Advances in Mathematical Physics

Advances in Mathematical Physics / 2010 / Article
Special Issue

Quantum Information and Entanglement

View this Special Issue

Review Article | Open Access

Volume 2010 |Article ID 169710 |

M. Merkli, G. P. Berman, I. M. Sigal, "Resonant Perturbation Theory of Decoherence and Relaxation of Quantum Bits", Advances in Mathematical Physics, vol. 2010, Article ID 169710, 20 pages, 2010.

Resonant Perturbation Theory of Decoherence and Relaxation of Quantum Bits

Academic Editor: Shao-Ming Fei
Received31 Aug 2009
Accepted10 Feb 2010
Published06 May 2010


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 𝑑β‰₯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: β„πœ”>>π‘˜B𝑇, where 𝑇 is the temperature and πœ” is the qubit transition frequency. This condition is widely used in superconducting quantum computation and quantum measurement, when π‘‡βˆΌβ€‰β€‰10–20 mK and β„πœ”βˆΌβ€‰β€‰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 πœ‡m 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 𝑑β‰₯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 π”₯𝑠=ℂ𝑁 and a Hamiltonian𝐻S𝐸=diag1,…,𝐸𝑁.(2.1) The environment R is modelled by a bosonic thermal reservoir with Hamiltonian𝐻R=ξ€œβ„3π‘Žβˆ—(||π‘˜||π‘˜)π‘Ž(π‘˜)d3π‘˜,(2.2) acting on the reservoir Hilbert space π”₯R, and where π‘Žβˆ—(π‘˜) and π‘Ž(π‘˜) are the usual bosonic creation and annihilation operators satisfying the canonical commutation relations [π‘Ž(π‘˜),π‘Žβˆ—(𝑙)]=𝛿(π‘˜βˆ’π‘™). 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(π‘˜), a square integrable function of π‘˜βˆˆβ„3 (momentum representation), the smoothed-out creation and annihilation operators are defined as π‘Žβˆ—βˆ«(𝑓)=ℝ3𝑓(π‘˜)π‘Žβˆ—(π‘˜)d3π‘˜ and βˆ«π‘Ž(𝑓)=ℝ3𝑓(π‘˜)π‘Ž(π‘˜)d3π‘˜, respectively, and the hermitian field operator is1πœ™(𝑓)=√2ξ€Ίπ‘Žβˆ—ξ€».(𝑓)+π‘Ž(𝑓)(2.3) The total Hamiltonian, acting on π”₯SβŠ—π”₯R, has the form𝐻=𝐻SβŠ—πŸ™R+πŸ™SβŠ—π»R+πœ†π‘£,(2.4) where πœ† 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=𝜌0βŠ—πœŒR,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 π”₯S) and setβŸ¨π΄βŸ©π‘‘βˆΆ=TrSξ€·πœŒπ‘‘π΄ξ€Έ=TrS+Rξ€·πœŒπ‘‘ξ€·π΄βŠ—πŸ™Rξ€Έξ€Έ,(2.6) where πœŒπ‘‘ is the density matrix of S+R at time 𝑑 and πœŒπ‘‘=TrRπœŒπ‘‘(2.7) is the reduced density matrix of S. In our approach, the dynamics of the reduced density matrix πœŒπ‘‘ is expressed in terms of the resonance structure of the system. Under the noninteracting dynamics (πœ†=0), the evolution of the reduced density matrix elements of S, expressed in the energy basis {πœ‘π‘˜}π‘π‘˜=1 of 𝐻S, is given byξ€ΊπœŒπ‘‘ξ€»π‘˜π‘™=ξ«πœ‘π‘˜,eβˆ’i𝑑𝐻S𝜌0ei𝑑𝐻Sπœ‘π‘™ξ¬=eiπ‘‘π‘’π‘™π‘˜ξ€ΊπœŒ0ξ€»π‘˜π‘™,(2.8) where π‘’π‘™π‘˜=πΈπ‘™βˆ’πΈπ‘˜. As the interaction with the reservoir is turned on, the dynamics (2.8) undergoes two qualitative changes.(1)The β€œBohr frequencies” ξ€½π‘’βˆˆπΈβˆ’πΈξ…žβˆΆπΈ,πΈξ…žξ€·π»βˆˆspecSξ€Έξ€Ύ(2.9) in the exponent of (2.8) become complex, π‘’β†¦πœ€π‘’. It can be shown generally that the resonance energies πœ€π‘’ have nonnegative imaginary parts, Imπœ€π‘’β‰₯0. If Imπœ€π‘’>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 (π‘š,𝑛) belonging to a fixed energy difference πΈπ‘šβˆ’πΈπ‘› 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 π΄βˆˆβ„¬(π”₯S) be an observable of the system S. We show in [9, 10] that the ergodic averages⟨⟨𝐴⟩⟩∞∢=limπ‘‡β†’βˆž1π‘‡ξ€œπ‘‡0βŸ¨π΄βŸ©π‘‘d𝑑(2.10) exist, that is, βŸ¨π΄βŸ©π‘‘ converges in the ergodic sense as π‘‘β†’βˆž. Furthermore, we show that for any 𝑑β‰₯0 and for any 0<πœ”ξ…ž<2πœ‹/𝛽,βŸ¨π΄βŸ©π‘‘βˆ’βŸ¨βŸ¨π΄βŸ©βŸ©βˆž=ξ“πœ€β‰ 0eiπ‘‘πœ€π‘…πœ€(ξ€·πœ†π΄)+𝑂2eβˆ’[πœ”ξ…žβˆ’π‘‚(πœ†)]𝑑,(2.11) where the complex numbers πœ€ are the eigenvalues of a certain explicitly given operator 𝐾(πœ”β€²), lying in the strip {π‘§βˆˆβ„‚|0≀Im𝑧<πœ”β€²/2}. They have the expansionsπœ€β‰‘πœ€π‘’(𝑠)=𝑒+πœ†2𝛿𝑒(𝑠)ξ€·πœ†+𝑂4ξ€Έ,(2.12) where π‘’βˆˆspec(𝐻SβŠ—πŸ™Sβˆ’πŸ™SβŠ—π»S)=spec(𝐻S)βˆ’spec(𝐻S) and 𝛿𝑒(𝑠) are the eigenvalues of a matrix Λ𝑒, called a level-shift operator, acting on the eigenspace of 𝐻SβŠ—πŸ™Sβˆ’πŸ™SβŠ—π»S corresponding to the eigenvalue 𝑒 (which is a subspace of π”₯SβŠ—π”₯S). The π‘…πœ€(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 expansionπ‘…πœ€(𝐴)=(π‘š,𝑛)βˆˆπΌπ‘’πœ˜π‘š,π‘›π΄π‘š,π‘›ξ€·πœ†+𝑂2ξ€Έ,(2.13) where 𝐼𝑒 is the collection of all pairs of indices such that 𝑒=πΈπ‘šβˆ’πΈπ‘›, with πΈπ‘˜ being the eigenvalues of 𝐻S. Here, π΄π‘š,𝑛 is the (π‘š,𝑛)-matrix element of the observable 𝐴 in the energy basis of 𝐻S, and πœ˜π‘š,𝑛 are coefficients depending on the initial state of the system (and on 𝑒, but not on 𝐴 nor on πœ†).

