Proof Copy [at10041] 020610pra

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PROOF COPY [AT10041] 020610PRA PHYSICAL REVIEW A 74, 1 共2006兲 Decay and storage of multiparticle entangled states of atoms in collective thermostat A. M. Basharov,* V. N. Gorbachev, and A. A. Rodichkina† Laboratory for Quantum Information & Computation, Aerospace University, St.-Petersburg 190000 Bolshaia Morskaia 67, Russia 共Received 7 June 2006兲 O PR We derive a master equation describing the collective decay of two-level atoms inside a single mode cavity in the dispersive limit. By considering atomic decay in the collective thermostat, we found a decoherence-free subspace of the multiparticle entangled states of the W-like class. We present a scheme for writing and storing these states in collective thermostat. DOI: XXXX PACS number共s兲: 03.67.Pp, 03.65.Yz, 03.65.Ud and a broadband field. For this model Klimov et al. 关10兴 derived master equations for atoms and a collective relaxation operator in the dispersive limit, assuming a vacuum bath. We discuss a more general squeezed thermostat, for which a master equation is derived using a formalism of unitary transformations. There are some differences between relaxation operators due to methods of derivation of master equations. From the physical point of view, in the dispersive limit there is only an exchange of phase between atoms and cavity mode. This is described by the effective Hamiltonian we found, which is diagonal over atomic and field variables. However, the behavior of atoms becomes more complicated due to an interaction between the cavity mode and broadband field, which plays the role of a thermostat. Then one finds a coupling of atoms with the thermostat, providing atomic collective relaxation. In this aspect our results differ from that of Ref. 关10兴. In contrast to 关10兴 we use the idea developed in Ref. 关11兴; we first find the effective Hamiltonian of the total system including the thermostat and then derive the master equation. Using the master equation we consider the dynamics of a class of multiparticle entangled states, which is a slightly generalized W class introduced by Cirac et al. 关12兴. Some of the optical and atomic implementations of the presented states have been demonstrated experimentally by Weinfurter et al. 关13兴 and Schmidt-Kaler 关14兴. Some properties of these states, different schemes to generate them, and several applications have been considered in Ref. 关15兴. We also find that in the case when these entangled states are reduced to the Dicke states, they belong to DFS and are immune to collective decay. To explain this feature we use symmetry arguments. In fact, the total space of the Dicke states is represented by irreducible subspaces distinguished by their symmetry type. The collective interaction we consider does not mix the wave functions from different subspaces due to symmetry conservation. Using these properties, we present a model of a quantum memory for writing, storing, and reading information encoded in these entangled states. The paper is organized as follows. We first derive the master equation in the dispersive limit assuming a general model of thermostat. Then we introduce a set of multiparticle entangled states which can be reduced to the Dicke family and we consider their decay in squeezed and vacuum thermostats. Finally we present a scheme for