We generalize the standard quantum adiabatic approximation to the case of open quantum systems. We define the adiabatic limit of an open quantum system as the regime in which its dynamical superoperator can be decomposed in terms of independently evolving Jordan blocks. We then establish validity and invalidity conditions for this approximation and discuss their applicability to superoperators changing slowly in time. As an example, the adiabatic evolution of a two-level open system is analysed.
We formulate an adiabatic approximation for the imaginary-time Schroedinger equation. The obtained adiabatic condition consists of two inequalities, one of which coincides with the conventional adiabatic condition for the real-time Schroedinger equation, but the other does not. We apply this adiabatic approximation to the analysis of Markovian dynamics of the classical Ising model, which can be formulated as the imaginary-time Schrödinger equation, to obtain an asymptotic formula for the probability that the system reaches the ground state in the limit of a long annealing time in simulated ann…
In this paper, we present an invariant perturbation theory of the adiabatic process based on the concepts of U(1)-invariant adiabatic orbit and U(1)-invariant adiabatic expansion. As its application, we propose and discuss new adiabatic approximation conditions.
A simple proof of quantum adiabatic theorem is provided. Quantum adiabatic approximation is divided into two kinds. For Hamiltonian H(t/T), a relation between the size of the error caused by quantum adiabatic approximation and the parameter T is given.
Since the discovery of adiabatic quantum computing, a need has arisen for rigorously proven bounds for the error in the adiabatic approximation. We present in this paper, a rigorous and elementary derivation of upper and lower bounds on the error incurred from using the adiabatic approximation for quantum systems. Our bounds are often asymptotically tight in the limit of slow evolution for fixed Hamiltonians, and are used to provide sufficient conditions for the application of the adiabatic approximation. We show that our sufficiency criteria exclude the Marzlin--Sanders counterexample from th…
We show how a method inspired in renormalization group techniques can be useful for deriving Hamiltonians in the adiabatic approximation in a systematic way.
Adiabatic approximation for quantum evolution is investigated quantitatively with addressing its dependence on the Berry connections. We find that, in the adiabatic limit, the adiabatic fidelity may uniformly converge to unit or diverge manifesting the breakdown of adiabatic approximation, depending on the type of the singularity of the Berry connections as the functions of slowly-varying parameter $R$. When the Berry connections have a singularity of $1/R^σ$ type with $σ< 1$, the adiabatic fidelity converges to unit in a power-law; whereas when the singularity index $σ$ is larger than one, ad…
The adiabatic approximation in open systems is formulated through the effective Hamiltonian approach. By introducing an ancilla, we embed the open system dynamics into a non-Hermitian quantum dynamics of a composite system, the adiabatic evolution of the open system is then defined as the adiabatic dynamics of the composite system. Validity and invalidity conditions for this approximation are established and discussed. A High-order adiabatic approximation for open systems is introduced. As an example, the adiabatic condition for an open spin-$\frac 1 2$ particle in time-dependent magnetic fiel…
Recently there have been some controversies about the criterion of the adiabatic approximation. It is shown that an approximate diagonalization of the effective Hamiltonian in the second quantized formulation gives rise to a reliable and unambiguous criterion of the adiabatic approximation. This is illustrated for the model of Marzlin and Sanders and a model related to the geometric phase which can be exactly diagonalized in the present sense.
An alternative interpretation of the quantum adiabatic approximation is presented. This interpretation is based on the ideas originally advocated by David Bohm in his quest for establishing a hidden variable alternative to quantum mechanics. It indicates that the validity of the quantum adiabatic approximation is a sufficient condition for the separability of the quantum action function in the time variable. The implications of this interpretation for Berry's adiabatic phase and its semi-classical limit are also discussed.
In this paper, we discuss time evolution and adiabatic approximation in $PT$-symmetric quantum mechanics. we give the time evolving equation for a class of $PT$-symmetric Hamiltonians and some conditions of the adiabatic approximation for the class of $PT$-symmetric Hamiltonians.
