Nonadiabatic Terms Research Articles

Physics-informed neural networks for an optimal counterdiabatic quantum computation

A novel methodology that leverages physics-informed neural networks to optimize quantum circuits in systems with NQ qubits by addressing the counterdiabatic (CD) protocol is introduced. The primary purpose is to employ physics-inspired deep learning techniques for accurately modeling the time evolution of various physical observables within quantum systems. To achieve this, we integrate essential physical information into an underlying neural network to effectively tackle the problem. Specifically, the imposition of the solution to meet the principle of least action, along with the hermiticity condition on all physical observables, among others, ensuring the acquisition of appropriate CD terms based on underlying physics. This approach provides a reliable alternative to previous methodologies relying on classical numerical approximations, eliminating their inherent constraints. The proposed method offers a versatile framework for optimizing physical observables relevant to the problem, such as the scheduling function, gauge potential, temporal evolution of energy levels, among others. This methodology has been successfully applied to 2-qubit representing H2 molecule using the STO-3G basis, demonstrating the derivation of a desirable decomposition for non-adiabatic terms through a linear combination of Pauli operators. This attribute confers significant advantages for practical implementation within quantum computing algorithms.

Nonadiabatic Field on Quantum Phase Space: A Century after Ehrenfest.

Nonadiabatic transition dynamics lies at the core of many electron/hole transfer, photoactivated, and vacuum field-coupled processes. About a century after Ehrenfest proposed "Phasenraum" and the Ehrenfest theorem, we report a conceptually novel trajectory-based nonadiabatic dynamics approach, nonadiabatic field (NAF), based on a generalized exact coordinate-momentum phase space formulation of quantum mechanics. It does not employ the conventional Born-Oppenheimer or Ehrenfest trajectory in the nonadiabatic coupling region. Instead, in NAF the equations of motion of the independent trajectory involve a nonadiabatic nuclear force term in addition to an adiabatic nuclear force term of a single electronic state. A few benchmark tests for gas phase and condensed phase systems indicate that NAF offers a practical tool to capture the correct correlation of electronic and nuclear dynamics for processes where the states remain coupled all the time as well as for the asymptotic region where the coupling of electronic states vanishes.

Nonadiabatic transition probabilities for quantum systems in electromagnetic fields: Dephasing and population relaxation due to contact with a bath.

We contrast Dirac's theory of transition probabilities and the theory of nonadiabatic transition probabilities, applied to a perturbed system that is coupled to a bath. In Dirac's analysis, the presence of an excited state |k0⟩ in the time-dependent wave function constitutes a transition. In the nonadiabatic theory, a transition occurs when the wave function develops a term that is not adiabatically connected to the initial state. Landau and Lifshitz separated Dirac's excited-state coefficients into a term that follows the adiabatic theorem of Born and Fock and a nonadiabatic term that represents excitation across an energy gap. If the system remains coherent, the two approaches are equivalent. However, differences between the two approaches arise when coupling to a bath causes dephasing, a situation that was not treated by Dirac. For two-level model systems in static electric fields, we add relaxation terms to the Liouville equation for the time derivative of the density matrix. We contrast the results obtained from the two theories. In the analysis based on Dirac's transition probabilities, the steady state of the system is not an equilibrium state; also, the steady-state population ρkk,s increases with increasing strength of the perturbation and its value depends on the dephasing time T2. In the nonadiabatic theory, the system evolves to the thermal equilibrium with the bath. The difference is not simply due to the choice of basis because the difference remains when the results are transformed to a common basis.

Numerical simulations of two-fluid magnetoacoustic waves in the solar atmosphere

ABSTRACT We study vertical variations of wave periods of magnetoacoustic two-fluid waves in the partially ionized lower solar atmosphere, consisting of ion (proton) + electron and neutral (atomic hydrogen) fluids, which are coupled by ion–neutral collisions. The study allows finding the wave period cut-offs and their variations in the solar atmosphere, as well as establishing the role of these cut-offs in determining the wave propagation conditions. The atmosphere is permitted by a uniform vertical magnetic field. We perform numerical simulations in the framework of a one-dimensional (1D), two-fluid model in which plane waves are exited by a harmonic driver in the vertical ion and neutral velocities, operating at the bottom of the solar photosphere. We observe excitation of waves with cut-off wave periods in addition to waves set directly by the driver. We also see that some waves exited by that driver can reach the solar corona. Despite of its limitations such as the lack of non-adiabatic and non-ideal terms and a simple 1D structure, the developed two-fluid model of the solar atmosphere sheds a new light on the role of cut-offs in setting up the wave propagation conditions in the solar atmosphere and finding periods of waves that may carry their energy from the solar surface to the corona.

