DOCSLIB.ORG

A View on Stochastic Differential Equations Derived from Quantum Optics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A View on Stochastic Differential Equations Derived from Quantum Optics A VIEW ON STOCHASTIC DIFFERENTIAL EQUATIONS DERIVED FROM QUANTUM OPTICS FRANCO FAGNOLA AND ROLANDO REBOLLEDO Quantum Optics has had an impressive development during the last two decades. Laser–based devices are invading our every–day life: printers, pointers, a new surgical technology replacing the old scalpel, transport of information, among many other applications which give the idea of a well established theory. Even though, various theoretical problems remain open. In particular Quantum Optics continues to inspire mathematicians in their own research. This survey is aimed at giving a taste of some mathematical problems raised by Quantum Optics within the field of Stochastic Analysis. Espe- cially, we want to show the interplay between classical and non commutative stochastic differential equations associated to the so called master equations of Quantum Optics. Contents 1. Introduction 2 1.1. Notations 3 1.2. Deriving the master equation 4 2. The associated Quantum Stochastic Differential Equation 7 2.1. Quantum noises in continuous time 8 2.2. The master equation for the quantum cocycle 9 2.3. The Quantum Markov Flow 10 3. Unravelling: classical stochastic processes related to quantum flows 12 3.1. The algebra generated by the number operator 12 3.2. Position, momentum and their algebras 13 3.3. The evolution on AN , Aq, Ap. 15 4. Stationary state and the convergence towards the equilibrium 16 References 18 Research partially supported by FONDECYT grant 1960917 and the “C´atedraPresi- dencial en An´alisis Estoc´astico y F´ısica Matem´atica”. 1 2 FRANCO FAGNOLA AND ROLANDO REBOLLEDO 1. Introduction A major breakthrough of Probability Theory during the twentieth cen- tury is the advent of Stochastic Analysis with all its branches, which is being currently used in modelling phenomena in different sciences. Par- ticularly, in Physics, it is well known the probabilistic nature of Quantum Mechanics, since its very beginning. Moreover, it is dramatically striking that both, Probability, and Quantum Mechanics were given ground for the first time contemporaneously. Indeed, Kolmogorov in the thirties, provided the first rigourous axiomatic setting for Probability Theory. In parallel, von Neumann, inspired by the research of Planck, Schr¨odinger, Heisenberg, Bohr and all the founders of Quantum Mechanics, looked for another theory of probability which could cope with the experimental evidence stated in Heisenberg’s uncertainty principle. Quantum Optics is perhaps one of the fields where the probabilistic na- ture of quantum physics is most self evident. Indeed, mathematical models for quantum optics are often labelled as examples of the so called open sys- tems in Quantum Physics. In those models, it is considered that the main phenomenon is affected by the interaction of the “free” system with the thermal bath in which the system is embedded. Naturally, the existence of a thermal bath is immediately associated to stochastic noises. However, physicists use to describe the evolution of an open system by mimicking very laboriously the customary setting of a closed system evolution. Indeed, that approach is strongly based in determining the generator, or Hamiltonian of a unitary group of transformations on the state space. Unfortunately, such a Hamiltonian should consider the description of position and momentum for an infinite number of particles, which is far from being rigorously tractable. Several different procedures have been developed by physicists to reduce the complexity of the above Hamiltonian, deriving the so called master or Langevin equations (e.g. coarse-graining, weak coupling limit, spectral tech- niques). Perhaps one of the most interesting approaches is that of weak coupling limit and its extension to the concept of stochastic limit which has been deeply studied by Accardi, Frigerio and Lu in several articles (see eg. [7]). The current article being not primarily addressed to physicists but to classical probabilists, will choose the less sophisticated physical approach which is, in our opinion, that of the coarse graining, to derive the basic mas- ter equation. We are most interested in the interplay between classical and quantum stochastic differential equations associated to the aforementioned master equation. There are four basic operations related to particles which we are faced to describe in a suitable mathematical language: creation, annihilation, the gauge or number of particles and the time shift. In a classical probabilistic approach, these operations may be seen first by means of point or counting processes, for instance. Indeed, creation may STOCHASTIC DIFFERENTIAL EQUATIONS OF QUANTUM OPTICS 3 be thought of a positive jump on the total number of particles at a given time, annihilation, on the contrary, is