# Resonant Perturbation Theory of Decoherence and Relaxation of Quantum Bits

Authors: M. Merkli; G. P. Berman; I. M. Sigal
Journal: Advances in Mathematical Physics
 (2010)

---

## Abstract

We describe our recent results on the resonant perturbation theory of decoherence
and relaxation for quantum systems with many qubits. The approach
represents a rigorous analysis of the phenomenon of decoherence and relaxation
for generalN-level systems coupled to reservoirs of bosonic fields. We derive
a representation of the reduced dynamics valid for all times t≥0 and for small
but fixed interaction strength. Our approach does not involve master equation
approximations and applies to a wide variety of systems which are not explicitly
solvable.


---

## Body

## 1. Introduction

Quantum computers (QCs) with large numbers of quantum bits (qubits) promise to solve important problems such as factorization of larger integer numbers, searching large unsorted databases, and simulations of physical processes exponentially faster than digital computers. Recently, many efforts were made for designing scalable (in the number of qubits) QC architectures based on solid-state implementations. One of the most promising designs of a solid-state QC is based on superconducting devices with Josephson junctions and solid-state quantum interference devices (SQUIDs) serving as qubits (effective spins), which operate in the quantum regime:ℏω>>kBT, where T is the temperature and ω is the qubit transition frequency. This condition is widely used in  superconducting quantum computation and quantum measurement, when T~  10–20 mK and ℏω~  100–150 mK (in temperature units) [1–8] (see also references therein). The main advantages of a QC with superconducting qubits are: (i) the two basic states of a qubit are represented by the states of a superconducting charge or current in the macroscopic (several μm size) device. The relatively large scale of this device facilitates the manufacturing, and potential controlling and measuring of the states of qubits. (ii) The states of charge- and current-based qubits can be measured using rapidly developing technologies, such as a single electron transistors, effective resonant oscillators and microcavities with RF amplifiers, and quantum tunneling effects. (iii) The quantum logic operations can be implemented exclusively by switching locally on and off voltages on controlling microcontacts and magnetic fluxes. (iv) The devices based on superconducting qubits can potentially achieve large quantum dephasing and relaxation times of milliseconds and more at low temperatures, allowing quantum coherent computation for long enough times. In spite of significant progress, current devices with superconducting qubits only have one or two qubits operating with low fidelity even for simplest operations.One of the main problems which must be resolved in order to build a scalable QC is to develop novel approaches for suppression of unwanted effects such as decoherence and noise. This also requires to develop rigorous mathematical tools for analyzing the dynamics of decoherence, entanglement, and thermalization in order to control the quantum protocols with needed fidelity. These theoretical approaches must work for long enough times and be applicable to both solvable and not explicitly solvable (nonintegrable) systems.Here we present a review of our results [9–11] on the rigorous analysis of the phenomenon of decoherence and relaxation for general N-level systems coupled to reservoirs. The latter are described by the bosonic fields. We suggest a new approach which applies to a wide variety of systems which are not explicitly solvable. We analyze in detail the dynamics of an N-qubit quantum register collectively coupled to a thermal environment. Each spin experiences the same environment interaction, consisting of an energy conserving and an energy exchange part. We find the decay rates of the reduced density matrix elements in the energy basis. We show that the fastest decay rates of off-diagonal matrix elements induced by the energy conserving interaction are of order N2, while the one induced by the energy exchange interaction is of the order N only. Moreover, the diagonal matrix elements approach their limiting values at a rate independent of N. Our method is based on a dynamical quantum resonance theory valid for small, fixed values of the couplings, and uniformly in time for t≥0. We do not make Markov-, Born- or weak coupling (van Hove) approximations.


## 2. Presentation of Results

We consider anN-level quantum system S interacting with a heat bath R. The former is described by a Hilbert space 𝔥s=ℂN and a Hamiltonian(2.1)HS=diag(E1,…,EN).
The environment R is modelled by a bosonic thermal reservoir with Hamiltonian(2.2)HR=∫ℝ3a*(k)|k|a(k)d3k,
acting on the reservoir Hilbert space 𝔥R, and where a*(k) and a(k) are the usual bosonic creation and annihilation operators satisfying the canonical commutation relations [a(k),a*(l)]=δ(k-l). It is understood that we consider R in the thermodynamic limit of infinite volume (ℝ3) and fixed temperature T=1/β>0 (in a phase without condensate). Given a form factorf(k), a square integrable function of k∈ℝ3 (momentum representation), the smoothed-out creation and annihilation operators are defined as a*(f)=∫ℝ3f(k)a*(k)d3k and a(f)=∫ℝ3f(k)¯a(k)d3k, respectively, and the hermitian field operator is(2.3)ϕ(f)=12[a*(f)+a(f)].
The total Hamiltonian, acting on 𝔥S⊗𝔥R, has the form(2.4)H=HS⊗1R+1S⊗HR+λv,
where λ is a coupling constant and v is an interaction operator linear in field operators. For simplicity of exposition, we consider here initial states where S and R are not entangled, and where R is in thermal equilibrium. (Our method applies also to initially entangled states, and arbitrary initial states of R normal w.r.t. the equilibrium state; see [10]). The initial density matrix is thus (2.5)ρ0=ρ¯0⊗ρR,β,
where ρ¯0 is any state of S and ρR,β is the equilibrium state of R at temperature 1/β.LetA be an arbitrary observable of the system (an operator on the system Hilbert space 𝔥S) and set(2.6)〈A〉t:=TrS(ρ¯tA)=TrS+R(ρt(A⊗1R)),
where ρt is the density matrix of S+R at time t and (2.7)ρ¯t=TrRρt
is the reduced density matrix of S. In our approach, the dynamics of the reduced density matrix ρ¯t is expressed in terms of the resonance structure of the system. Under the noninteracting dynamics (λ=0), the evolution of the reduced density matrix elements of S, expressed in the energy basis {φk}k=1N of HS, is given by(2.8)[ρ¯t]kl=〈φk,e-itHSρ¯0eitHSφl〉=eitelk[ρ¯0]kl,
where elk=El-Ek. As the interaction with the reservoir is turned on, the dynamics  (2.8) undergoes two qualitative changes.(1)
The “Bohr frequencies”(2.9)e∈{E-E′:E,E′∈spec(HS)}
in the exponent of (2.8) become complex, e↦ɛe. It can be shown generally that the resonance energiesɛe have nonnegative imaginary parts, Imɛe≥0. If Imɛe>0, then the corresponding dynamical process is irreversible.(2)
The matrix elements do not evolve independently any more. Indeed, the effective energy ofS is changed due to the interaction with the reservoirs, leading to a dynamics that does not leave eigenstates of HS invariant. (However, to lowest order in the interaction, the eigenspaces of HS are left invariant and therefore matrix elements with (m,n) belonging to a fixed energy difference Em-En will evolve in a coupled manner.)Our goal is to derive these two effects from the microscopic (hamiltonian) model and to quantify them. Our analysis yields the thermalization and decoherence times of quantum registers.


### 2.1. Evolution of Reduced Dynamics of anN-Level System

LetA∈ℬ(𝔥S) be an observable of the system S. We show in [9, 10] that the ergodic averages(2.10)〈〈A〉〉∞:=limT→∞1T∫0T〈A〉tdt
exist, that is, 〈A〉t converges in the ergodic sense as t→∞. Furthermore, we show that for any t≥0 and for any 0<ω′<2π/β,(2.11)〈A〉t-〈〈A〉〉∞=∑ɛ≠0eitɛRɛ(A)+O(λ2e-[ω'-O(λ)]t),
where the complex numbers ɛ are the eigenvalues of a certain explicitly given operator K(ω'), lying in the strip {z∈ℂ|0≤Imz<ω'/2}. They have the expansions(2.12)ɛ≡ɛe(s)=e+λ2δe(s)+O(λ4),
where e∈spec(HS⊗1S-1S⊗HS)=spec(HS)-spec(HS) and δe(s) are the eigenvalues of a matrix Λe, called a level-shift operator, acting on the eigenspace of HS⊗1S-1S⊗HS corresponding to the eigenvalue e (which is a subspace of 𝔥S⊗𝔥S). The Rɛ(A) in  (2.11) are linear functionals of A and are given in terms of the initial state, ρ0, and certain operators depending on the Hamiltonian H. They have the expansion(2.13)Rɛ(A)=∑(m,n)∈Ieϰm,nAm,n+O(λ2),
where Ie is the collection of all pairs of indices such that e=Em-En, with Ek being the eigenvalues of HS. Here, Am,n is the (m,n)-matrix element of the observable A in the energy basis of HS, and ϰm,n are coefficients depending on the initial state of the system (and on e, but not on A nor on λ).


#### 2.1.1. Discussion

(i)
In the absence of interaction (λ=0), we have ɛ=e∈ℝ; see (2.12). Depending on the interaction, each resonance energy ɛ may migrate into the upper complex plane, or it may stay on the real axis, as λ≠0.(ii)
The averages〈A〉t approach their ergodic means 〈〈A〉〉∞ if and only if Imɛ>0 for all ɛ≠0. In this case, the convergence takes place on the time scale [Imɛ]-1. Otherwise; 〈A〉t oscillates. A sufficient condition for decay is that Imδe(s)>0 (and λ small, see  (2.12)).(iii)
The error term in (2.11) is small in λ, uniformly in t≥0, and it decays in time quicker than any of the main terms in the sum on the r.h.s.: indeed, Imɛ=O(λ2) while ω′-O(λ)>ω′/2 independent of small values of λ. However, this means that we are in the regime λ2≪ω′<2π/β (see before (2.11)), which implies that λ2 must be much smaller than the temperature T=1/β. Using a more refined analysis, one can get rid of this condition; see also remarks on page 376 of [10].(iv)
Relation (2.13) shows that to lowest order in the perturbation, the group of (energy basis) matrix elements of any observable A corresponding to a fixed energy difference Em-En evolve jointly, while those of different such groups evolve independently.It is well known that there are two kinds of processes which drive decay (or irreversibility) ofS: energy-exchange processes characterized by [v,HS]≠0 and energy preserving ones where [v,HS]=0. The former are induced by interactions having nonvanishing probabilities for processes of absorption and emission of field quanta with energies corresponding to the Bohr frequencies of S and thus typically inducing thermalization of S. Energy preserving interactions suppress such processes, allowing only for a phase change of the system during the evolution (“phase damping”, [12–18]).To our knowledge, energy-exchange systems have so far been treated using Born and Markov master equation approximations (Lindblad form of dynamics) or they have been studied numerically, while for energy conserving systems, one often can find an exact solution. The present representation (2.11) gives a detailed picture of the dynamics of averages of observables for interactions with and without energy exchange. The resonance energies ɛ and the functionals Rɛ can be calculated for concrete models, as illustrated in the next section. We mention that the resonance dynamics representation can be used to study the dynamics of entanglement of qubits coupled to local and collective reservoirs, see [19].The dynamical resonance method can be generalized to time-dependent Hamiltonians.  See [20, 21] for time-periodic  Hamiltonians.


