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 |ψi = H|ψi, (1) ~∂t

1 where H is some Hermitian operator. The Hermiticity of H ensures that ∂ H − H† hψ|ψi = hψ| |ψi = 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 measurement, 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 †(t, 0)U(t, 0)ρ(0)U †(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 classicality 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 = pi|ψihψ|, with pi = 1, hψ|ψi = δij, i i namely, this state is a mixture of several orthogonal state3, which has a classical counterpart. In our macroscopic world, nothing preserves the coherence. This can be understood by the famous gedanken experiment, Schrödinger’s Cat. In recent years, a “SchrödingerCat” 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 {| ↑i, | ↓i} 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

† 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 = |0ih0|, then

† ρS(t) = T rE(U(t)ρS ⊗ |0ih0|U (t)) X † = I ⊗ hi|U(t)ρS ⊗ |0ih0|U (t)I ⊗ |ii i X † (3) = hi|U(t)|0iρSh0|U (t)|ii i X † = Ei(t)ρSEi (t), i where the Kraus Operator Ei satisﬁes

X † 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 ω X X H = ~ σ + ω b† b + σ g b† + g∗b , (5) tot 2 z ~ k k k z k k k k k k

† with [bi, bk] = δik. Working in the interaction picture, we have

˜ X iωkt † ∗ −iωkt Hint(t) = σz gke bk + gke bk. k

3 Since X 2 [H˜int(t1), H˜int(t2)] = 2i |gk| sin(ωk(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)b† − 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 − eiωkt αk(t) = 2gk . ~ωk Note that the operator part of U˜ is similar to the so called displacement operator † ∗ 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)| ↑i|ξi = eiφ(t)| ↑i D ( )|ξi (7) k 2 k

Y αk(t) U˜(t)| ↓i|ξi = eiφ(t)| ↓i D (− )|ξi. (8) k 2 k

Thus for ρtot(0) = ρ ⊗ ρE, the system evolves as

† ρ˜(t) = T rE(U˜(t)ρ ⊗ ρEU˜ (t)), it can be easily veriﬁed that h↑ |ρ˜(t)| ↑i = h↑ |ρ| ↑i, h↓ |ρ˜(t)| ↓i = h↓ |ρ| ↓i, and Y h↑ |ρ˜(t)| ↓i = h↑ |ρ| ↓iT rE( Dk(αk(t))ρE) = h↑ |ρ| ↓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 |0ih0|,

X 1 − cos(ωjt) χ(t) = exp(− 4|g |2 ). k 2ω2 k ~ k

−1 For times that are short compared to the ﬁeld dynamics, t << ωk , 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 → ∞, 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 ∂ ∂ † 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. Next we deﬁne the dynamical map as

+ Wt : ρ(0) 7→ ρ(t), for t ∈ R0 , (11) with the conditions that Wt preserves trace, complete positive, i.e.

Wt ⊗ 1 > 0,

5 which guarantees that the system state remains positive even if it is the reduced part of a non-separable state evolving in a higher dimensional space, and haves convex linearity, namely 0 0 Wt(λρ + (1 − λρ ) = λWt(ρ)) + (1 − λ)Wt(ρ ), which ensures that the transformation of mixed states is consistent with the classical notion of ignorance. Furthermore, we make the semigroup assumption, namely

Wt2 ◦ Wt1 = Wt1+t2 ∀t1, t2 > 0.

1 Expanding W in the basis {Ei} of the Hilbert space over L(H) with E0 = √ 1, it dim(H) can be veriﬁed that

dim(H)2−1 † X † L = c1ρ + Bρ + ρB + αijEiρEj i,j=1 (12) dim(H)2−1 1 X 1 = [H, ρ] + γ (L ρL† − {L† L , ρ}), i k k k 2 k k ~ k=1

~ † where H = 2i (B − B ) is Hermitian, and with the diagonalization of the coeﬃcient matrix α one can ﬁnd its eigenvalues {γk} and corresponding Lk. With this formalism, we can treat the dynamics of the system in the sense of simu- lation of Markovian process with Monte Carlo methods, which is called Quantum Tra- jectory. For a single ‘trajectory’, quantum states evolve continuously according to the † von Neumann equation, but sometimes jump to LkρLk with some probability. In this view, one can simulate the discrete-in-time photon count of an atom ensemble.

