LECTURES ON OPEN QUANTUM SYSTEMS

Transcription

1 LECTURES ON OPEN QUANTUM SYSTEMS Sabrina Maniscalco February 22, 217 1

2 Contents 1 Contents 1 Open Quantum Systems: What and why? What Why Plan of the course References Preliminaries: quantum states, channels and dynamical maps The structure of quantum states Positive and completely positive maps How to describe quantum evolution Microscopic derivation of the Markovian master equation Microscopic approach Markovian semigroups 18 5 Divisible maps and commutative maps Commutative dynamics Markovian evolution divisible dynamical maps Projection operator techniques Nakajima-Zwanzing technique Time-convolutionless technique Exact approaches: The Jaynes-Cummings model with losses The model and its exact solution Lorentzian spectral density Markovian and strong coupling limits of the exact solution Appendix: Some useful properties of Laplace transform Quantum Brownian motion Perturbative Master Equation Exact Master Equation Part Part Summary Death of the Schrödinger cat: The quantum to classical transition The Monte Carlo Wave Function (MCWF) Quantum Jumps approach Quantum Jumps: Introduction The MCWF approach Example: two level atom driven by a classical field

3 1 Open Quantum Systems: What and why? 2 1 Open Quantum Systems: What and why? 1.1 What An open quantum system is a quantum system interacting with its surroundings. During the last few decades incredible technological advances have made it possible to perform experiments at the level of single atoms or photons. In these experiments coherent manipulation of individual quantum systems allows us to prepare them in a desired quantum state with incredible precision. However, no quantum system can be consider completely isolated from its surroundings. Ultimately all quantum systems will be subjected to coupling to what is known as their external environment. The type of environment that most affects the quantum system of interest of course depends on the physical context we are considering. For example, if we are interested in the dynamics of a single atom which has been prepared in a given excited electronic state (for example by means of a resonant laser pulse), then the environment is the electromagnetic (e.m.) field surrounding the atom. We know, indeed, that vacuum field fluctuations, and more precisely the coupling between the atom and the zero-temperature e.m. field, are at the origin of the well-known phenomenon of spontaneous emission. If, instead, our quantum system of interest is an atom in a solid crystal, then a possible environment will be made of phonons (vibrational excitations) describing the vibrational motion of the crystal matrix. In both the two examples above the quantum environment is bosonic and is described in terms of an infinite number of quantum harmonic oscillators representing the normal modes of the electromagnetic or vibrational field. Photons (phonons) are quantum excitations of quantum harmonic oscillators having different frequencies. As we will see, the frequency spectrum of the environment, and its density of modes, play a key role in the description of open quantum system dynamics. Which types of environments are generally considered in the Theory of Open Quantum Systems? There there are a number of important distinctions, first of all one can deal with quantum or classical environments. Within the quantum environment scenario one can have bosonic or fermonic environments, stationary or non-stationary environments, and within stationary environments there can be thermal environments (often called reservoirs) and non-thermal environments. During this course we will mostly deal with the case in which the environment has infinitely many degrees of freedom, as for the e.m. field example or the phonon environments mentioned above. However, in some cases one can also use open quantum systems approach to study finite-size environments. In the latter case the Poincare time is finite, that is the system, after a certain time, will return to its initial state. By contrast, for truly infinite environments, the Poincare time is infinite. Generally an open quantum system is by definition "what we are interested in", but to investigate its properties, e.g., its dynamics, we need to know the effects of the environment on the system. One should mention at this point that due to the system-environment coupling, generally the system also influences the environment and changes its evolution. Under some circumstances, the changes induced by the system into the environment are not significant and the environment can be, for all intents and purposes, described as stationary. This is, e.g., the case for a quantum system in contact with a reservoir, as often encountered in thermodynamics and statistical physics. The size and the energy of the reservoir are in this case so much greater than the ones of the system that its state can be considered unaffected by the system itself. The quantum system,

4 1 Open Quantum Systems: What and why? 3 by contrast, if prepared in a given quantum state such as its ground state, will evolve due to the presence of the reservoir until it reaches thermal equilibrium with the reservoir. It is worth mentioning, however, that some of the examples considered in this course will greatly differ from this scenario. This is particularly true for cold quantum environments such as the vacuum of the e.m. field. As an example we will consider in the course the case of an atom emitting in a photonics band gap material at zero temperature. The aim of this course is to study the time evolution of open quantum systems in a number of different situations of both fundamental and applicative importance. To do so one generally starts from the equation of motion for the dynamics of the quantum state. This is known as the master equation. Since by definition the system of interest is not isolated, its state has to be described in terms of a density operator, rather than a state vector, which takes into account the fact that the quantum system can be, and generally will be, in a statistical mixture due to the interaction with the environment. The master equation is the equation of motion of the density operator. There are several ways to approach the description of open quantum systems dynamics [1]. One distinction worth pointing out is between phenomenologic and microscopic approaches. With the former term we refer to cases in which, based on certain physical assumptions on both the system and its environment, one assumes a certain form for the master equation. Comparison with experimental data allows one to validate such an assumption or reject it and look for a different model. The microscopic approach, by contrast, starts from the modelization, in terms of a total Hamiltonian H, of both system, environment and interaction. The total system is assumed to be closed and therefore its dynamics is unitary. We can formally solve the dynamics of the total system ϱ T (t), but this contains much more information than we are actually interested in. Therefore, by performing a partial trace over the degrees of freedom of the environment we can obtain the reduced density operator ϱ S (t) = Tr E [ϱ T (t)] describing the time evolution of the open quantum system interacting with the environment. Alternatively, we can start from the equation of motion of the total system, i.e., the von-neumann equation i dϱ T dt = [H, ϱ T ], and then by partial trace and, often, performing a number of assumptions and approximation, we derive the master equation for the open system dynamics. The microscopic approach has certain advantages over the phenomenological approach. Since we start from a specific model for the environment and the interaction we are also often able to gain more insight into the physical phenomena underlying the time evolution of open quantum systems. In this lecture course we will mostly focus on microscopic approaches but we will also consider phenomenological master equations and techniques to solve them. 1.2 Why During the last decade the theory of open quantum systems has received renewed attention due to its importance to both fundamental and applicative quantum science. From a fundamental perspective, the theory of open quantum systems allows us to assess what is perhaps the most important question still unanswered in quantum theory: the quantum measurement problem, or the quantum to classical transition. In simple terms this question is about the elusive border between the quantum description of reality, which generally applies to microscopic objects such as atoms or electrons, and the classical description of reality which we are familiar with. We live

5 1 Open Quantum Systems: What and why? 4 in a classical world. The objects of our every day experience do not experience bizarre quantum phenomena such as quantum superpositions or entanglement. Chairs and tables are here or there, but never in a quantum superposition of here and there. But chairs are composed of individual atoms which do behave quantum mechanically, so it is natural to ask how does classical mechanics emerge from quantum mechanics? This question does not yet have a satisfactory answer! Why is it that macroscopic objects behave in a (perhaps) more boring and certainly more predictable way while microscopic ones behave so differently? And is it really simply a matter of size? In fact, it isn t just a question of size of a sample. In a beautiful experiment performed at the University of Vienna interference due to the wave nature of a beam of large organic molecules has been observed. In solid state physics, Josephson junctions have been prepared in superpositions of macroscopic currents flowing clockwise or counterclockwise. One of the most profound descriptions of the quantum to classical border has been given by a Polish scientists working at Los Alamos, W. Zurek. His theory goes by the name of environmentinduced decoherence [3]. The main idea comes from the observation that quantum superpositions of macroscopically distinguishable states (Schroedinger cat states), as well as other crucially quantum states such as entangled states, can be created in mesoscopic quantum systems provided that the systems are extremely well isolated from their external environment. It is indeed the presence of the environment that is the main culprit for the destruction of all quantum features. In other words the environment introduces noise into quantum systems destroying fragile coherent superpositions and transforming them into classical statistical mixtures. Moreover, it is found that, given a certain environment and coupling, the bigger the system, prepared, for example in a Schroedinger cat state, the faster the environment induced decoherence transforming it into a classical statistical mixture and destroying quantum coherence. Therefore the reason why we do not observe quantum superpositions of distinguishable states of macroscopic objects would lie ultimately in the extremely fast decoherence that would occur. The bigger the cat, the faster it dies due to the environment. The theory of open quantum systems, therefore, allows us to describe the elusive border between the quantum and the classical. The considerations made above naturally highlight the importance of the theory of open quantum systems for applications too. We are currently on the verge of what is known as the second quantum revolution. Why the second? Because there was already a first. The first quantum revolution is the one that led to a whole new range of technologies based on the understanding of the quantum structure of microscopic systems. Examples are the invention of the laser, which is now widely used in a number of technologies. To have a clearer idea of the impact of the birth of quantum physics to our technologies and society in general one could cite an estimate by Tegmark and Wheeler [4] according to which already more than two decades ago 3% of the gross national product of the United States came from inventions made possible by quantum physics. The second quantum revolution will occur when so-called quantum technologies become commercially distributed and available on the market. These technologies include quantum computers, quantum sensors, quantum cryptographic devices and so on and so forth. Quantum technologies stem from the most definitively quantum features of microscopic systems, such as quantum superpositions and entanglement. Most of them rely on a new emerging field in science lying at the intersection between information theory, computer science and quantum physics, namely quantum information and computation. Quantum devices operate by manipulating and transmitting quantum information carried by physical systems. But quantum information is delicate

6 1 Open Quantum Systems: What and why? 5 and, once again, is quickly destroyed by the noisy environment. Hence, for quantum technologies to reach the market it is crucial to identify and combat the sources of decoherence coming from the interaction with the environment. Once again this requires the use of the methods and techniques of the theory of open quantum systems. In order to have a realistic description of quantum information processing and quantum communication it is indeed necessary to use the formalism and tools that we will learn during this course. 1.3 Plan of the course The aim of this course is to introduce the main topics and concepts of the theory of open quantum systems. This theory requires sophisticated mathematical techniques but its beauty can only be fully appreciated when the mathematical formalism is used to describe physical systems, helping to improve our understanding of fascinating aspects such as the quantum measurement problem or the potential of quantum technologies. We will therefore attempt to find a proper balance between the abstract mathematical description and the physical description of the basic phenomena characterising those systems currently used in the experiments, such as trapped ions and photons in optical and microwave high quality resonators. Below is the plan of the course Lectures. Lecture 2 Preliminaries: Quantum states, channels and dynamical maps. This lecture will introduce the notation and the main theorems widely used during the course. It contains mostly mathematical formalism but we will also give physical examples. Lectures 3 and 4 Microscopic derivation of the master equation. We will look in detail at the general steps to derive a master equation from a general system-environment model. We will highlight the main approximations usually performed in the literature such as the weak coupling (Born-Markov) approximation and the rotating wave or secular approximation. Lectures 5 and 6 We will discuss the theory of Markovian semigroups and present some useful examples of Markovian master equations and their solutions. Lectures 7 and 8 Divisibility and Commutative maps. We will focus on the concept of divisibility and present a number of physical examples of divisible and non-divisible maps with specific emphasis on dephasing dynamics (pure decoherence). The exact master equation and solution for two-level systems undergoing pure dephasing will be discussed. Lectures 9 and 1 Nakajima-Zwanzig and time-convolutionless projection operator techniques. We will introduce and discuss the advantages and limitations of two very general approaches to derive generalised master equations from a microscopic description of system and environment. Lectures 11 and 12 Jaynes-Cummings model with losses. We continue the study of exact approaches to the description of open quantum systems by looking at a fundamental physical example, the interaction between a two-level atom and the electromagnetic field in its ground state. Lecture 13 Quantum Brownian Motion. We introduce another paradigmatic open quantum system model amenable to an exact solution. In this case the system of interest is a quantum harmonic oscillator bilinearly coupled to a bosonic reservoir. We will discuss the weak coupling limit of the quantum Brownian motion model and its exact solution. Lecture 14 The quantum to classical transition. Environment induced decoherence described using the quantum Brownian motion master equation and solution.

