Decoherence induced by interacting quantum spin baths
Davide Rossini,1 Tommaso Calarco,2,3Vittorio Giovannetti,1Simone Montangero,1 and Rosario Fazio1,4
1NEST-CNR-INFM & Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy 2
Dipartimento di Fisica, Università di Trento and BEC-CNR-INFM, I-38050 Povo, Italy 3
ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA 4
International School for Advanced Studies (SISSA), Via Beirut 2-4, I-34014 Trieste, Italy
共Received 23 November 2006; published 26 March 2007兲
We study decoherence induced on a two-level system coupled to a one-dimensional quantum spin chain. We consider the cases where the dynamics of the chain is determined by the Ising, XY, or Heisenberg exchange Hamiltonian. This model of quantum baths can be of fundamental importance for the understanding of deco-herence in open quantum systems, since it can be experimentally engineered by using atoms in optical lattices. As an example, here we show how to implement a pure dephasing model for a qubit system coupled to an interacting spin bath. We provide results that go beyond the case of a central spin coupled uniformly to all the spins of the bath, in particular showing what happens when the bath enters different phases, or becomes critical; we also study the dependence of the coherence loss on the number of bath spins to which the system is coupled and we describe a coupling-independent regime in which decoherence exhibits universal features, irrespective of the system-environment coupling strength. Finally, we establish a relation between decoherence and entanglement inside the bath. For the Ising and the XY models we are able to give an exact expression for the decay of coherences, while for the Heisenberg bath we resort to the numerical time-dependent density matrix renormalization group.
DOI:10.1103/PhysRevA.75.032333 PACS number共s兲: 03.67.Lx, 05.50.⫹q, 03.65.Yz, 05.70.Jk
I. INTRODUCTION
Decoherence refers to the process through which superpo-sitions of quantum states are irreversibly transformed into statistical mixtures. It is due to the unavoidable coupling of a quantum system with its surrounding environment which, as a consequence, leads to entanglement between the system and the bath. This loss of coherence may considerably reduce the efficiency of quantum information protocols关1兴 and it is crucial in describing the emergence of classicality in quan-tum systems关2兴.
Although desirable, it is not always possible to fully char-acterize the bath. Therefore it is necessary to resort to inge-nious, though realistic, modelizations. Paradigmatic models represent the environment as a many-body system, such as a set of bosonic harmonic oscillators关3,4兴 or of spin-1/2 par-ticles 关5兴. In some cases it is also possible to recover the effects of a many-body environment via the coupling to a single-particle bath, provided its dynamics is chaotic关6兴. In order to grasp all the subtleties of the entanglement between the system and its environment, it would be of great impor-tance to study engineered baths 共and system-bath interac-tions兲 that can be realized experimentally. In this paper we discuss a class of interacting spin baths which satisfy these requirements: a two-level system coupled to a one-dimensional array of spin-1/2 particles, whose free evolution is driven by a Hamiltonian which embraces Ising, XY, and Heisenberg universality classes. In several nontrivial cases we can solve the problem exactly. Moreover we show that it is possible to realize these system⫹bath Hamiltonians with cold bosons in optical lattices. We thus extend the approach of Jané et al.关7兴 showing that optical lattices can be useful as open quantum system simulators.
Our analysis can be framed in the context of the recently growing interest in the study of decoherence due to spin
baths, see Refs.关8–20兴. Starting from the seminal paper of Zurek关8兴, several papers analyzed the decoherence due to a collection of independent spins. Cucchietti, Paz, and Zurek 关9兴 derived several properties of the Loschmidt echo starting from fairly general assumptions for the distribution of the splittings of the bath spins. The effect of an infinite-range interaction among the spins was introduced in Ref.关10兴; the same model was further exploited by Dawson et al.关11兴 to relate decoherence to the entanglement present in the bath. Properties of decoherence in presence of symmetry breaking, and the effect of critical behavior of the bath were discussed in Refs.关12,13兴. Most of the works done so far are based on the so called “central spin model,” where the two-level sys-tem is coupled isotropically to all the spins of the bath. This assumption tremendously simplifies the derivation, but at the same time it may introduce some fictitious symmetries which are absent in realistic systems. Moreover it can be very hard to simulate it with engineered baths. A crucial feature of our work is the assumption that the two-level system interacts with only few spins of the bath. As we will show, this intro-duces qualitative differences as compared to the central spin model, moreover it is amenable to an experimental imple-mentation with optical lattices.
The paper is organized as follows. In the next section we introduce the system⫹bath model Hamiltonians that will be studied throughout this paper and we define the central quan-tity that will be used in order to characterize the decoherence of the system, i.e., the Loschmidt echo. We then show in Sec. III how optical lattices can be used to simulate the class of Hamiltonians introduced in Sec. II. The rest of the paper is devoted to the derivation and to the analysis of our results, both for a single system-bath link共Sec. IV兲, and for multiple links共Sec. V兲. When the two-level system is coupled to an
XY or XX model, it is possible to derive an exact result for
the Loschmidt echo. This is explained in Sec. IV A, where PHYSICAL REVIEW A 75, 032333共2007兲
we also discuss in detail its short- and long-time behavior and relate it to the critical properties of the chain. Further insight is obtained by perturbative calculations which agree very well, in the appropriate limits, with the exact results. In Sec. IV B we present our results for the Heisenberg bath. In this case an analytic approach is not possible. Here we solve the problem by means of the time-dependent density matrix renormalization group 共t-DMRG — see Appendix A兲. In Sec. IV C we analyze the possible relation between decoher-ence and entanglement properties of the environment: we relate the short-time decay of coherences to the two-site nearest-neighbor concurrence inside the bath. In Sec. V we extend our results to the case in which the system is coupled to an arbitrary number of bath spins; a regime in which the decoherence is substantially independent of the coupling strength between the system and the environment is dis-cussed in Sec. V A. Finally, in Sec. VI we draw our conclu-sions. In the Appendixes we give some technical details on the numerical DMRG approach共Appendix A兲, we provide explicit expressions for the Fermion correlation functions needed for the evaluation of the Loschmidt echo in the XY model共Appendix B兲 and we briefly discuss the central spin model 共Appendix C兲 to make our paper self-contained. A brief account of some of the results discussed in this paper is presented in Ref.关21兴.
II. THE MODEL
The model we consider consists of a two-level quantum object共qubit兲 S coupled to an interacting spin bath E com-posed by N spin-1 / 2 particles关see Fig.1共a兲兴: the idea is to study how the internal dynamics of E affects the decoherent evolution of S.
In our scheme the global system S + E is fully character-ized by a standard Hamiltonian of the form
H = HS+HE+Hint, 共1兲
withHS,Ebeing the free Hamiltonians of S and E and Hint being the coupling term. Without loss of generality we will assume the free Hamiltonian of the qubit to be of the form
HS= e
2 共1 −z兲 =e兩e典具e兩, 共2兲 with␣ being the Pauli matrices of S共␣= x , y , z兲, and 兩e典 its excited state共the ground state being represented by the vec-tor兩g典兲. On the other hand, the environment will be modeled by a one-dimensional quantum spin-1/2 chain described by the Hamiltonian HE= − J 2
兺
j 关共1 +␥兲j x j+1 x +共1 −␥兲j y j+1 y +⌬j z j+1 z + 2j z兴, 共3兲wherei␣ 共␣= x , y , z兲 are the Pauli matrices of the ith spin. The sum over j goes from 1 to N − 1 for open boundary conditions, or from 1 to N for periodic boundary conditions 共where we assume that N+1␣ ⬅1␣兲. The constants J, ⌬, ␥, and, respectively, characterize the interaction strength be-tween neighboring spins, the anisotropy parameter along z and in the xy plane, and an external transverse magnetic field. The Hamiltonian共3兲 has a very rich structure 关22兴. For the sake of simplicity, we shall consider the following para-digmatic cases.