Marzlin and Sanders \cite{marzlin} have shown rigorously that the adiabatic approximation can be very inaccurate when applied to a Hamiltonian $H(t)$ that generates the evolution $U^{\dagger} (t)$ even if it gives an excellent approximation to the evolution $U(t)$ generated by a dual Hamiltonian $h(t)$. We show that this is not inconsistent with the adiabatic theorem and find that in general even if $h(t)$ satisfies the conditions of the adiabatic theorem, $H(t)$ will likely violate those conditions.
We present straightforward proofs of estimates used in the adiabatic approximation. The gap dependence is analyzed explicitly. We apply the result to interpolating Hamiltonians of interest in quantum computing.
We give a sufficient condition for the quantum adiabatic approximation, which is quantitative and can be used to estimate error caused by this approximation. We also discuss when the traditional condition is sufficient.
We analyze the validity of the adiabatic approximation, and in particular the reliability of what has been called the "standard criterion" for validity of this approximation. Recently, this criterion has been found to be insufficient. We will argue that the criterion is sufficient only when it agrees with the intuitive notion of slowness of evolution of the Hamiltonian. However, it can be insufficient in cases where the Hamiltonian varies rapidly but only by a small amount. We also emphasize the distinction between the adiabatic {\em theorem} and the adiabatic {\em approximation}, two quite di…
The problem of spin precession in a time-dependent magnetic field is considered in the adiabatic approximation where the field direction or the angular velocity of its rotation is changing slowly. The precession angles are given by integrals in a way similar to the semi-classical approximation for the Schrödinger equation.
In this letter, we point out that the widely used quantitative conditions in the adiabatic theorem are insufficient in that they do not guarantee the validity of the adiabatic approximation. We also reexamine the inconsistency issue raised by Marzlin and Sanders (Phys. Rev. Lett. 93, 160408, 2004) and elucidate the underlying cause.
The adiabatic theorem shows that the instantaneous eigenstate is a good approximation of the exact solution for a quantum system in adiabatic evolution. One may therefore expect that the geometric phase calculated by using the eigenstate should be also a good approximation of exact geometric phase. However, we find that the former phase may differ appreciably from the latter if the evolution time is large enough.
We introduce a perturbative approach to solving the time dependent Schroedinger equation, named adiabatic perturbation theory (APT), whose zeroth order term is the quantum adiabatic approximation. The small parameter in the power series expansion of the time-dependent wave function is the inverse of the time it takes to drive the system's Hamiltonian from the initial to its final form. We review other standard perturbative and non-perturbative ways of going beyond the adiabatic approximation, extending and finding exact relations among them, and also compare the efficiency of those methods aga…
The usual quantitative condition has been widely used in the practical applications of the adiabatic theorem. However, it had never been proved to be sufficient or necessary before. It was only recently found that the quantitative condition is insufficient, but whether it is necessary remains unresolved. In this letter, we prove that the quantitative condition is necessary in guaranteeing the validity of the adiabatic approximation.
This paper shows that WKB wave function can be expressed in the form of an adiabatic expansion. To build a bridge between two widely invoked approximation schemes seems pedagogically instructive. Further "cubic-WKB" method that has been devised in order to overcome the divergence problem of WKB can be also presented in the form of an adiabatic approximation: The adiabatic expansion of a wave function contains a certain parameter. When this parameter is adjusted so as to make the next order correction vanish approximately, the adiabatic wave function becomes equivalent to that of the "cubic-WKB…
We investigate the origin of quantum geometric phases, gauge fields and forces beyond the adiabatic regime. In particular, we extend the notions of geometric magnetic and electric forces discovered in studies of the Born-Oppenheimer approximation to arbitrary quantum systems described by matrix valued quantum Hamiltonians. The results are illustrated by several physical relevant examples.
