Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1805.04856 · International Journal of Modern Physics A

Comments on the adiabatic theorem

Dmitrii A. Trunin

Published Aug 28, 2018• 12 citations
arXivHTMLPDF

Abstract

We consider the simplest example of a nonstationary quantum system which is quantum mechanical oscillator with varying frequency and [Formula: see text] self-interaction. We calculate loop corrections to the Keldysh, retarded/advanced propagators and vertices using Schwinger–Keldysh diagrammatic technique and show that there is no physical secular growth of the loop corrections in the cases of constant and adiabatically varying frequency. This fact corresponds to the well-known adiabatic theorem in quantum mechanics. However, in the case of nonadiabatically varying frequency we obtain strong IR corrections to the Keldysh propagator which come from the “sunset” diagrams, grow with time indefinitely and indicate energy pumping into the system. It reveals itself via the change in time of the level population and of the anomalous quantum average.

DOI
10.1142/s0217751x18501403

Related papers

Interpolation approach to Hamiltonian-varying quantum systems and the adiabatic theorem

Yu Pan, Zibo Miao, Nina H. Amini +2 more

Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this paper we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure resembles the excitation energy or excess work performed in thermodynamics, which can be taken as the error of adiabatic approximation. We prove that under certain conditions, this error can be estimated for an arbitrarily given interpolating function. This error estimation could be used as guideline to induce adiabatic evolution. According to our calculation, the adiabatic approximation error is not linearly proportional to the average speed of the variation of the system Hamiltonian and the inverse of the energy gaps in many cases. In particular, we apply this analysis to an example in which the applicability of the adiabatic theorem is questionable.

arXiv:1503.03249•EPJ Quantum Technology•Nov 25, 2015•5 citations

Oscillating potential well in the complex plane and the adiabatic theorem

Stefano Longhi

A quantum particle in a slowly changing potential well $V(x,t)=V(x\ensuremath{-}{x}_{0}(\ensuremath{\epsilon}t))$, periodically shaken in time at a slow frequency $\ensuremath{\epsilon}$, provides an important quantum mechanical system where the adiabatic theorem fails to predict the asymptotic dynamics over time scales longer than $\ensuremath{\sim}1/\ensuremath{\epsilon}$. Specifically, we consider a double-well potential $V(x)$ sustaining two bound states spaced in frequency by ${\ensuremath{\omega}}_{0}$ and periodically shaken in a complex plane. Two different spatial displacements ${x}_{0}(t)$ are assumed: the real spatial displacement ${x}_{0}(\ensuremath{\epsilon}t)=Asin(\ensuremath{\epsilon}t)$, corresponding to ordinary Hermitian shaking, and the complex one ${x}_{0}(\ensuremath{\epsilon}t)=A\ensuremath{-}Aexp(\ensuremath{-}i\ensuremath{\epsilon}t)$, corresponding to non-Hermitian shaking. When the particle is initially prepared in the ground state of the potential well, breakdown of adiabatic evolution is found for both Hermitian and non-Hermitian shaking whenever the oscillation frequency $\ensuremath{\epsilon}$ is close to an odd resonance of ${\ensuremath{\omega}}_{0}$. However, a different physical mechanism underlying nonadiabatic transitions is found in the two cases. For the Hermitian shaking, an avoided crossing of quasienergies is observed at odd resonances and nonadiabatic transitions between the two bound states, resulting in Rabi flopping, can be explained as a multiphoton resonance process. For the complex oscillating potential well, breakdown of adiabaticity arises from the appearance of Floquet exceptional points at exact quasienergy crossing.

arXiv:1710.04429•Physical Review A•Oct 2, 2017•7 citations

Evolution of Quantum Systems with a Discrete Energy Spectrum in an Adiabatically Varying External Field

Yury Belousov

We introduce a new approach for describing nonstationary quantum systems with a discrete energy spectrum. The essence of this approach is that we describe the evolution of a quantum system in a time-dependent basis. In a sense, this approach is similar to the description of the system in the interaction representation. However, the time dependence of the basic states of the representation is determined not by the evolution operator with a time-independent Hamiltonian but by the eigenstates of the time-dependent Hamiltonian defined at the current time. The time dependence of the basic states of the representation leads to the appearance of an additional term in the Schrödinger equation, which in the case of slowly changing parameters of the Hamiltonian can be considered as a small perturbation. The adiabatic representation is suitable in cases where it is impossible to apply the standard interaction representation. The application of the adiabatic representation is illustrated by the example of two spins connected by a magnetic dipole–dipole interaction in a slowly varying external magnetic field.

