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Proceedings of the National Academy of Sciences of Belarus Physics and Mathematics Series

CUSP-SINGULARITY OF THE TRAJECTORY OF THE QUBIT’S BLOCH VECTOR UNDER AN EXTERNAL PERIODIC LIGHT FIELD

A. V. Efimova

Published Apr 5, 2018• 0 citations
Publisher

Abstract

The quantum dynamics of a two-level quantum-mechanical system subjected to the external monochromatic action beyond the rotating wave approximation was investigated. It was shown that under the condition of exact resonance on the trajectories of the Bloch vectors, special points are manifested under different initial conditions. These points are classified as cusps singularities. It is revealed that at such points, the instantaneous rotation axis, relative to which the Bloch vector rotates, reverses its direction. There is a movement stop. For a nonzero frequency detuning, the cusp singularities vanish. A numerical analysis of the singularities of the trajectories of the Bloch vector without rotating wave approximation was supplemented by a study based on the use of the Floquet methods. Within the framework of this approach, recurrence relations for the spectral components of the probability amplitudes were obtained and analyzed. An analytic expression was found for the two values of quasi-energies within a fourth order of magnitude in the interaction energy. It was shown that to obtain a singular behavior of the trajectories of the Bloch vector, it is sufficient to confine by four spectral harmonics in the Floquet expansion. The results obtained are important for achieving the accuracy when performing coherent transformations with two-level systems in the cases where the rotating wave approximation is inapplicable.

