# Registered Data

## [00915] The mathematics of quantum interaction models

**Session Date & Time**:- 00915 (1/3) : 1C (Aug.21, 13:20-15:00)
- 00915 (2/3) : 1D (Aug.21, 15:30-17:10)
- 00915 (3/3) : 1E (Aug.21, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: Research on quantum interaction models, describing the interaction of matter with light, has recently gained traction because of applications including quantum information science/technology and quantum computation. In contrast, despite the discovery of surprising relations with contemporary mathematical theory, including representation theory, geometry and number theory, the rich mathematical structure underlying these models has yet to be properly recognized. In this minisymposium we introduce the field and give an overview of recent results with a focus on the quantum Rabi model, the most fundamental model for light-matter interaction, and discuss related models in quantum optics and solid-state physics.**Organizer(s)**: Daniel Braak, Fumio Hiroshima, Cid Reyes Bustos, Masato Wakayama**Classification**:__81V80__,__81Q05__,__47A10__,__11M41__,__81R40__,__Mathematical physics__**Speakers Info**:- Daniel Braak (University of Augsburg)
- Masato Wakayama (NTT Institute for Fundamental Mathematics, Kyushu University)
- Murray Batchelor (Australian National University)
- Fumio Hiroshima (Kyushu University)
- Sahel Ashhab (NICT)
- Kazufumi Kimoto (University of the Ryukyus)
- Iñigo Egusquiza (University of the Basque Country)
- Alberto Parmeggiani (University of Bologna)
- Linh Nguyen (Kyushu University IMI)
- Marcello Malagutti (University of Bologna)
- Blas Manuel Rodríguez Lara (Tecnologico de Monterrey (ITESM))
- Yasunari Suzuki (NTT Computer and Data Science Laboratories)

**Talks in Minisymposium**:**[03218] New mathematics and machine learning applications from qubits and oscillators****Author(s)**:**Sahel Ashhab**(National Institute of Information and Communications Technology (NICT))

**Abstract**: I will present some of our studies on the physics of qubits and oscillators that produced interesting results that go into the realms of mathematics and computer science. In studying the dynamics of strongly driven qubits, we obtained a new approximation for Bessel functions. Our studies on the Landau-Zener problem, a hard problem that has a simple solution, inspired us to explore the use of symbolic regression to solve theoretical physics and mathematics problems.

**[03421] Spectrum of the noncommutative harmonic oscillator and number theory****Author(s)**:**Kazufumi Kimoto**(University of the Ryukyus)

**Abstract**: The noncommutative harmonic oscillator (NCHO) is a system of differential equations defined by a certain matrix-valued operator, and it is connected to the quantum Rabi model via a confluent process in the Heun differential equation picture. In the talk, I will present number-theoretic aspects of the spectral zeta function of the NCHO, especially those arising from its special values (i.e. values at positive integers).

**[03426] The weak limit of renormalized Rabi Hamiltonian****Author(s)**:**Fumio Hiroshima**(Kyushu Univerity)

**Abstract**: The weak limit of the renormalized Rabi Hamiltonian with a symmetry breaking term is investigated. It is shown that the spectral zeta function converges to the Riemann zeta function as the coupling constant goes to infinity. Furthermore the asymptotic behavior of the expectation of the number operator is also discussed.

**[03466] On the Weyl spectral counting function of certain semiregular global systems****Author(s)**:**Alberto Parmeggiani**(University of Bologna)

**Abstract**: In this talk I will be discussing some recent work with Marcello Malagutti about the spectral asymptotics of certain global semiregular pseudodifferential systems. The class considered here contains important models such as the Jaynes-Cummings system, which is fundamental in Quantum Optics, but also models of geometric differential complexes over $\mathbb{R}^n$. We give the asymptotics of the Weyl spectral counting functions in terms of the principal, semiprincipal and subprincipal symbols of the system, along with (time permitting) quasi-clustering properties of the spectrum.

**[03491] Quantum computation and its viewpoint from spectral zeta functions****Author(s)**:**MASATO WAKAYAMA**(NTT Institute for Fundamental Mathematics)

**Abstract**: We discuss the spectrum of the quantum interaction models such as the quantum Rabi models, non-commutative harmonic oscillators and their important derived models from the viewpoints of quantum computation and number theory via the corresponding heat kernels, partition functions and spectral zeta functions.

**[03683] Design and optimization of fault-tolerant quantum computing****Author(s)**:**Yasunari Suzuki**(Nippon Telegraph and Telephone)

**Abstract**: To demonstrate scalable quantum computing, we need to suppress the high error rates of quantum devices. While they can be reduced with quantum error correction (QEC) technology, it requires large overheads on computing resources. Thus, optimization methods and co-design of hardware and software for fault-tolerant quantum computing are demanded. In this talk, I will explain the recent progress relevant to computer architecture and compiler optimization technologies based on the QEC framework.

**[03826] The spectral problem in Hilbert spaces of analytic functions****Author(s)**:**Daniel Braak**(University of Augsburg)

**Abstract**: In the standard Hilbert space, the spectral problem of Hamilton operators with one degree of freedom takes the form of a lateral connection problem for functions with diverging power series expansions. According to common lore, the solution would require to construct these functions on the whole real line which is usually impossible. It will be demonstrated that this brute-force approach can be avoided by employing Hilbert spaces of analytic functions.

**[03916] On The Spectral Zeta Function Of Second Order Semiregular Non-Commutative Harmonic Oscillators****Author(s)**:**Marcello Malagutti**(University of Bologna)

**Abstract**: In this talk we give a meromorphic continuation of the spectral zeta function for semiregular Non-Commutative Harmonic Oscillators (NCHO). By “semiregular system” we mean a pseudodifferential systems with a step $−j$ in the homogeneity of the $j$th-term in the asymptotic expansion of the symbol. As an application of our results, we first compute the meromorphic continuation of the Jaynes-Cummings (JC) model spectral zeta function. Then we compute the spectral zeta function of the JC generalization to a 3-level atom in a cavity. For both of them we show that it has only one pole in 1.

**[03971] PT-Symmetric Quantum Rabi Model****Author(s)**:**Murray Batchelor**(Australian National University)

**Abstract**: We explore a PT-symmetric qubit coupled to a quantized light field. The model is solved analytically using the adiabatic approximation (AA) in the parameter regime of interest. We investigate the static and dynamic properties, using both the AA and numerical diagonalization. A series of exceptional points vanish and revive depending on the light-matter coupling strength. This talk is based on arXiv:2212.06586 with X. Lu, H. Li, J.-K. Shi, L.-B. Fan, V. Mangazeev and Z.-M. Li.

**[04189] Energy spacing and time evolution for asymmetric quantum Rabi models****Author(s)**:**Linh Thi Hoai Nguyen**(Institute of Mathematics for Industry, Kyushu University)- Cid Reyes Bustos (NTT IFM)
- Masato Wakayama (NTT Institute for Fundamental Mathematics)

**Abstract**: In this study, we describe the methods for numerical computations of the energy spacing distribution and time evolution for the asymmetric (or biased) quantum Rabi model (AQRM). The first several tens of thousands of eigenvalues are achieved by using the Truncated Hamiltonians method. From that, we observe the periodicity and symmetry of the consecutive energy spacing distribution with respect to the bias parameter. The time evolution is studied based on an explicit heat kernel formula.