#### 2.1.2. Contrast with Weak Coupling Approximation

Our representation (2.11) of the true dynamics of S relies only on the smallness of the coupling parameter λ, and no approximation is made. In the absence of an exact solution, it is common to make a weak coupling Lindblad master equation approximation of the dynamics, in which the reduced density matrix evolves according to ρ¯t=etℒρ¯0, where ℒ is the Lindblad generator, [22–24]. This approximation can lead to results that differ qualitatively from the true behaviour. For instance, the Lindblad master equation predicts that the system S approaches its Gibbs state at the temperature of the reservoir in the limit of large times. However, it is clear that in reality, the coupled system S+R will approach equilibrium, and hence the asymptotic state of S alone, being the reduction of the coupled equilibrium state, is the Gibbs state of S only to first approximation in the coupling (see also illustration below, and [9, 10]). In particular, the system's asymptotic density matrix is not diagonal in the original energy basis, but it has off-diagonal matrix elements of O(λ2). Features of this kind cannot be captured by the Lindblad approximation, but are captured in our approach.It has been shown (see, e.g., [23–26]) that the weak coupling limit dynamics generated by the Lindblad operator is obtained in the regime λ→0, t→∞, with λ2t fixed. One of the strengths of our approach is that we do not impose any relation between λ and t, and our results are valid for all times t≥0, provided λ is small. It has been observed [25, 26] that for certain systems of the type S+R, the second-order contribution of the exponents ɛe(s) in  (2.12) correspond to eigenvalues of the Lindblad generator. Our resonance method gives the true exponents, that is, we do not lose the contributions of any order in the interaction. If the energy spectrum of HS is degenerate, it happens that the second-order contributions to Imɛe(s) vanish. This corresponds to a Lindblad generator having several real eigenvalues. In this situation, the correct dynamics (approach to a final state) can be captured only by taking into account higher-order contributions to the exponents ɛe(s); see [27]. To our knowledge, so far this can only be done with the method presented in this paper, and is beyond the reach of the weak coupling method.


#### 2.1.3. Illustration: Single Qubit

ConsiderS to be a single spin 1/2 with energy gap Δ=E2-E1>0. S is coupled to the heat bath R via the operator(2.14)v=[acc¯b]⊗ϕ(g),
where ϕ(g) is the Bose field operator  (2.3), smeared out with a coupling function (form factor) g(k), k∈ℝ3, and the 2×2 coupling matrix (representing the coupling operator in the energy eigenbasis) is hermitian. The operator  (2.14)—or a sum of such terms, for which our technique works equally well—is the most general coupling which is linear in field operators. We refer to [10] for a discussion of the link between  (2.14) and the spin-boson model. We take S initially in a coherent superposition in the energy basis,(2.15)ρ¯0=12[1111].
In [10] we derive from representation  (2.11) the following expressions for the dynamics of matrix elements, for all t≥0:
(2.16)[ρ¯t]m,m=e-βEmZS,β+(-1)m2tanh(βΔ2)eitɛ0(λ)+Rm,m(λ,t),m=1,2,(2.17)[ρ¯t]1,2=12eitɛ-Δ(λ)+R1,2(λ,t),
where the resonance energies ɛ are given by(2.18)ɛ0(λ)=iλ2π2|c|2ξ(Δ)+O(λ4),ɛΔ(λ)=Δ+λ2R+i2λ2π2[|c|2ξ(Δ)+(b-a)2ξ(0)]+O(λ4),ɛ-Δ(λ)=-ɛΔ(λ)¯,
with(2.19)ξ(η)=limϵ↓01π∫ℝ3d3kcoth(β|k|2)|g(k)|2ϵ(|k|-η)2+ϵ2,R=12(b2-a2)〈g,ω-1g〉+12|c|2P.V.∫ℝ×S2u2|g(|u|,σ)|2coth(β|u|2)1u-Δ.
The remainder terms in (2.17), (2.17) satisfy |Rm,n(λ,t)|≤Cλ2, uniformly in t≥0, and they can be decomposed into a sum of a constant and a decaying part, (2.20)Rm,n(λ,t)=〈〈pn,m〉〉∞-δm,ne-βEmZS,β+R′m,n(λ,t),
where |R'm,n(λ,t)|=O(λ2e-γt), with γ=min{Imɛ0,Imɛ±Δ}. These relations show the following.(i)
To second order inλ, convergence of the populations to the equilibrium values (Gibbs law), and decoherence occur exponentially fast, with rates τT=[Imɛ0(λ)]-1 and τD=[ImɛΔ(λ)]-1, respectively. (If either of these imaginary parts vanishes then the corresponding process does not take place, of course.) In particular, coherence of the initial state stays preserved on time scales of the order λ-2[|c|2ξ(Δ)+(b-a)2ξ(0)]-1; compare for example  (2.18).(ii)
The final density matrix of the spin isnot the Gibbs state of the qubit, and it is not diagonal in the energy basis. The deviation of the final state from the Gibbs state is given by limt→∞Rm,n(λ,t)=O(λ2). This is clear heuristically too, since typically the entire system S+R approaches its joint equilibrium in which S and R are entangled. The reduction of this state to S is the Gibbs state of S modulo O(λ2) terms representing a shift in the effective energy of S due to the interaction with the bath. In this sense, coherence in the energy basis of S is created by thermalization. We have quantified this in [10, Theorem 3.3].(iii)
In a markovian master equation approach, the above phenomenon (i.e., variations ofO(λ2) in the time-asymptotic limit) cannot be detected. Indeed in that approach one would conclude that S approaches its Gibbs state as t→∞.


### 2.2. Evolution of Reduced Dynamics of anN-Level System

In the sequel, we analyze in more detail the evolution of a qubit register of sizeN. The Hamiltonian is(2.21)HS=∑i,j=1NJijSizSjz+∑j=1NBjSjz,
where Jij are pair interaction constants and Bj is the value of a magnetic field at the location of spin j. The Pauli spin operator is(2.22)Sz=[100-1]
and Sjz is the matrix Sz acting on the jth spin only.We consider acollective coupling between the register S and the reservoir R: the distance between the N qubits is much smaller than the correlation length of the reservoir and as a consequence, each qubit feels the same interaction with the reservoir. The corresponding interaction operator is (compare with  (2.4))(2.23)λ1v1+λ2v2=λ1∑j=1NSjz⊗ϕ(g1)+λ2∑j=1NSjx⊗ϕ(g2).
Here g1 and g2 are form factors and the coupling constants λ1 and λ2 measure the strengths of the energy conserving (position-position) coupling, and the energy exchange (spin flip) coupling, respectively. Spin-flips are implemented by the Sjx in  (2.23), representing the Pauli matrix
(2.24)Sx=[0110]
acting on the jth spin. The total Hamiltonian takes the form  (2.4) with λv replaced by  (2.23). It is convenient to represent ρ¯t as a matrix in the energy basis, consisting of eigenvectors φσ̲ of HS. These are vectors in 𝔥S=ℂ2⊗⋯⊗ℂ2=ℂ2N indexed by spin configurations(2.25)σ̲={σ1,…,σN}∈{+1,-1}N,φσ̲=φσ1⊗⋯⊗φσN,
where(2.26)φ+=[10],φ-=[01],
so that(2.27)HSφσ̲=E(σ̲)φσ̲withE(σ̲)=∑i,j=1NJijσiσj+∑j=1NBjσj.
We denote the reduced density matrix elements as(2.28)[ρ¯t]σ̲,τ̲=〈φσ̲,ρ¯tφτ̲〉.
The Bohr frequencies  (2.9) are now(2.29)e(σ̲,τ̲)=E(σ̲)-E(τ̲)=∑i,j=1NJij(σiσj-τiτj)+∑j=1NBj(σj-τj),
and they become complex resonance energies ɛe=ɛe(λ1,λ2)∈ℂ under perturbation.Assumption of Nonoverlapping Resonances
The Bohr frequencies (2.29) represent “unperturbed” energy levels and we follow their motion under perturbation (λ1,λ2). In this work, we consider the regime of nonoverlapping resonances, meaning that the interaction is small relative to the spacing of the Bohr frequencies.We show in [10, Theorem 2.1], that for all t≥0, (2.30)[ρ¯t]σ̲,τ̲-〈〈[ρ¯∞]σ̲,τ̲〉〉=∑{e:ɛe≠0}eitɛe[∑σ′̲,τ̲′wσ̲,τ̲;σ̲′,τ̲′ɛe[ρ¯0]σ̲′,τ̲′+O(λ12+λ22)]+O((λ12+λ22)e-[ω′+O(λ)]t).
This result is obtained by specializing  (2.11) to the specific system at hand and considering observables A=|φτ̲〉〈φσ̲|. In (2.30), we have in accordance with (2.10), 〈〈[ρ¯∞]σ̲,τ̲〉〉=limT→∞(1/T)∫0T[ρ¯t]σ̲,τ̲dt. The coefficients w are overlaps of resonance eigenstates which vanish unless e=-e(σ̲,τ̲)=-e(σ̲′,τ̲′) (see point (2) after  (2.9)). They represent the dominant contribution to the functionals Rɛ in  (2.11); see also  (2.13). The ɛe have the expansion(2.31)ɛe≡ɛe(s)=e+δe(s)+O(λ14+λ24),
where the label s=1,…,ν(e) indexes the splitting of the eigenvalue e into ν(e) distinct resonance energies. The lowest order corrections δe(s) satisfy(2.32)δe(s)=O(λ12+λ22).
They are the (complex) eigenvalues of an operator Λe, called the level shift operator associated to e. This operator acts on the eigenspace of LS associated to the eigenvalue e (a subspace of the qubit register Hilbert space; see [10, 11] for the formal definition of Λe). It governs the lowest order shift of eigenvalues under perturbation. One can see by direct calculation that Imδe(s)≥0.


#### 2.2.1. Discussion

(i)
To lowest order in the perturbation, the group of reduced density matrix elements[ρ¯t]σ̲,τ̲ associated to a fixed e=e(σ̲,τ̲) evolve in a coupled way, while groups of matrix elements associated to different e evolve independently.(ii)
The density matrix elements of a given group mix and evolve in time according to the weight functionsw and the exponentials eitɛe(s). In the absence of interaction (λ1=λ2=0), all the ɛe(s)=e are real. As the interaction is switched on, the ɛe(s) typically migrate into the upper complex plane, but they may stay on the real line (due to some symmetry or due to an “inefficient coupling”).(iii)
The matrix elements[ρ¯t]σ̲,τ̲ of a group e approach their ergodic means if and only if all the nonzero ɛe(s) have strictly positive imaginary part. In this case, the convergence takes place on a time scale of the order 1/γe, where
(2.33)γe=min{Imɛe(s):s=1,…,ν(e)s.t.ɛe(s)≠0}
is the decay rate of the group associated to e. If an ɛe(s) stays real, then the matrix elements of the corresponding group oscillate in time. A sufficient condition for decay of the group associated to e is γe>0, that is, Imδe(s)>0 for all s, and λ1, λ2 small.


