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arXiv:2001.01274 · Physical Review A

Complete population inversion of maximally entangled states in 22N -level systems via Pythagorean-triple coupling

Muhammad Erew, Moshe Goldstein, Haim Suchowski

Published Aug 31, 2021• 0 citations
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Abstract

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 $\mathrm{SU}(2)$, we present in this work a family of ${2}^{2N}$-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 ${(2n)}^{2}$-level systems, ${(2n+1)}^{2}$-level systems, and other unitary groups, and give a geometrical description of the inversion via the Majorana sphere.

DOI
10.1103/physreva.104.022616

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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.
We present a novel approach for analytically reducing a family of time-dependent multi-state quantum control problems to two-state systems. The presented method translates between related n 2 -state systems and two-state systems, such that the former undergo complete population inversion (CPI) if and only if the latter reach specific states. For even n , the method translates any two-state CPI scheme to a family of CPI schemes in n 2 -state systems. In particular, facilitating CPI in a four-state system via real time-dependent nearest-neighbors couplings is reduced to facilitating CPI in a two-level system. Furthermore, we show that the method can be used for operator control, and provide conditions for producing several universal gates for quantum computation as an example. In addition, we indicate a basis for utilizing the method in optimal control problems.
High fidelity and robustness in population inversion is very desirable for many quantum control applications. We expand composite pulse schemes developed for two-level dynamics and present an analytic solution for the coherent evolution of an N-level quantum system with SU(2) symmetry, for achieving high fidelity and robust population inversion. Our approach offers a platform for accurate steering of the population transfer in physical multi-level systems, which is crucial for fidelity in quantum computation and achieving fundamental excitations in nuclear magnetic resonances and atomic physics. We also introduce and discuss the geometrical trajectories of these dynamics on the Majorana sphere as an interpretation, allowing us to gain physical insight on the dynamics of many-body or high-dimensional quantum systems.
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.
Accurate population transfer of uncoupled or weakly coupled spin states is crucial for many quantum-information-processing tasks. In this paper, we propose a fast and robust scheme for population transfer which combines invariant-based inverse engineering and geometric formalism for robust quantum control. Our scheme is not constrained by the adiabatic condition and therefore can be implemented quickly. It can also effectively suppress the dominant noise in spin systems, which together with the fast feature guarantees the accuracy of the population transfer. Moreover, the control parameters of the driving Hamiltonian in our scheme are easy to design because they correspond to the curvature and torsion of a three-dimensional visual space curve derived by using geometric formalism for robust quantum control. We test the efficiency of our scheme by numerically simulating the ground-state population transfer in 15N^{15}\mathrm{N}15N nitrogen-vacancy centers and comparing our scheme with stimulated Raman transition, stimulated Raman adiabatic passage, and conventional shortcuts to adiabaticity based schemes, three types of schemes popularly used for population transfer. The numerical results clearly show that our scheme is advantageous over these previous ones.
We propose and analyze a robust population-transfer protocol in a driven two-level system based on a sudden sign change of the detuning at the maximum of a smooth coupling pulse. Away from the jump the dynamics is adiabatic, while the sign flip produces a single nonadiabatic kick in the adiabatic basis. Within a simple stepwise adiabatic-sudden approximation we obtain a compact analytic expression for the final transition probability, identify the parameter regimes that yield high-fidelity inversion, and show that the result depends only on the change of the mixing angle across the detuning jump, i.e., solely on the ratio of the peak Rabi frequency to the detuning. Numerical simulations of the full time-dependent Schrödinger equation confirm the validity and robustness of this description over a broad parameter range. We then use the Majorana decomposition to extend the scheme to an SU(2)-symmetric three-state chain driven by the same coupling and detuning functions. In this setting the three-state propagator is expressed in closed form through the two-level Cayley-Klein parameters, which allows us to derive explicit transition probabilities for all three initial states. In particular, we show that for strong coupling the protocol yields almost complete population transfer between the two outer states, with only small transient population of the middle state, while retaining the same intrinsic robustness as in the underlying two-level model.