2.1.1. Discussion
(i)In the absence of interaction (πœ†=0), we have πœ€=π‘’βˆˆβ„; see (2.12). Depending on the interaction, each resonance energy πœ€ may migrate into the upper complex plane, or it may stay on the real axis, as πœ†β‰ 0.(ii)The averages βŸ¨π΄βŸ©π‘‘ approach their ergodic means ⟨⟨𝐴⟩⟩∞ if and only if Imπœ€>0 for all πœ€β‰ 0. In this case, the convergence takes place on the time scale [Imπœ€]βˆ’1. Otherwise; βŸ¨π΄βŸ©π‘‘ oscillates. A sufficient condition for decay is that Im𝛿𝑒(𝑠)>0 (and πœ† small, see (2.12)).(iii)The error term in (2.11) is small in πœ†, uniformly in 𝑑β‰₯0, and it decays in time quicker than any of the main terms in the sum on the r.h.s.: indeed, Imπœ€=𝑂(πœ†2) while πœ”ξ…žβˆ’π‘‚(πœ†)>πœ”ξ…ž/2 independent of small values of πœ†. However, this means that we are in the regime πœ†2β‰ͺπœ”ξ…ž<2πœ‹/𝛽 (see before (2.11)), which implies that πœ†2 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 πΈπ‘šβˆ’πΈπ‘› 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]β‰ 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 (β€œphase 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 πœ€ and the functionals π‘…πœ€ 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 πœ†, 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 πœŒπ‘‘=eπ‘‘β„’πœŒ0, where β„’ 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(πœ†2). 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 πœ†β†’0, π‘‘β†’βˆž, with πœ†2𝑑 fixed. One of the strengths of our approach is that we do not impose any relation between πœ† and 𝑑, and our results are valid for all times 𝑑β‰₯0, provided πœ† is small. It has been observed [25, 26] that for certain systems of the type S+R, the second-order contribution of the exponents πœ€π‘’(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 Imπœ€π‘’(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 πœ€π‘’(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 Ξ”=𝐸2βˆ’πΈ1>0. S is coupled to the heat bath R via the operator𝑣=π‘Žπ‘ξ‚Ήπ‘π‘βŠ—πœ™(𝑔),(2.14) where πœ™(𝑔) is the Bose field operator (2.3), smeared out with a coupling function (form factor) 𝑔(π‘˜), π‘˜βˆˆβ„3, and the 2Γ—2 coupling matrix (representing the coupling operator in the energy eigenbasis) is hermitian. The operator (2.14)β€”or a sum of such terms, for which our technique works equally wellβ€”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=12ξ‚Έξ‚Ή1111.(2.15) In [10] we derive from representation (2.11) the following expressions for the dynamics of matrix elements, for all 𝑑β‰₯0: ξ€ΊπœŒπ‘‘ξ€»π‘š,π‘š=eβˆ’π›½πΈπ‘šπ‘S,𝛽+(βˆ’1)π‘š2ξ‚΅tanh𝛽Δ2ξ‚Άeiπ‘‘πœ€0(πœ†)+π‘…π‘š,π‘š(πœ†,𝑑),π‘š=1,2,(2.16)ξ€ΊπœŒπ‘‘ξ€»1,2=12eiπ‘‘πœ€βˆ’Ξ”(πœ†)+𝑅1,2(πœ†,𝑑),(2.17) where the resonance energies πœ€ are given byπœ€0(πœ†)=iπœ†2πœ‹2|𝑐|2ξ€·πœ†πœ‰(Ξ”)+𝑂4ξ€Έ,πœ€Ξ”(πœ†)=Ξ”+πœ†2i𝑅+2πœ†2πœ‹2ξ€Ί|𝑐|2πœ‰(Ξ”)+(π‘βˆ’π‘Ž)2ξ€»ξ€·πœ†πœ‰(0)+𝑂4ξ€Έ,πœ€βˆ’Ξ”(πœ†)=βˆ’πœ€Ξ”(πœ†),(2.18) withπœ‰(πœ‚)=limπœ–β†“01πœ‹ξ€œβ„3d3𝛽||π‘˜||π‘˜coth2ξ‚Ά||||𝑔(π‘˜)2πœ–ξ€·||π‘˜||ξ€Έβˆ’πœ‚2+πœ–2,1𝑅=2𝑏2βˆ’π‘Ž2𝑔,πœ”βˆ’1𝑔+12|𝑐|2ξ€œP.V.ℝ×𝑆2𝑒2||||𝑔(|𝑒|,𝜎)2ξ‚΅coth𝛽|𝑒|2ξ‚Ά1.π‘’βˆ’Ξ”(2.19) The remainder terms in (2.17), (2.17) satisfy |π‘…π‘š,𝑛(πœ†,𝑑)|β‰€πΆπœ†2, uniformly in 𝑑β‰₯0, and they can be decomposed into a sum of a constant and a decaying part, π‘…π‘š,𝑛𝑝(πœ†,𝑑)=𝑛,π‘šξ¬ξ¬βˆžβˆ’π›Ώπ‘š,𝑛eβˆ’π›½πΈπ‘šπ‘S,𝛽+π‘…ξ…žπ‘š,𝑛(πœ†,𝑑),(2.20) where |π‘…β€²π‘š,𝑛(πœ†,𝑑)|=𝑂(πœ†2eβˆ’π›Ύπ‘‘), with 𝛾=min{Imπœ€0,Imπœ€Β±Ξ”}. These relations show the following.(i)To second order in πœ†, convergence of the populations to the equilibrium values (Gibbs law), and decoherence occur exponentially fast, with rates πœπ‘‡=[Imπœ€0(πœ†)]βˆ’1 and 𝜏𝐷=[Imπœ€Ξ”(πœ†)]βˆ’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 πœ†βˆ’2[|𝑐|2πœ‰(Ξ”)+(π‘βˆ’π‘Ž)2πœ‰(0)]βˆ’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 limπ‘‘β†’βˆžπ‘…π‘š,𝑛(πœ†,𝑑)=𝑂(πœ†2). 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(πœ†2) 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(πœ†2) in the time-asymptotic limit) cannot be detected. Indeed in that approach one would conclude that S approaches its Gibbs state as π‘‘β†’βˆž.

