Hybrid quantum systems consisting of a collection of N spin- 1 / 2 particles uniformly interacting with an electromagnetic field, such as one confined in a cavity, are important for the development of quantum information processors and will be useful for metrology, as well as tests of collective behavior. Such systems are often modeled by the Tavis-Cummings model, and having an accurate understanding of the thermal behaviors of this system is needed to understand their behavior in realistic environments. We quantitatively show in this work that the Dicke subspace approximation is at times invoked too readily. Specifically, we show that there is a temperature above which the degeneracies in the system become dominant and the Dicke subspace is minimally populated. This transition occurs at a lower temperature than previously considered. In such a temperature regime, the key constants of the motion are the total excitation count between the spin system and cavity and the collective angular momentum of the spin system. These enable perturbative expansions for thermal properties in terms of the energy shifts of dressed states, called Lamb shifts herein. They enable efficient numeric methods that scale in terms of the size of the spin system. Notably the runtime to obtain certain parameters of the system scales as O ( N ) , and is thus highly efficient. These provide methods for approximating, and bounding, properties of these systems as well as characterizing the dominant population regions, including under perturbative noise. In the regime of stronger spin-spin coupling, the perturbations outweigh the expansion series terms, and inefficient methods must likely be employed, removing the computational efficiency of simulating such systems. The results in this work can also be used for related systems such as coupled-cavity arrays, cavity-mediated coupling of collective spin ensembles, and collective spin systems.
Hybrid quantum systems consisting of a collection of N spin- 1 / 2 particles uniformly interacting with an electromagnetic field, such as one confined in a cavity, are important for the development of quantum information processors and will be useful for metrology, as well as tests of collective behavior. Such systems are often modeled by the Tavis-Cummings model, and having an accurate understanding of the thermal behaviors of this system is needed to understand their behavior in realistic environments. We quantitatively show in this work that the Dicke subspace approximation is at times invoked too readily. Specifically, we show that there is a temperature above which the degeneracies in the system become dominant and the Dicke subspace is minimally populated. This transition occurs at a lower temperature than previously considered. In such a temperature regime, the key constants of the motion are the total excitation count between the spin system and cavity and the collective angular momentum of the spin system. These enable perturbative expansions for thermal properties in terms of the energy shifts of dressed states, called Lamb shifts herein. They enable efficient numeric methods that scale in terms of the size of the spin system. Notably the runtime to obtain certain parameters of the system scales as O ( N ) , and is thus highly efficient. These provide methods for approximating, and bounding, properties of these systems as well as characterizing the dominant population regions, including under perturbative noise. In the regime of stronger spin-spin coupling, the perturbations outweigh the expansion series terms, and inefficient methods must likely be employed, removing the computational efficiency of simulating such systems. The results in this work can also be used for related systems such as coupled-cavity arrays, cavity-mediated coupling of collective spin ensembles, and collective spin systems.
We present a comprehensive theoretical and computational study of the Tavis–Cummings model in the collective-spin basis, addressing both its static spectral properties and dynamic behavior under unitary and open-system evolution. By exploiting the commutation of the total-excitation operator with the Hamiltonian, we construct block-diagonal representations for systems of two-level molecules coupled to a single quantized cavity mode. We derive analytic expressions for the degeneracies, block dimensions, and secular polynomials, introducing a family of orthogonal “Félix–Sanz-Vicario” new class of orthogonal polynomials (here baptized Félix-Sanz-Vicario polynomials) whose roots furnish the energy eigenvalues and whose finite-degree structure allows precise control over photon-number truncation in small manifolds. Numerical implementations in Python (QuTiP) confirm exact agreement with the original Tavis–Cummings solution in each excitation manifold, while also highlighting the role of lower-spin subspaces—absent in the bosonized highest-spin description—in mediating configurational molecular interactions via a rank-1 spherical tensor operator. Time-dependent simulations compare dynamics under the rotating wave aproximation (Tavis-Cummings model) and without it (Dicke model), demonstrating population transfer, Rabi oscillations, and collapse–revival phenomena. The dynamics of open quantum system, via a Lindblad master equation, illustrate cavity-loss effects on photon-number decay, von Neumann entanglement entropy, and coherent exchange under dissipation. Our results establish that the collective-spin framework, using a set of solutions in terms of orthogonal polynomials, provides both analytical insight and computational efficiency for spectroscopic modeling, ultrastrong coupling studies, and the design of low-photon-number quantum simulations.
We describe how sideband cooling techniques may be applied to large spin ensembles in magnetic resonance. Using the Tavis-Cummings model in the presence of a Rabi drive, we solve a Markovian master equation describing the joint spin-cavity dynamics to derive cooling rates as a function of ensemble size. Our calculations indicate that the coupled angular momentum subspaces of a spin ensemble containing roughly 10(11) electron spins may be polarized in a time many orders of magnitude shorter than the typical thermal relaxation time. The described techniques should permit efficient removal of entropy for spin-based quantum information processors and fast polarization of spin samples. The proposed application of a standard technique in quantum optics to magnetic resonance also serves to reinforce the connection between the two fields, which has recently begun to be explored in further detail due to the development of hybrid designs for manufacturing noise-resilient quantum devices.
