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Theoretical Studies of Dynamics of Density in Contact with Thermal Bath and

Haw Jing Yan University of Tokyo Research Internship Programme Summary Report

In this internship project, we revisit the of the with the standard projection techniques of non-equilibrium statistical presented in [T. Mori and S. Miyashita, J. Phys. Soc. Jpn., Vol.77, No. 12]. Under the case of weak system- environment coupling, one derives a time convolutionless equation of of the reduced density matrix. In long time limit, the equation of motion can be simplified, which eventually leads to the conventional quantum master equations. Next, we examine the steady states of the equations of by referring to the equilibrium state of the system itself and the modified equilibrium state due to interaction with environment. Lastly, we illustrate our studies by providing two explicit examples of -boson and boson system.

I. INTRODUCTION

The theoretical description of the relaxation phe- nomena of quantum dynamics in open quantum system can be obtained from the Bloch equation of the open quantum system, which consist of system, thermal bath and the system-bath interaction [1, 2]. However, memory effects in open quantum system often leads to non-Markovian dynamics. The systematic approach to such dynamics is the method of the projection operator, which is often deployed in the non-equilibrium statistical FIG. 1: Open quantum system characterized by ρT, which mechanics [3]. consists of the system ρS and thermal bath ρB. The evolution of the system, thermal bath and the total system are described In this present summary report, following [4, 5] we by HS, HB and HT respectively discuss the idea of using projection operator technique to obtain the equation of motion (EOM) of the reduced density matrix of the system. In an open quantum sys- tem, the reduced density matrix of the steady state is ∂ 1 ρT = [HT, ρT] ≡ iLρT. (2) modified from the equilibrium of the system itself due to ∂t i~ the presence of interaction with the thermal bath. We explicitly verify this modified solution with the equilib- where the between the HT and ρT, the total rium density matrix of the total system. The role of real system density matrix defines a linear operator denoted and imaginary terms in the is by iL operating on ρT. studied by examining their respective steady state solu- tions. We complete our discussion with the example of spin-boson and boson system. B. Projection Operator Scheme

The projection operator technique is based on the idea II. FORMULATION of elimination of degrees of freedom in order to give a simplified effective description of the system through a A. Open quantum system reduced set of variables. We first introduced the pro- jection superoperator P, which acts on the states of the total system and projects it ρT onto a tensor product We study the dynamics of the reduced density matrix states [1, 2] of the system HS in the open quantum system (Fig. 1), in which the Hamiltonian of the total system HT is PρT = ρS ⊗ ρB ≡ TrBρT ⊗ ρB (3)

HT = HS + HB + λHI. (1) where ρB is the equilibrium density matrix of the thermal bath and the reduced density of the system ρS is the par- where λ is a parameter that represent the coupling tial taken over the bath TrB defined strength between the system and the environment. The as equation of motion of the density matrix of the total system is given by the Bloch equation ρS = TrBρT (4) 2

This projection PρT is often referred to the relevant part where we introduced Ψ(t), the autocorrelation function of the density matrix. The projection operator P satisfies of Y in the thermal bath P2 = P and we also have Q = 1 − P which is related iHBt/ −iHBt/ the irrelevant part of the density matrix. With these Ψ(t) = TrBe ~Y e ~Y ρB ≡ hY (t)Y i. (11) properties, we can seperate Eq.(2) as  ∂ ∂t Pρ = P(iLρ) = PiLPρ + PiLQρ B. Time convolutionless Form ∂ (5) ∂t Qρ = Q(iLρ) = QiLPρ + QiLQρ. Solving the second equation and substituting back into The equation of motion Eq.(10) is an integro- the first equation, one obtains differential equation which is rather difficult to solve be- ∂ cause it contains a convolution. Nevertheless, since we ∂t Pρ = PiLPρ are in the in the weak coupling regime, we may use the + PiL R t e(t−τ)Q(iL)QiLPρ(τ)dτ (6) t0 replacement (t−t0)Q(iL) + PiLe Qρ(t0). e−iHSt/~ρ (t)eiHSt/~ ≈ e−iHSτ/~ρ (τ)eiHSτ/~ (12) This equation can be understood as such: the first S S terms represents the quantum dynamics due to the up to the second order of λ. With this, our equation of system Hamiltonian, the second term arises from the motion now does not depend on τ. Together with the non-Markov evolution due to the memory effects, i.e. the substitution u = t − τ, we obtain future states depends on the initial time t0 as well. The 2 last term exhibits dependence to the initial total density ∂ 1 λ Z t−t0 matrix ρT(t0). ρS = [HS, ρS] − TrB du ∂t i~ ~ 0 × [XX(−u)ρS(t)Ψ(u) − XρS(t)X(−u)Ψ(−u)

