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Dynamics of Bloch vector in thermal Jaynes-Cummings model

In this paper, we investigate the dynamics of the Bloch vector of a single two-level atom which interacts with a single quantized electromagnetic field mode according to the Jaynes-Cummings model, where the field is initially prepared in a thermal state. The time evolution of the Bloch vector S(t) seems to be in complete disorder because of the thermal distribution of the initial state of the field. Both the norm and the direction of S(t) oscillate hard and their periods seem infinite. We observe that the trajectory of the time evolution of S(t) in the two- or three-dimensional space does not form a closed path. To remove the fast frequency oscillation from the trajectory, we take the time-average of the Bloch vector S(t). We examine the histogram of {S_{z}(n\Delta t)|n=0,1,...,N} for small \Delta t and large N. It represents an absolute value of a derivative of the inverse function of S_{z}(t). (When the inverse function of y=S_{z}(t) is a multi-valued function, the histogram represents a summation of the absolute values of its derivatives at points whose real parts are equal to y on the Riemann surface.) We examine the dependence of the variance of the histogram on the temperature of the field. We estimate the lower bound of the entanglement between the atom and the field.


I. INTRODUCTION
The Jaynes-Cummings model (JCM) is a solvable quantum mechanical model of a single two-level atom in a single electromagnetic field mode [1,2,3,4,5]. This model is originally designed for studying a spontaneous emission. The interaction term is obtained by the rotating wave approximation. In this interaction, each photon creation causes an atomic de-excitation and each photon annihilation causes an atomic excitation.
If the photon number is sharply defined in the initial state of the field, the JCM shows the Rabi oscillations in the populations of the atomic levels. If the initial state of the field is a coherent state, the oscillation of the mean photon number collapses and revives in the JCM. In this way, the JCM reveals the quantum natures of the radiation.
Because the JCM is exactly solvable, it is investigated from various viewpoints. The JCM whose boson field is prepared in a thermal state is discussed in Refs. [6,7,8]. Thermodynamics of the JCM is discussed in Ref. [9]. In this analysis, the grand partition function of both the atom and the boson field is considered. Extended JCMs are studied as dissipative models [10,11].
Recently, the JCM has been used for describing the evolution of the entanglement between the atom and the field [12,13]. In Refs. [12,13], the electromagnetic field is assumed to be initially prepared in a thermal state. In these papers, the JCM is regarded as a source of the entanglement between the atom and the field. In Ref. [14], this idea is advanced and generation of the entanglement between two atoms interacting with a single-mode thermal field according to the JCM is discussed. In this model, the atoms which are initially in a separable state obtain the entanglement through the time evolution.
In Ref. [15], evolution of the entanglement between an atom and a single-mode field described with the JCM under phase damping is studied. In Ref. [16], an atom interacting with two cavity modes is considered and it is shown that this system can be reduced to the JCM. Using this system, generation of entangled coherent states is discussed. In Refs. [17,18], the entanglement in an extended JCM where two atoms interacts with a single-mode field is studied.
As an interesting phenomenon related with the time evolution of the entanglement in the JCM, so called sudden death effect is studied [19,20,21]. In these references, two isolated atoms, each of which is located in its own Jaynes-Cummings cavity, are assumed. If these atoms are in a certain entangled state initially, the entanglement disappears in a finite time. This phenomenon is experimentally demonstrated in Ref. [22]. In Ref. [23], an attempt to find invariant entanglement among atoms and fields in two isolated JCM is done.
In this paper, we consider the dynamics of the Bloch vector of a two-level atom interacting with a single mode boson according to the JCM, where the initial state of the boson is in thermal equilibrium. The time evolution of the Bloch vector S(t) seems to be in a state of complete disorder because of the thermal distribution of the initial state of the boson field. The trajectory of S(t) oscillates hard and seems to wander without a purpose. We try to find a property that characterizes this confused movement of S(t).
If S x (0) = 0 or S y (0) = 0, the time evolution of the Bloch vector S(t) draws a trajectory in the two-or threedimensional space. We observe that this trajectory does not form a closed path. To remove the fast frequency oscillation from the trajectory, we take the time-average of the S(t) as S = lim T →∞ (1/T ) T 0 dt S(t). We find that S x and S y are equal to zero ∀β, ω and ω 0 , where β is an inverse of the temperature for the initial thermal state of the field, ω is an angular frequency of the field andhω 0 is an energy gap of the two-level atom. S z takes a certain function of β, ω and ω 0 .
We take the histogram of samples {S z (n∆t)|n = 0, 1, ..., N } for small ∆t and large N . We understand that it represents an absolute value of a derivative of the inverse function of S z (t). (When the inverse function of y = S z (t) is a multi-valued function, the histogram represents a summation of the absolute values of its derivatives |(d/dy)S −1 z | at points whose real parts are equal to y on the Riemann surface, where S −1 z is the inverse function of y = S z (t).) We approximate obtained histograms at a high temperature by the probability function of the normal distribution. We examine the dependence of the variance of the samples on the temperature β.
We examine the time evolution of the entanglement between the atom and the field in our model by estimating the entanglement of formation of the density matrix which lies on the 2 × 2 dimensional projected subspace of the atom and the field. Because this projection is an local operation, the entanglement of formation computed in the reduced 2 × 2 dimensional subspace gives the lower bound of the entanglement of formation of the whole system (the atom and the field).
This paper is organized as follows: In Sec. II, we derive an equation which governs the time evolution of the Bloch vector S(t). In Sec. III, we examine the trajectory of S(t) and observe that it does not form a closed path. In Sec. IV, we take the time-average of S(t) for the case where the field is resonant with the atom. In Sec. V, we take the histogram of data of S z (t) sampled at intervals of ∆t. In Sec. VI, we take the time-average of S(t) for the non-resonant case. In Sec. VII, we consider the lower bound of the entanglement of formation between the atom and the field. In Sec. VIII, we give brief discussions.