7 2 Preliminaries: quantum states, channels and dynamical maps 6 Lectures 15 and 16 Quantum jumps. We will describe the Monte Carlo wave function approach for Markovian system and its interpretation in terms of a measurement scheme on the environment. We will describe recent experiments which have led to the observation of quantum jumps in trapped ion systems and high quality microwave resonators. Lectures 17 and 18 Beyond Markovian dynamics. Non-Markovian time evolution and information flow. 1.4 References 1) Reference book of the course: H.-P. Bruer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (26). 2) Review article on time-local master equations: Dariusz Chruschinski, On time-local generators of quantum evolution, Open Syst. Inf. Dyn. 21, 1444 (214). See also arxiv: ) Zurek s article on environment induced decoherence: W.H. Zurek Decoherence and the transition from quantum to classical - revisited, Los Alamos Science 27, (22); see also arxiv.org/pdf/quantph/ ) Short article about quantum information and environmental effects: Rainer Blatt, Delicate information, Nature 412, 773 (21). 2 Preliminaries: quantum states, channels and dynamical maps This section closely follows Dariusz Chruschinski s Lecture Notes presented at the school "Open Quantum Systems", Torun, June 2-24, 212 and published as On time-local generators of quantum evolution, Open Syst. Inf. Dyn. 21, 1444 (214). We begin by introducing basic notation and terminology. 2.1 The structure of quantum states Let us consider a quantum system living in a finite-dimensional Hilbert space H isomorphic to C n. Fixing an orthonormal basis {e 1,..., e n } in H any linear operator in H may be identified with an n n complex matrix, i.e. an element from M n (C). A mixed state of such system is represented by a density matrix, i.e. a matrix ρ from M n (C) such that ρ and Tr ρ = 1. The space of states S n of the n-level quantum system defines an (n 2 1)-dimensional convex set. Pure states correspond to rank-1 projectors ψ ψ and define extremal elements of S n. Any mixed state may therefore be decomposed as follows ρ = k w k ψ k ψ k, (2.1) with w k > and k w k = 1, i.e. w k provides a probability distribution. It should be stressed that the above decomposition is highly non-unique. To illustrate the concept of density operator let us consider the following

8 2 Preliminaries: quantum states, channels and dynamical maps 7 Example 1 A 2-level system (qubit) living in C 2. Any hermitian operator ρ may be decomposed as follows ρ = (I 2 + x k σ k ), (2.2) where x = (x 1, x 2, x 3 ) R 3 and {σ 1, σ 2, σ 3 } are Pauli matrices. As usual I n denotes the unit matrix in M n (C). It is, therefore, clear that ρ is entirely characterized by the Bloch vector x. This representation already guaranties that Tr ρ = 1. Hence, ρ represents a density operator if and only if the corresponding eigenvalues {λ, λ + } are non-negative. One easily finds k=1 λ = 1 2 (1 x ), λ + = 1 (1 + x ), (2.3) 2 and hence ρ if and only if x = x x2 2 + x This condition defines a unit ball in R3 known as a Bloch ball. A state is pure if ρ defines a rank-1 projector, i.e. λ = and λ + = 1. This shows that pure states belong to Bloch sphere corresponding to x = 1. Unfortunately, this simple geometric picture is much more complicated if n > 2. For any A M n (C) we denote by A 1 := Tr A = Tr AA the trace-norm of A. If λ 1,... λ n are the (necessarily nonnegative) eigenvalues of AA, then A 1 = λ λ n. The space of states S n is equipped with a natural metric structure: given two states ρ, σ S n one defines the corresponding distance D[ρ, σ] = 1 2 ρ σ 1. (2.4) This quantity measures distinguishability between ρ and σ. It is clear that D[ρ, σ] =, i.e. ρ and σ are indistinguishable, if and only if ρ = σ. Note, that if ρ and σ are orthogonally supported, then D[ρ, σ] = 1 2 ( ρ 1 + σ 1 ) = 1, since ρ 1 = 1 for any density matrix ρ. In this case ρ and σ are perfectly distinguishable. Hence D[ρ, σ] 1. (2.5) In particular, if ρ and σ are two states of a qubit and x and y are corresponding Bloch vectors then D[ρ, σ] = 1 x y, (2.6) 2 reproduces the standard Euclidean distance in R 3. For more information about the structure of quantum states see [7, 8].

9 2 Preliminaries: quantum states, channels and dynamical maps Positive and completely positive maps Consider now a linear map Φ : M n (C) M n (C) and let M + n (C) = {A M n (C) A } M n (C) be a convex subset of positive matrices. One calls a linear map Φ Hermicity-preserving iff Φ(A ) = [Φ(A)], positive iff Φ(M + n (C)) M + n (C), trace-preserving iff Tr Φ(A) = Tr A, unital iff Φ(I n ) = I n. It is easy to show that a positive map is necessarily Hermicity-preserving. Moreover, observing that S n = {A M + n (C) Tr A = 1}, it is clear that if Φ is positive and trace preserving then it maps density matrices into density matrices, i.e. Φ(S n ) S n. If Φ is a linear map then one defines a dual map Φ : M n (C) M n (C) by Tr[AΦ (B)] = Tr[Φ(A)B], (2.7) for all A, B M n (C). Φ is trace-preserving iff Φ is unital [9, 1, 11]. Example 2 Consider a transposition T n : M n (C) M n (C), i.e. T n (ρ) = ρ T. Since transposition does not affect the spectrum of A it is clear that A T whenever A. Note that T n is trace-preserving and unital. Positive trace-preserving maps possess the following fundamental property Proposition 1 ([9, 1, 11]) If Φ is positive and trace-preserving, then Φ(X) 1 X 1, (2.8) for all X M n (C), that is, Φ is a contraction in trace-norm. Hence D[Φ(ρ), Φ(σ)] D[ρ, σ], (2.9) which means that the distinguishability of ρ and σ never increases under the action of a positive and trace-preserving map. It turns out that the positivity property is not sufficient to describe the dynamics of open quantum systems. The reason stems from the properties of composed systems. Composing two systems living in H 1 and H 2, respectively, one obtains a system living in H = H 1 H 2. Let dim H 1 = n, dim H 2 = m and consider two linear maps Φ 1 : M n (C) M n (C), Φ 2 : M m (C) M m (C). Recalling that M n m (C) = M n (C) M m (C) one defines a tensor product Φ 1 Φ 2 : M n m (C) M n m (C),

10 2 Preliminaries: quantum states, channels and dynamical maps 9 as follows: for a fixed orthonormal basis {e 1,..., e n } in H 1 let us define e ij := e i e j M n (C). Elements {e ij } for i, j = 1,..., n define an orthonormal basis in M n (C) with respect to the standard inner product (A, B) = tr(a B). Now, any matrix A M n m (C) may be represented in the following block form n A = e ij A ij, (2.1) i,j=1 with A ij M m (C). For example if n = 2 one has A = 2 ( ) A11 A e ij A ij = 12. (2.11) A 21 A 22 i,j=1 Hence the action of Φ 1 Φ 2 is given by [Φ 1 Φ 2 ](A) := n Φ 1 (e ij ) Φ 2 (A ij ). (2.12) i,j=1 In particular if n = 2 and Φ 1 = 1l 2, where 1l n : M n (C) M n (C) denotes an identity map defined by 1l n (X) = X, then [1l 2 Φ](A) = 2 e ij Φ(A ij ) = i,j=1 ( Φ(A11 ) Φ(A 12 ) Φ(A 21 ) Φ(A 22 ) ). (2.13) Now comes a surprise: even if Φ 1 and Φ 2 are positive Φ 1 Φ 2 need not be a positive map. Example 3 Interestingly, the transposition map considered in Example 2 loses its positivity when tensored with another positive map. This map is evidently positive and trace-preserving. Consider 1l 2 T 2. It turns out that this maps is not positive in M 4 (C). Indeed, let P + 2 = e ij e ij = 1 2 i,j= , (2.14) be a state in C 2 C 2. Note that P + 2 = ψ+ 2 ψ+ 2 with ψ+ 2 = (e 1 e 1 + e 2 e 2 )/ 2 being one of the well-known Bell states of two qubits. One finds [1l 2 T 2 ](P + 2 ) = e ij e ji = 1 2 i,j= , (2.15) Note that [1l 2 T 2 ](P 2 + ) has one negative eigenvalue and hence it is not a positive map.