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=1𝐡𝑗𝑆𝑧𝑗,(2.21) where 𝐽𝑖𝑗 are pair interaction constants and 𝐡𝑗 is the value of a magnetic field at the location of spin 𝑗. The Pauli spin operator is𝑆𝑧=ξ‚Έξ‚Ή100βˆ’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))πœ†1𝑣1+πœ†2𝑣2=πœ†1𝑁𝑗=1π‘†π‘§π‘—ξ€·π‘”βŠ—πœ™1ξ€Έ+πœ†2𝑁𝑗=1𝑆π‘₯π‘—ξ€·π‘”βŠ—πœ™2ξ€Έ.(2.23) Here 𝑔1 and 𝑔2 are form factors and the coupling constants πœ†1 and πœ†2 measure the strengths of the energy conserving (position-position) coupling, and the energy exchange (spin flip) coupling, respectively. Spin-flips are implemented by the 𝑆π‘₯𝑗 in (2.23), representing the Pauli matrix 𝑆π‘₯=ξ‚Έξ‚Ή0110(2.24) acting on the 𝑗th spin. The total Hamiltonian takes the form (2.4) with πœ†π‘£ replaced by (2.23). It is convenient to represent πœŒπ‘‘ as a matrix in the energy basis, consisting of eigenvectors πœ‘πœŽ of 𝐻S. These are vectors in π”₯S=β„‚2βŠ—β‹―βŠ—β„‚2=β„‚2𝑁 indexed by spin configurations𝜎=ξ€½πœŽ1,…,πœŽπ‘ξ€Ύβˆˆ{+1,βˆ’1}𝑁,πœ‘πœŽ=πœ‘πœŽ1βŠ—β‹―βŠ—πœ‘πœŽπ‘,(2.25) whereπœ‘+=ξ‚Έ10ξ‚Ή,πœ‘βˆ’=ξ‚Έ01ξ‚Ή,(2.26) so that𝐻Sπœ‘πœŽξ€·πœŽ=πΈξ€Έπœ‘πœŽξ€·πœŽwith𝐸=𝑁𝑖,𝑗=1π½π‘–π‘—πœŽπ‘–πœŽπ‘—+𝑁𝑗=1π΅π‘—πœŽπ‘—.(2.27) We denote the reduced density matrix elements asξ€ΊπœŒπ‘‘ξ€»πœŽ,𝜏=ξ‚¬πœ‘πœŽ,πœŒπ‘‘πœ‘πœξ‚­.(2.28) The Bohr frequencies (2.9) are nowπ‘’ξ€·πœŽ,πœξ€Έξ€·πœŽ=πΈξ€Έξ€·πœβˆ’πΈξ€Έ=𝑁𝑖,𝑗=1π½π‘–π‘—ξ€·πœŽπ‘–πœŽπ‘—βˆ’πœπ‘–πœπ‘—ξ€Έ+𝑁𝑗=1π΅π‘—ξ€·πœŽπ‘—βˆ’πœπ‘—ξ€Έ,(2.29) and they become complex resonance energies πœ€π‘’=πœ€π‘’(πœ†1,πœ†2)βˆˆβ„‚ under perturbation.

Assumption of Nonoverlapping Resonances
The Bohr frequencies (2.29) represent β€œunperturbed” energy levels and we follow their motion under perturbation (πœ†1,πœ†2). 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 𝑑β‰₯0, ξ€ΊπœŒπ‘‘ξ€»πœŽ,πœβˆ’ξ€Ίξ‚¬ξ‚¬πœŒβˆžξ€»πœŽ,𝜏={π‘’βˆΆπœ€π‘’β‰ 0}eiπ‘‘πœ€π‘’βŽ‘βŽ’βŽ’βŽ£ξ“πœŽβ€²,πœβ€²π‘€πœ€π‘’πœŽ,𝜏;πœŽβ€²,πœβ€²ξ€ΊπœŒ0ξ€»πœŽβ€²,πœβ€²ξ€·πœ†+𝑂21+πœ†22ξ€ΈβŽ€βŽ₯βŽ₯βŽ¦ξ‚€ξ€·πœ†+𝑂21+πœ†22ξ€Έeβˆ’[πœ”β€²+𝑂(πœ†)]𝑑.(2.30) This result is obtained by specializing (2.11) to the specific system at hand and considering observables 𝐴=|πœ‘πœβŸ©βŸ¨πœ‘πœŽ|. In (2.30), we have in accordance with (2.10), ⟨⟨[𝜌∞]𝜎,𝜏⟩⟩=limπ‘‡β†’βˆžβˆ«(1/𝑇)𝑇0[πœŒπ‘‘]𝜎,𝜏d𝑑. The coefficients 𝑀 are overlaps of resonance eigenstates which vanish unless 𝑒=βˆ’π‘’(𝜎,𝜏)=βˆ’π‘’(πœŽξ…ž,πœξ…ž) (see point (2) after (2.9)). They represent the dominant contribution to the functionals π‘…πœ€ in (2.11); see also (2.13). The πœ€π‘’ have the expansionπœ€π‘’β‰‘πœ€π‘’(𝑠)=𝑒+𝛿𝑒(𝑠)ξ€·πœ†+𝑂41+πœ†42ξ€Έ,(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)ξ€·πœ†=𝑂21+πœ†22ξ€Έ.(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)β‰₯0.