Computational Protocol to Evaluate Electron-Phonon Interactions Within Density Matrix Perturbation Theory.

We present a computational protocol, based on density matrix perturbation theory, to obtain non-adiabatic, frequency-dependent electron–phonon self-energies for molecules and solids. Our approach enables the evaluation of electron–phonon interaction using hybrid functionals, for spin-polarized systems, and the computational overhead to include dynamical and non-adiabatic terms in the evaluation of electron–phonon self-energies is negligible. We discuss results for molecules, as well as pristine and defective solids.

Toroidal modeling of energetic passing particle drift kinetic effects on tearing mode stability

Drift kinetic effects of the neutral beam injection induced passing energetic particles (EPs) on the linear stability of the n = 1 tearing mode (TM) (with the dominant poloidal harmonic of m = 2) are numerically investigated utilizing the MARS-K code (Liu et al 2008 Phys. Plasmas 15 112503), in a tokamak plasma with finite equilibrium pressure and anisotropic thermal transport. In the low plasma pressure regime, it is found that co- (counter-) passing EPs stabilize (destabilize) the TM, agreeing with previous studies. However, as the plasma pressure increases beyond a critical value, it is found that co-passing EPs also destabilize the mode. An in-depth analysis reveals that the net effect of co-passing EPs is a result of competition between the stabilizing contribution from the non-adiabatic drift kinetic terms and the destabilizing contribution associated with adiabatic terms, with the latter becoming more dominant at higher equilibrium pressure. Non-perturbative magnetohydrodynamic-kinetic hybrid modeling also finds that co- and counter-passing EPs modify the TM eigenfunction differently, with the counter-passing EPs enhancing the sideband harmonics. Furthermore, effects of the plasma resistivity and toroidal rotation, as well as that of the equilibrium distribution of EPs in the particle pitch angle space, are also investigated, showing asymmetric results on the TM stability between the co- and counter-passing EPs. The first order finite orbit width correction is found to be stabilizing with co-passing EPs and destabilizing with counter-passing particles. Finally, drift resonances between passing EPs and the TM induce finite frequency to the mode and generate finite net torques inside the plasma, due to the neoclassical toroidal viscosity and the Reynolds stress associated with 3D perturbations.

Analytic calculation of the Berry curvature and diagonal Born-Oppenheimer correction for molecular systems in uniform magnetic fields.

The diagonal nonadiabatic term arising from the Born-Oppenheimer wave function ansatz contains contributions from a vector and scalar potential. The former is provably zero when the wave function can be taken to be real valued, and the latter, known as the diagonal Born-Oppenheimer correction (DBOC), is typically small in magnitude. Therefore, unless high accuracy is sought, the diagonal nonadiabatic term is usually neglected when calculating molecular properties. In the presence of a magnetic field, the wave function is generally complex, and the geometric vector potential gives rise to a screening force that is qualitatively important for molecular dynamics. This screening force is written in terms of the Berry curvature and is added to the bare Lorentz force acting on the nuclei in the presence of the field. In this work, we derive analytic expressions for the Berry curvature and DBOC using both first- and second-quantization formalisms for the case of generalized and restricted Hartree-Fock theories in a uniform magnetic field. The Berry curvature and DBOC are calculated as a function of the magnetic field strength and the bond distance for the ground-state singlets of H2, LiH, BH, and CH+. We also examine the stability and time-reversal symmetry of the underlying self-consistent field solutions. The character of the DBOC and Berry curvature is found to depend on the magnetic field and varies between molecules. We also identify instances of broken time-reversal symmetry for the dissociation curves of BH and CH+.