11 2 Preliminaries: quantum states, channels and dynamical maps 1 This example proves that quantum physics of composed systems needs a more refined notion of positivity. Consider again a linear map Φ : M n (C) M n (C). One calls Φ k-positive if 1l k Φ : M k (C) M n (C) M k (C) M n (C), (2.16) is positive. Clearly 1-positive is just positive and k-positivity implies l-positivity for l < k. Finally, Φ is called completely positive (CP) if it is k-positive for k = 1, 2,.... Interestingly, one has the following characterization Proposition 2 (Choi [12]) If dim H = n, then Φ is CP if and only if Φ is n-positive. Denoting by P k a convex set of k-positive maps one has the following chain of inclusions CP P n... P 2 P 1 Positive maps. Let {e 1,..., e n } be a fixed orthonormal basis in H and let ψ n + = 1 n e k e k, (2.17) n = ψ + n ψ + n denote the corre- denote a maximally entangled state in H H. Moreover, let P n + sponding rank-1 projector. Proposition 3 (Choi [12]) Φ is CP if and only if [1l n Φ](P + n ). k=1 This beautiful result states that in order to prove that Φ is CP, or equivalently that 1l n Φ is positive, it is enough to check whether 1l n Φ is positive on one particular projector P + n. Positivity of [1l n Φ](P + n ) guaranties that [1l n Φ](X) for all positive X M n (C) M n (C). Corollary 1 If Φ 1 and Φ 2 are CP maps, then Φ 1 Φ 2 is always CP as well. This analysis shows that the motivation to use CP maps is deeply rooted in physics and it is not just a mathematical trick! The very presence of quantum entangled states enforces us to use maps which are not only positive but also completely positive. The following result provides the most important characterization of CP maps. Theorem 1 ([9, 1, 12, 13]) A map Φ : M n (C) M n (C) is CP if and only if Φ(X) = α K α X K α, (2.18) for X M n (C). Formula (2.18) is usually called the Kraus or Operator Sum Representation of Φ and K α are called Kraus operators. Actually, the above formula appeared already in the Sudarshan et. al. paper [14]. It should be stressed that the Kraus representation is highly non unique. A completely positive trace preserving map (CPTP) is called a quantum channel. A CP map possessing a Kraus representation is trace-preserving iff K α K α = I n. (2.19) The following result shows the origin of a genuine quantum channel. α

12 2 Preliminaries: quantum states, channels and dynamical maps 11 Theorem 2 (Unitary dilation) Any quantum channel Φ may be constructed as follows Φ(ρ) = Tr E [ U(ρ ω)u ], (2.2) where U is a unitary operator in H H E, ω is a density operator in H E, and Tr E denotes the partial trace over H E. One usually interprets H E as a Hilbert space of the environment and ω as its fixed state. Let ω E k = λ k E k, with λ k. Moreover, let U = k,l U kl E k E l. Formula (2.2) implies Φ(ρ) = [ λ k Tr E (Uij E i E j )(ρ E k E k )(U mn E n E m ) ] m,n i,j k = λ k Tr[ E i E j E k E k E n E m ] U ij ρu mn. m,n i,j k Using Tr[ E i E j E k E k E n E m ] = δ im δ jk δ kn and introducing K α := K mn = λ n U mn one arrives at the Kraus representation Φ(ρ) = α K αρ K α which proves that Φ defined via formula (2.2) is completely positive. One easily proves that Φ is also trace preserving and hence defines a quantum channel. 2.3 How to describe quantum evolution If ρ is an initial state of an n-level quantum system, then by its evolution we mean a trajectory ρ t for t starting at ρ. The simplest example of quantum evolution is provided by the von Neumann equation (in units of ) i ρ t = [H, ρ t ], (2.21) with the corresponding solution where the map U t : M n (C) M n (C) is defined by ρ t = U t (ρ), (2.22) U t (ρ) := U t ρu t, (2.23) with U t = e iht. Note that (2.23) defines a family of quantum channels. Let us observe that the 1-parameter unitary group U t implies the following composition law U t U s = U t+s, (2.24) for all t, s R. Hence U t defines a 1-parameter group of CP maps. Equation (2.21) may be transformed into the following equation for U t where the generator L : M n (C) M n (C) is defined by for any X M n (C). Now come the natural questions: U t = L U t, U = 1l, (2.25) L(X) = i[h, X], (2.26)

13 2 Preliminaries: quantum states, channels and dynamical maps how to generalize the unitary evolution defined by (2.23), valid for closed systems, to open quantum systems? 2. how to generalize the corresponding equation of motion (2.25)? Definition 1 By a general quantum evolution we mean a dynamical map, i.e. a family of quantum channels Λ t : M n (C) M n (C) for t such that Λ = 1l n. A dynamical map Λ t maps an initial state ρ into a current state ρ t := Λ t (ρ) and hence provides the natural generalization of the unitary evolution ρ t = U t (ρ). Assuming that ρ t satisfies a linear equation and that the initial state ρ provides all necessary information to uniquely determine ρ t we expect that ρ t satisfies the following equation ρ t = L t (ρ t ), (2.27) or equivalently Λ t = L t Λ t, Λ = 1l n, (2.28) where L t : M n (C) M n (C) denotes a time-dependent generator. This equation provides a natural generalization of (2.25). The formal solution of (2.28) may be written as follows ( t ) Λ t = T exp L τ dτ, (2.29) where T denotes the chronological ordering operator. The above formula is defined by the following Dyson series Λ t = 1l n + provided that it converges. t dt 1 L t1 + t t1 dt 1 dt 2 L t1 L t2 +..., (2.3)

14 3 Microscopic derivation of the Markovian master equation 13 3 Microscopic derivation of the Markovian master equation This section is based on the Breuer-Petruccione book, see Ref. [1]. An open quantum system is a system S which is coupled to an environment B. It is therefore a subsystem of the total system S +B. The total system is usually considered a closed system, i.e., its evolution follows Hamiltonian dynamics. The state of the system S will change due to both its internal dynamics and the interaction with the environment B. Due to the establishment of system-environment correlations the dynamics of S cannot be considered unitary anymore. The dynamics of S induced by the total system dynamics is called reduced system dynamics and S is also known as reduced system. The density matrix of the reduced system ϱ S is obtained from the density matrix of the total system ϱ SB by performing a partial trace over the environmental degrees of freedom ϱ S Tr B [ϱ SB ]. If we indicate with H S and H B the Hilbert spaces of system and environment, respectively, and with { e j } an orthonormal basis of H B, then ϱ S Tr B [ϱ SB ] = j e j ϱ SB e j. (3.1) 3.1 Microscopic approach In the microscopic approach to open quantum systems dynamics we start by modeling the total closed system, whose Hilbert space is H S H B, by means of the microscopic Hamiltonian H = H S I B + H B I S + H I, (3.2) where H S and H B are the free Hamiltonians of the system and of the environment, respectively, and H I is the interaction term. Generally, the initial state of the total system, i.e. the state of the total system at t =, is assumed to be of the form ϱ SB () = ϱ S () ϱ B. This means that no correlations between system and environment are initially present. As the total system is closed, we can write its unitary evolution as ϱ SB (t) = U(t)ϱ SB ()U (t), (3.3) with U(t) = exp[ iht]. If we now take the partial trace over the environment in the equation above, we have: ϱ S (t) = Tr B {U(t)ϱ SB ()U (t)} = Tr B {U(t) ϱ S () ϱ B U (t)} Λ t ϱ S (). (3.4) where Λ t is the dynamical map. In the following we will describe the assumptions that allow us, starting from a microscopic description of system plus environment, to derive a master equation which is physically meaningful. These are not the minimal assumptions, however. Namely, there exist physically meaningful master equations that are more general than the type we are going to derive in the following. We will examine the more general cases later during the course. For now, let us focus on what are known as Markovian master equations and let us describe the assumptions that are needed to derive them following a microscopic approach. It is important to stress that Markovian master equations have played a prominent role in the advancement of fields of importance such as quantum optics and quantum information theory as they are very

15 3 Microscopic derivation of the Markovian master equation 14 good approximations in a number of physical scenarios (e.g., for trapped ions and cavity QED systems). Let us consider the dynamics of the overall density operator ϱ SB given by the von Neumann equation which, in units of and in the interaction picture, reads as follows dϱ SB (t) dt = i[h I (t), ϱ SB (t)], (3.5) where we omit for simplicity of notation the subscript I which we should use to indicate the density matrix in the interaction picture. The integral form of this equation is ϱ SB (t) = ϱ SB () i t ds[h I (s), ϱ SB (s)]. (3.6) Inserting Eq. (3.6) into Eq. (3.5) and taking the partial trace over the environmental degrees of freedom we get dϱ t S dt (t) = dstr B {[H I (t), [H I (s), ϱ SB (s)]]}, (3.7) where we have assumed Tr B [H I (t), ϱ SB ()] =. We assume now that system and environment are weakly coupled (Born approximation). This approximation amounts to assuming that the correlations established between system and environment are negligible at all times (remember that they were initially zero), i.e., ϱ SB (t) ϱ S (t) ϱ B Within this approximation we get a closed integro-differential equation for ϱ S (t) dϱ S (t) dt = t dstr B {[H I (t), [H I (s), ϱ S (s) ϱ B ]]} (3.8) Note that, in the equation above, the future evolution of the system, described by dϱs dt (t), depends on the past states of the system ϱ S (s) for times s < t through the integral. A further simplification to this equation is obtained by assuming that we can replace ϱ S (s) appearing inside the integral with its value at time t, ϱ S (t). This is possible if the density matrix does not change sensibly in the interval of time s t. But t here is a variable so one may think that this approximation gets worse and worse for bigger and bigger values of time t. This reasoning would be correct if the integrand of Eq. (3.8) were, e.g., constant in time. However, in many physical situations this integrand (or rather that part of it describing the environment correlations) quickly decays to zero after a short characteristic correlation time τ B. This time scale quantifies the memory time of the reservoir. Hence, if the density matrix of the system does not change sensibly in the correlation time τ B, then we can approximate ϱ S (s) with ϱ S (t) in Eq. (3.8). The resulting equation is known as the Redfield equation dϱ S (t) dt = t dstr B {[H I (t), [H I (s), ϱ S (t) ϱ B ]]}. (3.9) Equation (3.9) is local in time, i.e., the future evolution of the state of the system does not depend on its past state. However, it still retains memory of the initial state ϱ S ().