2.2 Microscopic derivation Above we have derived the general form for the dynamics of an open quantum system, i.e. the Lindblad master equation. But for any speciﬁc systems, the exact form of H and {γk,Lk} depends on the interaction and the state of the environment. We will see how to derive them for speciﬁc systems. As before, we consider

H = HS ⊗ 1 + 1 ⊗ HE + HSE, and takes the interaction term is weak enough for a perturbative treatment. Then we make the so-called Born approximation, namely

ρtot(t) ' ρ(t) ⊗ ρE (to 2nd order in HSE).

Just imagine that an excited atom in vacuum jump to ground state with the emission of a photon(in the view of Quantum Trajectory),this change to the vacuum states can be

6 ˜ ˜ P ˜ ˜ ignored. With the Schmidt decomposition, one can write HSE as HSE = k Ak ⊗ Bk, then Z t 1 ˜ 1 ˜ ˜ ∂tρ˜(t) = T rE( [HSE(0), ρ˜tot(0)] + 2 ds[HSE(t), [HSE(s), ρ˜tot(s)]]) i~ (i~) 0 1 X ' hB˜ (t)i [A˜ (t), ρ˜(0)] i k ρE k ~ k 1 X Z t + dshB˜ (t)B˜ (s)i {A˜ (t)A˜ (s)˜ρ(s) − A˜ (s)˜ρ(s)A˜ (t)} + H.c. (i )2 k l ρE k l l k ~ k,l 0 1 X Z t = hB˜ (t)i [A˜ (t), ρ˜(0)] + dsK(t − s)˜ρ(s) i k ρE k ~ k 0 (13)

˜ The ﬁrst term can be disregarded since we can always reformulate HE to make hBk(t)iρE vanish. For the second term, obviously it is not Markovian, thus we make the Born- Markov approximation:

Z t Z ∞ dsK(t − s)˜ρ(s) ' dsK(s)˜ρ(t), 0 0 which leads to a Markovian master equation. Other properties such as completely pos- itivity must be treated carefully, but we will not discuss them here for simplicity. Fur- thermore, if we make rotating-wave approximation, we can derive the Lindblad master equation as above, and the Lamb shift appears naturally as the renormalization of the system energies. REMAIN TO BE COMPLEMENTED

2.3 Quantum Trajectories 2.4 Examples 2.4.1 Dephasing The simplest choice is to take the Lindblad operator to be proportional to the Hamil- tonian of a discrete quantum system, i.e. to the generator of the unitary dynamics, √ L = γH. The Lindblad equation is

∂ 1 1 1 ρ = [H, ρ] + γ(HρH − H2ρ − ρH2). ∂t i~ 2 2 In the energy eigenbasis, we have

i γ 2 ρmn(t) = hm|ρ|ni = ρmn(0) exp(− (Em − En)t − (Em − En) t) ~ 2

7 2.4.2 Damped Harmonic Oscillator Next, we choose H to be the Hamiltonian of a harmonic oscillator, H = œ4megaa†a, √ ~ and take as Lindblad operator the ladder operator, L = γa. 4 The resulting Lindblad equation is known empirically to describe the quantum dynamics of a damped harmonic oscillator. It is easy to show that for a initial coherent state, it evolves as γ ρ(t) = |α(t)ihα(t)|, where α(t) = α(0) exp(−iωt − t). 2 Thus in this situation, γ describes the damping rates of the harmonic oscillator since it quantiﬁes the energy loss, i.e. hα(t)|H|α(t)i = e−γthα(0)|H|α(0)i. To see where PS lies in, we choose the initial state to be superposition of two coherent states, i.e. 1 |ψ i = (|α i + |β i), 0 N 0 0 with N = 2 + 2Rehα0|β0i then 1 ρ(t) = (|α(t)ihα(t)| + |β(t)ihβ(t)| + c(t)|α(t)ihβ(t)| + c∗(t)|β(t)ihα(t)|), N where 1 c(t) = c(0) exp([− |α(0) − β(0)|2 + iImα β∗](1 − e−γt)). 2 0 0 So in a show time scale compared to the relaxation time t γ−1, the coherence behaves as exponential decay γ |c(t)| = |c(0)| exp(− |α − β |2t) (14) 2 0 0 2 with a rate γdeco = γ2|α0 −β0| . Thus for macroscopically distinct superpositions, where the phase space distance of the quantum states is much larger than their uncertainties, i.e. |α0 − β0| 1, the decoherence rate is much larger than the relaxation rate. This quadratic dependence on the decoherence rate with the separation have been conﬁrmed in cavity QED experiments. So in this sense, it seems that the coherent states turn out to be the PS. 5 From my point of view, that’s one of the reasons that we call coherent states the most “classical” quantum states.

