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arXiv:1712.04758 · Journal of Experimental and Theoretical Physics Letters

Emission Spectrum of a Qubit under Its Deep Strong Driving in the High-Frequency Dispersive Regime

A. P. Saǐko, S. A. Markevich, Ryhor Fedaruk

Published Jan 1, 2018• 1 citation
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Abstract

We study the emission spectrum of a qubit under deep strong driving in the high-frequency dispersive regime when the driving frequency and strength exceed significantly the qubit transition frequency. Closed-form expressions for the steady-state first-order field correlation function and the multiphoton emission spectrum are obtained. The spectrum comprises a series of narrow delta-like lines that stem from coherent processes and Lorentzian peaks that result from the incoherent scattering of photons. The oscillating dependence of the widths of the emission lines on the driving strength is predicted. We show how the features of this dependence are governed by the qubit dephasing and relaxation rates.

DOI
10.1134/s0021364018020030

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We study the emission spectrum of a qubit under deep strong driving in the high-frequency dispersive regime when the driving frequency and strength exceed significantly the qubit transition frequency. Closed-form expressions for the steady-state first-order field correlation function and the multiphoton emission spectrum are obtained. The spectrum comprises a series of narrow delta-like lines that stem from coherent processes and Lorentzian peaks that result from the incoherent scattering of photons. The oscillating dependence of the widths of the emission lines on the driving strength is predicted. We show how the features of this dependence are governed by the qubit dephasing and relaxation rates.
We study the dissipative Rabi model under deep strong far-off-resonant driving when the driving frequency and strength exceed significantly the qubit transition frequency. We find analytical expressions for the density matrix, the steady-state first-order field correlation function and the multiphoton emission spectrum. This spectrum consists of a series of narrow delta-like lines, caused by coherent processes and of Lorenzian peaks resulted from the incoherent scattering of photons. It is shown that the width of emission lines as well as the decay rate of the time oscillations in the population difference have oscillating dependence on the driving strength. We show how the features of this dependence are controlled by the relative rates of energetic relaxation and pure dephasing.
A new regime of coherent quantum dynamics of a qubit is realized at low driving frequencies in the strong driving limit. Coherent transitions between qubit states occur via the Landau-Zener process when the system is swept through an energy-level avoided crossing. The quantum interference mediated by repeated transitions gives rise to an oscillatory dependence of the qubit population on the driving-field amplitude and flux detuning. These interference fringes, which at high frequencies consist of individual multiphoton resonances, persist even for driving frequencies smaller than the decoherence rate, where individual resonances are no longer distinguishable. A theoretical model that incorporates dephasing agrees well with the observations.
We study two-level systems (2LS) coupled at different points to a one-dimensional waveguide in which one end is open and the other is either open (infinite waveguide) or closed by a mirror (semi-infinite). Upon injection of two photons (corresponding to weak coherent driving), the resonance fluorescence and photon correlations are shaped by the effective qubit transition frequencies and decay rates, which are substantially modified by interference effects. In contrast to the well-known result in an infinite waveguide, photons reflected by a single 2LS coupled to a semi-infinite waveguide are initially bunched, a result that can be simply explained by stimulated emission. As the number of 2LS increases (up to 10 are considered here), rapid oscillations build up in the correlations that persist for a very long time. For instance, when the incoming photons are slightly detuned, the transmitted photons in the infinite waveguide are highly antibunched. On the other hand, upon resonant driving, incoherently reflected photons are mostly distributed within the photonic band gap and several sharp side peaks. These features can be explained by considering the poles of the single-particle Green function in the Markovian regime combined with the time delay. Our calculation is not restricted to the Markovian regime, and we obtain several fully non-Markovian results. We show that a single 2LS in a semi-infinite waveguide can not be decoupled by placing it at the node of the photonic field, in contrast to recent results in the Markovian regime. Our results illustrate the complexities that ensue when several qubits are strongly coupled to a bus (the waveguide) as might happen in quantum information processing.
We investigate the photon emission properties of the driven-dissipative Rabi model in the so-called ultrastrong- and deep strong-coupling regimes where the atom-cavity coupling rate ggg becomes comparable or larger than the cavity frequency ${\ensuremath{\omega}}_{c}$. By solving numerically the master equation in the dressed-state basis, we compute the output field correlation functions in the steady-state for a wide range of coupling rates. We find that, as the atom-cavity coupling strength increases, the system undergoes multiple transitions in the photon statistics. In particular, a first sharp antibunching-to-bunching transition, occurring at $g\ensuremath{\sim}0.45{\ensuremath{\omega}}_{c}$, leading to the breakdown of the photon blockade due to the counter-rotating terms, is shown to be the consequence of a parity shift in the energy spectrum. A subsequent revival of the photon blockade and the emergence of the quasicoherent statistics for even larger coupling rates are attributed to an interplay between the nonlinearity in the energy spectrum and the transition rates between the dressed states.
We explore the role of quantum fluctuations in the strong-coupling limit of the dissipative Jaynes–Cummings oscillator driven on resonance. For weak excitation, we derive analytical expressions for the spectrum and the intensity correlation function for the photons scattered by the two-state atom coupled to the coherently driven cavity mode. We do so by writing down a birth–death process adding the higher orders in the excitation strength needed to go beyond the pure-state factorization, following the method introduced in Carmichael (2008). Our results for the first and second-order correlation functions are complemented by the numerical investigation of the waiting-time distribution for the photon emissions directed sideways, and the comparison with ordinary resonance fluorescence. To close out our discussion, we increase the driving field amplitude and approach the critical point organizing a second-order dissipative quantum phase transition by depicting the excitation pathways in the intracavity field distribution for a finite system size.
We consider the quantum Rabi model as an open system that is subject to dissipation, dephasing, and sinusoidal qubit driving. One can change to an interaction picture where the qubit-driving term disappears at the expense of changing the free energy of the qubit which becomes time dependent. If the driving frequency is large with respect to the rest of the parameters with the exception of the driving strength, then one can obtain an effective Hamiltonian that accurately describes the dynamics of the system. The driving has two effects: the qubit-transition frequency is changed and the qubit has reduced dephasing. The driving strength can be chosen so that the qubit-transition frequency is reduced, made equal to zero, or even made negative so that the excited and ground states of the qubit are interchanged. Therefore, sinusoidal qubit driving offers another method to control the qubit-transition frequency and to reduce qubit dephasing. Adjusting the driving strength allows one to consider a qubit with degenerate energy levels. Not taking dissipation into account, the evolution operator of the qubit-harmonic oscillator system is given by a linear combination of the orthogonal projectors onto the eigenstates of ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{x}$ followed by the evolution operator of a forced harmonic oscillator, the harmonic oscillator can be prepared in such a way that it is always found in a Schr\"odinger cat state, and the transition probability of the qubit can exhibit a collapse-revival behavior. In addition, the Born-Markov-secular master equation is deduced and the effects of dissipation are presented. In particular, smaller ultrastrong-coupling values are preferable over larger ultrastrong-coupling values and deep strong-coupling values in order to have long-lived, easily distinguishable Schr\"odinger cat states because the decoherence rate is inversely proportional to the square of the coupling. Finally, the qubit-harmonic oscillator system can be prepared in highly entangled states that are stable under dissipation.