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CRC Press eBooks

From the Jaynes-Cummings model to collective interactions

V.V. Dodonov, V.I. Man'ko

Published Mar 13, 2003• 3 citations
Publisher

Abstract

E = iJ2 eky/2irnu;k/V (akefi* - a\e'iir) , k where ak, a\ are the annihilation and creation operators of photons with the wave vector k, frequency ujk and polarization ek, satisfying the common boson commutation relations [ak, a}] = 8ki\ V stands for the cavity volume. The free field Hamiltonian has the form k Consider a collection of identical two-level atoms with the upper energy level |1) and the lower energy level |0). The free atomic Hamiltonian reads where sZj — — |0j)(0j| and hcja is the energy separation between two levels. The electric dipole interaction of the cavity field with the atomic system is described by the Hamiltonian where Ej is the electric field at the point fj which corresponds to the position of the j-th atom and dj is the electric dipole operator of this atom given by Here s + i = | l j ) (0j | and s__, = |0 j ) ( l j | are the atomic transition operators and dj = e(lj\f\0j) is the matrix element of the electric dipole momentum (which are assumed to be the same for all the atoms). Note that the operators sZj, s±j obey the algebraic relations for the common Pauli matrices.

DOI
10.1201/9781482288223-12

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E = iJ2 eky/2irnu;k/V (akefi* - a\e'iir) , k where ak, a\ are the annihilation and creation operators of photons with the wave vector k, frequency ujk and polarization ek, satisfying the common boson commutation relations [ak, a}] = 8ki\ V stands for the cavity volume. The free field Hamiltonian has the form k Consider a collection of identical two-level atoms with the upper energy level |1) and the lower energy level |0). The free atomic Hamiltonian reads where sZj — — |0j)(0j| and hcja is the energy separation between two levels. The electric dipole interaction of the cavity field with the atomic system is described by the Hamiltonian where Ej is the electric field at the point fj which corresponds to the position of the j-th atom and dj is the electric dipole operator of this atom given by Here s + i = | l j ) (0j | and s__, = |0 j ) ( l j | are the atomic transition operators and dj = e(lj\f\0j) is the matrix element of the electric dipole momentum (which are assumed to be the same for all the atoms). Note that the operators sZj, s±j obey the algebraic relations for the common Pauli matrices.
Spin-boson interactions are at the heart of modern physics since they bridge the world of photons as described by quantum electrodynamics and that of quantum mechanics of massive objects, like atoms or ions. It is theoretically well understood that for an electric field which is polarized in e.g. x-direction, the electromagnetic dipole interaction produces a term , where is the electric field vector, is the position vector of an electron in the atom, a and are annihilation and creation operators of the electromagnetic field, respectively, and is the Pauli matrix in x-direction, in standard representation . Now, in highly intense light fields far off resonance (), as used in recent experiments, or where the underlying physics itself provides
Abstract This chapter presents the basic elements of quantum theory, including the Dirac bra and ket notation for quantum states, Hermitian operators and their eigenstates, commutators and the Schrödinger and Heisenberg equations of motion. Simple model atoms are described as an effective spin and their properties related to the familiar Pauli spin-half operators. The density matrix is introduced to describe incoherent processes. The electric-dipole coupling between and atom and the electromagnetic field is treated leading to coupled equations for the probability amplitudes. Finally, quantized field modes are introduced and their elementary properties described leading to the fundamental Jaynes–Cummings model of a single two-state atom interacting with a quantized field mode.
The interaction of a nonrelativistic, ``two-level'' atom with a single-frequency, quantized electromagnetic field is usually analyzed in the dipole approximation by applying the rotating-wave approximation (RWA) to obtain the exactly solvable Jaynes-Cummings model (JCM). In this paper, it is shown that for a spin-1/2 particle, at rest in a constant magnetic field and for a system that obeys the electric dipole \ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1 selection rules, a choice of a perturbing field that consists of circularly polarized photons can make the RWA unnecessary for the solution of the quantum-electrodynamic (QED) problem. For these cases, the only approximation necessary to obtain an exact solution of the QED problem is to neglect the spontaneous emission of photons with a sense of polarization opposite that of the perturbing field. The matrix elements corresponding to these ``virtual'' processes are independent of the number of photons n in the applied field and become relatively small, compared with those that correspond to energy-conserving processes, as n becomes large.However, a two-level system that obeys the \ensuremath{\Delta}m=0 selection rule will require the RWA in order to reduce it to the JCM, regardless of the choice of polarization of the perturbing field. For this case, the emission of the photons that do not conserve energy is an induced process that corresponds to matrix elements that increase as \ensuremath{\surd}n+1 . It is thus concluded that, provided the right choice of selection rules and polarization of the applied field is made, the JCM provides a more accurate description of some systems than the accuracy of the RWA. Furthermore, it is concluded that an experimental test of the validity of the RWA can be made by performing the same experiments on an electric dipole \ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1 transition, first with circularly polarized light and then with linearly polarized light of the same intensity. Any difference in response of the atomic system for these two experiments can be identified with terms that are neglected in the RWA. This experimental technique may prove useful for observing the optical Bloch-Siegert shift and for investigating the failure of the RWA and the chaotic quantum behavior predicted by Milonni, Ackerhalt, and Galbraith [Phys. Rev. Lett. 50, 966 (1983)].
The Jaynes-Cummings (JC) model has been studied extensively because of its simple and realistic representation of the dipole interaction between an atom and the quantized radiation field. It exhibits, without perturbative expansions or statistical decorrelations, many quantum mechanical effects in the case of strong or weak interaction, long or short time regime, on or off the resonance. The model is extended to a three-level atom for the investigation of physical phenomena associated with two-photon processes.
Using the technique of classical r-matrices and quantum Lax operators, we construct the most general form of the quantum integrable “n-level, many-mode” spin-boson Jaynes-Cummings-Dicke-type hamiltonians describing an interaction of a molecule of N n-level atoms with many modes of electromagnetic field and containing, in general, additional non-linear interaction terms. We explicitly obtain the corresponding quantum Lax operators and spin-boson analogs of the generalized Gaudin hamiltonians and prove their quantum commutativity. We investigate symmetries of the obtained models that are associated with the geometric symmetries of the classical r-matrices and construct the corresponding algebra of quantum integrals. We consider in detail three classes of non-skew-symmetric classical r-matrices with spectral parameters and explicitly obtain the corresponding quantum Lax operators and Jaynes-Cummings-Dicke-type hamiltonians depending on the considered r-matrix.