16 3 Microscopic derivation of the Markovian master equation 15 Until now we have assumed the the density matrix does not change much within the correlation time τ B. The next step will be to neglect such a change altogether, by performing a coarsegraining in time. This is mathematically achieved by replacing the upper limit of the integral in Eq. (3.9) with : dϱ S dt (t) = dstr B {[H I (t), [H I (t s), ϱ S (t) ϱ B ]]}, (3.1) where we have replaced for the sake of convenience s with t s. The two-step approximation described in Eqs. (3.9) and (3.1) is known as the Markov approximation. So we say that Eq. (3.1) is derived from a microscopic model under the Born-Markov approximation, i.e., for weak coupling and quickly decaying reservoir correlations (memoryless dynamics). Let us decompose the interaction Hamiltonian H I in terms of operators of the system and of the environment: H I = A α B α α with A α (B α ) Hermitian operators of the system (environment). Let us assume that H S has a discrete spectrum and let us indicate with ɛ the eigenvalues and with Π(ɛ) the corresponding projectors into the corresponding eigenspace. We define the eigenoperators of the system as follows A α (ω) = Π(ɛ)A α Π(ɛ ). (3.11) ɛ ɛ=ω These operators have some important properties [H S, A α (ω)] = ωa α (ω); (A α is an eigenoperator of H S with eigenvalue ω). [H S, A α(ω)] = ωa α(ω); (A α is an eigenoperator of H S with eigenvalue ω). Eigenoperators in the interaction picture: A I α(ω) = e iωt A α (ω), A I α (ω) = e iωt A α(ω). A α(ω) = A α ( ω). Completeness: ω A α(ω) = ω A α(ω) = A α. We can rewrite the interaction Hamiltonian in terms of eigenoperators of H S, and then pass to the interaction picture exploiting the fact that the system eigenoperators have a simple time dependency in this picture. The environment operator in the interaction picture are simply given by B α (t) = e ih Bt B α e ih Bt. After some algebra, we can rewrite the master equation in the following form dϱ s dt (t) = e i(ω ω)t Γ αβ (ω)[a β (ω)ϱ S (t)a α(ω ) A α(ω )A β (ω)ϱ S (t)] + h.c. (3.12) ω,ω α,β where we introduced Γ αβ (ω) dse iωs B α(t)b β (t s),

17 3 Microscopic derivation of the Markovian master equation 16 with the reservoir correlation functions given by B α(t)b β (t s) Tr B {B α(t)b β (t s)ϱ B }. Such correlation functions are homogeneous in time if the reservoir is stationary, i.e. B α(t)b β (t s) = B α(s)b β (). We now make the last approximation, known as the secular approximation. First we define τ S as the characteristic intrinsic evolution time of the system. This timescale is generally of the order of τ S ω ω 1, ω ω. We indicate with τ R the relaxation time of the open system. If τ S τ R we can neglect all the exponential terms oscillating at frequency ω ω as they oscillate very rapidly (averaging out to zero) over the time scale τ R over which ϱ S changes appreciably. We then decompose the environment correlation functions into its real and imaginary parts where, for fixed ω, form a positive matrix and Γ αβ (ω) = 1 2 γ αβ(ω) + is αβ (ω), + γ αβ (ω) = Γ αβ (ω) + Γ βα (ω) = dse iωs B α(s)b β (), S αβ (ω) = 1 2i [Γ αβ(ω) Γ βα (ω)], form a Hermitian matrix. With these definitions we finally arrive at the interaction picture master equation dϱ S dt (t) = i[h LS, ϱ S (t)] + Lϱ S (t) (3.13) where H LS = S αβ (ω)a α(ω)a β (ω) ω α,β is a Lamb-Shift term which provides a Hamiltonian contribution to the dynamics and Lϱ S = γ αβ [A β (ω)ϱ S A α(ω) 1 ] 2 {A α(ω)a β (ω), ϱ S }. ω α,β This form of the dissipator (generator of the dynamics) L is know as first standard form. Diagonalizing the real positive matrix γ αβ (ω) we get the Gorini, Kossakowski, Sudarshan, Lindblad (GKSL) Markovian master equation Lϱ S = [ γ α (ω) Ā α (ω)ϱ S Ā α(ω) 1 ] 2 {Ā α(ω)āα(ω), ϱ S }. ω α Examples 1) Two-level atom in the vacuum (spontaneous emission) dϱ dt = i[h, ϱ] + Γ [ σ ϱσ {σ +σ, ϱ} ] (3.14)

18 3 Microscopic derivation of the Markovian master equation 17 where H = ω σ z 2) Lossy quantum harmonic oscillator (electromagnetic field in a cavity) [ dϱ = i[h, ϱ] + γ(n + 1) aϱa 1 ] [ dt 2 {a a, ϱ} + γn a ϱa 1 ] 2 {aa, ϱ} (3.15) where H = ω a a

19 4 Markovian semigroups 18 4 Markovian semigroups Starting from a microscopic description of system, environment and their interaction, and performing a number of approximations, the most notable of which are the weak-coupling, the Markovian and the rotating-wave (secular) approximations, we have derived a master equation in a specific operatorial form, known as the Gorini, Kossakowski, Sudarshan, Lindblad (GKSL) form, often referred to as the Lindblad form. The importance of this master equation is connected to its particularly useful properties that guarantee, for a physical initial density operator, the time evolution remains physical at all subsequent times. In other words the dissipator L is the generator of a CPTP dynamical map. In this section we recall the celebrated result derived independently by Gorini, Kossakowski and Sudarshan [15] and Lindblad [16]. Consider the simplest case of a time independent generator L Λ t = LΛ t, Λ = 1l n. (4.1) The formal solution is given by Λ t = e tl for t and hence satisfies the following composition law Λ t Λ s = Λ t+s, (4.2) for t, s. This proves that Λ t provides a semigroup of linear maps (it is a semigroup since the inverse Λ 1 t need not be CP). The properties of L are summarized in the following Theorem 3 ([15, 16]) A linear map L : M n (C) M n (C) generates a legitimate dynamical semigroup if and only if L(ρ) = i[h, ρ] + Φ(ρ) 1 2 {Φ (I n ), ρ}, (4.3) where Φ : M n (C) M n (C) is CP, Φ denotes the dual map, and H = H M n (C). In what follows we call L satisfying (4.3) a GKSL generator. Remark 1 If Φ is only a positive map, then L generates a semigroup of positive maps Λ t which need not be CP. Let Φ(ρ) = k V k ρv k, (4.4) be a Kraus representation of Φ. Its dual is represented by Φ (X) = k V k XV k. Corollary 2 A generator of a dynamical semigroup can be written in the following form L(ρ) = i[h, ρ] + ( V k ρv k 1 ) 2 {V k V k, ρ}, (4.5) k or equivalently L(ρ) = i[h, ρ] k ( [V k, ρv k ] + [V kρ, V k ] ). (4.6)

20 4 Markovian semigroups 19 Remark 2 Note, that if Φ is CP and trace preserving (i.e. a quantum channel), then Φ (I n ) = I n and hence the formula (4.3) simplifies to L(ρ) = i[h, ρ] + Φ(ρ) ρ. (4.7) Note that the dual map L defines the generator of quantum evolution in the Heisenberg picture or using the Kraus representation (4.4) L (X) = i[h, X] + Φ (X) 1 2 {Φ (I n ), X}, (4.8) L (X) = i[h, X] k ( ) [V k, XV k] + [V k X, V k]. (4.9) Let us consider simple examples of Markovian semigroups Λ t and corresponding generators L. Example 4 Suppose that Φ : M n (C) M n (C) is a quantum channel such that Φ is a CP projector, i.e. Φ 2 = Φ, and consider L = γ(φ 1l n ), (4.1) where γ > and we have assumed H =. One finds the corresponding solution Λ t = e tl = e γt e γtφ = e (1l γt n + γtφ + 1 ) 2 (γt)2 Φ +... = e γt 1l n + (1 e γt )Φ, (4.11) that is, Λ t is a convex combination of two quantum channels: 1l n and Φ. A typical example of a completely positive projector is given by Φ(ρ) = n P k ρp k, (4.12) where P k = k k, i.e. Φ projects ρ onto the diagonal: Φ(ρ) = k ρ kkp k. Hence k=1 ρ t = e γt ρ + (1 e γt ) n P k ρ P k, (4.13) which shows that diagonal elements ρ kk remain invariant and off-diagonal ρ kl are multiplied by the damping factor e γt. It is, therefore, clear that these dynamics describe pure decoherence with respect to an orthonormal basis { 1,..., n }. Example 5 Suppose that Φ : M n (C) M n (C) is a quantum channel such that Φ 2 = 1l n and consider L defined by (4.1). One finds the corresponding solution Λ t = e tl = e γt e γtφ = e (1l γt n + γtφ (γt)2 1l n + 1 ) 3! (γt)3 Φ +... = e γt[ ] cosh(γt)1l n + sinh(γt)φ, (4.14) k=1

21 5 Divisible maps and commutative maps 2 or equivalently Λ t = 1 2 (1 + e 2γt )1l n (1 e 2γt )Φ, (4.15) which is another convex combination of 1l n and Φ. To illustrate this class consider n = 2 and let Φ(ρ) = σ z ρσ z which gives rise to the following generator L(ρ) = γ(σ z ρσ z ρ). (4.16) One finds Φ 2 (ρ) = σ z (σ z ρσ z )σ z = ρ and hence the corresponding evolution is given by ρ t = Λ t (ρ) = 1 2 (1 + e 2γt ) ρ (1 e 2γt ) σ z ρσ z. (4.17) Again this corresponds to a pure decoherence: ρ 12 e 2γt ρ 12 while the diagonal elements ρ 11 and ρ 22 remain invariant. 5 Divisible maps and commutative maps We have discussed how to describe the quantum evolution of open quantum systems and we have introduced the concept of dynamical map Λ t and its time-dependent generator L t. The latter quantity is also known as the dissipator and describes the form of the master equation. The connection between the two is given by Eq. (2.29). A very important problem in the theory of open quantum systems is the following: Problem 1 What are the properties of the local time-dependent generator L t that guarantee Λ t, as defined by the T-product exponential formula (2.29), defines a legitimate dynamical map? The formulation of our problem is pretty simple, but in general the answer is not known. In fact, it turns out that the answer is only known for the special case of time-independent generators, giving rise to Markovian semigroups. In this case the answer to this important problem is given by the GKSL theorem discussed in Sec. 4. But what about more general types of generators? Let us observe that if we knew a dynamical map Λ t that was invertible, i.e. there exists Λ 1 t : M n (C) M n (C) such that Λ 1 t Λ t = Λ t Λ 1 t = 1l n, then where we defined Λ t = Λ t Λ 1 t Λ t = L t Λ t, (5.1) L t := Λ t Λ 1 t. (5.2) It should be stressed that the inverse of Λ t need not be CP. One may prove that if Λ t is CP then Λ 1 t is CP if and only if Λ t (ρ) = U t ρu t with unitary U t. Example 6 Consider a unitary dynamical map U t defined in (2.23). It is clear that U t is invertible and Ut 1 = U t is CP. One finds for the corresponding generator L t (ρ) = [ U t U t ](ρ) = U t (U t ρu t)u t + U t(u t ρu t) U t, (5.3) and hence recalling that U t satisfies the Schrödinger equation U t = ihu t, one obtains L t (ρ) = i[h, ρ].