a negative jump of the same process. Someone could perhaps argue that from the point of view of a (classical) mechanical description, what counts is a couple of data, the position and velocity of each individual particle. In that case, the use of diffusion pro- cesses should be more appropriate to write the evolution of a single particle. From there on, one has to consider the interaction between particles as they perform their own evolution, measure-valued processes arising then in a very natural way to characterize the motion of the system as a whole. However, interactions in quantum theory may be of more fundamental a nature, since they can deeply change the observed objects. This fact naturally leads to deal with non-commuting operations, that is, to work with linear operators defined on a given Hilbert space. Furthermore, the seminal work of Wiener on chaos decomposition, provided a deep understanding of the important role played by some classes of Hilbert spaces which contain, intrinsically, a model for Brownian motion or other remarkable stochastic “noises”. Therefore, to built up stochastic models for quantum physics one needs an additional amount of functional analysis and algebra. After all, from a very heuristic point of view, Itˆocalculus could be algebraically interpreted through three relationships on “differentials”: dtdWt = 0 = dWtdt, dtdt = 0, dWtdWt = dt, where W denotes a Brownian motion. As well as Canonical Commutation Relations (CCR) in quantum theory have a precise algebraic statement in terms of the commutator of the position and the momentum operators (that is the commutator of position and momentum is equal to i~, where ~ is the Planck constant). So that, intuitively, a mathematical structure supporting at least a non commutative version of Itˆocalculus as well as the CCR is needed. To start with, we consider an elementary Hilbert space and will introduce therein four important operators which will support our first approach to modelling stochastic noises. 2 1.1. Notations. We fix h = l (Z) as the initial space, with its canonical basis (en). B(h) denotes the space of bounded linear operators acting on h. Definition 1.1. We define the following fundamental operators on h: (1) The annihilation operator (√ n en−1 , if n > 0 (1) aen = 0 , if n ≤ 0, for all n ∈ Z. The domain of a is D(a) = {x = (xn)n ∈ h : P 2 n |n||xn| < ∞} 4 FRANCO FAGNOLA AND ROLANDO REBOLLEDO (2) The creation operator (√ † n + 1 en+1 , if n ≥ 0, (2) a en = 0 , if n < 0, † † for all n ∈ Z. The domain of a , is also D(a ) = D(a). (3) The number operator ( nen , if n > 0, (3) Nen = 0 , if n ≤ 0, for all n ∈ Z. The domain of N is D(N) = {x = (xn)n ∈ h : P 2 2 n |n| |xn| < ∞}. (4) The right–shift operator (4) Sen = en+1, (n ∈ Z). The above operators are related by the following relationship: (5) a† = N 1/2S = S(N + 1)1/2, (6) a = (N + 1)1/2S∗ = S∗N 1/2. Furthermore, given any function ϕ : N → C, we define a function of the number operator as follows: ( ϕ(n)en , if n > 0 (7) ϕ(N) en = ϕ(0)en , if n ≤ 0 Here below, we use the above operators to deal with the so called “coarse graining method” for deriving the master equation of quantum optics. It is worth noticing that for a classical probabilist, an effort is demanded to change his mind about elementary notions like events (which will now be assimilated to projections on a given Hilbert space), random variables (which become observables or self-adjoint operators on a Hilbert space), probability measures (replaced by states, which means positive trace-class operators with unit trace). 1.2. Deriving the master equation. The statement of master equations of quantum optics follows some general principles which we make explicit below for better understanding: • The device is based upon the emission of photons due to the inter- action of an electromagnetic field and a collection of atoms. The electromagnetic field acts within a cavity, STOCHASTIC DIFFERENTIAL EQUATIONS OF QUANTUM OPTICS 5 • The atoms are injected in the cavity at different times. They interact with the electromagnetic field, but they do not interact with each other, • The energy level of each atom is affected by the interaction with the field. This is a the basic framework for a microscopic approach to laser and maser description(see [36]). We are now interested in deriving (at least heuristically) an equation describing the evolution of the field state. What we mean by the state at time t is a positive trace-class operator ρ(t) with unit trace, defined over the Hilbert space h introduced before. This type of equation is named after Langevin. The first known research on this type of equations in quantum theory is due to Senitzky, and further used by Haken [27] and Lax [33] to study quantum optics. The Master Equation for the Laser was obtained by Scully and Lamb by the so called coarse–grained approximation [50], [49], which we recall briefly below. Let (ti; i ∈ N) denote the sequence of atom injection times inside the laser cavity.
