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arXiv:2404.06937 · Physical Review A

Phenomenon of a stronger trapping behavior in Λ -type quantum systems with symmetry

Boris Olegovich Volkov, Anastasia Aleksandrovna Myachkova, Alexander Nikolaevich Pechen

Published Feb 18, 2025• 7 citations
arXivHTMLPDF

Abstract

$\mathrm{\ensuremath{\Lambda}}$, $V$, $\mathrm{\ensuremath{\Xi}}$ (ladder), and other three-level quantum systems with one forbidden transition (referred to here as $\mathrm{\ensuremath{\Lambda}}$-type systems) play an important role in quantum physics. Various applications require manipulation of such systems using as a control a shaped laser field. In this work, we study how the degeneracy of energy states or of Bohr frequencies in these systems affects the efficiency or difficulty of finding the optimal shape of the control field. For this, we adopt the notion of higher-order traps, which was introduced in Pechen and Tannor, Phys. Rev. Lett. 106, 120402 (2011), where second-/third-order traps were discovered for a $\mathrm{\ensuremath{\Lambda}}$-atom with one forbidden transition and with nondegenerate energy levels. We theoretically study the control of such systems with and without degeneracy in their eigenstates and Bohr frequencies, and we investigate numerically using GRAPE and l-BFGS algorithms how this degeneracy influences the efficiency of optimizing the control laser field. We find that the degeneracy of the Bohr frequencies in the $\mathrm{\ensuremath{\Xi}}$ system, which makes the system energy levels symmetrically distributed, leads to the appearance of a seventh-order trap with a more significant attracting domain resulting in a more difficult optimization, while the degeneracy of energy states in generic $\mathrm{\ensuremath{\Lambda}}$-type systems does not lead to an increase of the order of the zero control trap compared to the nondegenerate case. We also find that when not only are the Bohr frequencies degenerate in the system $\mathrm{\ensuremath{\Xi}}$ but also the dipole moments for the two allowed transitions coincide (in this case, the $\mathrm{\ensuremath{\Xi}}$ system is not controllable), then true traps arise in the quantum control landscape. In particular, the constant zero control becomes a trap.