3 Quantum Origin of the Classical

In this section, we are going to talk about the emergence of classicality from pure quantum mechanics, where the role of decoherence begun to be appreciated and environment- induced superselection or einselection, environment-assisted invariance or envariance are recognized as the key to it.

4Here we merely accept it as an assumption. But in cavity QED, this operator can be derived from the microscopic point of view with Jaynes-Cummings model, which looks like the dipole interaction between an atom and light.REMAIN TO BE COMPLEMENTED. 5Actually coherent states is a set of overcomplete bases, namely R dαdα∗|αihα| = π1.

8 To be more precisely, the first question is: whta is quantum, classical and difference between them? Simply speaking, in our intuition, classical objects can be described by a set of definite properties independent of representations, like a trajectory in phase space, or more generally, an ensemble, i.e. mixture of such objects. However, this is not the case for quantum states since there is superposition. This can be seen from the fact that a linear space has much freedom to choose its basis. For instance, if we take a quantum state represented by density matrix ρ, then if we randomly choose a basis {|aii}, the property of this state, namely the diagonal terms or populations can be obtained. But if we choose another basis, generally the populations will change, so which basis corresponds to reality? Obviously this two bases are equivalent in mathematics. One may argue that the eigenbasis of ρ is more preferred, but this basis is state-dependent, which often varies with time evolution. So the problem is: why is there PS? One may also argue that, during unitary time evolution, eigenbasis of Hamiltonian is more preferred, but the energy eigenbasis is often contrary to our classical intuition, such as Hydrogen and harmonic oscillator, it is coherent states instead of energy eigenbasis that is quantum counterpart of classical objects. So as an conclusion, our problem is to justify why is there PS and how to figure out PS for specific systems. On the other hand, measurement is also closely connected to the PS problem. A general measurement process can be seen as strong interaction between system and apparatus, and after this process, the measured basis shows some kinds of classicality, namely it turns out to be the PS. In this section, we will talk about einselection and envariance, which justifies the existence of PS and why is there Born’s rule.

3.1 Quantum Measurement: information transfer to environment To realize measurement, we need to give it a schematic description.

3.1.1 System and Apparatus Firstly, avoiding the ill-understood anthropic attributes of observers such as conscious- ness, we should focus on the information-processing underpinnings of the “observership”. To consider the interplay between a system and an apparatus represented by S and S A A, respectively, we consider Hilbert space H = H ⊗ H with states {|sii}, {|Aii} in S A H , H , respectively. Besides, {|sii} are eigenbases of the observable S to be measured. To establish the correlation between system and apparatus, consider the conditional dynamics ! X X |Ψ(0)i = |ψi|A0i = ci|sii |A0i → ci|sii|Aii = |Ψ(t)i (15) i i which is called a premeasurement. Yet it is not enough to claim that a measurement has been achieved since operationally this EPR nature of the state emerging from the premeasurement can be made more explicit by re-writing the sum in a diﬀerent basis: X X |Ψ(t)i = ci|sii|Aii = di|rii|Bii. (16) i i

9 This freedom of basis choice – basis ambiguity – is guaranteed by the principle of superposition. Therefore, there is still a PS problem before enquiring about the speciﬁc outcome of the measurement. In the descriptions above, it seems that information of the system is transferred to the apparatus To quantify the information transfer, we ﬁrst consider the simple case that dim HS = dim HS = 2 and the conditional dynamics is represented by a CNOT gate, with the system being the control bit. Then the truth table reads:

|00i → |00i, |01i → |01i, |10i → |11i, |11i → |10i. (17)

However in the complementary basis, i.e. 1 |±i = √ (|0i ± |1i), (18) 2 the truth table reads |±i|+i → |±i|+i, |±i|−i → |∓i|−i, (19) that is, the state of apparatus becomes the control bit. Therefore, while the information about the observable with the eigenstates {|0i, |1i} travels from the system to the apparatus, in the complementary {|+i, |−i} basis it seems that the apparatus is measured by the system. In general case, dim HS = dim HS = N a pre measurement can be described by C-shifts, which can be written as:

|sji|Aki → |sji|Aj+ki (20) it seems that information is transferred from the system to the apparatus. The complementary basis of A can be deﬁned by iN ∂ B = (21) 2π ∂A in the basis {|Aii}, which is just the translation operator in this basis. If we deﬁne

exp(−iHte/~)|sji|Aki = |sji|A{k+Gj}N i, (24) where the index {k + Gj}N is evaluated modulo N. In conclusion, as the apparatus measures a certain observable of the system, the system simultaneously “measures” phases between the possible outcome states of the apparatus.