reading and storing entangled states in the collective thermostat. OF I. INTRODUCTION PY CO When information is encoded in a quantum state of a physical system, the robustness of the state is an important factor for successful communication. Due to decoherence, i.e., interaction with the environment, the state of the system can degrade and lose its quantum correlations. One of the possible solutions of the decoherence problem is to use decoherence-free subspaces 共DFSs兲; these include wave functions immune to decoherence 关1兴. The first DFS has been introduced by Zanardi et al. 关2兴 for two-level atoms interacting with an electromagnetic field playing the role of environment. The wave functions belonging to the DFS are annihilated by the interaction Hamiltonian and, therefore, are left invariant during evolution. Examples of DFSs for various physical systems, in particular, for light, have been proposed by several authors 共see, for example, 关3–5兴兲. Weinfurter et al. have demonstrated experimentally decoherence-free quantum communication based on four-photon polarized states 关6兴. Simple observations show that quantum correlations between particles can be produced and maintained in collective processes. These are interesting for DFSs and a large number of physical systems with collective interactions can be found. For the Dicke model with a single resonant mode, Bonifatio et al. have found a master equation describing a collective atomic decay when illuminated with a resonant mode 关7兴. Palma and Knight 关8兴 have shown that two-atom decay can result in pure entangled states in a collective squeezed thermostat. However, entanglement of two atoms can be achieved in collective decay with vacuum thermostat, when atoms are placed inside a cavity, as it has been shown by Basharov 关9兴. There is a simple reason for collective decay in the cavity scheme. When atoms interact with a single cavity mode, atomic relaxation arises because radiation leaves the cavity. This can be modeled as an interaction between the mode and an external broadband field, which plays role of a thermostat. Therefore, the atoms, being coupled with the single mode, have a collective decay. The aim of this paper is to investigate the collective decay of atoms in the entangled state. To achieve this we consider a simple model of two-level atoms inside a single mode cavity 61 20 ]0 41 00 T1 [A RA 0P II. INITIAL EQUATIONS *Permanent address: Laboratory for nonlinear optics, RRC “Kur- By considering the interaction between atoms and a field, one can obtain a master equation for one of the systems. This chatov Institute,” Moscow 123182, Russia. † Electronic address: [email protected] 1050-2947/2006/74共4兲/1共0兲 PROOF COPY [AT10041] 020610PRA 1-1 ©2006 The American Physical Society PROOF COPY [AT10041] 020610PRA PHYSICAL REVIEW A 74, 1 共2006兲 BASHAROV, GORBACHEV, AND RODICHKINA equation is also known as kinetic and often has a Lindblad form. It describes irreversible processes including atomic relaxation, absorption or amplification of light, and other phenomena which can be reduced to the Lindblad equations. Hg = − g 兺 ⌫␻共R+b␻ + R−b␻† 兲. ប⌬ ␻ 共7兲 where the Hamiltonians of free atoms, cavity mode, and broadband field are, respectively, Ha = ប␻0R3, Hc = ប␻cc†c, Hb = 兺␻ប␻b␻† b␻; here R3 = 兺 j共兩1典 j具1 兩 −兩0典 j具0 兩 兲, 0, 1 label the lower and upper levels of atom, and c , c† , b␻ , b␻† are creation and annihilation operators for photons of the cavity mode and broadband field, respectively. The interaction between atoms and cavity mode V1 has the form In contrast to the second term in V2 the obtained Hamiltonian Hg describes the collective interaction of atoms. Indeed, in the usual case of the dispersive limit there is no energy exchange between atoms and light, and this is in accordance with the effective Hamiltonian He, which is similar to Ref. 