DOI
10.1103/physreva.111.022617

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\ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ, VVV, \ensuremathΞ\mathrm{\ensuremath{\Xi}}\ensuremathΞ (ladder), and other three-level quantum systems with one forbidden transition (referred to here as \ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ-type systems) play an important role in quantum physics. Various applications require manipulation of such systems using as a control a shaped laser field. In this work, we study how the degeneracy of energy states or of Bohr frequencies in these systems affects the efficiency or difficulty of finding the optimal shape of the control field. For this, we adopt the notion of higher-order traps, which was introduced in Pechen and Tannor, Phys. Rev. Lett. 106, 120402 (2011), where second-/third-order traps were discovered for a \ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ-atom with one forbidden transition and with nondegenerate energy levels. We theoretically study the control of such systems with and without degeneracy in their eigenstates and Bohr frequencies, and we investigate numerically using GRAPE and l-BFGS algorithms how this degeneracy influences the efficiency of optimizing the control laser field. We find that the degeneracy of the Bohr frequencies in the \ensuremathΞ\mathrm{\ensuremath{\Xi}}\ensuremathΞ system, which makes the system energy levels symmetrically distributed, leads to the appearance of a seventh-order trap with a more significant attracting domain resulting in a more difficult optimization, while the degeneracy of energy states in generic \ensuremathΛ\mathrm{\ensuremath{\Lambda}}\ensuremathΛ-type systems does not lead to an increase of the order of the zero control trap compared to the nondegenerate case. We also find that when not only are the Bohr frequencies degenerate in the system \ensuremathΞ\mathrm{\ensuremath{\Xi}}\ensuremathΞ but also the dipole moments for the two allowed transitions coincide (in this case, the \ensuremathΞ\mathrm{\ensuremath{\Xi}}\ensuremathΞ system is not controllable), then true traps arise in the quantum control landscape. In particular, the constant zero control becomes a trap.
Quantum control is necessary for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems. An important problem in quantum control is to establish whether the control objective functional has trapping behaviour or no, namely if it has or no traps -- controls from which it is difficult to escape by local search optimization methods. Higher order traps were previously introduced in [A. N. Pechen, D. J. Tannor, "Are there traps in quantum control landscapes?", Phys. Rev. Lett., 106 (2011), 120402], where 3-rd order traps were found. In this note we show that traps of arbitrarily high order exist for controllable quantum systems with special symmetry in the Hamiltonian.
Abstract An important problem in quantum control is the analysis of quantum control landscapes and its trapping features. The notion of a trap in quantum control landscape was introduced in [H. Rabitz, M. Hsieh, C. Rosenthal, Quantum optimally controlled transition landscapes, Science 303, 1998 (2004)]. Later, the notion of higher-order traps was introduced [A. Pechen and D.J. Tannor, Are there traps in quantum control landscapes?, Phys. Rev. Lett. 106, 120402 (2011)] and such higher-order traps were shown to exist for a variety of important applications for physical systems such as for example Λ-atoms widely used in quantum optics and superconducting transmon qubit systems used as one of the potential platform for quantum computation. In this note, we provide several results on the existence of higher-order traps in quantum control landscapes.
Abstract We show that the second‐order traps in the control landscape for a three‐level Λ‐system found in our previous work (Phys. Rev. Lett.2011,106, 120402) are not local maxima: there exist directions in the space of controls in which the objective grows. The growth of the objective is slow — at best 4th order for weak variations of the control. This implies that simple gradient methods would be problematic in the vicinity of second‐order traps, where more sophisticated algorithms that exploit the higher order derivative information are necessary to climb up the control landscape efficiently. The theory is supported by a numerical investigation of the landscape in the vicinity of theε(t)=0 second‐order trap, performed using the GRAPE and BFGS algorithms.
In this work, we perform an analytical and numerical analysis of quantum landscapes for controlling special four-level quantum systems for which we prove that the null control is a five-order trap: a V−VV-VV−V system and an anharmonic system. As a control goal, an observable optimization is considered. The rigorous theoretical analysis is followed by the numerical experiments based on the GRadient Ascent Pulse Engineering (GRAPE) algorithm and Gradient Projection Method (GPM), performed to investigate the behavior of the efficiency of optimization for unconstrained (using GRAPE) and constrained (using GPM) controls. As the main result, we observe an interesting phenomenon with a diverse behavior of the optimization efficiency depending on the system Hamiltonian -- sharp increase of the optimization efficiency up to 100% at certain distance from the null control for a V-V system, while much slower and less significant increase (and even small decrease) for a system with the chain interaction. This sharp difference might be related with the fine structure of the subspace of controls where second derivative of the objective functional is zero.
In this work, we consider a general method for numerically comparing the efficiency of optimization over control landscapes with different trapping features, both for unconstrained (using GRAPE) and constrained (using gradient projection method) optimization. Traps are controls which locally appear optimal while not being globally optimal. Such controls can significantly affect the efficiency of finding controls close to optimal with local search, e.g. gradient-based, algorithms. As an explicit example, we apply the general method to study unconstrained control landscapes for two three-level quantum Ξ\Xi Ξ systems which are widely used in quantum optics and known to have different trapping features: a non-degenerate Ξ\Xi Ξ -system for which the null control is a third-order trap, and a degenerate Ξ\Xi Ξ -system for which the null control is a seventh-order trap. Using the general approach, we perform a comparative study of the impact of the null control on the efficiency of optimization and estimate the dependence of this impact on the maximal allowed number of iterations to check how the optimization is sensitive to this number. The results show that while quantitative behavior is different, the qualitative difference between the two control landscapes remains similar.
Quantum control is necessary for a variety of modern quantum technologies as it allows to optimally manipulate quantum systems. An important problem in quantum control is to establish whether the control objective functional has trapping behaviour or no, namely if it has or no traps -- controls from which it is difficult to escape by local search optimization methods. Higher order traps were previously introduced in [A. N. Pechen, D. J. Tannor, "Are there traps in quantum control landscapes?", Phys. Rev. Lett., 106 (2011), 120402], where 3-rd order traps were found. In this note we show that traps of arbitrarily high order exist for controllable quantum systems with special symmetry in the Hamiltonian.