#### 2.2.2. Decoherence Rates

We illustrate our results on decoherence rates for a qubit register withJij=0 (the general case is treated in [11]). We consider generic magnetic fields defined as follows. For nj∈{0,±1,±2}, j=1,…,N, we have
(2.34)∑j=1NBjnj=0⇔nj=0∀j.
Condition  (2.34) is satisfied generically in the sense that it does not hold only for very special choices of Bj (one such special choice is Bj=constant). For instance, if the Bj are chosen to be independent, and uniformly random from an interval [Bmin,Bmax], then  (2.34) is satisfied with probability one. We show in [11, Theorem 2.3], that the decoherence rates  (2.33) are given by(2.35)γe={λ12y1(e)+λ22y2(e)+y12(e),e≠0λ22y0,e=0}+O(λ14+λ24).
Here, y1 is contributions coming from the energy conserving interaction; y0 and y2 are due to the spin flip interaction. The term y12 is due to both interactions and is of O(λ12+λ22). We give explicit expressions for y0, y1, y2, and y12 in [11, Section 2]. For the present purpose, we limit ourselves to discussing the properties of the latter quantities.(i)
Properties ofy1(e):y1(e) vanishes if either e is such that e0:=∑j=1n(σj-τj)=0 or the infrared behaviour of the coupling function g1 is too regular (in three dimensions g1∝|k|p with p>-1/2). Otherwise, y1(e)>0. Moreover, y1(e) is proportional to the temperature T.(ii)
Properties ofy2(e):y2(e)>0 if g2(2Bj,Σ)≠0 for all Bj (form factor g2(k)=g2(|k|,Σ) in spherical coordinates). For low temperatures, T, y2(e)∝T, for high temperatures y2(e) approaches a constant.(iii)
Properties ofy12(e): if either of λ1, λ2 or e0 vanish, or if g1 is infrared regular as mentioned above, then y12(e)=0. Otherwise, y12(e)>0, in which case y12(e) approaches constant values for both T→0,∞.(iv)
Full decoherence: if γe>0 for all e≠0, then all off-diagonal matrix elements approach their limiting values exponentially fast. In this case, we say that full decoherence occurs. It follows from the above points that we have full decoherence if λ2≠0 and g2(2Bj,Σ)≠0 for all j, and provided λ1,λ2 are small enough (so that the remainder term in  (2.35) is small). Note that if λ2=0, then matrix elements associated to energy differences e such that e0=0 will not decay on the time scale given by the second order in the perturbation (λ12). We point out that generically, S+R will reach a joint equilibrium as t→∞, which means that the final reduced density matrix of S is its Gibbs state modulo a peturbation of the order of the interaction between S and R; see [9, 10]. Hence generically, the density matrix of S does not become diagonal in the energy basis as t→∞.(v)
Properties ofy0:y0 depends on the energy exchange interaction only. This reflects the fact that for a purely energy conserving interaction, the populations are conserved [9, 10, 17]. If g2(2Bj,Σ)≠0 for all j, then y0>0 (this is sometimes called the “Fermi Golden Rule Condition”). For small temperatures T, y0∝T, while y0 approaches a finite limit as T→∞.In terms of complexity analysis, it is important to discuss thedependence ofγeon the register sizeN.(i)
We show in [11] that y0 is independent of N. This means that the thermalization time, or relaxation time of the diagonal matrix elements (corresponding to e=0), is O(1) in N.(ii)
To determine the order of magnitude of the decay rates of the off-diagonal density matrix elements (corresponding toe≠0) relative to the register size N, we assume the magnetic field to have a certain distribution denoted by 〈·〉. We show in [11] that
(2.36)〈y1〉=y1∝e02,〈y2〉=CB𝔇(σ̲-τ̲),〈y12〉=cB(λ1,λ2)N0(e),
where CB and cB=cB(λ1,λ2) are positive constants (independent of N), with cB(λ1,λ2)=O(λ12+λ22). Here, N0(e) is the number of indices j such that σj=τj for each (σ̲,τ̲) s.t. e(σ̲,τ̲)=e, and
(2.37)𝔇(σ̲-τ̲):=∑j=1N|σj-τj|
is the Hamming distance between the spin configurations σ̲ and τ̲ (which depends on e only).(iii)
Considere≠0. It follows from  (2.35)–(2.37) that for purely energy conserving interactions (λ2=0), γe∝λ12e02=λ12[∑j=1N(σj-τj)]2, which can be as large as O(λ12N2). On the other hand, for purely energy exchanging interactions (λ1=0), we have γe∝λ22D(σ̲-τ̲), which cannot exceed O(λ22N). If both interactions are acting, then we have the additional term 〈y12〉, which is of order O((λ12+λ22)N). This shows the following: The fastest decay rate of reduced off-diagonal density matrix elements due to the energy conserving interaction alone is of orderλ12N2, while the fastest decay rate due to the energy exchange interaction alone is of the orderλ22N. Moreover, the decay of the diagonal matrix elements is of orderλ12, that is, independent ofN.Remark 2.2.2 s.
(1) Forλ2=0, the model can be solved explicitly [17], and one shows that the fastest decaying matrix elements have decay rate proportional to λ12N2. Furthermore, the model with a noncollective, energy-conserving interaction, where each qubit is coupled to an independent reservoir, can also be solved explicitly [17]. The fastest decay rate in this case is shown to be proportional to λ12N.
(2) As mentioned at the beginning of this section, we take the coupling constantsλ1, λ2 so small that the resonances do not overlap. Consequently, λ12N2 and λ22N are bounded above by a constant proportional to the gradient of the magnetic field in the present situation; see also [11]. Thus the decay rates γe do not increase indefinitely with increasing N in the regime considered here. Rather, γe are attenuated by small coupling constants for large N. They are of the order γe~Δ. We have shown that modulo an overall, common (N-dependent) prefactor, the decay rates originating from the energy conserving and exchanging interactions differ by a factor N.
(3) Collective decoherence has been studied extensively in the literature. Among the many theoretical, numerical, and experimental works, we mention here only [12, 14, 17, 28, 29], which are closest to the present work. We are not aware of any prior work giving explicit decoherence rates of a register for not explicitly solvable models, and without making master equation technique approximations.


## 2.1. Evolution of Reduced Dynamics of anN-Level System

LetA∈ℬ(𝔥S) be an observable of the system S. We show in [9, 10] that the ergodic averages(2.10)〈〈A〉〉∞:=limT→∞1T∫0T〈A〉tdt
exist, that is, 〈A〉t converges in the ergodic sense as t→∞. Furthermore, we show that for any t≥0 and for any 0<ω′<2π/β,(2.11)〈A〉t-〈〈A〉〉∞=∑ɛ≠0eitɛRɛ(A)+O(λ2e-[ω'-O(λ)]t),
where the complex numbers ɛ are the eigenvalues of a certain explicitly given operator K(ω'), lying in the strip {z∈ℂ|0≤Imz<ω'/2}. They have the expansions(2.12)ɛ≡ɛe(s)=e+λ2δe(s)+O(λ4),
where e∈spec(HS⊗1S-1S⊗HS)=spec(HS)-spec(HS) and δe(s) are the eigenvalues of a matrix Λe, called a level-shift operator, acting on the eigenspace of HS⊗1S-1S⊗HS corresponding to the eigenvalue e (which is a subspace of 𝔥S⊗𝔥S). The Rɛ(A) in  (2.11) are linear functionals of A and are given in terms of the initial state, ρ0, and certain operators depending on the Hamiltonian H. They have the expansion(2.13)Rɛ(A)=∑(m,n)∈Ieϰm,nAm,n+O(λ2),
where Ie is the collection of all pairs of indices such that e=Em-En, with Ek being the eigenvalues of HS. Here, Am,n is the (m,n)-matrix element of the observable A in the energy basis of HS, and ϰm,n are coefficients depending on the initial state of the system (and on e, but not on A nor on λ).


### 2.1.1. Discussion

(i)
In the absence of interaction (λ=0), we have ɛ=e∈ℝ; see (2.12). Depending on the interaction, each resonance energy ɛ may migrate into the upper complex plane, or it may stay on the real axis, as λ≠0.(ii)
The averages〈A〉t approach their ergodic means 〈〈A〉〉∞ if and only if Imɛ>0 for all ɛ≠0. In this case, the convergence takes place on the time scale [Imɛ]-1. Otherwise; 〈A〉t oscillates. A sufficient condition for decay is that Imδe(s)>0 (and λ small, see  (2.12)).(iii)
The error term in (2.11) is small in λ, uniformly in t≥0, and it decays in time quicker than any of the main terms in the sum on the r.h.s.: indeed, Imɛ=O(λ2) while ω′-O(λ)>ω′/2 independent of small values of λ. However, this means that we are in the regime λ2≪ω′<2π/β (see before (2.11)), which implies that λ2 must be much smaller than the temperature T=1/β. Using a more refined analysis, one can get rid of this condition; see also remarks on page 376 of [10].(iv)
Relation (2.13) shows that to lowest order in the perturbation, the group of (energy basis) matrix elements of any observable A corresponding to a fixed energy difference Em-En evolve jointly, while those of different such groups evolve independently.It is well known that there are two kinds of processes which drive decay (or irreversibility) ofS: energy-exchange processes characterized by [v,HS]≠0 and energy preserving ones where [v,HS]=0. The former are induced by interactions having nonvanishing probabilities for processes of absorption and emission of field quanta with energies corresponding to the Bohr frequencies of S and thus typically inducing thermalization of S. Energy preserving interactions suppress such processes, allowing only for a phase change of the system during the evolution (“phase damping”, [12–18]).To our knowledge, energy-exchange systems have so far been treated using Born and Markov master equation approximations (Lindblad form of dynamics) or they have been studied numerically, while for energy conserving systems, one often can find an exact solution. The present representation (2.11) gives a detailed picture of the dynamics of averages of observables for interactions with and without energy exchange. The resonance energies ɛ and the functionals Rɛ can be calculated for concrete models, as illustrated in the next section. We mention that the resonance dynamics representation can be used to study the dynamics of entanglement of qubits coupled to local and collective reservoirs, see [19].The dynamical resonance method can be generalized to time-dependent Hamiltonians.  See [20, 21] for time-periodic  Hamiltonians.


