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 general N-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.

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 an 𝑁-level quantum system S interacting with a heat bath R. The former is described by a Hilbert space 𝔥𝑠=ℂ𝑁 and a Hamiltonian𝐻S𝐸=diag1,…,𝐸𝑁.(2.1) The environment R is modelled by a bosonic thermal reservoir with Hamiltonian𝐻R=ℝ3ğ‘Žâˆ—(||𝑘||𝑘)ğ‘Ž(𝑘)d3𝑘,(2.2) acting on the reservoir Hilbert space 𝔥R, and where ğ‘Žâˆ—(𝑘) and ğ‘Ž(𝑘) are the usual bosonic creation and annihilation operators satisfying the canonical commutation relations [ğ‘Ž(𝑘),ğ‘Žâˆ—(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 factor 𝑓(𝑘), a square integrable function of 𝑘∈ℝ3 (momentum representation), the smoothed-out creation and annihilation operators are defined as ğ‘Žâˆ—âˆ«(𝑓)=ℝ3𝑓(𝑘)ğ‘Žâˆ—(𝑘)d3𝑘 and âˆ«ğ‘Ž(𝑓)=ℝ3𝑓(𝑘)ğ‘Ž(𝑘)d3𝑘, respectively, and the hermitian field operator is1𝜙(𝑓)=√2î€ºğ‘Žâˆ—î€».(𝑓)+ğ‘Ž(𝑓)(2.3) The total Hamiltonian, acting on 𝔥S⊗𝔥R, has the form𝐻=𝐻S⊗𝟙R+𝟙S⊗𝐻R+𝜆𝑣,(2.4) 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 𝜌0=𝜌0⊗𝜌R,𝛽,(2.5) where 𝜌0 is any state of S and 𝜌R,𝛽 is the equilibrium state of R at temperature 1/𝛽.

Let 𝐴 be an arbitrary observable of the system (an operator on the system Hilbert space 𝔥S) and set⟨𝐴⟩𝑡∶=TrS𝜌𝑡𝐴=TrS+R𝜌𝑡𝐴⊗𝟙R,(2.6) where 𝜌𝑡 is the density matrix of S+R at time 𝑡 and 𝜌𝑡=TrR𝜌𝑡(2.7) is the reduced density matrix of S. In our approach, the dynamics of the reduced density matrix 𝜌𝑡 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}𝑁𝑘=1 of 𝐻S, is given by𝜌𝑡𝑘𝑙=𝜑𝑘,e−i𝑡𝐻S𝜌0ei𝑡𝐻S𝜑𝑙=ei𝑡𝑒𝑙𝑘𝜌0𝑘𝑙,(2.8) 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” î€½ğ‘’âˆˆğ¸âˆ’ğ¸î…žâˆ¶ğ¸,ğ¸î…žî€·ğ»âˆˆspecS(2.9) in the exponent of (2.8) become complex, 𝑒↦𝜀𝑒. It can be shown generally that the resonance energies 𝜀𝑒 have nonnegative imaginary parts, Im𝜀𝑒≥0. If Im𝜀𝑒>0, then the corresponding dynamical process is irreversible.(2)The matrix elements do not evolve independently any more. Indeed, the effective energy of S 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 an 𝑁-Level System