10 Besides, in a more realistic case, N 1 and we choose a subspace of HS with dimension n N to be the physical Hilbert space of the system. Then one can perform amplification in the sense that |sji is attributed to redundant apparatus states |al+Gki, where X |ali = αl(k)|Aki. k Amplification can protect measurement outcomes from noise(interaction with environment) through redundancy. The above model of amplification is unitary. Yet, it contains seeds of irreversibility from the fact that disentanglement occurs after tRes = Nte = 2pi~/g, so recurrence is rare for large N. In addition, quantitative investigation about information can be quantified by von Neumann entropy, mutual information and action. For the state after premeasurement above, |Ψ(t)i, the information about the subsystems available to the observer locally decreases, this is quantified by the increase of the entropies:

X 2 2 HA = −T rρA log ρA = 0 → − |ci | log |ci| . (25) i As the evolution of the whole SA is unitary, the increase of entropies in the subsystems is compensated by the increase of the mutual information:

X 2 2 I(S : A) = HS + HA − HSA = −2 |ci | log |ci| . (26) i And an often raised question concerns the price of information in units of some other “physical currency”. Here we shall establish that the least action necessary to transfer one bit is of the order of a fraction of ~ for quantum systems. For the premeasurement process above, the expectation value of the action involved is no less than:

X 2 I = |cj| arccos |hA0|Aji|. (27) j

When {|Aji} are mutually orthogonal, I = π/2, then the least action per bit of information decreases with the increase of N: I π i = ≈ (28) log2 N 2 log2 N This may be one reason why information appears “free” in the macroscopic domain, but expensive (close to ~/bit) in the quantum case of small Hilbert spaces.

3.1.2 Three points of view on observers Deﬁnite outcomes we perceive appear to be at odds with the principle of superposition. Here we introduce three points of view on the role of observers which may diﬀers in between quantum cases and classical cases, namely insider, discoverer and outsider.

11 For the case of insider - an observer aware of the initial state of the system, such a priori knowledge can be represented by the preexisting record |Aii, which is only corroborated by an additional measurement:

|Aii|A0i|σii → |Aii|Aii|σii. (29) Any element of surprise (any use of probabilities) must be therefore blamed on partial ignorance, thus a description without knowledge of exact initial state should be an ensemble including a list of possible states {σi} and possibilities pi. In short, such process is a confirmation of the preexisting data. For the case of discoverer observer finds out which of the potential outcomes consistent with his prior (lack of) information actually happens. This act of acquisition of information changes physical state of the observer – the state of his memory: The initial memory state containing description AρS of an ensemble and a “blank” A0, is transformed into record of a specific outcome, i.e. X |AρS ihAρS ||A0ihA0| pi|σiihσi| → |AρS ihAρS ||AiihAi||σiihσi|. (30) i Such a process is a “collapse” associated with the information gain, and with the entropy decrease translated into algorithmic randomness of the acquired data. Last not least, an outsider – someone who knows about the measurement, but (in contrast to the insider) not about the initial state of the system nor (in contrast to the discoverer) about the measurement outcome, will describe the same process still differently: ! X X |AρS ihAρS ||A0ihA0| pi|σiihσi| → |AρS ihAρS | pi|AiihAi||σiihσi| . (31) i i This view of the outsider combines one-toone classical correlation of the states of the system and the records with the indefiniteness of the outcome. This process is an entropy-preserving establishment of a correlation. In classical physics the insider view always exists in principle. In quantum physics it does not. Besides, given a set value of ~ – information storage resources of any finite physical system are finite. Hence, in quantum physics observers remain largely ignorant of the detailed state of the Universe. For discoverer, the disappearance of all the potential alternatives save for one that becomes a “reality” is the essence of the collapse. REMAIN TO BE REVISED

