- Email: [email protected]
Violation of the quantum regression theorem and the Leggett–Garg inequality in an exactly solvable model
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Violation of the quantum regression theorem and the Leggett–Garg inequality in an exactly solvable model
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Article history: Received 21 March 2017 Accepted 16 May 2017 Available online xxxx Communicated by P.R. Holland
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Masashi Ban
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Keywords: Quantum regression theorem Leggett–Garg inequality Non-Markovianity Jaynes–Cummings model
Violation of the quantum regression theorem and the Leggett–Garg inequality is studied by means of the exactly solvable multi-mode Jaynes–Cummings model. An exact expression of a two-time correlation function is compared with that derived by the quantum regression theorem. It is found that the quantum regression theorem is not valid even if the reduced time evolution of the qubit is Markovian. Furthermore, it is shown that if the quantum regression theorem is applied in this model, the Leggett–Garg inequality is satisfied while it is violated by the exact correlation function. © 2017 Elsevier B.V. All rights reserved.
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1. Introduction
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A quantum system in a realistic situation is inevitably influenced by its surrounding environment. Such an open quantum system undergoes irreversible time evolution from a prepared initial state to a stationary state [1–5], during which quantumness such as coherence, discord and entanglement is degraded. A state of a quantum system at time t is described by a statistical operator ρ (t ), the time-evolution of which is determined by several methods such as the quantum master equation and the stochastic Schödinger equation [5]. Statistical properties of a quantum system, which depends on a single time, can be derived from the statistical operator ρ (t ). However, it is not sufficient to explain all properties of a quantum system. For instance, a two-time correlation function is necessary to explain how a quantum system responds to an external field applied to the system [4,5]. If the reduced time-evolution of a quantum system is described by the Markovian dynamical map with the semigroup property, thanks to the quantum regression theorem (QRT) [5], a two-time correlation function can be calculated if one finds the time-evolution of the statistical operator ρ (t ). However, it has been pointed out that the QRT is no longer valid if the reduced time-evolution is non-Markovian [6–14]. Recently the violation of the QRT has been investigated in detail by means of an exactly solvable pure dephasing model, and the relation to the non-Markovianity of the reduced time-evolution has been discussed [14].
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E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physleta.2017.05.036 0375-9601/© 2017 Elsevier B.V. All rights reserved.
A two-time correlation function is of essential importance in the foundations of quantum mechanics as well as the practical applications. For instance, one needs to calculate two-time correlation functions of a dichotomous variable in order to examine whether the Leggett–Garg inequality (LGI) is satisfied or not for a quantum system [16–27]. Non-invasive measurability and macrorealism lead to the LGI which is considered as a temporal version of the Bell inequality [28,29]. The violation of the LGI implies a manifestation of quantumness. So it is important to investigate whether the LGI is fulfilled or not for a quantum system which is placed in under the influence of an environment. If the QRT is applicable, one can use a statistical operator ρ (t ) or its equation of motion to examine the LGI. Recently the violation of the LGI has been shown by means of the QRT and the Markovian quantum master equation [27]. In this paper, using an exactly solvable model, we investigate whether the QRT and the LGI are valid or not for an open quantum system. The model considered in this paper is that a qubit (a two-level system) interacts with a set of harmonic oscillators and there is at most a single excitation in the whole system. Although the QRT has been examined by means of the exactly solvable pure dephasing model [14], the model used in this paper includes an energy relaxation. For this model, it is well-known that the exact statistical operator of the qubit can be obtained [5]. Recently it has been found that two-time correlation functions are also exactly calculated even if there is initial correlation between the qubit and the environment [15]. Hence this model is suitable for examining the violation of the QRT and the LGI. It will be found that the LGI is satisfied if the QRT is used to calculate a two-time corre-
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lation function while an exact one violates the LGI. In Section 2, we briefly explain the model and the relations among the timedependent parameters that are used to calculate two-time correlation functions. In Section 3, the exact two-time correlation function is compared with that obtained by making use of the QRT. The violation of the LGI is examined. In Section 4, we give concluding remarks.