− X(−u)ρS(t)XΨ(u) + ρS(t)X(−u)XΨ(−u)] III. EQUATION OF MOTION (t−t0)iL0 + λTrBiLI e QρT(t0) Z 1 Here, we will be studying the case where the system 2 (t−t0)iL0 + λ TrBiLIe dx and the environment are weakly coupled to each other, in 0 which only up to second order of the interaction strength −x(t−t0)iL0 x(t−t0)iL0 × Qe (t − t0)iLIQe Qρ(t0), λ will be considered. With the relations in Appendix, (13) Eq. (6) takes the form of which is a time-convolutionless form of equation of mo- tion of the reduced density matrix. We shall denoted this ∂ 2 R t (t−τ)iL ρS = iLSρS + λ TrBiLI e iLI ρBρS(τ)dτ as ∂t t0 0 (t−t0)iL0 + λTrBiLI e QρT(t0) ∂ 1 (2) 2 (t−t0)iL0 R ρS ≡ L (ρS) (14) + λ TrBiLIe 0 dx ∂t −x(t−t0)iL0 x(t−t0)iL0 × Qe (t − t0)iLIQe Qρ(t0), (7) where the superscript indicates the equation of motion is where up to the second order of λ. 1 1 [HS + HB, ρ] ≡ iL0ρ, and [HI, ρ] ≡ iLIρ. (8) i~ i~ C. Conventional Quantum Master Equations

A. Explicit Interaction The thermal bath in consideration satisfies the follow- ing properties: We now adopt the following form of the interaction 1. Equilibrium density matrix of thermal bath Hamiltonian e−βHB HI = XY, (9) ρB = ,Z = Tr e−βHB (15) Z B B where X denotes an operator from the system and Y is B the operator belong to the bath[6]. The second term can 2. Long time limit correlation be expanded as t λ 2 R −i(t−τ)HS i(t−τ)HS lim Ψ(t) = 0 (16) −( ) { Ψ(t − τ)Xe XρS(τ)e ~ t0 t→∞ −i(t−τ)HS i(t−τ)HS − Ψ(−t + τ)Xe ρS(τ)Xe −i(t−τ)HS i(t−τ)HS − Ψ(t − τ)e XρS(τ)e X 3. Kubo-Martin-Schwinger relation −i(t−τ)HS i(t−τ)HS + Ψ(−t + τ)e ρS(τ)Xe X}dτ. (10) Ψ(t) = Ψ(−t − i~β) (17) 3

In the long time limit where t0 → ∞, the last two terms V. STEADY STATE SOLUTIONS involving ρT(t0) in Eq. (13) vanishes due to relation Eq. (16) and we have We now trying to find the solutions for the equation of motions that satisfies ∂ 1 λ2 Z ∞ ρS = [HS, ρS] − TrB du ∂ 1 ∂t i~ ~ 0 ρ = [H, ρ] = 0 (27) ∂t i~ × [XX(−u)ρ (t)Ψ(u) − Xρ (t)X(−u)Ψ(−u) S S by using the equilibrium states obtained from previous − X(−u)ρS(t)XΨ(u) + ρS(t)X(−u)XΨ(−u)] sections. The results are summarized in the Table 2 (18) We shall denote this form of quantum master equation as ∂ ρ = L(2) (ρ ) (19) ∂t S RG S This equation can be further evaluated with the identity