3.2 Loss of Correspondence: why is open system necessary? For a Hamiltonian P 2 H = + V (X), 2m one can easily derive the quantum counterpart of equation of motion, namely ∂ i ∂V hP i = h [H,P ]i = −h i. (32) ∂t ~ ∂X

12 To compare it to the classical motion, we can choose our quantum states to be local in phase space, i.e. its support in position or momentum representation should be small enough, like a Gaussian wavepacket. Then we should identify some point in the support of wave function the same as a point particle in classical phase space, such as (hXi, hP i). Its equation of motion follows: ∂ i ∂V hP i = h [H,P ]i = − (hXi). (33) ∂t ~ ∂x Obviously these two equations6 are diﬀerent, but if the support of the quantum state is local enough, namely ∂ V ∆x ∼ ξ ' sqrt x , ∂xxxV where ξ is the nonlinearity scale over which the gradient of the potential changes sig- niﬁcantly, then the two equations above behave the same approximately, that’s where quantum-classical correspondence lies in. On the other hand, it is known that for classical chaos systems, there is exponential divergence of the trajectories, which is characterized by Lyapunov exponent Λ > 0. Then the interplay between quantum interference and chaotic exponential instability leads to the rapid loss of the quantum-classical correspondence. More precisely, classical probability distribution will develop structures on the scale

∆p ∼ ∆p0 exp(−Λt) resulting in the wavepacket of the quantum counterpart state spread over phase space like ∆x ∼ ~ exp(Λt), ∆p0 which deﬁnitely makes the wave function nonlocal and cause the loss of the correspondence. The time scale for the wavepacket to become nonlocal enough7 can be estimated as −1 ∆p0ξ −1 LP −1 I t~ ' Λ ln ≤ tr = Λ ln ' Λ ln , (34) ~ ~ ~ where L and P give the range of values of the coordinate and momentum in phase space 8 of the system and I is action of the system. tr ≥ t~ since L ' ξ and P > ∆p0. The above derivations now allow us to make a prediction for a real macroscopic object. The example is Hyperion, chaotically tumbling moon of Saturn. Hyperion has a prolate shape of a potato and moves on an eccentric orbit with a period tO = 21 days.

6Equation of motion of hXi are omitted since they are the same. 7This is exactly the Ehrenfest time – the time over which a quantum system that has started in a localized state will continue to be suﬃciently localized for the quantum corrections to the equations of motion obeyed by its expectation values to be negligible. 8 α In a regular system, tr ∼ (I/~) since phase space patches of regular systems undergo stretching with a power of time, which is much larger.

13 Interaction between its gravitational quadrupole and the tidal ﬁeld of Saturn leads to chaotic tumbling with Lyapunov time Λ−1 = 42 days. Then it turns out that

Hyperion 77 tr ' 42[days] ln 10 ' 20[yrs]. (35) After approximately 20 years Hyperion would be in a coherent superposition of orien- tations that diﬀer by 2π! Thus the logarithmic dependence of action is very small even for macroscopic objects. In conclusion, wavepacket becomes rapidly delocalized in a chaotic system, and the correspondence between classical and quantum is quickly lost. For a more rigorous treatment, we need to describe the quantum states using Wigner function, which evolves under Moyal bracket instead of Poisson bracket. REMAIN OT BE COMPLEMENTED.

3.3 Einselection 3.3.1 Scheme of Einselection From the examples above, it is now clear that why can there be some speciﬁc PS. How- ever for the more “classical” measurement, we need to ﬁgure out how can the classical correlation being established between measured system and the apparatus. Generally speaking, it is caused by premeasurement of the apparatus by environment, denoted by E. In fact, such correlations can be established as follows:

(α| ↑i|A1i + β| ↓i|A0i)|0i → α| ↑i|A1i|1i + β| ↓i|A0i|0i. (36) In this situation, the basis ambiguity of SA disappears. Obviously, providing that h0|1i = 0, the state of S − A pair loses coherence and contains only classical correlation. An example of it is aﬀorded by a two-state apparatus A interacting with an environment of N other spins, with Hamiltonians:

X A (k) HAE = gkσz ⊗ σz k and initial state Y |Φ(0)i = (a| ↑i + b| ↓i) (αk| ↑i + βk| ↓i) k evolves into |Φ(t)i = a| ↑i|E↑(t)i + b| ↓i|E↓(t)i. Then it can be veriﬁed that

Y 2 2 r(t) = hE↑(t)|E↓(t)i = [cos 2gkt + i(|αk| − |βk| ) sin 2gkt]. (37) k Since the fourier transform of r(t) have a distribution over a large frequency range, it will be quickly crumble to h|r(t)|2i ∼ 2−N .