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2. Single excitation model for quantum system and environment
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We briefly explain the exactly solvable model of a qubit and environmental system [15], which is used to calculate two-time correlation functions of the qubit. We suppose that a qubit interacts with a multi-mode environmental system. We denote | g and |e as ground and excited states of the qubit. Then the excitation number is given by N q = |e e |. We also denote the excitation number of the environment as N e = k N k where N k represents the excitation number of the kth mode of the environment. We assume that the Hamiltonian H of the qubit and the environment conserves the total excitation number N = N q + N e , that is, [ H , N ] = 0. Furthermore we assume that there is at most a single excitation in the whole system at an initial time. Then the whole system can be described by a Hilbert space spanned by {| g |0, | g |1k , |e |0}, where |0 is the ground state of the environment (N k |0 = 0 for all k) and |1k is a single-excited state in which only the kth mode has one excitation, that is, N k |1l = δkl |1l . Since the total excitation number is conserved during the time evolution, the basic vectors evolve as follows:
e
−i Ht /¯h
| g |0 = | g |0,
e −i Ht /¯h |e |0 = A (t )| g |0 +
e −i Ht /¯h |0|1k = B k (t )|e |0 +
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A (t 1 + t 2 ) = A (t 1 ) A (t 2 ) +
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B k (t 1 + t 2 ) = B k (t 1 ) A (t 2 ) +
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A l (t 1 ) B lk (t 2 ),
(5)
B km (t 1 ) B m (t 2 ),
B kl (t 1 + t 2 ) = B k (t 1 ) A l (t 2 ) +
(6)
B km (t 1 ) B mk (t 2 ).
(7)
m
A (t 1 ) = A (t 1 + t 2 ) A ∗ (t 2 ) +
A k (t 1 + t 2 ) A k∗ (t 2 ),
(8)
k
∗
B k (t 1 ) = B k (t 1 + t 2 ) A (t 2 ) +
+
h¯ ω
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h¯ (λk σ+ ak + λk σ
† − ak ),
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where σ+ = (σ− )† = |e g | and ak and are annihilation and creation operators of the kth mode of the environmental oscillator with angular frequency ωk . In this case, the total excitation num † ber is given by N = σ+ σ− + k ak ak which satisfies the commutation relation [ N , H ] = 0. Furthermore we assume the Lorentzian spectral density of the qubit-environmental coupling [5],
J () =
2π
λ2 , ( − ω − )2 + λ2
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λ + i
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−i ωt − 12 (λ+i )
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A (t ) = e
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In the resonance case ( = 0), if 2γ > λ, the reduced timeevolution of the qubit is non-Markovian and otherwise it is Markovian [30,31]. 3. Quantum regression theorem and Leggett–Garg inequality
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In this section, we compare an exact two-time correlation function with that obtained by making use of the QRT. For this purpose, we suppose that the qubit is prepared in an arbitrary state ρ and the environment is in the ground state ρe = |00| at an initial time t = 0. Then the statistical operator ρ (t ) of the qubit reduced state at time t is given by [5]
ρ (t ) = (1 − ρee | A (t )|2 )| g g | + ρee | A (t )|2 |ee| ∗
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ρee + ρ gg = 1. The sta-
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(9)
(15)
V (t )|e e | = | A (t )|2 |e e | + (1 − | A (t )|2 )| g g |,
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(19)
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If there is no initial correlation between the qubit and the environment, the time evolution of the statistical operator and two-time correlation functions is determined only by the parameter A (t ).
| A (t )|2
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A (t ) 1
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where the map V (t ) is defined through the following relations:
V −1 (t )|e e | =
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Furthermore we also have the relations,
† k ak ak
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B kl (t )| g |1l ,
with the initial condition A (0) = 1, A k (0) = B k (0) = 0 and B kl (0) = 2 2 2 δ kl . The equalities | A (t )| + k | A k (t )| = 1 and | B k (t )| + 2 l | B kl (t )| = 1 hold due to the normalization condition. The timedependent parameters A (t ), A k (t ), B k (t ), B kl (t ) completely determine the dynamics of the system. These parameters satisfy several useful relations, the derivation of which has been given in Ref. [15]. Here we only summarize some of them that are used to calculate a two-time correlation function of the qubit. First the decomposition formulas for the parameters are given by
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|e g |,
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(21)
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| g e |.
