Steady-state entanglement and heat current of two coupled qubits in two heat baths

Optik - International Journal for Light and Electron Optics 182 (2019) 1074–1081

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Original research article

Steady-state entanglement and heat current of two coupled qubits in two heat baths Mei-Jiao Wang (王美姣)a,b, Yun-Jie Xia (夏云杰)a,b, a b

T

⁎

College of Physics and Engineering, Qufu Normal University, Qufu, 273165, China Shandong Provincial Key Laboratory of Laser Polarization and Information Technology, Qufu Normal University, Qufu, 273165, China

A R T IC LE I N F O

ABS TRA CT

PACS: 03. 63. Yz 05 30. -d 05. 70. Ln

We study the steady-state entanglement and heat current of two coupled qubits, in which one qubit is connected with either two baths with the different temperature or a bath. We construct the master equation in the eigenstate representation of two coupled qubits to describe the dynamics and derive the solutions in the steady-state with strong coupling regime between them. We have demonstrated the variations of the steady-state entanglement with respect to various parameters of the system in both equilibrium and nonequilibrium cases. We find that the coupling strengths and the energy detuning as well as the temperature gradient are beneficial to the enhancement of the entanglement in equilibrium case. In the nonequilibrium case, the temperature gradient has different effects on entanglement due to the temperature of the bath(s) which the qubits touch. We also study the heat current and their variations with the energy detuning, coupling strengths and diverse low temperature. The energy detuning has a positive (negative) effect on the heat current in the low (high) temperature; heat current also have different trends with coupling strengths increases for a given temperature. The temperature of cool bath is lower, the heat current is larger.

Keywords: Steady-state entanglement Nonequilibrium baths Heat current

1. Introduction Quantum entanglement [1,2] contained in a composite system is a key resource in the realization of quantum communication and quantum computation [3]. However, the systems are inevitably in contact with the external environment in the practical applications and the prepared entanglement may vanish because of the decoherence effect [4] of environments. A more practical situation for the open quantum system is in nonequilibrium thermal baths with a non-vanishing temperature difference. The effects of nonequilibrium environments have been studied within various physical systems, for example, the electron spin qubits [5], and applying the temperature gradient, many quantum thermal devices are proposed, the thermal rectifiers [6–10], thermal diode [11,12], thermal transistors [13], the self-contained refrigerators [14–19] and quantum Otto cycle [20–22]. It is shown that the temperature gradient of involved baths may be beneficial for the steady-state thermal entanglement of the systems of interest [23–28]. Focusing on nonequilibrium environments, we study the quantum entanglement of two coupled qubits in the steady-state in contrast to the generation of transient entanglement. In addition to that, the conventional quantum thermalization works for the situations wherein one quantum system is connected with just one environment at thermal equilibrium. However, the individual subsystems of the coupled systems are difficult to be isolated to contact with their own local baths. The composite quantum system, which is constructed with many subsystems and

⁎

Corresponding author. E-mail address: [email protected] (Y.-J. Xia (夏云杰)).

https://doi.org/10.1016/j.ijleo.2019.01.108 Received 5 January 2019; Accepted 30 January 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.

Optik - International Journal for Light and Electron Optics 182 (2019) 1074–1081

M.-J. Wang (王美姣) and Y.-J. Xia (夏云杰)

Fig. 1. The schematic diagram of the physical model with two coupled qubits A and B interacting with a non-equilibrium environment consisted of bath-a and bath-b .

connected with many environments [29–33] have been studied. In fact, quantum thermalization of a composite quantum system is a very complex problem. On one hand, from the viewpoint of the environments, the composite quantum system could be connected with many independent environments or a common environment. At the same time, the temperatures of these environments could be the same or different. On the other hand, the coupling strengths among these subsystems can affect the physical picture to describe the quantum thermalization of the composite quantum system. Focusing on the above-mentioned situation, in this paper, we study the coupled qubits, in which one qubit is connected with either two baths with different temperature or a bath(we analyze two cases with respect to the configuration about the interactions between qubits and heat baths.), and the coupling strength among two subsystems is stronger than the system-bath couplings. When the coupling strength is stronger than the system-bath couplings, the composite quantum system can be regarded as a single system. The evolution process to the steady-state can be modeled by the quantum master equation, which is the most popular approach to describe the open quantum system [34,35]. As another figure of merit, we also consider the heat current with respect to a bath of two qubits, observing how the heat current changes with the energy detuning, coupling strengths and diverse lower temperature. 2. Physics model As described in the schematic diagram in Fig. 1, we consider the total model consists of two coupled qubits A, B and a nonequilibrium environment of two heat baths which are bath-a having the temperature T1 and bath-b having the temperature T2 . With system-environment interaction, the total Hamiltonian of the system and two baths can be divided into three parts: (1)

H = HS + HB + HI .