14 In this sense, decoherence is exponentially effective.9 Geometry of flows induced by decoherence in the Bloch sphere exhibits characteristics encountered in general: 1. The classical set of the einselected pointer states ({| ↑]rangle, | ↓i} in our case). Pointer states are the pure states least affected by decoherence. 2. Classical domain consisting of all the pointer states and their mixtures. In our case it corresponds to the section [−1, +1] of z-axis. 3. The quantum domain – the rest of the volume of the Bloch sphere – consisting of more general density matrices. Establishment of the measurement-like correlation between the apparatus and the P environment changes the density matrix from the premeasurement ρSA to the decohered D ρSA, and it is obvious that einselection is accompanied by the increase of entropy as before. Furthermore, we can investigate the change of mutual information. Considering P the general case that |Ψi = i ci|sii|Aii, then

P P P P X 2 2 I (S : A) = H(ρS ) + H(ρA) − H(ρSA) = −2 |ci | log |ci| , i (38) D D D D X 2 2 I (S : A) = H(ρS ) + H(ρA) − H(ρSA) = − |ci | log |ci| . i The decrease in the mutual information is due to the increase of the joint entropy H(ρSA):

P D D P ∆I(S : A) = I (S : A) − I (S : A) = H(ρSA) − H(ρSA) = ∆H(ρSA). Classically, equivalent deﬁnition of the mutual information obtains from the asymmetric formula: JA(S : A) = H(ρS ) − H(ρS|A) with the help of the conditional entropy H(ρS|A). Above, subscript A indicates the member of the correlated pair that will be the source of the information about its partner. D P 2 2 However, if we impose I (S : A) = JA(S : A), it follows H(ρS|A) = i |ci | log |ci| < 0! We shall simply regard this fact as an illustration of the strength of quantum correlations (i. e., entanglement), which allow I(S : A) to violate the inequality:

I(S : A) ≤ min(HS ,HA). Moreover, now the conditional entropy can be deﬁned in the classical pointer basis as the average of partial entropies computed from the conditional ρD over the probabilities S|Aii of diﬀerent outcomes: X H(ρ ) = p H(ρD ). S|A |Aii S|Aii i

9Although the coherence can be retained under certain initial condition, such kind of special environment state is, however, unlikely in realistic circumstances, or of measure zero.

15 On the other hand, a useful suﬃcient condition for the classicality of correlations is then the existence of an apparatus basis that allows quantum versions of the two classically identical expressions for the mutual information to coincide, or the discord

δIA(S|A) = I(S : A) − JA(S : A) must vanish. Additionally, we need to distinguish einselection and dephasing caused by noise. Clas- sical noise can cause dephasing when the observer does not know the time-dependent classical perturbation Hamiltonian responsible for this unitary, but unknown evolution. For example, in the pre-decoherence state vector, random phase noise will cause a tran- sition: X X (n) (n) |ΨSAi = ci|sii|Aii → ci exp(iφi )|sii|Aii = |ΨSAi. (39) i i The dephasing Hamiltonian can be chosen to be

(n) X ˙(n) HD = φi (t)|AiihAi|. i In contrast to interactions causing premeasurements, entanglement, and decoherence, HD cannot inﬂuence the nature or the degree of the SA correlations: HD does not imprint the state of S or A anywhere else in the Universe. However, in absence of detailed information of noise, one is often forced to represent SA by the density matrix averaged over the ensemble of noise realizations:

X 2 X (n) (n) ∗ ρ¯SA = h|ΨSAihΨSA|i = |ci| |siihsi||AiihAi|+ hexp(iφi −iφj )incicj |siihsj||AiihAj| i i,j (40) In this phase - averaged density matrix off-diagonal terms may disappear. Nevertheless, each member of the ensemble may exist in a state as pure as it was before dephasing. NMR offers examples of dephasing (which can be reversed using spin echo). Dephasing is a loss of phase coherence due to noise in phases. It does not result in an information transfer to the environment. Finally, we need to find a criterion for finding PS. An example is predictability sieve, since evolution of a quantum system prepared in a classical state should emulate classical evolution that can be idealized as a “trajectory” – a predictable sequence of objectively existing states. One can measure the loss of predictability caused by the evolution for every pure state |Ψi by von Neumann entropy or some other measure of predictability such as the purity: 2 ςΨ(t) = T rρΨ(t). (41) Pointer states are obtained by maximizing predictability functional over |Ψi. When decoherence leads to classicality, good pointer states exist, and the answer is robust.