The XY model in a transverse field — see Sec. IV A. Here one has⌬=0 and ,␥generic. For 0⬍␥ⱕ1, Eq. 共3兲 belongs to the Ising universality class, and it has a critical point at 兩c兩 =1; for ␥= 0 it reduces to the XX universality class, which is critical for兩兩 ⱕ1.
The XXZ anisotropic Heisenberg model — see Sec. IV B. Here one has,␥= 0 and ⌬ generic. In this case the Hamil-tonian 共3兲 is critical for −1ⱕ⌬ⱕ1 while it has ferromag-netic or antiferromagferromag-netic order for⌬⬎1 or ⌬⬍−1, respec-tively.
Finally, the qubit S is coupled to the spin bath through a dephasing interaction of the form
Hint= −⑀
兺
j=j1 jm 兩e典具e兩j z , 共4兲where ⑀ is the coupling constant, and the link number m counts the number of environmental spins 共labeled by
j1, . . . , jm兲 to which S is coupled — Fig. 1共a兲 refers to the case where S is interacting with the first spin of an open-boundary chain, i.e., m = 1, j1= 1.
By varying the parameters m ,⌬,␥,, and ⑀, the above Hamiltonians allow us to analyze several nontrivial S + E sce-narios. Moreover we will see in Sec. III that it is possible to use optical lattices manipulation techniques关7,23兴 to experi-mentally simulate the resulting dynamical evolution.
Decoherence: The Loschmidt Echo. With the choice of the
coupling共4兲 the populations of the ground and excited states of the qubit do not evolve in time, since关z,H兴=0. Conse-FIG. 1. 共Color online兲 共a兲 A sketch of the system-plus-bath
model we consider in this work. The two-level system共at position zero兲 is coupled to thezcomponent of the first spin of the chain
that acts as a spin bath. Atoms in an optical lattice can simulate this controlled decoherence by means of series of lasers共b兲 and lattice displacements共c兲, which allow one to realize both the interaction of the bath with an external magnetic field and the anisotropic ex-change coupling present in Eq.共3兲.
ROSSINI et al. PHYSICAL REVIEW A 75, 032333共2007兲
quently, no dissipation takes place in the model and the qubit evolution is purely decoherent: the system S loses its coher-ence without exchanging energy with the bath 关24,25兴. To study this effect of pure phase decoherence, we suppose that at the beginning the qubit is completely disentangled from the environment — namely, at time t = 0 the global system wave function is given by
兩⌿共0兲典 = 兩共0兲典S丢兩共0兲典E, 共5兲 where兩共0兲典S= cg兩g典+ce兩e典 is a generic superposition of the ground and the excited state of S and兩共0兲典E⬅兩G典E is the ground state of the environment HamiltonianHE. The global evolution of the composite system under the Hamiltonian共1兲 will then split into two terms: one where E evolves with the unperturbed Hamiltonian Hg⬅HE and the other where E evolves with the perturbed Hamiltonian
He⬅ HE+具e兩Hint兩e典. 共6兲
As a result, one has
兩⌿共0兲典 → 兩⌿共t兲典 = cg兩g典丢兩g共t兲典E+ cee−iet兩e典丢兩e共t兲典E, 共7兲 where the two branches of the environment are 兩g共t兲典E = e−iHgt兩共0兲典E and 兩e共t兲典E= e−iHet兩共0兲典E. Therefore the evolution of the reduced density matrix⬅TrE兩⌿典具⌿兩 of the two-level system corresponds to a pure dephasing process. In the basis of the eigenstates兵兩g典,兩e典其, the diagonal termsgg andeedo not evolve in time. Instead the off-diagonal terms will decay according to
eg共t兲 =eg共0兲e−ietD共t兲, where
D共t兲 ⬅ 具g共t兲兩e共t兲典 = 具共0兲兩eiHgte−iHet兩共0兲典 共8兲 is the decoherence factor. The decoherence of S can then be characterized by the so-called “Loschmidt echo,” i.e., by the real quantity
L共t兲 ⬅ 兩D共t兲兩2=兩具G兩e−i共HE+具e兩Hint兩e典兲t兩G典兩2, 共9兲
where in the equality term we used the fact that兩共0兲典=兩G典 is the ground state of the Hamiltonian HE; more precisely, the quantity in Eq.共9兲 expresses the “survival probability” of the ground state ofHEevolved by the perturbed Hamiltonian
He. On one hand, values ofL共t兲 close to 1 indicate a weak interaction between the environment and the qubit关the case
L共t兲=1 corresponds to total absence of interaction兴. On the
other hand, values ofL共t兲 close to 0 correspond instead to a strong suppression of the qubit coherence due to the interac-tion with E 关for L共t兲=0 the qubit is maximally entangled with the environment, and its density matrix becomes di-agonal兴. In the following we will analyze the time evolution of the Loschmidt echo to study how the internal dynamics of the environment E affects the decoherence of the system S. We will start with the case in which Sis coupled to a single spin of the chain E共i.e., m=1, see Sec. IV兲. The case with multiple links will be instead discussed in Sec. V.
For the sake of completeness, we finally notice that, since the function 共9兲 measures the overlap between the time-evolved states of the same initial configuration under two slightly different Hamiltonians, one can use it as an indicator of the stability of motion. This kind of analysis has been performed in Ref. 关26兴, where the fidelity between the ground states of quadratic Fermi Hamiltonians has been ana-lyzed in connection with quantum phase transitions.
III. SIMULATION OF OPEN QUANTUM SYSTEMS BY OPTICAL LATTICES
Before analyzing in detail the temporal evolution of the Loschmidt echo 共9兲, in this section we present a method which would allow one to experimentally simulate the dy-namics induced by the Hamiltonian of Eq.共1兲 in a realistic setup. The S + E system introduced in Sec. II can be seen as an “inhomogeneous” spin network with N + 1 sites, where one of the spins共say, the first兲 plays the role of the system of interest S while the remaining N play the role of the environ-ment E. This immediately suggests the possibility of simu-lating the dynamical evolution of such a system on optical lattices by employing the techniques recently developed in Refs.关7,23兴. An important aspect of our scheme, however, is the fact that we assume the coupling between the system S and E to be independent from the couplings among the N spins which compose the environment. Analogously the free Hamiltonian of S is assumed to be different from the on-site terms of the free Hamiltonian of E. On one hand, this allows us to study different environment Hamiltonians without af-fecting the coupling between E and S. On the other hand, this also allows us to analyze different S + E coupling regimes 共e.g., strong, weak兲 without changing the internal dynamics of the bath.
In order to include these crucial ingredients we found more convenient to follow the scheme developed by Jané et
al. in Ref.关7兴 共the optical lattice manipulation schemes de-veloped by Duan et al. in Ref.关23兴 seem to be less adequate to our purposes兲. The key advantage is that we can realize the system-plus-bath setup by using a single one-dimensional lattice in which the quantum system is placed on a given lattice site关for example, the first one, as in Fig. 1共b兲兴. The different Hamiltonians for the system and for the bath are realized by specific pulse sequences which simulate the dy-namics of the model. The same holds for the coupling Hamiltonian of the two-level system with the bath, which is different from the couplings within the bath. In Fig. 1 the leftmost atom simulates the two-level system, the coupling to the second site is the interaction between the quantum sys-tem and the environment, the rest of the chain is the inter-acting spin environment.
