Author's personal copy

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: properties of the pseudo- 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- tem size. Spontaneous symmetry breaking Mesoscopic superconductivity

Fermions undergoing BCS type pairing interaction can be trea- H0 ¼gSþS; 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 EðsÞ¼ðg=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 , 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 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 2s2i: 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 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 ¼g 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 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 , UðtÞ, is given by Author's personal copy

Z t possible final state as the system evolves in time. Even though UðtÞ¼exp i dt0Hðt0Þ : ð9Þ 0 we don’t have a general analytical solution yet, our calculations show that total probability of conservation of exchange symmetry In Ref. [5], using Kraus representation [7–9] is 1=d at least for small systems. Evaluation of probabilities for pos- XM sible broken symmetry states for higher d values will shed light on y qðtÞ¼ KlðtÞqð0ÞKlðtÞ; ð10Þ stability of superconducting state. l¼1 general time evolution has been evaluated and it has been shown Acknowledgements that not all possible final states This work has been partially supported by the Scientific and y KlðtÞqð0ÞKlðtÞ Technological Research Council of Turkey (TUBITAK) under grant ð11Þ y 107T530. Authors would like to thank the Institute of Theoretical trðKlðtÞqð0ÞKlðtÞÞ and Applied Physics at Turunç where part of this research has been have exchange symmetry. The same result has been reproduced for done. using the decoherence Hamiltonian [10]

XN1 XN2 References H1 ¼ s1z hx1kr1kz þ s2z hx2kr2kz; ð12Þ k¼1 k¼1 [1] P.W. Anderson, J. Phys. Chem. Solids 11 (1959) 26. [2] Z. Gedik, Solid State Commun. 124 (2002) 473. where z-component operators s1z and s2z are coupled to bath spins [3] B.R. Mottelson, The many-body problem, Ecole d’Eté de Physique Théorique represented by r . Here n ¼ 1; 2 labels the baths and Les Houches, Wiley, New York, 1959. nkz [4] I.O. Kulik, H. Boyaci, Z. Gedik, Physica C 352 (2001) 46. k ¼ 1; 2; 3; ...; Nn labels the individual spins in the baths. For ex- [5] G. Karpat, Z. Gedik, Optics Commun. 282 (2009) 4460. change symmetric case parameters of the two baths are taken to [6] Ö. Bozat, Z. Gedik, Solid State Commun. 120 (2001) 487. be identical. [7] K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory, Springer-Verlag, Berlin, 1983. In conclusion, exchange symmetry of a state, undergoing an [8] D. Salgado, J.L. Sáanchez-Gómez, Open Syst. Inf. Dyn. 12 (2005) 55. external noise having the same symmetry, can be spontaneously [9] M.D. Choi, Lin. Alg. and Appl. 10 (1975) 285. broken as a result of decoherence due to local interactions. A nat- [10] Z. Gedik, Solid State Comm. 138 (2006) 82. ural question is the maximum probability of finding a symmetric