22 5 Divisible maps and commutative maps 21 In this course instead of analyzing this problem in full generality we restrict ourselves to study special classes of dynamical maps and corresponding local generators. Specifically we analyze 3 important classes of generators C 1 a class of time-independent generators giving rise to Markovian semigroups, C 2 a class of time-dependent generators giving rise to commutative dynamics, C 3 a class of time-dependent generators giving rise to the so-called divisible dynamical maps. 5.1 Commutative dynamics We call a dynamical map Λ t commutative if [Λ t, Λ u ] = for all t, u. It means that for each A M n (C) one has Λ t (Λ u (A)) = Λ u (Λ t (A)). (5.4) It is easy to show that commutativity of Λ t is equivalent to commutativity of the local generator [L t, L u ] =, (5.5) for any t, u. Note that in this case the formula (2.3) considerably simplifies: the T product drops out and the solution is fully controlled by the integral t L udu: ( t ) t Λ t = exp L u du = 1l n + L u du + 1 ( t 2 L u du) (5.6) 2 Now, it follows from Theorem 3 that if Λ = e M, then Λ is a quantum channel if M is a GKSL generator. Therefore, one has the following Theorem 4 If L t satisfies (5.5), then L t is a legitimate generator if t L τ dτ is a GKSL generator for all t. Note that, if L t = L is time independent, then t L udu = tl and the above theorem reproduces Theorem 3. It is clear that if L is a legitimate GKSL generator and f : R + R an arbitrary function, then L t = f(t)l generates a commutative dynamical map Λ t iff t f(u)du for all t. A typical example of commutative dynamics is provided by L t = ω(t)l + a 1 (t)l a N (t)l N, (5.7) where [L α, L β ] = with L (ρ) = i[h, ρ], and for α > the generators L α are purely dephasing, that is, L α (ρ) = Φ α (ρ) 1 2 {Φ α(i), ρ}. One has for the corresponding dynamical map with Ω(t) = Λ t = e Ω(t)L e A 1(t)L1... e A N (t)l N, (5.8) t ω(u)du ; A α (t) = It is clear that Λ t is CP iff A α (t) for all α = 1,..., N. t a α (u)du.

23 5 Divisible maps and commutative maps 22 Example 7 Consider a qubit generator L (ρ) = i[σ 3, ρ] together with L 1, L 2, L 3 defined in (5.27). One easily proves and Define the time-dependent commutative generator with µ 1, µ 2 and µ 1 + µ 2 = 1. Defining Ω(t) = [L, L α ] = [L 3, L α ] = ; α = 1, 2, 3, (5.9) [L 1, L 2 ] = L 1 L 2. (5.1) L t = ω(t) 2 L + δ(t) 2 (µ 1L 1 + µ 2 L 2 ) + γ(t) 2 L z, (5.11) t ω(u)du ; (t) = t δ(u)du ; Γ(t) = t γ(u)du, (5.12) one finds that if (t) and Γ(t), then L t is a legitimate generator. The time evolution of ρ has the following form: the off-diagonal elements evolve according to and diagonal elements ρ 12 e Ω(t)+ 1 2 (t)+γ(t) ρ 12, ρ 11 ρ 11 e (t) + µ 1 [1 e (t)], ρ 22 ρ 22 e (t) + µ 2 [1 e (t)]. If (t) for t, then the dynamics possess an equilibrium state ( ) µ1 ρ t. µ 2 Example 8 (Random unitary qubit dynamics) Consider the following time-dependent generator L t (ρ) = 1 3 γ k (t)(σ k ρ σ k ρ), (5.13) 2 k=1 where {σ 1, σ 2, σ 3 } are Pauli matrices. It is easy to prove that [L t, L u ] = and hence L t generates a legitimate dynamical map iff Γ 1 (t) ; Γ 2 (t) ; Γ 3 (t), where Γ k (t) = t γ k(u)du. One finds that the corresponding dynamical map Λ t is given by Λ t (ρ) = 3 p α (t)σ α ρ σ α, (5.14) α=

24 5 Divisible maps and commutative maps 23 where σ = I 2 and with p (t) = 1 4 [1 + λ 3(t) + λ 2 (t) + λ 1 (t)], p 1 (t) = 1 4 [1 λ 3(t) λ 2 (t) + λ 1 (t)], p 2 (t) = 1 4 [1 λ 3(t) + λ 2 (t) λ 1 (t)], p 3 (t) = 1 4 [1 + λ 3(t) λ 2 (t) λ 1 (t)], λ 1 (t) = e Γ 2(t) Γ 3 (t), and similarly for λ 2 (t) and λ 3 (t). Interestingly Λ t (σ k ) = λ k (t)σ k. The formula (5.14) defines so-called random unitary dynamics. Note that p (t) + p 1 (t) + p 2 (t) + p 3 (t) = 1. Moreover, p α (t) for α =, 1, 2, 3 iff Γ k (t) for k = 1, 2, 3. Note that Λ t is unital. Actually, in the case of qubits any unital dynamical map is random unitary, i.e. Λ t (ρ) = k p k (t)u k (t)ρu k (t), (5.15) where p k (t) defines time-dependent probability distribution and U k (t) is a family of time-dependent unitary matrices. It is no longer true for n-level systems with n > Markovian evolution divisible dynamical maps We call a dynamical map Λ t divisible if for any t s one has the following decomposition Λ t = V t,s Λ s, (5.16) with a completely positive propagator V t,s. Note, that if Λ t is invertible then and hence V t,s satisfies an inhomogeneous composition law V t,s = Λ t Λ 1 s, (5.17) V t,s V s,u = V t,u, (5.18) for any t s u. The above formula provides a generalization of the semi-group composition law (4.2). Here we assume the following extended definition of Markovian evolution: a dynamical map Λ t corresponds to Markovian evolution if and only if it is divisible. Interestingly, the property of being Markovian (or divisible) is fully characterized in terms of the local generator L t. Note, that if Λ t satisfies (2.28) then V t,s satisfies d dt V t,s = L t V t,s, V s,s = 1l, (5.19)

25 5 Divisible maps and commutative maps 24 and the corresponding solution reads ( t ) V t,s = T exp L u du. (5.2) s It is clear that Λ t = V t, which shows that divisibility puts very strong requirements upon the dynamical map Λ t. One proves the following Theorem 5 The map Λ t is divisible if and only if L t is a GKSL generator for all t, that is, L t (ρ) = i[h(t), ρ] k ( [V k (t), ρv k (t)] + [V k(t)ρ, V k (t)] ) with time-dependent Hamiltonian H(t) and noise operators V k (t). Remark 3 If, (5.21) L t = ω(t)l + a 1 (t)l a N (t)l N, (5.22) where L (ρ) = i[h, ρ], and for α > the generators L α are purely dissipative and linearly independent, then L t generates Markovian evolution if and only if a 1 (t),..., a N (t). Example 9 Consider a qubit generator L t (ρ) = 1 2 γ(t)l 3(ρ) = 1 2 γ(t)(σ zρσ z ρ), (5.23) and introduce Γ(t) = t γ(τ)dτ, then it is clear that Λ t (ρ) = 1 [ 1 + e Γ(t)] ρ + 1 [ 1 e Γ(t)] σ z ρσ z, (5.24) 2 2 and hence 1. L t is a legitimate generator iff Γ(t), 2. L t generates Markovian evolution iff γ(t), 3. L t generates a Markovian semigroup iff γ(t) = const. >. Example 1 Let us consider a qubit generator defined by H = ω 2 σ z and the following CP map Φ(ρ) = γ 1 σ + ρ σ + γ 2 σ ρ σ + + γσ z ρ σ z, (5.25) where σ + = 2 1 and σ = 1 2 = σ + are standard qubit raising and lowering operators. The corresponding generator reads L(ρ) = i[h, ρ] + L D (ρ) with the dissipative part L D = γ 1 2 L 1 + γ 2 2 L 2 + γ 2 L z, (5.26)

26 5 Divisible maps and commutative maps 25 where L 1 (ρ) = [σ +, ρσ ] + [σ + ρ, σ ], L 2 (ρ) = [σ, ρσ + ] + [σ ρ, σ + ], (5.27) L 3 (ρ) = σ z ρσ z ρ. L 1 corresponds to pumping (heating) process, L 2 corresponds to relaxation (cooling), and L 3 is responsible for pure decoherence. To solve the master equation ρ t = Lρ t let us parameterize ρ t as follows ρ t = p 1 (t)p 1 + p 2 (t)p 2 + α(t)σ + + α(t)σ, (5.28) with P k = k k. Using the following relations L(P 1 ) = γ 1 (P 2 P 1 ) = γ 1 σ 3, L(P 2 ) = γ 2 (P 1 P 2 ) = γ 2 σ 3, L(σ + ) = (iω η) σ +, L(σ ) = ( iω η) σ, where η = γ 1+γ 2 2 +γ, one finds the following Pauli master equations for the probability distribution (p 1 (t), p 2 (t)) ṗ 1 (t) = γ 1 p 1 (t) + γ 2 p 2 (t), (5.29) ṗ 2 (t) = γ 1 p 1 (t) γ 2 p 2 (t), (5.3) together with α(t) = e (iω η)t α(). The corresponding solution reads [ ] p 1 (t) = p 1 () e (γ 1+γ 2 )t + p 1 1 e (γ 1+γ 1 )t, (5.31) [ ] p 2 (t) = p 2 () e (γ 1+γ 2 )t + p 2 1 e (γ 2+γ 2 )t, (5.32) where we introduced p 1 = γ 1 γ 1 + γ 2, p 2 = γ 2 γ 1 + γ 2. (5.33) Hence, we have purely classical evolution of the probability vector (p 1 (t), p 2 (t)) on the diagonal of ρ t and very simple evolution of the off-diagonal element α(t). Note that asymptotically one obtains a completely decohered density operator ( ) p ρ t 1 p. 2 In particular if γ 1 = γ 2 a state ρ t relaxes to the maximally mixed state (a state becomes completely depolarized).