Recently, the authors of Ref.1[arXiv:1004.3100] claimed that they have proven the traditional adiabatic condition is a necessary condition. Here, it is claimed that there are some mistakes and an artificial over-strong constraint in [1], making its result inconvincible.
Non-adiabatic transitions in multilevel systems appear in various fields of physics, but it is not easy to analyze their dynamics in general. In this paper, we propose to extend the adiabatic impulse approximation to multilevel systems. This approximation method is shown to be equivalent to a series of unitary evolutions and facilitates to evaluate the dynamics numerically. In particular, we analyze the dynamics of the Landau-Zener grid model and the multilevel Landau-Zener-Stückelberg-Majorana interference model, and confirm that the results are in good agreement with the exact dynamics evalu…
In various applications one is interested in quantum dynamics at intermediate evolution times, for which the adiabatic approximation is inadequate. Here we develop a quasi-adiabatic approximation based on the WKB method, designed to work for such intermediate evolution times. We apply it to the problem of a single qubit in a time-varying magnetic field, and to the Hamiltonian Grover search problem, and show that already at first order, the quasi-adiabatic WKB captures subtle features of the dynamics that are missed by the adiabatic approximation. However, we also find that the method is sensit…
We present a system composed of two flux qubits and a transmission-line resonator. Instead of using the rotating wave approximation (RWA), we analyse the system by the adiabatical approximation methods under two opposite extreme conditions. Basic properties of the system are calculated and compared under these two different conditions. Energy-level spectrum of the system in the adiabatical displaced oscillator basis is shown, and the theoretical result is compared with the numerical solution.
A relativistic analogue of the quantum adiabatic approximation is developed for Klein-Gordon fields minimally coupled to electromagnetism, gravity and an arbitrary scalar potential. The corresponding adiabatic dynamical and geometrical phases are calculated. The method introduced in this paper avoids the use of an inner product on the space of solutions of the Klein-Gordon equation. Its practical advantages are demonstrated in the analysis of the relativistic Landau level problem and the rotating cosmic string.
A gapped quantum system that is adiabatically perturbed remains approximately in its eigenstate after the evolution. We prove that, for constant gap, general quantum processes that approximately prepare the final eigenstate require a minimum time proportional to the ratio of the length of the eigenstate path to the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We also give a necessary condition for the adiabatic approximation that depends on local properties of the path, which is appropriate when the gap varies.
We examine the validity of the rotating wave approximation (RWA) in non-adiabatic holonomic single-qubit gates [New J. Phys. {\bf 14}, 103035 (2012)]. We demonstrate that the adoption of RWA may lead to a sharp decline in fidelity for rapid gate implementation and small energy separation between the excited and computational states. The validity of the RWA in the recent experimental realization [Nature (London) {\bf 496}, 482 (2013)] of non-adiabatic holonomic quantum computation for a superconducting qubit is examined.
We generalize the adiabatic approximation to the case of open quantum systems, in the joint limit of slow change and weak open system disturbances. We show that the approximation is ``physically reasonable'' as under wide conditions it leads to a completely positive evolution, if the original master equation can be written on a time-dependent Lindblad form. We demonstrate the approximation for a non-Abelian holonomic implementation of the Hadamard gate, disturbed by a decoherence process. We compare the resulting approximate evolution with numerical simulations of the exact equation.
We examine the quantitative condition which has been widely used as a criterion for the adiabatic approximation but was recently found insufficient. Our results indicate that the usual quantitative condition is sufficient for a special class of quantum mechanical systems. For general systems, it may not be sufficient, but it along with additional conditions is sufficient. The usual quantitative condition and the additional conditions constitute a general criterion for the validity of the adiabatic approximation, which is applicable to all $N-$dimensional quantum systems. Moreover, we illustrat…
I revisit a longstanding question in quantum optics; When is the rotating wave approximation justified? In terms of the Jaynes-Cummings and Rabi models I demonstrate that the approximation in general breaks down in the adiabatic limit regardless of system parameters. This is explicitly shown by comparing Berry phases of the two models, where it is found that this geometrical phase is strictly zero in the Rabi model contrary to the non-trivial Berry phase of the Jaynes-Cummings model. The source of this surprising result is traced back to different topologies in the two models.
