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arXiv:1807.08674 · Scientific Reports

A continued fraction based approach for the Two-photon Quantum Rabi Model

Elena Lupo, Anna Napoli, Antonino Messina, Enrique Solano, I. L. Egusquiza

Published Mar 11, 2019• 18 citations
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Abstract

We study the Two Photon Quantum Rabi Model by way of its spectral functions and survival probabilities. This approach allows numerical precision with large truncation numbers, and thus exploration of the spectral collapse. We provide independent checks and calibration of the numerical results by studying an exactly solvable case and comparing the essential qualitative structure of the spectral functions. We stress that the large time limit of the survival probability provides us with an indicator of spectral collapse, and propose a technique for the detection of this signal in the current and upcoming quantum simulations of the model.

DOI
10.1038/s41598-019-39281-4

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We study the Two Photon Quantum Rabi Model by way of its spectral functions and survival probabilities. This approach allows numerical precision with large truncation numbers, and thus exploration of the spectral collapse. We provide independent checks and calibration of the numerical results by studying an exactly solvable case and comparing the essential qualitative structure of the spectral functions. We stress that the large time limit of the survival probability provides us with an indicator of spectral collapse, and propose a technique for the detection of this signal in the current and upcoming quantum simulations of the model.
We study the Two Photon Quantum Rabi Model by way of its spectral functions and survival probabilities. This approach allows numerical precision with large truncation numbers, and thus exploration of the spectral collapse. We provide independent checks and calibration of the numerical results by studying an exactly solvable case and comparing the essential qualitative structure of the spectral functions. We stress that the large time limit of the survival probability provides us with an indicator of spectral collapse, and propose a technique for the detection of this signal in the current and upcoming quantum simulations of the model.
Techniques based on continued fractions to numerically compute the spectrum of the quantum Rabi model are reviewed. They are of two essentially different types. In the first case, the spectral condition is implemented using a representation in the infinite-dimensional Bargmann space of analytic functions. This approach is shown to approximate the correct spectrum of the full model if the continued fraction is truncated at sufficiently high order. In the second case, one considers the limit of a sequence of models defined in finite-dimensional state spaces. In contrast to the first, the second approach is ambiguous and can be justified only through recourse to the analyticity argument from the first method.
Abstract The two-photon quantum Rabi model is analyzed using the Z 4 -symmetry and su ( 1 , 1 ) -algebra. We derive the G -function whose zeros give the exact eigenvalues of the Hamiltonian. The singularity structure of the G -function allows to draw conclusions about the distribution of these eigenvalues along the real axis and we derive the spectral collapse phenomenon at critical coupling g c , found numerically before. The spectrum at g c consists of a discrete and a continuous part: the ground state is always separated from the continuum by a finite excitation gap, ruling out a quantum phase transition in the usual sense. For large qubit splitting, also other low lying states split off from the continuum. However, perturbation theory predicts the vanishing of the gap to all orders, demonstrating its non-perturbative nature. We corroborate this result with a variational calculation for the ground state.
We have investigated the eigenenergy spectrum of the two-photon Rabi model with a full quadratic coupling, particularly the special feature "spectral collapse". The critical coupling strength is reduced by half from that of the two-photon Rabi model, implying that the spectral collapse can now occur at a more attainable value of the critical coupling. At the critical coupling some discrete eigenenergy levels still survive below the continuous energy spectrum, i.e. an incomplete spectral collapse, and the set of discrete eigenenergies has a one-to-one mapping with that of a particle of variable effective mass in a finite potential well. Since the energy difference between the two atomic levels specifies the depth of the potential well, the number of bound states available (or the extent of the "spectral collapse") can be straightforwardly monitored. Obviously, this bears a great resemblance to the spectral collapse of the two-photon Rabi model, at least qualitatively. Moreover, since the full quadratic coupling includes an extra term proportional to the photon number operator only, our analysis indicates that one may manipulate the critical coupling of the two-photon Rabi model by incorporating an adjustable proportionality constant to this extra term.
Spectral collapse, the transition from a discrete to a continuous spectrum, is a characteristic in quantum Rabi models. We explore this phenomenon in the two-photon quantum Rabi model using optical phase space, and we find that, in the so-called degenerate qubit regime, the collapse is similar to the transition from a harmonic to an inverted oscillator with the free-particle potential as a critical transition point. In the degenerate qubit regime, we construct Dirac-normalizable eigenfunctions with well-defined parity for the model. In the general model, we use parity to diagonalize the system in the qubit basis and numerically find that the qubit frequency does not change the critical point where spectral collapse occurs. We numerically confirm the existence of an exceptional state at the critical coupling, and we argue its analytic provenance from both a Born-Oppenheimer and a variational approximation.
Abstract Applications of quantum systems with non‐linear light–matter interaction have to deal with a peculiar instability in the ultra‐ and deep strong coupling regime, the so‐called “spectral collapse.” To solve this problem, the present work investigates a generalized quantum Rabi model (QRM) with two‐ and four‐photon terms in view of applications for critical quantum metrology. In the introduced model, the spectral collapse occurring in the standard two‐photon QRM is stabilized by the presence of the quartic potential. The collapse is then transformed into a quantum phase transition, which occurs in the low‐frequency limit of the light mode, whose remnant at finite ratio between qubit and mode frequencies can be applied to critically enhanced quantum metrology. It is found that the four‐photon term entails a much higher measurement precision compared to the standard two‐photon QRM. The mechanism behind the higher precision can be traced to the different behavior of the ground state wave function as the system is tuned through the transition. As the standard two‐photon QRM, despite the absence of the spectral collapse, the proposed model allows for a finite preparation time for the probe state (PTPS).