DOI
10.29235/1561-2430-2018-54-1-62-68

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The quantum dynamics of a two-level quantum-mechanical system subjected to the external monochromatic action beyond the rotating wave approximation was investigated. It was shown that under the condition of exact resonance on the trajectories of the Bloch vectors, special points are manifested under different initial conditions. These points are classified as cusps singularities. It is revealed that at such points, the instantaneous rotation axis, relative to which the Bloch vector rotates, reverses its direction. There is a movement stop. For a nonzero frequency detuning, the cusp singularities vanish. A numerical analysis of the singularities of the trajectories of the Bloch vector without rotating wave approximation was supplemented by a study based on the use of the Floquet methods. Within the framework of this approach, recurrence relations for the spectral components of the probability amplitudes were obtained and analyzed. An analytic expression was found for the two values of quasi-energies within a fourth order of magnitude in the interaction energy. It was shown that to obtain a singular behavior of the trajectories of the Bloch vector, it is sufficient to confine by four spectral harmonics in the Floquet expansion. The results obtained are important for achieving the accuracy when performing coherent transformations with two-level systems in the cases where the rotating wave approximation is inapplicable.
The Bloch vector's path of a two-level system exposed to a monochromatic field exhibits, in the regime of strong coupling, complex corkscrew trajectories. By considering the infinitesimal evolution of the two-level system when the field is treated as a classical object, we show that the Bloch vector's rotation speed oscillates between zero and twice the rotation speed predicted by the rotating wave approximation. Cusps appear when the rotation speed vanishes. We prove analytically that in correspondence to cusps the curvature of the Bloch vector's path diverges. On the other hand, numerical data show that the curvature is very large even for a quantum field in the deep quantum regime with mean number of photons $\overline{n}\ensuremath{\lesssim}1$. We finally compute numerically the typical error size in a quantum gate when the terms beyond rotating wave approximation are neglected.
We investigate the dynamics of a driven two-level system (classical Rabi model) using the counter-rotating-hybridized rotating-wave method (CHRW), which is a simple method based on a unitary transformation with a parameter $\ensuremath{\xi}$. This approach is beyond the traditional rotating-wave approximation (Rabi-RWA) and more importantly, remains the RWA form with a renormalized tunneling strength and a modified driving strength. The reformulated rotating-wave method not only possesses the same mathematical simplicity as the Rabi-RWA but also allows us to explore the effects of counter-rotating (CR) components. We focus on the properties of off-resonance cases for which the Rabi-RWA method breaks down. After comparing the results of different RWA schemes and those of the numerically exact method in a wide range of parameter regime, we show that the CHRW method gives the accurate driven dynamics which is in good agreement with the numerical method. Moreover, the other RWA methods appear as various limiting cases of the CHRW method. The CHRW method reveals the effects of the CR terms clearly by means of coherent destruction of tunneling and Bloch-Siegert shift. Our main results are as follows: (i) the dynamics of the coherent destruction of tunneling is explicitly given and its dependence on $\ensuremath{\Delta}$ is clarified, which is quantitatively in good agreement with the exact results; (ii) the CR modulated Rabi frequency and the Bloch-Siegert shift are analytically calculated, which is the same as the exact results up to fourth order; (iii) the validity of parameter regions of different RWA methods are given and the comparison of dynamics of these methods are shown. Since the CHRW approach is mathematically simple as well as tractable and physically clear, it may be extended to some complicated problems where it is difficult to do a numerical study.
The nonlinear dynamics of dissipative quantum systems in periodic fields is studied in the framework of a Gisin-like nonlinear Schr\"odinger equation with deterministic nonunitary quantum friction terms describing the system-bath couplings. The virtue of this nonunitary evolution is that it is compatible with Dirac's superposition principle and the Hilbert-space structure of quantum kinematics. Floquet theory and the generalized Van Vleck nearly degenerate perturbation method are used to facilitate both analytical and numerical solutions. Closed-form analytic solutions can be obtained in the long-time average approximation or within the rotating-wave approximation. The methods are applied to the study of dissipative quantum dynamics of two-level systems driven by intense periodic fields. It is found that the system asymptotically approaches a limit cycle (whose orientation is subject to the quantum friction constraint), regardless of the strength of the perturbed fields and the nonlinearity constant, indicating quantum suppression of classical chaos. Further, each point of the limit cycle is found to be an attractor and \ensuremath{\psi}(t) exhibits a fractal-like evolution pattern in the course of time. The structure of the limit cycle depends strongly upon field intensity and frequency as well as the order of nonlinear multiphoton transitions. The power spectrum of the Bloch vector trajectory exhibits a dynamical symmetry inherent in the dissipative system and in the asymptotic limit cycle. A theoretical analysis is presented for the understanding of the origin and the role of the dynamical symmetry.
We present the time evolution of the electronic two-level model in rotating wave approximation and calculate the Aharonov–Anandan phase of the vector states with cyclic evolution either within the period of an external monochromatic electric field or the period corresponding to the generalized Rabi frequency. The Aharonov–Anandan phase in the most general case is shown to be dependent on the initial vector state, unless the system evolves in the adiabatic regime; in the latter case, the Aharonov–Anandan phase recovers Berry's phase. In the quasi-resonant regime of the electronic two-level model, the different relations between Γ and the frequency ω of the electric field correspond to distinct regimes of the coupling between this external field and matter.
Monochromatically driven two-level systems (i.e., Rabi models) are ubiquitous in various fields of physics. Though they have been exactly solved, the physical pictures in these exact solutions are not clear. Recently, approximate analytical solutions with neat physics have been obtained by using the counter-rotating hybridized rotating wave (CHRW) method, which has been proven to be effective over a wider range of parameters than the previous analytical solutions. However, the CHRW depends on a parameter $\ensuremath{\xi}$, which has no solution in some regimes. Here, we combine the double-unitary-transformation approach with the generalized Van Vleck nearly degenerate perturbation theory, and present approximate analytical results with clear physics for almost all parameter regimes, which agree well with the numerical solutions and the previous experimental results. Moreover, the dynamic frequencies of the Rabi model are regular, and the frequency with the highest Fourier amplitude changes from the Rabi frequency to $2n\ensuremath{\omega}$ with driving frequency $\ensuremath{\omega}$ and integer nnn, as the driving intensity increases from weak to deep-strong. In addition, we further explore the Floquet dynamics of the dissipative open Rabi model. Remarkably, the dissipations are tunable in the rotating frame, and the approximate analytical results obtained by our method are in good agreement with the numerical results in the strong driving regime. These results pave the way to quantum control using strong and deep-strong driving with applications in quantum technologies.
In a direct two-level qubit system, when the Rabi frequency is comparable to the resonance frequency, the rotating wave approximation is not appropriate. In this case, the Rabi oscillation is accompanied by another oscillation at twice the frequency of the driving field, the so called Bloch-Siegert oscillation (BSO), which depends on the initial phase of the driving field. This oscillation may restrict the precise rotation of a qubit made of direct two-level system. Here, we show that in case of an effectively two-level lambda system, the BSO is inherently negligible, implying a greater precision for rotation of a qubit made of such a lambda system when compared to a direct two-level qubit in a strong driving field.