27 6 Projection operator techniques 26 6 Projection operator techniques This section is heavily based on Ref. [1]. The projection operators techniques that we will describe in this section aim at deriving the dynamics of an open quantum system starting from the total unitary evolution. We are going to show two possible techniques that go beyond the Born-Markov approximation, allowing us to describe open systems even when system-environment correlations are not negligible. These two techniques are 1) the Nakajima-Zwanzing technique; 2) the TCL (Time-ConvolutionLess) technique. They both rely on the idea of describing the partial trace operation in terms of a formal projection operator P acting on the total density operator ϱ ϱ = Pϱ(relevant part) + Qϱ(unrelevant part), P + Q = I. 6.1 Nakajima-Zwanzing technique We start from the Hamiltonian of the total system H = H +αh I, with H the free Hamiltonian of system and environment, and H I the interaction Hamiltonian. The total density operator evolves in time according to the von Neumann equation (in the interaction picture and in units of ) dϱ dt (t) = iα[h I(t), ϱ(t)] = αl(t)ϱ(t). (6.1) We define the projection operator P, describing the relevant part of the total system (i.e., what we are interested in) Pϱ Tr B [ϱ] ϱ B, (6.2) where ϱ B is a fixed environment state. We also define the projection operator Qϱ = ϱ Pϱ, describing the irrelevant part of the total system. Being projectors, P and Q satisfy the properties: P + Q = I, P 2 = P, Q 2 = Q. [P, Q] = We now make a technical assumption: we require that the odd moments of H I vanish with respect to the environment reference state, i.e., Tr B [H I (t 1 ) H I (t 2n+1 )ϱ B ] = PL(t 1 ) L(t 2n+1 )P = n. (6.3) Our aim is to derive a closed equation for Pϱ. By applying the projectors to the von Neumann equation we get { Pϱ = αplϱ, t t Qϱ = αqlϱ. Remembering the first of the P and Q properties one obtains { tpϱ = αplpϱ + αplqϱ, tqϱ = αqlpϱ + αqlqϱ.

28 6 Projection operator techniques 27 Given an initial state ϱ(t ), we get the following formal solution for the second equation where we have defined t Qϱ(t) = G(t, t )Qϱ(t ) + α dsg(t, s)ql(s)pϱ(s), t (6.4) [ t ] G(t, t ) T exp α dsql(s), t and where T is the chronological time-ordering operator. By using this equation we obtain the following closed equation for Pϱ Pϱ t t (t) = αpl(t)g(t, t )Qϱ(t ) + α 2 t dspl(t)g(t, s)ql(s)pϱ(s). (6.5) The equation above is called the Nakajima-Zwanzig master equation. It is worth noticing that this is an exact equation for Pϱ. We call the integrand in the second term the memory kernel K(t, s) = PL(t)G(t, s)ql(s)p. This term describes the memory effects due to non-negligible correlations between system and environment. Note that the Nakajima-Zwanzig equation, contrary to all master equations studied so far, is an integro-differential equation, namely the future dynamics of the open system depends on its past history and not only on its initial state. The first term in Eq. (6.5) is known as the inhomogeneity term. This term takes into account the effect of initial correlations between system and environment. Assuming factorized initial conditions we get Pϱ(t ) = ϱ(t ) and Qϱ(t ) =, hence the inhomogeneity term vanishes in the absence of correlations between system and environment at the initial time. Note that, if K(t, s) δ(t s), then the Nakajima-Zwanzig equation becomes time-local, i.e., there are no memory effects. Let us now consider approximations of the Nakajma-Zwanzig equation for small couplings α. Expanding to second order in α and for factorised initial conditions we get Pϱ t (t) = α2 dspl(t)ql(s)pϱ(s). t t After some manipulations, and remembering the expression for P and L, we obtain ϱ t S t (t) = α2 Tr B { ds[h I (t), [H I (s), ϱ S (s) ϱ B ]]} (6.6) t which is Eq. (3.9) of the microscopic derivation in Sec Time-convolutionless technique The main idea behind the time-convolutionless (TCL) projection operator technique is to eliminate the memory kernel in the Nakajima-Zwanzig equation, transforming it into a time-local equation. The reason why this would be useful is that generally differential equations are simpler to handle than integro-differential equations. Of course, in doing so, we should ask ourselves: Are we performing any additional approximations? In other words are the Nakajima-Zwanzig and the TCL final forms of master equations both exact?

29 6 Projection operator techniques 28 Let us begin by again considering the equation for the irrelevant part of the system t Qϱ(t) = G(t, t )Qϱ(t ) + α dsg(t, s)ql(s)pϱ(s). t We introduce the time-reverse propagator from s < t to t for the total (closed) system [ Ḡ(t, s) T exp α t s ] ds L(s ). (6.7) We rewrite ϱ(s) = Ḡ(t, s)(p + Q)ϱ(t) and we use this expression in the equation for Qϱ(t), getting t Qϱ(t) = G(t, t )Qϱ(t ) + α dsg(t, s)ql(s)pḡ(t, s)(p + Q)ϱ(t). t We define the super-operator t Σ(t) α dsg(t, s)ql(s)pḡ(t, s) t (6.8) thus getting Qϱ(t) = G(t, t )Qϱ(t ) + Σ(t)Pϱ(t) + Σ(t)Qϱ(t), [I Σ(t)]Qϱ(t) = G(t, t )Qϱ(t ) + Σ(t)Pϱ(t). The operator Σ(t) has the following properties: Since it contains both G(t, t ) and order; Ḡ(t, s) it does not have a well defined chronological Σ(t ) = ; Σ(t) α= =. For values of t t small enough we can therefore assume that the inverse of the operator I Σ(t) exists. Hence we have Qϱ(t) = [I Σ(t)] 1 G(t, t )Qϱ(t ) + [I Σ(t)] 1 Σ(t)Pϱ(t). The first term on the r.h.s. is zero for factorised initial conditions, thus we get a time-local equation for the irrelevant part Qϱ(t) = [I Σ(t)] 1 Σ(t)Pϱ(t). Inserting this new expression in the equation for the relevant part Pϱ, we get the TCL equation Pϱ t (t) = αpl(t)[i Σ(t)] 1 Σ(t)Pϱ(t) (6.9)

30 6 Projection operator techniques 29 with time-local kernel K(t) αpl(t)[i Σ(t)] 1 Σ(t)P. We expand [I Σ(t)] 1 as follows [I Σ(t)] 1 = + n= [Σ(t)] n, so we get the following expansions + K(t) = αpl(t) [Σ(t)] n P = n=1 + n=1 α n K n (t) Σ(t) = + k= α k Σ k (t). It is instructive to look at the first terms of the expansion, where K 1 (t) = PL(t)P = K 2 (t) = PL(t)Σ 1 (t)p Σ 1 (t) = t dsql(s)p Using this approximation it is easy to show that the 2nd-order TCL master equation (TCL2) coincides with Eq. (6.6), which was obtained in Sec. 3. Note that both the Nakajima-Zwanzig and the TCL master equations are exact at the same order in α, however the latter one is clearly simpler.

31 7 Exact approaches: The Jaynes-Cummings model with losses 3 7 Exact approaches: The Jaynes-Cummings model with losses 7.1 The model and its exact solution We will now study a paradigmatic example of an open quantum system amenable to an exact solution. Solving the dynamics exactly, to all orders in the coupling constant, allows us to compare the exact approach with perturbative TCL solutions. The physical model we investigate is a two-level atom in a bosonic reservoir at zero temperature. The total Hamiltonian is ( = 1) H = H S + H E + H I where H S = ω 2 σ z, H E = ω k b k b k, k (7.1) H I = k (g k σ + b k + g k σ b k ), where σ z is the Pauli matrix, σ ± the raising and lowering operators, b k, b k the bosonic creation/annihilation operators of the environment, the index {k} labels the different field modes of the reservoir with frequencies ω k, and g k are the coupling constants. The interaction Hamiltonian describes a process where an excitation is exchanged between the system and the environment. It is important to remark that this type of interaction conserves the total number of excitations initially present in the total system, and indeed the existence of this constant of motion is crucial for the derivation of an exact solution. In the following we will derive the exact solution for the total closed system and, from this solution, we will obtain the dynamics of the open system. Assuming that only one excitation is present in the total system, the total Hilbert space is spanned by the following states { ψ = g, ψ 1 = e, ψ k = g 1 k }, where = k k is the multimode vacuum state of the reservoir, 1 k denotes the state with one photon in mode k of the environment and e and g are the excited and ground states of the two-level atom, respectively. Let us consider the Schroedinger equation in the interaction picture where d dt ψ(t) = ih I(t) ψ(t) H I (t) = exp[i(h S + H E )t]h I exp[ i(h S + H E )t] = k g k σ + (t)b k (t) + g k σ (t)b k (t), with σ ± (t) = σ ± ()e ±iωt and b k (t) = b k e iωkt. Since the total number of excitations N, given by N = σ z + b k b k k

32 7 Exact approaches: The Jaynes-Cummings model with losses 31 is a constant of motion, i.e. [H, N] =, the total state of the system ψ(t) can be written as ψ(t) = c (t) ψ + c 1 (t) ψ 1 + k c k (t) ψ k. (7.2) Noting that H I (t) ψ =, one can see that the amplitude c (t) = c is constant. Inserting (7.2) in the Schroedinger equation we get the following set of coupled differential equations for the coefficients { ċ 1 = i k g ke i(ω ω k )t c k ċ k = ig k e i(ω ω k )t c 1 (7.3) We can formally solve the equation for the c k s, assuming that c k () =, c k (t) = ig k and, inserting Eq. (7.4) in the equation for c 1 (t), we obtain t dt e i(ω ω k )t c 1 (t ) (7.4) ċ 1 (t) = k t g k 2 dt e i(ω ω k )(t t ) c 1 (t ). (7.5) Passing to the continuum limit, i.e. replacing the discrete spectral distribution with a continuous spectral density, g k 2 dωj(ω) we can define the reservoir correlation function f(t t ) as follows f(t t ) = dωj(ω)e i(ω ω)(t t ). k It is a simple but useful exercise to show that, if the environment is in the vacuum state ϱ B =, then dωj(ω)e iω(t t ) = Tr B [B(t)B (t )ϱ B ], where B(t) = k g kb k exp( iω k t). Moreover, if J(ω) is an integrable function, we can exchange the order of integration in Eq. (7.5) obtaining ċ 1 (t) = t dt f(t t )c 1 (t ). (7.6) The reduced density matrix of the two-level system is obtained from the total system pure state by tracing over the reservoir degrees of freedom: ϱ S (t) = Tr B { ψ(t) ψ(t) } = ψ(t) ψ(t) + k 1 k ψ(t) ψ(t) 1 k = c 2 g g + c c 1 (t) g e + c 1(t)c e g + k c k (t) 2 g g + c 1 (t) 2 e e. Since c 2 + c 1 (t) 2 + k c k(t) 2 = 1 we can rewrite ϱ S (t) as follows ( c1 (t) ϱ S (t) = 2 c c ) 1(t) c c 1 (t) 1 c 1(t) 2. (7.7)