### 2.1.2. Contrast with Weak Coupling Approximation

Our representation (2.11) of the true dynamics of S relies only on the smallness of the coupling parameter λ, and no approximation is made. In the absence of an exact solution, it is common to make a weak coupling Lindblad master equation approximation of the dynamics, in which the reduced density matrix evolves according to ρ¯t=etℒρ¯0, where ℒ is the Lindblad generator, [22–24]. This approximation can lead to results that differ qualitatively from the true behaviour. For instance, the Lindblad master equation predicts that the system S approaches its Gibbs state at the temperature of the reservoir in the limit of large times. However, it is clear that in reality, the coupled system S+R will approach equilibrium, and hence the asymptotic state of S alone, being the reduction of the coupled equilibrium state, is the Gibbs state of S only to first approximation in the coupling (see also illustration below, and [9, 10]). In particular, the system's asymptotic density matrix is not diagonal in the original energy basis, but it has off-diagonal matrix elements of O(λ2). Features of this kind cannot be captured by the Lindblad approximation, but are captured in our approach.It has been shown (see, e.g., [23–26]) that the weak coupling limit dynamics generated by the Lindblad operator is obtained in the regime λ→0, t→∞, with λ2t fixed. One of the strengths of our approach is that we do not impose any relation between λ and t, and our results are valid for all times t≥0, provided λ is small. It has been observed [25, 26] that for certain systems of the type S+R, the second-order contribution of the exponents ɛe(s) in  (2.12) correspond to eigenvalues of the Lindblad generator. Our resonance method gives the true exponents, that is, we do not lose the contributions of any order in the interaction. If the energy spectrum of HS is degenerate, it happens that the second-order contributions to Imɛe(s) vanish. This corresponds to a Lindblad generator having several real eigenvalues. In this situation, the correct dynamics (approach to a final state) can be captured only by taking into account higher-order contributions to the exponents ɛe(s); see [27]. To our knowledge, so far this can only be done with the method presented in this paper, and is beyond the reach of the weak coupling method.


### 2.1.3. Illustration: Single Qubit

ConsiderS to be a single spin 1/2 with energy gap Δ=E2-E1>0. S is coupled to the heat bath R via the operator(2.14)v=[acc¯b]⊗ϕ(g),
where ϕ(g) is the Bose field operator  (2.3), smeared out with a coupling function (form factor) g(k), k∈ℝ3, and the 2×2 coupling matrix (representing the coupling operator in the energy eigenbasis) is hermitian. The operator  (2.14)—or a sum of such terms, for which our technique works equally well—is the most general coupling which is linear in field operators. We refer to [10] for a discussion of the link between  (2.14) and the spin-boson model. We take S initially in a coherent superposition in the energy basis,(2.15)ρ¯0=12[1111].
In [10] we derive from representation  (2.11) the following expressions for the dynamics of matrix elements, for all t≥0:
(2.16)[ρ¯t]m,m=e-βEmZS,β+(-1)m2tanh(βΔ2)eitɛ0(λ)+Rm,m(λ,t),m=1,2,(2.17)[ρ¯t]1,2=12eitɛ-Δ(λ)+R1,2(λ,t),
where the resonance energies ɛ are given by(2.18)ɛ0(λ)=iλ2π2|c|2ξ(Δ)+O(λ4),ɛΔ(λ)=Δ+λ2R+i2λ2π2[|c|2ξ(Δ)+(b-a)2ξ(0)]+O(λ4),ɛ-Δ(λ)=-ɛΔ(λ)¯,
with(2.19)ξ(η)=limϵ↓01π∫ℝ3d3kcoth(β|k|2)|g(k)|2ϵ(|k|-η)2+ϵ2,R=12(b2-a2)〈g,ω-1g〉+12|c|2P.V.∫ℝ×S2u2|g(|u|,σ)|2coth(β|u|2)1u-Δ.
The remainder terms in (2.17), (2.17) satisfy |Rm,n(λ,t)|≤Cλ2, uniformly in t≥0, and they can be decomposed into a sum of a constant and a decaying part, (2.20)Rm,n(λ,t)=〈〈pn,m〉〉∞-δm,ne-βEmZS,β+R′m,n(λ,t),
where |R'm,n(λ,t)|=O(λ2e-γt), with γ=min{Imɛ0,Imɛ±Δ}. These relations show the following.(i)
To second order inλ, convergence of the populations to the equilibrium values (Gibbs law), and decoherence occur exponentially fast, with rates τT=[Imɛ0(λ)]-1 and τD=[ImɛΔ(λ)]-1, respectively. (If either of these imaginary parts vanishes then the corresponding process does not take place, of course.) In particular, coherence of the initial state stays preserved on time scales of the order λ-2[|c|2ξ(Δ)+(b-a)2ξ(0)]-1; compare for example  (2.18).(ii)
The final density matrix of the spin isnot the Gibbs state of the qubit, and it is not diagonal in the energy basis. The deviation of the final state from the Gibbs state is given by limt→∞Rm,n(λ,t)=O(λ2). This is clear heuristically too, since typically the entire system S+R approaches its joint equilibrium in which S and R are entangled. The reduction of this state to S is the Gibbs state of S modulo O(λ2) terms representing a shift in the effective energy of S due to the interaction with the bath. In this sense, coherence in the energy basis of S is created by thermalization. We have quantified this in [10, Theorem 3.3].(iii)
In a markovian master equation approach, the above phenomenon (i.e., variations ofO(λ2) in the time-asymptotic limit) cannot be detected. Indeed in that approach one would conclude that S approaches its Gibbs state as t→∞.


## 2.1.1. Discussion

(i)
In the absence of interaction (λ=0), we have ɛ=e∈ℝ; see (2.12). Depending on the interaction, each resonance energy ɛ may migrate into the upper complex plane, or it may stay on the real axis, as λ≠0.(ii)
The averages〈A〉t approach their ergodic means 〈〈A〉〉∞ if and only if Imɛ>0 for all ɛ≠0. In this case, the convergence takes place on the time scale [Imɛ]-1. Otherwise; 〈A〉t oscillates. A sufficient condition for decay is that Imδe(s)>0 (and λ small, see  (2.12)).(iii)
The error term in (2.11) is small in λ, uniformly in t≥0, and it decays in time quicker than any of the main terms in the sum on the r.h.s.: indeed, Imɛ=O(λ2) while ω′-O(λ)>ω′/2 independent of small values of λ. However, this means that we are in the regime λ2≪ω′<2π/β (see before (2.11)), which implies that λ2 must be much smaller than the temperature T=1/β. Using a more refined analysis, one can get rid of this condition; see also remarks on page 376 of [10].(iv)
Relation (2.13) shows that to lowest order in the perturbation, the group of (energy basis) matrix elements of any observable A corresponding to a fixed energy difference Em-En evolve jointly, while those of different such groups evolve independently.It is well known that there are two kinds of processes which drive decay (or irreversibility) ofS: energy-exchange processes characterized by [v,HS]≠0 and energy preserving ones where [v,HS]=0. The former are induced by interactions having nonvanishing probabilities for processes of absorption and emission of field quanta with energies corresponding to the Bohr frequencies of S and thus typically inducing thermalization of S. Energy preserving interactions suppress such processes, allowing only for a phase change of the system during the evolution (“phase damping”, [12–18]).To our knowledge, energy-exchange systems have so far been treated using Born and Markov master equation approximations (Lindblad form of dynamics) or they have been studied numerically, while for energy conserving systems, one often can find an exact solution. The present representation (2.11) gives a detailed picture of the dynamics of averages of observables for interactions with and without energy exchange. The resonance energies ɛ and the functionals Rɛ can be calculated for concrete models, as illustrated in the next section. We mention that the resonance dynamics representation can be used to study the dynamics of entanglement of qubits coupled to local and collective reservoirs, see [19].The dynamical resonance method can be generalized to time-dependent Hamiltonians.  See [20, 21] for time-periodic  Hamiltonians.


## 2.1.2. Contrast with Weak Coupling Approximation

Our representation (2.11) of the true dynamics of S relies only on the smallness of the coupling parameter λ, and no approximation is made. In the absence of an exact solution, it is common to make a weak coupling Lindblad master equation approximation of the dynamics, in which the reduced density matrix evolves according to ρ¯t=etℒρ¯0, where ℒ is the Lindblad generator, [22–24]. This approximation can lead to results that differ qualitatively from the true behaviour. For instance, the Lindblad master equation predicts that the system S approaches its Gibbs state at the temperature of the reservoir in the limit of large times. However, it is clear that in reality, the coupled system S+R will approach equilibrium, and hence the asymptotic state of S alone, being the reduction of the coupled equilibrium state, is the Gibbs state of S only to first approximation in the coupling (see also illustration below, and [9, 10]). In particular, the system's asymptotic density matrix is not diagonal in the original energy basis, but it has off-diagonal matrix elements of O(λ2). Features of this kind cannot be captured by the Lindblad approximation, but are captured in our approach.It has been shown (see, e.g., [23–26]) that the weak coupling limit dynamics generated by the Lindblad operator is obtained in the regime λ→0, t→∞, with λ2t fixed. One of the strengths of our approach is that we do not impose any relation between λ and t, and our results are valid for all times t≥0, provided λ is small. It has been observed [25, 26] that for certain systems of the type S+R, the second-order contribution of the exponents ɛe(s) in  (2.12) correspond to eigenvalues of the Lindblad generator. Our resonance method gives the true exponents, that is, we do not lose the contributions of any order in the interaction. If the energy spectrum of HS is degenerate, it happens that the second-order contributions to Imɛe(s) vanish. This corresponds to a Lindblad generator having several real eigenvalues. In this situation, the correct dynamics (approach to a final state) can be captured only by taking into account higher-order contributions to the exponents ɛe(s); see [27]. To our knowledge, so far this can only be done with the method presented in this paper, and is beyond the reach of the weak coupling method.