Let 𝐴∈ℬ(𝔥S) be an observable of the system S. We show in [9, 10] that the ergodic averagesâŸ¨âŸ¨ğ´âŸ©âŸ©âˆžâˆ¶=limğ‘‡â†’âˆž1𝑇𝑇0⟨𝐴⟩𝑡d𝑡(2.10) exist, that is, ⟨𝐴⟩𝑡 converges in the ergodic sense as ğ‘¡â†’âˆž. Furthermore, we show that for any 𝑡≥0 and for any 0<ğœ”î…ž<2𝜋/𝛽,âŸ¨ğ´âŸ©ğ‘¡âˆ’âŸ¨âŸ¨ğ´âŸ©âŸ©âˆž=𝜀≠0ei𝑡𝜀𝑅𝜀(𝜆𝐴)+𝑂2e−[ğœ”î…žâˆ’ğ‘‚(𝜆)]𝑡,(2.11) where the complex numbers 𝜀 are the eigenvalues of a certain explicitly given operator 𝐾(𝜔′), lying in the strip {𝑧∈ℂ|0≤Im𝑧<𝜔′/2}. They have the expansions𝜀≡𝜀𝑒(𝑠)=𝑒+𝜆2𝛿𝑒(𝑠)𝜆+𝑂4,(2.12) 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 𝑅𝜀(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 expansion𝑅𝜀(𝐴)=(𝑚,𝑛)∈𝐼𝑒𝜘𝑚,𝑛𝐴𝑚,𝑛𝜆+𝑂2,(2.13) 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 𝜀=𝑒∈ℝ; see (2.12). Depending on the interaction, each resonance energy 𝜀 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 âŸ¨âŸ¨ğ´âŸ©âŸ©âˆž if and only if Im𝜀>0 for all 𝜀≠0. In this case, the convergence takes place on the time scale [Im𝜀]−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, Im𝜀=𝑂(𝜆2) while ğœ”î…žâˆ’ğ‘‚(𝜆)>ğœ”î…ž/2 independent of small values of 𝜆. However, this means that we are in the regime 𝜆2â‰ªğœ”î…ž<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) of S: 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 𝜀 and the functionals 𝑅𝜀 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 𝜌𝑡=e𝑡ℒ𝜌0, 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, ğ‘¡â†’âˆž, 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 𝜀𝑒(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 Im𝜀𝑒(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 𝜀𝑒(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

Consider S 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𝑣=ğ‘Žğ‘î‚¹ğ‘ğ‘âŠ—ğœ™(𝑔),(2.14) 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,𝜌0=121111.(2.15) In [10] we derive from representation (2.11) the following expressions for the dynamics of matrix elements, for all 𝑡≥0: 𝜌𝑡𝑚,𝑚=e−𝛽𝐸𝑚𝑍S,𝛽+(−1)𝑚2tanh𝛽Δ2ei𝑡𝜀0(𝜆)+𝑅𝑚,𝑚(𝜆,𝑡),𝑚=1,2,(2.16)𝜌𝑡1,2=12ei𝑡𝜀−Δ(𝜆)+𝑅1,2(𝜆,𝑡),(2.17) where the resonance energies 𝜀 are given by𝜀0(𝜆)=i𝜆2𝜋2|𝑐|2𝜆𝜉(Δ)+𝑂4,𝜀Δ(𝜆)=Δ+𝜆2i𝑅+2𝜆2𝜋2|𝑐|2𝜉(Δ)+(ğ‘âˆ’ğ‘Ž)2𝜆𝜉(0)+𝑂4,𝜀−Δ(𝜆)=−𝜀Δ(𝜆),(2.18) with𝜉(𝜂)=lim𝜖↓01𝜋ℝ3d3𝛽||𝑘||𝑘coth2||||𝑔(𝑘)2𝜖||𝑘||−𝜂2+𝜖2,1𝑅=2𝑏2âˆ’ğ‘Ž2𝑔,𝜔−1𝑔+12|𝑐|2P.V.ℝ×𝑆2𝑢2||||𝑔(|𝑢|,ğœŽ)2coth𝛽|𝑢|21.𝑢−Δ(2.19) 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, 𝑅𝑚,𝑛𝑝(𝜆,𝑡)=𝑛,ğ‘šî¬î¬âˆžâˆ’ğ›¿ğ‘š,𝑛e−𝛽𝐸𝑚𝑍S,𝛽+ğ‘…î…žğ‘š,𝑛(𝜆,𝑡),(2.20) where |𝑅′𝑚,𝑛(𝜆,𝑡)|=𝑂(𝜆2e−𝛾𝑡), with 𝛾=min{Im𝜀0,Im𝜀±Δ}. These relations show the following.(i)To second order in 𝜆, convergence of the populations to the equilibrium values (Gibbs law), and decoherence occur exponentially fast, with rates 𝜏𝑇=[Im𝜀0(𝜆)]−1 and 𝜏𝐷=[Im𝜀Δ(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)+(ğ‘âˆ’ğ‘Ž)2𝜉(0)]−1; compare for example (2.18).(ii)The final density matrix of the spin is not 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 limğ‘¡â†’âˆžğ‘…ğ‘š,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 of 𝑂(𝜆2) in the time-asymptotic limit) cannot be detected. Indeed in that approach one would conclude that S approaches its Gibbs state as ğ‘¡â†’âˆž.

