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Conical Intersections and Geometric Phases in the Asymmetric Quantum Rabi Model

Zi-Min Li, Devid Ferri, David L. Tilbrook, Murray T. Batchelor

Published Jul 23, 2020• 3 citations
OpenAlex

Abstract

Conical intersections (CIs) between energy surfaces arise in various physical contexts. A rich example, involving the interaction of a qubit with a bosonic field, is the energy landscape of the asymmetric quantum Rabi model (AQRM). Conventional approximations for this model, including the slow-qubit approximation, fail to capture the correct location of the CIs. To go beyond this limitation, we introduce a relatively simple solvable light-matter interaction model, which by construction, shares the same CIs as the AQRM. As a demonstration of the use of this simplified model we calculate the geometric phases around CIs in the energy landscape. We find that the geometric phases acquired by closed loops in parameter space around CIs are multiples of $\pi$, depending on the CIs encircled by the trajectories considered. These results are seen to be in agreement with numerical results for the corresponding geometric phases of the AQRM.

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Conical intersections (CIs) between energy surfaces arise in various physical contexts. A rich example, involving the interaction of a qubit with a bosonic field, is the energy landscape of the asymmetric quantum Rabi model (AQRM). Conventional approximations for this model, including the slow-qubit approximation, fail to capture the correct location of the CIs. To go beyond this limitation, we introduce a relatively simple solvable light-matter interaction model, which by construction, shares the same CIs as the AQRM. As a demonstration of the use of this simplified model we calculate the geometric phases around CIs in the energy landscape. We find that the geometric phases acquired by closed loops in parameter space around CIs are multiples of π\piπ, depending on the CIs encircled by the trajectories considered. These results are seen to be in agreement with numerical results for the corresponding geometric phases of the AQRM.
The quantum Rabi model (QRM), describing a two-level system coupled to a quantum harmonic oscillator, is one of the simplest and most ubiquitous interaction models in quantum mechanics. Despite its simplicity, the QRM has found applications in various fields of physics. The most exciting realization of the QRM nowadays is arguably the circuit quantum electrodynamics (QED) system, where an on-chip artificial atom (or a qubit) is coupled to a transmission line or a waveguide. The qubits in circuit QED systems contain a bias term, which breaks the Z_2 symmetry in the QRM. Therefore, the precise model realized in circuit QED systems is referred to as the asymmetric quantum Rabi model (AQRM). This thesis explores theoretical methods to treat the AQRM and develops a complete approach to describe its energy landscape. The hidden symmetry of the AQRM is investigated in detail, and seen to be a rather general phenomenon in light-matter interaction models. The equivalence between the generalised Poschl-Teller potential and the spectrum of the AQRM is also established. Chapter 1 is dedicated to reviewing the analytic solutions and conventional approximations to the QRM/AQRM. Regarding the analytic solutions, the analytic results in terms of the transcendental functions determining the eigenspectrum of the AQRM are collected. Special emphasis is placed on the isolated closed-form exceptional solutions to the AQRM. Due to the lack of simple closed-form expressions for the general eigenvalues, some well-known approximations are considered. In Chapter 2, a generalised adiabatic approximation (GAA) is developed to calculate the eigenvalues of the excited states of the QRM and apply it to the AQRM. The GAA is based on the similarity between the exact exceptional solutions and the Laguerre polynomials. By construction, the GAA always has exact exceptional energy and approximates the regular spectrum with rather good agreement in a large parameter regime. Furthermore, the GAA is applied to the AQRM and shown to correctly recover the conical intersections in the energy landscape. The geometric phases around these conical intersections are calculated analytically as an illustrative example. In Chapter 3, a physically motivated variational wave function is presented for the ground state of the AQRM. The wave function is a weighted superposition of squeezed coherent states entangled with non-orthogonal qubit states and relies only on three variational parameters in the regimes of interest. The variational expansion describes the ground state remarkably well in almost all parameter regimes, especially with arbitrary bias. The results show that the variational expansion is a significant improvement over the existing approximations for the AQRM. In Chapter 4, the hidden symmetry and tunnelling dynamics in the AQRM and several related models are analysed in detail. The AQRM has a broken Z_2 symmetry, with generally a non-degenerate eigenvalue spectrum. In some special cases of the related models, stable level crossings typical of the Z_2-symmetric quantum Rabi model are recovered. It is thus shown that this hidden symmetry is not limited to the AQRM, but exists in various related light-matter interaction models. In Chapter 5, starting with the Gaudin-like Bethe ansatz equations associated with the quasi-exactly solved (QES) exceptional points of the AQRM, a spectral equivalence is established with QES hyperbolic Schrodinger potentials. This leads to particular QES Poschl-Teller potentials. The complete spectral equivalence is then established between the AQRM and generalised Poschl-Teller potentials. Conclusions and future directions are presented in the final Chapter 6.
We explore the simulation of conical intersections (CIs) on quantum devices, setting the groundwork for potential applications in nonadiabatic quantum dynamics within molecular systems. The intersecting potential energy surfaces of H3+_{3}^{+}3+​ are computed from a variance-based contracted quantum eigensolver. We show how the CIs can be correctly described on quantum devices using wavefunctions generated by the anti-Hermitian contracted Schr{ö}dinger equation ansatz, which is a unitary transformation of wavefunctions that preserves the topography of CIs. A hybrid quantum-classical procedure is used to locate the seam of CIs. Additionally, we discuss the quantum implementation of the adiabatic to diabatic transformation and its relation to the geometric phase effect. Results on noisy intermediate-scale quantum devices showcase the potential of quantum computers in dealing with problems in nonadiabatic chemistry.
Abstract We examine the energy surfaces of the driven Rabi model, also known as the biased or generalized Rabi model, as a function of the coupling strength and the driving term. The energy surfaces are plotted numerically from the known analytic solution. The resulting energy landscape consists of an infinite stack of sheets connected by conical intersection points located at the degenerate Juddian points in the eigenspectrum. Trajectories encircling these points are expected to exhibit a nonzero geometric phase.
are computed from a variance-based contracted quantum eigensolver. We show how the CIs can be correctly described on quantum devices using wavefunctions generated by the anti-Hermitian contracted Schrödinger equation ansatz, which is a unitary transformation of wavefunctions that preserves the topography of CIs. A hybrid quantum-classical procedure is used to locate the seam of CIs. Additionally, we discuss the quantum implementation of the adiabatic to diabatic transformation and its relation to the geometric phase effect. Results on noisy intermediate-scale quantum devices showcase the potential of quantum computers in dealing with problems in nonadiabatic chemistry.
Conical intersections are common in molecular physics and photochemistry, and are often invoked to explain observed reaction products. A conical intersection can occur when an excited electronic potential energy surface intersects with the ground electronic potential energy surface in the coordinate space of the nuclear positions. Theory predicts that the conical intersection will result in a geometric phase for a wavepacket on the ground potential energy surface. Although conical intersections have been observed experimentally, the geometric phase has not been observed in a molecular system. Here we use a trapped atomic ion system to perform a quantum simulation of a conical intersection. The internal state of a trapped atomic ion serves as the electronic state and the motion of the atomic nuclei are encoded into the normal modes of motion of the ions. The simulated electronic potential is constructed by applying state-dependent forces to the ion with a near-resonant laser. We experimentally observe the geometric phase on the ground-state surface using adiabatic state preparation followed by motional state measurement. Our experiment shows the advantage of combining spin and motion degrees of freedom in a quantum simulator.