Load more

Recommended publications
  • 2020 Volume 16 Issues
    Issues 1-2 2020 Volume 16 The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics PROGRESS IN PHYSICS “All scientists shall have the right to present their scientific research results, in whole or in part, at relevant scientific conferences, and to publish the same in printed scientific journals, electronic archives, and any other media.” — Declaration of Academic Freedom, Article 8 ISSN 1555-5534 The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics PROGRESS IN PHYSICS A quarterly issue scientific journal, registered with the Library of Congress (DC, USA). This journal is peer reviewed and included in the abstracting and indexing coverage of: Mathematical Reviews and MathSciNet (AMS, USA), DOAJ of Lund University (Sweden), Scientific Commons of the University of St. Gallen (Switzerland), Open-J-Gate (India), Referativnyi Zhurnal VINITI (Russia), etc. Electronic version of this journal: http://www.ptep-online.com Advisory Board Dmitri Rabounski, Editor-in-Chief, Founder Florentin Smarandache, Associate Editor, Founder April 2020 Vol. 16, Issue 1 Larissa Borissova, Associate Editor, Founder Editorial Board CONTENTS Pierre Millette [email protected] Dorda G. The Interpretation of the Hubble-Effect and of Human Vision Based on the Andreas Ries Differentiated Structure of Space . .3 [email protected] Tselnik F. Predictability Is Fundamental . .10 Gunn Quznetsov [email protected] Dorda G. The Interpretation of Sound on the Basis of the Differentiated Structure of Ebenezer Chifu Three-Dimensional Space . 15 [email protected] Nyambuya G. G. A Pedestrian Derivation of Heisenberg’s Uncertainty Principle on Stochastic Phase-Space .
    [Show full text]
  • Pip-2020-01.Pdf
    Issue 1 2020 Volume 16 The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics PROGRESS IN PHYSICS “All scientists shall have the right to present their scientific research results, in whole or in part, at relevant scientific conferences, and to publish the same in printed scientific journals, electronic archives, and any other media.” — Declaration of Academic Freedom, Article 8 ISSN 1555-5534 The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics PROGRESS IN PHYSICS A quarterly issue scientific journal, registered with the Library of Congress (DC, USA). This journal is peer reviewed and included in the abstracting and indexing coverage of: Mathematical Reviews and MathSciNet (AMS, USA), DOAJ of Lund University (Sweden), Scientific Commons of the University of St. Gallen (Switzerland), Open-J-Gate (India), Referativnyi Zhurnal VINITI (Russia), etc. Electronic version of this journal: http://www.ptep-online.com Advisory Board Dmitri Rabounski, Editor-in-Chief, Founder Florentin Smarandache, Associate Editor, Founder April 2020 Vol. 16, Issue 1 Larissa Borissova, Associate Editor, Founder Editorial Board CONTENTS Pierre Millette [email protected] Dorda G. The Interpretation of the Hubble-Effect and of Human Vision Based on the Andreas Ries Differentiated Structure of Space . .3 [email protected] Tselnik F. Predictability Is Fundamental . .10 Gunn Quznetsov [email protected] Dorda G. The Interpretation of Sound on the Basis of the Differentiated Structure of Ebenezer Chifu Three-Dimensional Space . 15 [email protected] Nyambuya G. G. A Pedestrian Derivation of Heisenberg’s Uncertainty Principle on Stochastic Phase-Space .
