Open quantum systems i; the hamiltonian approach

Lecture Notes in Mathematics Editors: J.-M Morel, Cachan F Takens, Groningen B Teissier, Paris 1880 S Attal • A Joye • C.-A Pillet (Eds.) Open Quantum Systems I The Hamiltonian Approach BC A www.pdfgrip.com Editors StØphane Attal Institut Camille Jordan Universit ØClaude Bernard Lyon 21 av Claude Bernard 69622 Villeurbanne Cedex France e-mail: attal@math.univ-lyon1.fr Alain Joye Institut Fourier Universit Øde Grenoble BP 74 38402 Saint-Martin d'HŁres Cedex France e-mail: alain.joye@ujf-grenoble.fr Claude-Alain Pillet CPT-CNRS, UMR 6207 UniversitØdu Sud Toulon-Var BP 20132 83957 La Garde Cedex France e-mail: pillet@univ-tln.fr Library of Congress Control Number: 2006923432 Mathematics Subject Classification (2000): 37A60, 37A30, 47A05, 47D06, 47L30, 47L90, 60H10, 60J25, 81Q10, 81S25, 82C10, 82C70 ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 ISBN-10 3-540-30991-8 Springer Berlin Heidelberg New York ISBN-13 978-3-540-30991-8 Springer Berlin Heidelberg New York DOI 10.1007/b128449 This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, 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Heidelberg Printed on acid-free paper SPIN: 11602606 V A 41/3100/ SPI www.pdfgrip.com 543210 Preface This is the rst in a series of three volumes dedicated to the lecture notes of the Summer School Open Quantum Systems which took place at the Institut Fourier in Grenoble from June 16th to July 4th 2003 The contributions presented in these volumes are revised and expanded versions of the notes provided to the students during the School Closed vs Open Systems By denition, the time evolution of aclosedphysical systemS is deterministic It is usually described by a differential equation x t = X (x t ) on the manifoldM of possible congurations of the system If the initial conguration x M is known then the solution of the corresponding initial value problem yields the conguration x t at any future timet This applies to classical as well as to quantum systems In the classical case M is the phase space of the system and x t describes the positions and velocities of the various components (or degrees of freedom) S at of timet In the quantum case, according to the orthodox interpretation of quantum mechanics, M is a Hilbert space and x t a unit vector — the wave function — describing the quantum state of the system at time t In both cases the knowledge of the state x t allows to predict the result of any measurement madeSon at timet Of course, what we mean by the result of a measurement depends on whether the system is classical or quantum, but we should not be concerned with this distinction here The only S relevant point is thatx t carries the maximal amount of information on the system at timet which is compatible with the laws of physics In principle any physical system S that is not closed can be considered as part of a larger but closed system It sufces to consider with S the setR of all systems which interact, in a way or another, with S The joint systemS R is closed and from the knowledge of its state x t at timet we can retrieve all the information on its subsystemS In this case we say that the system S is openand thatR is its environment.There are however some practical problems with this simple picture Since the joint systemS R can be really big e.g.,the ( entire universe) it may be difcult, if not impossible, to write down its evolution equation There is no solution to www.pdfgrip.com VI Preface this problem The pragmatic way to bypass it is to neglect parts of the environment R which, a priori, are supposed to be of negligible effect on the evolution of the subsystemS For example, when dealing with the motion of a charged particle it is often reasonable to neglect all but the electromagnetic interactions and suppose that the environment consists merely in the electromagnetic eld Moreover, if the particle moves in a very sparse environment like intergalactic space then we can consider that it is the only source in the Maxwell equations which governs the evolution of R Assuming that we can write down and solve the evolution equation of the joint systemS R we nevertheless hit a second problem: how to choose the initial conguration of the environment ? R If has a very largee.g.,innite) ( number of degrees of freedom then it ispractically impossible to determine its conguration at some initial time t = Moreover, the dynamics of the joint system is very likely to be chaotic,i.e., to display some sort of instability or sensitive dependence on the initial condition The slightest error in the initial conguration will be rapidly amplied and ruin our hope to predict the state of the system at some later time Thus, instead of specifying a single initial conguration ofR we should provide a statistical ensemble of typical congurations Accordingly, the best we can hope for is a statistical information on the state of our open system S at some later timet The resulting theory of open systems is intrinsically probabilistic It can be considered as a part of statistical mechanics at the interface with the ergodic theory of stochastic processes and dynamical systems The paradigm of this statistical approach to open systems is the theory of Brownian motion initiated by Einstein in one of his celebrated 1905 papers [3] (see also [4] for further developments) An account on this theory can be found in almost any textbook on statistical mechanics (see for example [9]) Brownian motion had a deep impact not only on physics but also on mathematics, leading to the development of the theory of stochastic processes (see for