and I. M. Sigal

Volume 2010, Article ID 169710,20pages doi:10.1155/2010/169710

Review Article

Resonant Perturbation Theory of Decoherence and Relaxation of Quantum Bits

M. Merkli,

G. P. Berman,

and I. M. Sigal

1Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NF, Canada A1C 5S7

2Theoretical Division, MS B213, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

3Department of Mathematics, University of Toronto, Toronto, ON, Canada M5S2E4

Correspondence should be addressed to M. Merkli,merkli@mun.ca Received 31 August 2009; Accepted 10 February 2010

Academic Editor: Shao-Ming Fei

Copyrightq2010 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.

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 timest ≥ 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 computersQCswith large numbers of quantum bitsqubitspromise 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 devicesSQUIDsserving as qubitseffective spins, which operate in the quantum regime:ω >> kBT, whereT is the temperature andω is the qubit transition frequency. This condition is widely used in superconducting quantum computation and quantum measurement, whenT ∼10–20 mK andω ∼100–150 mKin 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 macroscopicseveralμm sizedevice. 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 offvoltages on controlling microcontacts and magnetic fluxes.ivThe 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 solvablenonintegrablesystems.

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 anN-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 ofN. Our method is based on a dynamical quantum resonance theory valid for small, fixed values of the couplings, and uniformly in time fort ≥ 0. We do not make Markov-, Born- or weak couplingvan 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 spacehsC^Nand a Hamiltonian

H_S diagE1, . . . , E_N. 2.1

The environment R is modelled by a bosonic thermal reservoir with Hamiltonian

H_R

R³a^∗k|k|akd³k, 2.2 acting on the reservoir Hilbert space hR, and where a^∗k and ak are the usual bosonic creation and annihilation operators satisfying the canonical commutation relations ak, a^∗l δk−l. It is understood that we consider R in the thermodynamic limit of infinite volumeR³and fixed temperatureT 1/β > 0 in a phase without condensate.

Given a form factorfk, a square integrable function ofk∈R³momentum representation, the smoothed-out creation and annihilation operators are defined asa^∗f

R³fka^∗kd³k andaf

R³fkakd³k, respectively, and the hermitian field operator is φ

f 1

√2 a^∗

f a

f

. 2.3

The total Hamiltonian, acting onhS⊗hR, has the form

HHS⊗1R1S⊗HRλv, 2.4

whereλis a coupling constant andvis 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; see10. The initial density matrix is thus

ρ₀ρ₀⊗ρ_R,β, 2.5

whereρ₀is any state of S andρ_R,βis the equilibrium state of R at temperature 1/β.

Let Abe an arbitrary observable of the system an operator on the system Hilbert spacehSand set

A _t:Tr_S ρ_tA

Tr_SR

ρ_tA⊗1R

, 2.6

whereρ_tis the density matrix of SR at timetand

ρ_tTr_Rρ_t 2.7

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}^N_k1ofH_S, is given by

ρ_t

kl

ϕ_k,e^−itHSρ₀e^itHSϕ_l e^itelk ρ₀

kl, 2.8

wheree_lk E_l−E_k. As the interaction with the reservoir is turned on, the dynamics2.8 undergoes two qualitative changes.

1The “Bohr frequencies”

e∈

E−E:E, E∈specHS

2.9 in the exponent of2.8become complex,e→ε_e. It can be shown generally that the resonance energiesεehave nonnegative imaginary parts, Imεe≥0. If Imεe>0, then the corresponding dynamical process is irreversible.

2The 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 ofH_Sinvariant.However, to lowest order in the interaction, the eigenspaces of HS are left invariant and therefore matrix elements with m, nbelonging to a fixed energy difference Em −En will evolve in a coupled manner.

