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arXiv:2112.12514

Duality symmetry conjugates of the quantum Rabi model : effective bosonic, fermionic and coupling-only dynamical properties

Joseph Akeyo Omolo

Published Dec 23, 2021• 0 citations
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Abstract

Symmetry transformations have proved useful in determining the algebraic structure and internal dynamical properties of physical systems. In the quantum Rabi model, invariance under parity symmetry transformation has been used to obtain exact solutions of the eigenvalue equation and very good approximations of the internal dynamics of the interacting atom-light system. In this article, two symmetry operators, characterized as "duality" symmetry operators, have been introduced which transform the quantum Rabi Hamiltonian into duality conjugates. The parity and duality symmetry operators constitute an algebraically closed set of symmetry transformation operators of the quantum Rabi model. The closed $SU(2)$ Lie algebra provides the standard eigenvalues and eigenstates of the parity symmetry operator. It is established that Jaynes-Cummings and anti-Jaynes-Cummings operators are duality symmetry conjugates. Symmetric or antisymmetric linear combinations of the Rabi Hamiltonian and a corresponding duality conjugate yield the familiar spin-dependent force driven bosonic , coupling-only or quantized light mode quadrature-driven fermionic Hamiltonian. It is established that the effective bosonic, fermionic and coupling-only Hamiltonians are exact, not approximate forms of the quantum Rabi Hamiltonian as they have generally been interpreted. The effective bosonic form generates the dynamics of the light mode driven by the atomic spin-dependent force, while the fermionic form generates the dynamics of the atomic spin driven by the quantized light mode quadrature-dependent force, thus providing a complete picture of the quantum Rabi dynamics.