    [Show full text]
  • Using a Lindbladian Approach to Model Decoherence in Two Coupled Nuclear Spins Via Correlated Phase Damping and Amplitude Damping Noise Channels
    Pramana – J. Phys. (2020) 94:160 © Indian Academy of Sciences https://doi.org/10.1007/s12043-020-02027-3 Using a Lindbladian approach to model decoherence in two coupled nuclear spins via correlated phase damping and amplitude damping noise channels HARPREET SINGH1,2,ARVIND1 and KAVITA DORAI1 ,∗ 1Department of Physical Sciences, Indian Institute of Science Education and Research (IISER) Mohali, Sector 81 Knowledge City, P.O. Manauli, Mohali 140 306, India 2Fakultät Physik, Technische Universität Dortmund, 44221 Dortmund, Germany ∗Corresponding author. E-mail: [email protected] MS received 18 April 2020; revised 25 July 2020; accepted 18 August 2020 Abstract. In this work, we studied the relaxation dynamics of coherences of different orders present in a system of two coupled nuclear spins. We used a previously designed model for intrinsic noise present in such systems which considers the Lindblad master equation for Markovian relaxation. We experimentally created zero-, single- and double-quantum coherences in several two-spin systems and performed a complete state tomography and computed state fidelity. We experimentally measured the decay of zero- and double-quantum coherences in these systems. The experimental data fitted well to a model that considers the main noise channels to be a correlated phase damping (CPD) channel acting simultaneously on both spins in conjunction with a generalised amplitude damping channel acting independently on both spins. The differential relaxation of multiple-quantum coherences can be ascribed to the action of a CPD channel acting simultaneously on both the spins. Keywords. Nuclear magnetic resonance relaxation theory; Multiple quantum coherences; Lindblad master equation; Markovian noisy channels.
    [Show full text]
  • Efficient Quantum Algorithms for Simulating Lindblad Evolution
    Efficient Quantum Algorithms for Simulating Lindblad Evolution∗ Richard Cleveyzx Chunhao Wangyz Abstract We consider the natural generalization of the Schr¨odingerequation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an n-qubit system for time t within precision . If the Lindbladian consists of poly(n) operators that can each be expressed as a linear combination of poly(n) tensor products of Pauli operators then the gate cost of our algorithm is O(t polylog(t/)poly(n)). We also obtain similar bounds for the cases where the Lindbladian consists of local operators, and where the Lindbladian consists of sparse operators. This is remarkable in light of evidence that we provide indicating that the above efficiency is impossible to attain by first expressing Lindblad evolution as Schr¨odingerevolution on a larger system and tracing out the ancillary system: the cost of such a reduction incurs an efficiency overhead of O(t2/) even before the Hamiltonian evolution simulation begins. Instead, the approach of our algorithm is to use a novel variation of the \linear combinations of unitaries" construction that pertains to channels. 1 Introduction The problem of simulating the evolution of closed systems (captured by the Schr¨odingerequation) was proposed by Feynman [11] in 1982 as a motivation for building quantum computers. Since then, several quantum algorithms have appeared for this problem (see subsection 1.1 for references to these algorithms). However, many quantum systems of interest are not closed but are well-captured by the Lindblad Master equation [20, 12].
    [Show full text]
  • Arxiv:1701.02936V1 [Quant-Ph]
    Lindbladian purification Christian Arenz,1, 2 Daniel Burgarth,1 Vittorio Giovannetti,3 Hiromichi Nakazato,4 and Kazuya Yuasa4 1Institute of Mathematics, Physics, and Computer Science, Aberystwyth University, Aberystwyth SY23 2BZ, UK 2Frick Laboratory, Princeton University, Princeton NJ 08544, US 3NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy 4Department of Physics, Waseda University, Tokyo 169-8555, Japan (Dated: September 16, 2018) In a recent work [D. K. Burgarth et al., Nat. Commun. 5, 5173 (2014)] it was shown that a series of frequent measurements can project the dynamics of a quantum system onto a subspace in which the dynamics can be more complex. In this subspace even full controllability can be achieved, although the controllability over the system before the projection is very poor since the control Hamiltonians commute with each other. We can also think of the opposite: any Hamiltonians of a quantum system, which are in general noncommutative with each other, can be made commutative by embedding them in an extended Hilbert space, and thus the dynamics in the extended space becomes trivial and simple. This idea of making noncommutative Hamiltonians commutative is called “Hamiltonian purification.” The original noncommutative Hamiltonians are recovered by projecting the system back onto the original Hilbert space through frequent measurements. Here we generalize this idea to open-system dynamics by presenting a simple construction to make Lindbladians, as well as Hamiltonians, commutative on a larger space with an auxiliary system. We show that the original dynamics can be recovered through frequently measuring the auxiliary system in a non-selective way.