## 2.1.3. Illustration: Single Qubit

ConsiderS to be a single spin 1/2 with energy gap Δ=E2-E1>0. S is coupled to the heat bath R via the operator(2.14)v=[acc¯b]⊗ϕ(g),
where ϕ(g) is the Bose field operator  (2.3), smeared out with a coupling function (form factor) g(k), k∈ℝ3, and the 2×2 coupling matrix (representing the coupling operator in the energy eigenbasis) is hermitian. The operator  (2.14)—or a sum of such terms, for which our technique works equally well—is the most general coupling which is linear in field operators. We refer to [10] for a discussion of the link between  (2.14) and the spin-boson model. We take S initially in a coherent superposition in the energy basis,(2.15)ρ¯0=12[1111].
In [10] we derive from representation  (2.11) the following expressions for the dynamics of matrix elements, for all t≥0:
(2.16)[ρ¯t]m,m=e-βEmZS,β+(-1)m2tanh(βΔ2)eitɛ0(λ)+Rm,m(λ,t),m=1,2,(2.17)[ρ¯t]1,2=12eitɛ-Δ(λ)+R1,2(λ,t),
where the resonance energies ɛ are given by(2.18)ɛ0(λ)=iλ2π2|c|2ξ(Δ)+O(λ4),ɛΔ(λ)=Δ+λ2R+i2λ2π2[|c|2ξ(Δ)+(b-a)2ξ(0)]+O(λ4),ɛ-Δ(λ)=-ɛΔ(λ)¯,
with(2.19)ξ(η)=limϵ↓01π∫ℝ3d3kcoth(β|k|2)|g(k)|2ϵ(|k|-η)2+ϵ2,R=12(b2-a2)〈g,ω-1g〉+12|c|2P.V.∫ℝ×S2u2|g(|u|,σ)|2coth(β|u|2)1u-Δ.
The remainder terms in (2.17), (2.17) satisfy |Rm,n(λ,t)|≤Cλ2, uniformly in t≥0, and they can be decomposed into a sum of a constant and a decaying part, (2.20)Rm,n(λ,t)=〈〈pn,m〉〉∞-δm,ne-βEmZS,β+R′m,n(λ,t),
where |R'm,n(λ,t)|=O(λ2e-γt), with γ=min{Imɛ0,Imɛ±Δ}. These relations show the following.(i)
To second order inλ, convergence of the populations to the equilibrium values (Gibbs law), and decoherence occur exponentially fast, with rates τT=[Imɛ0(λ)]-1 and τD=[ImɛΔ(λ)]-1, respectively. (If either of these imaginary parts vanishes then the corresponding process does not take place, of course.) In particular, coherence of the initial state stays preserved on time scales of the order λ-2[|c|2ξ(Δ)+(b-a)2ξ(0)]-1; compare for example  (2.18).(ii)
The final density matrix of the spin isnot the Gibbs state of the qubit, and it is not diagonal in the energy basis. The deviation of the final state from the Gibbs state is given by limt→∞Rm,n(λ,t)=O(λ2). This is clear heuristically too, since typically the entire system S+R approaches its joint equilibrium in which S and R are entangled. The reduction of this state to S is the Gibbs state of S modulo O(λ2) terms representing a shift in the effective energy of S due to the interaction with the bath. In this sense, coherence in the energy basis of S is created by thermalization. We have quantified this in [10, Theorem 3.3].(iii)
In a markovian master equation approach, the above phenomenon (i.e., variations ofO(λ2) in the time-asymptotic limit) cannot be detected. Indeed in that approach one would conclude that S approaches its Gibbs state as t→∞.


## 2.2. Evolution of Reduced Dynamics of anN-Level System

In the sequel, we analyze in more detail the evolution of a qubit register of sizeN. The Hamiltonian is(2.21)HS=∑i,j=1NJijSizSjz+∑j=1NBjSjz,
where Jij are pair interaction constants and Bj is the value of a magnetic field at the location of spin j. The Pauli spin operator is(2.22)Sz=[100-1]
and Sjz is the matrix Sz acting on the jth spin only.We consider acollective coupling between the register S and the reservoir R: the distance between the N qubits is much smaller than the correlation length of the reservoir and as a consequence, each qubit feels the same interaction with the reservoir. The corresponding interaction operator is (compare with  (2.4))(2.23)λ1v1+λ2v2=λ1∑j=1NSjz⊗ϕ(g1)+λ2∑j=1NSjx⊗ϕ(g2).
Here g1 and g2 are form factors and the coupling constants λ1 and λ2 measure the strengths of the energy conserving (position-position) coupling, and the energy exchange (spin flip) coupling, respectively. Spin-flips are implemented by the Sjx in  (2.23), representing the Pauli matrix
(2.24)Sx=[0110]
acting on the jth spin. The total Hamiltonian takes the form  (2.4) with λv replaced by  (2.23). It is convenient to represent ρ¯t as a matrix in the energy basis, consisting of eigenvectors φσ̲ of HS. These are vectors in 𝔥S=ℂ2⊗⋯⊗ℂ2=ℂ2N indexed by spin configurations(2.25)σ̲={σ1,…,σN}∈{+1,-1}N,φσ̲=φσ1⊗⋯⊗φσN,
where(2.26)φ+=[10],φ-=[01],
so that(2.27)HSφσ̲=E(σ̲)φσ̲withE(σ̲)=∑i,j=1NJijσiσj+∑j=1NBjσj.
We denote the reduced density matrix elements as(2.28)[ρ¯t]σ̲,τ̲=〈φσ̲,ρ¯tφτ̲〉.
The Bohr frequencies  (2.9) are now(2.29)e(σ̲,τ̲)=E(σ̲)-E(τ̲)=∑i,j=1NJij(σiσj-τiτj)+∑j=1NBj(σj-τj),
and they become complex resonance energies ɛe=ɛe(λ1,λ2)∈ℂ under perturbation.Assumption of Nonoverlapping Resonances
The Bohr frequencies (2.29) represent “unperturbed” energy levels and we follow their motion under perturbation (λ1,λ2). In this work, we consider the regime of nonoverlapping resonances, meaning that the interaction is small relative to the spacing of the Bohr frequencies.We show in [10, Theorem 2.1], that for all t≥0, (2.30)[ρ¯t]σ̲,τ̲-〈〈[ρ¯∞]σ̲,τ̲〉〉=∑{e:ɛe≠0}eitɛe[∑σ′̲,τ̲′wσ̲,τ̲;σ̲′,τ̲′ɛe[ρ¯0]σ̲′,τ̲′+O(λ12+λ22)]+O((λ12+λ22)e-[ω′+O(λ)]t).
This result is obtained by specializing  (2.11) to the specific system at hand and considering observables A=|φτ̲〉〈φσ̲|. In (2.30), we have in accordance with (2.10), 〈〈[ρ¯∞]σ̲,τ̲〉〉=limT→∞(1/T)∫0T[ρ¯t]σ̲,τ̲dt. The coefficients w are overlaps of resonance eigenstates which vanish unless e=-e(σ̲,τ̲)=-e(σ̲′,τ̲′) (see point (2) after  (2.9)). They represent the dominant contribution to the functionals Rɛ in  (2.11); see also  (2.13). The ɛe have the expansion(2.31)ɛe≡ɛe(s)=e+δe(s)+O(λ14+λ24),
where the label s=1,…,ν(e) indexes the splitting of the eigenvalue e into ν(e) distinct resonance energies. The lowest order corrections δe(s) satisfy(2.32)δe(s)=O(λ12+λ22).
They are the (complex) eigenvalues of an operator Λe, called the level shift operator associated to e. This operator acts on the eigenspace of LS associated to the eigenvalue e (a subspace of the qubit register Hilbert space; see [10, 11] for the formal definition of Λe). It governs the lowest order shift of eigenvalues under perturbation. One can see by direct calculation that Imδe(s)≥0.


### 2.2.1. Discussion

(i)
To lowest order in the perturbation, the group of reduced density matrix elements[ρ¯t]σ̲,τ̲ associated to a fixed e=e(σ̲,τ̲) evolve in a coupled way, while groups of matrix elements associated to different e evolve independently.(ii)
The density matrix elements of a given group mix and evolve in time according to the weight functionsw and the exponentials eitɛe(s). In the absence of interaction (λ1=λ2=0), all the ɛe(s)=e are real. As the interaction is switched on, the ɛe(s) typically migrate into the upper complex plane, but they may stay on the real line (due to some symmetry or due to an “inefficient coupling”).(iii)
The matrix elements[ρ¯t]σ̲,τ̲ of a group e approach their ergodic means if and only if all the nonzero ɛe(s) have strictly positive imaginary part. In this case, the convergence takes place on a time scale of the order 1/γe, where
(2.33)γe=min{Imɛe(s):s=1,…,ν(e)s.t.ɛe(s)≠0}
is the decay rate of the group associated to e. If an ɛe(s) stays real, then the matrix elements of the corresponding group oscillate in time. A sufficient condition for decay of the group associated to e is γe>0, that is, Imδe(s)>0 for all s, and λ1, λ2 small.