Our goal is to derive these two effects from the microscopichamiltonianmodel 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∈ BhSbe an observable of the system S. We show in9,10that the ergodic averages

A _∞: lim

T→ ∞

1 T

_T

0

A _tdt 2.10

exist, that is,A _tconverges in the ergodic sense ast → ∞. Furthermore, we show that for anyt≥0 and for any 0< ω<2π/β,

A _t− A _∞

ε /0

e^itεR_εA O

λ²e^−ω−Oλt

, 2.11

where the complex numbersεare the eigenvalues of a certain explicitly given operatorKω, lying in the strip{z∈C | 0≤Imz < ω/2}. They have the expansions

ε≡ε^s_e eλ²δ_e^sO λ⁴

, 2.12

wheree∈specHS⊗1S−1S⊗HS specHS−specHSandδ_e^sare the eigenvalues of a matrix Λe, called a level-shift operator, acting on the eigenspace ofH_S⊗1S−1S⊗H_Scorresponding to the eigenvalueewhich is a subspace ofhS⊗hS. TheR_εAin2.11are linear functionals ofAand are given in terms of the initial state,ρ0, and certain operators depending on the HamiltonianH. They have the expansion

R_εA

m,n∈Ie

κm,nA_m,nO λ²

, 2.13

where Ie is the collection of all pairs of indices such that e Em−En, with Ek being the eigenvalues ofH_S. Here,A_m,nis them, n-matrix element of the observableAin the energy

basis ofHS, andκm,nare coefficients depending on the initial state of the systemand one, but not onAnor onλ.

2.1.1. Discussion

iIn the absence of interactionλ 0, we haveε e ∈R; see2.12. Depending on the interaction, each resonance energyεmay migrate into the upper complex plane, or it may stay on the real axis, asλ /0.

iiThe averagesA _t approach their ergodic meansA _∞ 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 _toscillates. A sufficient condition for decay is that Imδ^s_e > 0and λsmall, see2.12.

iiiThe error term in 2.11 is small inλ, uniformly int ≥ 0, and it decays in time quicker than any of the main terms in the sum on the r.h.s.: indeed, Imε Oλ² whileω−Oλ> ω/2 independent of small values ofλ. However, this means that we are in the regimeλ²ω<2π/βsee before2.11, which implies thatλ²must be much smaller than the temperatureT 1/β. Using a more refined analysis, one can get rid of this condition; see also remarks on page 376 of10.

ivRelation2.13shows that to lowest order in the perturbation, the group ofenergy 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 energiesεand the functionalsR_εcan 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, see19.

The dynamical resonance method can be generalized to time-dependent Hamiltoni- ans. See20,21for 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 parameterλ, 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 toρ_t e^tLρ₀, whereLis 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 SR 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 Oλ². Features of this kind cannot be captured by the Lindblad approximation, but are captured in our approach.

It has been shownsee, e.g.,23–26that the weak coupling limit dynamics generated by the Lindblad operator is obtained in the regimeλ → 0,t → ∞, withλ²tfixed. One of the strengths of our approach is that we do not impose any relation betweenλ andt, and our results are valid for all times t ≥ 0, providedλ is small. It has been observed25,26 that for certain systems of the type SR, the second-order contribution of the exponentsεe^s

in2.12correspond 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 ofHS is degenerate, it happens that the second-order contributions to Imε^s_e vanish. This corresponds to a Lindblad generator having several real eigenvalues. In this situation, the correct dynamicsapproach to a final statecan be captured only by taking into account higher-order contributions to the exponentsε^s_e ; see27. 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 gapΔ E₂−E₁> 0. S is coupled to the heat bath R via the operator

v a c

c b

⊗φ g

, 2.14

where φg is the Bose field operator 2.3, smeared out with a coupling function form factorgk,k∈R³, and the 2×2 coupling matrixrepresenting the coupling operator in the energy eigenbasisis hermitian. The operator2.14—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 to10for a discussion of the link between2.14and the spin-boson model. We take S initially in a coherent superposition in the energy basis,

ρ₀ 1 2

1 1 1 1

. 2.15

In10we derive from representation2.11the following expressions for the dynamics of matrix elements, for allt≥0:

