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Physical Review A

Pythagorean coupling: Complete population transfer in a four-state system

Haim Suchowski, Yaron R. Silberberg, Dmitry B. Uskov

Published Jul 18, 2011• 19 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.84.013414

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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.
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.
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.
A technique for complete population transfer between the two end states $|1\ensuremath{\rangle}$ and $|3\ensuremath{\rangle}$ of a three-state quantum system with a train of NNN pairs of resonant and coincident pump and Stokes pulses is introduced. A simple analytic formula is derived for the ratios of the pulse amplitudes in each pair for which the maximum transient population P2(t){P}_{2}(t)P2​(t) of the middle state $|2\ensuremath{\rangle}$ is minimized, ${P}_{2}^{\mathrm{max}}={\mathrm{sin}}^{2}(\ensuremath{\pi}/4N)$. It is remarkable that, even though the pulses are on exact resonance, P2(t){P}_{2}(t)P2​(t) is damped to negligibly small values even for a small number of pulse pairs. The population dynamics resembles generalized $\ensuremath{\pi}$ pulses for small NNN and stimulated Raman adiabatic passage for large NNN and therefore this technique can be viewed as a bridge between these well-known techniques.
We examine, both numerically and analytically, the limits of controllability and the dynamics of population transfer in systems with degenerate target states embedded in a finite manifold of states. An analysis of the dressed eigenstates of coupled field-matter systems is combined with the theory of the controllability of invariant systems on Lie groups to show the existence of a control scheme using a pair of shaped laser pulses acting upon a four-state system with intermediate degenerate states and a diamond-shaped array of transition moments $({\ensuremath{\mu}}_{12},{\ensuremath{\mu}}_{24},{\ensuremath{\mu}}_{13},{\ensuremath{\mu}}_{34}\ensuremath{\ne}0,{\ensuremath{\mu}}_{14},{\ensuremath{\mu}}_{23}=0).$ The limits of the controllability of these systems are linked to the symmetry of the transition dipole moments connecting the states. A broken symmetry of the transition dipole moments $({\ensuremath{\mu}}_{12}/{\ensuremath{\mu}}_{24}\ensuremath{\ne}{\ensuremath{\mu}}_{13}/{\ensuremath{\mu}}_{34})$ permits complete population transfer to one of the pair of degenerate states. Numerical simulations of the optimized population transfer to one of a pair of degenerate states in the system agree quantitatively with the analytically predicted limits of population transfer in different regimes in the joint material-field characteristics of the system. The applicability of the four-state model to selective population transfer in vibrationally mediated photodissociation experiments is discussed.
A zero-area pulse produces no excitation in a two-state quantum system on exact resonance. We show analytically and numerically that off resonance the transition probability may acquire significant values, and as the Rabi frequency increases, it approaches steadily unity, a feature reminiscent of adiabatic passage. The nature of the transition is not, however, of adiabatic nature. We show that complete population transfer occurs as a result of a quasiresonant transition between the adiabatic states due to a $\ensuremath{\delta}$-function-type interaction with an area of $\ensuremath{\pi}$.
In the previous paper on this topic it was shown how, for a pulse of arbitrary shape and duration, the drive frequency can be analytically optimized to maximize the amplitude of the population oscillations between the selected two levels in a general many level quantum system. It was shown how the standard Rabi theory can be extended beyond the simple two-level systems. Now, in order to achieve the quickest and as complete as possible population transfer between two pre-selected levels, driving pulse should be tailored so that it produces only a single half-oscillation of the population. In this paper, this second (and final) step towards the controlled population transfer using modified (i.e. many level system) Rabi oscillations is discussed. The results presented herein can be regarded as an extension of the standard πππ-pulse theory - also strictly valid only in the two level systems - to the coherently driven population oscillations in general many level systems.