium steady state vary with separation distance for \(a=10\) or \(M=10\). e, f The \(\kappa _{r}\) varies with separation distance for \(a=10\) or \(M=10\). One of the two atoms is stationary at \(1.006r_{+}\), while another is at \((1.006+\Delta r)r_{+}\)

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Under these considerations, we now investigate the quantum correlations of the nonequilibrium steady state on the bare basis. The \(\Delta r\) measures nonequilibrium. The difference in radius reflects the difference in the local acceleration or the local spacetime curvature. It shows the difference in temperatures through the Unruh–Hawking effect. Thus the case is similar to the two qubits coupled to individual baths separately with the different temperatures. Therefore, the system is in nonequilibrium. However, there is an interesting question. Are the quantum correlations in curved spacetime background the same or different from the case where the system is coupled to two corresponding baths? Addressing this issue can help us understand whether the effects of curved spacetime are equivalent to the temperature on the global correlation level. If there is only one field and only qubit–field interaction, as in Sect. 5, it has been shown that the spacetime curvature, the acceleration, and the temperature influence differently on the quantum correlations [7, 8]. Now let us go back and look at this case. Strictly speaking, the coupling of the inter-qubits K relies on the distance and the spacetime curvature. This directly reflects that the spacetime curvature, the acceleration, and the temperature influence differently on the system. However, we set K as a constant for simplicity. The effects of the spacetime curvature between the two point-like qubits on the quantum correlations are weakened in some sense. Even so, we can still perceive the different effects of the spacetime curvature and the temperature on the quantum correlations in our setup. We found the separation distance directly determines the property of the system due to the spacetime curvature. The system coupled to two corresponding baths, however, is not related to the separation distance between the two qubits. Moreover, the redshift effect caused by the local curvature can modify the energy levels of the system and further influence the quantum correlations of the system. These show the difference between the effects of the curved spacetime and the temperature on the global correlation level. Besides that, our system is very similar to the system coupled with two corresponding baths. To only consider the nonequilibrium effect induced by different locations, as mentioned before, we omit the redshift effect.

The final state contains no information about the initial state. We plot the correlations varying with \(\Delta r\) under different mass or different angular momentum. On the whole, the correlations arrive at a steady value when \(\Delta r\) is significant. For the entanglement, the discord and the mutual information in Figs. 12a, b, f, 13b, vary non-monotonously with \(\Delta r\). More importantly, they can be amplified by the nonequilibrium. The coherence and the von Neumann entropy monotonously decrease to a constant with \(\Delta r\) for both different masses and the angular momentum. The nonequilibrium appears to reduce the coherence and the correlation between the system and the environment.

The flux measures the energy exchange between the system and the environment. The energy flux from the ith field to the system at the steady state is given by \(I_{i}=Tr[\mathscr {L}_{i}(\rho _{sys})H_{sys}]\) (we can check \(I_{1}+I_{2}=0\), which satisfies flux conserved). The flux increases to a constant in Fig. 14a, b. This means that the energy exchange capacity of the system has an upper bound and is limited to the environment. We can also define an effective EPR: \(I(\frac{1}{T_{1}}-\frac{1}{T_{2}})\); the temperature is related to the local curvature or the acceleration \(k_r\). The EPR increases when the \(\Delta r\) increases, as we expect in Fig. 14c, d.

Fig. 15
figure 15

Quantum correlations at nonequilibrium steady state vary with \(\kappa _{r1}\) and \(\kappa _{r2}\)

