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Physical Review A

Interaction of a charged harmonic oscillator with a single quantized electromagnetic field mode

Michael D. Crisp

Published Jul 1, 1991• 18 citations
Publisher

Abstract

The interaction of an atomic system with a single mode of a quantized electromagnetic field is investigated in the dipole approximation. Two approximations that are almost always made in this analysis are the neglect of the diamagnetic term in the Hamiltonian and the dropping of the rotating-wave approximation (RWA) terms that correspond to processes that do not conserve energy. Both approximations involve the neglect of terms that are of the same order of magnitude so that consistency demands that they be considered together. It is demonstrated that, in the dipole approximation, a minor modification of the usual application of quantum electrodynamics to any atomic system can exactly incorporate the effects of the diamagnetic term into a new frequency of the electromagnetic cavity mode; this frequency is slightly different from that of the empty cavity. This result is then applied to the specific case of a charged harmonic oscillator interacting with a single mode. For this simple model an analytic solution can then be found that includes the effects of the diamagnetic term and does not use the RWA. It is found that, for off resonance, the normal modes of the coupled system oscillate at shifted frequencies which are in agreement with the second-order perturbation-theory calculation of the Lamb shift. When the cavity mode is tuned to the atomic oscillator frequency, the perturbation-theory Lamb-shift calculation is no longer valid, and the exact solution derived in this paper predicts a resonant Lamb shift that is significantly larger than the off-resonant single-mode Lamb shift. Furthermore, the frequency spectrum is qualitatively changed from the single frequency of the free atomic system to a doublet spectrum. An analytic solution of this harmonic-oscillator problem can also be found when the RWA terms are dropped from the Hamiltonian. Comparison of the two solutions reveals that the RWA terms contribute a frequency shift of magnitude independent of the field intensity in the simple case of a charged harmonic oscillator.When trajectories of the exact and RWA solutions are compared in phase space, it is found that one effect of the RWA terms is to lift a twofold degeneracy of the phase-space orbits that occurs when the RWA terms are neglected. The harmonic-oscillator problem has several features in common with the interaction of a two-level atom with a single quantized field mode. Both Hamiltonians exhibit inversion symmetry and hence have parity as a good quantum number. Both Hamiltonians contain RWA terms. When these RWA terms are neglected, both Hamiltonians have a new constant of motion that can be identified with the total number of excitations in the combined system.

DOI
10.1103/physreva.44.563

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Fock states and coherent states have been widely applied in analyses of quantum-optics experiments. In this paper, the application of a set of basis functions that consist of a product of displaced harmonic-oscillator states for the electromagnetic field and atomic states that are eigenfunctions of the momentum operator will be explored. For the case of a single mode, these states are the exact eigenfunctions in the limiting case where the electromagnetic-field mode frequency is much larger than the atomic transition frequency. This choice of basis functions will be used to analyze the interaction of a nonrelativistic, ``two-level'' atom with a single-frequency, quantized electromagnetic-field mode. Since in this basis the sign of the displacement of the electromagnetic-field state depends on the state of the atom, these new basis wave functions exhibit explicitly some aspects of atom-field correlations that are intrinsically quantum mechanical in nature. The analysis will be performed in the electric-dipole approximation for an atomic system that obeys the \ensuremath{\Delta}m=0 selection rule. The rotating-wave-approximation (RWA) terms cannot be made negligible through judicious choice of the field polarization for this system. The diamagnetic term, ${\mathit{e}}^{2}$${\mathbf{A}}^{2}$/2${\mathit{mc}}^{2}$, which is of the same order of magnitude as the RWA terms, is taken into account exactly by slightly modifying the traditional approach to quantizing the electromagnetic field. In this way the physical effects of the diamagnetic term are absorbed into a new electromagnetic cavity mode frequency.An approximate expression for the combined system's energy eigenvalues is found up to third order in the ratio of one-half the atomic transition frequency to the mode frequency. This expression indicates that inclusion of the RWA terms adds a small ``ripple'' on top of the smoothly varying energy eigenvalues when they are plotted as a function of the atom-field coupling constant. The ripple is interpreted as ``interference'' between two displaced oscillator-field states that represent the unperturbed state. When the Fock states are used as the basis for expanding the wave function of this system, a formidable five-term recurrence relation results. However, expanding the wave function in terms of the displaced harmonic-oscillator basis results in a less formidable three-term recurrence relation for the quantum coefficients.

