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Europhysics Letters (EPL)

Dynamical approach to shortcuts to adiabaticity for general two-level non-Hermitian systems

T. Z. Luan, Jia-Zheng Sun, H. Z. Shen

Published May 15, 2023• 5 citations
Publisher

Abstract

Abstract In this Letter, we apply a class of general two-level non-Hermitian Hamiltonians whose off-diagonal elements are neither equal nor conjugate of each other to study the dynamical approach to shortcut to adiabaticity which is based on engineering the Hamiltonian and dynamical properties of the system to remove any unwanted probability amplitude or implant the desired probability amplitude. Using the special characteristics that one of the eigenvalues of the general two-level non-Hermitian Hamiltonian is real and the other is complex, we can decay (amplify) the population exponentially in the undesired (desired) eigenstate, and maintain the conservation of probability amplitude of the other eigenstate. Applying this approach to general two-level non-Hermitian systems, we find that the efficient population transfer can be achieved by our method, which works even in almost instantaneous manner and has a lower cost compared with the common approach of shortcut to adiabaticity.

DOI
10.1209/0295-5075/acd57a

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Abstract In this Letter, we apply a class of general two-level non-Hermitian Hamiltonians whose off-diagonal elements are neither equal nor conjugate of each other to study the dynamical approach to shortcut to adiabaticity which is based on engineering the Hamiltonian and dynamical properties of the system to remove any unwanted probability amplitude or implant the desired probability amplitude. Using the special characteristics that one of the eigenvalues of the general two-level non-Hermitian Hamiltonian is real and the other is complex, we can decay (amplify) the population exponentially in the undesired (desired) eigenstate, and maintain the conservation of probability amplitude of the other eigenstate. Applying this approach to general two-level non-Hermitian systems, we find that the efficient population transfer can be achieved by our method, which works even in almost instantaneous manner and has a lower cost compared with the common approach of shortcut to adiabaticity.
We introduce a class of non-Hermitian Hamiltonians that offers a dynamical approach to a shortcut to adiabaticity (DASA). In particular, in our proposed 2×2 Hamiltonians, one eigenvalue is absolutely real and the other one is complex. This specific form of eigenvalues helps us to exponentially decay the population in an undesired eigenfunction or amplify the population in the desired state while keeping the probability amplitude in the other eigenfunction conserved. This provides us with a powerful method to have a diabatic process with the same outcome as its corresponding adiabatic process. In contrast to standard shortcuts to adiabaticity, our Hamiltonians have a much simpler form with a lower thermodynamic cost. Furthermore, we show that DASA can be extended to higher dimensions using the parameters associated with our 2×2 Hamiltonians. Our proposed Hamiltonians not only have application in DASA but also can be used for tunable mode selection and filtering in acoustics, electronics, and optics.
Shortcuts to adiabaticity are alternative fast processes which reproduce the same final state as the adiabatic process in a finite or even shorter time, which have been extended from Hermitian systems to non-Hermitian systems in recent years, but they are barely explored for general non-Hermitian systems where off-diagonal elements of the Hamiltonian are not Hermitian. In this paper, we propose a shortcuts to adiabaticity technique which is based on a transitionless quantum driving algorithm to realize population transfer for general two-level non-Hermitian systems and give both exact and approximate analytical solutions of the corresponding counteradiabatic driving Hamiltonian, where the latter can be extended to the zeroth-order and first-order terms by applying perturbative theory. We find that the first-order correction term is different from the previous results, which is caused by the non-Hermiticity of the off-diagonal elements. We work out an exact expression for the control function and present examples consisting of a general two-level system with gain and loss to show the theory. The results suggest that the high-fidelity population transfer can be implemented in general non-Hermitian systems by our method, which works even with strong non-Hermiticity and without rotating wave approximation. Furthermore, we show that the general Hamiltonian the off-diagonal elements of which are not conjugate to each other can be implemented in many physical systems with the present experimental technology, such as an atom-light interaction system and whispering-gallery microcavity, which might have potential applications in quantum information processing.
The technique of shortcuts to adiabaticity (STA) has attracted broad attention due to their possible applications in quantum information processing and quantum control. However, most studies published so far have been only focused on Hermitian systems under the rotating-wave approximation (RWA). In this paper, we propose a modified shortcuts to adiabaticity technique to realize population transfer for a non-Hermitian system without RWA. We work out an exact expression for the control function and present examples consisting of two-and three-level systems with decay to show the theory. The results suggest that the shortcuts to adiabaticity technique presented here is robust for fast passages. We also find that the decay has small effect on the population transfer in the three-level system. To shed more light on the physics behind this result, we reduce the quantum three-level system to an effective two-level one with large detunings. The shortcuts to adiabaticity technique of effective two-level system is studied. Thereby the high-fidelity population transfer can be implemented in non-Hermitian systems by our method, and it works even without RWA.
Abstract It is still a challenge to experimentally realize shortcuts to adiabaticity (STA) for a non‐Hermitian quantum system since a non‐Hermitian quantum system's counterdiabatic driving Hamiltonian contains some unrealizable auxiliary control fields. In this paper, we relax the strict condition in constructing STA and propose a method to redesign a realizable supplementary Hamiltonian to construct non‐Hermitian STA. The redesigned supplementary Hamiltonian can be eithersymmetric or asymmetric. For the sake of clearness, we apply this method to an Allen‐Eberly model as an example to verify the validity of the optimized non‐Hermitian STA. The numerical simulation demonstrates that a ultrafast population inversion could be realized in a two‐level non‐Hermitian system.