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American Journal of Physics

Bose particles in an external electromagnetic field as a one-particle problem

C. Avilez, Pier A. Mello

Published Oct 1, 1977• 0 citations
Publisher

Abstract

The statement is often made that the second quantized Hamiltonian for a system of noninteracting particles in an external weak electromagnetic field gives rise essentially to a one-particle problem, to which there corresponds a dressed vacuum different from the bare one. In this paper we implement that statement by explicitly constructing the dressed vacuum, together with the quasiparticle creation and annihilation operators associated with the one-particle problem. The process of diagonalizing the Hamiltonian is illustrated step by step in a simple model.

DOI
10.1119/1.10855

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The statement is often made that the second quantized Hamiltonian for a system of noninteracting particles in an external weak electromagnetic field gives rise essentially to a one-particle problem, to which there corresponds a dressed vacuum different from the bare one. In this paper we implement that statement by explicitly constructing the dressed vacuum, together with the quasiparticle creation and annihilation operators associated with the one-particle problem. The process of diagonalizing the Hamiltonian is illustrated step by step in a simple model.
A unitary operator, which transforms free states into the exact eigenstates for a one-dimensional 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.
Summary of the lectures: 1. Creation and annihilation operators for the system of indistinguishable particles 1.1 The permutation group and the states of a system of indistinguishable particles 1.2 Dimension of the Hilbert space of a system of indistinguishable particles 1.3 Definition and properties of the creation and annihilation operators 1.4 TheFockspace 1.5 The representations of state vectors and operators 1.5.1 N-particlewave-functions 1.5.2 The second-quantization representation of operators 1.5.3 Examples 1.6 Evolution of operators in the Heisenberg picture 1.7 Statistical operators of indistinguishable particles 1.7.1 The averages of the s-particle operators 1.7.2 The general structure of the one-particle statistical operator 2 Quadratic Hamiltonian and the diagonalization 2.1 Diagionalization of the Hamiltonian quadratic in the fermion operators 2.2 Diagionalization of the Hamiltonian quadratic in the boson operators 2.2.1 Diagonalization of the quadratic bosonic Hamiltonian possessing a zero mode 2.3 Long range order, condensation and the Bogoliubov spectrum of weakly interacting Bose gas 2.3.1 The excitation spectrum of weakly interacting Bose gas inabox 2.3.2 The excitation spectrum of an interacting Bose gas in an externalpotential 2.4 The Jordan-Wigner transformation: fermionization of interacting 1D spin chains
The many-body problem for a system of composite particles is formulated in a way which takes explicit account of the composite nature of these particles and allows a clear separation between interatomic and intraatomic interactions. A second-quantization representation, fully equivalent to the conventional representation in which nuclei and electrons appear explicitly, is developed in terms of atomic annihilation and creation operators satisfying elementary Bose or Fermi commutation relations. All effects of the composite nature of the atoms are exactly contained in the interatomic and intraatomic matrix elements and in certain exchange integrals. An application is made to the problem of Bose condensation of fermion pairs.
The Hamiltonian most often used to describe a system of electrons interacting with lattice vibrations has the asymmetry that the electrons are represented by the creation and annihilation operators which arise from quantum field theory whereas the lattice vibrations are represented by the one-particle ladder operators of the one-dimensional harmonic-oscillator problem. Although the equal spacing of the harmonic-oscillator energy levels and the commutation relations satisfied by the oscillator ladder operators have led to an interpretation of this formalism in terms of the "creation and annihilation" of phonons, the phonons thus defined are not the quasiparticles which are obtained by a field-theoretical formulation of the lattice Hamiltonian in the harmonic approximation. A Hamiltonian is derived in this paper in which both the electrons and the lattice (in the harmonic approximation) are represented in field-theoretical formalism. A lattice creation operator in this formulation actually creates a vibrational mode (in a given state) from the physical vacuum, in contrast to a ladder operator which merely generates transitions between two states of an already existing vibrational mode. As a consequence, the number operator for the lattice quasiparticles commutes with the Hamiltonian, a property which is useful in certain applications. It is shown that even though our Hamiltonian is formally different from the ordinary one, it yields equivalent results for the state vectors of physical interest.
In terms of excitation creation and annihilation operators of the Jaynes-Cummings model, acting in the representation of dressed states, the Hamiltonian is written which describes the character of the spectrum of excitations of two modes, representing a quantum analog of the classical behavior of two interacting one-dimensional anharmonic oscillators, namely, the field and atomic oscillators. The anharmonicity is caused by the nonlinearity of the oscillator interaction and manifests itself in the dependence of the frequencies of both modes on the number of excitations, i.e., on the energy. It is shown that an external deterministic force, acting on the system during a certain time t 0, transfers it from a vacuum state to a coherent state or from one of the coherent states to another coherent state. The probability of the transition from the vacuum state to the coherent state with a given number of excitations represents the Poissonian distribution for the number of excitations formed in the (atom + field) system by the end of action of the external force. It was found to be proportional to the excitation time t 0.
The desirability of considering "dressed" rather than "bare" states of atoms in strong electromagnetic fields has been noted by others. It is suggested here that such a treatment is conveniently carried out by performing a unitary transformation of the Hamiltonian so that the "dressed" states in the new basis are precisely the "bare" states of the usual basis. A perturbative method for constructing such a transformation, closely related to a procedure described by Heitler, is proposed. The method is applied through second order to a two-level atom model, and it is verified that the transformation makes the dressed ground atomic state stable, and the excited state unstable with the proper decay probability; furthermore, the energies of the two states are renormalized by the correct amounts. The method is then compared with the "nonperturbative" momentum-translation approximation, and shown to contain already in first order the entire operator content of that approximation, but to yield different transition probabilities. Finally, the transformation method is applied through second order to the standard scalar-field model and shown to produce the usual dressing transformation, mass renormalization, and induced nucleon-nucleon potential of that model, a result which tends to substantiate the hypothesis that the transformation produces the correct dressed states of a physical system.