Note on Decoherence Theory Yixuan Wang May 30, 2020 Contents 1 Introduction 1 1.1 Motivation . 1 1.2 Open Quantum Systems . 3 1.3 An Exactly Solvable Model . 3 2 Markovian Dynamics of Open Quantum Systems 5 2.1 Lindblad Master Equation . 5 2.2 Microscopic derivation . 6 2.3 Quantum Trajectories . 7 2.4 Examples . 7 2.4.1 Dephasing . 7 2.4.2 Damped Harmonic Oscillator . 8 3 Quantum Origin of the Classical 8 3.1 Quantum Measurement: information transfer to environment . 9 3.1.1 System and Apparatus . 9 3.1.2 Three points of view on observers . 11 3.2 Loss of Correspondence: why is open system necessary? . 12 3.3 Einselection . 14 3.3.1 Scheme of Einselection . 14 3.3.2 Einselection in Phase Space . 17 1 Introduction 1.1 Motivation As we all know, in standard quantum mechanics, dynamics of quantum systems are governed by Schr¨odingerequation, namely @ i j i = Hj i; (1) ~@t 1 where H is some Hermitian operator. The Hermiticity of H ensures that @ H − Hy h j i = h j j i = 0: @t i~ Thus, time evolution operator U(t; t0) is unitary, so the coherence of quantum states, simply speaking the non-diagonal elements of density matrix, does not disappear during time evolution. However, in practice, we do see the loss of coherence. For example, after a measure- ment, a pure quantum state loses the coherence and becomes a mixture of several pure states, namely, the non-diagonal terms in the measured observable vanishes. This can be described by X ρ ! ρAfterMeasurement = ΠiρΠi; i 1 where Πis are projection operator onto the eigenspaces of the measured observable. Such a process cannot be described by a unitary evolution. This is manifested since T r(ρ2) = 1 for pure states, T r(ρ2) < 1 for mixed states and T r(ρ(t)2) = T r(U(t; 0)ρ(0)U y(t; 0)U(t; 0)ρ(0)U y(t; 0)) = T r(ρ(0)2); i.e. unitary evolution preserves purity. Furthermore, von Neumann entropy S(ρ) = T r(ρ ln ρ) is also a constant of motion. It seems that if one can include other systems to make it an isolated system, thus the evolution of the total system becomes unitary again. But this is not practical in realistic situations due to the huge size of environment and our ignorance, such as electromagnetic vacuum. On the other hand, the problem of realism arises since one cannot directly see clas- sicality in the realm of Quantum Mechanics. As presented above, the vanishment of the off-diagonal term in some basis means that the classical definiteness emerges, where this basis is called preferred states or pointer basis(PS).2 This can be seen from that if we have such a state, then it can be decomposed into X X ρ = pij ih j; with pi = 1; h j i = δij; i i namely, this state is a mixture of several orthogonal state3, which has a classical coun- terpart. In our macroscopic world, nothing preserves the coherence. This can be un- derstood by the famous gedanken experiment, Schr¨odinger's Cat. In recent years, a \Schr¨odingerCat" state had been realized in several experiments. REMAIN TO BE COMPLEMENTED. Decoherence theory can gives a satisfactory answer to the origin of classicality. 1They can be generalized to Kraus operator in describing the dynamics of a subsystem, s well as generalized measurements. 2In mathematics, there is no such a special basis that makes it more sense to be PS. But in experiment, physicists can intuitively choose such a basis, such as fj "i; j #ig in Stern-Gerlach experiment, although theoretically there is no reasonable argument. 3you can always do that by diagonalizing the density matrix. This is not in contrary with previous footnote since this basis is state-dependent. In general this basis varies with time. 2 1.2 Open Quantum Systems In general, the quantum system which we consider consists of two parts, the system of our interest and its environment. To avoid confusion, we often call the former one as Principle system. We denote their Hilbert space as HS andHE, and the total Hamiltonian as H = HS ⊗ 1 + 1 ⊗ HE + HSE: Suppose the density operator at time t = 0 is uncorrelated, that is ρ(0) = ρS ⊗ ρE; then the principle system evolves with the unitary evolution operator U(t) as y ρS(t) = T rE(U(t)ρS ⊗ ρEU (t)); (2) where T rE represents partial trace over H. If we further assume the environment is a pure state initially, i.e.