ρ_t

m,m e^−βEm

ZS,β −1^m 2 tanh

βΔ 2

e^itε0λR_m,mλ, t, m1,2, 2.16 ρ_t

1,2 1

2e^itε−ΔλR1,2λ, t, 2.17 where the resonance energiesεare given by

ε₀λ iλ²π²|c|²ξΔ O λ⁴

, ε_Δλ Δ λ²R i

2λ²π²

|c|²ξΔ b−a²ξ0 O

λ⁴ , ε_−Δλ −εΔλ,

2.18

with

ξ η

lim

↓0

1 π

R³d³kcoth β|k|

2

gk² |k| −η₂

², R 1

2

b²−a² g, ω⁻¹g 1

2|c|²P.V.

R×S²u²g|u|, σ²coth β|u|

2 1

u−Δ.

2.19

The remainder terms in2.17,2.17satisfy|Rm,nλ, t| ≤ Cλ², uniformly int≥ 0, and they can be decomposed into a sum of a constant and a decaying part,

R_m,nλ, t p_n,m

∞−δ_m,ne^−βEm

Z_S,β R_m,nλ, t, 2.20 where |Rm,nλ, t| Oλ²e^−γt, with γ min{Imε₀,Imε_±Δ}. These relations show the following.

iTo second order in λ, convergence of the populations to the equilibrium values Gibbs law, and decoherence occur exponentially fast, with ratesτ_T Imε₀λ⁻¹ andτ_D Imε_Δλ⁻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|²ξΔ b− a²ξ0⁻¹; compare for example2.18.

iiThe 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 lim_{t→ ∞}Rm,nλ, t Oλ². This is clear heuristically too, since typically the entire system SR approaches its joint equilibrium in which S and R

are entangled. The reduction of this state to S is the Gibbs state of S moduloOλ² 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 in10, Theorem 3.3.

iiiIn a markovian master equation approach, the above phenomenoni.e., variations ofOλ²in the time-asymptotic limitcannot be detected. Indeed in that approach one would conclude that S approaches its Gibbs state ast → ∞.

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 size N. The Hamiltonian is

HS ^N

i,j1

JijS^z_iS^z_j ^N

j1

BjS^z_j, 2.21

whereJ_ijare pair interaction constants andB_jis the value of a magnetic field at the location of spinj. The Pauli spin operator is

S^z 1 0

0 −1

2.22

andS^z_j is the matrixS^zacting on thejth 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 iscompare with2.4

λ1v1λ2v2λ1

N j1

S^z_j ⊗φ g1

λ2

N j1

S^x_j ⊗φ g2

. 2.23

Hereg₁andg₂are form factors and the coupling constantsλ₁andλ₂ measure the strengths of the energy conserving position-position coupling, and the energy exchange spin flip coupling, respectively. Spin-flips are implemented by theS^x_j in2.23, representing the Pauli matrix

S^x 0 1

1 0

2.24

acting on thejth spin. The total Hamiltonian takes the form2.4withλvreplaced by2.23.

It is convenient to representρ_tas a matrix in the energy basis, consisting of eigenvectorsϕ_σ ofH_S. These are vectors inhS C²⊗ · · · ⊗C²C^2N indexed by spin configurations

σ{σ1, . . . , σN} ∈ {1,−1}^N, ϕσ ϕσ1⊗ · · · ⊗ϕσN, 2.25

where

ϕ 1

0

, ϕ₋ 0

1

, 2.26

so that

HSϕσ E σ

ϕσ with E σ

^N

i,j1

Jijσiσj^N

j1

Bjσj. 2.27

We denote the reduced density matrix elements as ρ_t

σ,τ

ϕ_σ, ρ_tϕ_τ

. 2.28

The Bohr frequencies2.9are now

e σ, τ

E σ

−E τ

^N

i,j1

Jij

σiσj−τiτj

^N

j1

Bj

σj−τj

, 2.29

and they become complex resonance energiesεeεeλ1, λ2∈Cunder perturbation.

