Specific Heat and Thermal Entanglement in an Open Quantum system B. Lari1,*, and H. Hassanabadi2 1 Department of Physics, Ahvaz Branch, Islamic Azad University, Ahvaz, Iran 2 Physics Department, Shahrood University of Technology, Shahrood, Iran, P. O. Box 3619995161-316 E-mail: 1 [email protected]* and 2 [email protected] Abstract In this paper, the density matrix is obtained using the Non-Markovian master equation method for a system consisting of two qubit modeling with the Heisenberg XXZ chain, which involves the Dzyaloshinskii–Moriya interaction and exposed to the bosonic baths. Also, using the proposed formula for calculating the specific heat through the density matrix of the open quantum system, the specific heat is calculated. The quantum entanglement behavior with time and coupling constant is investigated. It is observed that, the specific heat at low temperatures becomes negative when the system is exposed to the environment. The time behavior of quantum entanglement for this model showed that the entanglement decreases to zero with increasing time. Keywords: Open Quantum System, Dzyaloshinskii–Moriya Interaction, Born-approximation, Entanglement, Specific Heat. 1. Introduction To fabricate quantum gates and memories [1-4], the quantum entanglement of the system and proper choice of the material according to environmental parameters is of great importance. The engaged parameters in the studies include the specific heat and the constant magnetic coupling [5]. In this regards, some researchers have tried to find a relationship between the heat capacity and the maximum quantum entanglement for closed systems [6-7]. Others considered the entanglement area law from specific heat capacity and derived the relation between exponential decay of specific heat capacity at low temperature and the entanglement at low-energy state [8]. In this paper we consider one of the standard models, namely, the anisotropic Heisenberg XXZ regime for two qubit system with the Dzyaloshinskii–Moriya (DM) interaction [9-10]. The model arises from the spin–orbit coupling where each of the qubits are exposed to separate bosonic baths. Many scientific reports have investigated the entanglement [11-15] and its specific heat [16-17] in different open quantum systems. Nevertheless, it seems useful to present a formula to calculate the specific heat (as a determinative thermodynamics parameter of solids) using density matrix and its eigenvalues for an open quantum system. In order to calculate the time evolution of the density matrix of system, we use a unitary transform to the interaction picture and solve the Von-Neumann equation with the Born-approximation [18-19]. It is assumed that, the bosonic baths obey the Cauchy- Lorentz distribution of frequencies. The time evolution of entanglement with the effective parameters in the Hamiltonian of system, such