example [12]) Open systems appeared quite early in the development of quantum mechanics Indeed, to explain the nite lifetime of the excited states of an atom and to compute the width of the corresponding spectral lines it is necessary to take into account the interaction of the electrons with the electromagnetic eld Einsteins seminal paper [5] on atomic radiation theory can be considered as the rst attempt to use a Markov process — or more precisely a master equation — to describe the dynamics of a quantum open system The theory of master equations and its application to radiation theory and quantum statistical mechanics was subsequently developed by Pauli [8], Wigner and Weisskopf [13], and van Hove [11] The mathematical theory of the quantum Markov semigroups associated with these master equations started to develop more than 30 years later, after the works of Davies [2] and Lindblad [7] It further led to the development of quantum stochastic processes To illustrate the philosophy of the modern approach to open systems let us consider a simple, classical, microscopic model of Brownian motion Even though this model is not realistic from a physical point of view it has the advantage of being exactly solvable In fact such models are often used in the physics literature (see [10, 6, 1]) www.pdfgrip.com Preface VII Brownian Motion: A Simple Microscopic Model In a cubic crystal denote by qx the deviation of an atom from its equilibrium position = { N, , N } Z and bypx the corresponding momentum Suppose x N that the inter-atomic forces are harmonic and only acts between nearest neighbors of the crystal lattice In appropriate units the Hamiltonian of the crystal is then p2x + x N xy Z3 x,y qy ) , (qx where = xy if |x y| = 1; otherwise; and Dirichlet boundary conditions are imposed by setting qx = for x If the atom at sitex = is replaced by a heavy impurity of mass M Hamiltonian becomes H x N p2x + 2m x xy x,y Z3 qy ) , (qx Z3 \ N then the where mx = M if x = 0; otherwise We shall consider the heavy impurity xat = as an open system S whose environmentR is made of the(2N +1) remaining atoms of the crystal To write down the equation of motion in a convenient form let us introduce some notation We set N = N \ { 0} , q = ( qx ) x N , p = ( px ) x N , Q = q0 , P = p0 For x Z we denote by x the Kronecker delta function at x and by|x| the Euclidean norm of S R is governed by x We also set = |x |=1 x The motion of the joint system the following linear system q = p, p= M Q = P, P = 0q + 0Q Q, (1) + ( , q ), 2 where is the discrete Dirichlet Laplacian onN and = According to the open system philosophy described in the previous paragraph we should supply some appropriate statistical ensemble of initial states of the environment To motivate the choice of this ensemble suppose that in the remote past the impurity was pinned at some xed position, sayQ = P = , and that at timet = the resulting system has reached thermal equilibrium at some temperature T > The positions and momenta in the crystal will be distributed according to the Gibbs-Boltzmann canonical ensemble corresponding to the pinned Hamiltonian H = H |Q = P =0 , H0 = (p, p) + ( q, www.pdfgrip.com q) VIII Preface This ensemble is given by the Gaussian measure d = Z H ( q,p) e dqdp, whereZ is a normalization factor and = /k B T with kB the Boltzmann constant At time t = we release the impurity The subsequent evolution of the system is determined by the Cauchy problem for Equ (1) The evolution of the environment can be expressed by means of the Duhamel formula q(t) = cos( t)q(0) + t) sin( t sin( p(0) + 0 (t s)) Q (s) ds 0 Inserting this relation into the equation of motion for Q leads to Ô= MQ t 0Q + K (t s)Q(s) ds + (t), (2) where the integral kernel K is given by K (t) = ( , sin( t) ), (3) and (t) = , cos( t)q(0) + t) sin( p(0) Sinceq(0), p(0) are jointly Gaussian random variables, (t) is a Gaussian stochastic process It is a simple exercise to compute its mean and covariance E( (t)) = , E( (t) (s)) = C(t s) = (, cos( (t s)) ) (4) We note in particular that this process is stationary The term (t) in Equ (2) is the noise generated by the uctuations of the environment It vanishes if the environment is initially at rest The integral in Equ (2) is the force exerted by the environment on the impurity in reaction to its motion Note that this dissipative term is independent of the state of the environment The dissipative and the uctuating forces are related by the so calleductuation-dissipation theorem K (t) = t C(t) (5) The solutionzt = ( Q(t), P (t)) of the random integro-differential equation (2) denes a family of stochastic processes indexed by the initial condition z0 These processes provide a statistical description of the motion of our open system An inR3 R3 such that variant measure for the processzt is a measure on f (zt ) d (z0 ) = f (z) d (z), www.pdfgrip.com Preface IX holds for all reasonable functions f and all t R Such a measure describes a steady state of the system If one can show that for any initial distribution which is absolutely continuous with respect to Lebesgue measure one has t lim f (zt ) d (z0 ) = f (z) d (z), (6) then the steady state provides a good statistical description of the dynamics on large time scales One of the main problem in the theory of open systems is to show that such a natural steady state exists and to study its properties The Hamiltonian Approach Remark that in our example, such a steady state fails to exist since the motion of the joint system is clearly quasi-periodic However, in a real situation the number of • 1023 atoms in the crystal is very large, of