Assumption of Nonoverlapping Resonances

The Bohr frequencies 2.29 represent “unperturbed” energy levels and we follow their motion under perturbationλ1, 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 in10, Theorem 2.1, that for allt≥0, ρ_t

σ,τ−

ρ_∞

σ,τ

{e:εe/0}

e^itεe

⎡

⎣

σ,τ

w^ε_σ,τ;σ^e ,τ

ρ₀

σ,τO

λ²₁λ²₂⎤

⎦ O

λ²₁λ²₂

e^−ωOλt .

2.30

This result is obtained by specializing2.11to the specific system at hand and considering observables A |ϕτ ϕσ|. In 2.30, we have in accordance with 2.10, ρ_∞σ,τ lim_{T→ ∞}1/T_T

0ρ_tσ,τdt. The coefficients w are overlaps of resonance eigenstates which vanish unlesse−eσ, τ −eσ, τ see point2after2.9. They represent the dominant contribution to the functionalsRεin2.11; see also2.13. Theεehave the expansion

ε_e ≡ε^s_e eδ^s_e O

λ⁴₁λ⁴₂

, 2.31

where the label s 1, . . . , νe indexes the splitting of the eigenvalue e into νedistinct resonance energies. The lowest order correctionsδ^s_e satisfy

δ^s_e O

λ²₁λ²₂

. 2.32

They are thecomplexeigenvalues of an operatorΛe, called the level shift operator associated toe. This operator acts on the eigenspace ofL_Sassociated to the eigenvalueea subspace of the qubit register Hilbert space; see10,11for the formal definition ofΛe. It governs the lowest order shift of eigenvalues under perturbation. One can see by direct calculation that Imδ^s_e ≥0.

2.2.1. Discussion

iTo lowest order in the perturbation, the group of reduced density matrix elements ρ_tσ,τ associated to a fixede eσ, τevolve in a coupled way, while groups of matrix elements associated to differenteevolve independently.

iiThe density matrix elements of a given group mix and evolve in time according to the weight functionswand the exponentials e^itεse . In the absence of interaction λ1 λ2 0, all theε_e^s eare real. As the interaction is switched on, the ε_e^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”.

iiiThe matrix elements ρ_{tσ ,τ} of a group e approach their ergodic means if and only if all the nonzero ε^s_e have strictly positive imaginary part. In this case, the convergence takes place on a time scale of the order 1/γe, where

γ_emin

Imε_e^s:s1, . . . , νes.t. ε^s_e /0

2.33 is the decay rate of the group associated toe. If anε^s_e stays real, then the matrix elements of the corresponding group oscillate in time. A sufficient condition for decay of the group associated toeisγ_e > 0, that is, Imδ^s_e >0 for alls, andλ₁,λ₂ small.

2.2.2. Decoherence Rates

We illustrate our results on decoherence rates for a qubit register withJij0the general case is treated in11. We consider generic magnetic fields defined as follows. Forn_j∈ {0,±1,±2}, j1, . . . , N, we have

N j1

Bjnj 0⇐⇒nj0 ∀j. 2.34

Condition2.34is satisfied generically in the sense that it does not hold only for very special choices ofB_j one such special choice isB_j constant. For instance, if theB_jare chosen to

be independent, and uniformly random from an intervalBmin, Bmax, then2.34is satisfied with probability one. We show in 11, Theorem 2.3, that the decoherence rates2.33are given by

γe

λ²₁y₁e λ²₂y₂e y₁₂e, e /0 λ²₂y0, e0

O

λ⁴₁λ⁴₂

. 2.35

Here,y1is contributions coming from the energy conserving interaction;y0andy2are due to the spin flip interaction. The termy₁₂is due to both interactions and is ofOλ²₁λ²₂. We give explicit expressions fory0,y1,y2, andy12in11, Section 2. For the present purpose, we limit ourselves to discussing the properties of the latter quantities.

iProperties of y₁e:y₁evanishes if eithereis such thate₀: ⁿ_j1σj−τ_j 0 or the infrared behaviour of the coupling functiong₁ is too regularin three dimensions g1 ∝ |k|^pwithp > −1/2. Otherwise,y1e >0. Moreover,y1eis proportional to the temperatureT.