Here, HS is the Hamiltonian of two coupled qubits, HB is the Hamiltonian of two heat baths, HI is the interaction Hamiltonian between two qubits and two heat baths. The Hamiltonian HS reads(we takeℏ = 1)

HS = ωA 2 σAz + ωB 2 σBz + ξ (σA+σB− + σA−σB+), σAz (B)

(2) (σA−(B)

σA+(B)

= |1〉A (B) 〈1| − |0〉A (B) 〈0| is the Pauli operator, = |0〉A (B) 〈1|) is the raising (lowering) operator for = |1〉A (B) 〈0| where two qubits A and B . ωA and ωB are the bare frequencies of two qubits. They are coupled with each other by the dipole-dipole interaction of strength ξ . The Hilbert space of two coupled qubits may be spanned by the following four bare states: η1 = |11〉, η2 = |10〉, η3 = |01〉 and η4 = |00〉, which are eigenstates of the free Hamiltonian ωA 2 σAz + ωB 2 σBz , with the corresponding eigenenergies Eη1 = −Eη4 = ωm and Eη2 = −Eη3 = Δω 2. Here, we have introduced the mean energy separation ωm = (ωA + ωB ) 2 and the energy detuning Δω = ωA − ωB . Due to the dipole-dipole interaction and the strong coupling regime between two qubits, we construct the master equation describing the system,s evolution in the eigenstates representation. We can solve the eigenequation HS |λn〉 = En |λn〉 (n = 1, 2, 3, 4) to obtain the following four eigenstates and eigenvalues: 1075

Optik - International Journal for Light and Electron Optics 182 (2019) 1074–1081

M.-J. Wang (王美姣) and Y.-J. Xia (夏云杰)

|λ1〉 = |η1〉, |λ2〉 = cos( θ 2 )|η2〉 + sin( θ 2 )|η3〉, |λ3〉 = −sin( θ 2 )|η2〉 + cos( θ 2 )|η3〉, |λ 4〉 = |η4 〉, E1 = E4 = −ωm , E2 = −E3 =

(Δω)2 4 + ξ 2 ,

(3)

where the mixing angle θ (0 < θ < π 2) is given by tan θ = 2g Δω. Hence, the eigenstructure for the system Hamiltonian is given by 4

HS =

∑ Ei |λi 〉〈λi|.

(4)

i=1

In this paper, we consider the two harmonic-oscillator heat baths. The Hamiltonian HB of two baths reads as

HB =

∑ ωaj a+j aj + ∑ ωbk bk+bk , j

(5)

k

a+j (bk+)

and aj (bk ) are the creation and annihilation operators of the j-th(k -th) harmonic oscillator with frequency ωaj (ωbk ) . The where interaction Hamiltonian HI between two qubits and two heat baths is

HI =

⎛ g a σ+ + ⎜∑ Aj j A j ⎝

⎞

∑ gBj aj σB++ ∑ gBk bk σB+⎟ + h. c. , j

k

(6)

⎠

where gAj , gBj , gBk are the interaction strengths between qubit-bath. For simplicity, we assume that they are real numbers. The Hamiltonian HI can be rewritten as

HI′ = ∑n = 1,2 ∑j gAj [V A+ (ωn ) aj + VA (ωn ) a+j ] + ∑n = 1,2 ∑j gBj [V B+ (ωn ) aj + VB (ωn ) a+j ] + ∑n = 1,2 ∑k gBk [V B+ (ωn ) bk + VB (ωn ) bk+].