16 3.3.2 Einselection in Phase Space Einselection in phase space should lead to phase space points, trajectories, and to classical (Newtonian) dynamics. We will illustrate it with Quantum Brownian motion model. It consists of an environment E – a collection of harmonic oscillators (coordinates qn, masses mn, frequencies ωn, and coupling constants cn) interacting with the system S (coordinate x), with a mass M and a potential V (x). We shall often consider harmonic 2 2 V (x) = MΩ0x /2, so that the whole SE is linear and one can obtain an exact solution. The Lagrangian of the system-environment entity is:

L(x, qn) = LS (x) + LSE (x, {qn}), (42) and the system alone has the Lagrangian M M L (x) = x˙ 2 − V (x) = (x ˙ 2 − Ω2x2). (43) S 2 2 0 The eﬀect of the environment is modelled by the sum of the Lagrangians of individual oscillators and of the system-environment interaction terms: X mn cnx L = q˙2 − ω2(q − )2 (44) SE 2 n n n m ω2 n n n An important characteristic of the model is the spectral density of the environment:

X c2 C(ω) = n δ(ω − ω ) (45) 2m ω n n n n The eﬀect of the environment can be expressed through the propagator acting on the reduced ρ : S Z 0 0 0 0 0 ρS (x, x , t) = dx0dx0J(x, x , t|x0, x0, t0)ρS (x0, x0, t0). (46) We focus on the case when the system and the environment are initially statistically independent, so that their density matrices in a product state:

ρSE = ρS ρE . (47) Besides, environment in thermal equilibrium provides a useful and tractable model for the initial state. The density matrix of the n’th mode of the thermal environment is:

mnωn mnωn 2 02 ~ωn 0 ρEn = ω × exp −{ ω × [(qn + q n) cosh( ) − 2qnqn]}. (48) 2π sinh( ~ n ) 2π sinh( ~ n ) kBT ~ kB T ~ kB T 10 Assuming ρE keeps in thermal equilibrium at kBT = 1/β, then the calculation of J through path integral have a Gaussian form. The result can be conveniently written in

10Note that this does not mean we ignore the correlation between system and environment. Actually in the derivation of some Lindblad master equation, such an approximation is also assumed, it is valid since the nonlocal(in time) ﬂuctuations are averaged to generate eﬀectively Markovian dynamics. REMAIN TO BE CONFIRMED

17 terms of the diagonal and oﬀdiagonal coordinates of the density matrix in the position representation, X = x + x0,Y = x − x0:

b3 exp i(b1XY + b2X0Y − b3XY0 − b4X0Y0) J(X, Y, t|X0,Y0, t0) = 2 2 . (49) 2π exp(a11Y + 2a12YY0 + a22Y0 )

The time-dependent coefficients bk and aij are computed from the noise kernels ν(s) and dissipation kernels η(s) defined in terms of spectral density, that reflect the properties of the environment: Z ∞ ωβ ν(s) = dωC(ω) coth(~ ) cos(ωs) 0 2 Z ∞ (50) η(s) = dωC(ω) sin(ωs) 0 Defining the auxiliary equation: Z s 2 0 0 u¨(s) + Ω0u(s) + 2 dsη(s − s )u(s ) = 0, (51) 0 and two such solutions with boundary conditions u1(0) = u2(t) = 1 and u1(t) = u2(0) = 0 can yield the coefficients above as:

b1(2)(t) =u ˙ 2(1)(t)/2, b3(4)(t) =u ˙ 2(1)(0)/2 Z t Z t (52) 1 0 0 aij(t) = ds ds ui(s)uj(s;)ν(s − s ). 1 + δij 0 0 After some calculation, we can obtain the master equation in position representation. A further modiﬁcation yields the master equation in operator form: i iγ(t) f(t) ρ˙S = − [Hren, ρS ] − [x, {p, ρS }] − D(t)[x, [x, ρS ]] − [x, [p, ρS ]], (53) ~ ~ ~ 2 2 where Hren is renormalized Hamiltonian Hren = HS + MΩ˜ (t)x /2. Those coeﬃcients are given by spectral density as:

