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Transport of Quantum States of Periodically Driven Systems H.P Transport of quantum states of periodically driven systems H.P. Breuer, K. Dietz, M. Holthaus To cite this version: H.P. Breuer, K. Dietz, M. Holthaus. Transport of quantum states of periodically driven systems. Journal de Physique, 1990, 51 (8), pp.709-722. 10.1051/jphys:01990005108070900. jpa-00212403 HAL Id: jpa-00212403 https://hal.archives-ouvertes.fr/jpa-00212403 Submitted on 1 Jan 1990 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. J. Phys. France 51 (1990) 709-722 15 AVRIL 1990, 709 Classification Physics Abstracts 03.65 - 32.80 - 42.50 Transport of quantum states of periodically driven systems H. P. Breuer, K. Dietz and M. Holthaus Physikalisches Institut, Universität Bonn, Nussallee 12, D-5300 Bonn 1, F.R.G. (Reçu le 17 août 1989, accepté sous forme définitive le 23 novembre 1989) Résumé. 2014 On traite des holonomies quantiques sur des surfaces de quasi-énergie de systèmes soumis à une excitation périodique, et on établit leur topologie globale non triviale. On montre que cette dernière est causée par des transitions diabatiques entre niveaux au passage d’anti- croisements serrés. On donne brièvement quelques conséquences expérimentales concernant le transport adiabatique et les transitions de Landau-Zener entre états de Floquet. Abstract. 2014 We discuss the transport of quantum states on quasi-energy surfaces of periodically driven systems and establish their non-trivial structure. The latter is shown to be caused by diabatic transitions at lines of narrow avoided crossings. Some experimental consequences pertaining to adiabatic transport and Landau-Zener transitions among Floquet states are briefly sketched. 1. Introduction. Periodically driven quantum systems are of very broad physical interest and are experimen- tally readily realised by studying laser/maser - matter interactions. Among the many facets of related phenomena we shall address theoretical problems and their experimental repercussions which specifically pertain to the regime of strong laser/maser - fields ; strong field - matter interaction cannot be described by perturbation theory, non-perturbative methods have to be applied. The particular problem we shall treat in this note is the transport of quantum states [1] by variation of external parameters, i. e. their adiabatic - diabatic motion. In order to get a clear-cut physical picture let us consider a strong single - mode laser field interacting with a quantum system (atom, molecule or, generally speaking, any mesoscopic system) which is described by the following set of Hamiltonians where Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01990005108070900 710 we explicitly indicated the parameter dependence of H : À --+- strength of the laser/maser field to laser/maser frequency e1 cp = arg - phase controlling the transition from linearly to circularly polarised plane -e2 wave fields ; Ho describes the static system, the dipole approximation is not assumed at this state of argument, D is the matter current density. Introducing now the région A in which the external parameters À, (J), cp can be varied (in a given experiment) and we can formulate our problem as follows : a time-variation in an interval of length Ta of external parameters is represented as a curve r in parameter space the problem of quantum transport is now to study the behaviour of states, i.e. of solutions of the Schrôdinger equation under motions on the curve r. A particularly transparent description emerges if we use Floquet states [2, 3] as a basis [4]. We write the general solution of the Schrôdinger equation as a superposition of Floquet states where the « stationary » states Ua (t) obey Of course, ua and the quasi-energies Ea do depend on the parameters R adiabatic motion of the state gi is characterised by the requirement that the probabilities 1 aa 12 in (1.5) remain constant for a sufficiently slow change of parameters along T and are thus independent of R. Speaking in terms of experiments, parameter variations on curves T are well known : the simplest example is a laser/maser pulse where 711 curves necessarily embedded in higher dimensional parameter regions appear if in addition to a varying field strength the frequency cv and/or the phase cp are modulated. Closed curves T are of particular interest if at least two of the three parameters vary in time such that a non- vanishing surface is enclosed. The ad libitum realisation of such curves is doubtlessly an experimental challenge, we believe, however, and hope to convince the reader that new physics can be uncovered in experiments of this type. The requirement of periodicity for the « stationary » states ua