Spin transfer torque bias (STTB) due to domain wall resistance in an infinitely long ferromagnetic nanowire

The shift of a magnetization loop along the magnetic field axis for a ferromagnetic (FM)/anti-ferromagnetic (AFM) system when it is cooled through Néel temperature of AFM layer is called exchange anisotropy or exchange bias. Here, using micromagnetic simulations we propose that spin transfer torque (STT) mechanism would indeed be helpful in realizing the shift of the magnetization loop along magnetic field axis through domain wall (DW) resistance for an infinitely long FM nanowire without having AFM layer, which we call as spin transfer torque bias (STTB). Essentially, STTB is realized on both positive and negative magnetic field axes by varying the angle between spin polarized current and Zeeman field from 0° to 180° respectively and the origin is attributed to helical motion of the DW. However, we do not see STTB at 90° due to coherent rotation of domain. We also ascertain that STTB is also a function of magnetic anisotropy, current density, polarization strength and non-adiabatic STT term. Variation in STTB for different FM systems such as Fe2CoSi, Ni80Fe20 and Fe is attributed to a change in DW width. We believe that present results would lead to a new dimension in the field of spintronics.

On the Landau-Lifshitz-Bloch equation with spin torque effects

A mathematical model for current-induced magnetization dynamics in ferromagnets at elevated temperatures is considered. The model is represented by a classical Landau-Lifshitz-Bloch equation containing adiabatic and non-adiabatic torques. In the case of a ferromagnet confined to a bounded domain of R3, we prove existence of a weak solution by using Faedo-Galerkin approach and a compactness method. This paper extends the work done by Le et al. (2016) in the absence of adiabatic and non-adiabatic terms. Furthermore, uniqueness is proved in the case of dimension one and two. Finally, the limiting behaviour in large time is studied for the non-adiabatic case.

A Synthetic Study of the Position Difference of the Southern Branch Trough of the Qinghai-Tibet Plateau Based on Objective Identification

The southern branch trough (SBT) mainly appears in the winter half year (November to May of the following year), using the 4 times daily NCEP/NCAR re-analysis data nearly 41 years (1979-2019) to analyze the differences of the SBT distribution of spatial location, frequency in winter and spring, then selects the “eastern type” and “western type” of the 10 most typical SBT, using simplified vertical vorticity tendency equation, using simplified vertical vorticity tendency equation to diagnosis of the SBT in power, heat, water vapor and wave energy in different positions. The results show that: 1) The location of the SBT is more eastward in winter, and more westward in spring. 2) The diagnosis results of the vorticity equation show that the vorticity of the southern branch of the “western type” is mainly contributed by advection term; the vorticity of the “eastern type” south branch is mainly contributed by the non-adiabatic heating term. 3) The SBT of the “eastern type” has more obvious vorticity advection than the southern branch of the “western type”, and the dynamic action is stronger. The “western type” SBT has stronger Q1, specific humidity advection and water vapor flux than the “eastern type” SBT, which is greatly affected by thermal action and water vapor. When the “eastern type” and “western type” SBT occur, the T-N wave activity flux appears obvious abnormal energy fluctuation propagation.

Variance of the energy of a quantum system in a time-dependent perturbation: Determination by nonadiabatic transition probabilities.

For a quantum system in a time-dependent perturbation, we prove that the variance in the energy depends entirely on the nonadiabatic transition probability amplitudes bk(t). Landau and Lifshitz introduced the nonadiabatic coefficients for the excited states of a perturbed quantum system by integrating by parts in Dirac's expressions for the coefficients ck (1)(t) of the excited states to first order in the perturbation. This separates ck (1)(t) for each state into an adiabatic term ak (1)(t) and a nonadiabatic term bk (1)(t). The adiabatic term follows the adiabatic theorem of Born and Fock; it reflects the adjustment of the initial state to the perturbation without transitions. If the response to a time-dependent perturbation is entirely adiabatic, the variance in the energy is zero. The nonadiabatic term bk (1)(t) represents actual excitations away from the initial state. As a key result of the current work, we derive the variance in the energy of the quantum system and all of the higher moments of the energy distribution using the values of |bk(t)|2 for each of the excited states along with the energy differences between the excited states and the ground state. We prove that the same variance (through second order) is obtained in terms of Dirac's excited-state coefficients ck(t). We show that the results from a standard statistical analysis of the variance are consistent with the quantum results if the probability of excitation Pk is set equal to |bk(t)|2, but not if the probability of excitation is set equal to |ck(t)|2. We illustrate the differences between the variances calculated with the two different forms of Pk for vibration-rotation transitions of HCl in the gas phase.

