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Dynamics of 4 level systems & Pythagorean triples coupling

We have theoretically studied the dynamics of four-state system, which is of particular interest in quantum information processing technology, where two qubit quantum logic gates serve as elementary building blocks for designing fully functional scalable devices.

In general, solutions of dynamic coupled equations are difficult to analyze, and even for the simpler case of a two-state system, realized by a spin-1/2 particle or a two-state atomic system, only a handful of analytical solutions are known. The task of finding schemes for complete population transfer between selected states becomes increasingly difficult in multi-state coupled systems. We have performed qualitative physical analysis of the resulting set of two-state equations, An analytical solution of the evolution of the two separate pairs of states were derived, where instead of one Rabi frequency that governs the dynamics of the two-state system case, we found two such frequencies, which provide a natural generalization of the Rabi two-state solution for the four-state system. The solution emerges from a common geometric character of both solutions.

With this scheme, we analytically derive the requirements for efficient population transfer in a four-state quantum system. These requirements impose certain analytical relations on the coupling coefficients, and we found that these relations have special algebraic character are closely linked to number theory: we discovered that complete population transfer occurs if ratios between coupling coefficients exactly match a set of Pythagorean triples, which are a set of three integer numbers a, b and c, which do not possess a common factor and satisfy the equation:

Recently, the first experimental demonstration of the Pythagorean dynamical control scheme have been realized with flux-biased Josephson phase qudit. The experiments, which were done by the group of Prof. Natav Katz, have verified the theoretical predictions for successive complete inversions in a four-level system using the family of Pythagorean triple couplings (see figure below). Also, it was shown that such experimental system in the strongly interacting regime supports a richer dynamical system with further control knobs in the transition from SO(4) to SU(4) dynamical systems, by utilizing detunings, anharmonicity and decoherences (of the mechanism of Cartan decomposition). More information can be found in our recent article.