2.2.1. Discussion
(i)To lowest order in the perturbation, the group of reduced density matrix elements [πœŒπ‘‘]𝜎,𝜏 associated to a fixed 𝑒=𝑒(𝜎,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 𝑀 and the exponentials eiπ‘‘πœ€π‘’(𝑠). In the absence of interaction (πœ†1=πœ†2=0), all the πœ€π‘’(𝑠)=𝑒 are real. As the interaction is switched on, the πœ€π‘’(𝑠) typically migrate into the upper complex plane, but they may stay on the real line (due to some symmetry or due to an β€œinefficient coupling”).(iii)The matrix elements [πœŒπ‘‘]𝜎,𝜏 of a group 𝑒 approach their ergodic means if and only if all the nonzero πœ€π‘’(𝑠) have strictly positive imaginary part. In this case, the convergence takes place on a time scale of the order 1/𝛾𝑒, where 𝛾𝑒=minImπœ€π‘’(𝑠)βˆΆπ‘ =1,…,𝜈(𝑒)s.t.πœ€π‘’(𝑠)≠0(2.33) is the decay rate of the group associated to 𝑒. If an πœ€π‘’(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 πœ†1, πœ†2 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 π‘›π‘—βˆˆ{0,Β±1,Β±2}, 𝑗=1,…,𝑁, we have 𝑁𝑗=1𝐡𝑗𝑛𝑗=0βŸΊπ‘›π‘—=0βˆ€π‘—.(2.34) Condition (2.34) is satisfied generically in the sense that it does not hold only for very special choices of 𝐡𝑗 (one such special choice is 𝐡𝑗=constant). For instance, if the 𝐡𝑗 are chosen to be independent, and uniformly random from an interval [𝐡min,𝐡max], then (2.34) is satisfied with probability one. We show in [11, Theorem 2.3], that the decoherence rates (2.33) are given by𝛾𝑒=ξƒ―πœ†21𝑦1(𝑒)+πœ†22𝑦2(𝑒)+𝑦12πœ†(𝑒),𝑒≠022𝑦0ξƒ°ξ€·πœ†,𝑒=0+𝑂41+πœ†42ξ€Έ.(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(πœ†21+πœ†22). 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 𝑒0βˆ‘βˆΆ=𝑛𝑗=1(πœŽπ‘—βˆ’πœπ‘—)=0 or the infrared behaviour of the coupling function 𝑔1 is too regular (in three dimensions 𝑔1∝|π‘˜|𝑝 with 𝑝>βˆ’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(2𝐡𝑗,Ξ£)β‰ 0 for all 𝐡𝑗 (form factor 𝑔2(π‘˜)=𝑔2(|π‘˜|,Ξ£) in spherical coordinates). For low temperatures, 𝑇, 𝑦2(𝑒)βˆπ‘‡, for high temperatures 𝑦2(𝑒) approaches a constant.(iii)Properties of 𝑦12(𝑒): if either of πœ†1, πœ†2 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 πœ†2β‰ 0 and 𝑔2(2𝐡𝑗,Ξ£)β‰ 0 for all 𝑗, and provided πœ†1,πœ†2 are small enough (so that the remainder term in (2.35) is small). Note that if πœ†2=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 (πœ†21). We point out that generically, S+R will reach a joint equilibrium as π‘‘β†’βˆž, 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 π‘‘β†’βˆž.(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(2𝐡𝑗,Ξ£)β‰ 0 for all 𝑗, then 𝑦0>0 (this is sometimes called the β€œFermi Golden Rule Condition”). For small temperatures 𝑇, 𝑦0βˆπ‘‡, while 𝑦0 approaches a finite limit as π‘‡β†’βˆž.

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 βŸ¨β‹…βŸ©. We show in [11] that βŸ¨π‘¦1⟩=𝑦1βˆπ‘’20,βŸ¨π‘¦2⟩=πΆπ΅π”‡ξ€·πœŽβˆ’πœξ€Έ,βŸ¨π‘¦12⟩=π‘π΅ξ€·πœ†1,πœ†2𝑁0(𝑒),(2.36) where 𝐢𝐡 and 𝑐𝐡=𝑐𝐡(πœ†1,πœ†2) are positive constants (independent of 𝑁), with 𝑐𝐡(πœ†1,πœ†2)=𝑂(πœ†21+πœ†22). Here, 𝑁0(𝑒) is the number of indices 𝑗 such that πœŽπ‘—=πœπ‘— for each (𝜎,𝜏) s.t. 𝑒(𝜎,𝜏)=𝑒, and π”‡ξ€·πœŽβˆ’πœξ€ΈβˆΆ=𝑁𝑗=1||πœŽπ‘—βˆ’πœπ‘—||(2.37) is the Hamming distance between the spin configurations 𝜎 and 𝜏 (which depends on 𝑒 only).(iii)Consider 𝑒≠0. It follows from (2.35)–(2.37) that for purely energy conserving interactions (πœ†2=0), π›Ύπ‘’βˆπœ†21𝑒20=πœ†21[βˆ‘π‘π‘—=1(πœŽπ‘—βˆ’πœπ‘—)]2, which can be as large as 𝑂(πœ†21𝑁2). On the other hand, for purely energy exchanging interactions (πœ†1=0), we have π›Ύπ‘’βˆπœ†22𝐷(πœŽβˆ’πœ), which cannot exceed 𝑂(πœ†22𝑁). If both interactions are acting, then we have the additional term βŸ¨π‘¦12⟩, which is of order 𝑂((πœ†21+πœ†22)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 πœ†21𝑁2, while the fastest decay rate due to the energy exchange interaction alone is of the order πœ†22𝑁. Moreover, the decay of the diagonal matrix elements is of order πœ†21, that is, independent of 𝑁.

Remark 2.2.2 s. (1) For πœ†2=0, the model can be solved explicitly [17], and one shows that the fastest decaying matrix elements have decay rate proportional to πœ†21𝑁2. 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 πœ†21𝑁.
(2) As mentioned at the beginning of this section, we take the coupling constants πœ†1, πœ†2 so small that the resonances do not overlap. Consequently, πœ†21𝑁2 and πœ†22𝑁 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 π›Ύπ‘’βˆΌΞ”. 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=π‘…ξ“π‘Ÿ=1πœ†π‘ŸπΊπ‘Ÿξ€·π‘”βŠ—πœ™π‘Ÿξ€Έ,(3.1) where each πΊπ‘Ÿ is a hermitian 𝑁×𝑁 matrix, the π‘”π‘Ÿ(π‘˜) are form factors, and the πœ†π‘Ÿβˆˆβ„ are coupling constants. Fix any phase πœ’βˆˆβ„ and defineπ‘”π‘Ÿ,𝛽(𝑒,𝜎)∢=𝑒1βˆ’eβˆ’π›½π‘’|𝑒|1/2ξƒ―π‘”π‘Ÿ(𝑒,𝜎)if𝑒β‰₯0,βˆ’eiπœ’π‘”π‘Ÿ(βˆ’π‘’,𝜎)if𝑒<0,(3.2) where π‘’βˆˆβ„ and πœŽβˆˆπ‘†2. The phase πœ’ is a parameter which can be chosen appropriately as to satisfy the following condition.(A) The map πœ”β†¦π‘”π‘Ÿ,𝛽(𝑒+πœ”,𝜎) has an analytic extension to a complex neighbourhood {|𝑧|<πœ”β€²} of the origin, as a map from β„‚ to 𝐿2(ℝ3,d3π‘˜).