(7)

In Eq. (7), VA (B) (ωn ) denotes the eigenoperators of the Hamiltonian HS , such that [HS , VAB (ωn )] = −ωn VAB (ωn ) and ωn stands for the eigenfrequency of the total system due to the effect of the baths. ω1 = E1 − E2 = E3 − E4 = ωm − (Δω)2 4 + ξ 2 corresponding to the transitions |λ1〉 ↔ |λ2〉 and |λ3〉 ↔ |λ 4〉, while ω2 = E1 − E3 = E2 − E4 = ωm + (Δω)2 4 + ξ 2 corresponding to the transitions |λ1〉 ↔ |λ3〉 and |λ2〉 ↔ |λ 4〉. Explicitly, the form of VA (B) (ωn ) are obtained as follows

VA (ω1) = sin (θ 2)(|λ2 〉〈λ1| − |λ 4 〉〈λ3|), VA (ω2) = cos (θ 2)(|λ3 〉〈λ1| + |λ 4 〉〈λ2|), VB (ω1) = cos (θ 2)(|λ2 〉〈λ1| + |λ 4 〉〈λ3|), VB (ω2) = sin (θ 2)(|λ 4 〉〈λ2| − |λ3 〉〈λ1|).

(8)

V A+(B) (ωn ) .

In the above equations, we have presented the forms of VA (B) (ωn ) when ωn > 0 , otherwise VA (B) (−ωn ) = In the presence of the weak coupling between qubits and baths, the equation of motion of the qubits can be derived within the framework of the Born-Markov approximation as

ρ˙ = −i [HS , ρ] + ζ Aa [ρ] +

∑

a ζ BR [ρ] + ζ AB [ρ].

(9)

R =a,b

The Lindblad operator ζ AR(B) [ρ] represents the dissipation of qubit A (B ) due to the heat bath- R and takes the form as

ζ AR(B) [ρ] = ∑n = 1,2 γAR(B) (ωn )[n¯ R (ωn ) + 1][2VA (B) (ωn ) ρV A+(B) (ωn ) − {V A+(B) (ωn ) VA (B) (ωn ), ρ}] + ∑n = 1,2 γAR(B) (ωn ) n¯ R (ωn )[2V A+(B) (ωn ) ρVA (B) (ωn ) − {VA (B) (ωn ) V A+(B) (ωn ), ρ}],

(10)

a ζ AB [ρ]

reflects the collective behavior of two qubits as a single entity to emit or absorb photons induced by the common bath a where being of the forms a a ζ AB [ρ] = ∑n = 1,2 γAB (ωn )[n¯ a (ωn ) + 1][2VA (ωn ) ρV B+ (ωn ) − {V B+ (ωn ) VA (ωn ), ρ}] a + ∑n = 1,2 γAB (ωn ) n¯ a (ωn )[2V A+ (ωn ) ρVB (ωn ) − {VB (ωn ) V A+ (ωn ), ρ}] a + ∑n = 1,2 γAB (ωn )[n¯ a (ωn ) + 1][2VB (ωn ) ρV A+ (ωn ) − {V A+ (ωn ) VB (ωn ), ρ}] a + ∑n = 1,2 γAB (ωn ) n¯ a (ωn )[2V B+ (ωn ) ρVA (ωn ) − {VA (ωn ) V B+ (ωn ), ρ}],

γ R (ω)

(11)

|2 δ (ω

= 2π ∑i |gR, i − ωR, i ) denotes the spectral coupling density of the bath R at the frequency ω . For simplicity, we where suppose that γAR(B) (ωn ) = γAR(B) are frequency-independent throughout the paper, then the collective damping rate is a γAB (ωn ) = γAa (ωn ) γBa (ωn ) = Bose distribution as

a γAa γBa = γAB . The average photon number n¯ R (ωn ) of the bath R depends on the temperature TR and takes

n¯ R (ωn ) = 1 [ exp (ωn TR) − 1] .

(12)

Taking Eqs. (10)–(12) into the equation of motion (9), we can get the master equation 1076

Optik - International Journal for Light and Electron Optics 182 (2019) 1074–1081

M.-J. Wang (王美姣) and Y.-J. Xia (夏云杰)

1 1 2 2 ρ˙ S = −i [HS , ρS ] + Γ↑− L τ 21 (ρ) + Γ↓+ L τ12 (ρ) + Γ↑− L τ31 (ρ) + Γ↓− L τ13 (ρ) 2 2 1 1 + Γ↑+ L τ42 (ρ) + Γ↓+ L τ 24 (ρ) + Γ↑+ L τ43 (ρ) + Γ↓− L τ34 (ρ)

+ 2(Λ↑1 τ21 ρτ34 + Λ↓1 τ12 ρτ43 + Λ↑2 τ42 ρτ13 + Λ↓2 τ24 ρτ31 + h. c.), where LX (ρS ) ≡ 2XρS

