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arXiv:1309.4847 · Physical Review A

Coherent states of composite bosons

Su‐Yong Lee, Jayne E. Thompson, Paweł Kurzyński, Akihito Soeda, Dagomir Kaszlikowski

Published Dec 3, 2013• 11 citations
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Abstract

We present a systematic analysis of coherent states of composite bosons consisting of two distinguishable particles. By defining an effective composite boson (coboson) annihilation operator, we derive its eigenstate and commutator. Depending on the elementary particles comprising the composite particles, we gauge the resemblance between this eigenstate and traditional coherent states through typical measures of nonclassicality, such as quadrature variances and Mandel's $Q$ parameter. Furthermore, we show that the eigenstate of the coboson annihilation operator is useful in estimating the maximum eigenvalue of the coboson number operator.

DOI
10.1103/physreva.88.063602

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We present a systematic analysis of coherent states of composite bosons consisting of two distinguishable particles. By defining an effective composite boson (coboson) annihilation operator, we derive its eigenstate and commutator. Depending on the elementary particles comprising the composite particles, we gauge the resemblance between this eigenstate and traditional coherent states through typical measures of nonclassicality, such as quadrature variances and Mandel's QQQ parameter. Furthermore, we show that the eigenstate of the coboson annihilation operator is useful in estimating the maximum eigenvalue of the coboson number operator.
We consider composite bosons (cobosons) comprised two elementary particles, fermions or bosons, in an entangled state. First, we show that the effective number of cobosons implies the level of correlation between the two constituent particles. For the maximum level of correlation, the effective number of cobosons is the same as the total number of cobosons, which can exhibit the original Bose–Einstein condensation (BEC). In this context, we study a model of BEC for indistinguishable cobosons with a controllable parameter, i.e., entanglement between the two constituent particles. We find that bi-fermions behave in a predictable way, i.e., the effective number of the ground state coboson is an increasing function of entanglement between a pair of constituent fermions. Interestingly, bi-bosons exhibit the opposite behavior—the effective number of the ground state coboson is a decreasing function of entanglement between a pair of constituent bosons.
The theory of composite bosons (cobosons) made of two fermions [C. K. Law, Phys. Rev. A 71, 034306 (2005); M. C. Tichy et al., Phys. Rev. Lett. 109, 260403 (2012)] converges to ordinary structureless bosons in the limit of infinitely strong entanglement between the fermionic constituents. For finite entanglement, the annihilation operator $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{c}$ of a composite boson couples the NNN-coboson Fock state not only to the $(N\ensuremath{-}1)$-coboson state---as for ordinary bosons---but also to a component which is orthogonal to the Fock-state ladder of cobosons. Coupling with states orthogonal to the Fock ladder arises also in dynamical processes of cobosons. Here, with a Gedanken experiment involving both mode splitting and collective Hong-Ou-Mandel-like interference, we derive the characteristic physical signature of the states orthogonal to the Fock ladder generated in the splitting process. This allows us to extract the microscopic properties of many-fermion wave functions from the collective coboson behavior. We show that consecutive beam-splitter dynamics increases the deviation from the ideal bosonic behavior pattern, which opens up a rigorous approach to the falsification of coboson theory.
The quantum nature of elementary bosons can be completely erased by using coherent states known as Glauber states. Here, we consider composite bosons (cobosons) made of two fermions and look for the possibility of erasing the bosonic quantum nature of the field and the fermionic quantum nature of its constituents, when the distribution of the coboson Schmidt decomposition is either flat like Frenkel excitons or localized like Wannier excitons. We show that for Frenkel-like cobosons, complete erasure of the field quantum nature is possible up to a density that is a sizable fraction of the number of fermion-pair states making up the coboson at hand. At a higher density, the Pauli exclusion principle between fermionic constituents shows up dramatically. It induces (i) a decrease in the number-operator eigenvalues down to 1 and an increase in the number-state second-order correlation functions up to 2; (ii) the disappearance of the usual sharp peak in the coherent-state distribution and an increase in the coherent-state second-order correlation function up to 2. It also is possible to construct number states and coherent states for Wannier-like cobosons, but their forms are far more complex. Finally, we show that Pauli blocking makes the coboson coherent states qualitatively different from Anderson's ansatz.
We investigate squeezed states of composite bosons (cobosons) formed by pairs of spin-1/21/21/2 fermions, with emphasis on Frenkel-like cobosons. While squeezing for standard bosonic modes is well established, its extension to cobosons requires accounting for Pauli blocking and the resulting non-canonical commutation algebra. Building on earlier constructions of coboson coherent states, we define squeezed cobosons as eigenstates of a Bogoliubov transformed coboson operator and derive explicit expressions for the associated quadrature variances. We show that the underlying fermionic structure leads to state-dependent modifications of the Heisenberg--Robertson uncertainty bound, which may fall below the canonical bosonic limit without implying any violation of uncertainty principles. Numerical results based on finite-dimensional matrix representations illustrate how these effects constrain the attainable squeezing. Our framework is relevant to composite boson systems such as tightly bound electron-hole pairs and provides a physically transparent setting to probe compositeness through observable quadrature fluctuations.
Abstract This chapter defines and determines the properties of the important (though non-physical) boson and fermion coherent states, which are specified in terms of c-number (bosons) or Grassmann (fermions) variables and their complex conjugates, and defined via unitary displacement operators acting on the vacuum state. Important non-orthogonality, overcompleteness and coherence properties of coherent states are obtained. Coherent states are shown to be eigenstates of mode annihilation operators, with eigenvalues given by the c-number or Grassmann variable specifying the state. Related (unnormalised) Bargmann states are defined, depending only on the variable and not its complex conjugate, and are used to obtain useful representations of physical states and canonical forms for operators in terms of Bargmann state projectors, as well as obtaining important results such as for traces of operator products in terms of phase space integrals. Mode creation operators acting on Bargmann states are shown as equivalent to differentiation.
Coherent states (CS) for non-Hermitian systems are introduced as eigenstates of pseudo-Hermitian boson annihilation operators. The set of these CS includes two subsets which form bi-normalized and bi-overcomplete system of states. The subsets consist of eigenstates of two complementary lowering pseudo-Hermitian boson operators. Explicit constructions are provided on the example of one-parameter family of pseudo-boson ladder operators. The wave functions of the eigenstates of the two complementary number operators, which form a bi-orthonormal system of Fock states, are found to be proportional to new polynomials, that are bi-orthogonal and can be regarded as a generalization of standard Hermite polynomials.