Examples of 𝑔 satisfying (A) are given by 𝑔(π‘Ÿ,𝜎)=π‘Ÿπ‘eβˆ’π‘Ÿπ‘šπ‘”1(𝜎), where 𝑝=βˆ’1/2+𝑛, 𝑛=0,1,…, π‘š=1,2, and 𝑔1(𝜎)=eiπœ™π‘”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 functions 𝑔(π‘˜), which have to behave appropriately in the infrared regime so that the parts of (3.2) fit nicely together at 𝑒=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 πœ”ξ…ž<2πœ‹/𝛽.

It is convenient to introduce the doubled Hilbert space β„‹S=π”₯SβŠ—π”₯S, whose normalized vectors accommodate any state on the system S (pure or mixed). The trace state, or infinite temperature state, is represented by the vectorΞ©S=1βˆšπ‘π‘ξ“π‘—=1πœ‘π‘—βŠ—πœ‘π‘—(3.3) viaℬπ”₯Sξ€Έβˆ‹π΄βŸΌβŸ¨Ξ©S,(π΄βŠ—πŸ™)Ξ©S⟩.(3.4) Here πœ‘π‘— are the orthonormal eigenvectors of 𝐻S. This is just the Gelfand-Naimark-Segal construction for the trace state. Similarly, let β„‹R and Ξ©R,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 β„‹R=β„±βŠ—β„±, where β„± is the bosonic Fock space over the one-particle space 𝐿2(ℝ3,d3π‘˜) and Ξ©R,𝛽=Ξ©βŠ—Ξ©, with Ξ© being the Fock vacuum of β„± (see also [10, 11] for more detail). Let πœ“0βŠ—Ξ©R,𝛽 be the vector in β„‹SβŠ—β„‹R representing the density matrix at time 𝑑=0. It is not difficult to construct the unique operator in π΅βˆˆπŸ™SβŠ—π”₯S satisfying 𝐡ΩS=πœ“0.(3.5) (See also [10] for concrete examples.) We define the reference vector Ξ©ref∢=Ξ©SβŠ—Ξ©R,𝛽(3.6) and set πœ†=maxπ‘Ÿ=1,…,𝑅||πœ†π‘Ÿ||.(3.7)

Theorem 3.1 (Dynamical resonance theory [9–11]). Assume condition (A) with a fixed πœ”ξ…ž satisfying 0<πœ”ξ…ž<2πœ‹/𝛽. There is a constant 𝑐0 s.t.; if πœ†β‰€π‘0/𝛽, then the limit ⟨⟨𝐴⟩⟩∞, (2.10), exists for all observables π΄βˆˆβ„¬(π”₯S). Moreover, for all such 𝐴 and for all 𝑑β‰₯0, we have βŸ¨π΄βŸ©π‘‘βˆ’βŸ¨βŸ¨π΄βŸ©βŸ©βˆž=𝑒,π‘ βˆΆπœ€π‘’(𝑠)β‰ 0𝜈(𝑒)𝑠=1eiπ‘‘πœ€π‘’(𝑠)ξ‚¬ξ€·π΅βˆ—πœ“0ξ€ΈβŠ—Ξ©R,𝛽,𝑄𝑒(𝑠)ξ€·π΄βŠ—πŸ™Sξ€ΈΞ©refξ‚­ξ‚€πœ†+𝑂2eβˆ’[πœ”β€²+𝑂(πœ†)]𝑑.(3.8) The πœ€π‘’(𝑠) are given by (2.12), 1β‰€πœˆ(𝑒)≀mult(𝑒) counts the splitting of the eigenvalue 𝑒 into distinct resonance energies πœ€π‘’(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 𝑀) from (3.8) by analyzing the projections 𝑄𝑒(𝑠) in more detail. Let us explain how to link the overlap ⟨(π΅βˆ—πœ“0)βŠ—Ξ©R,𝛽,𝑄𝑒(𝑠)(π΄βŠ—πŸ™S)Ξ©ref⟩ to its initial value for a nondegenerate Bohr energy 𝑒, and where 𝐴=|πœ‘π‘›βŸ©βŸ¨πœ‘π‘š|. (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 πΎπœ† associated with the eigenvalue πœ€π‘’(𝑠); see (3.19) (In reality, we consider a spectral deformation πΎπœ†(πœ”), where πœ” is a complex parameter. This is a technical trick to perform our analysis. Physical quantities do not depend on πœ” 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 πœ€π‘’ bifurcating out of 𝑒, as πœ†β‰ 0. In this case, the projection 𝑄𝑒≑𝑄𝑒(𝑠) has rank one, 𝑄𝑒=|πœ’π‘’βŸ©βŸ¨ξ‚πœ’π‘’|, where πœ’π‘’ and ξ‚πœ’π‘’ are eigenvectors of πΎπœ† and its adjoint, with eigenvalue πœ€π‘’ and its complex conjugate, respectively, and βŸ¨πœ’π‘’,ξ‚πœ’π‘’βŸ©=1. From perturbation theory, we obtain πœ’π‘’=ξ‚πœ’π‘’=πœ‘π‘˜βŠ—πœ‘π‘™βŠ—Ξ©R,𝛽+𝑂(πœ†), where 𝐻Sπœ‘π‘—=πΈπ‘—πœ‘π‘— and πΈπ‘˜βˆ’πΈπ‘™=𝑒. The overlap in the sum of (3.8) becomesπ΅ξ«ξ€·βˆ—πœ“0ξ€ΈβŠ—Ξ©R,𝛽,π‘„π‘’ξ€·π΄βŠ—πŸ™Sξ€ΈΞ©ref=π΅ξ«ξ€·βˆ—πœ“0ξ€ΈβŠ—Ξ©R,𝛽,||πœ‘π‘˜βŠ—πœ‘π‘™βŠ—Ξ©R,π›½πœ‘ξ¬ξ«π‘˜βŠ—πœ‘π‘™βŠ—Ξ©R,𝛽||ξ€·π΄βŠ—πŸ™Sξ€ΈΞ©refξ¬ξ€·πœ†+𝑂2ξ€Έ=ξ«π΅βˆ—πœ“0,||πœ‘π‘˜βŠ—πœ‘π‘™βŸ©βŸ¨πœ‘π‘˜βŠ—πœ‘π‘™||ξ€·π΄βŠ—πŸ™Sξ€ΈΞ©Sξ¬ξ€·πœ†+𝑂2ξ€Έ.(3.9) The choice 𝐴=|πœ‘π‘›βŸ©βŸ¨πœ‘π‘š| in (2.6) gives βŸ¨π΄βŸ©π‘‘=[πœŒπ‘‘]π‘š,𝑛, the reduced density matrix element. With this choice of 𝐴, the main term in (3.9) becomes (see also (3.3))ξ«π΅βˆ—πœ“0,||πœ‘π‘˜βŠ—πœ‘π‘™βŸ©βŸ¨πœ‘π‘˜βŠ—πœ‘π‘™||ξ€·π΄βŠ—πŸ™Sξ€ΈΞ©S=1βˆšπ‘π›Ώπ‘˜π‘›π›Ώπ‘™π‘šβŸ¨π΅βˆ—πœ“0,πœ‘π‘›βŠ—πœ‘π‘šβŸ©=π›Ώπ‘˜π‘›π›Ώπ‘™π‘šξ«π΅βˆ—πœ“0,ξ€·||πœ‘π‘›βŸ©βŸ¨πœ‘π‘š||βŠ—πŸ™Sξ€ΈΞ©S=π›Ώπ‘˜π‘›π›Ώπ‘™π‘šξ«πœ“0,ξ€·||πœ‘π‘›βŸ©βŸ¨πœ‘π‘š||βŠ—πŸ™S𝐡ΩS=π›Ώπ‘˜π‘›π›Ώπ‘™π‘šξ€ΊπœŒ0ξ€»π‘šπ‘›.(3.10) In the second-last step, we commute 𝐡 to the right through |πœ‘π‘›βŸ©βŸ¨πœ‘π‘š|βŠ—πŸ™S, since 𝐡 belongs to the commutant of the algebra of observables of S. In the last step, we use 𝐡ΩS=πœ“0.