X+

−

{X+X ,

ρS } with X an arbitrary system operator, τpq = |λp 〉〈λq|, and

+

(−1)n cos θ (γAR − γBR )

¯ 2 ⎤ [n

n Γ↓± = ∑R =a,b ⎡ (γA + γB ) 2 + ⎣

(−1)n cos θ (γAR − γBR )

¯ 2⎤ n

n Γ↑± =

R R ∑R =a,b ⎡ (γA + γB ) 2 ⎣ R

Λ↑n = ∑R =a,b ⎡ (−1) ⎣ Λ↓n =

R

(13)

n (γ R − γ R ) A B 2

R n R ∑R =a,b ⎡ (−1) (γA − γB ) 2 ⎣

⎦ ⎦

R (ω ) n

R (ω ) n

+

cos θ (γAR + γBR )

+

cos θ (γAR + γBR ) ¯ R (ωn ). 2⎤ n

¯ 2 ⎤ [n ⎦

R (ω ) n

+ 1] ± (−1)n sin θ γAa γBa [n¯ a (ωn ) + 1],

± (−1)n sin θ γAa γBa n¯ a (ωn ), + 1], (14)

⎦

ρ SS

of the master Eq. (13). By In order to study the stationary regime of the model, we need to solve the steady-state solution SS |λ 〉 with l ≠ l′ are zero, while the diagonal making ρ˙ SS = 0 in Eq. (13), we obtain that all off-diagonal elements ρllSS l′ ′ = 〈λl | ρ elements can be obtained as 1 2 2 1 2 1 1 2 1 2 2 1 ρ11SS = (Γ↓+ Γ↓− Γ↓+ + Γ↓+ Γ↓+ Γ↑+ + Γ↓+ Γ↓− Γ↓− + Γ↓− Γ↑+ Γ↓−) A, SS 1 2 2 2 2 1 1 2 1 1 2 1 ρ22 = (Γ↑+ Γ↓− Γ↓+ + Γ↑− Γ↓+ Γ↑+ + +Γ↑− Γ↓+ Γ↑+ + Γ↑− Γ↓− Γ↓−) A, SS 1 2 2 1 2 1 2 2 1 1 2 1 ρ33 = (Γ↓+ Γ↑− Γ↓+ + Γ↓+ Γ↑− Γ↓− + Γ↑− Γ↑+ Γ↓− + Γ↑− Γ↑+ Γ↓−) A, 1 2 2 1 2 1 2 2 1 1 2 1 SS ρ44 = (Γ↑− Γ↓− Γ↑+ + Γ↓+ Γ↑− Γ↑+ + Γ↑− Γ↑+ Γ↑+ + Γ↑− Γ↑+ Γ↑+) A,

where

ρijss

= 〈λi |

ρ SS

(15)

|λj〉 and

1 2 2 1 2 1 1 2 1 2 2 1 A = Γ↓+ Γ↓− Γ↓+ + Γ↓+ Γ↓+ Γ↑+ + Γ↓+ Γ↓− Γ↓− + Γ↓− Γ↑+ Γ↓− 1 2 2 2 2 1 1 2 1 1 2 1 + Γ↑+ Γ↓− Γ↓+ + Γ↑− Γ↓+ Γ↑+ + Γ↑− Γ↓+ Γ↑+ + Γ↑− Γ↓− Γ↓− 1 2 2 1 2 1 2 2 1 1 2 1 + Γ↓+ Γ↑− Γ↓+ + Γ↓+ Γ↑− Γ↓− + Γ↑− Γ↑+ Γ↓− + Γ↑− Γ↑+ Γ↓− 1 2 2 1 2 1 2 2 1 1 2 1 + Γ↑− Γ↓− Γ↑+ + Γ↓+ Γ↑− Γ↑+ + Γ↑− Γ↑+ Γ↑+ + Γ↑− Γ↑+ Γ↑+.