    [Show full text]
  • Efficient Quantum Algorithms for Simulating Lindblad Evolution
    Efficient Quantum Algorithms for Simulating Lindblad Evolution∗† Richard Cleve1 and Chunhao Wang2 1 Institute for Quantum Computing, University of Waterloo, Waterloo, Canada; and Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada; and Canadian Institute for Advanced Research, Toronto, Canada [email protected] 2 Institute for Quantum Computing, University of Waterloo, Waterloo, Canada; and Cheriton School of Computer Science, University of Waterloo, Canada [email protected] Abstract We consider the natural generalization of the Schrödinger equation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an n-qubit system for time t within precision . If the Lindbladian consists of poly(n) operators that can each be expressed as a linear combination of poly(n) tensor products of Pauli operators then the gate cost of our algorithm is O(t polylog(t/)poly(n)). We also obtain similar bounds for the cases where the Lindbladian consists of local operators, and where the Lindbladian consists of sparse operators. This is remarkable in light of evidence that we provide indicating that the above efficiency is impossible to attain by first expressing Lindblad evolution as Schrödinger evolution on a larger system and tracing out the ancillary system: the cost of such a reduction incurs an efficiency overhead of O(t2/) even before the Hamiltonian evolution simulation begins. Instead, the approach of our algorithm is to use a novel variation of the “linear combinations of unitaries” construction that pertains to channels. 1998 ACM Subject Classification F.1.1 Models of Computation Keywords and phrases Quantum algorithms, open quantum systems, Lindblad simulation Digital Object Identifier 10.4230/LIPIcs.ICALP.2017.17 1 Introduction The problem of simulating the evolution of closed systems (captured by the Schrödinger equation) was proposed by Feynman [12] in 1982 as a motivation for building quantum computers.
    [Show full text]
  • Existence, Uniqueness and Approximation for Stochastic Schrodinger Equation: the Poisson Case Clement Pellegrini
    Existence, uniqueness and approximation for stochastic Schrodinger equation: the Poisson case Clement Pellegrini To cite this version: Clement Pellegrini. Existence, uniqueness and approximation for stochastic Schrodinger equation: the Poisson case. 2009. hal-00174357v2 HAL Id: hal-00174357 https://hal.archives-ouvertes.fr/hal-00174357v2 Preprint submitted on 6 Mar 2009 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. Existence, Uniqueness and Approximation of a Stochastic Schr¨odinger Equation: the Poisson Case Cl´ement PELLEGRINI Institut C.Jordan Universit´eC.Bernard, lyon 1 21, av Claude Bernard 69622 Villeurbanne Cedex France e-mail: [email protected] March 6, 2009 Abstract In quantum physics, recent investigations deal with the so-called ”quantum trajec- tory” theory. Heuristic rules are usually used to give rise to “stochastic Schr¨odinger equations” which are stochastic differential equations of non-usual type describing the physical models. These equations pose tedious problems in terms of mathematical and physical justifications: notion of solution, existence, uniqueness... In this article, we concentrate on a particular case: the Poisson case. Random Measure theory is used in order to give rigorous sense to such equations. We prove existence and uniqueness of a solution for the associated stochastic equation.