Combining (3.9) and (3.10) with Theorem 3.1 we obtain, in case 𝑒=πΈπ‘šβˆ’πΈπ‘› is a simple eigenvalue, ξ€ΊπœŒπ‘‘ξ€»π‘šπ‘›βˆ’ξ«ξ«ξ€ΊπœŒβˆžξ€»π‘šπ‘›={𝑒,π‘ βˆΆπœ€π‘’(𝑠)β‰ 0}eiπ‘‘πœ€π‘’(𝑠)ξ€Ίπ›Ώπ‘˜π‘›π›Ώπ‘™π‘šξ€ΊπœŒ0ξ€»π‘šπ‘›ξ€·πœ†+𝑂2ξ‚€πœ†ξ€Έξ€»+𝑂2eβˆ’[πœ”β€²+𝑂(πœ†)]𝑑.(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 π‘˜,𝑙 satisfy πΈπ‘˜βˆ’πΈπ‘™=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∈𝐡(π”₯S). We haveβŸ¨π΄βŸ©π‘‘=TrSξ€ΊπœŒπ‘‘π΄ξ€»=TrS+Rξ€ΊπœŒπ‘‘π΄βŠ—πŸ™Rξ€»=ξ«πœ“0,eiπ‘‘πΏπœ†ξ€Ίπ΄βŠ—πŸ™SβŠ—πŸ™Rξ€»eβˆ’iπ‘‘πΏπœ†πœ“0.(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 πœ“0 (in particular, the Hilbert space of the small system becomes π”₯SβŠ—π”₯S); 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 πœ“0=Ξ©SβŠ—Ξ©R,𝛽, where Ξ©S is the trace state of S, (3.4), ⟨ΩS,(π΄βŠ—πŸ™S)Ξ©S⟩=(1/𝑁)Tr(𝐴), and where Ξ©R,𝛽 is the equilibrium state of R at a fixed inverse temperature 0<𝛽<∞. The dynamics is implemented by the group of automorphisms eiπ‘‘πΏπœ†β‹…eβˆ’iπ‘‘πΏπœ†. The self-adjoint generator πΏπœ† is called the Liouville operator. It is of the form πΏπœ†=𝐿0+πœ†π‘Š, where 𝐿0=𝐿S+𝐿R represents the uncoupled Liouville operator, and πœ†π‘Š is the interaction (3.1) represented in the GNS Hilbert space. We refer to [10, 11] for the specific form of π‘Š.

We borrow a trick from the analysis of open systems far from equilibrium: there is a (nonself-adjoint) generator πΎπœ† s.t. eiπ‘‘πΏπœ†π΄eβˆ’iπ‘‘πΏπœ†=eiπ‘‘πΎπœ†π΄eβˆ’iπ‘‘πΎπœ†πΎforallobservables𝐴,𝑑β‰₯0,andπœ†πœ“0=0.(3.13)

πΎπœ† can be constructed in a standard way, given πΏπœ† and the reference vector πœ“0. πΎπœ† is of the form πΎπœ†=𝐿0+πœ†πΌ, where the interaction term undergoes a certain modification (π‘Šβ†’πΌ); 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 πœ“0 will then just disappear due to the invariance of πœ“0. One can carry out this procedure in a rigorous manner, obtaining the following resolvent representation [10]βŸ¨π΄βŸ©π‘‘1=βˆ’ξ€œ2πœ‹iβ„βˆ’iξ‚¬πœ“0,ξ€·πΎπœ†(ξ€Έπœ”)βˆ’π‘§βˆ’1ξ€Ίπ΄βŠ—πŸ™SβŠ—πŸ™Rξ€»πœ“0ξ‚­ei𝑑𝑧d𝑧,(3.14) where πΎπœ†(πœ”)=𝐿0(πœ”)+πœ†πΌ(πœ”), 𝐼 is representing the interaction, and πœ”β†¦πΎπœ†(πœ”) is a spectral deformation (translation) of πΎπœ†. The latter is constructed as follows. There is a deformation transformation π‘ˆ(πœ”)=eβˆ’iπœ”π·, where 𝐷 is the (explicit) self-adjoint generator of translations [10, 11, 30] transforming the operator πΎπœ† asπΎπœ†(πœ”)=π‘ˆ(πœ”)πΎπœ†π‘ˆ(πœ”)βˆ’1=𝐿0+πœ”π‘+πœ†πΌ(πœ”).(3.15)