Jané et al. 关7兴 showed that atoms loaded in an optical lattice can simulate the evolution of a generic spin Hamil-tonian in a stroboscopic way when subjected to appropriate laser pulses, Fig. 1共b兲, and controlled displacements, Fig. 1共c兲, which allow us to implement the single-site and two-site contributions to the Hamiltonian. The key point is that in our case the sequences of gates need to allow for discrimi-nating between the system and the bath. The types of baths
DECOHERENCE INDUCED BY INTERACTING QUANTUM… PHYSICAL REVIEW A 75, 032333共2007兲
that one can simulate by these means embrace Ising, XY and Heisenberg exchange Hamiltonian. Therefore, by varying the parameters of the optical lattice, it is possible to test the impact of the different phases共critical, ferromagnetic, anti-ferromagnetic, etc.兲 of the environment on the decoherence of the two-level system.
Following the idea of a universal quantum simulator de-scribed in Ref. 关7兴, the time evolution operator associated withH over a time t can be simulated by decomposing it into a product of operators acting on very short timest:
e−iHt= lim n→⬁
冋
U0z共e兲U0,1 zz 共⑀兲兿
j=1 N Uj z冉
J 2冊
⫻ Uj,j+1xx冉
J共1 +␥兲 2冊
Uj,j+1 yy冉
J共1 −␥兲 2冊
Uj,j+1 zz冉
J⌬ 2冊
册
n , 共10兲 where= t / n, Uj z共兲⬅eizj, Uj,k␣共兲⬅eij␣k, and the index 0 labels the two-level system S. For ␣苸兵x,y其 one can write
Uj,k␣␣= Vj␣Vk␣Uj,k zz
Vk␣†V␣†j , where V␣j=共1−ij␣兲/
冑
2 are fast ho-mogeneous local unitary operations. These can be realized with single atoms trapped in an optical lattice关7兴, each hav-ing two relevant electronic levels共兩0典j,兩1典j兲 interacting with a resonant laser according toHj L
=⍀共ei兩1典j具0兩 + e−i兩0典j具1兩兲. 共11兲 The evolution under the Hamiltonian 共11兲, UjL共t,兲 ⬅e−iHj
L
t, yields the single-qubit operations V j x = ULj
共
4⍀, 0兲, Vj y = Uj L共
2⍀, 0兲Uj L共
3 4⍀,2兲
and Uj z共兲=UjL共
2⍀,+兲
Uj L共
2⍀, 0兲, whence Uj z共
−2兲
= ij z , while Uj L共
2⍀兲
= ij x . These operations can be made very fast by simply increasing the laser inten-sity and thereby the Rabi frequency. They can be performed either simultaneously on all qubits, by shining the laser ho-mogeneously onto all atoms, or selectively on some of them, by focusing it appropriately关see Fig.1共b兲兴. For our purposes the individual addressing is needed only for the atom in po-sition 0, which represents the quantum system; this is any-way the minimal physical requirement for being able to monitor its state during the evolution.Two-qubit operations can be performed by displacing the lattice in a state-selective way 关27兴, so that state 兩0典j兩1典j+1 acquires a phase factor e−i, as experimentally realized in Ref.关28兴. The resulting gate Gj,j+1共兲 can be composed with x rotations to yield U j,j+1 zz 共兲=ei关Gj,j+1共2兲 j x j+1 x 兴2. This will affect all atoms from 0 to N. Since we want a different coupling for the 兵01其 pair than for all others, we need to erase the effect of the interaction for that specific pair using only local operations on atom 0, as in the sequences 关0z U01xx共兲兴2=关0z U01yy共兲兴2=关0x U01zz共兲兴2=1. Defining U 丢 ␣␣共兲 ⬅兿j=1 N
Uj,j+1␣␣ 共兲, we can generate each simulation step in Eq.
共10兲 as Un⬅
再
0 x U丢zz冋
冉
⑀−J⌬ 2冊
2册
冎
2冋
0z U丢xx冉
−J共␥+ 1兲 4冊
册
2 ⫻冋
0z Uyy丢冉
J共␥− 1兲 4冊
册
2 U丢zz共−⑀兲U0z共e兲U丢z冉
J 2冊
共12兲 involving only global lattice displacements, global laser-induced rotations and local addressing of atom 0.We point out that an alternative scheme exists关23兴, based on tunnel coupling between neighboring atoms rather than on lattice displacements, which can attain the simulation de-scribed here for the special case ␥= 0 and would require some additional stroboscopic steps in order to reproduce the general case. We also note that, beyond the possibility of using optical lattices, trapped ions interacting with lasers may also be good candidates as quantum simulators of inho-mogeneous spin systems 关29兴, thus making it possible to implement our open quantum system model.
Apart from 1D spin baths, this approach can be extended to other types of environment. For example, it would be quite interesting to consider, as an engineered bath, a 3D optical lattice. In addition to being feasible from an experi-mental point of view, this could be useful in studying, for instance, the situation found in solid-state NMR 关30兴. It would also be intriguing to study the Bose-Hubbard model as a bath, which would make the experimental realization even simpler 关31兴. Here we just focus on one-dimensional baths since, in several cases, they are amenable to an exact solu-tion.
IV. RESULTS: THE SINGLE-LINK SCENARIO In this section we analyze the time evolution of the Loschmidt echo from Eq. 共9兲 for several distinct scenarios where the qubit S is coupled to just one spin of the chain, i.e.,
m = 1 in Eq.共4兲. In this case, in the thermodynamic limit the interactionHintbetween the system S and the environment E does not affect the description of the bath HamiltonianHE, since it is local; therefore it can be considered in all senses as a small perturbation of the environment. The bath is effec-tively treated as a reservoir, which is in contact with the system through just one point. In the following we study the cases in which the bath is described by a one dimensional spin-1/2 Ising, an XY 共Sec. IV A兲 and a Heisenberg 共Sec. IV B兲 chain.
A. XY bath
Here we focus on the case of a spin bath E characterized by a free Hamiltonian 共3兲 of the XY form, i.e., with null anisotropy parameter along z 共⌬=0兲. In this case the envi-ronment is a one-dimensional spin-1/2 XY model, which is analytically solvable 关32兴. Below we show that also the Loschmidt echo can be evaluated exactly关21兴, both for open and for periodic boundary conditions.
In the case m⬍N in which the system S is coupled to just some of the spins of the bath E, the perturbed Hamiltonian
Heof Eq. 共6兲 is the Hamiltonian of an XY chain in a
non-ROSSINI et al. PHYSICAL REVIEW A 75, 032333共2007兲
uniform magnetic field. In this circumstance one cannot em-ploy the approach of Ref.关13兴, and in general the dynamical evolution of the system has to be solved numerically. In-stead, the derivation we present here is analytical and it ap-plies for all values of m = 1 , . . . , N. In the following we will present it for the case of a generic m but, in the rest of the section, we will explicitly discuss its results only for the single-link case共i.e., m=1兲.