iiProperties of y₂e:y₂e>0 ifg₂2Bj,Σ/0 for allB_jform factorg₂k g₂|k|,Σ in spherical coordinates. For low temperatures,T,y2e∝T, for high temperatures y₂eapproaches a constant.

iiiProperties of y₁₂e: if either of λ₁,λ₂ or e₀ vanish, or if g₁ is infrared regular as mentioned above, then y12e 0. Otherwise, y12e > 0, in which case y12e approaches constant values for bothT → 0,∞.

ivFull decoherence: ifγ_e>0 for alle /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 ifλ2/0 and g₂2Bj,Σ/0 for allj, and providedλ₁, λ₂are small enoughso that the remainder term in 2.35 is small. Note that if λ2 0, then matrix elements associated to energy differencesesuch thate0 0 will not decay on the time scale given by the second order in the perturbationλ²₁.

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; see9,10. Hence generically, the density matrix of S does not become diagonal in the energy basis as t → ∞.

vProperties ofy₀:y₀depends on the energy exchange interaction only. This reflects the fact that for a purely energy conserving interaction, the populations are conserved 9,10,17. Ifg22Bj,Σ/0 for allj, theny0>0this is sometimes called the “Fermi Golden Rule Condition”. For small temperaturesT,y₀∝T, whiley₀approaches a finite limit asT → ∞.

In terms of complexity analysis, it is important to discuss the dependence of γe on the register sizeN.

iWe show in11thaty₀ is independent ofN. This means that the thermalization time, or relaxation time of the diagonal matrix elementscorresponding toe 0, isO1inN.

iiTo determine the order of magnitude of the decay rates of the off-diagonal density matrix elementscorresponding toe /0relative to the register sizeN, we assume the magnetic field to have a certain distribution denoted by · . We show in11 that

y1

y1∝e²₀, y2

CBD σ−τ

, y12

cBλ1, λ2N0e, 2.36

where C_B and c_B c_Bλ1, λ₂ are positive constants independent of N, with cBλ1, λ2 Oλ²₁λ²₂. Here,N0eis the number of indicesj such thatσj τj

for eachσ, τs.t.eσ, τ e, and

D σ−τ

:^N

j1

σj−τj 2.37

is the Hamming distance between the spin configurationsσ andτ which depends oneonly.

iiiConsider e /0. It follows from 2.35–2.37 that for purely energy conserving interactions λ2 0,γ_e ∝ λ²₁e²₀ λ²₁ ^N_j1σj −τ_j², which can be as large as Oλ²₁N². On the other hand, for purely energy exchanging interactionsλ1 0, we haveγ_e ∝ λ²₂Dσ−τ, which cannot exceedOλ²₂N. If both interactions are acting, then we have the additional termy12 , which is of orderOλ²₁ λ²₂N.

This shows the following:

The fastest decay rate of reduced off-diagonal density matrix elements due to the energy conserving interaction alone is of orderλ²₁N², 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 orderλ²₁, that is, independent of N.

Remarks. 1Forλ20, the model can be solved explicitly17, and one shows that the fastest decaying matrix elements have decay rate proportional toλ²₁N². Furthermore, the model with a noncollective, energy-conserving interaction, where each qubit is coupled to an independent reservoir, can also be solved explicitly17. The fastest decay rate in this case is shown to be proportional toλ²₁N.

2As mentioned at the beginning of this section, we take the coupling constantsλ₁,λ₂ so small that the resonances do not overlap. Consequently,λ²₁N²andλ²₂Nare bounded above by a constant proportional to the gradient of the magnetic field in the present situation; see also11. Thus the decay ratesγ_edo not increase indefinitely with increasingNin the regime considered here. Rather,γeare attenuated by small coupling constants for largeN. They are of the orderγe∼Δ. We have shown that modulo an overall, commonN-dependentprefactor, the decay rates originating from the energy conserving and exchanging interactions differ by a factorN.