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The above nontrivial phenomena can also be understood from the dependence of the local curvature \(\kappa _{r}\) on the mass and the angular momentum. The difference is that the system is now determined by not only \(\kappa _{r1}\), but also \(\kappa _{r2}\). This leads to a different effect compared to the previous nonequilibrium model. We plot the quantum correlations, the flux, and EPR varying with both \(\kappa _{r1}\) and \(\kappa _{r2}\) in Fig. 15. These figures are symmetric along the line \(\kappa _{r1}=\kappa _{r2}\). We can see that the entanglement, the discord, and the mutual information in Fig. 15a, c, d show the non-monotonic behaviors as the results before, while the coherence, the von Neumann entropy, the flux, and the EPR show the monotonic behaviors. The above figures explain Figs. 12, 13, and 14 well according to the behaviors of \(\kappa _{r}\) in Fig. 14e, f. The non-monotonic behaviors of entanglement, discord, and mutual information can also be understood as a competition between the populations and the coherence [51]. The concurrence in this model can be formulated as \(\mathscr {C}=Max(0,\mathscr {C}_{l_{1}}-\sqrt{\rho _{11}\rho _{22}})\), where \(\mathscr {C}_{l_{1}}\) is the coherence. Thus, the concurrence is directly dependent on the coherence and the population. We see that both the coherence and the population vary non-monotonically. Although the discord and mutual information cannot be derived with a similar formula, we believe that this competition perspective still holds for the discord and the mutual information because they measure the quantum correlations with certain similar parts in some sense. The \(\kappa _{r}\) amplifies the von Neumann entropy. The reason is apparent: the higher temperature leads to strengthening the interaction between the system and the field. Therefore, this makes it easier to produce the correlation. The coherence is representation-dependent; it vanishes on the eigenenergy basis, but is non-vanishing on the bare basis. The non-vanishing coherence in the bare basis is induced by the interaction of inter-qubits and is proportional to \(|\rho _{33}-\rho _{44}|\). One can also understand the behavior of coherence from a competition relationship. On the one hand, when the temperature is low, only the ground state \(|\lambda _{1}\rangle \) and the first excited state \(|\lambda _{4}\rangle \) are significantly occupied; then, the coherence is proportional to \(\rho _{44}\) and increases with temperature. As the temperature increases, the second excited state \(|\lambda _{3}\rangle \) also starts to be occupied. The coherence is then proportional to \(|\rho _{33}-\rho _{44}|\), which shows a competition relationship. As long as the temperature is high enough, the coherence decreases and vanishes at the infinite temperature.

Here, we see that the information of the two separate qubits is encoded in a spacetime structure again. Unlike the previous nonequilibrium case, the qubit system here involves two different locations so that the intrinsic nonequilibrium emerges where detailed balanced is explicitly broken. As a result, the energy flux and associated dissipative cost emerge. They are used to support sustaining and survival of the quantum correlations for a long time (at the steady state). Therefore, nonequilibrium can contribute to forming the quantum correlation of the system.

8 Conclusion

In this paper, we focus on the quantum correlations of curved spacetime near the horizon of the Kerr black hole by using dimensional reduction and the Born–Markov master equation. We quantify the quantum correlations of the two-qubit system and the entanglement between the system and the environment. In the equilibrium model, we can obtain a steady state containing the initial partial information. It is possible to harvest the quantum correlations from the Unruh vacuum. We investigate how the quantum correlations vary with the mass and the angular momentum. We found that the quantum correlations in the system decrease at first and then increase to a constant as the mass increases from a value close to the angular momentum per mass. The growth of angular momentum can amplify the quantum correlations. The entanglement between the system and environment behaves oppositely to the correlations within the system. Importantly, we found that the increase of the local spacetime curvature can reduce the correlations in the system due to the thermal Unruh effect but enhance the entanglement between the system and environment.

In the second nonequilibrium transient model, we found that the information scrambles inevitably to the environment. The angular momentum weakens the scrambling, but the mass relates to the correlations non-monotonously seen from the decay rates of the correlations. The entanglement between the system and environment also behaves oppositely to the correlations in the system. At the fixed time, the quantum correlations are very similar to the equilibrium case: the quantum correlations in a two-qubit system vary non-monotonically with the mass of the black hole and are amplified by the angular momentum; the increase of the spacetime curvature will suppress the quantum correlations in the system. We quantify the system’s EPR (entropy production rate) and found that it decreases in time. The EPR decreases at first and then increases to a constant with respect to the increase of mass and increases when the angular momentum increases. Besides, the EPR decreases monotonically with the local curvature. We also found that the local curvature can enhance the decay rates of quantum correlations, but reduce the decay rate of the von Neumann entropy, which is negative growth. The \(\kappa _{r}\) stands for the local curvature of spacetime and the thermal nature of the black hole. The similar behaviors of the quantum correlations in the above two scenarios are because the system state is determined by \(\kappa _{r}\). More profoundly speaking, the features and characteristics of the system information are encoded in the spacetime structure.

In the third nonequilibrium model, we investigate the quantum correlations of the nonequilibrium steady state. The quantum correlations survive and sustain at a steady when \(\Delta r\) (which measures the nonequilibrium) is significant. The entanglement, discord, and mutual information behave non-monotonically under certain parameters. This means that nonequilibrium can amplify the quantum correlations. The coherence monotonically decreases to a constant. The von Neumann entropy decreases to a constant, which means that the nonequilibrium reduces the correlation between the system and the environment. The flux that measures the degree of the detailed balance breaking increases to a constant. The EPR as a nonequilibrium thermodynamics dissipative cost increases when the \(\Delta r\) increases, as we expect. We can qualitatively understand the above nontrivial behaviors by checking the system’s dependence on local curvatures or accelerations \(\kappa _{r1}\) and \(\kappa _{r2}\). The information of the quantum correlations of two separate qubits system is encoded in the spacetime structure and from the nonequilibrium contribution in this model.