3. The steady-state entanglement and heat current 3.1. The steady-state entanglement In this work, we are interested in the steady-state entanglement between two qubits after the total system has reached the stationary state. As a figure of merit, we apply the concurrence [36] to quantify the steady-state entanglement in the following sections. Since the concurrence is defined in the bare-state representation, the evolution of the system is expressed in the eigenstate representation. Therefore, we need to obtain the transformation between the two representations as follows: SS SS = ρ44 μ11SS = ρ11SS , μ44 , SS SS SS = cos2 (θ 2) ρ22 + sin2 (θ 2) ρ33 μ22 , SS SS SS = sin2 (θ 2) ρ22 + cos2 (θ 2) ρ33 μ33 , SS SS SS SS = μ32 = sin θ (ρ22 − ρ33 μ23 ) 2,

μijss

(16)

ρ SS

= 〈ηi | |ηj 〉. Hence, for the so-called X-class state, the density matrix (expressed in the bare-state representation where {|11〉, |10〉, |01〉, |00〉 } ) is given as [37] SS

μSS

⎛ μ11 ⎜ 0 =⎜ ⎜ 0 ⎜ 0 ⎝

0

0

SS μ22 SS μ32

SS μ23 SS μ33

0

0

0 ⎞ 0 ⎟ ⎟. 0 ⎟ SS ⎟ μ44 ⎠

(17)

The concurrence is SS C = 2 max {0, |μ23 |−

{

= 2 max 0,

1 2

ss μ11ss μ44 }

ss sin θ (ρ22 − ρ33ss ) −

}

ss ρ11ss ρ44 .

(18)

Obviously, C = 0 means the entanglement is zero and C = 1 stands for the maximal entanglement. 1077

Optik - International Journal for Light and Electron Optics 182 (2019) 1074–1081

M.-J. Wang (王美姣) and Y.-J. Xia (夏云杰)

Fig. 2. In thermal equilibrium baths with T1 = T2 = T , the steady-state concurrence as a function of the scaled temperature of thermal baths T for various coupling strengths ξ in (a) and energy detunings Δω in (b). The critical temperature Tc as a function of the energy detunings Δω for different ξ is shown in (c). We set Δω = 2γ in (a) and ξ = 10γ in (b). Other parameters are set as γAa = γAb = γBa = γBb = γ and ωm = 20γ .

In the following, we first study the steady-state entanglement of two coupled qubits in the case of T1 = T2 = T . In Fig. 2(a)(b), we plot the steady-state concurrence as a function of the scaled temperature thermal baths T for various coupling strengths ξ and energy detunings Δω, respectively. The concurrence at T = 0 is always zero since at which the steady-state of the system is in the ground state |00〉 with no any correlation. As the temperature increases, the steady-state entanglement increases and reaches to a maximal value. Then, the entanglement decreases with a further increase in the temperature and vanishes eventually. In fact, the state of two coupled qubits at high temperature is near to the completely mixed state which contains no entanglement. Fig. 2 shows that the more steady-state entanglement can be created and the larger region of T can sustain the entanglement for a stronger coupling ξ in (a) and a larger energy detunings Δω in (b). We also see the sudden death of the concurrence when the bath temperature increases up to a critical value Tc . The dependences of Tc on the energy detunings Δω and coupling strengths ξ are shown in Fig. 2(c). In which one can see that Tc increases with Δω (ξ ) for a given ξ ( Δω) being consistent with the results in Fig. 2(a)(b). Therefore, by increasing the energy detuning and coupling strengths one can enhance the steady-state entanglement in the high temperature regimes. Next, we consider the steady-state entanglement of two coupled qubits in the nonequilibrium case with T1 > T2 , supposing T1 = Tm + ΔT 2 and T2 = Tm − ΔT 2 , where Tm denotes the average temperature and ΔT stands for the gradient of two thermal baths. We note that the concurrence increases with the average temperature Tm increases for ΔT = 0 , similar to the results in Fig. 2(a)(b). In the nonequilibrium regime, we are more interested in the effect of the temperature gradient ΔT on the steady-state entanglement. As shown in Fig. 3(a), the concurrence decreases with the temperature gradient ΔT increases. The concurrence may also decrease firstly and then increases with the temperature gradient ΔT increases, e.g., for Tm = 2γ . Therefore, we get that the temperature gradient ΔT has the disadvantage for entanglement when the qubit B interacts with two baths and qubit A only interacts with bath-a . The optimal entanglement appears at the equilibrium condition with ΔT = 0 . Here, we also concerned with the critical value ΔTc in which leads to the entanglement sudden death. The dependences of ΔTc on the average temperature Tm is shown in Fig. 3(b). In which one can see that ΔTc decreases with Tm increases when ξ is smaller, e.g., for ξ = 6γ , ξ = 8γ , while ΔTc increases firstly and then decreases with Tm increases, e.g., ξ = 10γ . That is, in the low temperature, the temperature gradient has a wider range of changes so that it may be useful for improvement of concurrence as shown in Fig. 2(a). In Fig. 4, we consider the nonequilibrium case with T1 < T2 , supposing T1 = Tm − ΔT 2 and T2 = Tm + ΔT 2 . In the nonequilibrium regime, we note that the concurrence firstly increases and then decreases with the average temperature Tm increases for ΔT = 0 as shown in Fig. 4(a) and the optimal concurrence approximately arises in Tm = 6γ , which are consistent with Fig. 2(a)(b). The concurrence increases with the temperature gradient ΔT increases when the average temperature Tm is relatively small(e.g., for Tm = 4γ , Tm = 5γ , Tm = 5.5γ and Tm = 6γ ), while the concurrence increases firstly and then decreases with ΔT increases for the larger Tm (e.g. for Tm = 8γ in Fig. 4(a), Tm = 10γ and Tm = 11γ in Fig. 4(b)), which implies that nonequilibrium condition with ΔT ≠ 0 is beneficial to the concurrence under the low average temperature, whereas for the higher temperature the optimal entanglement appears at ΔT = 10γ . With a further increase in the average temperature Tm , the entanglement is zero at the beginning(e.g., for Tm = 12γ , Tm = 13γ , Tm = 14γ , Tm = 15γ , Tm = 16γ and Tm = 17γ in Fig. 4(b)). As the temperature gradient ΔT increases, the entanglement begins to appear. Here, we also concern with the critical value ΔTr in which the entanglement occurs. In Fig. 4(c), we note that ΔTr increases with Tm increases whatever the value of ξ is. This means that the higher average temperature, the greater the temperature gradient required for entanglement to occur. Similarly to the case T1 > T2 , for the low temperature, the temperature gradient has a wider range of changes. 3.2. The heat current As another figure of merit, we consider the heat current with respect to a bath Rμ defined as