3Collective decoherence has been studied extensively in the literature. Among the many theoretical, numerical, and experimental works, we mention here only12,14,17,28, 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 representation2.11, 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 operator

v^R

r1

λ_rG_r ⊗φ g_r

, 3.1

where eachG_r is a hermitianN×Nmatrix, theg_rkare form factors, and theλ_r ∈ Rare coupling constants. Fix any phaseχ∈Rand define

gr,βu, σ:

! u

1−e^−βu|u|^1/2

⎧⎨

⎩

gru, σ if u≥0,

−e^iχg_r−u, σ if u <0, 3.2

whereu∈Randσ∈S². The phaseχis a parameter which can be chosen appropriately as to satisfy the following condition.

AThe mapω→g_r,βuω, σhas an analytic extension to a complex neighbourhood {|z|< ω}of the origin, as a map fromCtoL²R³,d³k.

Examples ofgsatisfyingAare given bygr, σ r^pe^−rmg1σ, wherep −1/2n, n0,1, . . .,m1,2, andg₁σ e^iφg₁σ.

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 functionsgk, which have to behave appropriately in the infrared regime so that the parts of3.2fit nicely together atu0, to allow for an analytic continuation. On the other hand, the square root in3.2must be analytic as well, which implies the conditionω<2π/β.

It is convenient to introduce the doubled Hilbert spaceHShS⊗hS, whose normalized vectors accommodate any state on the system Spure or mixed. The trace state, or infinite temperature state, is represented by the vector

ΩS 1

√N N

j1

ϕj⊗ϕj 3.3

via

BhSA−→ ΩS,A⊗1ΩS . 3.4

Here ϕ_j are the orthonormal eigenvectors of H_S. This is just the Gelfand-Naimark-Segal construction for the trace state. Similarly, letHRandΩR,βbe the Hilbert space and the vector representing the equilibrium state of the reservoirs at inverse temperatureβ. In the Araki- Woods representation of the field, we haveHR F ⊗ F, whereFis the bosonic Fock space

over the one-particle spaceL²R³,d³kandΩR,β Ω⊗Ω, withΩbeing the Fock vacuum of Fsee also10,11for more detail. Letψ₀⊗ΩR,βbe the vector inHS⊗ HRrepresenting the density matrix at timet0. It is not difficult to construct the unique operator inB∈1S⊗hS

satisfying

BΩSψ0. 3.5

See also10for concrete examples.We define the reference vector

Ωref: ΩS⊗ΩR,β 3.6

and set

λ max

r1,...,R|λr|. 3.7

Theorem 3.1 Dynamical resonance theory 9–11. Assume condition (A) with a fixed ω satisfying 0 < ω < 2π/β. There is a constantc0 s.t.; ifλ ≤ c0/β, then the limitA _∞,2.10, exists for all observablesA∈ BhS. Moreover, for all suchAand for allt≥0, we have

A _t− A _∞

e,s:εe^s/0 νe

s1

e^itεse B^∗ψ₀

⊗ΩR,β, Q^s_e A⊗1SΩref

O

λ²e^−ωOλt .

3.8

Theε^s_e are given by2.12, 1≤νe≤multecounts the splitting of the eigenvalueeinto distinct resonance energiesε^s_e , and theQ^s_e 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 inSection 2.2. We obtain2.30 in particular, the overlap functionswfrom3.8by analyzing the projectionsQ_e^sin more detail. Let us explain how to link the overlap B^∗ψ0⊗ΩR,β, Q^s_e A⊗1SΩref to its initial value for a nondegenerate Bohr energye, and whereA|ϕn ϕm|.The latter observables used in2.11 give the matrix elements of the reduced density matrix in the energy basis.