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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)].

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A dressed-state perturbation theory beyond the rotating waveapproxi-mation (RWA) is presented to investigate the interaction between a two level electronic transition of the polar molecules and a quantized cavity field. Analytical expressions can be explicitly derived for both the ground- and excited-state-energy spectrums and wave functions of the system, where the contribution of permanent dipole moments (PDM) and the counter-rotating wave term (CRT) can be shown separately. The validity of these explicit results is discussed by comparing with the direct numerical simulation. Comparing to CRT coupling, PDM results in the coupling of more dressed states and the energy shift proportional to the square of the normalized permanent dipole difference, and a greater Bloch-Siegert shift could be produced in giant dipole molecule cavity QED. In addition, our method could also be extended to the solution of two-level atom Rabi model Hamiltonian beyond the RWA.

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The interaction of an atomic system with a single mode of a quantized electromagnetic field is investigated in the dipole approximation. Two approximations that are almost always made in this analysis are the neglect of the diamagnetic term in the Hamiltonian and the dropping of the rotating-wave approximation (RWA) terms that correspond to processes that do not conserve energy. Both approximations involve the neglect of terms that are of the same order of magnitude so that consistency demands that they be considered together. It is demonstrated that, in the dipole approximation, a minor modification of the usual application of quantum electrodynamics to any atomic system can exactly incorporate the effects of the diamagnetic term into a new frequency of the electromagnetic cavity mode; this frequency is slightly different from that of the empty cavity. This result is then applied to the specific case of a charged harmonic oscillator interacting with a single mode. For this simple model an analytic solution can then be found that includes the effects of the diamagnetic term and does not use the RWA. It is found that, for off resonance, the normal modes of the coupled system oscillate at shifted frequencies which are in agreement with the second-order perturbation-theory calculation of the Lamb shift. When the cavity mode is tuned to the atomic oscillator frequency, the perturbation-theory Lamb-shift calculation is no longer valid, and the exact solution derived in this paper predicts a resonant Lamb shift that is significantly larger than the off-resonant single-mode Lamb shift. Furthermore, the frequency spectrum is qualitatively changed from the single frequency of the free atomic system to a doublet spectrum. An analytic solution of this harmonic-oscillator problem can also be found when the RWA terms are dropped from the Hamiltonian. Comparison of the two solutions reveals that the RWA terms contribute a frequency shift of magnitude independent of the field intensity in the simple case of a charged harmonic oscillator.When trajectories of the exact and RWA solutions are compared in phase space, it is found that one effect of the RWA terms is to lift a twofold degeneracy of the phase-space orbits that occurs when the RWA terms are neglected. The harmonic-oscillator problem has several features in common with the interaction of a two-level atom with a single quantized field mode. Both Hamiltonians exhibit inversion symmetry and hence have parity as a good quantum number. Both Hamiltonians contain RWA terms. When these RWA terms are neglected, both Hamiltonians have a new constant of motion that can be identified with the total number of excitations in the combined system.
Fock states and coherent states have been widely applied in analyses of quantum-optics experiments. In this paper, the application of a set of basis functions that consist of a product of displaced harmonic-oscillator states for the electromagnetic field and atomic states that are eigenfunctions of the momentum operator will be explored. For the case of a single mode, these states are the exact eigenfunctions in the limiting case where the electromagnetic-field mode frequency is much larger than the atomic transition frequency. This choice of basis functions will be used to analyze the interaction of a nonrelativistic, ``two-level'' atom with a single-frequency, quantized electromagnetic-field mode. Since in this basis the sign of the displacement of the electromagnetic-field