The use of the adiabatic approximation in practical applications, as in adiabatic quantum computation, demands an assessment of the errors made in finite-time evolutions. Aiming at such scenarios, we derive bounds relating error and evolution time in the adiabatic approximation that go beyond typical scaling relations. Using the Adiabatic Perturbation Theory, we obtain leading-order expressions valid for long evolution time $T$, while explicitly determining the shortest time $T$ and the largest error $\varepsilon$ for which they are valid. In this validity regime, we can make clear and precise…
For adiabatic controls of quantum systems, the non-adiabatic transitions are reduced by increasing the operation time of processes. Perfect quantum adiabaticity usually requires the infinitely slow variation of control parameters. In this paper, we propose the dynamical quantum geometric tensor, as a metric in the control parameter space, to speed up quantum adiabatic processes and reach quantum adiabaticity in relatively short time. The optimal protocol to reach quantum adiabaticity is to vary the control parameter with a constant velocity along the geodesic path according to the metric. For …
We give a mathematical procedure to obtain the adiabatic approximation for the generalized quantum Rabi Hamiltonian both without and with a quadratic interaction. We consider the Hamiltonian as the energy of a model describing the interaction system of a two-level artificial atom and a one-mode microwave photon in circuit QED. In the case without the quadratic interaction, we show in the adiabatic approximation that whether each bare eigenstate forms a Schrodinger-cat-like entangled state or not depends on whether the energy bias of the atom is zero or non-zero, and then, the effect of the tun…
Recently, Marzlin and Sanders (2004) demonstrated an inconsistency when the adiabatic approximation was applied to specific, "inverse" time-evolving systems. Following that, Tong et al. (2005) showed that the widely used traditional adiabatic conditions are insufficient to guarantee the validity of the adiabatic approximation for this class of systems. In this article we explore the origin of these observations by a perturbative approach and find that in first order approximation certain nonzero terms appear in the solution which gives rise to the breakdown of the adiabatic approximation (desp…
In this paper we study up to which extent we can apply adiabatic control strategies to a quantum control model obtained by rotating wave approximation. In particular, we show that, under suitable assumptions on the asymptotic regime between the parameters characterizing the rotating wave and the adiabatic approximations, the induced flow converges to the one obtained by considering the two approximations separately and by combining them formally in cascade. As a consequence, we propose explicit control laws which can be used to induce desired populations transfers, robustly with respect to par…
The scattering properties of any complex scattering potential, v:R -> C, can be obtained from the dynamics of a particular non-unitary two-level quantum system S_v. The application of the adiabatic approximation to S_v yields a semiclassical treatment of the scattering problem. We examine the adiabatic series expansion for the evolution operator of S_v and use it to obtain corrections of arbitrary order to the semiclassical formula for the transfer matrix of v. This results in a high-energy approximation scheme that unlike the semiclassical approximation can be applied for potentials with larg…
We consider a finite-time quantum Otto cycle with single and two-spin-$1/2$ systems as its working medium. In order to mimic adiabatic dynamics at a finite-time, we employ a shortcut-to-adiabaticity technique and evaluate the performance of the engine including the cost of the shortcut. We compare our results with the true adiabatic and non-adiabatic performances of the same cycle. Our findings clearly indicate that the use of shortcut-to-adiabaticity scheme significantly enhances the performance of the quantum Otto engine as compared to its adiabatic and non-adiabatic counterparts for differe…
We study the adiabatic-impulse approximation (AIA) as a tool to approximate the time evolution of quantum states, when driven through a region of small gap. The AIA originates from the Kibble-Zurek theory applied to continuous quantum phase transitions. The Kibble-Zurek mechanism was developed to predict the power-law scaling of the defect density across a continuous quantum phase transition. Instead here, we quantify the accuracy of the AIA via the trace norm distance with respect to the exact evolved state. As expected, we find that for short times/fast protocols, the AIA outperforms the sim…