2.2. Evolution of Reduced Dynamics of an 𝑁-Level System

In the sequel, we analyze in more detail the evolution of a qubit register of size 𝑁. The Hamiltonian is𝐻S=𝑁𝑖,𝑗=1𝐽𝑖𝑗𝑆𝑧𝑖𝑆𝑧𝑗+𝑁𝑗=1𝐵𝑗𝑆𝑧𝑗,(2.21) 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 is𝑆𝑧=100−1(2.22) and 𝑆𝑧𝑗 is the matrix 𝑆𝑧 acting on the 𝑗th spin only.

We consider a collective 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))𝜆1𝑣1+𝜆2𝑣2=𝜆1𝑁𝑗=1𝑆𝑧𝑗𝑔⊗𝜙1+𝜆2𝑁𝑗=1𝑆𝑥𝑗𝑔⊗𝜙2.(2.23) 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 𝑆𝑥𝑗 in (2.23), representing the Pauli matrix 𝑆𝑥=0110(2.24) acting on the 𝑗th spin. The total Hamiltonian takes the form (2.4) with 𝜆𝑣 replaced by (2.23). It is convenient to represent 𝜌𝑡 as a matrix in the energy basis, consisting of eigenvectors ğœ‘ğœŽ of 𝐻S. These are vectors in 𝔥S=ℂ2⊗⋯⊗ℂ2=ℂ2𝑁 indexed by spin configurationsğœŽ=î€½ğœŽ1,…,ğœŽğ‘î€¾âˆˆ{+1,−1}𝑁,ğœ‘ğœŽ=ğœ‘ğœŽ1âŠ—â‹¯âŠ—ğœ‘ğœŽğ‘,(2.25) where𝜑+=10,𝜑−=01,(2.26) so that𝐻Sğœ‘ğœŽî€·ğœŽ=ğ¸î€¸ğœ‘ğœŽî€·ğœŽwith𝐸=𝑁𝑖,𝑗=1ğ½ğ‘–ğ‘—ğœŽğ‘–ğœŽğ‘—+𝑁𝑗=1ğµğ‘—ğœŽğ‘—.(2.27) We denote the reduced density matrix elements asî€ºğœŒğ‘¡î€»ğœŽ,𝜏=î‚¬ğœ‘ğœŽ,𝜌𝑡𝜑𝜏.(2.28) The Bohr frequencies (2.9) are nowğ‘’î€·ğœŽ,ğœî€¸î€·ğœŽ=𝐸𝜏−𝐸=𝑁𝑖,𝑗=1ğ½ğ‘–ğ‘—î€·ğœŽğ‘–ğœŽğ‘—âˆ’ğœğ‘–ğœğ‘—î€¸+𝑁𝑗=1ğµğ‘—î€·ğœŽğ‘—âˆ’ğœğ‘—î€¸,(2.29) and they become complex resonance energies 𝜀𝑒=𝜀𝑒(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, î€ºğœŒğ‘¡î€»ğœŽ,ğœâˆ’î€ºî‚¬î‚¬ğœŒâˆžî€»ğœŽ,𝜏={𝑒∶𝜀𝑒≠0}eiğ‘¡ğœ€ğ‘’âŽ¡âŽ¢âŽ¢âŽ£î“ğœŽâ€²,ğœâ€²ğ‘¤ğœ€ğ‘’ğœŽ,𝜏;ğœŽâ€²,𝜏′𝜌0î€»ğœŽâ€²,𝜏′𝜆+𝑂21+𝜆22î€¸âŽ¤âŽ¥âŽ¥âŽ¦î‚€î€·ğœ†+𝑂21+𝜆22e−[𝜔′+𝑂(𝜆)]𝑡.(2.30) This result is obtained by specializing (2.11) to the specific system at hand and considering observables 𝐴=|ğœ‘ğœâŸ©âŸ¨ğœ‘ğœŽ|. In (2.30), we have in accordance with (2.10), ⟨⟨[ğœŒâˆž]ğœŽ,𝜏⟩⟩=limğ‘‡â†’âˆžâˆ«(1/𝑇)𝑇0[𝜌𝑡]ğœŽ,𝜏d𝑡. The coefficients 𝑤 are overlaps of resonance eigenstates which vanish unless 𝑒=−𝑒(ğœŽ,𝜏)=−𝑒(ğœŽî…ž,ğœî…ž) (see point (2) after (2.9)). They represent the dominant contribution to the functionals 𝑅𝜀 in (2.11); see also (2.13). The 𝜀𝑒 have the expansion𝜀𝑒≡𝜀𝑒(𝑠)=𝑒+𝛿𝑒(𝑠)𝜆+𝑂41+𝜆42,(2.31) 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) satisfy𝛿𝑒(𝑠)𝜆=𝑂21+𝜆22.(2.32) 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 [𝜌𝑡]ğœŽ,𝜏 associated to a fixed 𝑒=𝑒(ğœŽ,𝜏) 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 functions 𝑤 and the exponentials ei𝑡𝜀𝑒(𝑠). In the absence of interaction (𝜆1=𝜆2=0), all the 𝜀𝑒(𝑠)=𝑒 are real. As the interaction is switched on, the 𝜀𝑒(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 [𝜌𝑡]ğœŽ,𝜏 of a group 𝑒 approach their ergodic means if and only if all the nonzero 𝜀𝑒(𝑠) have strictly positive imaginary part. In this case, the convergence takes place on a time scale of the order 1/𝛾𝑒, where 𝛾𝑒=minIm𝜀𝑒(𝑠)∶𝑠=1,…,𝜈(𝑒)s.t.𝜀𝑒(𝑠)≠0(2.33) is the decay rate of the group associated to 𝑒. If an 𝜀𝑒(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 with 𝐽𝑖𝑗=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 𝑁𝑗=1𝐵𝑗𝑛𝑗=0⟺𝑛𝑗=0∀𝑗.(2.34) 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 by𝛾𝑒=𝜆21𝑦1(𝑒)+𝜆22𝑦2(𝑒)+𝑦12𝜆(𝑒),𝑒≠022𝑦0𝜆,𝑒=0+𝑂41+𝜆42.(2.35) 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(𝜆21+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 of 𝑦1(𝑒): 𝑦1(𝑒) vanishes if either 𝑒 is such that 𝑒0∑∶=𝑛𝑗=1(ğœŽğ‘—âˆ’ğœğ‘—)=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 of 𝑦2(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 of 𝑦12(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 (𝜆21). We point out that generically, S+R will reach a joint equilibrium as ğ‘¡â†’âˆž, 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 ğ‘¡â†’âˆž.(v)Properties of 𝑦0: 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 ğ‘‡â†’âˆž.