The Q^s_e is the spectral Riesz projection of an operator K_λ associated with the eigenvalueεe^s; see3.19 In reality, we consider a spectral deformationKλω, whereωis a complex parameter. This is a technical trick to perform our analysis. Physical quantities do not depend onωand therefore, we do not display this parameter here. If a Bohr energye,2.9, is simple, then there is a single resonance energyεebifurcating out ofe, asλ /0. In this case, the projectionQe≡Q^s_e has rank one,Qe|χe χ%e|, whereχeandχ%eare eigenvectors ofKλ

and its adjoint, with eigenvalueε_e and its complex conjugate, respectively, andχe,χ%_e 1.

From perturbation theory, we obtainχeχ%e ϕk⊗ϕl⊗ΩR,βOλ, whereHSϕjEjϕjand E_k−E_le. The overlap in the sum of3.8becomes

B^∗ψ₀

⊗ΩR,β, Q_eA⊗1SΩref

B^∗ψ₀

⊗ΩR,β,ϕ_k⊗ϕ_l⊗ΩR,β

ϕ_k⊗ϕ_l⊗ΩR,βA⊗1SΩref

O λ²

B^∗ψ₀,ϕ_k⊗ϕ_l

ϕ_k⊗ϕ_lA⊗1SΩS

O λ²

.

3.9

The choiceA |ϕn ϕm|in2.6givesA _t ρ_tm,n, the reduced density matrix element.

With this choice ofA, the main term in3.9becomessee also3.3 B^∗ψ0,ϕk⊗ϕl

ϕk⊗ϕlA⊗1SΩS

1

√Nδknδlm

B^∗ψ0, ϕn⊗ϕm

δ_knδ_lm

B^∗ψ₀,ϕ_n

ϕ_m⊗1S

ΩS

δknδlm

ψ0,ϕn

ϕm⊗1S

BΩS

δknδlm

ρ₀

mn.

3.10

In the second-last step, we commuteBto the right through|ϕn ϕm| ⊗1S, sinceBbelongs to the commutant of the algebra of observables of S. In the last step, we useBΩS ψ₀.

Combining3.9and3.10withTheorem 3.1we obtain, in caseeEm−Enis a simple eigenvalue,

ρ_t

mn−

ρ_∞

mn

{e,s:ε^se /0}

e^itεse δ_knδ_lm

ρ₀

mnO

λ² O

λ²e^−ωOλt

. 3.11

This explains the form2.30for a simple Bohr energye. The case of degenerateei.e., where several different pairs of indicesk, lsatisfyEk−Eleis analyzed along the same lines; see 11for details.

3.1. Mechanism of Dynamical Resonance Theory, Outline of Proof ofTheorem 3.1

Consider any observableA∈BhS. We have A _tTr_S

ρ_tA Tr_SR

ρ_tA⊗1R

ψ0,e^itLλA⊗1S⊗1Re^−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

Isolated eigenvalues Imω

Im Continuous

spectrum

Re 0

Figure 1: Spectrum ofK0ω.

Hilbert space of the small system becomeshS⊗hS; see also before3.3,3.4. 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 vectorψ₀ ΩS⊗ΩR,β, whereΩSis the trace state of S,3.4, ΩS,A⊗1SΩS 1/NTrA, and whereΩR,βis the equilibrium state of R at a fixed inverse temperature 0< β <∞. The dynamics is implemented by the group of automorphisms e^itLλ· e^−itLλ. The self-adjoint generatorL_λis called the Liouville operator. It is of the formL_λL₀λW, whereL0 LS LR represents the uncoupled Liouville operator, andλW is the interaction 3.1represented in the GNS Hilbert space. We refer to10,11for the specific form ofW.