Here, 𝑁=𝑁1βŠ—πŸ™+πŸ™βŠ—π‘1 is the total number operator of a product of two bosonic Fock spaces β„±βŠ—β„± (the Gelfand-Naimark-Segal Hilbert space of the reservoir), and where 𝑁1 is the usual number operator on β„±. 𝑁 has spectrum β„•βˆͺ{0}, where 0 is a simple eigenvalue (with vacuum eigenvector Ξ©R,𝛽=Ξ©βŠ—Ξ©). For real values of πœ”, π‘ˆ(πœ”) is a group of unitaries. The spectrum of πΎπœ†(πœ”) depends on Imπœ” and moves according to the value of Imπœ”, whence the name β€œspectral deformation”. Even though π‘ˆ(πœ”) becomes unbounded for complex πœ”, the r.h.s. of (3.15) is a well-defined closed operator on a dense domain, analytic in πœ” 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(πœ”) is infinitesimally small with respect to the number operator 𝑁. Hence we use perturbation theory in πœ† to examine the spectrum of πΎπœ†(πœ”).

The point of the spectral deformation is that the (important part of the) spectrum of πΎπœ†(πœ”) is much easier to analyze than that of πΎπœ†, because the deformation uncovers the resonances of πΎπœ†. We have (see Figure 1) 𝐾spec0ξ€Έ=𝐸(πœ”)π‘–βˆ’πΈπ‘—ξ€Ύπ‘–,𝑗=1,…,π‘ξšπ‘›β‰₯1{πœ”π‘›+ℝ},(3.16) because 𝐾0(πœ”)=𝐿0+πœ”π‘, 𝐿0 and 𝑁 commute, and the eigenvectors of 𝐿0=𝐿S+𝐿R are πœ‘π‘–βŠ—πœ‘π‘—βŠ—Ξ©R,𝛽. Here, we have 𝐻Sπœ‘π‘—=πΈπ‘—πœ‘π‘—. The continuous spectrum is bounded away from the isolated eigenvalues by a gap of size Imπœ”. For values of the coupling parameter πœ† small compared to Imπœ”, we can follow the displacements of the eigenvalues by using analytic perturbation theory. (Note that for Imπœ”=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 Imπœ” s.t. 0<Imπœ”<πœ”ξ…ž (where πœ”ξ…ž is as in Condition (A)). There is a constant 𝑐0>0 s.t. if |πœ†|≀𝑐0/𝛽 then, for all πœ” with Iπ‘šπœ”>7πœ”ξ…ž/8, the spectrum of πΎπœ†(πœ”) in the complex half-plane {Im𝑧<πœ”ξ…ž/2} is independent of πœ” and consists purely of the distinct eigenvalues ξ‚†πœ€π‘’(𝑠)ξ€·πΏβˆΆπ‘’βˆˆspecS,𝑠=1,…,𝜈(𝑒),(3.17) where 1β‰€πœˆ(𝑒)≀mult(𝑒) counts the splitting of the eigenvalue 𝑒. Moreover, limπœ†β†’0||πœ€π‘’(𝑠)||(πœ†)βˆ’π‘’=0(3.18) for all 𝑠, and we have Imπœ€π‘’(𝑠)β‰₯0. Also, the continuous spectrum of πΎπœ†(πœ”) lies in the region {Im𝑧β‰₯3πœ”ξ…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=β„βˆ’i into the line 𝑧=ℝ+iπœ”ξ…ž/2, thereby picking up the residues of poles of the integrand at πœ€π‘’(𝑠) (all 𝑒, 𝑠). Let π’žπ‘’(𝑠) be a small circle around πœ€π‘’(𝑠), not enclosing or touching any other spectrum of πΎπœ†(πœ”). We introduce the generally nonorthogonal Riesz spectral projections𝑄𝑒(𝑠)=𝑄𝑒(𝑠)(1πœ”,πœ†)=βˆ’ξ€œ2πœ‹iπ’žπ‘’(𝑠)ξ€·πΎπœ†(ξ€Έπœ”)βˆ’π‘§βˆ’1d𝑧.(3.19)

It follows from (3.14) thatβŸ¨π΄βŸ©π‘‘=ξ“π‘’πœˆ(𝑒)𝑠=1eiπ‘‘πœ€π‘’(𝑠)ξ‚¬πœ“0,𝑄𝑒(𝑠)ξ€Ίπ΄βŠ—πŸ™SβŠ—πŸ™Rξ€»πœ“0ξ‚­ξ‚€πœ†+𝑂2eβˆ’πœ”β€²π‘‘/2.(3.20) Note that the imaginary parts of all resonance energies πœ€π‘’(𝑠) are smaller than πœ”ξ…ž/2, so that the remainder term in (3.20) is not only small in πœ†, 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[πœ”ξ…žβˆ’π‘‚(πœ†)], 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 πœ€π‘’(𝑠)β‰ 0 will vanish in the ergodic mean limit, so ⟨⟨𝐴⟩⟩∞=limπ‘‡β†’βˆž1π‘‡ξ€œπ‘‡0βŸ¨π΄βŸ©π‘‘ξ“d𝑑=π‘ βˆΆπœ€0(𝑠)=0ξ‚¬πœ“0,𝑄0(𝑠)ξ€Ίπ΄βŠ—πŸ™RβŠ—πŸ™Rξ€»πœ“0ξ‚­.(3.21) We now see that the linear functionals (2.13) are represented asπ‘…πœ€π‘’(𝑠)ξ‚¬πœ“(𝐴)=0,𝑄𝑒(𝑠)ξ€Ίπ΄βŠ—πŸ™SβŠ—πŸ™Rξ€»πœ“0ξ‚­.(3.22) This concludes the outline of the proof of Theorem 3.1.


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.