Qμ = Tr {ζμ [ρ] HS }.

(19)

From the side of a bath, a positive heat current means heat release from the bath to the system, while a negative value implies heat absorption of the bath from the system. Therefore, a change of sign of the heat current can be used as a witness for a crossover between heat absorption and heat release or vice versa. According to the Eqs. (4),(10),(11) and (14), we can derive from (19) the explicit expressions of heat currents for the two baths as 1a SS 1a SS 1a SS 1a SS 2a SS 2a SS 2a SS 2a SS Qa = 2(Γ↓+ ρ22 + Γ↓− ρ44 − Γ↑− ρ11 − Γ↑+ ρ33 ) ω1 + 2(Γ↓− ρ33 + Γ↓+ ρ44 − Γ↑− ρ11 − Γ↑+ ρ22 ) ω2 ,

1078

(20)

Optik - International Journal for Light and Electron Optics 182 (2019) 1074–1081

M.-J. Wang (王美姣) and Y.-J. Xia (夏云杰)

Fig. 3. In thermal nonequilibrium baths with T1 > T2 , the steady-state concurrence as a function of the scaled temperature gradient of two thermal baths ΔT γ for different average temperatures Tm in (a). The critical temperature Tc as a function of the average temperatures Tm for different ξ is shown in (b). We set ξ = 10γ in (a). Other parameters are set as γAa = γAb = γBa = γBb = γ , ωm = 20γ and Δω = 10γ .

Fig. 4. In thermal nonequilibrium baths with T1 < T2 , the steady-state concurrence as a function of the scaled temperature gradient of two thermal baths ΔT γ for different average temperatures Tm in (a) and (b). The critical temperature Tc as a function of the average temperatures Tm for different ξ is shown in (c). We set ξ = 10γ in (a) and (b). Other parameters are set as γAa = γAb = γBa = γBb = γ , ωm = 20γ and Δω = 10γ . 1b SS 1b SS 1b SS 1b SS 2b SS 2b SS 2b SS 2b SS Q b = 2(Γ↓+ ρ22 + Γ↓− ρ44 − Γ↑− ρ11 − Γ↑+ ρ33 ) ω1 + 2(Γ↓− ρ33 + Γ↓+ ρ44 − Γ↑− ρ11 − Γ↑+ ρ22 ) ω2 .

