The interaction Hamiltonian of an electron and a quasi-monochromatic pulse of a strong quantized electromagnetic field is examined. Canonical transformations of the field variables are found that allow the division of the system's Hamiltonian in two parts. The first one describes the interaction between an electron and a single collective mode of the field. The properties of this mode are defined by the superposition of the modes corresponding to the pulse wave packet. The second part describes the field fluctuations relatively to the collective mode. The field intensity is estimated for which a single-mode approximation can be used for the system's description.
The interaction Hamiltonian of an electron and a quasi-monochromatic pulse of a strong quantized electromagnetic field is examined. Canonical transformations of the field variables are found that allow the division of the system's Hamiltonian in two parts. The first one describes the interaction between an electron and a single collective mode of the field. The properties of this mode are defined by the superposition of the modes corresponding to the pulse wave packet. The second part describes the field fluctuations relatively to the collective mode. The field intensity is estimated for which a single-mode approximation can be used for the system's description.
Abstract We investigate the applicability of the two major approximations which are most commonly employed in the study of the quantum Rabi model, namely the description of a resonant cavity mode as a single-mode quantized field and the use of the rotating wave approximation. Starting from the Hamiltonian of a two-level system interacting with a multi-mode quantized field, we perform the canonical transformation of the field operators. This allows one to partition the Hamiltonian of the system into two parts. The first part is the interaction of the two-level system with a single collective field mode, while the second one describes the interaction with field fluctuations. The first part is usually associated with the resonant cavity mode. This division enables us to determine the applicability condition of the single-mode approximation. In addition we identify simple approximate relations for the description of the eigenstates, eigenfunctions and the time evolution of the quantum Rabi model beyond the rotating wave approximation.
We investigate the dynamics of a charged particle interacting with a multimode quantized electromagnetic field and obtain an analytic solution for the full electron--field system. This framework enables the calculation of position expectation values and uncertainties for arbitrary wave packets and field states, allowing us to identify quantum corrections to the corresponding classical motion. While the corrections to the position expectation value are weak and largely insensitive to the quantum state of the field, the wave packet broadening exhibits a pronounced dependence on the field state. In particular, the quantum uncertainty of the radiation is directly imprinted onto the spatial uncertainty of the particle. We illustrate these effects for Gaussian wave packets interacting with coherent, Fock, and squeezed states, including bright squeezed vacuum. The interaction with a finite-duration laser pulse is also analyzed as a multimode example. Our results provide a transparent analytic route toward understanding how quantum fluctuations of light influence electron dynamics in strong-field settings.
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.
In this contribution we investigate quantum electrodynamical many-mode aspects by exploring the simplest possible situation in this context, namely the interaction of a single atom, modeled by a simple two-level system, with many-mode one-photon quantum states of the electromagnetic field. In particular, we concentrate on the question how the engineering of electromagnetic field modes influences the atom-field interaction. An interesting problem in this context is the engineering of ideal one-photon multi-mode quantum states which may excite a single atom perfectly [M. Stobińska, G. Alber, and G. Leuchs, EPL {\bf 86}, 14007 (2009)]. For purposes of quantum information processing such one-photon multi-mode states are particularly well suited for transmitting quantum information and for storing it in a material memory again.
A general approximation method is developed for the treatment of bound quantum systems in the presence of an intense external electromagnetic field of long wavelength. The method is based upon a unitary transformation which approximately removes the electromagnetic field from the problem. The accuracy of the technique increases with the number of photons involved in any given process; and in the case of large photon number, the results have a simple, concise form resembling first-order perturbation theory. It is possible to handle intense-field problems with small numbers of photons also, but then the formalism takes on a complexity analogous to the corresponding perturbation expressions for the same number of photons. The theory is extended to the case where two electromagnetic field modes are present, one or both of which may be intense. Bound-bound transitions in a hydrogenlike atom are calculated as an illustration of the application of the method. One of the explicit nonlinear effects to appear is an intensity-dependent deviation of the transition probability behavior from In for n-photon transitions, where I is the field intensity. It is demonstrated that intensity effects are much more probable with bound electrons than with free electrons.
Photons and electrons One of the great technical achievements of the second half of the 20th century was the invention of the laser. With its help extremely intense light could be produced. Furthermore, it even became posssible to excite a single resonator mode so that both the frequency and the propagation direction of the emitted radiation field are extraordinarily sharp. The mere existence of such high-intensity single-mode fields – the real problem that was ingeniously solved was to generate them – is no matter of surprise. We know from classical electromagnetic theory that the amplitude of a monochromatic electromagnetic field can take on arbitrarily large values. This is a direct consequence of the linearity of Maxwell's equations. What does this mean for quantized radiation fields? Certainly, the mentioned property of classical fields must also be retained in the quantum mechanical description. Since the electromagnetic energy is quantized in ‘packets’ of magnitude h ν (photons), the number of photons in a given mode must be unlimited, in principle. All this is very well known. However, it becomes noteworthy when we compare the behaviour of photons with that of electrons. Amazingly, it turns out that an electronic quantum state can be occupied by one electron, at maximum. (Strictly speaking, in the description of the electronic state the spin variable must be included.) This drastic difference in the collective behaviour of photons and electrons could, in fact, not be expected from what we know from single photons and electrons. Both of them have particle-like as well as wave-like properties. However, when many of them are present, they exhibit quite different a demeanour.