In multi-level systems, the commonly used adiabatic elimination is a method for approximating the dynamics of the system by eliminating irrelevant, non-resonantly coupled levels. This procedure is, however, somewhat ambiguous and it is not clear how to improve on it systematically. We use an integro-differential equation for the probability amplitudes of the levels of interest, which is equivalent to the original Schrodinger equation for all probability amplitudes. In conjunction with a Markov approximation, the integro-differential equation is then used to generate a hierarchy of approximatio…
This report presents a new approach for treating the coupling of electrons and nuclei in quantum mechanical calculations for molecules and condensed matter. It includes the standard "Born-Oppenheimer approximation" as a special case but treats both adiabatic and non-adiabatic corrections using perturbation theory. The adiabatic corrections include all terms that do not explicitly involve the nuclear wavefunctions, so that the nuclei move on a single electronic potential surface. The non-adiabatic corrections, which allow the nuclei to move on more than one potential surface, include coupling b…
We consider the dynamics of a massless scalar field with time-dependent sources in the adiabatic limit. This is an example of an adiabatic problem without spectral gap. The main goal of our paper is to illustrate the difference between the error of the adiabatic approximation and the concept of non-adiabatic transitions for gapless systems. In our example the non-adiabatic transitions correspond to emission of free bosons, while the error of the adiabatic approximation is dominated by a velocity-dependent deformation of the ground state of the field. In order to capture these concepts precisel…
The asymmetric quantum Rabi model (AQRM), which describes the interaction between a quantum harmonic oscillator and a biased qubit, arises naturally in circuit quantum electrodynamic circuits and devices. The existence of hidden symmetry in the AQRM leads to a rich energy landscape of conical intersections (CIs) and thus to interesting topological properties. However, current approximations to the AQRM fail to reproduce these CIs correctly. To overcome these limitations we propose a generalized adiabatic approximation (GAA) to describe the energy spectrum of the AQRM. This is achieved by combi…
We show that in a quantum adiabatic evolution, even though the adiabatic approximation is valid, the total phase of the final state indicated by the adiabatic theorem may evidently differ from the actual total phase. This invalidates the application of the linearity and the adiabatic approximation simultaneously. Besides, based on this observation we point out the mistake in the traditional proof for the adiabatic theorem. This mistake is the root of the troubles that the adiabatic theorem has met. We also show that a similar mistake remains in some recent modifications of the traditional adia…
A new and intuitive perturbative approach to time-dependent quantum mechanics problems is presented, which is useful in situations where the evolution of the Hamiltonian is slow. The state of a system which starts in an instantaneous eigenstate of the initial Hamiltonian is written as a power series which has a straightforward diagrammatic representation. Each term of the series corresponds to a sequence of "adiabatic" evolutions, during which the system remains in an instantaneous eigenstate of the Hamiltonian, punctuated by transitions from one state to another. The first term of this series…
We present a detailed semiclassical study on the propagation of a pair of optical fields in resonant media with and without adiabatic approximation. In the case of near and on resonance excitation, we show detailed calculation, both analytically and numerically, on the extremely slowly propagating probe pulse and the subsequent regeneration of a pulse via a coupling laser. Further discussions on the adiabatic approximation provide many subtle understandings of the process including the effect on the band width of the regenerated optical field. Indeed, all features of the optical pulse regenera…
In this paper, we discuss the compatibility between the rotating-wave and the adiabatic approximations for controlled quantum systems. Although the paper focuses on applications to two-level quantum systems, the main results apply in higher dimension. Under some suitable hypotheses on the time scales, the two approximations can be combined. As a natural consequence of this, it is possible to design control laws achieving transitions of states between two energy levels of the Hamiltonian that are robust with respect to inhomogeneities of the amplitude of the control input.