### 2.2.2. Decoherence Rates

We illustrate our results on decoherence rates for a qubit register withJij=0 (the general case is treated in [11]). We consider generic magnetic fields defined as follows. For nj∈{0,±1,±2}, j=1,…,N, we have
(2.34)∑j=1NBjnj=0⇔nj=0∀j.
Condition  (2.34) is satisfied generically in the sense that it does not hold only for very special choices of Bj (one such special choice is Bj=constant). For instance, if the Bj are chosen to be independent, and uniformly random from an interval [Bmin,Bmax], then  (2.34) is satisfied with probability one. We show in [11, Theorem 2.3], that the decoherence rates  (2.33) are given by(2.35)γe={λ12y1(e)+λ22y2(e)+y12(e),e≠0λ22y0,e=0}+O(λ14+λ24).
Here, y1 is contributions coming from the energy conserving interaction; y0 and y2 are due to the spin flip interaction. The term y12 is due to both interactions and is of O(λ12+λ22). We give explicit expressions for y0, y1, y2, and y12 in [11, Section 2]. For the present purpose, we limit ourselves to discussing the properties of the latter quantities.(i)
Properties ofy1(e):y1(e) vanishes if either e is such that e0:=∑j=1n(σj-τj)=0 or the infrared behaviour of the coupling function g1 is too regular (in three dimensions g1∝|k|p with p>-1/2). Otherwise, y1(e)>0. Moreover, y1(e) is proportional to the temperature T.(ii)
Properties ofy2(e):y2(e)>0 if g2(2Bj,Σ)≠0 for all Bj (form factor g2(k)=g2(|k|,Σ) in spherical coordinates). For low temperatures, T, y2(e)∝T, for high temperatures y2(e) approaches a constant.(iii)
Properties ofy12(e): if either of λ1, λ2 or e0 vanish, or if g1 is infrared regular as mentioned above, then y12(e)=0. Otherwise, y12(e)>0, in which case y12(e) approaches constant values for both T→0,∞.(iv)
Full decoherence: if γe>0 for all e≠0, then all off-diagonal matrix elements approach their limiting values exponentially fast. In this case, we say that full decoherence occurs. It follows from the above points that we have full decoherence if λ2≠0 and g2(2Bj,Σ)≠0 for all j, and provided λ1,λ2 are small enough (so that the remainder term in  (2.35) is small). Note that if λ2=0, then matrix elements associated to energy differences e such that e0=0 will not decay on the time scale given by the second order in the perturbation (λ12). We point out that generically, S+R will reach a joint equilibrium as t→∞, which means that the final reduced density matrix of S is its Gibbs state modulo a peturbation of the order of the interaction between S and R; see [9, 10]. Hence generically, the density matrix of S does not become diagonal in the energy basis as t→∞.(v)
Properties ofy0:y0 depends on the energy exchange interaction only. This reflects the fact that for a purely energy conserving interaction, the populations are conserved [9, 10, 17]. If g2(2Bj,Σ)≠0 for all j, then y0>0 (this is sometimes called the “Fermi Golden Rule Condition”). For small temperatures T, y0∝T, while y0 approaches a finite limit as T→∞.In terms of complexity analysis, it is important to discuss thedependence ofγeon the register sizeN.(i)
We show in [11] that y0 is independent of N. This means that the thermalization time, or relaxation time of the diagonal matrix elements (corresponding to e=0), is O(1) in N.(ii)
To determine the order of magnitude of the decay rates of the off-diagonal density matrix elements (corresponding toe≠0) relative to the register size N, we assume the magnetic field to have a certain distribution denoted by 〈·〉. We show in [11] that
(2.36)〈y1〉=y1∝e02,〈y2〉=CB𝔇(σ̲-τ̲),〈y12〉=cB(λ1,λ2)N0(e),
where CB and cB=cB(λ1,λ2) are positive constants (independent of N), with cB(λ1,λ2)=O(λ12+λ22). Here, N0(e) is the number of indices j such that σj=τj for each (σ̲,τ̲) s.t. e(σ̲,τ̲)=e, and
(2.37)𝔇(σ̲-τ̲):=∑j=1N|σj-τj|
is the Hamming distance between the spin configurations σ̲ and τ̲ (which depends on e only).(iii)
Considere≠0. It follows from  (2.35)–(2.37) that for purely energy conserving interactions (λ2=0), γe∝λ12e02=λ12[∑j=1N(σj-τj)]2, which can be as large as O(λ12N2). On the other hand, for purely energy exchanging interactions (λ1=0), we have γe∝λ22D(σ̲-τ̲), which cannot exceed O(λ22N). If both interactions are acting, then we have the additional term 〈y12〉, which is of order O((λ12+λ22)N). This shows the following: The fastest decay rate of reduced off-diagonal density matrix elements due to the energy conserving interaction alone is of orderλ12N2, while the fastest decay rate due to the energy exchange interaction alone is of the orderλ22N. Moreover, the decay of the diagonal matrix elements is of orderλ12, that is, independent ofN.Remark 2.2.2 s.
(1) Forλ2=0, the model can be solved explicitly [17], and one shows that the fastest decaying matrix elements have decay rate proportional to λ12N2. Furthermore, the model with a noncollective, energy-conserving interaction, where each qubit is coupled to an independent reservoir, can also be solved explicitly [17]. The fastest decay rate in this case is shown to be proportional to λ12N.
(2) As mentioned at the beginning of this section, we take the coupling constantsλ1, λ2 so small that the resonances do not overlap. Consequently, λ12N2 and λ22N are bounded above by a constant proportional to the gradient of the magnetic field in the present situation; see also [11]. Thus the decay rates γe do not increase indefinitely with increasing N in the regime considered here. Rather, γe are attenuated by small coupling constants for large N. They are of the order γe~Δ. We have shown that modulo an overall, common (N-dependent) prefactor, the decay rates originating from the energy conserving and exchanging interactions differ by a factor N.
(3) Collective decoherence has been studied extensively in the literature. Among the many theoretical, numerical, and experimental works, we mention here only [12, 14, 17, 28, 29], which are closest to the present work. We are not aware of any prior work giving explicit decoherence rates of a register for not explicitly solvable models, and without making master equation technique approximations.


## 2.2.1. Discussion

(i)
To lowest order in the perturbation, the group of reduced density matrix elements[ρ¯t]σ̲,τ̲ associated to a fixed e=e(σ̲,τ̲) evolve in a coupled way, while groups of matrix elements associated to different e evolve independently.(ii)
The density matrix elements of a given group mix and evolve in time according to the weight functionsw and the exponentials eitɛe(s). In the absence of interaction (λ1=λ2=0), all the ɛe(s)=e are real. As the interaction is switched on, the ɛe(s) typically migrate into the upper complex plane, but they may stay on the real line (due to some symmetry or due to an “inefficient coupling”).(iii)
The matrix elements[ρ¯t]σ̲,τ̲ of a group e approach their ergodic means if and only if all the nonzero ɛe(s) have strictly positive imaginary part. In this case, the convergence takes place on a time scale of the order 1/γe, where
(2.33)γe=min{Imɛe(s):s=1,…,ν(e)s.t.ɛe(s)≠0}
is the decay rate of the group associated to e. If an ɛe(s) stays real, then the matrix elements of the corresponding group oscillate in time. A sufficient condition for decay of the group associated to e is γe>0, that is, Imδe(s)>0 for all s, and λ1, λ2 small.


## 2.2.2. Decoherence Rates

We illustrate our results on decoherence rates for a qubit register withJij=0 (the general case is treated in [11]). We consider generic magnetic fields defined as follows. For nj∈{0,±1,±2}, j=1,…,N, we have
(2.34)∑j=1NBjnj=0⇔nj=0∀j.
Condition  (2.34) is satisfied generically in the sense that it does not hold only for very special choices of Bj (one such special choice is Bj=constant). For instance, if the Bj are chosen to be independent, and uniformly random from an interval [Bmin,Bmax], then  (2.34) is satisfied with probability one. We show in [11, Theorem 2.3], that the decoherence rates  (2.33) are given by(2.35)γe={λ12y1(e)+λ22y2(e)+y12(e),e≠0λ22y0,e=0}+O(λ14+λ24).
Here, y1 is contributions coming from the energy conserving interaction; y0 and y2 are due to the spin flip interaction. The term y12 is due to both interactions and is of O(λ12+λ22). We give explicit expressions for y0, y1, y2, and y12 in [11, Section 2]. For the present purpose, we limit ourselves to discussing the properties of the latter quantities.(i)
Properties ofy1(e):y1(e) vanishes if either e is such that e0:=∑j=1n(σj-τj)=0 or the infrared behaviour of the coupling function g1 is too regular (in three dimensions g1∝|k|p with p>-1/2). Otherwise, y1(e)>0. Moreover, y1(e) is proportional to the temperature T.(ii)
Properties ofy2(e):y2(e)>0 if g2(2Bj,Σ)≠0 for all Bj (form factor g2(k)=g2(|k|,Σ) in spherical coordinates). For low temperatures, T, y2(e)∝T, for high temperatures y2(e) approaches a constant.(iii)
Properties ofy12(e): if either of λ1, λ2 or e0 vanish, or if g1 is infrared regular as mentioned above, then y12(e)=0. Otherwise, y12(e)>0, in which case y12(e) approaches constant values for both T→0,∞.(iv)
Full decoherence: if γe>0 for all e≠0, then all off-diagonal matrix elements approach their limiting values exponentially fast. In this case, we say that full decoherence occurs. It follows from the above points that we have full decoherence if λ2≠0 and g2(2Bj,Σ)≠0 for all j, and provided λ1,λ2 are small enough (so that the remainder term in  (2.35) is small). Note that if λ2=0, then matrix elements associated to energy differences e such that e0=0 will not decay on the time scale given by the second order in the perturbation (λ12). We point out that generically, S+R will reach a joint equilibrium as t→∞, which means that the final reduced density matrix of S is its Gibbs state modulo a peturbation of the order of the interaction between S and R; see [9, 10]. Hence generically, the density matrix of S does not become diagonal in the energy basis as t→∞.(v)
Properties ofy0:y0 depends on the energy exchange interaction only. This reflects the fact that for a purely energy conserving interaction, the populations are conserved [9, 10, 17]. If g2(2Bj,Σ)≠0 for all j, then y0>0 (this is sometimes called the “Fermi Golden Rule Condition”). For small temperatures T, y0∝T, while y0 approaches a finite limit as T→∞.In terms of complexity analysis, it is important to discuss thedependence ofγeon the register sizeN.(i)
We show in [11] that y0 is independent of N. This means that the thermalization time, or relaxation time of the diagonal matrix elements (corresponding to e=0), is O(1) in N.(ii)
To determine the order of magnitude of the decay rates of the off-diagonal density matrix elements (corresponding toe≠0) relative to the register size N, we assume the magnetic field to have a certain distribution denoted by 〈·〉. We show in [11] that
(2.36)〈y1〉=y1∝e02,〈y2〉=CB𝔇(σ̲-τ̲),〈y12〉=cB(λ1,λ2)N0(e),
where CB and cB=cB(λ1,λ2) are positive constants (independent of N), with cB(λ1,λ2)=O(λ12+λ22). Here, N0(e) is the number of indices j such that σj=τj for each (σ̲,τ̲) s.t. e(σ̲,τ̲)=e, and
(2.37)𝔇(σ̲-τ̲):=∑j=1N|σj-τj|
is the Hamming distance between the spin configurations σ̲ and τ̲ (which depends on e only).(iii)
Considere≠0. It follows from  (2.35)–(2.37) that for purely energy conserving interactions (λ2=0), γe∝λ12e02=λ12[∑j=1N(σj-τj)]2, which can be as large as O(λ12N2). On the other hand, for purely energy exchanging interactions (λ1=0), we have γe∝λ22D(σ̲-τ̲), which cannot exceed O(λ22N). If both interactions are acting, then we have the additional term 〈y12〉, which is of order O((λ12+λ22)N). This shows the following: The fastest decay rate of reduced off-diagonal density matrix elements due to the energy conserving interaction alone is of orderλ12N2, while the fastest decay rate due to the energy exchange interaction alone is of the orderλ22N. Moreover, the decay of the diagonal matrix elements is of orderλ12, that is, independent ofN.Remark 2.2.2 s.
(1) Forλ2=0, the model can be solved explicitly [17], and one shows that the fastest decaying matrix elements have decay rate proportional to λ12N2. Furthermore, the model with a noncollective, energy-conserving interaction, where each qubit is coupled to an independent reservoir, can also be solved explicitly [17]. The fastest decay rate in this case is shown to be proportional to λ12N.
(2) As mentioned at the beginning of this section, we take the coupling constantsλ1, λ2 so small that the resonances do not overlap. Consequently, λ12N2 and λ22N are bounded above by a constant proportional to the gradient of the magnetic field in the present situation; see also [11]. Thus the decay rates γe do not increase indefinitely with increasing N in the regime considered here. Rather, γe are attenuated by small coupling constants for large N. They are of the order γe~Δ. We have shown that modulo an overall, common (N-dependent) prefactor, the decay rates originating from the energy conserving and exchanging interactions differ by a factor N.
(3) Collective decoherence has been studied extensively in the literature. Among the many theoretical, numerical, and experimental works, we mention here only [12, 14, 17, 28, 29], which are closest to the present work. We are not aware of any prior work giving explicit decoherence rates of a register for not explicitly solvable models, and without making master equation technique approximations.