The first step of the analytical derivation is a Jordan-Wigner transformation共JWT兲, in order to map both Hamil-toniansHgandHeonto a free-Fermion model, described by the quadratic form关32兴
H =
兺
i,j冋
ci † Ai,jcj+ 1 2共ci † Bi,jcj †+ H.c.兲册
+1 2兺
i Ai,i, 共13兲 where ci, ci †are the annihilation and creation operators for the spinless Jordan-Wigner fermions, defined by
ck= exp
冉
i兺
j=1 k−1j+−j
冊
k−. The two matrices A, B are given by关A兴j,k= − J共␦k,j+1+␦j,k+1兲 − 2共 +⑀j兲␦j,k, 共14兲 关B兴j,k= −␥J共␦k,j+1−␦j,k+1兲, 共15兲 where⑀j= 0 for Hg, while
⑀j=
再
⑀ if S is coupled to the jth spin,
0 elsewhere, 共16兲
forHe. A generic quadratic form, such as Eq.共13兲 共where A is a Hermitian matrix, due to the Hermiticity ofH, and B is antisymmetric, due to the anticommutation rules among the
ci兲, can be diagonalized in terms of the normal-mode spinless
Fermi operators兵k,k †其: H =
兺
k Ek冉
k † k− 1 2冊
, 共17兲wherek=兺i共gk,ici+ hk,ici†兲, or in matrix form
ជ= g · cជ+ h · cជ†. 共18兲 If we rewrite the two change-of-basis matrices g and h as
gk,i⬅ 1
2共k,i+k,i兲 and hk,i⬅ 1
2共k,i−k,i兲, we eventually ar-rive at the following coupled linear equations, whose solu-tion permits to find the eigenbasis of the nonuniform Hamil-tonian in Eq.共13兲:
ជk共A − B兲 = Ekជk,
ជk共A + B兲 = Ekជk. 共19兲 Since A is symmetric and B is antisymmetric, all of the Ek are real; also the gk,iand the hk,ican be chosen to be real. The canonical commutation rules for the normal-mode operators impose the constraints g · gT+ h · hT=1; g · hT− h · gT= 0.
It is convenient to rewrite the spin-bath Hamiltonian-plus-interactionHein the form
He= 1 2⌿ †C⌿, 共20兲 where⌿†=共c 1 †¯cN†c
1¯cN兲 共ci are the corresponding spin-less Jordan-Wigner fermion operators兲 and C=z丢A + iy
丢B is a tridiagonal block matrix.
The Loschmidt echo, Eq. 共9兲, can then be evaluated by means of the following formula关33兴:
L共t兲 = 兩具e−it兺i,jCi,j⌿i
†⌿
j典兩 = 兩det共1 − r + re−iCt兲兩, 共21兲 where the elements of the matrix r are simply the two-point correlation functions of the spin chain rij=具G兩⌿i†⌿j兩G典, where兩G典 is the ground state of Hg. An explicit expression of the correlators rijas well as of the matrix e−iCtis given in Appendix B.
Equation 共21兲 provides an explicit formula for the Loschmidt echo in terms of the determinant of a 2N⫻2N matrix, whose entries are completely determined by the di-agonalization of the two linear systems given by Eq. 共19兲. This is one of the central results of our work. It allows us to go beyond the central spin model where all the spins of E are uniformly coupled with S 共m=N兲, whose solution 共at least for periodic boundary conditions兲 was discussed in Refs. 关13–15兴 and, for the sake of completeness, has been re-viewed in Appendix C关see Eq. 共C6兲兴. Notice that, similarly to the central spin model, this formula allows to study a system composed of a large number of spins in the bath N ⬃102− 103, as it only requires manipulations of matrices whose size scales linearly, and not exponentially, with N.
1. Ising bath: General features
The generic behavior ofL as a function of time for dif-ferent values of, and fixed coupling constant⑀, is shown in Fig.2. For⬍1 the echo oscillates with a frequency propor-tional to ⑀, while for ⬎1 the oscillation amplitudes are drastically reduced. The Loschmidt echo reaches its mini-mum value at the critical pointc= 1, thus revealing that the decoherence is enhanced by the criticality of the environ-ment. Since the chain is finite, at long times there are reviv-als of coherence 关34兴; in the thermodynamic limit N→⬁ they completely disappear. In any case, as it can be seen from the figure, already for N = 300 spins there is a wide interval where the asymptotic behavior at long times can be analyzed.
2. Short-time behavior
At short times the Loschmidt echoL decays as a Gauss-ian关35兴:
L共t兲 ⬃ e−␣t2
, 共22兲
as can be seen in Fig.3共a兲, which shows a magnification of the curves from Fig.2for small times. This behavior can be predicted within a second-order time perturbation theory in the coupling⑀between the system and the bath: if⑀is small as compared to the interaction J between neighboring spins in the bath共⑀J兲, the decoherence factor D共t兲 of Eq. 共8兲 can be expanded in series of⑀:
DECOHERENCE INDUCED BY INTERACTING QUANTUM… PHYSICAL REVIEW A 75, 032333共2007兲
具eiHgt e−iHet典 =
冓
T冋
exp冉
− i冕
0 t dt⬘
eiHgt⬘We−iHgt⬘冊
册
冔
⯝ 1 +⑀1+⑀22, 共23兲whereT is the time ordered product and W=⑀1z accounts for the interaction of the two-level system with the spin chain. The above expression has to be evaluated on the ground state ofHg, therefore it is useful to rewrite the inter-actionW in terms of the normal mode operatorsk共g兲ofHg:
W =⑀
冋
2兺
i,j共gi,1i共g兲†+ hi,1i共g兲兲共gj,1共g兲j + hj,1共g兲†j 兲 − 1
册
. The first-order term then reads1= − it
冉
2兺
i兩hi,1兩2− 1
冊
, 共24兲 while the second-order term is given by2= −
冕
0 t dt⬘
冕
0 t⬘ dt⬙
冋
4兺
i⫽j 关共gi,1hj,1兲2− gi,1gj,1hi,1hj,1兴e−i共Ei+Ej兲共t⬘−t⬙兲+
冉
2兺
i兩hi,1兩2− 1
冊
2册
. 共25兲 The Loschmidt echo is then evaluated by taking the square modulus of the decoherence factorL共t兲 ⯝ 1 − 4⑀2t2
兺
i⫽j关共gi,1hj,1兲2− gi,1gj,1hi,1hj,1兴. 共26兲 In Fig. 3共b兲 the initial Gaussian rate ␣ is plotted as a function of ; circles represent numerical data, while the solid curve is the perturbative estimate obtained from second-order perturbation theory, given by Eq.共26兲. In Fig.4 we analyze the behavior of the first derivative of the rate ␣共,⑀兲 as a function of the distance from criticality ⌬⬅ −c, for a fixed number N of spins in the chain. As predicted by the perturbative estimate,␣scales as⑀2; most remarkably, its first derivative with respect to the transverse field diverges if the environment is at the critical pointc. In the inset we show that ␣ diverges logarithmically on approaching the critical value, as
␣
= c1ln兩 − c兩 + const. 共27兲 This is a universal feature, entirely due to the underlying criticality of the Ising model.
Our results show that at short times the Loschmidt echo is regular even in the presence of a bath undergoing a phase transition. The critical properties of the bath manifest in the changes ofL when the bath approaches the critical point.
3. Long-time behavior
At long times, for⬎1 the Loschmidt echo approaches
⑀= 0.25. The minimum value L0c reached by the Loschmidt echo decays with the size N as
L0 c
= L0
1 +ln共N兲. 共33兲
Notice also that revivals are due to the finiteness of the en-vironmental size共as in Fig.2兲 关34兴. We expect that the co-herence loss should go to zero at the thermodynamic limit
N→⬁; this is hard to see numerically, since the decay is
logarithmic, and the actual value ofL0cis still quite far from zero, even for N = 2000 spins.
5. XY baths with arbitrary xy anisotropy␥
The properties described in the previous sections are typi-cal of the Ising universality class, indeed they remain quali-tatively the same as far as⌬=0,␥⫽0 in Eq. 共3兲: in particular this class of models has one critical point at=1. Following our previous analysis of the Ising model共␥= 1兲, we focus on the two distinct regions of short- and long-time behavior, whose features are depicted in Fig.9.
