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

Molecular Physics

Non-adiabatic transitions for random processes in a two-level system

Igor Vasilevich Alexandrov, L.G. Karamjan

Published Jan 1, 1971• 16 citations
Publisher

Abstract

Non-adiabatic transitions in two-level systems are investigated theoretically for a random time dependence of ħω, the energy difference, between the levels. We assumed that ω = ω(x) and the coordinate x = x(t) is a random function of time. Diffusion and Poisson processes (both homogeneous and with a source) for x(t) were assumed. The cases of linear crossing terms (ω = γx) and non-linear terms (ω = ω e exp (- αx) + ω0) were considered. Values of the non-adiabatic transition probability per unit time were obtained by perturbation theory for ω1τ c ≪ 1 where τ c is the correlation time and ω1 is the off-diagonal matrix element.

DOI
10.1080/00268977100101861

Related papers

Full asymptotic expansion of transition probabilities in the adiabatic limit

Alain Joye, C. -E. Pfister

The authors consider a two-level quantum mechanical system driven by an analytic time-dependent Hamiltonian of the form H( in t). In the adiabatic limit, in <<1, the transition probability P(+,-) from one energy level (labelled by -) at time t=- infinity to the other (labelled by +) at time t=+0OI is known to behave as P(+,-)=exp(- alpha -1 in -1 ) exp( alpha 0 )(1+O( in )). Using a simple iterative procedure generating Hamiltonians H 0 =H,H 1 ,. . .H N+1 , they compute the full asymptotic expansion of the transition probability ln P(+,-)=- alpha -1 in -1 + alpha 0 + j=0 N alpha j in j +O ( in N+1 ) For all N>or=0.

Journal of Physics A Mathematical and General•Feb 21, 1991•42 citations

Non-adiabatic and adiabatic transitions at level crossing with decay: two- and three-level systems

M. B. Kenmoe, S. E. Mkam Tchouobiap, C. Kenfack-Sadem +2 more

We investigate the Landau–Zener (LZ) like dynamics of decaying two- and three-level systems with decay rates Γ 1 and Γ 2 for levels with minimum and maximum spin projection. Non-adiabatic and adiabatic transition probabilities are calculated from diabatic and adiabatic bases for two- and three-level systems. We extend the familiar two-level model of atoms with decay from the excited state out of the system into the hierarchy of three-level models which can be solved analytically or computationally in a non-perturbative manner. Exact analytical solutions are obtained within the framework of an extended form of the proposed procedure which enables to take into account all possible initial moments rather than large negative time t 0 = − ∞ as in standard LZ problems. We elucidate the applications of our results from a unified theoretical basis that numerically analyzes the dynamics of a system as probed by experiments.

Journal of Physics A Mathematical and Theoretical•Feb 10, 2015•12 citations

Adiabatic and nonadiabatic contributions to the energy of a system subject to a time-dependent perturbation: Complete separation and physical interpretation

Anirban Mandal, Katharine L. C. Hunt

When a time-dependent perturbation acts on a quantum system that is initially in the nondegenerate ground state ∣0> of an unperturbed Hamiltonian H(0), the wave function acquires excited-state components ∣k> with coefficients c(k)(t) exp(-iE(k)t/ℏ), where E(k) denotes the energy of the unperturbed state ∣k>. It is well known that each coefficient c(k)(t) separates into an adiabatic term a(k)(t) that reflects the adjustment of the ground state to the perturbation--without actual transitions--and a nonadiabatic term b(k)(t) that yields the probability amplitude for a transition to the excited state. In this work, we prove that the energy at any time t also separates completely into adiabatic and nonadiabatic components, after accounting for the secular and normalization terms that appear in the solution of the time-dependent Schrödinger equation via Dirac's method of variation of constants. This result is derived explicitly through third order in the perturbation. We prove that the cross-terms between the adiabatic and nonadiabatic parts of c(k)(t) vanish, when the energy at time t is determined as an expectation value. The adiabatic term in the energy is identical to the total energy obtained from static perturbation theory, for a system exposed to the instantaneous perturbation λH'(t). The nonadiabatic term is a sum over excited states ∣k> of the transition probability multiplied by the transition energy. By evaluating the probabilities of transition to the excited eigenstates ∣k'(t)> of the instantaneous Hamiltonian H(t), we provide a physically transparent explanation of the result for E(t). To lowest order in the perturbation parameter λ, the probability of finding the system in state ∣k'(t)> is given by λ(2) ∣b(k)(t)∣(2). At third order, the transition probability depends on a second-order transition coefficient, derived in this work. We indicate expected differences between the results for transition probabilities obtained from this work and from Fermi's golden rule.