the order of Avogadros number NA In this case the recurrence time of the system becomes so large that it makes sense to take the limitN In this limit becomes the discrete Dirichlet Laplacian on the innite lattice Z \ { 0} This is a well dened, bounded, negative operator on the Hilbert space2 (Z ) Thus, Equ (2),(3), (4) and (5) still make sense in this limit In the sequel we only consider the resulting innite system We distinguish two main approaches to the study of open systems The rst one, the Hamiltonian approach, deals directly with the dynamics of the joint system SR We briey discuss the second one, the Markovian approach, in the next paragraph In the Hamiltonian approach we rewrite the equation of motion (1) as Z = i Z, where = m 1/ 2 m 1/ with m = I +( M 1) ( , •) the operator of multi2 is the discrete Laplacian on Z The complex variable Z is plication bym x and 1/ m 1/ p andq = ( qx ) x Z , p = ( px ) x Z It folgiven byZ = 1/ m 1/ q+ i lows from elementary spectral analysis thatM for> the operator is self-adjoint with purely absolutely continuous spectrum ( ) = ac ( ) = [0 , ] on (Z ) 2 shows A simple argument involving the scattering theory for the pair 0 /M , that the systemS has a unique steady statesuch that (6) holds for all which are absolutely continuous with respect to Lebesgue measure Moreover, is the marginal on S of the innite dimensional Gaussian measure Z e H dpdqdP dQwhich describes the thermal equilibrium state of the joint system at temperature T = /k B This is easily computed to be the Gaussian measure (dP, dQ) = N e ( P / 2M + whereN is a normalization factor and = ( 0, 0) www.pdfgrip.com Q / 2) dP dQ, X Preface The Markovian Approach A remarkable feature of Equ (2) is the memory effect induced by the kernel K As a result the process zt is non-Markovian,i.e., for s > 0, zt + s does not only depend on zt and{ (u) | u [t, t + s]} but also on the full history{ zu | u [0, t]} The only way to avoid this effect is to have K proportional to the derivative of a delta function By Relation (5) this means thatshould be a white noise This is certainly not the case with our choice of initial conditions However, as we shall see, it is possible to obtain a Markov process in some particular scaling limits This is not a uniquely dened procedure: different scaling limits correspond to different physical regimes and lead to distinct Markov processes As a simple illustration let us consider the particular scaling limit M 1/ Q(M 1/ t), QM (t) M of our model For nite M the equation of motion forQM reads t ÔM (t) = Q QM (t) + K M (t s)QM (s) ds + M (t), where K M (t) M 1/ K (M 1/ t), M 1/ (M 1/ t) has covariance and the scaled process M (t) CM (t) M 1/ C(M 1/ t) One can show that C(t) is in L (R) and that = distributional sense, lim CM (t) = M (t), C(t) dt > It follows that, in lim K M (t) = M We conclude that the limiting equation for Q is Ô = Q(t) Q(t) 1/ + where is white noise,i.e., E( (t) (s)) = (t Markov process onR3 R3 with generator L = 2 P P• Q (t), s) The solution(Q(t), Q(t)) is a + 0Q • P It is a simple exercise to show that the unique invariant measure of this process is the Lebesgue measure Moreover, one can show that for any initial condition (Q0 , P0 ) and any functionf L (R3 R3 , dQdP) one has t lim E(f (Q(t), Q(t))) = f (Q, P ) dQdP, a scaled version of return to equilibrium www.pdfgrip.com Preface XI It is worth pointing out that in many instances of classical or quantum open systems the dynamics of the joint system S R is too complicated to be controlled analytically or even numerically Thus, the Hamiltonian approach is inefcient and the Markovian approximation becomes the only available option The study of the Markovian dynamics of open systems is the main subject of the second volume in this series The third volume is devoted to applications of the techniques introduced in the rst two volumes It aims at leading the reader to the front of the current research on open quantum systems Organization of this Volume This rst volume is devoted to the Hamiltonian approach Its purpose is to develop the mathematical framework necessary to dene and study the dynamics and thermodynamics of quantum systems with innitely many degrees of freedom The rst two lectures by A Joye provide a minimal background in operator theory and statistical mechanics The third lecture by S Attal is an introduction to the theory of operator algebras which is the natural framework for quantum mechanics of many degrees of freedom Quantum dynamical systems and their ergodic theory are the main object of the fourth lecture by C.-A Pillet The fth lecture by M Merkli deals with the most common instances of environments in quantum physics, the ideal Bose and Fermi gases Finally the last lecture by V si« cJak introduces one of the main tool in the study of quantum dynamical systems: spectral analysis Lyon, Grenoble and Toulon, September 2005 St« ephane Attal Alain Joye Claude-Alain Pillet www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com www.pdfgrip.com ... the study of open systems The rst one, the Hamiltonian approach, deals directly with the dynamics of the joint system SR We briey discuss the second one, the Markovian approach, in the next paragraph... in the rst two volumes It aims at leading the reader to the front of the current research on open quantum systems Organization of this Volume This rst volume is devoted to the Hamiltonian approach. .. mechanics at the interface with the ergodic theory of stochastic processes and dynamical systems The paradigm of this statistical approach to open systems is the theory of Brownian motion initiated