In terms of complexity analysis, it is important to discuss the dependence of 𝛾𝑒 on the register size 𝑁.(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 to 𝑒≠0) relative to the register size 𝑁, we assume the magnetic field to have a certain distribution denoted by ⟨⋅⟩. We show in [11] that ⟨𝑦1⟩=𝑦1∝𝑒20,⟨𝑦2⟩=ğ¶ğµğ”‡î€·ğœŽâˆ’ğœî€¸,⟨𝑦12⟩=𝑐𝐵𝜆1,𝜆2𝑁0(𝑒),(2.36) where 𝐶𝐵 and 𝑐𝐵=𝑐𝐵(𝜆1,𝜆2) are positive constants (independent of 𝑁), with 𝑐𝐵(𝜆1,𝜆2)=𝑂(𝜆21+𝜆22). Here, 𝑁0(𝑒) is the number of indices 𝑗 such that ğœŽğ‘—=𝜏𝑗 for each (ğœŽ,𝜏) s.t. 𝑒(ğœŽ,𝜏)=𝑒, and ğ”‡î€·ğœŽâˆ’ğœî€¸âˆ¶=𝑁𝑗=1||ğœŽğ‘—âˆ’ğœğ‘—||(2.37) is the Hamming distance between the spin configurations ğœŽ and 𝜏 (which depends on 𝑒 only).(iii)Consider 𝑒≠0. It follows from (2.35)–(2.37) that for purely energy conserving interactions (𝜆2=0), 𝛾𝑒∝𝜆21𝑒20=𝜆21[∑𝑁𝑗=1(ğœŽğ‘—âˆ’ğœğ‘—)]2, which can be as large as 𝑂(𝜆21𝑁2). On the other hand, for purely energy exchanging interactions (𝜆1=0), we have 𝛾𝑒∝𝜆22𝐷(ğœŽâˆ’ğœ), which cannot exceed 𝑂(𝜆22𝑁). If both interactions are acting, then we have the additional term ⟨𝑦12⟩, which is of order 𝑂((𝜆21+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 order 𝜆21𝑁2, while the fastest decay rate due to the energy exchange interaction alone is of the order 𝜆22𝑁. Moreover, the decay of the diagonal matrix elements is of order 𝜆21, that is, independent of 𝑁.