    [Show full text]
  • The Lindbladian Form and the Reincarnation of Felix Bloch's
    The Lindbladian Form And The Reincarnation of Felix Bloch’s Generalized Theory of Relaxation Thomas M. Barbara Advanced Imaging Research Center Oregon Health and Sciences University Portland, OR 97239 Nov 10th, 2020 Email: [email protected] 1 Abstract The relationship between the classic magnetic resonance density matrix relaxation theories of Bloch and Hubbard, and the modern Lindbladian master equation methods are explored. These classic theories are in full agreement with the latest results obtained by the modern methods. A careful scrutiny shows that this also holds true for Redfield’s later treatment, offered in 1965. The early contributions of Bloch and Hubbard to rotating frame relaxation theory are also highlighted. Taken together, these seminal efforts of Bloch and Hubbard can enjoy a new birth of contemporary relevance in magnetic resonance. 2 1. Introduction In a recent and important publication [1], Bengs and Levitt formalize relaxation theory for systems that deviate significantly from the equilibrium state, conditions that invalidate the high temperature, weak ordering approximation, a corner stone in the commonly used Redfield theory [2]. Bengs and Levitt employed very modern methods that arose in the late 1970’s and are now fundamental in the topic of open quantum systems. These methods are often referred to eponymously as the “Lindbladian Form”. In addition to [1] a useful tutorial on this topic can be found in the work of Manzano [3]. Even though many, if not all of the classic papers are cited in their offering of their own Lindbladian analysis, the authors of [1], were not aware that a result identical to theirs was already offered by Bloch in 1957 [4], and masterly expounded upon by Hubbard in 1961 [5].
    [Show full text]
  • Dissipation in Linear and Non-Linear Quantum Systems
    ספריות הטכניון The Technion Libraries בית הספר ללימודי מוסמכים ע"ש ארווין וג'ואן ג'ייקובס Irwin and Joan Jacobs Graduate School © All rights reserved to the author This work, in whole or in part, may not be copied (in any media), printed, translated, stored in a retrieval system, transmitted via the internet or other electronic means, except for "fair use" of brief quotations for academic instruction, criticism, or research purposes only. Commercial use of this material is completely prohibited. © כל הזכויות שמורות למחבר/ת אין להעתיק (במדיה כלשהי), להדפיס, לתרגם, לאחסן במאגר מידע, להפיץ באינטרנט, חיבור זה או כל חלק ממנו, למעט "שימוש הוגן" בקטעים קצרים מן החיבור למטרות לימוד, הוראה, ביקורת או מחקר. שימוש מסחרי בחומר הכלול בחיבור זה אסור בהחלט. Library Central Elyachar Dissipation Models of Quantum Systems Technology, of in Linear and Nonlinear Amit Tsabary Institute Israel - Technion © Library Central Elyachar Technology, of Institute Israel - Technion © Library Dissipation in Linear and Nonlinear Models of Quantum Systems Central Research Thesis Elyachar Submitted in partial fulfillment of the requirements for the degree of Master of Science in Physics Technology, of Amit Tsabary Institute Israel - Submitted to the Senate of the Technion — Israel Institute of Technology Nisan 5777 Haifa March 2017 Technion © Library Central Elyachar Technology, of Institute Israel - Technion © This research was carried out under the supervision of Prof. Joseph Avron, in the Faculty of Physics. Library Acknowledgements I would like to thank Prof. Avron for his guidance and dedication to this research. I would like to thank Dr. Oded Kenneth for many fruitful discussions and for making significant contributions to this work.
    [Show full text]
  • Arxiv:1112.1303V1
    Quantum Fluctuation Relations for the Lindblad Master Equation R. Chetrite Laboratoire J. A. Dieudonn´e, UMR CNRS 6621, Universit´ede Nice Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France K. Mallick Institut de Physique Th´eorique, CEA Saclay, 91191 Gif-sur-Yvette, France (Dated: May 24, 2018) An open quantum system interacting with its environment can be modeled under suitable as- sumptions as a Markov process, described by a Lindblad master equation. In this work, we derive a general set of fluctuation relations for systems governed by a Lindblad equation. These identities provide quantum versions of Jarzynski-Hatano-Sasa and Crooks relations. In the linear response regime, these fluctuation relations yield a fluctuation-dissipation theorem (FDT) valid for a station- ary state arbitrarily far from equilibrium. For a closed system, this FDT reduces to the celebrated Callen-Welton-Kubo formula. PACS numbers: I. INTRODUCTION Fluctuations in non-equilibrium systems have been shown to satisfy various remarkable relations [20, 21, 36, 41, 47, 48, 54] discovered during the last twenty years. These results have lead to fierce discussions concerning the nature of heat, work and entropy, raising the fundamental issue of understanding the interactions between a given system and its environment (e.g., a thermal bath). In the classical realm, these problems have been progressively clarified whereas they are still under investigation in the quantum world. Historically, quantum fluctuation relations were first studied by Callen and Welton in 1950 [12]. These authors derived a Fluctuation-Dissipation Theorem for a closed quantum system isolated from its environment, initially in thermal equilibrium and suddenly perturbed by a small time-dependent term added to the time-independent Hamiltonian H.