The effect of anisotropic stress and non-adiabatic pressure perturbations on the evolution of the comoving curvature perturbation

We derive the equation for the evolution of the curvature perturbation on the comoving time slice, , in the presence of anisotropic and non-adiabatic terms in the energy–momentum tensor of matter fields. The equation is obtained by manipulating the perturbed Einstein’s equations in the comoving time slice. It could be used to study the evolution of the comoving curvature perturbations for systems with an anisotropic energy–momentum tensor, such as in the presence of vector fields, in the presence of entropy, such as in a multi-field system, or in modified gravity theories. As a simple application, after checking that the comoving time slice for a multi-field system does not coincide with the uniform field time slice in general, we use the equation in the case of two minimally coupled scalar fields and derive a closed set of equations for the curvature and entropy perturbations on the comoving time slice.

Damping of slow magnetoacoustic oscillations by the misbalance between heating and cooling processes in the solar corona

Context. Rapidly decaying slow magnetoacoustic waves are regularly observed in the solar coronal structures, offering a promising tool for a seismological diagnostics of the coronal plasma, including its thermodynamical properties. Aims. The effect of damping of standing slow magnetoacoustic oscillations in the solar coronal loops is investigated accounting for field-aligned thermal conductivity and a wave-induced misbalance between radiative cooling and some unspecified heating rates. Methods. The non-adiabatic terms were allowed to be arbitrarily large, corresponding to the observed values. The thermal conductivity was taken in its classical form, and a power-law dependence of the heating function on the density and temperature was assumed. The analysis was conducted in the linear regime and in the infinite magnetic field approximation. Results. The wave dynamics is found to be highly sensitive to the characteristic timescales of the thermal misbalance. Depending on certain values of the misbalance, timescales three regimes of the wave evolution were identified, namely the regime of a suppressed damping, enhanced damping in which the damping rate drops down to observational values, and acoustic over-stability. The specific regime is determined by the dependences of the radiative cooling and heating functions on thermodynamical parameters of the plasma in the vicinity of the perturbed thermal equilibrium. Conclusions. The comparison of the observed and theoretically derived decay times and oscillation periods allows us to constrain the coronal heating function. For typical coronal parameters, the observed properties of standing slow magnetoacoustic oscillations could be readily reproduced with a reasonable choice of the heating function.

Nonadiabatic transition probabilities in a time-dependent Gaussian pulse or plateau pulse: Toward experimental tests of the differences from Dirac's transition probabilities.

For a quantum system subject to a time-dependent perturbing field, Dirac's analysis gives the probability of transition to an excited state |k⟩ in terms of the norm square of the entire excited-state coefficient ck(t) in the wave function. By integrating by parts in Dirac's equation for ck(t) at first order, Landau and Lifshitz separated ck (1)(t) into an adiabatic term ak (1)(t) that characterizes the gradual adjustment of the ground state to the perturbation without transitions and a nonadiabatic term bk (1)(t) that depends explicitly on the time derivative of the perturbation at times t' ≤ t. Landau and Lifshitz stated that the probability of transition in a pulsed perturbation is given by |bk(t)|2, rather than by |ck(t)|2. We use the term "transition probability" to refer to the probability that a true excited-state component is present in the time-evolved wave function, as opposed to a smooth modification of the initial state. In recent work, we have examined the differences between |bk(t)|2 and |ck(t)|2 when a system is perturbed by a harmonic wave in a Gaussian envelope. We showed that significant differences exist when the frequency of the harmonic wave is off-resonance with the transition frequency. In this paper, we consider Gaussian perturbations and pulses that rise via a half Gaussian shoulder to a level plateau and later return to zero via a down-going half Gaussian. While the perturbation is constant, the transition probability |bk(t)|2 does not change. By contrast, |ck(t)|2 continues to oscillate while the perturbation is constant, and its time averaged value differs from |bk(t)|2. We suggest a general type of experiment to prove that the transition probability is given by |bk(t)|2, not |ck(t)|2. We propose a ratio test that does not require accurate knowledge of transition matrix elements or absolute field intensities.

Coupled-Trajectory Mixed Quantum-Classical Algorithm: A Deconstruction

We analyze a mixed quantum-classical algorithm recently derived from the exact factorization equations [Min, Agostini, Gross, PRL 115, 073001 (2015)] to show the role of the different terms in the algorithm in bringing about decoherence and wavepacket branching. The algorithm has the structure of Ehrenfest equations plus a "coupled-trajectory" term for both the electronic and nuclear equations, and we analyze the relative roles played by the different nonadiabatic terms in these equations, including how they are computed in practice. In particular, we show that while the coupled-trajectory term in the electronic equation is essential in yielding accurate dynamics, that in the nuclear equation has a much smaller effect. A decoherence time is extracted from the electronic equations and compared with that of augmented fewest-switches surface-hopping. We revisit a series of nonadiabatic Tully model systems to illustrate our analysis.