关10兴. In the same time the effective Hamiltonian Hg shows that atoms and the thermostat field exchange excitations. This is a particular feature of the dispersive limit due to the initial interaction 共3兲. A close analogy is parametric down conversion in transparent nonlinear media, in which the virtual transitions result in an interaction between photons. Now we have a problem specified by H⬘ = Ha + Hb + Hc + He + Hg + V2, where broadband fields can be considered as a thermostat in a given state. Assume the thermostat is ␦-correlated and its state is squeezed with a center frequency ⍀: V1 = g共c†R− + cR+兲, 具b␻† b␻⬘典 = N共␻兲␦␻,␻⬘ , A. Hamiltonian O PR We consider n two-level atoms inside a high-finesse optical cavity, a single cavity mode, and a broadband field outside the cavity. We assume the Hamiltonian of the system has the form 共1兲 H = Ha + Hc + Hb + V1 + V2 , OF CO 共2兲 PY where collective atomic operators are given by R± = 兺 jR±共j兲 , R+共j兲 = 关R−共j兲兴† = 兩1典 j具0兩. The term V2 describes two processes: 共1兲 an interaction between the broadband field and the cavity mode due to nonzero transmittance of the output mirror; 共2兲 an interaction between atoms and the broadband field due to nonideal sidewalls of the cavity. It reads 册 具b␻b␻⬘典 = M共␻兲␦2⍀,␻+␻⬘ , 具b␻† b␻⬘典 = M *共␻兲␦2⍀,␻+␻⬘ , † 共3兲 T1 j † [A 冋 V2 = 兺 b␻ ⌫␻c† + 兺 K␻ jR+共j兲 + H.c. ␻ 具b␻b␻⬘典 = „N共␻兲 + 1…␦␻,␻⬘ , where the photon numbers N共␻兲 and M are related as 兩M共␻兲 兩 艋 冑N共␻兲关N共␻兲 + 1兴. A physical model of this thermostat can be represented by a light generated in parametric down conversion process. Its simple nondegenerate version † † is described by the Hamiltonian H = 兺␻共k␻b⍀+ ␻b⍀−␻ + H.c.兲, where 2 ⍀ is the pump frequency and ␻ belongs to a frequency band h given by phase matching conditions. The photon numbers N and M have the form N共␻兲 = sinh2 r␻ , M共␻兲 = exp共i arg k␻兲cosh r␻ sinh r␻, where r␻ ⬃ 兩k␻兩 is a squeezing parameter. For a squeezed vacuum r Ⰶ 1 and N ⬇ 0, while M ⬇ exp共i arg k␻兲r␻. The generated light is broadband if h is much bigger than all representative frequencies of the problem, like the atomic and cavity mode decay rates. More precisely, assume that the width of the squeezed broadband field given by Eqs. 共8兲 is much bigger than the detuning ⌬ as in Eq. 共4兲. Then following the standard procedure of replacing a finite bandwidth system with white noise 关16兴 we can make all parameters of squeezed light to be independent from the frequency: N共␻兲 = N , M共␻兲 = M. Assume the squeezed thermostat is modeled by a parametric down conversion source. Then its bandwidth is determined by the phase matching conditions and can be experimentally varied on a wide range. The next step is switching on the broadband field. This can be achieved by several methods based on projection operator techniques, stochastic differential equations, and others. In any case we need a Markovian approximation to obtain a closed