In the last couple of years much effort has been dedicated to the development and the investigation of hybrid quantum devices, which combine the advantages of very dissimilar quantum systems for the realization of efficient quantum computation and communication technologies. Particularly interesting in this regard are solid-state quantum memories based on ensembles of spins (e.g. nitrogen vacancy defects in diamond or rare-earth doped crystals) coupled to superconducting microwave cavities. Here the spin-ensemble acts as a robust memory where the collective coupling to the cavity mode allows for the coherent transfer of quantum information. The major downside of ensembles inside solid-state systems is their natural tendency to exhibit inhomogeneous broadening of the transition frequencies, which makes experiments as well as their theoretical description very challenging. On the theoretical side it is primarily the computational complexity due to the exponential growth of the associated Hilbert space with the system size that inevitably calls for approximation schemes. To date, the theoretical description of large spin-cavity systems is mainly limited to weak driving fields, where the number of spin-excitations in the ensemble is negligible, corresponding to a Holstein-Primakoff approximation, or to mean-field approaches, where correlations within the system are neglected. In this thesis we aim to develop a model that accurately accounts for the dynamics of large inhomogeneously broadened spin ensembles, coupled to a single cavity mode, even in the case of strong driving fields. In particular we employ the generalized cumulant expansion method to account for correlations within the system, which allows us to go beyond the Holstein-Primakoff and mean-field approximations. As a first step, we demonstrate the applicability of the cumulant expansion technique to the well-studied Jaynes-Cummings model and investigate its main limitations. We then move on to the Tavis-Cummings model with multiple spins inside the cavity and generalize our approach to ensembles containing a very large number of spins (ca. 10 12). Our model aims not only to provide a proper description of ensemble-based quantum memories, but also to serve as a novel tool for the investigation of interesting new physics arising from cooperative effects in inhomogeneous ensembles.
Confined electromagnetic modes strongly couple to collective excitations in ensembles of quantum emitters, producing light–matter hybrid states known as polaritons. Under such conditions, the discrete multilevel spectrum of molecular systems offers an appealing playground for exploring multiphoton processes. This work contrasts predictions from the Tavis–Cummings model in which the material is a collection of two-level systems, with the implications of considering additional energy levels with harmonic and anharmonic structures. We discuss the exact eigenspectrum, up to the second excitation manifold, of an arbitrary number N of oscillators collectively coupled to a single cavity mode in the rotating-wave approximation. Elaborating on our group-theoretic approach [New J. Phys. 23, 063081 (2021)], we simplify the brute-force diagonalization of N2 × N2 Hamiltonians to the eigendecomposition of, at most, 4 × 4 matrices for arbitrary N. We thoroughly discuss the eigenstates and the consequences of weak and strong anharmonicities. Furthermore, we find resonant conditions between bipolaritons and anharmonic transitions where two-photon absorption can be enhanced. Finally, we conclude that energy shifts in the polaritonic states induced by anharmonicities become negligible for large N. Thus, calculations with a single or few quantum emitters qualitatively fail to represent the nonlinear optical response of the collective strong coupling regime. Our work highlights the rich physics of multilevel anharmonic systems coupled to cavities absent in standard models of quantum optics. We also provide concise tabulated expressions for eigenfrequencies and transition amplitudes, which should serve as a reference for future spectroscopic studies of molecular polaritons.
We present a fast, memory-efficient, unitarity-preserving numerical method beyond the rotating-wave approximation for the closed Tavis-Cummings model in which a multilevel spin system interacts with a cavity mode. This model can describe the interaction of an ensemble of spins with a cavity mode in which the spin frequency and other parameters are time-dependent. The method exploits the fact that, while the Tavis-Cummings model is not tri-diagonal, it can be brought into tri-diagonal form by a change of basis that can be implemented purely by re-indexing (permuting basis elements), which is a fast operation. By truncating the Fock basis of the cavity mode, the computational complexity of the method is linear in the total dimension of the coupled system, both in time and memory. The method can be employed to simulate any closed quantum system whose Hamiltonian terms can be brought into tri-diagonal form.
Abstract The interaction between anharmonic quantum emitters (e.g. molecular vibrations) and confined electromagnetic fields gives rise to quantum states with optical and chemical properties that are different from those of their precursors. The exploration of these properties has been typically constrained to the first excitation manifold, the harmonic approximation, ensembles of two-level systems [Tavis–Cummings (TC) model], or the anharmonic single-molecule case. The present work studies, for the first time, a collective ensemble of identical multi-level anharmonic emitters and their dipolar interaction with a photonic cavity mode, which is an exactly solvable many-body problem. The permutational properties of the system allow identifying symmetry classified submanifolds in the energy spectrum. Notably, in this approach, the number of particles, typically in the order of several millions, becomes only a parameter from the operational standpoint, and the size of the dimension of the matrices to diagonalize is independent of it. The formalism capabilities are illustrated by showing the energy spectrum structure, up to the third excitation manifold, and the calculation of the photon contents as a permutationally invariant quantity. Emphasis is placed on (a) the collective (superradiant) scalings of light–matter couplings and the various submanifolds of dark (subradiant) states with no counterpart in the single-molecule case, as well as (b) the delocalized modes containing more than one excitation per molecule with no equivalent in the TC model. We expect these findings to be applicable in the study of non-linear spectroscopy and chemistry of polaritons.