The Journal of Chemical Physics•Oct 26, 2012•12 citations

Non-adiabatic transition probability with a moving $δ$ potential coupling

Diwaker, Aniruddha Chakraborty

The present work focuses on the calculation of a non-adiabatic transition probability between two states which may or may not cross with each other and are coupled to each other by a moving $δ$ function potential. Here, the time dependent Schrodinger equation is converted to time independent one by using a scaling factor which is function of time. This time independent Schrodinger equation is then considered for two potentials coupled by a moving $δ$ potential and an expression for non-adiabatic transition probability has been derived.

arXiv:1312.7504•Dec 29, 2013•0 citations

Quantum transition probabilities during a perturbing pulse: Differences between the nonadiabatic results and Fermi’s golden rule forms

Anirban Mandal, Katharine L. C. Hunt

For a perturbed quantum system initially in the ground state, the coefficient ck(t) of excited state k in the time-dependent wave function separates into adiabatic and nonadiabatic terms. The adiabatic term ak(t) accounts for the adjustment of the original ground state to form the new ground state of the instantaneous Hamiltonian H(t), by incorporating excited states of the unperturbed Hamiltonian H0 without transitions; ak(t) follows the adiabatic theorem of Born and Fock. The nonadiabatic term bk(t) describes excitation into another quantum state k; bk(t) is obtained as an integral containing the time derivative of the perturbation. The true transition probability is given by bk(t)2, as first stated by Landau and Lifshitz. In this work, we contrast bk(t)2 and ck(t)2. The latter is the norm-square of the entire excited-state coefficient which is used for the transition probability within Fermi’s golden rule. Calculations are performed for a perturbing pulse consisting of a cosine or sine wave in a Gaussian envelope. When the transition frequency ωk0 is on resonance with the frequency ω of the cosine wave, bk(t)2 and ck(t)2 rise almost monotonically to the same final value; the two are intertwined, but they are out of phase with each other. Off resonance (when ωk0 ≠ ω), bk(t)2 and ck(t)2 differ significantly during the pulse. They oscillate out of phase and reach different maxima but then fall off to equal final values after the pulse has ended, when ak(t) ≡ 0. If ωk0 < ω, bk(t)2 generally exceeds ck(t)2, while the opposite is true when ωk0 > ω. While the transition probability is rising, the midpoints between successive maxima and minima fit Gaussian functions of the form a exp[−b(t − d)2]. To our knowledge, this is the first analysis of nonadiabatic transition probabilities during a perturbing pulse.

The Journal of Chemical Physics•May 17, 2018•8 citations

On the probability of non-adiabatic transitions in multiple level crossings

Yōsuke Kayanuma, Susumu Fukuchi

The authors calculate the probability of nonadiabatic transitions in a multiple-crossing Landau-Zener model, in which a diabatic state mod 0) crosses a set of diabatic states mod k) (k=1, 2, . . ., N) with a constant velocity. It is shown that the probability P that the system stays in mod 0) after the multiple crossings under the condition it starts from mod 0) is exactly given by P=exp(-2 pi Sigma k=1 N J k 2 /h(cross) mod u mod ) irrespective of the configuration of mod k), where J k is the off-diagonal coupling constant between mod 0) and mod k( and u is the velocity of the change of the energy differences.