as spin-orbit coupling 퐷푧 and coupling constant 퐽푖 (where 푖 ∈ {푥, 푦, 푧}), is useful in practical fabrication of graphene-based (or GaAs-based) quantum gates [5], [20-25]. Therefore, in order to achieve our goals, this paper has been organized as follows. In Sec. 2, using the Hamiltonian of anisotropic Heisenberg XXZ model for two qubits with the DM Interaction exposed to separate bosonic baths, we obtain the initial density matrix of the system as a function of temperature and solve the Non-Markovian Master equation for our model. In Sec. 3, we have proposed a formula to calculate the specific heat using density matrix of the open quantum system. Also, the entanglement of formation (EOF) is computed. In Sec. 4, the obtained results will be summarized. 2. Non-Markovian Approach The Hamiltonian of anisotropic Heisenberg XXZ model for two-qubit system with D-M interaction, such that each qubit be exposed to Bosonic bath, can be defined as follows, 퐻 = 퐻푠 + 퐻푏 + 퐻푠푏 (1) In which 푖 푖 푥 푦 푦 푥 퐻푠 = ∑푖 퐽푖 휎1휎2 + 퐷푧(휎1 휎2 − 휎1 휎2 ) , 푖 ∈ {푥, 푦, 푧}) (2) 퐻푏 = ∑푛1 휔푛1푏푛1 푏푛1 + ∑푛2 휔푛2푏푛2 푏푛2 (3) ∑2 + ∑ 퐻푠푏 = 푖=1 휎푖 푛푖 푔푛푖푏푛푖 + ℎ. 푐. (4) 퐻푠 is Hamiltonian of the system with DM interaction. 퐻푏 shows the Hamiltonian of baths including 푛푖 mode with frequency 휔푛푖. 퐻푠푏 denotes the Hamiltonian of the interaction between the 푖 system and the baths with strength interaction 푔푛푖. "푖" labels the first and the second qubit and 휎…is the Pauli matrix. The coupling constant 퐽푖 > 0 corresponds to the antiferromagnetic case and 퐽푖 < 0 corresponds to the ferromagnetic case. The model is called Heisenberg XXZ model if 퐽푥 = 퐽푦 = 퐽 푎푛푑 퐽푧 ≠ 퐽. The density matrix of system 휌푠(푇), is obtained in terms of the standard basis {|00⟩, |10⟩, |01⟩, |11⟩} as 푒−훽퐻푠 휌 (푡 = 0, 푇) = 푠 푍 1 −훽퐽 푖휃 −푖휃 −훽퐽 = {푒 푧|00⟩⟨00|+푢|10⟩⟨10| + 푣푒 |01⟩⟨10| + 푣푒 |10⟩⟨01| + 푢|01⟩⟨01| + 푒 푧|11⟩⟨11| } (5) 푍 1 where 훽 = (푘퐵 is Boltzmann constant) and 푘퐵푇 1 훽(퐽 −2휂) 4훽휂 푢 = 푒 푧 ( 1 + 푒 ) 2 1 −훽(퐽 −2휂) 4훽휂 푣 = 푒 푧 ( 1 − 푒 ) 2 푍 = 2 푒−훽퐽푧 [1 + 푒2훽퐽푧 퐶표푠ℎ(2훽휂)] (6) 2 2 퐷 휂 = √퐽 + 퐷 ; 휃 = 푡푎푛−1 ( 푧) 푧 퐽 The density matrix defined in Eq. (5), has the X-type form [26]. We start with Von-Liouville equation and find the time evolution of density matrix 휕휌 (푡) 푖 푠푏 = − [퐻(푡), 휌 (푡)] (7) 휕푡 ћ 푠푏 In the rest of the paper, we set ћ = 1. To obtain the Non-Markovian master equation we apply the Born-approximation 휕휌퐼(푡) 푡 푠 = −푖 푡푟 {[퐻퐼 (푡), 휌퐼 (0)]} − ∫ 푑푡́푡푟 {[퐻퐼 (푡), [퐻퐼 (푡́), 휌퐼 (푡́)]]} (8) 휕푡 푏 푠푏 푠푏 0 푏 푠푏 푠푏 푠푏 Where the superscript "퐼", denotes the interaction picture. The first term in Eq. (8) is considered as zero (see the Appendix and [27-28]). Therefore, we have 휕휌퐼(푡) 푡 푠 = − ∫ 푑푡́푡푟 {[퐻퐼 (푡), [퐻퐼 (푡́), 휌퐼(푡) ⊗ 휌퐼 (0)]]} (9) 휕푡 0 푏 푠푏 푠푏 푠 푏 퐼 where 퐻푠푏(푡) = exp{푖(퐻푠 + 퐻푏)푡} 퐻푠푏 exp{−푖(퐻푠 + 퐻푏)푡} (10) 퐼 휌푠(푡) = exp{푖퐻푠푡} 휌푠(푡) exp{−푖퐻푠푡} (11) 퐼 휌푏(0) = 휌푏(0) (12) We rewrite Eq. (4) in interaction picture as 퐼 ∑2 + ∑ 퐻푠푏 = 푗=1 휎푗 푛푗 푔푛푗 푏푛푗 exp { −푖휔푛푗푡 } + ℎ. 