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Scientific Reports

Deciphering the spectral collapse in two-photon Rabi model

Chi‐Fai Lo

Published May 20, 2021• 3 citations
Publisher

Abstract

In this communication, based upon a squeezed-state trial wave function, we have performed a simple variational study of the spectral collapse of the two-photon Rabi model. Our analysis indicates that the light-matter interaction and the spin-flipping effectively constitute two competing impacts upon the radiation mode. Whilst the former tries to decrease the radiation mode frequency, the latter may counteract or reinforce it, contingent upon the state of the atomic system. The light-matter interaction appears to dominate the frequency modulation as its coupling strength goes beyond the critical value, leading to the emergence of the spectral collapse. However, at the critical coupling the dominance of the light-matter interaction is not complete, and incomplete spectral collapse appears. The extent of incomplete spectral collapse is found to depend upon the energy difference between the two atomic levels as well.

DOI
10.1038/s41598-021-90085-x

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In this communication, based upon a squeezed-state trial wave function, we have performed a simple variational study of the spectral collapse of the two-photon Rabi model. Our analysis indicates that the light-matter interaction and the spin-flipping effectively constitute two competing impacts upon the radiation mode. Whilst the former tries to decrease the radiation mode frequency, the latter may counteract or reinforce it, contingent upon the state of the atomic system. The light-matter interaction appears to dominate the frequency modulation as its coupling strength goes beyond the critical value, leading to the emergence of the spectral collapse. However, at the critical coupling the dominance of the light-matter interaction is not complete, and incomplete spectral collapse appears. The extent of incomplete spectral collapse is found to depend upon the energy difference between the two atomic levels as well.
In this communication, based upon a squeezed-state trial wave function, we have performed a simple variational study of the spectral collapse in the anisotropic two-photon Rabi model. Our analysis indicates that the light-matter interaction and the spin-flipping (together with the anisotropy) effectively constitute two competing impacts upon the radiation mode. Whilst the former tries to decrease the radiation mode frequency, the latter may counteract or reinforce it. The light-matter interaction appears to dominate the frequency modulation as its coupling strengths go beyond the critical values, leading to the emergence of the spectral collapse. However, at the critical couplings the dominance of the light-matter interaction is not complete, and incomplete spectral collapse appears. Accordingly, at the critical couplings the eigenenergy spectrum comprises both a set of discrete energy levels and a continuous energy spectrum. The discrete eigenenergy spectrum can be derived via a simple one-to-one mapping to the bound state problem of a particle of variable effective mass in a finite potential well, and the number of bound states available is determined by the energy difference between the two atomic levels. Each of these eigenenergies has a twofold degeneracy corresponding to the spin degree of freedom.
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.
We have shown that the smallest possible singel-qubit critical coupling strength of the N-qubit two-photon Rabi model is only 1/N times that of the two-photon Rabi model. The spectral collapse can thus occur at a more attainable value of the critical coupling. For both of the two-qubit and three-qubit cases, we have also rigorously demonstrated that at the critical coupling the system not only has a set of discrete eigenenergies but also a continuous energy spectrum. The discrete eigenenergy spectrum can be derived via a simple one-to-one mapping to the bound state problem of a particle of variable effective mass in the presence of a finite potential well and a nonlocal potential. The energy difference of each qubit, which specifies both the depth of the finite potential well and the strength of the nonlocal potential, determines the number of bound states available, implying that the extent of the incomplete spectral collapse can be monitored in a straightforward manner.
We have investigated the eigenenergy spectrum of the two-photon Rabi model at the critical coupling, particularly the special feature "spectral collapse", by means of an elementary quantum mechanics approach. The eigenenergy spectrum is found to consist of both a set of discrete energy levels and a continuous energy spectrum. Each of these eigenenergies has a two-fold degeneracy corresponding to the spin degree of freedom. The discrete eigenenergy spectrum has a one-to-one mapping with that of a particle in a "Lorentzian function" potential well, and the continuous energy spectrum can be derived from the scattering problem associated with a potential barrier. The number of bound states available at the critical coupling is determined by the energy difference between the two atomic levels so that the extent of the "spectral collapse" can be monitored in a straightforward manner.
Abstract This paper presents a proof of the existence of novel bound states of the two-photon quantum Rabi model at the collapse point. The two-photon Rabi model is interesting not only for its important role on non-linear light–matter interaction, but also for the exhibition of the many-energy-levels degenerating process called the ‘spectral collapse’. The squeezing property of the two-photon annihilation and creation operators is the origin for this phenomenon which is well studied without the energy-slitting term ω 0 . However, many numerical studies have pointed out that with the presence of ω 0 , some low-level isolated states exist while other high energy states collapse to E = − ω 2 , which is known as incomplete spectral collapse. From the eigenvalue equation in real space, a pair of second order differential equations, which are similarly to the Schrödinger equation, are derived at the collapse point. These differential equations provide an explanation to the existence of isolated bound states below E = − ω 2 with the presence of the spin slitting ω 0 and a better numerical method to generate those bound states.
Spectral collapse in the two-photon quantum Rabi model (tpQRM) has long been regarded as insufficient to establish quantum criticality because, under generic conditions, the lowest excitation gap remains finite. We show that, at a special qubit frequency, spectral collapse in the anisotropic tpQRM constitutes a continuous quantum phase transition governed by a single soft mode. The same-parity gap, set by the vanishing effective oscillator frequency, closes with exponent zν=1/2zν= 1/2zν=1/2, whereas the lower different-parity gap arises from symmetry-induced splitting. Parity symmetry excludes the different-parity excitation from the quantum Fisher information response and restricts Kibble-Zurek dynamics to excitations within the ground-state parity sector. Together, these results identify the same-parity excitation as the soft mode and its gap as the characteristic energy scale. The corresponding critical exponents, ν=1/4ν=1/4ν=1/4 and z=2z=2z=2, place the anisotropic tpQRM in the same universality class as the standard QRM. Our results establish spectral collapse as a potential route to experimentally accessible quantum criticality in a few-body system and show that universality is determined by the soft-mode structure rather than by the closing of the lowest excitation gap.