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

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.

1. Braak D. Integrability of the Rabi model. Physical Review Letters, 2011, vol. 107, no. 10, pp. 100401-1–100401-4. Doi: 10.1103/physrevlett.107.100401

2. Irish E. K. Generalized rotating-wave approximation for arbitrarily large coupling. Physical Review Letters, 2007, vol. 99, no. 17, pp. 173601-1–173601-4. Doi: 10.1103/physrevlett.99.173601

3. Allen L., Eberly J. Optical resonance and two-level atoms. John Wiley and Sons LTD, 1975. 233 p.

4. Mogilevtsev D. S., Kilin S. Ya. Quantum optics methods for structured reservoirs. Minsk, Belaruskaya navuka Publ., 2007. 174 p. (in Russian).

5. Abdel-Aty M., Metwally N., Obama A.-S. F. Dynamics of Bloch vectors and channel capacity of two non-identical charge qubits. Physics Letters A, 2009, vol. 373, no. 10, pp. 927–933. Doi: 10.1016/j.physleta.2009.01.017

6. Dandoloff R., Mosseri R. Geometry of entangled states, Bloch spheres and Hopf fibrations. Journal of Physics A: Mathematical and General, 2001, vol. 34, no. 47, pp. 10243–10252. Doi: 10.1088/0305-4470/34/47/324

7. Benenti G., Siccardi S., Strini G. Nonperturbative interpretation of the Bloch vector’s path beyond the rotating-wave approximation. Physical Review A, 2013, vol. 88, pp. 033814-1–033841-8. Doi: 10.1103/physreva.88.033814

8. Mathews J., Walker R. Mathematical Methods of Physics. New York, Benjamin, 1964. 475 p.

9. Dittrich T., Hänggi P., Ingold G.-L., Kramer B., Schoen G., Zwerger W. Quantum Transport and Dissipation. Chapter 5. Driven quantum systems. New York, WILEY-VCN, 1999, pp. 249–286.

10. Davydov A. S. Quantum mechanics. 3nd ed. Saint Petersburg: BKhV-Peterburg Publ., 2011. 703 p. (in Russian).

11. Scully M. O., Zubairy M. S. Quantum Optics. Cambridge University Press, 1997. 630 p. Doi: 10.1017/CBO9780511813993

12. Efimova A. V. Singularity of Bloch vector’s paths without using the rotating wave approximation. Sovremennye prob lemy fiziki: Mezhdunarodnaya shkola-konferentsiya molodykh uchenykh i spetsialistov. Minsk, 8–10 iyunya 2016 g. [Modern problems of physics, international school-conf. of young scientists and specialists, Minsk, 8–10 June 2016]. Minsk, 2016, pp. 41–45 (in Russian).

13. Zel’dovich Ya. B. The quasienergy of an quantum-mechanical system subjected to a periodic action. Journal of Experimental and Theoretical Physics, 1967, vol. 24, no. 5, pp. 1006–1008.

14. Ritus V. I. Shift and splitting of atomic energy levels by the field of an electromagnetic wave. Journal of Experimental and Theoretical Physics, 1967, vol. 24, no. 5, pp. 1041–1044.

15. Feranchuk I. D., Komarov L. I., Ulyanenkov A. P. Two-level system in a one-mode quantum field: numerical solution on the basis of the operator method. Journal of Physics A: Mathematical and General, 1996, vol. 29, no. 14, pp. 4035–4047. Doi: 10.1088/0305-4470/29/14/026