Remark 2.2.2 s. (1) For 𝜆2=0, the model can be solved explicitly [17], and one shows that the fastest decaying matrix elements have decay rate proportional to 𝜆21𝑁2. 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 𝜆21𝑁.
(2) As mentioned at the beginning of this section, we take the coupling constants 𝜆1, 𝜆2 so small that the resonances do not overlap. Consequently, 𝜆21𝑁2 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.

The 𝑁-level system is coupled to the reservoir (see also (2.1), (2.2)) through the operator𝑣=𝑅𝑟=1𝜆𝑟𝐺𝑟𝑔⊗𝜙𝑟,(3.1) 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 define𝑔𝑟,𝛽(𝑢,ğœŽ)∶=𝑢1−e−𝛽𝑢|𝑢|1/2𝑔𝑟(𝑢,ğœŽ)if𝑢≥0,−ei𝜒𝑔𝑟(−𝑢,ğœŽ)if𝑢<0,(3.2) where 𝑢∈ℝ and ğœŽâˆˆğ‘†2. The phase 𝜒 is a parameter which can be chosen appropriately as to satisfy the following condition.(A) The map 𝜔↦𝑔𝑟,𝛽(𝑢+𝜔,ğœŽ) has an analytic extension to a complex neighbourhood {|𝑧|<𝜔′} of the origin, as a map from ℂ to 𝐿2(ℝ3,d3𝑘).

Examples of 𝑔 satisfying (A) are given by 𝑔(𝑟,ğœŽ)=𝑟𝑝e−𝑟𝑚𝑔1(ğœŽ), where 𝑝=−1/2+𝑛, 𝑛=0,1,…, 𝑚=1,2, and 𝑔1(ğœŽ)=ei𝜙𝑔1(ğœŽ).

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 functions 𝑔(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 ğœ”î…ž<2𝜋/𝛽.

It is convenient to introduce the doubled Hilbert space ℋS=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 vectorΩS=1√𝑁𝑁𝑗=1𝜑𝑗⊗𝜑𝑗(3.3) viaℬ𝔥S∋𝐴⟼⟨ΩS,(𝐴⊗𝟙)ΩS⟩.(3.4) 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 𝐵ΩS=𝜓0.(3.5) (See also [10] for concrete examples.) We define the reference vector Ωref∶=ΩS⊗ΩR,𝛽(3.6) and set 𝜆=max𝑟=1,…,𝑅||𝜆𝑟||.(3.7)

Theorem 3.1 (Dynamical resonance theory [9–11]). Assume condition (A) with a fixed ğœ”î…ž satisfying 0<ğœ”î…ž<2𝜋/𝛽. There is a constant 𝑐0 s.t.; if 𝜆≤𝑐0/𝛽, then the limit âŸ¨âŸ¨ğ´âŸ©âŸ©âˆž, (2.10), exists for all observables 𝐴∈ℬ(𝔥S). Moreover, for all such 𝐴 and for all 𝑡≥0, we have âŸ¨ğ´âŸ©ğ‘¡âˆ’âŸ¨âŸ¨ğ´âŸ©âŸ©âˆž=𝑒,𝑠∶𝜀𝑒(𝑠)≠0𝜈(𝑒)𝑠=1ei𝑡𝜀𝑒(𝑠)𝐵∗𝜓0⊗ΩR,𝛽,𝑄𝑒(𝑠)𝐴⊗𝟙SΩref𝜆+𝑂2e−[𝜔′+𝑂(𝜆)]𝑡.(3.8) The 𝜀𝑒(𝑠) are given by (2.12), 1≤𝜈(𝑒)≤mult(𝑒) counts the splitting of the eigenvalue 𝑒 into distinct resonance energies 𝜀𝑒(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 the 𝑁-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.)