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Fig. 1. Time evolution of the correlation function C zz (τ ) = σz (t + τ )σz (t ) with γ t = 1. The green solid line (red dotted line) is the real (imaginary) part of the exact correlation function, and the blue dashed line is the correlation function derived by the QRT.
Then unless A (t ) = 0, the map V (t , s) = V (t ) V −1 (s) with t ≥ s, which transforms ρ (s) into ρ (t ), becomes
| A (t )|2 | A (t )|2 | g g |, V (t , s)|e e | = |e e | + 1 − | A (s)|2 | A (s)|2 V (t , s)| g g | = | g g |, A (t ) |e g |, V (t , s)|e g | = A (s) A ∗ (t ) V (t , s)| g e | = ∗ | g e |. A (s)
(24) (25) (26) (27)
If the map V (t , s) is divisible, the reduced time-evolution of the qubit is Markovian [30,31]. We assume that the QRT is valid for the reduced time-evolution of the qubit. Then a two-time correlation function of the qubit can be calculated in terms of the map V (t , s) [5,27]. For instance, a two-time correlation function of the Pauli operator σz is
σz (t )σz (s)QRT = Tr[σz V (t , s)σz V (s, 0)ρ ] 2
2
= 1 + (1 + σz )(| A (t )| − | A (s)| ),
(28)
where σz = Tr[σz ρ ] and Tr stands for the trace operation over the qubit Hilbert space. Here we note that the correlation function σz (t )σz (s)QRT is real. In the same way, we obtain the correlation function, ∗
∗
σx (t )σz (s)QRT = ρeg A (t ) − ρeg A (t ).
(29)
It should be noted that σx (t )σz (s)QRT does not depend on the earlier time s. On the other hand, using the relations given by Eqs. (4)–(9), we can obtain exact expressions for these two-time correlation functions [15], 2
σz (t )σz (s) = 1 − (1 + σz )[| A (t )| + | A (s)|
2
− 2 A ∗ (t ) A (t − s) A (s)],
(30)
∗ ∗ σx (t )σz (s) = ρeg [2 A (t − s) A (s) − A (t )] − ρeg A (t ),
(31)
which are clearly different from the correlation functions given by Eqs. (28) and (29). For instance, the exact correlation function
σz (t )σz (s) takes a complex value while σz (t )σz (s)QRT is real, and σx (t )σz (s) depend on both t and s while σx (t )σz (s)QRT does only on t. This means that the QRT is not satisfied in general. It should be noted that the result is independent of an explicit form of the time-dependent parameter A (t ). Comparing Eqs. (28) and (29) with Eqs. (30) and (31), we find that the QRT is established if and only if the equality A (t ) = A (t − s) A (s) or A (t 1 + t 2 ) = A (t 1 ) A (t 2 ) holds, which means that A (t ) is an exponential function of time t. This condition is stronger than the Markovianity of the reduced time-evolution. In order to see the violation of the QRT, we consider the case that the time-dependent parameter A (t ) is given by Eq. (12). The correlation functions σz (t + τ )σz (t ) and σz (t + τ )σz (t )QRT are plotted in Fig. 1. Note that the exact correlation function becomes real only if there is no detuning between the qubit and the environment. When the reduced time-evolution is Markovian (2γ < λ), the real part of the exact correlation function and one derived from the QRT decay monotonously with time. For the non-zero detuning, the imaginary part of the exact one does not vanishes even in this case. It is important to note that the QRT does not hold even in the Markovian time-evolution, however small the difference is. The Markovian condition (2γ < λ) is not sufficient to yield the equality A (t 1 + t 2 ) = A (t 1 ) A (t 2 ) and so Markovianity does not guarantee the QRT. This equality is established only in the limit λ/2γ → ∞. In the case of the non-Markovian time-evolution, the correlation functions show the oscillatory behavior due to the backflow of information from the environment to the qubit [30,31]. The difference between the exact correlation function and the one obtained by the QRT becomes more significant than that in the Markovian time-evolution. Although the difference between the correlation functions is not so large and the QRT may be approximately established for the Markovian time-evolution, the difference has an important physical meaning as explained below. In order to see the qualitative difference between the exact correlation function and the one derived by making use of the QRT, we consider the LGI [16] for the quit. Let us denote as Q (t ) a dichotomous observable which takes the values ±1 at time t. As-
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Fig. 2. Time evolution of the parameter K 3 (τ ), where K 3 (τ ) > 1 means the violation of the LGI. The blue dashed line stands for λ/γ = 0.2, the green dot-dashed line for λ/γ = 0.5, the red dotted line for λ/γ = 1.5 and the brown solid line for λ/γ = 4.0.