ρE = j0ih0j, then y ρS(t) = T rE(U(t)ρS ⊗ j0ih0jU (t)) X y = I ⊗ hijU(t)ρS ⊗ j0ih0jU (t)I ⊗ jii i X y (3) = hijU(t)j0iρSh0jU (t)jii i X y = Ei(t)ρSEi (t); i where the Kraus Operator Ei satisfies X y Ei (t)Ei(t) = I: (4) i Obviously, this is not a unitary evolution for the principle system locally. It can be easily generalized to the case that ρE is a mixed state. However, since the existence of Schmidt decomposition, we can always see it as a part of a larger environment and redefine the environment to make it a pure state. 1.3 An Exactly Solvable Model Consider a two-level system, or qubit, and model the environment as a collection of bosonic field modes. The Hamiltonian reads ! X X H = ~ σ + ! by b + σ g by + g∗b ; (5) tot 2 z ~ k k k z k k k k k k y with [bi; bk] = δik. Working in the interaction picture, we have ~ X i!kt y ∗ −i!kt Hint(t) = σz gke bk + gke bk: k 3 Since X 2 [H~int(t1); H~int(t2)] = 2i jgkj sin(!k(t2 − t1)); k which is a c-number, the time evolution operator can be derived from Dyson series that 1 X U~(t) = eiφ(t)exp( σ α(t)by − α∗(t)b ); (6) 2 z k k k k where Z t Z t i ~ ~ φ(t) = 2 dt1 dt2Θ(t1 − t2)[Hint(t1); Hint(t2)] 2~ 0 0 and 1 − ei!kt αk(t) = 2gk : ~!k Note that the operator part of U~ is similar to the so called displacement operator y ∗ Dk(α) = exp(αbk − α bk), thus the dynamics of the total system is in the sense that the environment bosonic modes are displaced under the control of the system. To make it clearer, note that Y αk(t) U~(t)j "ijξi = eiφ(t)j "i D ( )jξi (7) k 2 k Y αk(t) U~(t)j #ijξi = eiφ(t)j #i D (− )jξi: (8) k 2 k Thus for ρtot(0) = ρ ⊗ ρE; the system evolves as y ρ~(t) = T rE(U~(t)ρ ⊗ ρEU~ (t)); it can be easily verified that h" jρ~(t)j "i = h" jρj "i; h# jρ~(t)j #i = h# jρj #i; and Y h" jρ~(t)j #i = h" jρj #iT rE( Dk(αk(t))ρE) = h" jρj #iχ(t): (9) k Then by calculating χ(t), we can account for the decoherence effect cause by coupling Q to the environment. For the vacuum state ρE = k j0ih0j, X 1 − cos(!jt) χ(t) = exp(− 4jg j2 ): k 2!2 k ~ k −1 For times that are short compared to the field dynamics, t << !k , one observes a Gaussian decay of the coherences. But this is not always the case, in some specific situation, the suppression of the spectral density at low frequency can lead to a non-zero coherence even with t ! 1, which is called super-ohmic coupling. 4 2 Markovian Dynamics of Open Quantum Systems In general, we does not know the exact state of the big environment, and a simply partial trace can creates a non-closed equation, which is nonlocal in time, thus we need to make the Markovian assumption in the following. In this section, we'll introduce the basic concepts of Decoherence Theory and derive the standard formalism of it, i.e. the Lindblad master equation, and introduce some examples. 2.1 Lindblad Master Equation For an isolated system, it evolves under the von Neumann equation, i.e. @ 1 ρ = [H; ρ]; @t i~ above all, we have show that a principle system which interacts with an environment evolves as @ @ y 1 ρS(t) = T rE(U(t)ρtotU (t)) = T rE([Htot; ρtot]): (10) @t @t i~ However, this equation is not closed, and in general the environment is fairly large and it is hard to tell about the state of the environment. Besides, since there is correlation between the principle system and environment, the evolution of the system must depend of its all history, i.e. @ ρ(t) = K[ρ(τ): τ < t]; @t where K is some superoperaor. An interpretation of this dependence on the system's past is that the environment has a memory, since it affects the system in a way which depends on the history of the system-environment interaction. One may hope that on a coarse-grained time-scale, which is large compared to the inter-environmental correlation times, these memory effects might become irrelevant. In this case, a proper master equation might be appropriate, where the infinitesimal change of ρ depends only on the instantaneous system state, through a Liouville super-operator L, @t = Lρ Master equations of this type are also called Markovian, because of their resemblance to the differential Chapman-Kolmogorov equation for a classical Markov process.