(21)

Fig. 5(a) shows the variations of heat current Qa and Qb as a function of the temperature T1. For the temperature T1 < 2 , Qb is positive while Qa is negative, which implies that the heat flows from bath-b into bath-a . At the critical temperature T1 = 2 (a small diagram in Fig. 5(a)), the total system reaches thermal equilibrium and all heat current become zero. When T1 > 2 , the heat of bath-a is transferred into bath-b . In Fig. 5(b), we show the heat current Qa as a function of the temperatures T1 for different energy detunings Δω. The figures show that the larger Δω is beneficial on the heat current when T1 < 13, while for T1 > 13, the larger Δω leads to a negative effect. In Fig. 5(c), we observe that heat current Qa decreases with coupling strengths ξ increases when T1 > T2 ; for T1 < T2 , heat current Qa increases slowly with coupling strengths ξ increases. For a given ξ , Qa increases with T1 increases, which is consistent with Fig. 5(a). In Fig. 5(d), Qa becomes zero the moment the corresponding temperature of the two baths is the same and the low temperature is smaller, heat current is larger. 1079

Optik - International Journal for Light and Electron Optics 182 (2019) 1074–1081

M.-J. Wang (王美姣) and Y.-J. Xia (夏云杰)

Fig. 5. Heat currents Qa and Qb as a function of the high temperatures T1 in (a)； Qa as a function of the high temperature T1 for different energy detunings Δω in (b) and diverse low temperature T2 in (d)； Qa as a function of various coupling strengths ξ for different high temperatures T1 in (c). We set Δω = 2 in (a)(c)(d), ξ = 10 in (a)(b)(d) and T2 = 2 in (a)(b)(c). Other parameters are set as γAa = γAb = γBa = γBb = γ = 0.01 and ωm = 20 .

4. Conclusion In conclusion, we have investigated the steady-state entanglement and heat current of two coupled qubits in bath-a and bath-b with temperatures T1 and T2 , respectively. We construct the master equation in the eigenstate representation of two coupled qubits to describe the dynamics of the total system and derive the solutions in the steady-state with strong coupling regime between them. In thermal equilibrium baths with T1 = T2 = T , the steady-state entanglement firstly increases and then decreases withT increases in a finite region. It vanishes eventually with a further increase temperatureT . For a given T , the entanglement is proportional to the coupling strengths ξ and the energy detunings Δω. In the nonequilibrium case with T1 > T2 , the concurrence decreases with the temperature gradient ΔT increases, so we get that the temperature gradient ΔT has the disadvantage for entanglement. The optimal entanglement appears at the equilibrium condition with ΔT = 0 . For T1 < T2 , the entanglement is enhanced by the temperature gradient ΔT for the smaller average temperature Tm . Whlie the concurrence increases firstly and then decreases with ΔT increases for the higher average temperature Tm , and that the optimal entanglement appears at ΔT = 10γ . We also consider the ciritical temperature Tc in equilibrum case and the critical temperature difference ΔT in the nonequilibrium case. In equilibrium case, Tc increases with the energy detunings Δω(coupling strengths ξ )for a given ξ ( Δω). In the nonequilibrium case, for T1 > T2 , the critical temperature difference ΔTc decreases with the average temperature Tm increases when ξ is smaller, while ΔTc increases firstly and then decreases with Tm increases when ξ is relatively bigger; for T1 < T2 , ΔTr increases with Tm . Subsequently, we study the heat current Qμ of two coupled qubits. For T1 < 2 , the heat current flows from bath- b into bath-a . At the critical temperature T1 = 2 , the total system reaches thermal equilibrium and all heat current become zero. For various low temperature T2 , T2 is smaller, heat current is larger. When T1 > 2 , the heat of bath-a is transferred into bath- b . The energy detuning Δω has a positive (negative) effect on the heat current in the low (high) temperature; heat current Qa decreases(increases) with coupling strengths ξ increases for a given temperature whenT1 > T2 (T1 < T2 ). We can conclude that the two qubits are in contact with the two baths with different temperature, the evolution is different and the impact on the heat current is also different. The results of this work might be useful for the quantum information tasks in thermal environments.

Acknowledgment We gratefully acknowledges the support of the National Natural Science Foundation of China grant 61675115.

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Optik - International Journal for Light and Electron Optics 182 (2019) 1074–1081

M.-J. Wang (王美姣) and Y.-J. Xia (夏云杰)

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