Journal of Physics B Atomic and Molecular Physics•Oct 28, 1985•45 citations
Non-adiabatic transitions in two-level systems are investigated theoretically for a random time dependence of ħω, the energy difference, between the levels. We assumed that ω = ω(x) and the coordinate x = x(t) is a random function of time. Diffusion and Poisson processes (both homogeneous and with a source) for x(t) were assumed. The cases of linear crossing terms (ω = γx) and non-linear terms (ω = ω e exp (- αx) + ω0) were considered. Values of the non-adiabatic transition probability per unit time were obtained by perturbation theory for ω1τ c ≪ 1 where τ c is the correlation time and ω1 is the off-diagonal matrix element.
The authors consider a two-level quantum mechanical system driven by an analytic time-dependent Hamiltonian of the form H( in t). In the adiabatic limit, in <<1, the transition probability P(+,-) from one energy level (labelled by -) at time t=- infinity to the other (labelled by +) at time t=+0OI is known to behave as P(+,-)=exp(- alpha -1 in -1 ) exp( alpha 0 )(1+O( in )). Using a simple iterative procedure generating Hamiltonians H 0 =H,H 1 ,. . .H N+1 , they compute the full asymptotic expansion of the transition probability ln P(+,-)=- alpha -1 in -1 + alpha 0 + j=0 N alpha j in j +O ( in N+1 ) For all N>or=0.
We investigate the Landau–Zener (LZ) like dynamics of decaying two- and three-level systems with decay rates Γ 1 and Γ 2 for levels with minimum and maximum spin projection. Non-adiabatic and adiabatic transition probabilities are calculated from diabatic and adiabatic bases for two- and three-level systems. We extend the familiar two-level model of atoms with decay from the excited state out of the system into the hierarchy of three-level models which can be solved analytically or computationally in a non-perturbative manner. Exact analytical solutions are obtained within the framework of an extended form of the proposed procedure which enables to take into account all possible initial moments rather than large negative time t 0 = − ∞ as in standard LZ problems. We elucidate the applications of our results from a unified theoretical basis that numerically analyzes the dynamics of a system as probed by experiments.
When a time-dependent perturbation acts on a quantum system that is initially in the nondegenerate ground state ∣0> of an unperturbed Hamiltonian H(0), the wave function acquires excited-state components ∣k> with coefficients c(k)(t) exp(-iE(k)t/ℏ), where E(k) denotes the energy of the unperturbed state ∣k>. It is well known that each coefficient c(k)(t) separates into an adiabatic term a(k)(t) that reflects the adjustment of the ground state to the perturbation--without actual transitions--and a nonadiabatic term b(k)(t) that yields the probability amplitude for a transition to the excited state. In this work, we prove that the energy at any time t also separates completely into adiabatic and nonadiabatic components, after accounting for the secular and normalization terms that appear in the solution of the time-dependent Schrödinger equation via Dirac's method of variation of constants. This result is derived explicitly through third order in the perturbation. We prove that the cross-terms between the adiabatic and nonadiabatic parts of c(k)(t) vanish, when the energy at time t is determined as an expectation value. The adiabatic term in the energy is identical to the total energy obtained from static perturbation theory, for a system exposed to the instantaneous perturbation λH'(t). The nonadiabatic term is a sum over excited states ∣k> of the transition probability multiplied by the transition energy. By evaluating the probabilities of transition to the excited eigenstates ∣k'(t)> of the instantaneous Hamiltonian H(t), we provide a physically transparent explanation of the result for E(t). To lowest order in the perturbation parameter λ, the probability of finding the system in state ∣k'(t)> is given by λ(2) ∣b(k)(t)∣(2). At third order, the transition probability depends on a second-order transition coefficient, derived in this work. We indicate expected differences between the results for transition probabilities obtained from this work and from Fermi's golden rule.
Non-adiabatic transition probability with a moving δδδ potential coupling
The present work focuses on the calculation of a non-adiabatic transition probability between two states which may or may not cross with each other and are coupled to each other by a moving δδδ function potential. Here, the time dependent Schrodinger equation is converted to time independent one by using a scaling factor which is function of time. This time independent Schrodinger equation is then considered for two potentials coupled by a moving δδδ potential and an expression for non-adiabatic transition probability has been derived.
For a perturbed quantum system initially in the ground state, the coefficient ck(t) of excited state k in the time-dependent wave function separates into adiabatic and nonadiabatic terms. The adiabatic term ak(t) accounts for the adjustment of the original ground state to form the new ground state of the instantaneous Hamiltonian H(t), by incorporating excited states of the unperturbed Hamiltonian H0 without transitions; ak(t) follows the adiabatic theorem of Born and Fock. The nonadiabatic term bk(t) describes excitation into another quantum state k; bk(t) is obtained as an integral containing the time derivative of the perturbation. The true transition probability is given by bk(t)2, as first stated by Landau and Lifshitz. In this work, we contrast bk(t)2 and ck(t)2. The latter is the norm-square of the entire excited-state coefficient which is used for the transition probability within Fermi’s golden rule. Calculations are performed for a perturbing pulse consisting of a cosine or sine wave in a Gaussian envelope. When the transition frequency ωk0 is on resonance with the frequency ω of the cosine wave, bk(t)2 and ck(t)2 rise almost monotonically to the same final value; the two are intertwined, but they are out of phase with each other. Off resonance (when ωk0 ≠ ω), bk(t)2 and ck(t)2 differ significantly during the pulse. They oscillate out of phase and reach different maxima but then fall off to equal final values after the pulse has ended, when ak(t) ≡ 0. If ωk0 < ω, bk(t)2 generally exceeds ck(t)2, while the opposite is true when ωk0 > ω. While the transition probability is rising, the midpoints between successive maxima and minima fit Gaussian functions of the form a exp[−b(t − d)2]. To our knowledge, this is the first analysis of nonadiabatic transition probabilities during a perturbing pulse.
The authors calculate the probability of nonadiabatic transitions in a multiple-crossing Landau-Zener model, in which a diabatic state mod 0) crosses a set of diabatic states mod k) (k=1, 2, . . ., N) with a constant velocity. It is shown that the probability P that the system stays in mod 0) after the multiple crossings under the condition it starts from mod 0) is exactly given by P=exp(-2 pi Sigma k=1 N J k 2 /h(cross) mod u mod ) irrespective of the configuration of mod k), where J k is the off-diagonal coupling constant between mod 0) and mod k( and u is the velocity of the change of the energy differences.