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arXiv:0911.3012

Complete population transfer in 4-level system via Pythagorean triple coupling

Haim Suchowski, Dmitry B. Uskov

Published Nov 16, 2009• 1 citation
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Abstract

We describe a relation between the requirement of complete population transfer in a four-mode system and the generating function of Pythagorean triples from number theory. We show that complete population transfer will occur if ratios between coupling coefficients exactly match one of the Pythagorean triples (a; b; c) in Z, c^{2} = a^{2} + b^{2}. For a four-level ladder system this relation takes a simple form (V12; V23; V34) ~ (c; b; a), where coefficients Vij describe the coupling between modes. We find that the structure of the evolution operator and the period of complete population transfer are determined by two distinct frequencies. A combination of these frequencies provides a generalization of the two-mode Rabi frequency for a four-mode system.

DOI
10.48550/arxiv.0911.3012

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We describe a relation between the requirement of complete population transfer in a four-mode system and the generating function of Pythagorean triples from number theory. We show that complete population transfer will occur if ratios between coupling coefficients exactly match one of the Pythagorean triples (a; b; c) in Z, c^{2} = a^{2} + b^{2}. For a four-level ladder system this relation takes a simple form (V12; V23; V34) ~ (c; b; a), where coefficients Vij describe the coupling between modes. We find that the structure of the evolution operator and the period of complete population transfer are determined by two distinct frequencies. A combination of these frequencies provides a generalization of the two-mode Rabi frequency for a four-mode system.
Complete population transfer in a four-coupled-modes system is analyzed from a geometrical point of view. An analytical solution of the dynamics is written by the use of two distinct frequencies, the generalization of the single Rabi frequency of the two-state dynamics. We also present its visualization on two separate Bloch spheres with two independent torque equations. With this scheme we analytically derive the requirements for complete population transfer in a four-state quantum system. Interestingly, the solutions are found to be linked to fundamental number theory, whereas complete population transfer occurs only if the ratios between coupling coefficients exactly match a set of Pythagorean triples.
Maximally entangled states play a central role in quantum information processing. Despite much progress throughout the years, robust protocols for manipulations of such states in many-level systems are still scarce. Here we present a control scheme that allows efficient manipulation of complete population inversion between two maximally entangled states. Exploiting the self-duality of SU(2)\mathrm{SU}(2)SU(2), we present in this work a family of 22N{2}^{2N}22N-level systems with couplings related to Pythagorean triples that make a complete population inversion from one state to another (orthogonal) state, using very few couplings and generators. We relate our method to the recently developed retrograde-canon scheme and derive a more general complete transfer recipe. We also discuss the cases of (2n)2{(2n)}^{2}(2n)2-level systems, (2n+1)2{(2n+1)}^{2}(2n+1)2-level systems, and other unitary groups, and give a geometrical description of the inversion via the Majorana sphere.
For the purpose of an examination of coherence control, we study the dynamics of population inversion in the four-level λ system where initial populations are arbitrary coherently distributed in two lower levels. It is shown that the coherence control, which causes population transfer between the lower levels to be forbidden, is possible by using the ratio of Rabi frequencies. Under this condition, it is remarkable that the adiabatic following and the optimal detuning methods are useful to invert the four-level system completely, and it is shown that the two-photon coherences of the coupled three-level ladders are dispersive and absorptive in the former and the latter methods, respectively. From the geometrical interpretation, the Bloch vectors, describing the four-level inversions, could be defined as ones in a two-level system.
The population transfer in a four-level quasi-lambda system driven by unmatched Gaussian pulses is investigated.By using the matrix equation theory,evolutions of the populations of the four states Vs time is investigated.Numerical simulations show that the Rabi frequency intensity and detuning of pumping field have different effects on the population transfer comparing with the probe and coupling field;and spontaneously generated coherence(SGC)have little effect on population transfer.If the parameters are chosen appropriately,population can be transferred almost completely to the target state.
The population transfer in a ladder-type atomic system driven by linearly polarized sech-shape femtosecond laser pulses is investigated by numerically solving Schrödinger equation without including the rotating wave approximation (RWA). It is shown that population transfer is mainly determined by the Rabi frequency (strength) of the driving laser field and the chirp rate, and that the ratio of the dipole moments and the pulse width also have a prominent effect on the population transfer. By choosing appropriate values of the above parameters, complete population transfer can be realized.
The utilization of qudits in quantum systems has led to significant advantages in quantum computation and information processing. Therefore, qudits have gained increased attention in recent research for their precise and efficient operations. In this work, we demonstrate the complete population transfer between the next-adjacent energy levels of a transmon qudit using the Pythagorean coupling method and energy level mapping. We achieve a |0〉 to |2〉 transfer with a process fidelity of 97.76% in the subspace spanned by |0〉 to |2〉. Moreover, the transfer operation is achieved within a remarkably fast timescale, as short as 20 ns. This study may present a promising avenue for enhancing the operation flexibility and efficiency of qudits in future implementations.