33 7 Exact approaches: The Jaynes-Cummings model with losses 32 The equation above shows that the dynamics of the reduced system only depends on the coefficient c 1 (t) and on the initial condition c. Let us define ] [ċ1 (t) S(t) = 2I, (7.8) c 1 (t) ] [ċ1 (t) γ(t) = 2R, (7.9) c 1 (t) i.e. the time-dependent Lamb shift S(t) and the decay rate γ(t) are the imaginary and real part, respectively, of the ratio between the derivative of c 1 (t) and the coefficient itself. By deriving the solution for the reduced density matrix, i.e. Eq. (7.7), it is straightforward to show that the master equation is given by dϱ S (t) = is(t) [σ z, ϱ S (t)] + γ(t) dt 2 [ σ ϱ S (t)σ + 1 ] 2 {σ +σ, ϱ S (t)}. (7.1) This is the exact master equation for the system dynamics. However, for the equation above to be considered a proper master equation, the Lamb shift S(t) and the decay rate γ(t) should be properties of the open system only and should not depend on the initial state of the system. This doesn t seem to be the case from Eqs. (7.8)-(7.9). One should notice that, as Eq. (7.6) is a first order integro-differential equation, it admits one independent solution only and in particular it admits a solution of the form c 1 (t) = cf(t), with f(t) a time dependent function and c a constant. Hence, the general solution can be written as c 1 (t) = c 1 ()f(t), which eliminates the dependence on the initial condition in both S(t) and γ(t). Another important remark is that Eq. (7.1) is exact; however, it is in time local form, i.e. it does not contain memory kernels. One could argue whether this equation always exists (remember our discussion on the existence of the inverse superoperator (I Σ(t)) 1 in the TCL projection operator derivation). It is easy to see that the time-dependent decay rate diverges if c 1 (t) = (and ċ 1 (t) ) for certain time instants. In this case, even if the solution of the master equation is well defined, the master equation is not. In other words a TCL master equation does not exist. Finally, we note that, in order to apply the method described in this section, we have assumed that the spectral density function is integrable. This is not the case, for example, in certain models of photonic band gap materials, wherein J(ω) = c ω ωe θ(ω ω e ). (7.11) 7.2 Lorentzian spectral density A paradigmatic example of an exact model for an open quantum system is the case of a two level atom interacting with the electromagnetic field in the vacuum, with a Lorenztian spectral density J(ω) = 1 2π γ λ 2 (ω ω) 2 + λ 2. (7.12) A Lorentzian spectrum describes, for example the quantized field inside a single mode high Q optical cavity. For this type of spectrum, the correlation function f(t t ) = dωj(ω)e i(ω ω)(t t ) =

34 7 Exact approaches: The Jaynes-Cummings model with losses 33 γ λ 2 e λ(t t ) describes an exponential decay with rate λ. If we now take the Laplace transform of Eq. (7.6), namely s c(s) c() = f(s) c(s), we get [ ] 1 c(s) = s + f(s) c(). Performing the inverse Laplace transform leads to [ ] Lap 1 1 s + f(s) = ( 1 (n k 1)! k Hence, we get to the following solution c 1 (t) = c 1 ()e λt/2 [cosh d (n k 1) ds ) e st s + f(s) (s p k) n k ( ) ( ) s+ + λ = e s +t s + λ + e s t. s + s s s + ( ) λt 1 2R R sinh s=pk ( )] λt 1 2R. (7.13) 2 where R = γ /λ. From this equation we can calculate the reduced density matrix elements, e.g. the excited state probability ϱ ee (t) = c 1 (t) 2. Weak and strong coupling We can switch from weak to strong coupling by varying the R parameter, which quantifies the competitive effects of coupling and dissipation. Fig.1 shows the change in the dynamics of ρ ee (t) in the two regimes: R < 1/2, R > 1/2 corresponding to weak (red curve) and strong (blue curve) coupling, respectively. The existence of zeros in ρ ee (t) for strong couplings signals that a TCL master equation does not exist Figure 1: ρ ee (t) for different choices of R: R < 1/2 (red) corresponds to weak coupling, whereas R < 1/2 (blue) corresponds to strong coupling.

35 7 Exact approaches: The Jaynes-Cummings model with losses 34 Master equation Recalling Eq.(7.9) we can derive the time dependent decay rate γ(t) for the Lorentzian spectrum γ(t) = 2R } {ċ1 (t) = c 1 (t) 1 2R cosh ( λt 2 2γ sinh ( ) λt 2 1 2R ) ( 1 2R + sinh λt 2 1 2R ) which diverges in the strong coupling limit. Formally, this means that the superoperator [I Σ(t))] 1 is not defined. TCL2 Using the TCL projection operator technique one can derive the following time-local master equation [ ] dϱ s (t) = α n K n (t) ϱ s (t). (7.14) dt n=1 One can show that the operatorial form of the master equation looks the same at any expansion order (neglecting Lamb-Shift terms) [ ] dϱ s (t) = α 2n γ 2n (t) Lϱ s (t) (7.15) dt n=1 where L is the GKSL operator. To the second order in the coupling constant α we get For the Lorentzian spectrum we find γ 2 (t) + is 2 (t) = 2 t dt 1 f(t t 1 ). S 2 (t) =, γ 2 (t) = γ (1 e λt ). (7.16) 7.3 Markovian and strong coupling limits of the exact solution Recalling Eq.(7.13) ϱ 11 (t) = c 1()e λt/2 [ cosh ( ) λt 1 2R R sinh ( )] λt 2 1 2R, 2 where the correlation time of the reservoir is τ = 1/λ. In the Markovian limit λ but λγ is constant (γ ). In other words the Markovian limit in this model corresponds at taking both a flat reservoir spectrum λ and very weak coupling γ, the two assumptions are not independent. By expanding the above equation in the R limit we get c 1 (t) c 1 ()e λt 2 [ e (1 2R)λt/2 2 + (1 + 2R)e(1 2R)λt/2 2 = c 1 ()e λt 2 e (1 2R)λt/2 c 1 ()e Rλt 2 = c 1 ()e γ t 2. ]

36 7 Exact approaches: The Jaynes-Cummings model with losses 35 Hence, ϱ Mark 11 (t) = ϱ 11 ()e 2γt. (7.17) This limit also leads to J(ω) γ /2π, f(τ) γ δ(τ). In the opposite limit, known as the strong coupling limit, where γ and λ we have approximately unitary dynamics (Jaynes-Cummings type) ( ) ϱ JC λγ 11 (t) ϱ 11 () cos 2 t. (7.18) 7.4 Appendix: Some useful properties of Laplace transform Given a function f we define its Laplace transform F as F (s) = Lap[f(t)] = + Below are some important properties to remember Lap[αf(t) + βg(t)] = α f(s) + β g(s), [ ] Lap df dt (t) = s f(s) f(), dte st f(t) = f(s) s C. Lap[f g(t)] = f(s) g(s), f g(t) = t dt 1f(t t 1 )g(t 1 ). Also, some important and useful Laplace transforms to keep in mind Lap[α] = α s, Lap[t] = 1 s 2, Lap[e αt ] = 1 s α Re[s] > Re[α]. The inverse Laplace transform is defined as Lap 1 [ f(s)] = f(t) = 1 2πi γ+i γ i In order to compute it we need the Residue theorem n dzf(z) = 2πi Res[f, p k ], γ k=1 dse st f(s). where γ is a continuous path in the complex plane and p k is a n k order pole for the function f. The residues are defined as ( ) 1 (nk 1) [ Res[f, p k ] = lim f(z)(z p k ) nk]. (n k 1)! z p k z It immediately follows that Lap 1 [ f(s)] = n k=1 ( ) 1 (nk 1) [ e st f(s)(s pk ) nk]. (n k 1)! s

37 8 Quantum Brownian motion 36 8 Quantum Brownian motion The archetype of an open system is a Brownian particle suspended in some environment at temperature T. Classically, we can observe the erratic trajectory of such a Brownian particle. Quantum mechanically, such a classical trajectory ought to be some semiclassical limit of the dynamics of a localised wave packet. The quantum Brownian motion model is the quantum generalisation of the Brownian motion and it is therefore a paradigmatic model of the theory of open quantum system. This model was used by Zurek and collaborators to propose environment induced decoherence as a way to understand the transition between the quantum and the classical worlds. In the following we will follow Zurek s approach to quantum Brownian motion. 8.1 Perturbative Master Equation This Section is mostly taken from Ref. [18] The system of interest is a quantum particle, which moves in a one dimensional space (generalization to higher dimensions is immediate). The environment is an ensemble of harmonic oscillators interacting bilinearly through position with the system. Thus, the complete Hamiltonian is H = H S + H E + V where H E = ( 1 p 2 n + 1 ) 2m n n 2 m nωnq 2 n 2 and V = n λ nq n x. The Hamiltonian of the system will be left unspecified for the moment (we will concentrate later on the case of a harmonic oscillator). The initial state of the environment will be assumed to be a thermal equilibrium state at temperature T = 1/k B β. Under these assumptions the first-order term in the master equation disappears because T r E (Ṽ (t)ρ E) =, where Ṽ (t) indicates the interaction Hamiltonian in the interaction picture. Therefore, the master equation in the Schröedinger picture and in weak coupling (second order in the coupling constant) can be shown to be of the form ρ = 1 i [H S, ρ] 1 t ) dt 1 (ν(t 1 )[x, [x( t 1 ), ρ]] iη(t 1 )[x, {x( t 1 ), ρ}]. (8.1) The two memory kernels appearing here are respectively called the noise and the dissipation kernel and are defined as ν(t) = 1 2 η(t) = i 2 λ 2 n {q n (t), q n ()} = n λ 2 n [q n (t), q n ()] = n dωj(ω) cos ωt(1 + 2N(ω)) dωj(ω) sin ωt, (8.2) where J(ω) = n λ2 nδ(ω ω n )/2m n ω n is the spectral density of the environment and N(ω) is the mean occupation number of the environmental oscillators (i.e., 1 + 2N(ω) = coth(β ω/2)). Equation (8.1) is already very simple but it can be further simplified if one assumes that the system is a harmonic oscillator. Thus, if we consider the Hamiltonian of the system to be H S = p 2 /2M +MΩ 2 x 2 /2, we can explicitly solve the Heisenberg equations for the system and determine