Tunneling criteria and a nonadiabatic term for strong-field ionization

We investigate tunneling ionization of a model helium atom in a strong circularly polarized short laser pulse using the classical backpropagation method and compare ten different tunneling criteria on the same footing, aiming for a consistent classical picture of the tunneling dynamics. These tunneling criteria are categorized into velocity-based, position-based, and energy-based criteria according to different notions of a tunnel exit. We find that velocity-based criteria give consistent tunneling exit characteristics with nonadiabatic effects fully included. Other criteria are either inconsistent or only able to include nonadiabatic effects partially. Furthermore, we construct a simple tunneling rate formula, identify a term in the rate responsible for the nonadiabatic effects, and demonstrate the importance of this term.

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

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.

Recent Advances and Perspectives on Nonadiabatic Mixed Quantum-Classical Dynamics.

Nonadiabatic mixed quantum-classical (NA-MQC) dynamics methods form a class of computational theoretical approaches in quantum chemistry tailored to investigate the time evolution of nonadiabatic phenomena in molecules and supramolecular assemblies. NA-MQC is characterized by a partition of the molecular system into two subsystems: one to be treated quantum mechanically (usually but not restricted to electrons) and another to be dealt with classically (nuclei). The two subsystems are connected through nonadiabatic couplings terms to enforce self-consistency. A local approximation underlies the classical subsystem, implying that direct dynamics can be simulated, without needing precomputed potential energy surfaces. The NA-MQC split allows reducing computational costs, enabling the treatment of realistic molecular systems in diverse fields. Starting from the three most well-established methods-mean-field Ehrenfest, trajectory surface hopping, and multiple spawning-this review focuses on the NA-MQC dynamics methods and programs developed in the last 10 years. It stresses the relations between approaches and their domains of application. The electronic structure methods most commonly used together with NA-MQC dynamics are reviewed as well. The accuracy and precision of NA-MQC simulations are critically discussed, and general guidelines to choose an adequate method for each application are delivered.

Topological Hall Effect from Strong to Weak Coupling

The topological Hall effect (THE) of electrons coupled to a noncoplanar spin texture has been studied so far for the strong- and weak-coupling regimes separately; the former in terms of the Berry phase and the latter by perturbation theory. In this letter, we present a unified treatment in terms of spin gauge field by considering not only the adiabatic (Berry phase) component of the gauge field but also the nonadiabatic component. While only the adiabatic contribution is important in the strong-coupling regime, it is completely canceled by a part of the nonadiabatic contribution in the weak-coupling regime, where the THE is governed by the remaining nonadiabatic terms. We found a new weak-coupling region that cannot be accessed by a simple perturbation theory, where the Hall conductivity is proportional to M, with 2M being the exchange splitting of the electron spectrum.

Density functional theory of electron transfer beyond the Born-Oppenheimer approximation: Case study of LiF.

We perform model calculations for a stretched LiF molecule, demonstrating that nonadiabatic charge transfer effects can be accurately and seamlessly described within a density functional framework. In alkali halides like LiF, there is an abrupt change in the ground state electronic distribution due to an electron transfer at a critical bond length R = Rc, where an avoided crossing of the lowest adiabatic potential energy surfaces calls the validity of the Born-Oppenheimer approximation into doubt. Modeling the R-dependent electronic structure of LiF within a two-site Hubbard model, we find that nonadiabatic electron-nuclear coupling produces a sizable elongation of the critical Rc by 0.5 bohr. This effect is very accurately captured by a simple and rigorously derived correction, with an M-1 prefactor, to the exchange-correlation potential in density functional theory, M = reduced nuclear mass. Since this nonadiabatic term depends on gradients of the nuclear wave function and conditional electronic density, ∇Rχ(R) and ∇Rn(r, R), it couples the Kohn-Sham equations at neighboring R points. Motivated by an observed localization of nonadiabatic effects in nuclear configuration space, we propose a local conditional density approximation-an approximation that reduces the search for nonadiabatic density functionals to the search for a single function y(n).