in the eigenvalue problem (1.6) entails energy conservation modulo W , i. e. and the quasi-energy spectrum is arranged in Brillouin zones. We assume a discrete spectrum of the static Hamiltonian and anticipate a choice for {e j} such that we then have Nonetheless, the quasi-energy spectrum {Ea} is, in general, dense. For our context this fact has as a consequence that the adiabatic theorem cannot be proven in its original form for periodically driven quantum systems : an essential part in its proof is the requirement that the state to be transported adiabatically be separated in energy from all other states of the system [5]. On the other hand, simple model calculations [6, 7] have clearly demonstrated the validity of adiabatic concepts ; more realistic calculations [8, 9] pertaining to the problem of microwave ionisation of excited hydrogen atoms [10] have shown the importance of adiabatic evolution for the interpretation of experimentally observed features. Numerical evidence of this kind and new experimental support [11] for the concept of diabatic - adiabatic transitions at avoided level crossings may be taken as sound guide - lines in the following argumentation. We start by dividing the quasi-energy spectrum into sets of specific symmetry of states, sets which we denote by { ê a } sym The point is here that levels belonging to the same symmetry class do not cross [12] in the generic case, avoided crossings (AC’s) appear as characteristic patterns. A typical feature of periodically driven systems is that AC’s emerge, in general, densely distributed : the set (s/) sym 8> JL contains dense patterns of AC’s. Not all of these AC’s are dynamically relevant for the description of the parameter motion (1.3). In a recent publication [13] we have described a simple selection procedure : given an experimental set-up and, hence, a time-scale Ta we determine all the AC’s for which 712 with 5 e = level distance in a Brillouin zone OT = time which the system spends in the vicinity of the AC (more precisely speaking OT = 15R 1/1 Ji1 where 1 5R1 is the parameter distance within which the level distance changes by a factor of J2). The levels connected by these AC’s determine a finite dimensional of states in a subspace {ua } eff which, very good approximation, effectively suffices to describe the diabatic-adiabatic motion of quantum states in experiments performed with the very set-up considered : where the aa do depend, in general, on time during the motion (1.3) on the parameter curve r. We are thus led to the discussiôn of a finite set of smooth effective quasi-energy surfaces It so happens that quasi-energy surfaces in this set get very close : they almost touch along curves of, in general, finite length. A cross section of this approach is a very narrow AC, very narrow in the scale set by Ta defined in (1.3) and used for the construction of these narrow AC’s are while on T in the sense that the { Ua } eff’ Dynamically, ignored moving system jumps across from one quasi-energy surface to the adjacent one with probability very close to one. Geometrically speaking, on the other hand, this line of narrow AC’s plays the role of a twist, the two surfaces should be visualised as manifolds cut along the line of narrow AC’s with cross-wise identified borders, the cross-wise structure being due to the purely diabatic nature of transitions at these narrow AC’s. To get this point unequivocally clear : having set the time-scale Ta by specifying experimental conditions we are led to a coarse- graining [13] of the quasi-energy spectrum and thus to the construction of an effective set of states {ua} cff’ narrow AC’s imply diabatic transitions and, hence, cross-identification of quasi-energy surfaces the geometric purport of which we express by taking the two surfaces diabatically connected as two branches of one geometrical object connected in the twisted manner just described. Needless to say, this is again an effective, however, consequent construction translating into geometry the dynamic notions of diabatic-adiabatic and purely adiabatic transitions in periodically driven quantum systems. for a M the set in the Clearly, sufficiently large parameter region {ua} cff will, generic case, appear as one geometrical entity in which all the states ua are connected to all the other states by the above described cut ; the assignment of quantum numbers for its various sheets u,,,, is a matter of convention fixed by the choice of parameters R. In the following, we shall describe the precise construction of such an « universal covering » in some detail and discuss, in particular, connections for adiabatic motion and their implications for generalised Berry phases [14].
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