The time-evolution operator for an explicitly time-dependent Hamiltonian is expressed as the product of a sequence of unitary operators. These are obtained by successive time-dependent unitary transformations of the Hilbert space followed by the adiabatic approximation at each step. The resulting adiabatic product expansion yields a generalization of the quantum adiabatic approximation. Furthermore, it leads to an infinite class of exactly solvable models.
We present a detailed derivation and numerical tests of a new mixed quantum-classical scheme to deal with non-adiabatic processes. The method is presented as the zero-th order approximation to the exact coupled dynamics of electrons and nuclei offered by the factorization of the electron-nuclear wave function [A. Abedi, N. T. Maitra and E. K. U. Gross, Phys. Rev. Lett., 105 (2010)]. Numerical results are presented for a model system for non-adiabatic charge transfer in order to test the performance of the method and to validate the underlying approximations.
The adiabatic approximation exhibits wide applicability in quantum mechanics, providing a simple approach for nontransitional dynamics in quantum systems governed by slowly varying time-dependent Hamiltonians. However, the standard adiabatic theorem is specifically derived for closed quantum systems. In a realistic open system scenario, the inevitable system-reservoir interaction must be taken into account, which strongly impacts the generalization of the adiabatic behavior. In this paper, we introduce sufficient conditions for the adiabatic approximation in open quantum systems. These conditi…
The quantum Rabi model (QRM) describes the interaction between a two-level system (qubit) and a quantum harmonic oscillator. In the limit where the qubit frequency is smaller than the harmonic frequency, the QRM can be well approximated by the adiabatic approximation (AA). The AA is widely used due to its simplicity and explicit physical interpretation. However, the level crossings in the spectrum of the QRM predicted by the AA are determined by the zeros of Laguerre polynomials, which deviate from the exact points. We propose a new approximation to the QRM that predicts the level crossings co…
In this letter we give a systematic derivation and justification of the semiclassical model for the slow degrees of freedom in adiabatic slow-fast systems first found by Littlejohn and Flynn [5]. The classical Hamiltonian obtains a correction due to the variation of the adiabatic subspaces and the symplectic form is modified by the curvature of the Berry connection. We show that this classical system can be used to approximate quantum mechanical expectations and the time-evolution of operators also in sub-leading order in the combined adiabatic and semiclassical limit. In solid state physics t…
We discuss applications of shortcuts to adiabaticity (STA) to adiabatic quantum computation. After reviewing the fundamental properties and the present status of STA from the author's personal point of view, we apply the method to the adiabatic algorithm of the Grover's problem. We discuss two possible implementations of STA for the adiabatic quantum computations: the method of quantum adiabatic brachistochrone and the Lewis-Riesenfeld invariant-based inverse engineering.
We examine and explain the stability properties of the ``atom diode'', a laser device that lets the ground state atom pass in one direction but not in the opposite direction. The diodic behavior and the variants that result by using different laser configurations may be understood with an adiabatic approximation. The conditions to break down the approximation, which imply also the diode failure, are analyzed.
We show that the quantum Fisher information attained in an adiabatic approach to critical quantum metrology cannot lead to the Heisenberg limit of precision and therefore regular quantum metrology under optimal settings is always superior. Furthermore, we argue that even though shortcuts to adiabaticity can arbitrarily decrease the time of preparing critical ground states, they cannot be used to achieve or overcome the Heisenberg limit for quantum parameter estimation in adiabatic critical quantum metrology. As case studies, we explore the application of counter-diabatic driving to the Landau-…
Duality is considered for the perturbation theory by deriving, given a series solution in a small parameter, its dual series with the development parameter being the inverse of the other. A dual symmetry in perturbation theory is identified. It is then shown that the dual to the Dyson series in quantum mechanics is given by a recent devised series having the adiabatic approximation as leading order. A simple application of this result is given by rederiving a theorem for strongly perturbed quantum systems.