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Q (t 2 ) Q (t 1 ) + Q (t 3 ) Q (t 2 ) − Q (t 3 ) Q (t 1 ) ≤ 1,
(32)
with t 3 ≥ t 2 ≥ t 1 . In the quantum mechanical case of Q = σz , the correlation function Q (t ) Q (t ) is replaced by Reσz (t )σz (t ) [24]. Hence we introduce
K 3 (t 3 , t 2 , t 1 ) = Reσz (t 2 )σz (t 1 ) + Reσz (t 3 )σz (t 2 )
− Reσz (t 3 )σz (t 2 ).
(33)
Then LGI is violated if K 3 (t 3 , t 2 , t 1 ) > 1. It is easy to see that when the QRT is applied, the substitution of Eq. (28) into Eq. (33) yields the equality K 3 (t 3 , t 2 , t 1 ) = 1 holds. This means that for the correlation function σz (t )σz (t ), the QRT ensures that the LGI is fulfilled. Assuming that the time-dependent parameter A (t ) is given by (12), we depict the parameter K 3 (τ ) = K 3 (t + 2τ , t + τ , t ) in Fig. 2. It is found from the figure that the violation of the LGI becomes smaller as the parameter λ/γ is larger. In the limit of λ/γ → ∞, where the QRT is valid, the LGI is fulfilled. Furthermore the violation becomes smaller as the detuning between the qubit and the environment is larger. The detuning weakens the qubit-reservoir interaction and if the interaction is negligible, we have σz (t )σz (s) ≈ 1 and so K 3 (t 3 , t 2 , t 1 ) ≈ 1. For the correlation function σz (t )σz (t ) of this model, the interaction with the environment is indispensable for violating the LGI. Although the correlation function derived by the QRT satisfies the LGI in this model, this is not true in general. In fact, the decoherence model that fulfills the QRT can violate the LGI [27]. 4. Concluding remarks
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suming macroscopic realism and non-invasive measurability [16, 24], one can derive the LGI,
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In this paper, we have considered the QRT and the LGI by means of the exactly solvable single-excitation model, where the qubit is influenced by its surrounding environment and there is at most one excitation in the whole system. We have shown that the QRT is not valid even if the reduced time-evolution of the qubit is Markovian. The validity of the QRT requires the more
stronger condition than that for the Markovianity. When the environment consists of independent harmonic oscillators and has the Lorentzian spectral density, the reduced time-evolution is Markovian if the inequality λ/2γ > 1 with = 0 holds. However the QRT is valid only in the limit of λ/2γ → ∞. Furthermore we have examined the LGI by means of the single excitation model. The exact correlation function σz (t + τ )σz (t ) (τ ≥ 0) takes a complex value and violates the LGI. On the other hand, the correlation function σz (t + τ )σz (t )QRT derived by means of the QRT is real and satisfies the LGI. Although we have considered a very simple system-environment model, the result might an important implication on the non-Markovianity, the QRT and the LGI.
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Graduate School of Humanities and Sciences, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan
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a r t i c l e
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Keywords: Quantum regression theorem Leggett–Garg inequality Non-Markovianity Jaynes–Cummings model
Violation of the quantum regression theorem and the Leggett–Garg inequality is studied by means of the exactly solvable multi-mode Jaynes–Cummings model. An exact expression of a two-time correlation function is compared with that derived by the quantum regression theorem. It is found that the quantum regression theorem is not valid even if the reduced time evolution of the qubit is Markovian. Furthermore, it is shown that if the quantum regression theorem is applied in this model, the Leggett–Garg inequality is satisfied while it is violated by the exact correlation function. © 2017 Elsevier B.V. All rights reserved.