equation. In our case this means that the evolution of the broadband light is given by the free Hamiltonian Hb only and the thermostat parameters N and M are 41 00 From the physical point of view the broadband field plays the role of a thermostat and causes the relaxation of atoms and cavity mode. Relaxation terms can be achieved by switching on this field. It can be done in different ways using various approximations. B. Dispersive limit 20 ]0 We assume that detuning ⌬ = 兩␻c − ␻0兩 is large and consider the dispersive limit, which can be justified when 关17兴 兩⌬兩 Ⰷ ng冑具c†c典 + 1. 共4兲 61 To derive the master equation let us introduce a transformation of Hamiltonian H given by a time independent unitary operator S 共5兲 0P H⬘ = e−iSHeiS = − i关S;H兴 − 共 21 兲†S;关S;H兴‡ + ¯ . R−R+ + cc†2R3 . ប⌬ RA Using perturbation theory over interactions V1 and V2 one finds the operator S, from which an effective Hamiltonian describing the interaction between atoms and cavity mode can be obtained. This Hamiltonian is diagonal over the field and atomic variables and has the form He = g2 共6兲 Under this approximation there is another effective Hamiltonian Hg which describes the interaction between atoms and broadband field 1-2 PROOF COPY [AT10041] 020610PRA 共8兲 PROOF COPY [AT10041] 020610PRA PHYSICAL REVIEW A 74, 1 共2006兲 DECAY AND STORAGE OF MULTIPARTICLE… frequency independent. As a result we find a master equation for the density matrix ␳ of atoms and cavity mode. The equation includes an effective Hamiltonian He and relaxation terms. In the dispersive limit and in the interaction picture the master equation has the form ␳˙ = − 共i/ប兲关He ; ␳兴 − T␳ , ˙f = − L f . a III. COLLECTIVE DECAY AND STORAGE ] OF ENTANGLED STATES O PR 共9兲 Considering decay of atoms in collective thermostats one finds that quantum correlations between particles can be supported, and their final or steady state depends on the initial one. where the relaxation operator T includes three terms of the Lindblad form: T = 兺 jL j + Lc + La. The first term describes the independent decay of atoms in the squeezed thermostat. When the atoms have the same coupling constant K␻ j = K␻ it reads OF A. Entangled Dicke states L j␳ = 共␥↓/2兲共R+共j兲R−共j兲␳ − 2R−共j兲␳R+共j兲 + ␳R+共j兲R−共j兲兲 + 共␥↑/2兲 We introduce the multiparticle entangled states, the slight modification of the W states discovered by Cirac 关12兴 ⫻共R−共j兲R+共j兲␳ − 2R+共j兲␳R−共j兲 + ␳R−共j兲R+共j兲兲 − 2MK2R+共j兲␳R+共j兲 ␩n共1兲 = q1兩10 ¯ 0典 + q2兩01 ¯ 0典 + ¯ qn兩00 . . . 1典, 共10兲 CO − 2M *K2R−共j兲␳R−共j兲 , 共14兲 where the decay rates of atomic levels are denoted by ␥↓ = 兩K兩2共N + 1兲, ␥↑ = 兩K兩2N, and 兩K兩2 = ប−2兺␻兩K␻兩2␦共␻0 − ␻兲. In free space one finds that 兩K兩2 reduces to the well-known formula for the spontaneous decay rate 4␻20d2 / 3បc3. Equation 共10兲 describes spontaneous decay of independent atoms in the squeezed thermostat, for which the transversal decay rate becomes slow because of squeezing: ␥⬜ = 共␥↓ + ␥↑兲 / 2 − Re兵MK2其. The second term of T is the relaxation of the cavity mode due to photons leaving the cavity, and has the form PY where 兺k兩qk兩2 = 1. Some of these states belong to the Dicke states 兩jma典 关18兴, specified by three quantum numbers j , m , a, where 兩m兩 艋 j = 0 , . . . , n / 2 − 1 , n / 2, n is a particle number, and parameter a describes the degeneracy and takes − Cn/2+j+1 values. The numbers j and m are eigenn j = Cn/2+j n n values of two commuting