II. THE TIME EVOLUTION OF THE BLOCH VECTOR
In this section, we give the equation of the time evolution of the Bloch vector of the atom. The Jaynes-Cummings model is a system that is described by the following Hamiltonian: where and [a, a † ] = 1. The Pauli matrices (σ i , i = x, y, z) are operators of the atom and a and a † are operators of the field. In this paper, we assume that g is a constant, so that it does not depend on ω 0 or ω. Let us divide H as follows [4,5]: where ∆ω = ω − ω 0 . We can confirm Because C 1 can be diagonalized at ease, we take the following interaction picture. We write a state vector of the whole system in the Schrödinger picture as |ψ(t) . A state vector in the interaction picture is defined by (We assume |ψ I (0) = |ψ(0) .) Because of Eq. (7), the time evolution of |ψ I (t) is given by where U (t) = exp(−iC 2 t).
We define the density operator of the initial state of the atom and the boson field as where and β is an inverse of the temperature. (The indices A and F imply the atom and the field, respectively.) The density operator of the atom in the interaction picture evolves according to where Tr F means a partial trace over the field. Because ρ A,10 (t) = ρ A,01 (t) * and ρ A,11 (t) = 1 − ρ A,00 (t), we examine ρ A,00 (t) and ρ A,01 (t) only. We can derive A kl,ij (t) as follows: The unitary evolution operator of the whole system U (t) given by Eqs. (6) and (10) is rewritten as where From Eqs. (17), (18), (19) and (20), we obtain (The trace over the field is taken by the basis vectors of the photon number states.) We introduce the Bloch vector S(t) = (S x (t), S y (t), S z (t)) which is given by where I is the identity operator and σ = (σ x , σ y , σ z ). Because ρ A (t) † = ρ A (t) and ρ A (t) ≥ 0, S(t) is a real vector and satisfies |S(t)| 2 ≤ 1. From Eqs. (16), (22) and (24), we obtain where L (1) This is the equation of the time evolution of S(t).  28) and (30), the summations of the index n is carried out up to 500.

III. THE TRAJECTORY OF THE BLOCH VECTOR
In this section, we observe the trajectory of the time evolution of S(t). To simplify the discussion, we concentrate on the case where the field is resonant with the atom, that is, ∆ω = 0. Furthermore, we regard ω 0 as a constant. Thus, the model has two variables, t and β. We replace |g|t with t and βhω(= βhω 0 ) with β, where we assume g = 0 and ω 0 = 0. They imply that the time t is in units of |g| −1 and the inverse of the temperature β is in units of (hω 0 ) −1 . From the above assumptions and Eqs. (21), (23), (25) and (26), we obtain the following equation: where  27), we understand that the trajectory (a curve in the three-dimensional space of S parameterized by t) is uniquely determined by the initial state S(0) for given β. Figure 1 shows a trajectory of S(t) whose initial state and temperature are given by S(0) = (1, 0, 0) and β = 0.5, respectively. In this case, S y (t) = 0 for t ≥ 0 and the curve of S(t) lies on the xz-plane. The trajectory of  L(t 2 ) = L(t 2 − t 1 )L(t 1 ) does not hold in general. In fact, we can observe an example where the trajectory intersects itself and does not form a closed path in Fig. 3.
If S x (0) = 0 or S y (0) = 0, S(t) draws a trajectory in the two-or three-dimensional space. From the above considerations, we understand that this trajectory does not form a closed loop in general.