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1. Introduction
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A quantum system in a realistic situation is inevitably influenced by its surrounding environment. Such an open quantum system undergoes irreversible time evolution from a prepared initial state to a stationary state [1–5], during which quantumness such as coherence, discord and entanglement is degraded. A state of a quantum system at time t is described by a statistical operator ρ (t ), the time-evolution of which is determined by several methods such as the quantum master equation and the stochastic Schödinger equation [5]. Statistical properties of a quantum system, which depends on a single time, can be derived from the statistical operator ρ (t ). However, it is not sufficient to explain all properties of a quantum system. For instance, a two-time correlation function is necessary to explain how a quantum system responds to an external field applied to the system [4,5]. If the reduced time-evolution of a quantum system is described by the Markovian dynamical map with the semigroup property, thanks to the quantum regression theorem (QRT) [5], a two-time correlation function can be calculated if one finds the time-evolution of the statistical operator ρ (t ). However, it has been pointed out that the QRT is no longer valid if the reduced time-evolution is non-Markovian [6–14]. Recently the violation of the QRT has been investigated in detail by means of an exactly solvable pure dephasing model, and the relation to the non-Markovianity of the reduced time-evolution has been discussed [14].
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E-mail address: [email protected]. http://dx.doi.org/10.1016/j.physleta.2017.05.036 0375-9601/© 2017 Elsevier B.V. All rights reserved.
A two-time correlation function is of essential importance in the foundations of quantum mechanics as well as the practical applications. For instance, one needs to calculate two-time correlation functions of a dichotomous variable in order to examine whether the Leggett–Garg inequality (LGI) is satisfied or not for a quantum system [16–27]. Non-invasive measurability and macrorealism lead to the LGI which is considered as a temporal version of the Bell inequality [28,29]. The violation of the LGI implies a manifestation of quantumness. So it is important to investigate whether the LGI is fulfilled or not for a quantum system which is placed in under the influence of an environment. If the QRT is applicable, one can use a statistical operator ρ (t ) or its equation of motion to examine the LGI. Recently the violation of the LGI has been shown by means of the QRT and the Markovian quantum master equation [27]. In this paper, using an exactly solvable model, we investigate whether the QRT and the LGI are valid or not for an open quantum system. The model considered in this paper is that a qubit (a two-level system) interacts with a set of harmonic oscillators and there is at most a single excitation in the whole system. Although the QRT has been examined by means of the exactly solvable pure dephasing model [14], the model used in this paper includes an energy relaxation. For this model, it is well-known that the exact statistical operator of the qubit can be obtained [5]. Recently it has been found that two-time correlation functions are also exactly calculated even if there is initial correlation between the qubit and the environment [15]. Hence this model is suitable for examining the violation of the QRT and the LGI. It will be found that the LGI is satisfied if the QRT is used to calculate a two-time corre-
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lation function while an exact one violates the LGI. In Section 2, we briefly explain the model and the relations among the timedependent parameters that are used to calculate two-time correlation functions. In Section 3, the exact two-time correlation function is compared with that obtained by making use of the QRT. The violation of the LGI is examined. In Section 4, we give concluding remarks.
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2. Single excitation model for quantum system and environment
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We briefly explain the exactly solvable model of a qubit and environmental system [15], which is used to calculate two-time correlation functions of the qubit. We suppose that a qubit interacts with a multi-mode environmental system. We denote | g and |e as ground and excited states of the qubit. Then the excitation number is given by N q = |e e |. We also denote the excitation number of the environment as N e = k N k where N k represents the excitation number of the kth mode of the environment. We assume that the Hamiltonian H of the qubit and the environment conserves the total excitation number N = N q + N e , that is, [ H , N ] = 0. Furthermore we assume that there is at most a single excitation in the whole system at an initial time. Then the whole system can be described by a Hilbert space spanned by {| g |0, | g |1k , |e |0}, where |0 is the ground state of the environment (N k |0 = 0 for all k) and |1k is a single-excited state in which only the kth mode has one excitation, that is, N k |1l = δkl |1l . Since the total excitation number is conserved during the time evolution, the basic vectors evolve as follows:
e
−i Ht /¯h
| g |0 = | g |0,
e −i Ht /¯h |e |0 = A (t )| g |0 +
e −i Ht /¯h |0|1k = B k (t )|e |0 +
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A (t 1 + t 2 ) = A (t 1 ) A (t 2 ) +
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B k (t 1 + t 2 ) = B k (t 1 ) A (t 2 ) +
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A l (t 1 ) B lk (t 2 ),
(5)
B km (t 1 ) B m (t 2 ),
B kl (t 1 + t 2 ) = B k (t 1 ) A l (t 2 ) +
(6)
B km (t 1 ) B mk (t 2 ).