Z ∞ i eiωudu = πδ(ω) + P . (20) 0 ω By discarding the principal value part of the integral, we arrive at the conventional quantum master equation FIG. 2: Steady state solutions (right) for the equation of mo- tions (left) ∂ 1  λ 2 ρS = [HS, ρS] + ∂t i~ i~ Here, we explicitly verified that the interaction- (2) † † modified equilibrium state satisfiesL (ρS). This implies [XRρS(t) − RρS(t)X − XρS(t)R + ρS(t)R X] (21) that the steady states of the EOM up to order of λ is in- in which in the of the eigenstate of the system deed different from the equilibrium state of the system alone ρeq. However, ρeq satisfies the conventional master Hamiltonian HS, matrix R read as S S equation.   El − Em ∂ Rlm = πXlmΨ , (22) ρ = L(2)(ρeq) = 0 (28) ~ ∂t S 0 S and we represent this quantum master equation by This means the conventional master equation causes the system to relax to equilibrium of the system without the ∂ trace of the effects of thermal bath. The interaction ρ = L(2)(ρ ) (23) ∂t S 0 S with the thermal bath can be recovered by the quan- tum master equations that incorporates the imaginary part (Eq.(18)). This imaginary part gives the renormal- IV. EQUILIBRIUM STATES ization of Hamiltonian of the system, which is known as “Lamb shift”. The equilibrium state of the total system is VI. EXAMPLES e−βHT ρeq = ,Z = Tre−βHT . (24) T Z As verification, we work out the equation of motions And the equilibrium state of the system itself is and the reduced density matrices of the total density ma- trices for the following system e−βHS eq −βHS ρS = ,ZS = TrSe . (25) ZS A. Spin-boson system To obtain the reduced density matrix of the equilibrium state, we take the of the thermal bath from Here we study the spin-boson system given by the total system (4) HS = ~ωC |eihe|, (29) eq(2) eq(2) ρS = TrBρT , (26) X † HB = ~ωαbαbα, (30) which results in the α 4 where ωC is the transition frequency of the of the system itself. The proper equation of motion to and the |ei. ωα are the frequencies for the deploy depends on the regime, i.e. time evolution within bosonic bath with {b†, b} denotes the creation and annihi- the thermal bath relaxation time or long time limit or in- lation operator of the bath. The interaction Hamiltonian terest of studies, i.e. relaxation to equilibrium of system is given by or effect of contact with thermal bath.

X † † † HI = λ (καbασ+ +καbασ−) = λ(Bσ+ +B σ−), (31) α Acknowledgement where The author would like to thank Professor Seiji X X B = κ b , and B† = κ† b† . (32) Miyashita and Mr. Takashi Mori for their valuable and α α α α insightful discussion, Mr. Sergio Andraus for useful ad- α α vices (and also Remo san, Takahashi san and Go san for attending my presentation twice per week). The author B. Boson system would also like to express this gratitude to the organizer of UTRIP. This work is supported by Graduate School of Science (GSS) of the University of Tokyo and Daiwa- For the boson system, basically we replace HS of pre- vious case with Securities Group Scholarship for Summer-School Partic- ipants. † Appendix HS = ~ωC a a, (33) where {a†, a} denotes the creation and annihilation The relations used to derive the equation of motion of operator of the system. the reduced density matrix Eq. (7) are

In both cases, by substituting the reduced density ma- PiLPρ = PiLPρBρS. trix backed to the equation of motion, we show that they PiLBf = ρBTrB[HB, f]/i~ = 0. are indeed the steady states for the spin-boson and boson iLBPf = [HB, ρBTrBf]/i~ = 0. system respectively. PiLSPf = ρBTrB[HS, ρBTrBf]/i~ = iLSPf. PiLIPf = ρB[TrB(HIρB), TrBf]/i~ = [HI, Pf]/i~ = iLIPf. VII. CONCLUSIONS QiLP = iλ(L − LI)P. PiLQ = iλP(L − LI). We explicitly verified that the interaction with system leads to modification from the equilibrium density matrix where f is some arbituary function.

[1] R. Kubo, M. Toda and N. Hashitsume, [4] T. Mori and S. Miyashita: J. Phys. Soc. Jpn. 77 (2008) II, (Springer-Verlag, New York, 1985). 124005. [2] W. H. Louisell, Quantum Statistical Properties of Radia- [5] T. Mori and S. Miyashita: unpublished. P tion, (Wiley, New York, 1973). [6] We may also extend this to a more general form of i XiYi [3] Breuer, Heinz-Peter; F. Petruccione (2007). The Theory of Open Quantum Systems, Oxford University Press