The 𝑄𝑒(𝑠) is the spectral (Riesz) projection of an operator 𝐾𝜆 associated with the eigenvalue 𝜀𝑒(𝑠); see (3.19) (In reality, we consider a spectral deformation 𝐾𝜆(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 𝜀𝑒 bifurcating out of 𝑒, as 𝜆≠0. In this case, the projection 𝑄𝑒≡𝑄𝑒(𝑠) has rank one, 𝑄𝑒=|𝜒𝑒⟩⟨𝜒𝑒|, where 𝜒𝑒 and 𝜒𝑒 are eigenvectors of 𝐾𝜆 and its adjoint, with eigenvalue 𝜀𝑒 and its complex conjugate, respectively, and ⟨𝜒𝑒,𝜒𝑒⟩=1. From perturbation theory, we obtain 𝜒𝑒=𝜒𝑒=𝜑𝑘⊗𝜑𝑙⊗ΩR,𝛽+𝑂(𝜆), where 𝐻S𝜑𝑗=𝐸𝑗𝜑𝑗 and 𝐸𝑘−𝐸𝑙=𝑒. The overlap in the sum of (3.8) becomes𝐵∗𝜓0⊗ΩR,𝛽,𝑄𝑒𝐴⊗𝟙SΩref=𝐵∗𝜓0⊗ΩR,𝛽,||𝜑𝑘⊗𝜑𝑙⊗ΩR,𝛽𝜑𝑘⊗𝜑𝑙⊗ΩR,𝛽||𝐴⊗𝟙SΩref𝜆+𝑂2=𝐵∗𝜓0,||𝜑𝑘⊗𝜑𝑙⟩⟨𝜑𝑘⊗𝜑𝑙||𝐴⊗𝟙SΩS𝜆+𝑂2.(3.9) The choice 𝐴=|𝜑𝑛⟩⟨𝜑𝑚| in (2.6) gives ⟨𝐴⟩𝑡=[𝜌𝑡]𝑚,𝑛, the reduced density matrix element. With this choice of 𝐴, the main term in (3.9) becomes (see also (3.3))𝐵∗𝜓0,||𝜑𝑘⊗𝜑𝑙⟩⟨𝜑𝑘⊗𝜑𝑙||𝐴⊗𝟙SΩS=1√𝑁𝛿𝑘𝑛𝛿𝑙𝑚⟨𝐵∗𝜓0,𝜑𝑛⊗𝜑𝑚⟩=𝛿𝑘𝑛𝛿𝑙𝑚𝐵∗𝜓0,||𝜑𝑛⟩⟨𝜑𝑚||⊗𝟙SΩS=𝛿𝑘𝑛𝛿𝑙𝑚𝜓0,||𝜑𝑛⟩⟨𝜑𝑚||⊗𝟙S𝐵ΩS=𝛿𝑘𝑛𝛿𝑙𝑚𝜌0𝑚𝑛.(3.10) 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, î€ºğœŒğ‘¡î€»ğ‘šğ‘›âˆ’î«î«î€ºğœŒâˆžî€»ğ‘šğ‘›={𝑒,𝑠∶𝜀𝑒(𝑠)≠0}ei𝑡𝜀𝑒(𝑠)𝛿𝑘𝑛𝛿𝑙𝑚𝜌0𝑚𝑛𝜆+𝑂2𝜆+𝑂2e−[𝜔′+𝑂(𝜆)]𝑡.(3.11) 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 Theorem 3.1

Consider any observable 𝐴∈𝐵(𝔥S). We have⟨𝐴⟩𝑡=TrS𝜌𝑡𝐴=TrS+R𝜌𝑡𝐴⊗𝟙R=𝜓0,ei𝑡𝐿𝜆𝐴⊗𝟙S⊗𝟙Re−i𝑡𝐿𝜆𝜓0.(3.12) 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) generator 𝐾𝜆 s.t. ei𝑡𝐿𝜆𝐴e−i𝑡𝐿𝜆=ei𝑡𝐾𝜆𝐴e−i𝑡𝐾𝜆𝐾forallobservables𝐴,𝑡≥0,and𝜆𝜓0=0.(3.13)

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]⟨𝐴⟩𝑡1=−2𝜋iℝ−i𝜓0,𝐾𝜆(𝜔)−𝑧−1𝐴⊗𝟙S⊗𝟙R𝜓0ei𝑡𝑧d𝑧,(3.14) 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 as𝐾𝜆(𝜔)=𝑈(𝜔)𝐾𝜆𝑈(𝜔)−1=𝐿0+𝜔𝑁+𝜆𝐼(𝜔).(3.15)

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 of 𝐾𝜆(𝜔) is much easier to analyze than that of 𝐾𝜆, because the deformation uncovers the resonances of 𝐾𝜆. We have (see Figure 1) 𝐾spec0=𝐸(𝜔)𝑖−𝐸𝑗𝑖,𝑗=1,…,𝑁𝑛≥1{𝜔𝑛+ℝ},(3.16) 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!)