## 3. Resonance Representation of Reduced Dynamics

The goal of this section is to give a precise statement of the core representation  (2.11), and to outline the main ideas behind the proof of it.TheN-level system is coupled to the reservoir (see also  (2.1),  (2.2)) through the operator(3.1)v=∑r=1RλrGr⊗ϕ(gr),
where each Gr is a hermitian N×N matrix, the gr(k) are form factors, and the λr∈ℝ are coupling constants. Fix any phase χ∈ℝ and define(3.2)gr,β(u,σ):=u1-e-βu|u|1/2{gr(u,σ)ifu≥0,-eiχg¯r(-u,σ)ifu<0,
where u∈ℝ and σ∈S2. The phase χ is a parameter which can be chosen appropriately as to satisfy the following condition.(A)
The mapω↦gr,β(u+ω,σ) has an analytic extension to a complex neighbourhood {|z|<ω'} of the origin, as a map from ℂ to L2(ℝ3,d3k).Examples ofg satisfying (A) are given by g(r,σ)=rpe-rmg1(σ), where p=-1/2+n, n=0,1,…, m=1,2, and g1(σ)=eiϕg¯1(σ).This condition ensures that the technically simplest version of the dynamical resonance theory, based on complex spectral translations, can be implemented. The technical simplicity comes at a price: on one hand, it limits the class of admissible functionsg(k), which have to behave appropriately in the infrared regime so that the parts of (3.2) fit nicely together at u=0, to allow for an analytic continuation. On the other hand, the square root in  (3.2) must be analytic as well, which implies the condition ω′<2π/β.It is convenient to introduce the doubled Hilbert spaceℋS=𝔥S⊗𝔥S, whose normalized vectors accommodate any state on the system S (pure or mixed). The trace state, or infinite temperature state, is represented by the vector(3.3)ΩS=1N∑j=1Nφj⊗φj
via(3.4)ℬ(𝔥S)∋A↦〈ΩS,(A⊗1)ΩS〉.
Here φj are the orthonormal eigenvectors of HS. This is just the Gelfand-Naimark-Segal construction for the trace state. Similarly, let ℋR and ΩR,β be the Hilbert space and the vector representing the equilibrium state of the reservoirs at inverse temperature β. In the Araki-Woods representation of the field, we have ℋR=ℱ⊗ℱ, where ℱ is the bosonic Fock space over the one-particle space L2(ℝ3,d3k) and ΩR,β=Ω⊗Ω, with Ω being the Fock vacuum of ℱ (see also [10, 11] for more detail). Let ψ0⊗ΩR,β be the vector in ℋS⊗ℋR representing the density matrix at time t=0. It is not difficult to construct the unique operator in B∈1S⊗𝔥S satisfying (3.5)BΩS=ψ0.
(See also [10] for concrete examples.) We define the reference vector (3.6)Ωref:=ΩS⊗ΩR,β
and set (3.7)λ=maxr=1,…,R|λr|.Theorem 3.1 (Dynamical resonance theory [9–11]).
Assume condition (A) with a fixedω′ satisfying 0<ω′<2π/β. There is a constant c0 s.t.; if λ≤c0/β, then the limit 〈〈A〉〉∞,  (2.10), exists for all observables A∈ℬ(𝔥S). Moreover, for all such A and for all t≥0, we have
(3.8)〈A〉t-〈〈A〉〉∞=∑e,s:ɛe(s)≠0∑s=1ν(e)eitɛe(s)〈(B*ψ0)⊗ΩR,β,Qe(s)(A⊗1S)Ωref〉+O(λ2e-[ω′+O(λ)]t).
The ɛe(s) are given by  (2.12), 1≤ν(e)≤mult(e) counts the splitting of the eigenvalue e into distinct resonance energies ɛe(s), and the Qe(s) are (nonorthogonal) finite-rank projections.This result is the basis for a detailed analysis of the reduced dynamics of concrete systems, like theN-qubit register introduced in Section 2.2. We obtain  (2.30) (in particular, the overlap functions w) from  (3.8) by analyzing the projections Qe(s) in more detail. Let us explain how to link the overlap 〈(B*ψ0)⊗ΩR,β,Qe(s)(A⊗1S)Ωref〉 to its initial value for a nondegenerate Bohr energy e, and where A=|φn〉〈φm|. (The latter observables used in  (2.11) give the matrix elements of the reduced density matrix in the energy basis.)TheQe(s) is the spectral (Riesz) projection of an operator Kλ associated with the eigenvalue ɛe(s); see  (3.19) (In reality, we consider a spectral deformationKλ(ω), where ω is a complex parameter. This is a technical trick to perform our analysis. Physical quantities do not depend on ω and therefore, we do not display this parameter here). If a Bohr energy e,  (2.9), is simple, then there is a single resonance energy ɛe bifurcating out of e, as λ≠0. In this case, the projection Qe≡Qe(s) has rank one, Qe=|χe〉〈χ̃e|, where χe and χ̃e are eigenvectors of Kλ and its adjoint, with eigenvalue ɛe and its complex conjugate, respectively, and 〈χe,χ̃e〉=1. From perturbation theory, we obtain χe=χ̃e=φk⊗φl⊗ΩR,β+O(λ), where HSφj=Ejφj and Ek-El=e. The overlap in the sum of  (3.8) becomes(3.9)〈(B*ψ0)⊗ΩR,β,Qe(A⊗1S)Ωref〉=〈(B*ψ0)⊗ΩR,β,|φk⊗φl⊗ΩR,β〉〈φk⊗φl⊗ΩR,β|(A⊗1S)Ωref〉+O(λ2)=〈B*ψ0,|φk⊗φl〉〈φk⊗φl|(A⊗1S)ΩS〉+O(λ2).
The choice A=|φn〉〈φm| in (2.6) gives 〈A〉t=[ρ¯t]m,n, the reduced density matrix element. With this choice of A, the main term in  (3.9) becomes (see also  (3.3))(3.10)〈B*ψ0,|φk⊗φl〉〈φk⊗φl|(A⊗1S)ΩS〉=1Nδknδlm〈B*ψ0,φn⊗φm〉=δknδlm〈B*ψ0,(|φn〉〈φm|⊗1S)ΩS〉=δknδlm〈ψ0,(|φn〉〈φm|⊗1S)BΩS〉=δknδlm[ρ¯0]mn.
In the second-last step, we commute B to the right through |φn〉〈φm|⊗1S, since B belongs to the commutant of the algebra of observables of S. In the last step, we use BΩS=ψ0.Combining  (3.9) and  (3.10) with Theorem 3.1 we obtain, in case e=Em-En is a simple eigenvalue, (3.11)[ρ¯t]mn-〈〈[ρ¯∞]mn〉〉=∑{e,s:ɛe(s)≠0}eitɛe(s)[δknδlm[ρ¯0]mn+O(λ2)]+O(λ2e-[ω′+O(λ)]t).
This explains the form  (2.30) for a simple Bohr energy e. The case of degenerate e (i.e., where several different pairs of indices k,l satisfy Ek-El=e) is analyzed along the same lines; see [11] for details.


### 3.1. Mechanism of Dynamical Resonance Theory, Outline of Proof of Theorem3.1

Consider any observableA∈B(𝔥S). We have(3.12)〈A〉t=TrS[ρ¯tA]=TrS+R[ρtA⊗1R]=〈ψ0,eitLλ[A⊗1S⊗1R]e-itLλψ0〉.
In the last step, we pass to the representation Hilbert space of the system (the GNS Hilbert space), where the initial density matrix is represented by the vector ψ0 (in particular, the Hilbert space of the small system becomes 𝔥S⊗𝔥S); see also before  (3.3),  (3.4). As mentioned above, in this review we consider initial states where S and R are not entangled. The initial state is represented by the product vector ψ0=ΩS⊗ΩR,β, where ΩS is the trace state of S,  (3.4), 〈ΩS,(A⊗1S)ΩS〉=(1/N)Tr(A), and where ΩR,β is the equilibrium state of R at a fixed inverse temperature 0<β<∞. The dynamics is implemented by the group of automorphisms eitLλ·e-itLλ. The self-adjoint generator Lλ is called the Liouville operator. It is of the form Lλ=L0+λW, where L0=LS+LR represents the uncoupled Liouville operator, and λW is the interaction  (3.1) represented in the GNS Hilbert space. We refer to [10, 11] for the specific form of W.We borrow a trick from the analysis of open systems far from equilibrium: there is a (nonself-adjoint) generatorKλ s.t. (3.13)eitLλAe-itLλ=eitKλAe-itKλforallobservablesA,t≥0,andKλψ0=0.Kλ can be constructed in a standard way, given Lλ and the reference vector ψ0. Kλ is of the form Kλ=L0+λI, where the interaction term undergoes a certain modification (W→I); see for example [10]. As a consequence, formally, we may replace the propagators in  (3.12) by those involving K. The resulting propagator which is directly applied to ψ0 will then just disappear due to the invariance of ψ0. One can carry out this procedure in a rigorous manner, obtaining the following resolvent representation [10](3.14)〈A〉t=-12πi∫ℝ-i〈ψ0,(Kλ(ω)-z)-1[A⊗1S⊗1R]ψ0〉eitzdz,
where Kλ(ω)=L0(ω)+λI(ω), I is representing the interaction, and ω↦Kλ(ω) is a spectral deformation (translation) of Kλ. The latter is constructed as follows. There is a deformation transformation U(ω)=e-iωD, where D is the (explicit) self-adjoint generator of translations [10, 11, 30] transforming the operator Kλ as(3.15)Kλ(ω)=U(ω)KλU(ω)-1=L0+ωN+λI(ω).Here,N=N1⊗1+1⊗N1 is the total number operator of a product of two bosonic Fock spaces ℱ⊗ℱ (the Gelfand-Naimark-Segal Hilbert space of the reservoir), and where N1 is the usual number operator on ℱ. N has spectrum ℕ∪{0}, where 0 is a simple eigenvalue (with vacuum eigenvector ΩR,β=Ω⊗Ω). For real values of ω, U(ω) is a group of unitaries. The spectrum of Kλ(ω) depends on Imω and moves according to the value of Imω, whence the name “spectral deformation”. Even though U(ω) becomes unbounded for complex ω, the r.h.s. of  (3.15) is a well-defined closed operator on a dense domain, analytic in ω at zero. Analyticity is used in the derivation of  (3.14) and this is where the analyticity condition (A) after  (3.2) comes into play. The operator I(ω) is infinitesimally small with respect to the number operator N. Hence we use perturbation theory in λ to examine the spectrum of Kλ(ω).The point of the spectral deformation is that the (important part of the) spectrum ofKλ(ω) is much easier to analyze than that of Kλ, because the deformation uncovers the resonances of Kλ. We have (see Figure 1) (3.16)spec(K0(ω))={Ei-Ej}i,j=1,…,N⋃n≥1{ωn+ℝ},
because K0(ω)=L0+ωN, L0 and N commute, and the eigenvectors of L0=LS+LR are φi⊗φj⊗ΩR,β. Here, we have HSφj=Ejφj. The continuous spectrum is bounded away from the isolated eigenvalues by a gap of size Imω. For values of the coupling parameter λ small compared to Imω, we can follow the displacements of the eigenvalues by using analytic perturbation theory. (Note that for Imω=0, the eigenvalues are imbedded into the continuous spectrum, and analytic perturbation theory is not valid! The spectral deformation is indeed very useful!)Figure 1
Spectrum ofK0(ω).Theorem 3.2 (see [10] and Figure 2).
FixImω s.t. 0<Imω<ω′ (where ω′ is as in Condition (A)). There is a constant c0>0 s.t. if |λ|≤c0/β then, for all ω with Imω>7ω′/8, the spectrum of Kλ(ω) in the complex half-plane {Imz<ω′/2} is independent of ω and consists purely of the distinct eigenvalues
(3.17){ɛe(s):e∈spec(LS),s=1,…,ν(e)},
where 1≤ν(e)≤mult(e) counts the splitting of the eigenvalue e. Moreover,
(3.18)limλ→0|ɛe(s)(λ)-e|=0
for all s, and we have Imɛe(s)≥0. Also, the continuous spectrum of Kλ(ω) lies in the region {Imz≥3ω′/4}.Figure 2
Spectrum ofKλ(ω). Resonances ɛe(s) are uncovered.Next we separate the contributions to the path integral in  (3.14) coming from the singularities at the resonance energies and from the continuous spectrum. We deform the path of integration z=ℝ-i into the line z=ℝ+iω′/2, thereby picking up the residues of poles of the integrand at ɛe(s) (all e, s). Let 𝒞e(s) be a small circle around ɛe(s), not enclosing or touching any other spectrum of Kλ(ω). We introduce the generally nonorthogonal Riesz spectral projections(3.19)Qe(s)=Qe(s)(ω,λ)=-12πi∫𝒞e(s)(Kλ(ω)-z)-1dz.It follows from  (3.14) that(3.20)〈A〉t=∑e∑s=1ν(e)eitɛe(s)〈ψ0,Qe(s)[A⊗1S⊗1R]ψ0〉+O(λ2e-ω′t/2).
Note that the imaginary parts of all resonance energies ɛe(s) are smaller than ω′/2, so that the remainder term in  (3.20) is not only small in λ, but it also decays faster than all of the terms in the sum. (See also Figure 3.) We point out also that instead of deforming the path integration contour as explained before (3.19), we could choose z=ℝ+i[ω′-O(λ)], hence transforming the error term in  (3.20) into the one given in  (3.8).Figure 3
Contour deformation:∫ℝ-idz=∑e,s∫𝒞e(s)dz+∫ℝ+iω'/2dz.Finally, we notice that all terms in  (3.20) with ɛe(s)≠0 will vanish in the ergodic mean limit, so (3.21)〈〈A〉〉∞=limT→∞1T∫0T〈A〉tdt=∑s:ɛ0(s)=0〈ψ0,Q0(s)[A⊗1R⊗1R]ψ0〉.
We now see that the linear functionals  (2.13) are represented as(3.22)Rɛe(s)(A)=〈ψ0,Qe(s)[A⊗1S⊗1R]ψ0〉.
This concludes the outline of the proof of Theorem 3.1.


