Title One-Dimensional Many Boson System . III : UnitaryTransformation and Exactly Dressed Bose Particle (original) (raw)
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One-Dimensional Many Boson System. III: Unitary Transformation and Exactly Dressed Bose Particle
Progress of Theoretical Physics, 1981
A unitary operator, which transforms free states into the exact eigenstates for a onedimensional many boson system with repulsive delta-function potential, is explicitly constructed. By the unitary operator, original bare creation and annihilation operators are transformed into new operators which are shown to create or annihilate the exactly dressed bosons with the interaction cloud. The total Hamiltonian, total number and total momentum in the system are expressed in the diagonalized form in terms of the dressed operators. It is shown that the exact ground state is the condensed state of all exactly dressed bosons with zero momentum.
One-Dimensional Many Boson System. IV: Condensation and Excitation Energy
Progress of Theoretical Physics, 1981
The analysis for a one-dimensional many boson system with a repulsive delta-function potential is continued. It is shown that the expectation value of the number ao * ao of bare bosons with zero momentum for the exact ground state 11f!0.0 ..... 0> is negligibly small compared with the total number n in the limit of an infinitely large coupling constant g; i.e., < 1f!o ..... olao*aol1f!o ..... o>/n->O(for n->CXJ, g->CXJ). This result shows a striking contrast to the result < 1f!o ..... oIAo * Aol1f!o ..... o> /n = 1 for an arbitrary g, where Ao * Aoindicates the number operator of exactly dressed bosons with zero momentum. It is clarified that the energy of a dressed boson strongly depends upon the number distribution {nq} of the other dressed bosons in the system. The excitation energy of a dressed boson has phonon character when no/n "r 0, where no denotes the number of dressed bosons with zero momentum. When no/n tends to zero, the phonon character disappears in a drastic way. Some characteristic phenomena in liquid helium II are discussed on the basis of the results of our previous papers and the present paper. N ow, in the same way as the derivation of the unitary operator U (see (3'1) ~(3'12) in III), the operator C(v) satisfies C(v)IO)=IO), C(V)lql, "', qn)= Cn(V)lql, "', qn), (n:;:;l) From (C•4) and (C'3), we can see that C*(v)G(v)lo)=lo), G(v)G*(v)IO)=IO),
Energy Spectrum of One-Dimensional Many Boson System
Progress of Theoretical Physics, 1989
An energy spectrum is exactly examined for a one•dimensional many boson system where bosons are interacting to each other through a delta• functional repulsive potential. ,After a unitary transfor• mation, the Hamiltonian is diagonalized to be the following compact form "2,p(p2/2m)np+(7rn!2mL) x "2,p,qip-qinpnq + (1/6m)(7rn!L)2[("2,pnp)3-"2,~np] at the infinitely large limit of the coupling constant. This form is non-linear with respect to the numbers np of quasi-particles. The non-linear terms are Galilean-invariant and produce a phonon-like spectrum. In a case of a finite coupling constant g, the total energy is expanded to power series of (l/g).
Boson description of collective states
Nuclear Physics A, 1971
Pauli principle. The problem of the separation of the "physical" and "unphysical" components has been solved by the introduction of a non-linear boson transformation. 145 146 D. JANSSEN et al.
Continuous unitary transformations in two-level boson systems
Physical Review C, 2005
Two-level boson systems displaying a quantum phase transition from a spherical (symmetric) to a deformed (broken) phase are studied. A formalism to diagonalize Hamiltonians with O(2L + 1) symmetry for large number of bosons is worked out. Analytical results beyond the simple mean-field treatment are deduced by using the continuous unitary transformations technique. In this scheme, a 1/N expansion for different observables is proposed and allows one to compute the finite-size scaling exponents at the critical point. Analytical and numerical results are compared and reveal the power of the present approach to compute the finite-size corrections in such a context.
Journal of Physics B: Atomic, Molecular and Optical Physics, 2013
We introduce unitary quantum phase operators for material particles. We carry out a model study on quantum phases of interacting bosons in a symmetric double-well potential in terms of unitary and commonly-used non-unitary phase operators and compare the results for different number of bosons. We find that the results for unitary quantum phase operators are significantly different from those for non-unitary ones especially in the case of low number of bosons. We introduce unitary operators corresponding to the quantum phase-difference between two single-particle states of fermions. As an application of fermionic phase operators, we study a simple model of a pair of interacting two-component fermions in a symmetric double-well potential. We also investigate quantum phase and number fluctuations to ascertain number-phase uncertainty in terms of unitary phase operators.
Strongly interacting one-dimensional Bose condensates in power law potentials
Europhysics Letters (EPL), 2000
We study the interaction effects on the condensates by considering a model of one-dimensional bosons. The power law type external potential allows for the formation of a condensate in these systems. Using a density-functional theory type formalism we obtain an equation describing the condensate wave function in the limit of very strong interactions between the bosons. The properties of the condensate in the model system with strong interactions are investigated. The equivalence of strongly interacting bosons to noninteracting spinless fermions is demonstrated.
Two-particle States in One-dimensional Coupled Bose-Hubbard Models
2022
We study dynamically coupled one-dimensional Bose-Hubbard models and solve for the wave functions and energies of two-particle eigenstates. Even though the wave functions do not directly follow the form of a Bethe Ansatz, we describe an intuitive construction to express them as combinations of Choy-Haldane states for models with intra- and inter-species interaction. We find that the two-particle spectrum of the system with generic interactions comprises in general four different continua and three doublon dispersions. The existence of doublons depends on the coupling strength ΩΩΩ between two species of bosons, and their energies vary with ΩΩΩ and interaction strengths. We give details on one specific limit, i.e., with infinite interaction, and derive the spectrum for all types of two-particle states and their spatial and entanglement properties. We demonstrate the difference in time evolution under different coupling strengths, and examine the relation between the long-time behavior...
Many-body excitations in trapped Bose gas: A non-Hermitian view
2021
We provide the analysis of a physically motivated model for a trapped dilute Bose gas with repulsive pairwise atomic interactions at zero temperature. Our goal is to describe aspects of the excited many-body quantum states by accounting for the scattering of atoms in pairs from the macroscopic state (condensate). We formally construct a many-body Hamiltonian, Happ, that is quadratic in the Boson field operators for noncondensate atoms. This Happ conserves the total number of atoms. Inspired by Wu (J. Math. Phys., 2:105–123, 1961), we apply a non-unitary transformation to Happ. Key in this non-Hermitian view is the pair-excitation kernel, which in operator form obeys a Riccati equation. In the stationary case, we develop an existence theory for solutions to this operator equation by a variational approach. We connect this theory to the one-particle excitation wave functions heuristically derived by Fetter (Ann. Phys., 70:67–101, 1972). These functions solve an eigenvalue problem for ...
Fermions and associated bosons of one-dimensional model
Communications in Mathematical Physics, 1967
The representation of the canonical commutation relations involved in the construction of boson operators from fermion operators according to the recipe of the neutrino theory of light is studied. Starting from a cyclic Fockrepresentation for the massless fermions the boson operators are reduced by the spectral projectors of two charge-operators and form an infinite direct sum of cyclic Fock-representations. Kronig's identity expressing the fermion kinetic energy in terms of the boson kinetic energy and the squares of the charge operators is verified as an identity for strictly self adjoint operators. It provides the key to the solubility of LTJTTINGER'S model. A simple sufficient condition is given for the unitary equivalence of the representations linked by the canonical transformation which diagonalizes the total Hamiltonian.