collective operators J3 and J2 = J21 + J22 + J23 [A J3兩jma典 = m兩jma典, J2兩jma典 = j共j + 1兲兩jma典, 共15兲 T1 where Jb obeys the commutation relations of the momentum operators 关Jb ; Jc兴 = i␧bcdJd, b , c , d = 1 , 2 , 3. In the considered case J1 = 共 21 兲共R− + R+兲, J2 = 共i / 2兲共R+ − R−兲. When Lc␳ = 兩⌫兩2关共N + 1兲共c†c␳ − 2c␳c† + ␳c†c兲 + N共cc†␳ − 2c†␳c + ␳cc†兲 + M共cc␳ − 2c␳c + ␳cc兲M *共c†c†␳ − 2c†␳c† 共11兲 ]0 20 共12兲 where 兩0典共1k兲 denotes a 兩0典 state of n − 2 particles 共without first and kth兲, ⌿− = 共1 / 冑2兲共兩01典 − 兩10典兲. Equation 共17兲 gives the structure of entanglement of ␩n共1兲; it tells that one of the particles, say 1, forms EPR pairs with all other particles 2 , . . . , n. This feature is invariant under particle permutations. Due to antisymmetric vectors ⌿− the collective evolution of n particles in the state 共17兲 involves only n − 2 particles. This has a simple reason. Two-particle collective operators R± and R3 annihilate ⌿−, this causes that for any evolution operator U depending on R± , R3 we have 2 61 RA 0P − − U兩␩n典具␩n兩 = 2 兺 兩⌿1k 典具⌿1s 兩U共1k;1s兲兩0典共1k兲共1s兲具0兩, 共18兲 ks where U共1k ; 1s兲兩0典共1k兲共1s兲具0兩 acts on all particles except 1, k and 1, s. These features allow us to get simple exact solutions for several collective decay problems of ␩n共1兲. 1-3 PROOF COPY [AT10041] 020610PRA 共17兲 k=2 where 兩␹兩 = 兩g⌫ / ប ⌬兩 ␶ , ␶ = L / c. As a result, in the dispersive limit there are three relaxation operators describing singleparticle and collective decays. They have a straightforward physical meaning and they differ from the relaxation operator in Ref. 关10兴, which has cross terms including products of the collective atomic operators by operators of the cavity mode. In order to consider the collective decay of atoms let us introduce the interaction picture ␳⬘ = exp共−iប−1Het兲␳ exp共iប−1Het兲 and assume the following approximations: Let the first term in He and single-particle relaxation be small, g2R−R+ / ប⌬, 兺 jL j Ⰶ cc†R3 / ប⌬, La, Lc. This is true if g2 / ប兩⌬兩, ␥↓,↑ Ⰶ 具cc†典g2 / 兩⌬兩, 兩␹兩2n, and 兩⌫兩2具c†c典 / n. Then we can neglect the difference between He and g22c†cR3 / ប⌬ so that the master equation for the atomic density matrix f = Trc␳⬘ is 2 n ␩n共1兲 = 冑2 兺 qk兩⌿−典1k 丢 兩0典共1k兲 , − 2R+␳R− + ␳R−R+兲 + M共R+R+␳ − 2R+␳R+ + ␳R+R+兲 + M 共R−R−␳ − 2R−␳R− + ␳R−R−兲兴, 共16兲 we have a set of the zero sum amplitude states discovered by Pati 关19兴. However, the wave functions ␩n共1兲 under condition 共16兲 belong to the Dicke family with j = m = n / 2 − 1 关20兴. The states have the next representation La␳ = 兩␹兩2关共N + 1兲共R+R−␳ − 2R−␳R+ + ␳R+R−兲 + N共R−R+␳ * 兺k qk = 0, 41 where 兩⌫兩2 = ប−2兺␻兩⌫␻兩2␦共␻c − ␻兲. If R is the reflectance of the output cavity mirror, then 兩⌫兩2 → c共1 − R兲 / 2L, where L is length of the cavity. The collective decay of atoms is represented by the operator La: 00 + ␳c†c†兲兴, 2 共13兲 PROOF COPY [AT10041] 020610PRA PHYSICAL REVIEW A 74, 1 共2006兲 BASHAROV, GORBACHEV, AND RODICHKINA L0兩1;n典具␩n兩 = 关L0兩␩n典具1;n兩兴† , In squeezed thermostat there is an interesting feature. It can produce or store quantum correlations between particles for several initial states. Considering a two-atom collective decay with initial density matrix f共0兲 = A兩00典具00兩 where + B兩⌿+典具⌿+兩 + C兩11典具00兩 + C*兩00典具11兩 + D兩11典具11兩, + 冑 ⌿ = 共1 / 2兲共兩01典 + 兩10典兲, A + B + D = 1, one finds the next pure steady state 关8兴: where Q = 兺kqq, 兩0典 = 兩00¯ 0典 is a ground state of atoms and 兩1 ; n典 is a fully symmetric state, the normalized version of which, Wn = 共1 / 冑n兲兩1 ; n典, is known as W state O PR B. Collective squeezed thermostat s = 共冑N + 1兩00典 + 冑N兩11典兲/共冑2N + 1兲. Wn = 共1/冑n兲共兩10 ¯ 0典 + 兩01 ¯ 0典 + ¯ 兩00 ¯ 1典兲. 