Symmetry•Nov 4, 2024•1 citation

Quantum adiabatic theorem with energy gap regularization

Nikolay Borisovich Ilin

The dynamics of a nonstationary quantum system whose Hamiltonian explicitly depends on time is called adiabatic if a system state that is an eigenstate of the Hamiltonian at the initial instant of time remains close to this eigenstate throughout the evolution. The degree of such closeness depends on the smallness of the parameter that determines the rate of change of the Hamiltonian. It is usually believed that one of the factors playing a decisive role for the stability of the adiabatic dynamics is the structure of the spectrum of the Hamiltonian. As the quantum adiabatic theorem states in its usual formulation, deviations from the adiabatic evolution can be estimated from above by the ratio of the rate of change of the Hamiltonian to the minimum distance between the energy of the state that approximates the adiabatic dynamics and the rest of the spectrum of the Hamiltonian. We analyze this dependence and prove theorems showing that the efficiency of the adiabatic approximation is more influenced by the rate of change of the Hamiltonian eigenvectors than by the dynamics of the spectrum. In a vast majority of physically meaningful cases, it turns out that controlling the dynamics of eigenvectors is sufficient for ensuring the adiabaticity, regardless of the dynamics of the spectrum as such.

Theoretical and Mathematical Physics•Apr 1, 2022•1 citation

Quantum adiabatic evolution is not necessary adiabatic: Adiabatic evolution realized by dynamical decoupling

Zhenyu Wang

We decompose the quantum adiabatic evolution as the products of gauge invariant unitary operators and obtain the exact nonadiabatic correction in the adiabatic approximation. A necessary and sufficient condition that leads to adiabatic evolution with geometric phases is provided and we determine that in the adiabatic evolution, while the eigenstates are slowly varying, the eigenenergies and degeneracy of the Hamiltonian can change rapidly. We exemplify this result by the example of the adiabatic evolution driven by parametrized pulse sequences. For driving fields that are rotating slowly with the same average energy and evolution path, fast modulation fields can have smaller nonadiabatic errors than obtained under the traditional approach with a constant amplitude.

•Oct 16, 2012•0 citations

Generalized Adiabatic Theorems: Quantum Systems Driven by Modulated Time-Varying Fields

Amro Dodin, Paul Brumer

We present generalized adiabatic theorems for closed and open quantum systems that can be applied to slow modulations of rapidly varying fields, such as oscillatory fields that occur in optical experiments and light-induced processes.The generalized adiabatic theorems show that a sufficiently slow modulation conserves the dynamical modes of time-dependent reference Hamiltonians.In the limiting case of modulations of static fields, the standard adiabatic theorems are recovered.Applying these results to periodic fields shows that they remain in Floquet states rather than in energy eigenstates.More generally, these adiabatic theorems can be applied to transformations of arbitrary time-dependent fields, by accounting for the rapidly varying part of the field through the dynamical normal modes, and treating the slow modulation adiabatically.As examples, we apply the generalized theorem to (a) predict the dynamics of a two-level system driven by a frequency-modulated resonant oscillation, a pathological situation beyond the applicability of traditional adiabatic theorems, and (b) to show that open quantum systems driven by slowly turned-on incoherent light, such as biomolecules under natural illumination conditions, can display only coherences that survive in the steady state.

