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Entanglement of hard-core bose gas in degenerate levels under local noise
Z. Gedik *, G. Karpat
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, Istanbul 34956, Turkey article info abstract
Article history: Quantum entanglement properties of the pseudo-spin representation of the BCS model is investigated. In Accepted 23 November 2009 case of degenerate energy levels, where wave functions take a particularly simple form, spontaneous Available online 26 November 2009 breaking of exchange symmetry under local noise is studied. Even if the Hamiltonian has the same sym- metry, it is shown that there is a non-zero probability to end up with a non-symmetric final state. For Keywords: small systems, total probability for symmetry breaking is found to be inversely proportional to the sys- Quantum decoherence tem size. Spontaneous symmetry breaking Mesoscopic superconductivity
Fermions undergoing BCS type pairing interaction can be trea- H0 ¼ gSþS ; ð2Þ ted as hard-core bosons in real or momentum space depending P where ~S ~s is the total spin and S are the corresponding rais- on whether they interact strongly or weakly. In both cases, lattice ¼ n n points (or energy levels) are occupied by pairs or else they are ing and lowering operators. Energy eigenvalues are given by empty. Therefore, the system can be described by pseudo-spin EðsÞ¼ ðg=4ÞðN sÞð2d s N þ 2Þ; ð3Þ variables [1]. Quantum entanglement and superconducting order parameter of such systems have been found to be closely related where seniority number s ¼ d 2S can take values s ¼ 0; 2; 4; ...; N [2]. In case of degenerate energy levels, wave functions take a par- when the number of fermions N is even. The ground state, which is ticularly simple form and hence all possible eigenstates can be invariant under Cooper pair exchange, has the maximum total-S va- written down easily [3]. Such a degeneracy can for example be lue d=2. Degeneracy of each state can be found easily as [6] due to parabolic energy bands in mesoscopic and nanoscopic superconducting particles [4]. Xs=2 d s þ 2i d s þ 2i d XðsÞ¼ 2s 2i: ð4Þ Recently, it has been shown that exchange symmetry of some i¼0 i i 1 s 2i entangled states can be spontaneously broken under local noise even if the Hamiltonian has the same symmetry [5]. Considering Now, let us consider an exchange symmetric local external noise the symmetry of the initial state and the Hamiltonian, this is a very described by, for d ¼ 2, interesting result. Since BCS model of superconductivity can be written in terms of pseudo-spins or (in the language of quantum H1ðtÞ¼w1ðtÞðs1z IÞþw2ðtÞðI s2zÞ; ð5Þ information theory) in terms of qubits, one might ask conse- quences of this kind of symmetry breaking on superconducting where w1ðtÞ and w2ðtÞ are stochastic noise fields that lead to statis- state. tically independent Markov processes satisfying Our starting point is a single, d-fold degenerate energy level with N spin 1/2 fermions. Assuming that fermions in the same state hwnðtÞi ¼ 0; ð6Þ 0 0 are paired, the model Hamiltonian becomes hwnðtÞwnðt Þi ¼ andðt t Þ: ð7Þ
Xd In case of real space pairing, such a noise might originate from an y y H0 ¼ g an0"an0#an#an"; ð1Þ external disturbance localized in space in a region of Cooper pair n;n0¼1 size. Generalization to arbitrary d is obvious. Let the system be in state qðt ¼ 0Þ, say the ground state. The time evolution of the sys- where ay ða Þ creates (annihilates) a fermion in state n with spin nr nr tem’s density matrix can be obtained as ~ y y r. Introducing pseudo-spins sn defined by snþ ¼ an"an# and y y y snz ¼ð1=2Þ an"an" þ an#an# 1 , we can rewrite the Hamiltonian as qðtÞ¼hUðtÞqð0ÞU ðtÞi; ð8Þ
* Corresponding author. Tel.: +90 2164839610; fax: +90 2164839550. where ensemble averages are evaluated over the two noise fields E-mail address: [email protected] (Z. Gedik). w1ðtÞ and w2ðtÞ and the time evolution operator, UðtÞ, is given by Author's personal copy