DOI
10.48550/arxiv.2112.12514

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Symmetry transformations have proved useful in determining the algebraic structure and internal dynamical properties of physical systems. In the quantum Rabi model, invariance under parity symmetry transformation has been used to obtain exact solutions of the eigenvalue equation and very good approximations of the internal dynamics of the interacting atom-light system. In this article, two symmetry operators, characterized as "duality" symmetry operators, have been introduced which transform the quantum Rabi Hamiltonian into duality conjugates. The parity and duality symmetry operators constitute an algebraically closed set of symmetry transformation operators of the quantum Rabi model. The closed SU(2)SU(2)SU(2) Lie algebra provides the standard eigenvalues and eigenstates of the parity symmetry operator. It is established that Jaynes-Cummings and anti-Jaynes-Cummings operators are duality symmetry conjugates. Symmetric or antisymmetric linear combinations of the Rabi Hamiltonian and a corresponding duality conjugate yield the familiar spin-dependent force driven bosonic , coupling-only or quantized light mode quadrature-driven fermionic Hamiltonian. It is established that the effective bosonic, fermionic and coupling-only Hamiltonians are exact, not approximate forms of the quantum Rabi Hamiltonian as they have generally been interpreted. The effective bosonic form generates the dynamics of the light mode driven by the atomic spin-dependent force, while the fermionic form generates the dynamics of the atomic spin driven by the quantized light mode quadrature-dependent force, thus providing a complete picture of the quantum Rabi dynamics.
The quantum Rabi model is the simplest and most important theoretical description of light–matter interaction for all experimentally accessible coupling regimes. It can be solved exactly and is even integrable due to a discrete symmetry, the Z 2 or parity symmetry. All qualitative properties of its spectrum, especially the differences to the Jaynes–Cummings model, which possesses a larger, continuous symmetry, can be understood in terms of the so-called “G-functions” whose zeroes yield the exact eigenvalues of the Rabi Hamiltonian. The special type of integrability appearing in systems with discrete degrees of freedom is responsible for the absence of Poissonian level statistics in the spectrum while its well-known “Juddian” solutions are a natural consequence of the structure of the G-functions. The poles of these functions are known in closed form, which allows drawing conclusions about the global spectrum.
We use the standard Jaynes-Cummings model with the field initially in the vacuum state and the atom in an arbitrary superposition state to illustrate the concept of dynamic symmetry in quantum optics. By dynamic symmetry we mean a situation in which a Hamiltonian is expressible as a linear combination of operators obeying a Lie algebra. This circumstance permits a psuedospin (or coherence vector) treatment to be given of the dynamics of the system. We give two such interpretations of the Jaynes-Cummings model. In the first we regard the atom and the field as a pair of interacting two-level systems and formulate a SU(2)-SU(2) dynamic symmetry. In the second we interpret the composite atom-field system is a three-level system and formulate a SU(3) dynamic symmetry. The former model illustrates the ideas of Fano1 and the latter those of Hioe and Eberly.2 We give a complete pseudospin description from both points of view and also obtain all the constants of motion.
The scientific interest in the analytical solution of the quantum Rabi model is due to the widespread use of this simple model in quantum optics, quantum computing, cavity QED, and nanoelectromechanical systems. This interest is related to the need for the theoretical description of the interaction of a two-level system with a quantum oscillator in the case when the rotating wave approximation fails. In this Letter, we present an approach to the exact diagonalization of the quantum Rabi Hamiltonian. This approach is based on the properties of the Pauli operators and allows us to readily solve the stationary Schrodinger equation for a two-level system. First, we demonstrate the applicability of the approach to the Jaynes-Cummings Hamiltonian to get the well-known solution. Then, we obtain the eigenvalues and eigenstates for the quantum Rabi Hamiltonian using the proposed approach. It is shown that the obtained eigenstates can be represented in the basis of the eigenstates of the Jaynes-Cummings Hamiltonian.
The single-mode spin-boson model exhibits behavior not included in the rotating-wave approximation (RWA) in the ultra and deep-strong coupling regimes, where counter-rotating contributions become important. We introduce a symmetric rotating-wave approximation that treats rotating and counter-rotating terms equally, preserves the invariances of the Hamiltonian with respect to its parameters, and reproduces several qualitative features of the spin-boson spectrum not present in the original rotating-wave approximation both off-resonance and at deep-strong coupling. The symmetric rotating-wave approximation allows for the treatment of certain ultra- and deep-strong coupling regimes with similar accuracy and mathematical simplicity as does the RWA in the weak-coupling regime. Additionally, we symmetrize the generalized form of the rotating-wave approximation to obtain the same qualitative correspondence with the addition of improved quantitative agreement with the exact numerical results. The method is readily extended to higher accuracy if needed. Finally, we introduce the two-photon parity operator for the two-photon Rabi Hamiltonian and obtain its generalized symmetric rotating-wave approximation. The existence of this operator reveals a parity symmetry similar to that in the Rabi Hamiltonian as well as another symmetry that is unique to the two-photon case, providing insight into the mathematical structure of the two-photon spectrum, significantly simplifying the numerics, and revealing some interesting dynamical properties.
The quantum Rabi model is a standard effective Hamiltonian in studies of light-matter interaction, capturing the simplest nontrivial setting in which a qubit couples to a single harmonic oscillator. Within the broader Rabi family, we focus on two special cases: the Jaynes-Cummings (JC) model, which carries an explicit U(1)U(1)U(1) symmetry, and the asymmetric quantum Rabi model (AQRM), which possesses a parameter-dependent "hidden'' symmetry that appears only at integer bias, ε/ω∈Z\varepsilon/ω\in\mathbb{Z}ε/ω∈Z, and is not manifest in the Hamiltonian. We use the time-averaged entangling power as an operator-level diagnostic of these symmetry structures. Since the oscillator Hilbert space is infinite-dimensional, we compare two finite input ensembles: a Haar average after Fock-space truncation and a coherent-state average at fixed mean occupation nˉ\bar{n}nˉ. Both diagnostics show peaks at the integer-bias points of the AQRM, where the hidden symmetry resides. In contrast, the manifest U(1)U(1)U(1) symmetry at the JC point instead gives a weak dip. Thus, the time-averaged entangling power responds to both the hidden symmetry and the U(1)U(1)U(1) symmetry in the Rabi family, with the sign of the response indicating how the symmetry reorganizes the spectral expansion. These results demonstrate that the entangling power can serve as an operator diagnostic to reveal the presence and properties of hidden and manifest symmetries in light-matter systems.