(7)
m
A (t 1 ) = A (t 1 + t 2 ) A ∗ (t 2 ) +
A k (t 1 + t 2 ) A k∗ (t 2 ),
(8)
k
∗
B k (t 1 ) = B k (t 1 + t 2 ) A (t 2 ) +
+
h¯ ω
k
∗
h¯ (λk σ+ ak + λk σ
† − ak ),
k
where σ+ = (σ− )† = |e g | and ak and are annihilation and creation operators of the kth mode of the environmental oscillator with angular frequency ωk . In this case, the total excitation num † ber is given by N = σ+ σ− + k ak ak which satisfies the commutation relation [ N , H ] = 0. Furthermore we assume the Lorentzian spectral density of the qubit-environmental coupling [5],
J () =
2π
λ2 , ( − ω − )2 + λ2
× cosh
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λ + i
2
at
+
1 a
cosh
λ + i
2
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at
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,
(12)
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with
a=
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−i ωt − 12 (λ+i )
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(11)
where γ represents the strength of the qubit-environment coupling, λ−1 is a correlation time of the environmental oscillators and represents a detuning between the qubit and the environment. Under these assumptions, the time-dependent parameter A (t ) is given by
A (t ) = e
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† ak
γ
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(10)
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1−
2γ λ
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(λ + i )2
(13)
.
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In the resonance case ( = 0), if 2γ > λ, the reduced timeevolution of the qubit is non-Markovian and otherwise it is Markovian [30,31]. 3. Quantum regression theorem and Leggett–Garg inequality
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In this section, we compare an exact two-time correlation function with that obtained by making use of the QRT. For this purpose, we suppose that the qubit is prepared in an arbitrary state ρ and the environment is in the ground state ρe = |00| at an initial time t = 0. Then the statistical operator ρ (t ) of the qubit reduced state at time t is given by [5]
ρ (t ) = (1 − ρee | A (t )|2 )| g g | + ρee | A (t )|2 |ee| ∗
∗
+ ρeg A (t )|e g | + ρeg A (t )| g e |, where we have set ρ jk = j |ρ |k and used tistical operator ρ (t ) can be rewritten into
(14)
ρee + ρ gg = 1. The sta-
∗
B kl (t 1 + t 2 ) A l (t 2 ).
(9)
(15)
V (t )|e e | = | A (t )|2 |e e | + (1 − | A (t )|2 )| g g |,
(16)
V (t )| g g | = | g g |,
(17)
V (t )|e g | = A (t )|e g |,
(18)
V (t )| g e | = A ∗ (t )| g e |.
(19)
l
If there is no initial correlation between the qubit and the environment, the time evolution of the statistical operator and two-time correlation functions is determined only by the parameter A (t ).
| A (t )|2
|e e | + 1 −
V −1 (t )| g g | = | g g |, V −1 (t )|e g | = V −1 (t )| g e | =
A (t ) 1
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where the map V (t ) is defined through the following relations:
V −1 (t )|e e | =
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If A (t ) = 0, the inverse of this map is given by
Furthermore we also have the relations,
† k ak ak
ρ (t ) = V (t )ρ ,
m
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(4)
l
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A k (t 1 ) B k (t 2 ),
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(3)
k
A k (t 1 + t 2 ) = A (t 1 ) A k (t 2 ) +
H = h¯ σ+ σ− +
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B kl (t )| g |1l ,
with the initial condition A (0) = 1, A k (0) = B k (0) = 0 and B kl (0) = 2 2 2 δ kl . The equalities | A (t )| + k | A k (t )| = 1 and | B k (t )| + 2 l | B kl (t )| = 1 hold due to the normalization condition. The timedependent parameters A (t ), A k (t ), B k (t ), B kl (t ) completely determine the dynamics of the system. These parameters satisfy several useful relations, the derivation of which has been given in Ref. [15]. Here we only summarize some of them that are used to calculate a two-time correlation function of the qubit. First the decomposition formulas for the parameters are given by
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(2)
l
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A k (t )| g |1k ,
k
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(1)
To obtain an explicit form of the parameter A (t ), we assume that the Hamiltonian of the system [5] is given by
|e g |,
A ∗ (t )
1
| A (t )|2
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| g g |,
124
(20)
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(21)
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(22)
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| g e |.