IV.
THE TIME-AVERAGE OF THE BLOCH VECTOR FOR ∆ω = 0 In this section, we take the time-average of the Bloch vector for the case where the field is resonant with the atom (∆ω = 0). As shown in Fig. 1, the evolution of S(t) is in complete disorder. To remove the fast frequency oscillation, we consider the time-average of S(t) as From Eqs. (27), (28), (29) and (30), we obtain Let us take the limit of T → ∞ in Eqs. (33), (34) and (35). First, we evaluate L 1 . We can obtain the following relation about L 1 from Eq. (33): where From Eq. (37), we can derive the following relation: we obtain Thus, if β = 0, V 1 = 0 and L 1 = 0. Moreover, we can derive L 1 = 0 for β = 0 from Eq. (33) in direct. Hence, we obtain L 1 = 0 ∀β. Second, we evaluate L 3 . We can obtain the following relation about L 3 from Eq. (34): where Moreover, we can derive the following relation: Because ∞ n=1 e −nβ = 1/(e β − 1), we obtain Thus, we obtain By a similar derivation, we can obtain V.
We investigate the time evolution of L 4 (t) by the following way. Fixing β at a certain value and defining a small time interval ∆t, we collect (N + 1) samples of L 4 (n∆t) for n = 0, 1, ..., N , where N is large enough. Next, we make a histogram of these (N + 1) samples {L 4 (0), L 4 (∆t), ..., L 4 (N ∆t)}. We adjust the class interval of bins of the histogram, so that the line graph of the histogram approaches a smooth curve.
This histogram represents an absolute value of a derivative of the inverse function of L 4 (t). As shown in Fig. 4, the probability that there is a sample L 4 (n∆t) in the range of the bin of the histogram [y, y + ∆y) is proportional to where t = L −1 4 (y) is an inverse function of y = L 4 (t) and ∆y is a class interval of the bin. (Here, we assume L −1 4 (y) is not a multi-valued function.) If we take the limit of small ∆y, this probability reaches If L −1 4 (y) is a multi-valued function, the probability is proportional to a summation of |(d/dy)L −1 4 | at points whose real parts are equal to y on the Riemann surface.
Let us evaluate the histogram of L 4 (t) in the low temperature limit. In the low temperature limit, we can describe L 4 (t) as Thus, the histogram is proportional to Next, we consider the histogram of L 4 (t) for the high temperature. If β ≪ 1, we can expect that L 4 (t) varies at random around a mean with a certain variance. Hence, we can approximate the histogram by the probability function of the normal distribution, where µ and σ 2 are a mean and a variance of samples, that is, and If N ∆t is large enough, µ is nearly equal to In Fig. 8, we plot the variance σ 2 of samples against β for 0.01 ≤ β ≤ 10. For small β, we can approximate plotted points by σ 2 (β) = c 1 β c2 for 0.01 ≤ β ≤ 0.1, where c 1 = 0.0516 and c 2 = 2.95. The author cannot find a reason why the function of σ 2 (β) for small β has such a simple form.

VI. THE TIME-AVERAGE OF THE BLOCH VECTOR FOR ∆ω = 0
In this section, we evaluate the time-average of the Bloch vector for the non-resonant case (∆ω = 0). Here, we put h = 1 (β is in units ofh). We fix g and ω 0 and regard them as constants. Thus, the model has three variables, ω, β and t. From Eqs. (21), (25), (26) and (31), we obtain the following relation: Using the method shown in Sec. IV, we can take the limit T → ∞ in Eq. (55) and we obtain and lim β→∞ L (4) The above results imply lim β→0 S = 0 ∀S(0) and From Eq. (60), we can say the following: If we start from S(0) = 0 [the complete mixed state ρ A (0) = (1/2)I] in the low temperature limit, we can expect to obtain a slightly purified state of S = (0, 0, −2g 2 /[(∆ω) 2 + 4g 2 ]) on average after the enough time evolution.