共24兲 It follows from Eqs. 共23兲 that the Lindblad operator L0 maps the set of states 兵兩␩n典具␩n兩 , 兩0典具0兩 , 兩1 ; n典具1 ; n兩 , 兩1 ; n典具␩n兩 , ⫻兩␩n典具1 ; n兩其 into itself. This observation allows us to get an exact solutions for density matrix 共19兲 This state is entangled. Note that this solution is correct for the initially symmetric state f共0兲. Consider the collective decay of the two entangled states ␩3 and ␩4 described by 共13兲. Under conditions 共16兲 the wave functions read OF f共t兲 = A共t兲兩1;n典具␩n兩 + A*共t兲兩␩n典具1;n兩 + B共t兲兩1;n典具1;n兩 + S共t兲 ⫻兩0典具0兩 + D兩␩n典具␩n兩, where the normalization condition reads A共t兲Q + A共t兲*Q + B共t兲n + S共t兲 + D共t兲 = 1 and coefficients obey equations CO A˙ = − ␬共An + DQ兲, 共20兲 B˙ = − ␬共AQ* + A*Q兲 − ␥nB, PY According to Eq. 共18兲 the evolution of the density matrix 兩␩3典具␩3兩 reduces to the dynamics of the single-particle state 兩0典 具0兩 for which there is a simple solution 兩0典具0兩 → ␭兩0典具0兩 + 共1 − ␭兲兩1典具1兩. The ␭ = N / 共2N + 1兲 is the occupation number of the lower atomic level. One finds that ␩3 decays into a mixed state with complex structure. This fact can be explained using a symmetry argument, which tells that under the single-particle decay the symmetry of the initial state is not conserved. In contrast to dynamics of ␩3, the dynamics of ␩4 has other features. To obtain the solution we use 共19兲 and find that the final state is obtained by replacing 兩00典 →s S˙ = − 2␬关nS + n共1 − D兲 + 兩Q兩2D兴, ˙ = 0. D [A Similarly to the squeezed thermostat there is a steady state solution, if t → ⬁ T1 共21兲 f ss = D关− 共Q/n兲兩1;n典 + 兩␩n典兴关− 共Q*/n兲具1;n兩 + 具␩n兩兴 + 关共1 − D兲 which depends on the initial state through parameter D. If D = 0, then f ss = 兩0典具0兩. If D = 1 one finds evolution of ␩n共1兲 41 兩␩n典具␩n兩 → 共Q/冑n兲共e−n␬t − 1兲兩Wn典具␩n兩 + H.c. + 共兩Q兩2/n兲 From this equation it follows that the state is pure, has a more complicated entanglement structure, but as before one of the atoms forms EPR pairs with all other atoms. ]0 ⫻共1 − e−n␬t兲2兩Wn典具Wn兩 + 共兩Q兩2/n兲共1 − e−2n␬t兲 ⫻兩0典具0兩 + 兩␩n典具␩n兩. C. Vacuum thermostat 20 共22兲 61 where ␬ = 兩␹兩 . This equation describes a collective decay in the vacuum thermostat conserving quantum correlations. The simplest example is the two-particle antisymmetric function ⌿− belonging to DFS and immune to decay. The more interesting examples, introduced by Zanardi 关2兴, are DFSs of multiatom states, products of ⌿−. Suppose the atoms inside the cavity are prepared in the state ␩n共1兲, then they evolve according to Eq. 共22兲 which can be solved exactly. It is easy to verify that the Lindblad operator L0 f = R+R− f − R−FR+ + H.c. in Eq. 共22兲 has the following properties: 2 RA 0P L0兩␩n典具␩n兩 = Q兩1;n典具␩n兩 − 兩Q兩2兩0典具0兩 + Q*兩␩n典具1;n兩, L0兩␩n典具1;n兩 = Q兩1;n典具1;n兩 − 2Qn兩0典具0兩 + n兩1;n典具␩n兩, 1-4 PROOF COPY [AT10041] 020610PRA 共27兲 It follows from 共27兲 that under condition 共16兲 ␩n共1兲 is a Dicke state and has immunity to the collective decay. However, this result can be obtained without any calculations due to its annihilation by the Lindblad operator L0. In contrast to ␩n共1兲, the fully symmetric Wn state degrades: Wn → 兩0典. The robustness of ␩n entangled states can be clear from the symmetry argument. In the considered