푐. (13) To derive the above equation, we have used the following relations 푖퐻 푡 −푖퐻 푡 휎+(푡) = 푇푟 {exp ( 푠 ) 휎+(0) exp ( 푠 )} 푗 푖 ћ 푗 ћ + + + 푓표푟 푖, 푗 ∈ {1,2} 푎푛푑 푖 ≠ 푗 ⟹ 휎푗 (푡) = 휎푗 (0) = 휎푗 푑푏푛́ (푡) 푗 = 푖 [퐻 , 푏 (푡) ] ⟹ 푏 (푡) = 푏 exp { −푖휔 푡} 푑푡 푏 푛́ 푗 푛푗 푛푗 푛푗 Apart from the physical dimensions, we can choose the following initial state to the baths. 휌 (0) = |푁; … , 푛 , … , 푛 , … ⟩ ⟨푁; … , 푛 , … , 푛 , …| 푏 푗 푝 푏 푗 푝 Finally, after tracing over the baths and using the above definition for 휌푏(0), we have 2 퐼 푡 푑휌 (푡) 2 −푖휔 ( 푡−푡́) 푠 − 퐼( ) + ́ 푛푗 = ∑{ [휎푗 휌푠 푡 , 휎푗 ] ∑ |푔푛푗 | 푛푗 ∫ 푑푡 푒 푑푡 0 푗=1 푛푗 푡 2 −푖휔 ( 푡−푡́) + 퐼( ) − ́ 푛푗 + [휎푗 , 휌푠 푡 휎푗 ] ∑ |푔푛푗| ( 푛푗 + 1 ) ∫ 푑푡 푒 + ℎ. 푐. } 0 푛푗 2 퐼 (14) = ∑푗=1 풯푗(푡)휌푠(푡) If we suppose the bath is fixed on zero temperature, then the initial state has the form|푁; 0, … , 푛푖 = 0, … ,0⟩푏. Therefore, the first term in Eq. (14) is neglected. Finally we have 푑휌퐼(푡) 2 푡 −푖휔 ( 푡−푡́) 푠 = ∑2 { [휎+, 휌퐼(푡)휎−] ∑ |푔 | ∫ 푑푡́ 푒 푛푗 + ℎ. 푐. } = ∑2 풯 (푡)휌퐼(푡) (15) 푑푡 푗=1 푗 푠 푗 푛푗 푛푗 0 푗=1 푗 푠 풯푗(푡) is a super operator. If we suppose the baths are bosonic systems with Cauchy-Lorentz distribution, we have 푑휌퐼(푡) 푠 = 푅(푡) ∑2 [휎+, 휌퐼(푡)휎−] + ℎ. 푐. = ∑2 풯(푡)휌퐼(푡) (16) 푑푡 푗=1 푗 푠 푗 푗=1 푗 푠 푡 +∞ 훾 With 푅(푡) = ∫ 푑푡́ ∫ 푑휔 퐽(휔) 푒푖휔푡́ = 0 ( 1 − 푒−훾푡 ) 0 −∞ 2 The Markovian limit can be constructed by taking the limit 푡 → ∞. To derive 푅(푡), one can 2 + substitute the ∑ |푔 | by 푑휔 퐽(휔) and then use the relation 푛푗 푛푗 ∫− 훾 훾2 퐽(휔) = 0 [ ] 휋 휔2 + 훾2 Where 훾 is the scale parameter which specifies the half-width at half-maximum. Finally, transforming back to the Schrodinger picture we obtain the master equation 푑휌 (푡) 푠 = −푖[퐻 , 휌 (푡)] + ∑2 풯 (푡) 휌 (푡) (17) 푑푡 푠 푠 푗=1 푗 푠 After a little algebra, we can obtain three independent differential equations on components of 휌푖푗(t): −훾푡 푑 휌14(푡, 푇) −훾0(1 − 푒 ) 0 휌14(푡, 푇) ( ) = ( ) ( ) (18) 푑푡 −훾푡 휌41(푡, 푇) 0 −훾0(1 − 푒 ) 휌41(푡, 푇) −2푖퐽 − 3훾0 (1 − 푒−훾푡) 푖(2퐽 − 2푖퐷 ) 푧 2 푧 0 0 휌12(푡, 푇) 휌12(푡, 푇) 푖(2퐽 + 2푖퐷 ) −2푖퐽 − 3훾0 (1 − 푒−훾푡) 0 0 푑 휌13(푡, 푇) 푧 푧 2 휌13(푡, 푇) ( ) = 훾 ( ) (19) 푑푡 휌24(푡, 푇) 0 −훾푡 휌24(푡, 푇) 0 훾 (1 − 푒−훾푡) 2푖퐽푧 − (1 − 푒 ) −푖(2퐽 + 2푖퐷푧) 휌 (푡, 푇) 0 2 휌 (푡, 푇) 34 −훾푡 훾0 −훾푡 34 훾0(1 − 푒 ) 0 −푖(2퐽 − 2푖퐷 ) −2푖퐽 − (1 − 푒 ) ( 푧 푧 2 ) 4훾 − 0 (1 − 푒−훾푡) 0 0 0 0 0 휌11(푡, 푇) 2 휌11(푡, 푇) −훾푡 −훾푡 휌22(푡, 푇) 훾 (1 − 푒 ) −훾 (1 − 푒 ) 0 0 푖(2퐽 − 2푖퐷푧) −푖(2퐽 + 2푖퐷푧) 휌22(푡, 푇) 0 0 푑 휌 (푡, 푇) −훾푡 −훾푡 0 −푖(2퐽 − 2푖퐷푧) 푖(2퐽 − 2푖퐷푧) 휌 (푡, 푇) 33 = 훾0(1 − 푒 ) 0 −훾0(1 − 푒 ) 33 (20) 푑푡 휌 (푡, 푇) −훾푡 −훾푡 휌 (푡, 푇) 44 0 훾 (1 − 푒 ) 훾 (1 − 푒 ) 0 0 0 44 0 0 휌23(푡, 푇) −훾푡 휌23(푡, 푇) 0 푖(2퐽 + 2푖퐷푧) −푖(2퐽 + 2푖퐷푧) 0 −훾0(1 − 푒 ) 0 (휌32(푡, 푇)) −훾푡 (휌32(푡, 푇)) ( 0 −푖(2퐽 − 2푖퐷푧) 푖(2퐽 − 2푖퐷푧) 0 0 −훾0(1 − 푒 )) We can rewrite the above equations in as 푑 |훼(푡)⟩ = 푀̂ (푡)|훼(푡)⟩ , 푖 ∈ {1,2,3} (21) 푑푡 푖 푖 푖 The “i” labels each of the above differential equations.