Gauge freedom, quantum measurements, and time-dependent interactions in cavity QED

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Gauge freedom, quantum measurements, and time-dependent interactions in cavity QED

Alessio Settineri, Omar Di Stefano, David Zueco, Stephen Hughes, Salvatore Savasta, and Franco Nori

Phys. Rev. Research 3, 023079 – Published 28 April 2021

Abstract

The interaction between quantized electromagnetic fields in cavities and natural or artificial atoms has played a crucial role in developing our understanding of light-matter interactions and quantum technologies. Recently, new regimes beyond the weak and strong light-matter coupling of cavity-QED have been explored in several settings, wherein the light-matter coupling rate becomes comparable to (ultrastrong coupling) or even exceeds (deep-strong coupling) the photon frequency. These ultrastrong coupling regimes can give rise to new physical effects and applications, and they challenge our understanding of cavity QED; fundamental issues like the proper definition of subsystems, their quantum measurements, the structure of light-matter ground states, and the analysis of time-dependent interactions are subject to gauge ambiguities that lead to even qualitatively distinct predictions. The resolution of these ambiguities is important for understanding and designing next-generation quantum devices that can operate in extreme coupling regimes. Here we discuss and provide solutions to these ambiguities by adopting an approach based on operational procedures involving measurements on the individual light and matter components of the interacting system.

Received 27 December 2019
Revised 23 December 2020
Accepted 5 April 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.023079

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Josephson effect Optical transient phenomena Polaritons Quantum optics

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Alessio Settineri¹, Omar Di Stefano^1,2, David Zueco ³, Stephen Hughes ⁴, Salvatore Savasta^1,*, and Franco Nori ^2,5

¹Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Università di Messina, I-98166 Messina, Italy
²Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
³Instituto de Ciencia de Materiales y Nanociencia de Aragón and Departamento de Física de la Materia Condensada, CSIC-Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain
⁴Department of Physics, Engineering Physics, and Astronomy, Queen's University, Kingston, Ontario, Canada K7L 3N6
⁵Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA

^*Corresponding author: ssavasta@unime.it

Article Text

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Vol. 3, Iss. 2 — April - June 2021

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Images

Figure 1
Cavity QED setup. Schematic view of a typical cavity QED system consisting of an atom (depicted as an effective spin) embedded in an optical cavity. The symbols $γ$ and $κ$ represent the atom and cavity decay rates, respectively.
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Figure 2
Quantum Rabi model. (a) Normalized energy levels differences between the lowest excited levels and the ground energy level of the quantum Rabi Hamiltonian ${\hat{H}}_{C}$ for the case of zero detuning ( $ω_{c} = ω_{0}$ ) as a function of the normalized coupling strength $η$ ; (b) modulus squared of the transition matrix elements of the electric-field operator, $W_{\tilde{1} \pm, \tilde{0}}$ , accounting for the transitions between the two lowest excited levels and the ground state of the quantum Rabi Hamiltonian, vs $η$ . For comparison, the panel also reports the wrong matrix elements $W_{\tilde{1} \pm, \tilde{0}}^{'}$ (see text).
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Figure 3
Schematic view of dispersive readout of a two-level atom. The resonance frequency of the readout resonator is conditioned by the qubit state. Thus, by probing the resonance frequency of the resonator, e.g., by looking at the reflectivity of a readout tone, the qubit state can be determined. (a) Standard readout scheme of a bare two-level atom. (b) the same scheme for a two-level atom interacting with an additional resonator in the USC regime.
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Figure 4
Readout of a strongly coupled qubit. Qubit excitation probabilities for the system in its ground state (black curves) and in the first excited state (red curves), calculated in both the Coulomb (solid curves) and the dipole (dotted-dashed) gauges as a function of the normalized coupling strength $η$ . Note that ${〈 {\hat{σ}}_{+} {\hat{σ}}_{-} 〉}_{D}$ corresponds to what is measured via dispersive readout of the qubit (see Sec. 4). On the contrary, the photon rate released by the qubit after a sudden switch off of the light-matter interaction is proportional to ${〈 {\hat{σ}}_{+} {\hat{σ}}_{-} 〉}_{C}$ (see Sec. 5).
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Figure 5
Vacuum emission. (a) Mean photon number calculated in the Coulomb (solid curve) and in the wrong dipole (dot-dashed curve) gauges as a function of $η$ for the system prepared in the ground state of the quantum Rabi model. (inset) Vacuum emission (mean photon number) after the switch off evaluated for $η = 0.8$ . (b) Qubit entropies (which quantifies the qubit-oscillator entanglement) for the ground states (black curves) calculated in both the Coulomb (solid curves) and the wrong dipole (dotted-dashed) gauges as a function of the normalized coupling strength $η$ .
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Figure 6
Gauge-invariant emission of a TLS coupled to a single-mode resonator (quantum Rabi Hamiltonian) induced by sudden switching on and off the light-matter interaction, calculated for three normalized coupling strengths. (a) Displays the switching function $λ (t)$ . The system is initially prepared in its ground state: $| ψ_{C} (t_{in}) 〉 = | g, 0 〉$ . At $t = t_{1}$ , the interaction is suddenly switched on, and it is finally switched off at $t = t_{2}$ . (b)–(d) display $〈 ψ_{C} (t) | {\hat{Y}}^{(-)} {\hat{Y}}^{(+)} | ψ_{C} (t) 〉$ for three different coupling strengths.
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