arXiv:2105.01140•PRX Quantum•Jul 2, 2021•11 citations
We consider the simplest example of a nonstationary quantum system which is quantum mechanical oscillator with varying frequency and [Formula: see text] self-interaction. We calculate loop corrections to the Keldysh, retarded/advanced propagators and vertices using Schwinger–Keldysh diagrammatic technique and show that there is no physical secular growth of the loop corrections in the cases of constant and adiabatically varying frequency. This fact corresponds to the well-known adiabatic theorem in quantum mechanics. However, in the case of nonadiabatically varying frequency we obtain strong IR corrections to the Keldysh propagator which come from the “sunset” diagrams, grow with time indefinitely and indicate energy pumping into the system. It reveals itself via the change in time of the level population and of the anomalous quantum average.
Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this paper we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure resembles the excitation energy or excess work performed in thermodynamics, which can be taken as the error of adiabatic approximation. We prove that under certain conditions, this error can be estimated for an arbitrarily given interpolating function. This error estimation could be used as guideline to induce adiabatic evolution. According to our calculation, the adiabatic approximation error is not linearly proportional to the average speed of the variation of the system Hamiltonian and the inverse of the energy gaps in many cases. In particular, we apply this analysis to an example in which the applicability of the adiabatic theorem is questionable.
A quantum particle in a slowly changing potential well $V(x,t)=V(x\ensuremath{-}{x}_{0}(\ensuremath{\epsilon}t))$, periodically shaken in time at a slow frequency $\ensuremath{\epsilon}$, provides an important quantum mechanical system where the adiabatic theorem fails to predict the asymptotic dynamics over time scales longer than $\ensuremath{\sim}1/\ensuremath{\epsilon}$. Specifically, we consider a double-well potential V(x)V(x)V(x) sustaining two bound states spaced in frequency by ${\ensuremath{\omega}}_{0}$ and periodically shaken in a complex plane. Two different spatial displacements x0(t){x}_{0}(t)x0​(t) are assumed: the real spatial displacement ${x}_{0}(\ensuremath{\epsilon}t)=Asin(\ensuremath{\epsilon}t)$, corresponding to ordinary Hermitian shaking, and the complex one ${x}_{0}(\ensuremath{\epsilon}t)=A\ensuremath{-}Aexp(\ensuremath{-}i\ensuremath{\epsilon}t)$, corresponding to non-Hermitian shaking. When the particle is initially prepared in the ground state of the potential well, breakdown of adiabatic evolution is found for both Hermitian and non-Hermitian shaking whenever the oscillation frequency $\ensuremath{\epsilon}$ is close to an odd resonance of ${\ensuremath{\omega}}_{0}$. However, a different physical mechanism underlying nonadiabatic transitions is found in the two cases. For the Hermitian shaking, an avoided crossing of quasienergies is observed at odd resonances and nonadiabatic transitions between the two bound states, resulting in Rabi flopping, can be explained as a multiphoton resonance process. For the complex oscillating potential well, breakdown of adiabaticity arises from the appearance of Floquet exceptional points at exact quasienergy crossing.
We introduce a new approach for describing nonstationary quantum systems with a discrete energy spectrum. The essence of this approach is that we describe the evolution of a quantum system in a time-dependent basis. In a sense, this approach is similar to the description of the system in the interaction representation. However, the time dependence of the basic states of the representation is determined not by the evolution operator with a time-independent Hamiltonian but by the eigenstates of the time-dependent Hamiltonian defined at the current time. The time dependence of the basic states of the representation leads to the appearance of an additional term in the Schrödinger equation, which in the case of slowly changing parameters of the Hamiltonian can be considered as a small perturbation. The adiabatic representation is suitable in cases where it is impossible to apply the standard interaction representation. The application of the adiabatic representation is illustrated by the example of two spins connected by a magnetic dipole–dipole interaction in a slowly varying external magnetic field.
The dynamics of a nonstationary quantum system whose Hamiltonian explicitly depends on time is called adiabatic if a system state that is an eigenstate of the Hamiltonian at the initial instant of time remains close to this eigenstate throughout the evolution. The degree of such closeness depends on the smallness of the parameter that determines the rate of change of the Hamiltonian. It is usually believed that one of the factors playing a decisive role for the stability of the adiabatic dynamics is the structure of the spectrum of the Hamiltonian. As the quantum adiabatic theorem states in its usual formulation, deviations from the adiabatic evolution can be estimated from above by the ratio of the rate of change of the Hamiltonian to the minimum distance between the energy of the state that approximates the adiabatic dynamics and the rest of the spectrum of the Hamiltonian. We analyze this dependence and prove theorems showing that the efficiency of the adiabatic approximation is more influenced by the rate of change of the Hamiltonian eigenvectors than by the dynamics of the spectrum. In a vast majority of physically meaningful cases, it turns out that controlling the dynamics of eigenvectors is sufficient for ensuring the adiabaticity, regardless of the dynamics of the spectrum as such.
We decompose the quantum adiabatic evolution as the products of gauge invariant unitary operators and obtain the exact nonadiabatic correction in the adiabatic approximation. A necessary and sufficient condition that leads to adiabatic evolution with geometric phases is provided and we determine that in the adiabatic evolution, while the eigenstates are slowly varying, the eigenenergies and degeneracy of the Hamiltonian can change rapidly. We exemplify this result by the example of the adiabatic evolution driven by parametrized pulse sequences. For driving fields that are rotating slowly with the same average energy and evolution path, fast modulation fields can have smaller nonadiabatic errors than obtained under the traditional approach with a constant amplitude.
We present generalized adiabatic theorems for closed and open quantum systems that can be applied to slow modulations of rapidly varying fields, such as oscillatory fields that occur in optical experiments and light-induced processes.The generalized adiabatic theorems show that a sufficiently slow modulation conserves the dynamical modes of time-dependent reference Hamiltonians.In the limiting case of modulations of static fields, the standard adiabatic theorems are recovered.Applying these results to periodic fields shows that they remain in Floquet states rather than in energy eigenstates.More generally, these adiabatic theorems can be applied to transformations of arbitrary time-dependent fields, by accounting for the rapidly varying part of the field through the dynamical normal modes, and treating the slow modulation adiabatically.As examples, we apply the generalized theorem to (a) predict the dynamics of a two-level system driven by a frequency-modulated resonant oscillation, a pathological situation beyond the applicability of traditional adiabatic theorems, and (b) to show that open quantum systems driven by slowly turned-on incoherent light, such as biomolecules under natural illumination conditions, can display only coherences that survive in the steady state.