  1. G. Wendin and V. S. Shumeiko, β€œQuantum bits with Josephson junctions,” Low Temperature Physics, vol. 33, no. 9, pp. 724–744, 2007. View at: Publisher Site | Google Scholar
  2. D. Kinion and J. Clarke, β€œMicrostrip superconducting quantum interference device radio-frequency amplifier: scattering parameters and input coupling,” Applied Physics Letters, vol. 92, no. 17, Article ID 172503, 3 pages, 2008. View at: Publisher Site | Google Scholar
  3. Y. Makhlin, G. SchΓΆn, and A. Shnirman, β€œQuantum-state engineering with Josephson-junction devices,” Reviews of Modern Physics, vol. 73, no. 2, pp. 357–400, 2001. View at: Publisher Site | Google Scholar
  4. Y. Yu, S. Han, X. Chu, S.-I. Chu, and Z. Wang, β€œCoherent temporal oscillations of macroscopic quantum states in a Josephson junction,” Science, vol. 296, no. 5569, pp. 889–892, 2002. View at: Publisher Site | Google Scholar
  5. M. H. Devoret and J. M. Martinis, β€œImplementing qubits with superconducting integrated circuits,” Quantum Information Processing, vol. 3, no. 1–5, pp. 163–203, 2004. View at: Publisher Site | Google Scholar
  6. M. Steffen, M. Ansmann, R. McDermott et al., β€œState tomography of capacitively shunted phase qubits with high fidelity,” Physical Review Letters, vol. 97, no. 5, Article ID 050502, 4 pages, 2006. View at: Publisher Site | Google Scholar
  7. N. Katz, M. Neeley, M. Ansmann et al., β€œReversal of the weak measurement of a quantum state in a superconducting phase qubit,” Physical Review Letters, vol. 101, no. 20, Article ID 200401, 4 pages, 2008. View at: Publisher Site | Google Scholar
  8. A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, and R. J. Schoelkopf, β€œIntroduction to quantum noise, measurement and amplifcation,” Reviews of Modern Physics, vol. 82, pp. 1155–1208, 2010. View at: Google Scholar
  9. M. Merkli, I. M. Sigal, and G. P. Berman, β€œDecoherence and thermalization,” Physical Review Letters, vol. 98, no. 13, Article ID 130401, 2007. View at: Publisher Site | Google Scholar
  10. M. Merkli, I. M. Sigal, and G. P. Berman, β€œResonance theory of decoherence and thermalization,” Annals of Physics, vol. 323, no. 2, pp. 373–412, 2008. View at: Publisher Site | Google Scholar
  11. M. Merkli, G. P. Berman, and I. M. Sigal, β€œDynamics of collective decoherence and thermalization,” Annals of Physics, vol. 323, no. 12, pp. 3091–3112, 2008. View at: Publisher Site | Google Scholar
  12. G. P. Berman, D. I. Kamenev, and V. I. Tsifrinovich, β€œCollective decoherence of the superpositional entangled states in the quantum Shor algorithm,” Physical Review A, vol. 71, no. 3, Article ID 032346, 5 pages, 2005. View at: Publisher Site | Google Scholar
  13. D. A. R. Dalvit, G. P. Berman, and M. Vishik, β€œDynamics of open bosonic quantum systems in coherent state representation,” Physical Review A, vol. 73, no. 1, Article ID 013803, 2006. View at: Publisher Site | Google Scholar
  14. L.-M. Duan and G.-C. Guo, β€œReducing decoherence in quantum-computer memory with all quantum bits coupling to the same environment,” Physical Review A, vol. 57, no. 2, pp. 737–741, 1998. View at: Google Scholar
  15. E. Joos, H. D. Zeh, C. Kiefer, D. Giulini, J. Kupsch, and I. O. Stamatescu, Decoherence and The Appearence of a Classical World in Quantum Theory, Lecture Notes in Mathematics, Springer, Berlin, Germany, 2nd edition, 2003.
  16. D. Mozyrsky and V. Privman, β€œAdiabatic decoherence,” Journal of Statistical Physics, vol. 91, no. 3–4, pp. 787–799, 1998. View at: Google Scholar
  17. G. M. Palma, K.-A. Suominen, and A. K. Ekert, β€œQuantum computers and dissipation,” Proceedings of the Royal Society A, vol. 452, no. 1946, pp. 567–584, 1996. View at: Google Scholar
  18. J. Shao, M.-L. Ge, and H. Cheng, β€œDecoherence of quantum-nondemolition systems,” Physical Review E, vol. 53, no. 1, pp. 1243–1245, 1996. View at: Google Scholar
  19. M. Merkli, G. P. Berman, F. Borgonovi, and K. Gebresellasie, β€œEvolution of entanglement of two qubits interacting through local and collective environments,” Quantum Physics, preprint, View at: Google Scholar
  20. M. Merkli and S. Starr, β€œA resonance theory for open quantum systems with time-dependent dynamics,” Journal of Statistical Physics, vol. 134, pp. 871–898, 2009. View at: Google Scholar
  21. V. Bach, M. Merkli, W. Pedra, and I. M. Sigal, β€œDecoherence control,” in preparation. View at: Google Scholar
  22. H.-P Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, Oxford, UK, 2002.
  23. E. B. Davies, β€œMarkovian master equations,” Communications in Mathematical Physics, vol. 39, no. 2, pp. 91–110, 1974. View at: Publisher Site | Google Scholar
  24. E. B. Davies, β€œMarkovian master equations. II,” Mathematische Annalen, vol. 219, no. 2, pp. 147–158, 1976. View at: Publisher Site | Google Scholar
  25. J. DereziΕ„ski and R. FrΓΌboes, Fermi Golden Rule and Open Quantum Systems, vol. 1882 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2006.
  26. V. JakΕ‘iΔ‡ and C. A. Pillet, β€œFrom resonances to master equations,” Annales de l'Institut Henri PoincarΓ©, vol. 67, no. 4, pp. 425–445, 1997. View at: Google Scholar
  27. M. Merkli, β€œLevel shift operators for open quantum systems,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 376–399, 2007. View at: Publisher Site | Google Scholar
  28. J. B. Altepeter, P. G. Hadley, S. M. Wendelken, A. J. Berglund, and P. G. Kwiat, β€œExperimental investigation of a two-qubit decoherence-free subspace,” Physical Review Letters, vol. 92, no. 14, Article ID 147901, 4 pages, 2004. View at: Publisher Site | Google Scholar
  29. A. Fedorov and L. Fedichkin, β€œCollective decoherence of nuclear spin clusters,” Journal of Physics Condensed Matter, vol. 18, no. 12, pp. 3217–3228, 2006. View at: Publisher Site | Google Scholar
  30. M. Merkli, M. MΓΌck, and I. M. Sigal, β€œInstability of equilibrium states for coupled heat reservoirs at different temperatures,” Journal of Functional Analysis, vol. 243, no. 1, pp. 87–120, 2007. View at: Publisher Site | Google Scholar

Copyright © 2010 M. Merkli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.