VII. ENTANGLEMENT BETWEEN THE ATOM AND THE THERMAL FIELD
In this section, we consider the evolution of the lower bound of the entanglement of formation between the atom and the thermal field in the JCM. In our model defined in Sec. II, the initial state of the atom and the field is separable. However, because of the Jaynes-Cummings interaction, we can expect that the entanglement is generated between the atom and the field during the time evolution and their bipartite state becomes inseparable. Such a mechanism of entanglement generation is discussed in Refs. [12,13], as well. Recently, many researchers have been regarded the JCM as a source of the entanglement [14,15,16,17,18,19,20,21,22,23].
Let us pursue the time evolution of the entanglement between the atom and the field in our model. To simplify the discussion, we assume ∆ω = 0 and g > 0. Furthermore, we assume the atom initially to be in a pure state of (1/ √ 2)(|0 A + |1 A ), which implies the Bloch vector S(0) = (1, 0, 0) and the density matrix The time evolution of the whole state is described as where ρ A (0) is given by Eq. (61), ρ F is given by Eq. (13), and U (t) is given by Eq. (18).
Here, we are interested in studying the entanglement for the mixed state of the bipartite system AF (the atom and the field). [Because ρ F given by Eq. (13) is a mixed state, ρ AF (t) in Eq. (62) is also a mixed state in general.] Moreover, although the dimension of the system A is finite (the two-dimensional system), the dimension of the system F is infinite. Entanglement for such a system is difficult to define.
Some measures of entanglement are proposed at present, for example, the relative entropy of entanglement, entanglement of formation, and so on. However, because analytical methods are not found for these measures of the entanglement in general, it is difficult to compute the value of the entanglement for an arbitrary bipartite state. However, exceptionally, an explicit formula of the entanglement of formation for an arbitrary 2 × 2 state is obtained [24]. Thus, as the measure of the entanglement, we choose the entanglement of formation. where ρ * AB is the complex conjugate of ρ AB that is represented in the basis vectors of {|i A |j B : i, j ∈ {0, 1}}. We write the eigenvalues of ρ ABρAB as λ 1 , λ 2 , λ 3 and λ 4 , where λ 1 ≥ λ 2 ≥ λ 3 ≥ λ 4 (≥ 0). The concurrence C(ρ AB ) is given by The entanglement of formation of ρ AB is given by E(C(ρ AB )), where Here, we want to estimate the entanglement of formation of R AF (t) = p AF (t)σ AF (t), which is not a normalized density matrix. The entanglement of formation of R AF (t), E(R AF (t)), should be written as E(R AF (t)) = p AF (t)E(C(σ AF (t))). Because the analytical form of σ AF (t) obtained from Eqs. (64), (65) and (68) is very complicated, it is difficult to obtain C(σ AF (t)) in an explicit formula. Thus, we estimate C(σ AF (t)) [and E(R AF (t))] numerically. The variations of E(R AF (t)) against t ∈ [0, 2π] with fixed β = 10, 2 and 1 are shown in Fig. 9.
In Fig. 9, E(R AF (t)) against t seems to vary periodically. Because we concentrate on the photon number states |0 F and |1 F only, the effect of the thermal distribution of the field is neglected. Looking at Fig. 9, we notice that if β takes a large value (at a low temperature), a certain amount of the entanglement arises between the atom and the field. As β becomes smaller (as the temperature becomes higher), the amplitude of oscillation of E(R AF (t)) becomes smaller. However, this observation does not imply that the entanglement of formation of the entire system (the atom and the field) becomes smaller as the temperature becomes higher, because E(R AF (t)) is just the lower bound of the entanglement of formation of ρ AF (t). From these results, we may regard the thermal JCM as the source of the entanglement.

VIII. DISCUSSIONS
In this paper, we examine the dynamics of the Bloch vector of the two-level atom in the thermal Jaynes-Cummings model (JCM). In the evolution of the Bloch vector, for example, if ∆ω = 0, infinite summation of cos( √ n + 1t) cos( √ nt)e −nβ and cos(2 √ nt)e −nβ for n = 0, 1, 2, ... appears and this makes it difficult to treat the problem exactly.
In our model, the thermal effects are introduced only in the initial state of the boson field. To discuss the thermodynamics of the JCM strictly, we have to think the grand partition function of the whole system (the atom and the boson field) and pursue its non-equilibrium time evolution. Although the JCM has been studied by many researchers, understanding about the thermal JCM seems not to be enough.
In this paper, we try to obtain a global property that characterizes the confused behavior of the Bloch vector. We observe the trajectory of the Bloch vector and take its time-average. We take the histogram of the z components of the Bloch vector sampled at intervals of ∆t. However, the author wonders whether these results are good global aspects of the trajectory that is in complete disorder.
Recently, entanglement generation during the evolution of JCM has been studied from the viewpoint of the quantum information theory [12,13,14,15,16,17,18]. On the other hand, the entanglement sudden death in the JCM is also discussed [19,20,21,22,23]. These matters are explained in Sec. I. In Sec. VII, we consider the evolution of the entanglement of formation between the atom and the thermal field. The thermal JCM may become an important source of the entanglement in future.