2 Z t Ω˜ 2(t) = − ds cos(Ωs)η(s) M 0 2 Z t γ(t) = ds sin(Ωs)η(s) MΩ 0 (54) 1 Z t D(t) = ds cos(Ωs)ν(s) ~ 0 1 Z t f(t) = − ds cos(Ωs)ν(s). MΩ 0 To see decoherence in phase space, we consider in the case of high temperature, and in the macroscopic limit (that is, when ~ is small compared to other quantities

18 p 0 2 with dimensions of action, such as 2MkBT i(x − x ) i). The master equation is then dominated by 0 0 x − x 2 0 ∂tρS (x, x , t) = −γ( ) ρS (x, x , t), (55) λT ~ where λT = √ is thermal de Broglie wavelength. Then it is obvious the density 2MkB T matrix loses oﬀ-diagonal terms as: 0 0 0 x − x 2 ρS (x, x , t) = ρS (x, x , 0) exp −γt( ) (56) λT Thus quantum coherence decays exponentially at the rate given by the relaxation rate times the square of the distance measured in units of thermal de Broglie wavelength. In classical regime, for a mass 1g at room temperature and for the separation x0 − x =1cm, the equation above predicts decoherence approximately 1040 times faster than relaxation! For a more rigorous treatment of dynamics in phase space, we should focus on evolution of Wigner function. In the high temperature limit, it follows

∂tW = {Hren, W }MB + 2γ∂p(pW ) + D∂ppW (57) For Wigner function which is initially coherent superposition of to Gaussian wavepacket, i.e. 2 2 2 1 p ξ x 2x0p W (x, p) = G(x + x0, p) + G(x − x0, p) + exp − − cos , π~ ~2 ξ2 ~ where 2 2 2 1 (x ∓ x0) (p − p0) ξ G(x ± x0, p − p0) = exp − − π~ ξ2 ~2 is a Gaussian wavepacket. Since the dominant term in interference term is cos 2x0p , ~ which is an eigenfunction of the diﬀusion operator, the interference term evolves as

2 D(2x0) W˙ int ∼ − Wint, ~2 which indicates an exponential decay as above. From the discussions above, it seems that eigenstate of x is PS. But we will see this is not true compared to the results from predictability sieve. The master equation leads to the loss of purity at the rate

2 4ηkBT 2 2 2 2 ∂tT rρ = − T r(ρ x − (ρx) ) + 2γT rρ (58) ~2 For predictability sieve the second term is usually unimportant, as for a vast major- ity of the initially pure states its eﬀect will be negligible when compared to the ﬁrst, decoherence - related term. Thus for pure initial state this equation reduced to

2 4ηkBT 2 2 ∂tT rρ = − (hx i − hxi ). (59) ~2

19 Therefore, the instantaneous loss of purity is minimized for perfectly localized states. To ﬁnd most predictable states relevant for dynamics consider entropy increase over a period of the oscillator, i.e.

∆p2 ∆ς|2π/Ω = −2D(∆x2 + ) (60) 0 M 2Ω2 The loss of purity will be smallest when MΩ ∆x2 = ~ , ∆p2 = ~ . (61) 2MΩ 2 Coherent quantum states are selected by the predictability sieve in an underdamped harmonic oscillator. Therefore, we conclude that for an underdamped harmonic oscillator coherent Gaussians are the best quantum theory has to offer as an approximation to a classical point. Finally, we will talk about classical limit. The first approach is to make ~ → 0. In this limit, for sufficient large t Wint → 0. Simultaneously,

lim G(x − x0, p − p0) = δ(x − x0, p − p0). ~→0

But actually ~ is a constant, therefore, a physically more reasonable approach increases the size of the object, and, hence, its susceptibility to decoherence. Assume that density of the object is independent of its size R, and that the environment quanta scatter from its surface (as would photons or air molecules). Then M ∼ R3 and η ∼ R2, where D = 2MγkBT = ηkBT Hence η 1 η ∼ O(R2) → ∞, γ = ∼ O( ) → 0 as R → ∞ 2M R According to the master equation of Wigner function, localization in phase space and reversibility can be simultaneously achieved in a macroscopic limit. REMAIN TO BE COMPLEMENTED ABOUT SECOND LAW.