Theorem 3.2 (see [10] and Figure 2). Fix Im𝜔 s.t. 0<Im𝜔<ğœ”î…ž (where ğœ”î…ž is as in Condition (A)). There is a constant 𝑐0>0 s.t. if |𝜆|≤𝑐0/𝛽 then, for all 𝜔 with I𝑚𝜔>7ğœ”î…ž/8, the spectrum of 𝐾𝜆(𝜔) in the complex half-plane {Im𝑧<ğœ”î…ž/2} is independent of 𝜔 and consists purely of the distinct eigenvalues 𝜀𝑒(𝑠)𝐿∶𝑒∈specS,𝑠=1,…,𝜈(𝑒),(3.17) where 1≤𝜈(𝑒)≤mult(𝑒) counts the splitting of the eigenvalue 𝑒. Moreover, lim𝜆→0||𝜀𝑒(𝑠)||(𝜆)−𝑒=0(3.18) for all 𝑠, and we have Im𝜀𝑒(𝑠)≥0. Also, the continuous spectrum of 𝐾𝜆(𝜔) lies in the region {Im𝑧≥3ğœ”î…ž/4}.

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+iğœ”î…ž/2, thereby picking up the residues of poles of the integrand at 𝜀𝑒(𝑠) (all 𝑒, 𝑠). Let ğ’žğ‘’(𝑠) be a small circle around 𝜀𝑒(𝑠), not enclosing or touching any other spectrum of 𝐾𝜆(𝜔). We introduce the generally nonorthogonal Riesz spectral projections𝑄𝑒(𝑠)=𝑄𝑒(𝑠)(1𝜔,𝜆)=−2𝜋iğ’žğ‘’(𝑠)𝐾𝜆(𝜔)−𝑧−1d𝑧.(3.19)

It follows from (3.14) that⟨𝐴⟩𝑡=𝑒𝜈(𝑒)𝑠=1ei𝑡𝜀𝑒(𝑠)𝜓0,𝑄𝑒(𝑠)𝐴⊗𝟙S⊗𝟙R𝜓0𝜆+𝑂2e−𝜔′𝑡/2.(3.20) Note that the imaginary parts of all resonance energies 𝜀𝑒(𝑠) are smaller than ğœ”î…ž/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[ğœ”î…žâˆ’ğ‘‚(𝜆)], hence transforming the error term in (3.20) into the one given in (3.8).

Finally, we notice that all terms in (3.20) with 𝜀𝑒(𝑠)≠0 will vanish in the ergodic mean limit, so âŸ¨âŸ¨ğ´âŸ©âŸ©âˆž=limğ‘‡â†’âˆž1𝑇𝑇0⟨𝐴⟩𝑡d𝑡=𝑠∶𝜀0(𝑠)=0𝜓0,𝑄0(𝑠)𝐴⊗𝟙R⊗𝟙R𝜓0.(3.21) We now see that the linear functionals (2.13) are represented as𝑅𝜀𝑒(𝑠)𝜓(𝐴)=0,𝑄𝑒(𝑠)𝐴⊗𝟙S⊗𝟙R𝜓0.(3.22) This concludes the outline of the proof of Theorem 3.1.

Acknowledgments

This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract no. DE-AC52-06NA25396 and by Lawrence Livermore National Laboratory under Contract no. DE-AC52-07NA27344. This research was funded by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA). All statements of fact, opinion or conclusions contained herein are those of the authors and should not be constructed as representing the official views or polices of IARPA, the ODNI, or the U.S. Government. I. M. Sigal also acknowledges a support by NSERC under Grant NA 7901. M. Merkli also acknowledges a support by NSERC under Grant 205247.