At small times the decay is again Gaussian, as in Eq.共22兲, and the first derivative of the decay rate␣with respect to diverges at c 关see Fig. 9共a兲兴. As far as the xy anisotropy
decreases, the system approaches a limiting case in which −␣= −⬁, while
+␣= 0. The asymptotic value L
⬁ at long times shows a cusp at=1; notice that, as the anisotropy␥ decreases, despite the fact that the decay of coherences at short times is always reduced, their saturation valueL⬁ be-comes lower for⬍1, while gets higher for ⬎1. Therefore decoherence is asymptotically suppressed only in the ferro-magnetic phase, while it is enhanced for ⬍1. As we said before, when␥→0 the system is driven towards a disconti-nuity at=1. This is due to the fact that for ⬎1 the ground state approaches a fully polarized ferromagnetic state. Fi-nally we analyzed the scaling of the decoherence with the size of the environment at criticality. As in the Ising model, the minimum valueL0creached by the Loschmidt echo atc depends on the chain length N as in Eq. 共33兲: L0c=L0/关1 +ln共N兲兴.
The system behavior dramatically changes when it be-comes isotropic, i.e., at␥= 0. In this case the environment is an XX spin chain and it does not belong any more to the Ising universality class. In particular it exhibits a critical be-havior over the whole parameter range 兩兩 ⱕ1, while it is
Configuration 1
Configuration 2
FIG. 6. 共Color online兲 Qubit 共red兲 coupled to two noncommu-nicating N-site chains 共configura-tion 1兲 and to two spins of a 2N-site chain 共configuration 2兲. Here, as an example, we set N = 11. 0 10 20 30 40 50 60
J t
10-15 10-12 10-9 10-6 10-3 100|
δL|
λ=0.99 λ=1 λ=1.01 λ=0.9 λ=1.1 λ=0.5 λ=1.5FIG. 7. 共Color online兲 Absolute differences in the Loschmidt echo decay between configuration 1共qubit coupled to two noncom-municating open N = 200 spin Ising chains兲 and configuration 2 共qu-bit coupled to two opposite spins of a periodic N = 400 spin Ising chain兲. Different curves are for various values of the transverse magnetic field:=0.5 共dotted black兲, 0.9 共dashed red兲, 0.99 共green兲, 1共blue兲, 1.01 共brown兲, 1.1 共magenta兲, 1.5 共cyan兲.
ferromagnetic 共anti-ferromagnetic兲 for ⬎1 共⬍−1兲. The decay of the Loschmidt echo共shown in Fig. 10兲 reflects its critical properties: indeed we found that it behaves as in Eq. 共33兲 over the whole range 兩兩 ⱕ1. In the ferromagnetic case instead the coupled qubit does not decohere at all 关i.e.,
L共t兲=1兴, since the ground state of the XX model for ⬎1 is
the fully polarized state with all spins parallel to the external field. The first derivative of the initial decay rate ␣ is monotonically negative in the critical region by increasing and diverges for →1−, while it is strictly zero for ⬎1.
Also the plateauL⬁as a function of presents a discontinu-ity in =1, since it drops to zero for →1−, and it equals one for⬎1.
B. XXZ-Heisenberg chain bath
In this section we consider a single-link共m=1兲 XXZ an-isotropic Heisenberg chain关=␥= 0 in Eq. 共3兲兴 with aniso-tropy parameter ⌬. We resort to the numerical t-DMRG to compute the Loschmidt echo 共9兲, since this specific spin model is not integrable. A brief introduction to the static and the time-dependent DMRG algorithm is given in Appendix A. We consider open boundary conditions, since DMRG with periodic boundaries intrinsically gives much less accu-rate results关36兴. The evaluation of the Loschmidt echo with the t-DRMG is more time and memory consuming than its exact calculation in the solvable cases, therefore at present we could not study the behavior ofL共t兲 at long times. None-theless we are able to fully analyze the short-time behavior in systems with environment size of up to N⬃102spins. In the following we show the results concerning the coupling of the two-level system to one spin in the middle of the chain: this corresponds to the case with less border effects. We numeri-cally checked that the results are not qualitatively affected from this choice, but changing the system-environment link position results in a faster appearance of finite size effects due to the open boundaries.
In Fig.11 we plot the decay of the Loschmidt echo as a function of time, for various values of the anisotropy⌬ and fixed coupling strength ⑀= 0.1. In the ferromagnetic zone outside of the critical region共⌬ⱖ1兲 L does not decay at all, as a consequence of the ground state full polarization. In the critical region and for long times, the Loschmidt echo decay is proportional to the modulus of⌬ and for ⌬⬍−1 it slows
0 0.5 1 1.5 2
λ
0 0.01 0.02 0.03 0.04 0.05 0.06α
0 0.5 1 1.5 2λ
0.875 0.9 0.925 0.95 0.975 1L
∞a)
b)
FIG. 9. 共Color online兲 Single-link XY models 共m=1,⌬=0兲: Short- and long-time behavior of the Loschmidt echo for a qubit coupled to a periodic N = 300 spin XY chain, with a coupling strength⑀=0.25. 共a兲 Small times 共initial Gaussian decay rate ␣兲; 共b兲 long times共saturation values L⬁兲. The various curves are for dif-ferent anisotropy values␥=0 共XX model, full black circles兲, 0.1 共red squares兲, 0.25 共green diamonds兲, 0.5 共blue triangles up兲, 0.75 共brown triangles down兲, 0.9 共magenta triangles left兲, 1 共Ising, cyan triangles right兲. 0 0.25 0.5 0.75 1 1.25 1.5
λ
-0.8 -0.6 -0.4 -0.2 0d
λα
0 0.25 0.5 0.75 1 1.25 1.5λ
0.4 0.5 0.6 0.7 0.8 0.9 1L
∞
down until it is completely suppressed in the perfectly anti-ferromagnetic regime 共⌬→−⬁兲. The short-time decay is again Gaussian at short times and the rate␣is shown in Fig. 12for various sizes of the bath. We notice two qualitatively different behaviors at the boundaries of the critical region: at ⌬= +1 there is a sharp discontinuity, while at ⌬=−1 the curve is continuous. In the critical region −1ⱕ⌬ⱕ1 the ini-tial decay rate␣is constant and reaches its maximum value due to the presence of low energy modes, while in the ferro-magnetic phase⌬ⱖ1 it is strictly zero. In the figure, finite size effects are evident: indeed, contrary to the XY model,
the decay rate ␣ changes with the bath size N outside the critical region. Nonetheless, as the system approaches the thermodynamic limit, the dependence on N weakens; more-over, even if the sizes of the simulated system do not permit us to guarantee what exactly happens in the limit N→⬁, numerical data seem to indicate that, while at ⌬=−1 the curve␣共⌬兲 appears to remain continuous, its first derivative with respect to⌬ tends to diverge. On the contrary, the fer-romagnetic transition ⌬=1 is discontinuous, independently of N.
C. Decoherence and entanglement
We propose here to establish a link between decoherence effects on the system and entanglement inside the environ-ment. This study can be justified by the fact that decoherence properties of the qubit system seem to be intrinsically related to quantum correlations of the bath, as the proximity to criti-cal points reveals. On the other hand, entanglement quanti-fies the amount of these correlations that do not have classi-cal counterparts, and it has been widely studied in the recent years, especially in connection with the onset of quantum phase transitions关37–40兴.
In Ref.关11兴 it was shown that two-party entanglement in the environmental bath of a central spin model can suppress decoherence; this effect has been explained as a consequence of entanglement sharing, and it was supposed to be common to any system whose environment maintains appreciable in-ternal entanglement, while evolving in time. We now char-acterize a more complex case of system-plus-bath coupling, given by Eq.共1兲, with a richer structure in the ground state entanglement which suggests the following picture, valid at short times for Jt1.
