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arXiv:2507.04829

Multi-Photon Quantum Rabi Models with Center-of-Mass Motion

Sabrina Hartmann, Nicolija Momčilović, Alexander Friedrich

Published Jul 7, 2025• 0 citations
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Abstract

The precise control of light-matter interaction serves as a cornerstone for diverse quantum technologies, spanning from quantum computing to advanced atom optics. Typical experiments often feature multi-mode fields and require atoms with intricate internal structures to facilitate e.g., multi-photon interactions or cooling. Moreover, the development of high-precision cold-atom sensors necessitates the use of entangled atoms. We introduce a rigorous, second-quantized framework for describing multi-$Λ$-atoms in a cavity, encompassing their center-of-mass motion and interaction with a two-mode electromagnetic field. A key feature of our approach is the systematic application of a Hamiltonian averaging theory to the atomic field operators, enabling the derivation of a compact and physically insightful effective Hamiltonian governing the system's time evolution. This effective Hamiltonian not only incorporates the AC-Stark and Bloch-Siegert shifts, augmented by contributions from all involved ancillary states, but also includes effects from counter-rotating terms, offering a more complete description than conventional RWA-based models. A significant finding is the emergence of a particle-particle interaction mediated by ancillary states, stemming directly from the combined quantum nature of the coupled electromagnetic field and atomic system. We illustrate the practical implications of our model through an analysis of atomic Raman diffraction, determining the time evolution for a typical initial state and demonstrating the resulting Rabi oscillations. Furthermore, our results confirm that both the atomic and optical Hong-Ou-Mandel effect can be observed within this Raman configuration, utilizing a specific two-particle input state, thereby extending the utility of Raman diffraction for quantum interference phenomena and providing a microscopic model of the effect.