collective processes the particles permutation operator is an integral of motion, so that state symmetry is conserved. Therefore the antisymmetric wave function ⌿− is robust to decay because the transition ⌿− → 兩0典 is forbidden, but the fully symmetric W state can transform into 兩0典. In the case of ␩n共1兲 the situation is more complicated, nevertheless symmetry plays a principal role here also. As it is known the space of Dicke states is represented by irreducible subspaces distinguished by their symmetry type over particles permutations. Under the condition 共16兲 the wave functions ␩n共1兲 belong to 关n , n − 1兴 irreducible representation of the Dicke states in contrast to W and ground state 兩0典, which belong to the 关n , 0兴 Assuming a simpler thermostat model for which M = N = 0 the master equation 共13兲 reduces to ˙f = − ␬共R R f − R fR + fR R 兲, + − − + + − 共26兲 + 兩Q兩2D/n兴兩0典具0兩 00 − − − ␩4 → q2⌿12 兩s典23 + q3⌿13 兩s典24 + q4⌿14 兩s典23 . 共25兲 * − − ␩3 = q2⌿12 兩0典3 + q3⌿13 兩0典2 , − − − ␩4 = q2⌿12 兩0典23 + q3⌿13 兩0典24 + q4⌿14 兩0典23 . 共23兲 PROOF COPY [AT10041] 020610PRA PHYSICAL REVIEW A 74, 1 共2006兲 DECAY AND STORAGE OF MULTIPARTICLE… simplicity assume exp共ijr j兲 ⬇ 1, then the evolution is given by one. Due to symmetry conservation the subspaces of different symmetry type do not mix. This point is in accordance with the fact that in the dynamics of the wave functions the final states at t → ⬁ are not usual steady states but depend on their subspace and initial conditions. Robustness of ␩n共1兲 is a natural basis for a quantum memory. Memory includes writing, storing, and reading of information encoded by a quantum state. By choosing ␩n共1兲 states to encode information, a model of quantum memory can be designed. Writing and reading are achieved by swapping: a 丢 b → b 丢 a. A particular case of swapping of two mode light in a Fock state into atomic ensemble has been considered in Ref. 关21兴. Here we introduce a scheme for writing and storing multiparticle states. Assume that an interaction between atoms inside cavity and light, represented by its spacial modes with wave vectors j, has the form exp共− iប−1Vt兲兩0典a 丢 兩␩n典b = cos共ft兲兩0典a 丢 兩␩n典b + sin共ft兲兩␩n典a 丢 IV. CONCLUSIONS Being collective properties of a physical system, quantum correlations between particles and entanglement can be produced and stored in the collective processes. These processes can describe interaction between the physical system and its environment, which often plays a role of thermostat. In contrast to the usual thermostat the collective thermostat supports quantum correlations and it is possible to find a DFS, which is a natural basis for quantum memory. For the considered example of collective decay of atoms inside a cavity, we found that a set of entangled states of the W-like class is decoherence free and therefore is suitable to encode quantum information for storing it in the collective thermostat. OF CO j 共29兲 If sin共ft兲 = 1, the state of light 兩␩n典b is swapped into atoms. Under condition 共16兲 it can be stored in the collective thermostat. Due to the unitarity of transformation 共29兲, we can achieve a reading of the atomic state. O PR V = iប 兺 f关R+j a j exp共ijr j兲 − H.c.兴, 兩0典b . 共28兲 PY where r j is a position of jth atom. This Hamiltonian describes an exchange of excitation between a single atom and a single mode. 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