We borrow a trick from the analysis of open systems far from equilibrium: there is a nonself-adjointgeneratorKλs.t.

e^itLλAe^−itLλ e^itKλAe^−itKλ for all observablesA, t≥0, and

K_λψ₀0. 3.13

K_λ can be constructed in a standard way, givenL_λand the reference vectorψ₀.K_λis of the formKλ L0λI, where the interaction term undergoes a certain modificationW → I;

see for example10. As a consequence, formally, we may replace the propagators in3.12 by those involvingK. The resulting propagator which is directly applied toψ₀will then just disappear due to the invariance ofψ0. One can carry out this procedure in a rigorous manner, obtaining the following resolvent representation10

A _t− 1 2πi

R−i

ψ0,Kλω−z⁻¹A⊗1S⊗1Rψ0 e^itzdz, 3.14

whereK_λω L₀ω λIω,Iis representing the interaction, andω→K_λωis a spectral deformationtranslationofKλ. The latter is constructed as follows. There is a deformation transformationUω e^−iωD, whereDis theexplicitself-adjoint generator of translations 10,11,30transforming the operatorK_λas

Kλω UωKλUω⁻¹L0ωNλIω. 3.15

Here,NN1⊗11⊗N1is the total number operator of a product of two bosonic Fock spacesF ⊗Fthe Gelfand-Naimark-Segal Hilbert space of the reservoir, and whereN₁is the

usual number operator onF.Nhas spectrumN∪ {0}, where 0 is a simple eigenvaluewith vacuum eigenvectorΩR,β Ω⊗Ω. For real values ofω,Uωis a group of unitaries. The spectrum ofK_λωdepends on Imωand moves according to the value of Imω, whence the name “spectral deformation”. Even thoughUωbecomes unbounded for complexω, the r.h.s. of3.15 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 Aafter 3.2 comes into play. The operator Iω is infinitesimally small with respect to the number operatorN. Hence we use perturbation theory inλto examine the spectrum of K_λω.

The point of the spectral deformation is that the important part of the spectrum ofKλωis much easier to analyze than that ofKλ, because the deformation uncovers the resonances ofK_λ. We haveseeFigure 1

specK0ω

E_i−E_j

i,j1,...,N

&

n≥1

{ωnR}, 3.16

becauseK₀ω L₀ ωN,L₀ andN commute, and the eigenvectors of L₀ L_S L_Rare 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 Imω0, the eigenvalues are imbedded into the continuous spectrum, and analytic perturbation theory is not valid! The spectral deformation is indeed very useful!

Theorem 3.2see10andFigure 2. Fix Imωs.t. 0 < Imω < ω(whereωis as in Condition (A)). There is a constantc₀ >0 s.t. if|λ| ≤ c₀/βthen, for allωwith Im ω > 7ω/8, the spectrum ofK_λωin the complex half-plane{Imz < ω/2}is independent ofω and consists purely of the distinct eigenvalues

ε_e^s:e∈specLS, s1, . . . , νe

, 3.17

where 1≤νe≤multecounts the splitting of the eigenvaluee. Moreover,

λlim→0

ε^se λ−e0 3.18

for alls, and we have Imε^s_e ≥0. Also, the continuous spectrum ofKλωlies in the region{Imz≥ 3ω/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 integrationz R−i into the linez Riω/2, thereby picking up the residues of poles of the integrand atε^s_e alle,s. LetC^se be a small circle aroundε_e^s, not enclosing

Im

0

Re 3ω/4

Resonances

Location of continuous spectrum

Figure 2: Spectrum ofKλω. Resonancesεe^sare uncovered.

Im Location of continuous spectrum

Re zRiω/2

zR−i 0

C^se

Figure 3: Contour deformation:

R−idz e,s

C^se dz

Riω/2dz.

or touching any other spectrum ofK_λω. We introduce the generally nonorthogonal Riesz spectral projections

Q^s_e Q_e^sω, λ − 1 2πi

C^se

Kλω−z⁻¹dz. 3.19

It follows from3.14that

A _t

e νe

s1

e^itεse

ψ₀, Q^s_e A⊗1S⊗1Rψ0 O

λ²e^−ωt/2

. 3.20

Note that the imaginary parts of all resonance energiesε^s_e are smaller thanω/2, so that the remainder term in3.20is not only small inλ, but it also decays faster than all of the terms in the sum.See alsoFigure 3.We point out also that instead of deforming the path integration contour as explained before3.19, we could choosezRiω−Oλ, hence transforming the error term in3.20into the one given in3.8.