    [Show full text]
  • Stability of Local Quantum Dissipative Systems
    Communications in Mathematical Physics manuscript No. (will be inserted by the editor) Stability of local quantum dissipative systems Toby S. Cubitt1, Angelo Lucia2, Spyridon Michalakis3, David Perez-Garcia2 1 DAMTP, University of Cambridge, Cambridge CB3 0WA, UK, E-mail: [email protected] 2 Dpto. de An´alisis Matem´atico,Universidad Complutense de Madrid, 28040 Madrid, Spain, E-mail: [email protected] E-mail: [email protected] 3 Institute for Quantum Information and Matter, Caltech, Pasadena, CA 91125, USA, E-mail: [email protected] The date of receipt and acceptance will be inserted by the editor Abstract: Open quantum systems weakly coupled to the environment are mod- eled by completely positive, trace preserving semigroups of linear maps. The generators of such evolutions are called Lindbladians. In the setting of quantum many-body systems on a lattice it is natural to consider Lindbladians that de- compose into a sum of local interactions with decreasing strength with respect to the size of their support. For both practical and theoretical reasons, it is crucial to estimate the impact that perturbations in the generating Lindbladian, arising as noise or errors, can have on the evolution. These local perturbations are po- tentially unbounded, but constrained to respect the underlying lattice structure. We show that even for polynomially decaying errors in the Lindbladian, local observables and correlation functions are stable if the unperturbed Lindbladian has a unique fixed point and a mixing time which scales logarithmically with the system size. The proof relies on Lieb-Robinson bounds, which describe a finite group velocity for propagation of information in local systems.
    [Show full text]
  • Lecture Notes for Ph219/CS219: Quantum Information Chapter 3
    Lecture Notes for Ph219/CS219: Quantum Information Chapter 3 John Preskill California Institute of Technology Updated October 2018 Contents 3 Foundations II: Measurement and Evolution 4 3.1 Orthogonal measurement and beyond 4 3.1.1 Orthogonal Measurements 4 3.1.2 Generalized measurements 8 3.2 Quantum channels 11 3.2.1 The operator-sum representation 11 3.2.2 Reversibility 13 3.2.3 Quantum channels in the Heisenberg picture 14 3.2.4 Quantum operations 16 3.2.5 Linearity 17 3.2.6 Complete positivity 18 3.3 Channel-state duality and the dilation of a channel 20 3.3.1 Channel-state duality 20 3.3.2 Stinespring dilation 22 3.3.3 Axioms revisited 23 3.4 Three quantum channels 24 3.4.1 Depolarizing channel 24 3.4.2 Dephasing channel 27 3.4.3 Amplitude-damping channel 31 3.5 Master equations for open quantum systems 34 3.5.1 Markovian evolution 34 3.5.2 The Liouvillian 35 3.5.3 Damped harmonic oscillator 38 3.6 Non-Markovian noise 41 3.6.1 Gaussian phase noise 41 3.6.2 Spin echo 43 3.6.3 Qubits as noise spectrometers 44 3.6.4 Spin-boson model at nonzero temperature 45 3.7 Summary 47 2 Contents 3 3.8 Exercises 49 3 Foundations II: Measurement and Evolution 3.1 Orthogonal measurement and beyond In Chapter 2 we discussed how to describe the state of an open quantum system, one which is part of a larger system. In this Chapter we will extend the theory of open quantum systems further.
    [Show full text]