(23)
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Fig. 1. Time evolution of the correlation function C zz (τ ) = σz (t + τ )σz (t ) with γ t = 1. The green solid line (red dotted line) is the real (imaginary) part of the exact correlation function, and the blue dashed line is the correlation function derived by the QRT.
Then unless A (t ) = 0, the map V (t , s) = V (t ) V −1 (s) with t ≥ s, which transforms ρ (s) into ρ (t ), becomes
| A (t )|2 | A (t )|2 | g g |, V (t , s)|e e | = |e e | + 1 − | A (s)|2 | A (s)|2 V (t , s)| g g | = | g g |, A (t ) |e g |, V (t , s)|e g | = A (s) A ∗ (t ) V (t , s)| g e | = ∗ | g e |. A (s)
(24) (25) (26) (27)
If the map V (t , s) is divisible, the reduced time-evolution of the qubit is Markovian [30,31]. We assume that the QRT is valid for the reduced time-evolution of the qubit. Then a two-time correlation function of the qubit can be calculated in terms of the map V (t , s) [5,27]. For instance, a two-time correlation function of the Pauli operator σz is
σz (t )σz (s)QRT = Tr[σz V (t , s)σz V (s, 0)ρ ] 2
2
= 1 + (1 + σz )(| A (t )| − | A (s)| ),
(28)
where σz = Tr[σz ρ ] and Tr stands for the trace operation over the qubit Hilbert space. Here we note that the correlation function σz (t )σz (s)QRT is real. In the same way, we obtain the correlation function, ∗
∗
σx (t )σz (s)QRT = ρeg A (t ) − ρeg A (t ).
(29)
It should be noted that σx (t )σz (s)QRT does not depend on the earlier time s. On the other hand, using the relations given by Eqs. (4)–(9), we can obtain exact expressions for these two-time correlation functions [15], 2
σz (t )σz (s) = 1 − (1 + σz )[| A (t )| + | A (s)|
2
− 2 A ∗ (t ) A (t − s) A (s)],
(30)
∗ ∗ σx (t )σz (s) = ρeg [2 A (t − s) A (s) − A (t )] − ρeg A (t ),
(31)
which are clearly different from the correlation functions given by Eqs. (28) and (29). For instance, the exact correlation function
σz (t )σz (s) takes a complex value while σz (t )σz (s)QRT is real, and σx (t )σz (s) depend on both t and s while σx (t )σz (s)QRT does only on t. This means that the QRT is not satisfied in general. It should be noted that the result is independent of an explicit form of the time-dependent parameter A (t ). Comparing Eqs. (28) and (29) with Eqs. (30) and (31), we find that the QRT is established if and only if the equality A (t ) = A (t − s) A (s) or A (t 1 + t 2 ) = A (t 1 ) A (t 2 ) holds, which means that A (t ) is an exponential function of time t. This condition is stronger than the Markovianity of the reduced time-evolution. In order to see the violation of the QRT, we consider the case that the time-dependent parameter A (t ) is given by Eq. (12). The correlation functions σz (t + τ )σz (t ) and σz (t + τ )σz (t )QRT are plotted in Fig. 1. Note that the exact correlation function becomes real only if there is no detuning between the qubit and the environment. When the reduced time-evolution is Markovian (2γ < λ), the real part of the exact correlation function and one derived from the QRT decay monotonously with time. For the non-zero detuning, the imaginary part of the exact one does not vanishes even in this case. It is important to note that the QRT does not hold even in the Markovian time-evolution, however small the difference is. The Markovian condition (2γ < λ) is not sufficient to yield the equality A (t 1 + t 2 ) = A (t 1 ) A (t 2 ) and so Markovianity does not guarantee the QRT. This equality is established only in the limit λ/2γ → ∞. In the case of the non-Markovian time-evolution, the correlation functions show the oscillatory behavior due to the backflow of information from the environment to the qubit [30,31]. The difference between the exact correlation function and the one obtained by the QRT becomes more significant than that in the Markovian time-evolution. Although the difference between the correlation functions is not so large and the QRT may be approximately established for the Markovian time-evolution, the difference has an important physical meaning as explained below. In order to see the qualitative difference between the exact correlation function and the one derived by making use of the QRT, we consider the LGI [16] for the quit. Let us denote as Q (t ) a dichotomous observable which takes the values ±1 at time t. As-
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Fig. 2. Time evolution of the parameter K 3 (τ ), where K 3 (τ ) > 1 means the violation of the LGI. The blue dashed line stands for λ/γ = 0.2, the green dot-dashed line for λ/γ = 0.5, the red dotted line for λ/γ = 1.5 and the brown solid line for λ/γ = 4.0.