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Acta Physica Sinica (Overseas Edition)•Feb 1, 1999•1 citation
The precise control of light-matter interaction serves as a cornerstone for diverse quantum technologies, spanning from quantum computing to advanced atom optics. Typical experiments often feature multi-mode fields and require atoms with intricate internal structures to facilitate e.g., multi-photon interactions or cooling. Moreover, the development of high-precision cold-atom sensors necessitates the use of entangled atoms. We introduce a rigorous, second-quantized framework for describing multi-ΛΛΛ-atoms in a cavity, encompassing their center-of-mass motion and interaction with a two-mode electromagnetic field. A key feature of our approach is the systematic application of a Hamiltonian averaging theory to the atomic field operators, enabling the derivation of a compact and physically insightful effective Hamiltonian governing the system's time evolution. This effective Hamiltonian not only incorporates the AC-Stark and Bloch-Siegert shifts, augmented by contributions from all involved ancillary states, but also includes effects from counter-rotating terms, offering a more complete description than conventional RWA-based models. A significant finding is the emergence of a particle-particle interaction mediated by ancillary states, stemming directly from the combined quantum nature of the coupled electromagnetic field and atomic system. We illustrate the practical implications of our model through an analysis of atomic Raman diffraction, determining the time evolution for a typical initial state and demonstrating the resulting Rabi oscillations. Furthermore, our results confirm that both the atomic and optical Hong-Ou-Mandel effect can be observed within this Raman configuration, utilizing a specific two-particle input state, thereby extending the utility of Raman diffraction for quantum interference phenomena and providing a microscopic model of the effect.
The Rabi Hamiltonian, describing the interaction between a two-level atomic system and a single-cavity mode of the electromagnetic field, is one of the fundamental models in quantum optics. The model becomes exactly solvable by considering an atom without permanent dipole moments, whose excitation energy is quasiresonant with the cavity photon energy, and by neglecting the nonresonant (counter-rotating) terms. In this case, after including the decay of either the atom or the cavity mode to a continuum, one can derive the well-known phenomenology of quasiresonant transitions, including the fluorescence triplets. In this work we consider the most general Rabi model, incorporating the effects of permanent atomic electric dipole moments. Based on a perturbative analysis, we compare the intensities of emission lines induced by rotating terms, counter-rotating terms, and parity-symmetry-breaking terms in order to identify the parameter regimes in which these different contributions play a significant role. The analysis reveals that the emission strength related to the existence of permanent dipoles may surpass the one due to the counter-rotating interaction terms but is usually much weaker than the emission due to the main, resonant coupling. This ratio can be modified in systems with a reduced dimensionality or by engineering the energy spectral density of the continuum.
In cold atom, the coupling between "spin" (atomic internal hyperfine states) and "orbit" (atomic center-of-mass motion) can be induced by Raman transition, where different hyperfine states are coupled by a pair of Raman lasers. In recent years, this synthetic spin-orbit coupling has received tremendous attention, as it leads to a variety of novel quantum phenomena in precisely controllable cold atom systems. In this thesis, we first present a comprehensive analysis of one-, two- and many- particle physics of harmonically trapped atoms with spin-orbit coupling, followed by the study of "novel spin-orbit coupling" in two different systems: (1) cold atom spinor mixtures and (2) cold atoms in an optical cavity. In the first system, we consider a spinor mixture consisting of two species of cold atoms, where the spin-orbit coupling can be transmitted from one species to the other, and we discuss novel topological properties and the supersolid stripe phase in this mixture. In the second system, we consider the coupling among three parts: the cavity photon field, the atomic internal hyperfine states, and the atomic external center-of-mass motion, and we discuss how this coupling affects familiar quantum optical phenomena, such as Rabi oscillation and Dicke superradiance phase transition. Our novel systems contribute new and practical platforms for the research field of synthetic spin-orbit coupling in cold atoms.
We theoretically investigate quantum Rabi oscillations in a system consisting of a two-level atom (qubit) strongly coupled to a one-dimensional open waveguide. In contrast to conventional cavity quantum electrodynamics, the qubit interacts with a continuum of propagating modes, which gives rise to fundamentally different dynamical behavior. Within the rotating-wave approximation, we express the multimode Jaynes-Cummings Hamiltonian in terms of collective bosonic operators and show that the system possesses two integrals of motion, enabling an exact diagonalization of the Hamiltonian in the single-excitation subspace. We show that for a strongly interacting qubit-photon system, the dynamics are captured by a reduced two-level model. In this framework, each level is defined as the product of the atomic excited state and a specific field mode, which takes the form of a multiphoton Fock-like state. In this picture, the Rabi oscillations represent a collective phenomenon corresponding to oscillations between multiphoton states differing by a single photon. We then extend our analysis to multiphoton processes in which the initial field is a coherent state with a continuous spectrum. In this case, the Rabi frequency is shown to be sensitive to the spectral profile of the function that generates the coherent state.
The quantum nature of light makes it a basic component for models of quantum measurement and information exchange between disparate quantum modes, pioneered in the field of cavity quantum electrodynamics. The interaction of atomic ensembles with the mode of an optical cavity provides a flexible platform for exploring the coherent interaction of light with diverse macroscopic dynamics, such as collective motion and spin. This dissertation presents experimental results and theoretical models for continuous measurement and control of the center of mass motion and collective spin precession of an atomic ensemble, mediated by coupling to a high-finesse optical cavity. First, the theory of dispersive coupling between the cavity mode and the collective motion and spin of an atomic ensemble is derived, and then a general time-domain formalism is developed for theoretical analysis of multi-mode optodynamical systems. Single-mode optodynamical effects are introduced through experimental demonstrations of measurement and control of the collective atomic spin, providing a close analogy to cavity optomechanics.Next, multiple collective atomic modes are considered within a single cavity, in order to assemble optically mediated interactions within multi-mode optodynamical systems. A demonstration of optodynamical interactions between the center of mass motion of two atomic ensembles is presented, coupled through an optical spring mediated by the cavity mode. Then simultaneous coupling of the center of mass motion and total spin precession of a single ensemble of atoms is described, yielding an experimental realization of a negative-mass instability, facilitated by the novel resource of the spin ensembles inverted state. A theoretical analysis of the negative-mass instability is presented, which indicates the possibility of generating two-mode squeezed states in the absence of excess incoherent noise. Finally, linear state retrodiction from the optodynamical signals is discussed, providing background and supplemental material for a forthcoming manuscript.
In the framework of the nonsecular perturbation theory based on the Bogoliubov averaging method, the coherent dynamics of multiphoton Raman transitions in a two-level spin system driven by an amplitude-modulated microwave field is studied. Closed-form expressions for the Rabi frequencies of these transitions are obtained beyond the rotating-wave approximation for the low-frequency driving component. It is shown that spin states dressed by the high-frequency component of the driving field are shifted due to the Bloch-Siegert-like effect caused by antiresonant interactions with the strong low-frequency driving. We predict that with increasing the order of the Raman transition, the Rabi frequency decreases and the contribution of the Bloch-Siegert shift to this frequency becomes dominant. It is found that the amplitude and phase of the Rabi oscillations strongly depend on the initial phase of the low-frequency field as well as on detuning from multiphoton resonance. The recent experimental data for the second- and third-order Raman transitions observed for the nitrogen-vacancy center in diamond (Z. Shu et al., arXiv:1804.10492) are well described in the frame of our approach. Our results may provide new possibilities for coherent control of quantum systems.
The Hamiltonian is derived for the interaction of a Λ-type three-level atom with a two-mode quantum cavity field from the general interaction Hamiltonian between a multi-level atom and a multi-mode radiation field, and it is reduced to an effective two-mode Raman-coupled model under a large detuning condition. We propose a modified effective Hamiltonian for the two-mode Raman-coupled model, find the time-dependent state vectors, and present the validity conditions for the involved interaction Hamiltonians. It is shown that in the study of the two-mode Raman-coupled model it is not enough to retain only the usually used effective Hamiltonian, one must also take into account the ac Stark shift of the atomic levels (at least one of the levels). Finally, we study the atomic dynamics in the interaction of a Λ-type three-level atom with a two-mode quantum cavity field and in the two-mode Raman-coupled model. We find that the number of collapse-revivals, the collapse time and the revival time show new characteristics.