## 3.1. Mechanism of Dynamical Resonance Theory, Outline of Proof of Theorem3.1

Consider any observableA∈B(𝔥S). We have(3.12)〈A〉t=TrS[ρ¯tA]=TrS+R[ρtA⊗1R]=〈ψ0,eitLλ[A⊗1S⊗1R]e-itLλψ0〉.
In the last step, we pass to the representation Hilbert space of the system (the GNS Hilbert space), where the initial density matrix is represented by the vector ψ0 (in particular, the Hilbert space of the small system becomes 𝔥S⊗𝔥S); see also before  (3.3),  (3.4). As mentioned above, in this review we consider initial states where S and R are not entangled. The initial state is represented by the product vector ψ0=ΩS⊗ΩR,β, where ΩS is the trace state of S,  (3.4), 〈ΩS,(A⊗1S)ΩS〉=(1/N)Tr(A), and where ΩR,β is the equilibrium state of R at a fixed inverse temperature 0<β<∞. The dynamics is implemented by the group of automorphisms eitLλ·e-itLλ. The self-adjoint generator Lλ is called the Liouville operator. It is of the form Lλ=L0+λW, where L0=LS+LR represents the uncoupled Liouville operator, and λW is the interaction  (3.1) represented in the GNS Hilbert space. We refer to [10, 11] for the specific form of W.We borrow a trick from the analysis of open systems far from equilibrium: there is a (nonself-adjoint) generatorKλ s.t. (3.13)eitLλAe-itLλ=eitKλAe-itKλforallobservablesA,t≥0,andKλψ0=0.Kλ can be constructed in a standard way, given Lλ and the reference vector ψ0. Kλ is of the form Kλ=L0+λI, where the interaction term undergoes a certain modification (W→I); see for example [10]. As a consequence, formally, we may replace the propagators in  (3.12) by those involving K. The resulting propagator which is directly applied to ψ0 will then just disappear due to the invariance of ψ0. One can carry out this procedure in a rigorous manner, obtaining the following resolvent representation [10](3.14)〈A〉t=-12πi∫ℝ-i〈ψ0,(Kλ(ω)-z)-1[A⊗1S⊗1R]ψ0〉eitzdz,
where Kλ(ω)=L0(ω)+λI(ω), I is representing the interaction, and ω↦Kλ(ω) is a spectral deformation (translation) of Kλ. The latter is constructed as follows. There is a deformation transformation U(ω)=e-iωD, where D is the (explicit) self-adjoint generator of translations [10, 11, 30] transforming the operator Kλ as(3.15)Kλ(ω)=U(ω)KλU(ω)-1=L0+ωN+λI(ω).Here,N=N1⊗1+1⊗N1 is the total number operator of a product of two bosonic Fock spaces ℱ⊗ℱ (the Gelfand-Naimark-Segal Hilbert space of the reservoir), and where N1 is the usual number operator on ℱ. N has spectrum ℕ∪{0}, where 0 is a simple eigenvalue (with vacuum eigenvector ΩR,β=Ω⊗Ω). For real values of ω, U(ω) is a group of unitaries. The spectrum of Kλ(ω) depends on Imω and moves according to the value of Imω, whence the name “spectral deformation”. Even though U(ω) becomes unbounded for complex ω, the r.h.s. of  (3.15) is a well-defined closed operator on a dense domain, analytic in ω at zero. Analyticity is used in the derivation of  (3.14) and this is where the analyticity condition (A) after  (3.2) comes into play. The operator I(ω) is infinitesimally small with respect to the number operator N. Hence we use perturbation theory in λ to examine the spectrum of Kλ(ω).The point of the spectral deformation is that the (important part of the) spectrum ofKλ(ω) is much easier to analyze than that of Kλ, because the deformation uncovers the resonances of Kλ. We have (see Figure 1) (3.16)spec(K0(ω))={Ei-Ej}i,j=1,…,N⋃n≥1{ωn+ℝ},
because K0(ω)=L0+ωN, L0 and N commute, and the eigenvectors of L0=LS+LR are φi⊗φj⊗ΩR,β. Here, we have HSφj=Ejφj. The continuous spectrum is bounded away from the isolated eigenvalues by a gap of size Imω. For values of the coupling parameter λ small compared to Imω, we can follow the displacements of the eigenvalues by using analytic perturbation theory. (Note that for Imω=0, the eigenvalues are imbedded into the continuous spectrum, and analytic perturbation theory is not valid! The spectral deformation is indeed very useful!)Figure 1
Spectrum ofK0(ω).Theorem 3.2 (see [10] and Figure 2).
FixImω s.t. 0<Imω<ω′ (where ω′ is as in Condition (A)). There is a constant c0>0 s.t. if |λ|≤c0/β then, for all ω with Imω>7ω′/8, the spectrum of Kλ(ω) in the complex half-plane {Imz<ω′/2} is independent of ω and consists purely of the distinct eigenvalues
(3.17){ɛe(s):e∈spec(LS),s=1,…,ν(e)},
where 1≤ν(e)≤mult(e) counts the splitting of the eigenvalue e. Moreover,
(3.18)limλ→0|ɛe(s)(λ)-e|=0
for all s, and we have Imɛe(s)≥0. Also, the continuous spectrum of Kλ(ω) lies in the region {Imz≥3ω′/4}.Figure 2
Spectrum ofKλ(ω). Resonances ɛe(s) are uncovered.Next we separate the contributions to the path integral in  (3.14) coming from the singularities at the resonance energies and from the continuous spectrum. We deform the path of integration z=ℝ-i into the line z=ℝ+iω′/2, thereby picking up the residues of poles of the integrand at ɛe(s) (all e, s). Let 𝒞e(s) be a small circle around ɛe(s), not enclosing or touching any other spectrum of Kλ(ω). We introduce the generally nonorthogonal Riesz spectral projections(3.19)Qe(s)=Qe(s)(ω,λ)=-12πi∫𝒞e(s)(Kλ(ω)-z)-1dz.It follows from  (3.14) that(3.20)〈A〉t=∑e∑s=1ν(e)eitɛe(s)〈ψ0,Qe(s)[A⊗1S⊗1R]ψ0〉+O(λ2e-ω′t/2).
Note that the imaginary parts of all resonance energies ɛe(s) are smaller than ω′/2, so that the remainder term in  (3.20) is not only small in λ, but it also decays faster than all of the terms in the sum. (See also Figure 3.) We point out also that instead of deforming the path integration contour as explained before (3.19), we could choose z=ℝ+i[ω′-O(λ)], hence transforming the error term in  (3.20) into the one given in  (3.8).Figure 3
Contour deformation:∫ℝ-idz=∑e,s∫𝒞e(s)dz+∫ℝ+iω'/2dz.Finally, we notice that all terms in  (3.20) with ɛe(s)≠0 will vanish in the ergodic mean limit, so (3.21)〈〈A〉〉∞=limT→∞1T∫0T〈A〉tdt=∑s:ɛ0(s)=0〈ψ0,Q0(s)[A⊗1R⊗1R]ψ0〉.
We now see that the linear functionals  (2.13) are represented as(3.22)Rɛe(s)(A)=〈ψ0,Qe(s)[A⊗1S⊗1R]ψ0〉.
This concludes the outline of the proof of Theorem 3.1.



---

*Source: 169710-2010-05-06.xml*