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arXiv:1503.01665 · Journal of the Optical Society of America B

Multiphoton resonant manipulation of qubits by train of pulses

Gor A. Abovyan, Gagik Yu. Kryuchkyan

Published Apr 26, 2016• 1 citation
arXivHTMLPDF

Abstract

We study qubit manipulations by a train of pulses in a systematic approach based on the Magnus expansion and Furry representation in quantum electrodynamics. Furry–Magnus expansion gives approximate exponential representation of a time-evolution operator without the Dyson time-ordering procedure for interactions with external fields of arbitrary intensity and allows us to obtain general expression for single-qubit populations as a function of time-dependent Rabi frequency. The concrete calculations are performed for tunneling quantum dynamics, for multiphoton resonance interactions of qubits, as well as for off-resonance excitations of qubits driven by Gaussian pulses beyond rotating wave approximation. In this way, the populations of qubit states are investigated for various operational regimes, including single-pulse excitation and two-pulse excitation with a phase shift between pulse envelopes being the controlling parameter, and for excitation with sequential pulses. In the last case, we demonstrate the formation of quasi-energetic states and quasi-energies of qubits driven by a train of identical pulses. In this case the transition probability of the qubit generally exhibiting aperiodic oscillations becomes periodically regular for definite values of the quasi-energy.