We expect that when the decay of coherences is quadratic, only short range correlations in the bath are important, there-fore it seems natural to relate the short-time decay rate␣to the two-site nearest neighbor entanglement of the chain. All the information needed to analyze bipartite entanglement be-tween two spins inside the chain, say i and j, is contained in the two-qubit reduced density matrixij, obtained after trac-ing out all the other spins. We use the concurrence C共兩i − j兩兲 as an entanglement measure for arbitrary mixed stateij of two qubits, defined as关41兴
C共兩i − j兩兲 = max共1−2−3−4,0兲, 共34兲 where1 are the square roots of the eigenvalues of the ma-trixij˜ij, in decreasing order; the spin flipped matrix ofij is defined as ˜ij=共y丢y兲ij
*共
y丢y兲, and the complex conjugate is computed in the canonical basis 兵兩↑ ↑典,兩↑ ↓典,兩↓ ↑典,兩↓ ↓典其. The entanglement of formation of the mixed state ij is a monotonic function of the concur-rence.
We start our analysis by considering again the XY spin bath: the concurrence C共k兲 in terms of one-point and two-point spin-correlation functions can be analytically evaluated 关42,43兴. As long as ␥⫽0, this system belongs to the Ising universality class, for which it has been shown that the con-currence between two spins vanishes unless the two sites are at most next-to-nearest neighbor 关37,38兴. The
nearest-0 1 2 3 4
J t
0.994 0.996 0.998 1L
∆=0.9 ∆=-1 ∆=-0.9 ∆=-1.5 ∆=-0.5 ∆=0.5 ∆=0 ∆=-2 ∆=-2.5FIG. 11. 共Color online兲 Single-link XXZ model 共m=1,=␥ = 0兲: Loschmidt echo as a function of time for a qubit coupled to the central spin of an open ended N = 100 spin XXZ-Heisenberg chain, with coupling strength⑀=0.1. The various curves are for different values of the anisotropy:⌬=0.9 共black兲, 0.5 共red兲, 0 共green兲, −0.5 共blue兲, −0.9 共brown兲, −1 共magenta兲, −1.5 共violet兲, −2 共cyan兲, −2.5 共orange兲; curves corresponding to noncritical situations are dashed.
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4
∆
0 0.002 0.004 0.006 0.008 0.01α
critical regionFIG. 12. 共Color online兲 Scaling of the decay rate ␣ at short times for an open-ended XXZ-Heisenberg chain as a function of⌬. The various symbols are for different sizes of the chain: N = 10 共black circles兲, 20 共red squares兲, 50 共green diamonds兲, 100 共blue triangles兲. The coupling strength between the qubit and the chain is kept fixed at⑀=0.1.
ROSSINI et al. PHYSICAL REVIEW A 75, 032333共2007兲
neighbor concurrence C共1兲 presents a maximum that occurs at=m⬎1, and it is not related to the critical properties of the model. For⬎mit monotonically decreases to zero, as the ground state goes towards the ferromagnetic product state. For ⬍m the concurrence decreases and precisely reaches zero at K=
冑
共1+␥兲共1−␥兲, where the ground state exactly factorizes关44兴.In the proximity of the quantum phase transition, the en-tanglement obeys a scaling behavior关37兴; in particular at the thermodynamic limit the critical point is characterized by a logarithmic divergence of the first derivative of the concur-rence: C共1兲⬃ln兩−c兩. A similar divergence has been
found for the first derivative of the decay rate ␣ of the Loschmidt echo共see the inset of Fig.4兲. In Fig.13we ana-lyzed the dependence of ␣ as a function of the nearest-neighbor concurrence C共1兲. The behavior␣共C兲 is not mono-tonic, since␣monotonically decreases with 关see Fig.3共b兲兴, while C共1兲 has a nonmonotonic behavior. Indeed, for ⬎mand⬍Kthe two quantities are correlated, while they are anti-correlated form⬎⬎K.
A similar scenario has been found in the case of the Heisenberg chain as a spin bath, reported in Fig.14. As for the Ising model, the behavior of␣共C兲 is not monotonic, fol-lowing the concurrence behavior: Indeed C共1兲 is zero for ⌬ⱖ1 共in the fully polarized ground state兲, it monotonically increases while decreasing⌬ up to ⌬=−1, and then starts to decrease again, vanishing in the perfect antiferromagnetic limit ⌬→−⬁ 关45兴. In conclusion, the behavior ␣=␣共C兲 is monotonically increasing in the antiferromagnetic region, is constant at the criticality, and is strictly zero for the ferro-magnetic phase 共see also Fig.12兲, resulting in a direct cor-relation of␣and C共1兲 in the antiferromagnetic phase.
V. RESULTS: THE MULTIPLE-LINKS SCENARIO In this section we study the case in which the system S is nonlocally coupled to more than one spin of the bath E,
namely, we choose m⫽1 in the interaction Hamiltonian Eq. 共4兲. In general, given a number m of links, there are many different ways to couple the two-level system with the bath spins. In the following we will consider two geometrically different setups: a spin-symmetric configuration 共type A兲, where the qubit is linked to some spins of the chain which are equally spaced, and a nonsymmetric configuration共type B兲, where the spins to which the qubit is linked are nearest neighbor. We will also suppose that the interaction strength⑀ between the qubit and each spin of the bath is kept fixed and equal for all the links. A schematic picture of the two con-figurations is given in Fig.15. We will present results con-cerning an environment constituted by an Ising spin chain with periodic boundary conditions. Notice that this system-environment coupling can induce the emergence of criticality in E even for small coupling ⑀J in the central spin scheme, i.e., m = N关13–15兴. Indeed, in this case, a perturba-tion⑀results in a change of the external magnetic field of the chain from to +⑀. If the environment is characterized by ⬍1, at⑀⬎1− the coupling drives a quantum phase tran-sition in the bath关14兴.
共1兲 Short-time behavior. At short times the Loschmidt echo exhibits a Gaussian decay, as in Eq.共22兲, for both con-figurations. For the setup A the decay rate ␣, far from the critical region, scales as
␣⬀ m⑀2, 共35兲
as it can be seen from Fig.16共a兲. The scaling with the num-ber of links m is a consequence of the fact that the short-time behavior is dominated by the dynamics of the environment spins close to those linked to the qubit共see also Sec. IV C兲. Therefore, if the linked spins are not close to each other, they do not interact among them on the short time scale Jt1. Near the critical point this picture is not valid, since long-range correlations between the spins of the bath become im-portant, even at small times. In this region indeed the scaling ␣⬃m is less appropriate, as it can be seen in the inset of Fig. 16共a兲.
The scaling of Eq. 共35兲 does not hold any more in the setup B, as in this case collective modes influence the dy-namics of the system even at short times, since the spins linked to the central spin are close to each other and they are not independent any more. This is clearly visible in Fig. 16共b兲. We notice also that, as m / N increases, ␣ tends to remain constant forⱕ1 and then decreases for ⬎1. In the limiting case m = N,␣ is positive and strictly constant for ⱕc; the first derivative␣ presents a discontinuity at c, showing a divergence from the ferromagnetic zone→1+.
共2兲 Long-time behavior. Here we concentrate on the setup B, since the configuration A is less interesting: indeed for long times the Loschmidt echo behaves as if the qubit was coupled to an environment with a smaller number of spins.
