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arXiv:1609.08119 · Physical Review A

Effect of Pauli blocking on coherent states of composite bosons

Shiue-Yuan Shiau, Monique Combescot

Published Jan 24, 2017• 2 citations
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Abstract

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.

DOI
10.1103/physreva.95.013838

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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.
Any experimental realization of Bose-Einstein condensation (BEC) deals with composite bosons (cobosons), which are quasi-particles made of fermion pairs that suffer Pauli blocking. We evidence that the prime consequence of the Pauli exclusion principle is to prevent cobosons from having their centers of mass located at the same position within their spatial extension. We moreover predict a many-body effect induced by Pauli blocking that leads to high interference wave modes resulting from fermion exchanges between two colliding coboson condensates. These two effects can be detected by measuring spatial correlation functions. We here propose an original procedure to exactly derive such correlation functions for Frenkel-like cobosons characterized by just their momentum. We physically extend the obtained results to more complex Wannier-like cobosons that have an additional relative motion index. In addition to revealing the physics involved in results previously obtained for elementary boson condensates, we use Shiva diagrams to show how it is changed for cobosons. The predictions we here present, established for matter bosons, are directly applicable to experiments.
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.
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.
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 study statistical signatures of composite bosons made of two fermions by extending number states to these quantum particles. Two-particle correlations as well as the dispersion of the probability distribution are analyzed. We show that the particle composite nature reduces the antibunching effect predicted for elementary bosons. Furthermore, the probability distribution exhibits a dispersion that is greater for composite bosons than for elementary bosons. This dispersion corresponds to the one of sub-Poissonian processes, as for a quantum state but, unlike its elementary boson counterpart, it is not minimum. In general, our work shows that it is necessary to take into account the Pauli exclusion principle, which acts between fermionic components of composite bosons---along the line used here---to possibly extract statistical properties in a precise way.
Molecules made of two fermionic atoms are ``cobosons''---a contraction for ``composite bosons.'' These molecules ``feel'' each other not only through interactions between atoms, but also through the Pauli exclusion between their fermionic components. In order to point out the importance of this Pauli exclusion in cold atom physics, we have calculated here the energy change of NNN identical cobosonic molecules when a similar molecule, made of atoms having possibly different spin states, is added. Due to the difference in the number of fermion exchanges with a molecule having zero, one, or two atoms identical to the ones already present in the system, we may think that this energy change can take three different values, even for spin-independent interatomic interactions. We actually find that the energy change, in the Born approximation, is exactly the same at all orders in density, when the added molecule has two or just one atom identical to the ones already there. In other words, the scattering length is predicted to be the same if the added molecule can have exchanges with one set of fermionic components or with two sets. This unexpected equality, which of course holds for spin-independent interactions only, results from a subtle balance in the fermion exchanges. To prove it, we make use of a recent extension of the composite-exciton many-body theory we have constructed, to any type of cobosons, the physical understanding of the various exchange processes being enlightened by the Shiva diagram representation (so named because of the multiarm structure reminiscent of the Hindu god Shiva) of this many-body theory.