DOI
10.1364/josab.33.000971

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We study qubit manipulations by a train of pulses in a systematic approach based on the Magnus expansion and Furry representation in quantum electrodynamics. Furry–Magnus expansion gives approximate exponential representation of a time-evolution operator without the Dyson time-ordering procedure for interactions with external fields of arbitrary intensity and allows us to obtain general expression for single-qubit populations as a function of time-dependent Rabi frequency. The concrete calculations are performed for tunneling quantum dynamics, for multiphoton resonance interactions of qubits, as well as for off-resonance excitations of qubits driven by Gaussian pulses beyond rotating wave approximation. In this way, the populations of qubit states are investigated for various operational regimes, including single-pulse excitation and two-pulse excitation with a phase shift between pulse envelopes being the controlling parameter, and for excitation with sequential pulses. In the last case, we demonstrate the formation of quasi-energetic states and quasi-energies of qubits driven by a train of identical pulses. In this case the transition probability of the qubit generally exhibiting aperiodic oscillations becomes periodically regular for definite values of the quasi-energy.
Abstract The problem of speeding up quantum computations through controlling the states of qubits by short large-amplitude unipolar pulses is discussed. A new method is proposed to describe the dynamics of qubits that is based on the Magnus representation for the evolution operator (propagator) of qubits, which allows one to find a solution to the quantum control problem outside the framework of perturbation theory. The evolution of the states of a multiqubit system is represented as a motion on a special unitary group SU(N), similar to the rotation of the Bloch vector on the group SU(2) in the case of a single qubit. As an example, unitary transformations of single-qubit states under different excitation modes are considered. A general formula is found that establishes a relation between the shape of a pulse and the result of its action, similar to the well-known “area theorem” for Rabi pulses. The effect of unipolar pulses on a two-qubit system is analyzed in detail. Two situations are considered: first, a symmetric configuration, when the parameters of qubits do not differ and are acted upon by the same magnetic field from a fluxon, and second, when the configuration of levels arises in which, for one of the qubits, a transition to the nearest level becomes possible under perturbation. The general case of two interacting qubits—a four-level system—is considered analytically. A direct numerical simulation of the dynamics of a multiqubit system is performed to estimate the accuracy of approximation, and fidelity is used as a criterion of accuracy of operations. It is shown that the expressions obtained for the propagator allow one to formulate conditions for the pulse parameters that are necessary to perform logical operations.
In this article the dynamics of the qubits states based on solution of the time-dependent Schrödinger equation is investigated. Using the Magnus method we obtain an explicit interpolation representation for the propagator, which allows to find wave function at an arbitrary time. To illustrate the effectiveness of the approach, the population of the levels a single and two coupled qubits have been calculated by applying the Magnus propagator and the result have been compared with the numerical solution of the Schrödinger equation. As a measure of the approximation of the wave function, we calculate fidelity, which indicates proximity when the exact and approximate evolution operator acts on the initial state. We discuss the possibility of extending the developed methods to multi-qubits system, when high-speed calculation methods of the operators of evolution is particularly relevant.
We analyze the dynamics of a superconducting qubit and the phenomenon of multiorder Rabi oscillations in the presence of a time-modulated external field. Such a field is also presented as a bichromatic field consisting of two spectral components, which are symmetrically detuned from the qubit resonance frequency. This approach leads to obtaining qualitative quantum effects beyond those for the case of monochromatic excitation of qubits. We calculate Floquet states and quasienergies of the composite system ``superconducting qubit plus time-modulated field'' for various resonant regimes. We analyze the dependence of quasienergies from the amplitude of an external field, demonstrating the zeros of difference between quasienergies. We show that, as a rule, populations of qubit states exhibit aperiodic oscillations, but we demonstrate the specific important regimes in which dynamics of populations becomes periodically regular.
The multiple-linear-time-scale method is used to construct a perturbation theory for NNN-level quantum systems subjected to time-dependent perturbations. The secular and small-denominator terms which plague conventional time-dependent perturbation theory are avoided; consequently, the theory is useful for treating long-term behavior and resonant interactions. The introduction of a self-energy operator allows the shifts in energy levels to be displayed explicitly. When the perturbation is the dipole interaction with an electromagnetic field, the successive approximations yield the well-known rotating-wave approximation, corrections due to counter rotating terms, and the Bloch-Siegert frequency shift. The formalism is applicable to arbitrary pulse shapes, provided that $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}\ensuremath{\ll}\ensuremath{\hbar}\overline{\ensuremath{\omega}}$, where $\ensuremath{\parallel}{H}_{1}\ensuremath{\parallel}$ represents the inter-action strength and $\overline{\ensuremath{\omega}}$ is a characteristic frequency of the field. Rabi oscillations induced by multiphoton resonances are automatically included, and this effect is demonstrated in the case of a square pulse with frequency approximately one half the resonance frequency for a transition between two vibrational levels of a molecule. These calculations are compared to an exact numerical solution in order to find the limits of validity of the approximation. Even at high intensities ($I\ensuremath{\sim}{10}^{14}$ W/cm2{\mathrm{cm}}^{2}cm2) the shifted resonance frequency is quite accurate; however, the approximate and exact solutions are slightly out of phase, so that the approximate solution can only be trusted for a few Rabi cycles. For much lower intensities ($I\ensuremath{\sim}{10}^{10}$ W/cm2{\mathrm{cm}}^{2}cm2), the approximate solution is valid for many thousands of Rabi cycles.
We theoretically investigate quantum Rabi oscillations in a system consisting of a two-level atom (qubit) strongly coupled to a one-dimensional open waveguide. In contrast to conventional cavity quantum electrodynamics, the qubit interacts with a continuum of propagating modes, which gives rise to fundamentally different dynamical behavior. Within the rotating-wave approximation, we express the multimode Jaynes-Cummings Hamiltonian in terms of collective bosonic operators and show that the system possesses two integrals of motion, enabling an exact diagonalization of the Hamiltonian in the single-excitation subspace. We show that for a strongly interacting qubit-photon system, the dynamics are captured by a reduced two-level model. In this framework, each level is defined as the product of the atomic excited state and a specific field mode, which takes the form of a multiphoton Fock-like state. In this picture, the Rabi oscillations represent a collective phenomenon corresponding to oscillations between multiphoton states differing by a single photon. We then extend our analysis to multiphoton processes in which the initial field is a coherent state with a continuous spectrum. In this case, the Rabi frequency is shown to be sensitive to the spectral profile of the function that generates the coherent state.