In Fig.17we plotted the asymptotic value of the Loschmidt echoL⬁for the setup B. We observe that, as m increases, the coherence loss enhances, and the valley around the critical point c deepens and gets broader. Notice also that, when approaching the central spin limit, the asymptotic valueL⬁ reaches values very close to zero, even far from criticality. This situation is completely different from what occurs in the single-link scenario where, away from criticality, the Loschmidt echo remains very close to one even for non neg-ligible system-bath coupling strengths 共e.g., for ⑀= J / 4 we foundL⬁⬃0.99, as it can be seen, for example, from Figs.2 and5兲.
Strong coupling to the bath. Under certain conditions
de-coherence induced by the coupling with a bath manifests universal features: In Refs. 关14,20兴 it has been shown that when the coupling to the system drives a quantum phase transition in the environment, the decay of coherences in the system is Gaussian in time, with a width independent of the coupling strength. In particular, for a central spin coupled to an N-spin Ising chain, the Loschmidt echo in Eq. 共C6兲 is characterized by a Gaussian envelope modulating an oscil-lating term L共t兲 = 兩cos共⑀t兲兩N/2e−SN 2t2 , 共36兲 provided that ⬍ 1 and +⑀ 1. 共37兲
The oscillations are not universal, but the Gaussian width SN2 depends only on the properties of the environment Hamil-tonian, in particular it is independent of the coupling strength ⑀and of the transverse magnetic field, while it is propor-tional to the number N of spins in the bath. The case of the central spin model is illustrated in Fig.18共a兲, where the dif-ferent curves stand for various values of⑀.
0 0.5 1 1.5 2 2.5
λ
0 0.01 0.02 0.03 0.04 0.05 0.06α/m
0.96 1 1.04 0.03 0.035 0.04 0.045 0 0.5 1 1.5 2 2.5λ
0 0.01 0.02 0.03 0.04 0.05 0.06α/m
1.5 1.75 2 0.008 0.012 0.016 a) b)FIG. 16.共Color online兲 Decay rate␣ rescaled with respect to the number of links m, in a star symmetric configuration共type A兲 on the left, and in a nonsymmetric configuration共type B兲 on the right. The environment is an N = 300 spin Ising chain with periodic boundary conditions. The interaction strength⑀ is kept fixed for all the links between the qubit and the environment spins:⑀=0.25. Various sym-bols stand for different values of m: 1 共cyan crosses兲, 2 共orange triangles right兲, 3 共magenta triangles down兲, 5 共brown triangles left兲, 10 共blue triangles up兲, 30 共green diamonds兲, 100 共red squares兲, 150共black circles, left兲, 300 共dashed black line, right兲. The insets on the left figure show a magnification of the same plot, centered in proximity of the critical pointc= 1, and far from it.
0 0.5 1 1.5 2 2.5
λ
0 0.2 0.4 0.6 0.8 1L
∞
FIG. 17.共Color online兲 Asymptotic value of the Loschmidt echo for the setup B; the environment is a 300 spin Ising chain, with⑀ = 0.25. Various symbols stand for different values of m: 1 共cyan crosses兲, 2 共orange triangles right兲, 5 共magenta triangles down兲, 10 共brown triangles left兲, 30 共blue triangles up兲, 100 共green diamonds兲, 150共red squares兲, 300 共black circles兲.
ROSSINI et al. PHYSICAL REVIEW A 75, 032333共2007兲
This universal behavior for strong couplings is present in more general qubit-environment coupling scenarios, different from the central spin limit, provided that the same conditions in Eq. 共37兲 on and ⑀ hold. In the Ising bath model, we checked that it remains valid also for a number of system-bath links m⫽N and for different geometries, as it can be seen in Fig.18关panels 共b兲–共d兲兴. The fast oscillations remain proportional to⑀, since they are a consequence of the inter-action⬀⑀ zj
z
between the qubit and the spins of the bath. Instead, in general the envelope is Gaussian only for small times, but its width SN
2
is independent of⑀.
We found that, for the nonsymmetric configuration B, SN2 depends only on the number of links m, being proportional to it. No dependence from the perturbation strength⑀, the size
N of the bath, or the transverse magnetic field has been
observed, provided that mⲏ10, as can be seen from Fig. 19共b兲. Indeed in the setup B the qubit behaves similar to the central spin model: in the regime of strong coupling, the effect of the environment spins not directly linked to the qubit can be neglected since they interact weakly, as com-pared to the system-bath interactions. Therefore a system strongly coupled to m neighbor spins of an N-spin chain decoheres in the same way as if it was coupled to all the spins of an m-spin chain.
The decoherence effects are richer in the geometry A. In this setup the effect of the spins not directly coupled to the qubit cannot be ignored, indeed we found that SN
2
depends on the internal dynamics of the bath. In particular we found that
SN
2⬃ m2, 共38兲
where is the transverse magnetic field 关see Fig.19共a兲and Fig.20兴. Notice that, as it can be seen from Fig. 19共a兲, the
case=0 is rather peculiar, since the system does not deco-here at all, unless it is uniformly coupled to all the spins of the environment. This is due to the fact that an Ising chain in the absence of a transverse field is not capable of transmit-ting quantum information: in this system the density matrix of two non-neighboring spins evolves independently of the other spins.
VI. CONCLUSIONS
In this work we analyzed a model of environment consti-tuted by an interacting one-dimensional quantum spin-1/2
chain. We have shown that this type of baths can be experi-mentally engineered in a fully controllable and tunable way by means of optical lattices, and we provided a detailed de-scription of the pulse sequence needed to realize a pure dephasing model for a qubit system coupled to a spin bath. The open quantum system model we proposed here is sim-pler to implement in an experimental setup, with respect to the widely studied central spin model in which the system is uniformly coupled to all the spins of the environment. None-theless, we have analyzed the crossover from the single link to the central spin model by taking into account different geometries and different number of links, recovering, in the central spin limit, known results present in the literature, and characterizing new behaviors in between.
The coherence loss in the system has been quantified by means of the Loschmidt echo; the rich structure of the envi-ronment baths allowed us to analyze the impact of the dif-ferent phases and of the distance from criticality, and more generally to study the connection between entanglement in-side the bath and decoherence effects in the system. First we considered the case of one link between the qubit and the chain. We showed that at short times the Loschmidt echo decays as a Gaussian, while for long times it approaches an asymptotic value which strongly depends on the transverse field strength. Criticality emerges in both short- and long-time behaviors: the XY spin bath is characterized by the first derivative of the Gaussian width with respect to the field strength that diverges logarithmically atc, while the satura-tion values of the Loschmidt echo at long times exhibits a sharp cusp in correspondence to the critical point. The decay rate at short times due to a Heisenberg spin bath is maximum and constant throughout the whole critical region. We also showed that, contrary to the central spin model, if the envi-ronment is not critical, the decay of coherences in the system is independent of the bath size. Subsequently we related de-coherence properties of the system with quantum correla-tions inside the environment: we studied the Gaussian decay rate at short times as a function of the concurrence of two nearest neighbor spins in the bath. We then analyzed a non-local system-bath coupling in which the qubit is connected to the chain with multiple links, by considering two different setups: a spin-symmetric and a nonsymmetric one. Finally we reported the existence of a coupling-independent regime, with the only hypothesis of having a strong system-bath cou-pling.
ACKNOWLEDGMENTS
We thank D. Cory, G. De Chiara, and M. Rizzi for very useful discussions. We acknowledge support from EC共Grant Nos. QOQIP, RTNNANO, SCALA, SPINTRONICS, and EUROSQIP兲 and MIUR-PRIN. The present work has been partially supported by the National Science Foundation through a grant for the Institute for Theoretical Atomic, Mo-lecular and Optical Physics at Harvard University and Smith-sonian Astrophysical Observatory, and it has been performed within the “Quantum Information” research program of Cen-tro di Ricerca Matematica “Ennio De Giorgi” of Scuola Nor-male Superiore.