state depends on the state of the atom, these new basis wave functions exhibit explicitly some aspects of atom-field correlations that are intrinsically quantum mechanical in nature. The analysis will be performed in the electric-dipole approximation for an atomic system that obeys the \ensuremath{\Delta}m=0 selection rule. The rotating-wave-approximation (RWA) terms cannot be made negligible through judicious choice of the field polarization for this system. The diamagnetic term, e2{\mathit{e}}^{2}e2A2{\mathbf{A}}^{2}A2/2mc2{\mathit{mc}}^{2}mc2, which is of the same order of magnitude as the RWA terms, is taken into account exactly by slightly modifying the traditional approach to quantizing the electromagnetic field. In this way the physical effects of the diamagnetic term are absorbed into a new electromagnetic cavity mode frequency.An approximate expression for the combined system's energy eigenvalues is found up to third order in the ratio of one-half the atomic transition frequency to the mode frequency. This expression indicates that inclusion of the RWA terms adds a small ``ripple'' on top of the smoothly varying energy eigenvalues when they are plotted as a function of the atom-field coupling constant. The ripple is interpreted as ``interference'' between two displaced oscillator-field states that represent the unperturbed state. When the Fock states are used as the basis for expanding the wave function of this system, a formidable five-term recurrence relation results. However, expanding the wave function in terms of the displaced harmonic-oscillator basis results in a less formidable three-term recurrence relation for the quantum coefficients.
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)].
One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.
A dressed-state perturbation theory beyond the rotating wave approximation(RWA) is presented to investigate the interaction between a two-level electronic transition of polar molecules and a quantized cavity field. Analytical expressions can be explicitly derived for both the ground- and excited-state-energy spectrums and wave functions of the system, where the contribution of permanent dipole moments(PDM) and the counter-rotating wave term(CRT) can be shown separately.The validity of these explicit results is discussed by comparison with the direct numerical simulation. Compared to the CRT coupling, PDM results in the coupling of more dressed states and the energy shift is proportional to the square of the normalized permanent dipole difference, and a greater Bloch–Siegert shift can be produced in the giant dipole molecule cavity QED. In addition, our method can also be extended to the solution of the two-level atom Rabi model Hamiltonian beyond the RWA.更多还原
A dressed-state perturbation theory beyond the rotating wave approximation (RWA) is presented to investigate the interaction between a two level electronic transition of the polar molecules and a quantized cavity field.Analytical expressions can be explicitly derived for both the ground-and excited-state-energy spectrums and wave functions of the system, where the contribution of permanent dipole moments (PDM) and the counter-rotating wave term (CRT) can be shown separately.The validity of these explicit results is discussed by comparing with the direct numerical simulation.Comparing to CRT coupling, PDM results in the coupling of more dressed states and the energy shift proportional to the square of the normalized permanent dipole difference, and a greater Bloch-Siegert shift could be produced in giant dipole molecule cavity QED.In addition, our method could also be extended to the solution of two-level atom Rabi model Hamiltonian beyond the RWA.
A dressed-state perturbation theory beyond the rotating waveapproxi-mation (RWA) is presented to investigate the interaction between a two level electronic transition of the polar molecules and a quantized cavity field. Analytical expressions can be explicitly derived for both the ground- and excited-state-energy spectrums and wave functions of the system, where the contribution of permanent dipole moments (PDM) and the counter-rotating wave term (CRT) can be shown separately. The validity of these explicit results is discussed by comparing with the direct numerical simulation. Comparing to CRT coupling, PDM results in the coupling of more dressed states and the energy shift proportional to the square of the normalized permanent dipole difference, and a greater Bloch-Siegert shift could be produced in giant dipole molecule cavity QED. In addition, our method could also be extended to the solution of two-level atom Rabi model Hamiltonian beyond the RWA.