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Q (t 2 ) Q (t 1 ) + Q (t 3 ) Q (t 2 ) − Q (t 3 ) Q (t 1 ) ≤ 1,
(32)
with t 3 ≥ t 2 ≥ t 1 . In the quantum mechanical case of Q = σz , the correlation function Q (t ) Q (t ) is replaced by Reσz (t )σz (t ) [24]. Hence we introduce
K 3 (t 3 , t 2 , t 1 ) = Reσz (t 2 )σz (t 1 ) + Reσz (t 3 )σz (t 2 )
− Reσz (t 3 )σz (t 2 ).
(33)
Then LGI is violated if K 3 (t 3 , t 2 , t 1 ) > 1. It is easy to see that when the QRT is applied, the substitution of Eq. (28) into Eq. (33) yields the equality K 3 (t 3 , t 2 , t 1 ) = 1 holds. This means that for the correlation function σz (t )σz (t ), the QRT ensures that the LGI is fulfilled. Assuming that the time-dependent parameter A (t ) is given by (12), we depict the parameter K 3 (τ ) = K 3 (t + 2τ , t + τ , t ) in Fig. 2. It is found from the figure that the violation of the LGI becomes smaller as the parameter λ/γ is larger. In the limit of λ/γ → ∞, where the QRT is valid, the LGI is fulfilled. Furthermore the violation becomes smaller as the detuning between the qubit and the environment is larger. The detuning weakens the qubit-reservoir interaction and if the interaction is negligible, we have σz (t )σz (s) ≈ 1 and so K 3 (t 3 , t 2 , t 1 ) ≈ 1. For the correlation function σz (t )σz (t ) of this model, the interaction with the environment is indispensable for violating the LGI. Although the correlation function derived by the QRT satisfies the LGI in this model, this is not true in general. In fact, the decoherence model that fulfills the QRT can violate the LGI [27]. 4. Concluding remarks
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suming macroscopic realism and non-invasive measurability [16, 24], one can derive the LGI,
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In this paper, we have considered the QRT and the LGI by means of the exactly solvable single-excitation model, where the qubit is influenced by its surrounding environment and there is at most one excitation in the whole system. We have shown that the QRT is not valid even if the reduced time-evolution of the qubit is Markovian. The validity of the QRT requires the more
stronger condition than that for the Markovianity. When the environment consists of independent harmonic oscillators and has the Lorentzian spectral density, the reduced time-evolution is Markovian if the inequality λ/2γ > 1 with = 0 holds. However the QRT is valid only in the limit of λ/2γ → ∞. Furthermore we have examined the LGI by means of the single excitation model. The exact correlation function σz (t + τ )σz (t ) (τ ≥ 0) takes a complex value and violates the LGI. On the other hand, the correlation function σz (t + τ )σz (t )QRT derived by means of the QRT is real and satisfies the LGI. Although we have considered a very simple system-environment model, the result might an important implication on the non-Markovianity, the QRT and the LGI.
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