APPENDIX A: DMRG APPROACH
An analytic solution for a spin coupled to a Heisenberg
XXZ-model 关i.e., 兵⌬⫽0,=0其 in Eq. 共3兲兴 is not available, therefore a numerical approach is required. We resort to the recently developed time-dependent density matrix renormal-ization group 共t-DMRG兲 with open boundary conditions 关36,46–48兴, in order to calculate the overlap between the ground state兩G典 of Hg and the time evolution of the same state under the HamiltonianHe关see Eq. 共9兲兴.
First, the static DMRG algorithm allows us to evaluate the ground state兩G典 of Hg. This algorithm in its first formulation 关46兴 is an iterative numerical technique to find the ground state of a one-dimensional system constituted by sites which possess local and nearest-neighbor couplings, therefore it is well suited for systems such as the Hamiltonian in Eq. 共3兲. The key strategy of the DMRG is to construct a portion of the system, called the system block, and then recursively enlarge it, until the desired size is reached. At every step the basis of the corresponding Hamiltonian is truncated, so that the size of the Hilbert space remains manageable as the physical system grows. The truncation of the Hilbert space is performed by retaining the eigenstates corresponding to the
D highest eigenvalues of the block’s reduced density matrix.
The t-DMRG is then subsequently used, in order to simu-late the dynamics of 兩G典 under the Hamiltonian He. The t-DMRG algorithm关47兴 is an extension of the static DMRG, which follows the dynamics of a certain state兩典 under the Hamiltonian H of a nearest-neighbor one-dimensional sys-tem. The time evolution can be implemented by using a second-order Suzuki-Trotter decomposition关47兴 of the time evolution operator U = e−iHet:
e−iHet⬇ 共e−iF共␦t/2兲e−iG␦te−iF共␦t/2兲兲n. 共A1兲 Here we have first written the Hamiltonian as He=F+G whereF contains the on-site terms and the even bonds while
G is formed by the odd bonds; n=t/␦t is the time expressed
in number of Trotter steps. We also keep track of the state 兩G典 in the new truncated basis of e−iHet兩G典, so that we can straightforwardly evaluate the overlap in Eq.共9兲. During the evolution the wave function is changing, therefore the trun-cated basis chosen to represent the initial state has to be updated by repeating the DMRG renormalization procedure using the instantaneous state as the target state for the re-duced density matrix.
The results presented in this paper concerning Heisenberg spin baths have been obtained by performing time evolution within a second order Trotter expansion, with a Trotter slic-ing J␦t = 10−3 and a truncated Hilbert space of dimension
D = 100. The evaluation of the Loschmidt echo with the
t-DMRG is much more time and memory expensive than the analytical approach of Eq.共21兲, limiting the bath size in our simulations up to N⬃102spins.
APPENDIX B: FERMION CORRELATION FUNCTIONS We provide an explicit expression for the two-point cor-relation matrix 共r兲ij=具⌿i†⌿j典 of the operators ⌿† =共c1†¯cN†c
1¯cN兲 on the ground state of the system
Hamil-ROSSINI et al. PHYSICAL REVIEW A 75, 032333共2007兲
tonianHg. The 2N⫻2N matrix r is written in terms of the Jordan Wigner fermions兵ck, ck
†其 as r =
冉
具ci†cj典i,j=1,N 具ci†c j †典i,j=1,N 具cicj典i,j=1,N 具cicj
†典i,j=1,N
冊
. 共B1兲 Therefore it is sufficient to express兵ck, ck†其 in terms of the normal mode operators 兵k共g兲,k共g兲†其 which diagonalize the HamiltonianHg, since具共g兲j k共g兲†典=␦jk, the other expectation values of the’s on the ground state being zero. By inverting Eq.共18兲 we getc = gT·+ hT·†, 共B2兲
from which it directly follows that
r =
冉
h共g兲Th共g兲 h共g兲Tg共g兲
g共g兲Th共g兲 g共g兲Tg共g兲
冊
. 共B3兲The superscript共g兲stands for the change-of basis matrices g and h relative to the HamiltonianHg.
The last ingredient for the evaluation of the Loschmidt echo 共21兲 is the exponential eiCt. We introduce the vector ⌫†=共 1 共g兲†¯ N 共g兲† 1 共g兲¯ N
共g兲兲, so that ⌿=U†⌫, where U =
冉
g共e兲 h共e兲
h共e兲 g共e兲
冊
. 共B4兲We can therefore rewrite Eq.共20兲 as
He= 1 2⌿ †C⌿ =1 2⌫ †U · C · U†⌫ ⬅1 2⌫ †D⌫, 共B5兲 where D is a 2N⫻2N diagonal matrix, whose elements are the energy eigenvalues ofHeand their opposites:
D =
冉
E 共e兲 00 − E共e兲
冊
. 共B6兲It then follows that C = U†· D · U, from which one can easily calculate the exponential eiCt.
APPENDIX C: CENTRAL SPIN MODEL
If the qubit is uniformly coupled to all the spins of the chain, the effect of the interaction in Eq. 共4兲 is simply of renormalizing the transverse field strength in the bath Hamil-tonian: for⌬=0 both Hamiltonians HgandHecorrespond to an XY model with anisotropy ␥, uniform couplings J, and local magnetic field and +⑀, respectively. They can be diagonalized via a standard JWT, followed by a Bogoliubov
rotation 关32兴. The normal mode operators that diagonalize
He, satisfying the fermion anticommutation rules, are given by k共e兲=
兺
j e−2ijk/N冑
N兿
l⬍j l x共uk共e兲 j +− iv k 共e兲 j −兲, 共C1兲 where the coefficients uk共e兲= cos共k/ 2兲, vk共e兲= sin共k/ 2兲 depend on the angle
k共⑀兲 = arctan
冋
− sin共2k/N兲cos共2k/N兲 − 共 +⑀兲
册
. 共C2兲The corresponding single quasi-excitation energy is given by
Ek共e兲⬅Ek共⑀兲: Ek共⑀兲 = 2J
冑
冋
cos冉
2k N冊
−共 +⑀兲册
2 +␥2sin2冉
2k N冊
. 共C3兲 The HamiltonianHg can be diagonalized in a similar way; the corresponding normal-mode operators are connected to the previous ones by a Bogoliubov transformation±k共g兲= cos共␣k兲±k共e兲− i sin共␣k兲 ⫿k
共e兲 †, 共C4兲 with␣k=关k共0兲−k共⑀兲兴/2.
The spin chain is initially in the ground state兩G典 of Hg; this state can be rewritten as a BCS-like state
兩G典 =
兿
k⬎0关
cos共␣k兲 − i sin共␣k兲k共e兲†−k共e兲†
兴
兩Ge典, 共C5兲 where兩Ge典 is the ground state of He. This expression allows one to rewrite the Loschmidt echoL共t兲 in Eq. 共9兲 in a simple factorized formLm=N共t兲 =
兿
k=1 N/2关1 − sin2共2␣k兲sin2共Ekt兲兴. 共C6兲 Equation共C6兲 provides a straightforward formula in order to calculate the Loschmidt echo for a central spin coupled uniformly to all the spins of the bath. Nonetheless this model, although in some circumstances it can reveal the emergence of a quantum phase transition in the environment 关13–15兴, does not provide an effective physical description of a standard reservoir, since the coupling with the system is highly non-local and it can drive the evolution of the bath itself. However, it is useful throughout the paper to test the convergence of our results to the ones presented in Refs. 关13–15兴, in the limit m→N.
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DECOHERENCE INDUCED BY INTERACTING QUANTUM… PHYSICAL REVIEW A 75, 032333共2007兲