38 8 Quantum Brownian motion 37 the operator x(t) to be x(t) = x cos(ωt) + 1 expression for the master equation, ρ = i MΩ p sin(ωt). Inserting this into (8.1), we get the final [ HS M Ω 2 (t)x 2, ρ ] i γ(t)[ x, { p, ρ }] D(t) [ x, [ x, ρ ]] 1 f(t)[ x, [ p, ρ ]]. (8.3) Here the time dependent coefficients (the frequency renormalization Ω(t), the damping coefficient γ(t), and the two diffusion coefficients D(t) and f(t)) are Ω 2 (t) = 2 M D(t) = 1 t t dt cos(ωt )η(t ), γ(t) = 1 MΩ dt cos(ωt )ν(t ), f(t) = 1 MΩ t t dt sin(ωt )η(t ) dt sin(ωt )ν(t ). (8.4) From this equation it is possible to have a qualitative idea of the effects the environment produces on the system. First we observe that there is a frequency renormalization. Thus, the bare frequency of the oscillator is renormalized by Ω 2. This term does not affect the unitarity of the evolution. The terms proportional to γ(t), D(t) and f(t) bring about non unitary effects. The second term is responsible for producing friction (γ(t) plays the role of a time dependent relaxation rate). The last two are diffusion terms. The one proportional to D(t) is the main cause for decoherence. We will now make a very important observation about the master equation for quantum Brownian motion given by Eq. (8.3). From an exact treatment of the dynamics relying on path integral formalism (See Sec 8.2) one can prove that the exact master equation for this system has the same operatorial form of Eq. (8.3) with the only different being the time-dependent coefficients. In particular, in the exact case the time-dependent coefficients are not obtainable as closed analytical expressions but they do coincide with Eqs. (8.4) in the weak-coupling limit. Of course, the explicit time dependency of the coefficients can only be computed once we specify the spectral density of the environment. To illustrate their qualitative behavior, we will consider a typical Ohmic environment characterized by a spectral density of the form ω Λ 2 J(ω) = 2Mγ π Λ 2 + ω 2, (8.5) where Λ plays the role of a high frequency cutoff and γ is a constant characterizing the strength of the interaction. For this environment, it is rather straightforward to find the following exact expressions for the coefficients Ω(t) and γ(t): γ(t) = γ Λ 2 Λ 2 + Ω 2 ( 1 Ω 2 (t) = 2γ Λ Λ2 Λ 2 + Ω 2 (cos Ωt + ΛΩ sin Ωt ) exp( Λt) ) (8.6) ( 1 (cos Ωt ΩΛ ) ) sin Ωt exp( Λt). (8.7) From these equations we see that these coefficients are initially zero and grow to asymptotic values on a timescale that is fixed by the high frequency cutoff Λ.

39 8 Quantum Brownian motion 38 The time dependence of the diffusion coefficients can also be studied for the above environment. However, the form of the coefficients for arbitrary temperature is quite complicated. To analyze their qualitative behavior, it is convenient to evaluate them numerically. In Figure 3 one can see the dynamics of the coefficients (for both the long and short timescales) for several temperatures (high and low). We observe that both coefficients have an initial transient where they exhibit a behavior that is essentially temperature independent (over periods of time comparable with the one fixed by the cutoff). The direct diffusion coefficient D(t) after the initial transient rapidly settles into the asymptotic value given by D = Mγ Ω coth(β Ω/2)Λ 2 / (Λ 2 + Ω 2 ). The anomalous diffusion coefficient f(t) also approaches an asymptotic value (which for high temperatures is suppressed with respect to D by a factor of Λ), but the approach is algebraic rather than exponential. More general environments can be studied using our equation. In fact, the behavior of the coefficients is rather different for environments with different spectral content. It is interesting to mention that the master equation (8.3) (although it has been derived perturbatively) can be shown to be very similar to its exact counterpart whose derivation we will discuss later in this section. 8.2 Exact Master Equation The following section is not compulsory but is given here for completeness After presenting some simple perturbative master equations one may wonder under what circumstances are they a reasonable approximation. To partially address this issue, it is interesting to compare these equations with the ones that can be obtained for exactly solvable problems. In particular, we describe the master equation for a model that has been thoroughly studied in connection with decoherence, i.e., the linear quantum Brownian motion. Thus, because the Hamiltonian is quadratic both in the coordinates of the system and the environment, it is not surprising that it can be exactly solved. In this subsection, we will describe a simple derivation of the exact master equation, discuss its main features, and show that its functional form is the same as the one obtained by using perturbation theory. Indeed, the exact master equation has the same functional form as (8.3), the only difference being that the time dependence of the coefficients is different in general, as expected. It is interesting to note that the exact master equation for QBM has only been found quite recently in spite of the simplicity of the model (in particular, the fact that it can always be written as an equation that is local in time was not appreciated until Ref. [19]). The original derivation of the exact master equation is not so simple. Here we will present a simpler derivation of the exact master equation, which is done following the method proposed first in [2]. Previous studies of the master equation for QBM, obtained under various approximations, include the celebrated paper by Caldeira and Leggett [21] among others. The derivation will focus on properties of the evolution operator for the reduced density matrix. This operator will be denoted as J and is defined as the one that enables us to find the reduced density matrix (in position representation) at some arbitrary time from the initial one. Thus, by definition, this operator satisfies: ρ(x, x, t) = dx dx J(x, x, t; x, x, t )ρ(x, x, t ). (8.8)

40 8 Quantum Brownian motion 39 Figure 2: Time dependence of the diffusion coefficients of the perturbative master equation for quantum Brownian motion. Plots on the right show that the initial transient is temperature independent (different curves correspond to different temperatures, higher temperatures produce higher final values of the coefficients). Plots on the left show that the final values of the coefficients are strongly dependent on the temperature of the environment. The parameters used in the plot (where time is measured in units of 1/Ω) are γ/ω =.5, Λ/Ω = 1, k B T/ Ω = 1, 1,.1.

41 8 Quantum Brownian motion 4 The derivation of the exact master equation has two essential steps. The first step is to find an explicit form for the evolution operator of the reduced density matrix. The second step is to use this explicit form to obtain the master equation satisfied by the reduced density matrix. To make our presentation simpler, we postpone the proof of the first step, which will be done below using path integral techniques Part 1 Here, we first want to demonstrate how to obtain the master equation once we know the explicit form of the evolution operator. So, let us show what the evolution operator for the reduced density matrix looks like. For linear QBM we will show later that it can always be written as J(X, Y, t; X, Y, t ) = b 3 2π exp i (b 1XY + b 2 X Y b 3 XY b 4 X Y ) exp ( a 11 Y 2 a 12 Y Y a 22 Y 2 ), (8.9) where for notational convenience we are using sum and difference coordinates (i.e., X = x + x, Y = x x, etc) and the coefficients b i and a jl are time dependent functions whose explicit form will be given below (and depend on the properties of the environment). Thus, the evolution operator (8.9) is simply a Gaussian function of its arguments with time dependent coefficients. This comes as no surprise because the problem is linear. Knowing the propagator for the reduced density matrix, it is easy to obtain the master equation following the simple method described in [2]. This is the second step of the derivation of the master equation and is done as follows. We compute the temporal derivative of the propagator J noting that the only time dependence is through the coefficients b i and a jl. Thus, we obtain J = ( ḃ3 b 3 + i(ḃ1xy + ḃ2x Y + ḃ3xy + ḃ4x Y ) ȧ 11 Y 2 ȧ 12 Y Y ȧ 22 Y 2 ) J. (8.1) Using this equation, we can try to find the master equation through multiplying by the initial density matrix and integrating this over the initial coordinates. The master equation would be trivially obtained in this way if, after multiplying by the initial density matrix, we could integrate over all the initial coordinates. This is straightforward, with some of the terms appearing in (8.1) but it is not so obvious how to handle terms that explicitly depend upon the initial coordinates X and Y. Fortunately, there is a simple trick that we can use: because we know that the propagator (8.9) is Gaussian, we can make use of this fact to obtain the following simple relations: Y J = X J = ( b1 b 3 Y + i b 3 X ) J, and [ b 1 b 2 X i b 2 Y i ( 2a11 b 2 + a 12b 1 b 2 b 3 Y ) + a ] 12 X J. b 2 b 3 These two equations can be used in (8.1) and in this way we can express the right hand side of

42 8 Quantum Brownian motion 41 this equation entirely in terms of the reduced density matrix. The resulting master equation is ρ(x, x ) = 1 i x [H R(t), ρ] x γ(t)(x x )( x x)ρ(x, x ) D(t)(x x ) 2 ρ(x, x ) + if(t)(x x )( x + x)ρ(x, x ). (8.11) The coefficients appearing in this equation are determined by b i and a jl as follows: Ω 2 (t) = 2(ḃ2b 1 /b 2 ḃ1) γ(t) = ḃ2/2b 2 b 1 D(t) = ȧ 11 4a 11 b 1 + ȧ 12 b 1 /b 2 ḃ2(2a 11 + a 12 b 1 /b 3 )/b 2 2f(t) = ȧ 12 /b 3 ḃ2a 12 /b 2 b 3 4a 11. (8.12) Thus, we showed that the exact master equation is a simple consequence of the Gaussian form of the evolution operator (8.9). To complete our derivation of this equation we need to explicitly show how to obtain equation (8.9) and also find the explicit form of the time dependent coefficients (which is also required to simplify the expressions leading to the master equation (8.11) Part 2 To obtain the explicit form of the evolution operator we will follow a derivation based on the use of path integral techniques (see [19, 2]). To understand it, very little previous knowledge of path integrals is required. The main ingredient is the path integral expression for the evolution operator of the complete wave function. Thus, if the action of the combined system is S T [x, q], the matrix elements of the evolution operator U can be written as U(x, q, t; x, q, t ) = DxDq e is T [x,q],. (8.13) where the integration is over all paths that satisfy the boundary conditions, x() = x, x(t) = x, q() = q, q(t) = q. (8.14) In the above and following equations, to avoid the proliferation of sub-indices we use q to collectively denote all the coordinates of the oscillators q n (we will not write the subscript n that should be implicitly assumed). Using this equation, one can obtain a path integral representation of the evolution operator of the complete density matrix and, after taking the final trace over the environment, we find a path integral representation of the propagator for the reduced density matrix. It is clear that the resulting expression will involve a double path integral (one to evolve kets and another one to evolve bras). For a generic initial state ρ T, the propagator is a somewhat complicated looking expression. To simplify our presentation, we will only consider here factorizable initial states. Thus, if the initial state can be factored we can express the reduced density matrix at arbitrary times as a function of the reduced density matrix at initial time using a (state independent) propagator that has the following path integral representation: J(x, x, t; x, x, t ) = Dx Dx exp(is[x] is[x ])F [x, x ]. (8.15)

43 8 Quantum Brownian motion 42 where the integral is over paths satisfying the above boundary conditions, S[x] is the action for the system only, and F [x, x ] is the so called Influence Functional first introduced by Feynman and Vernon. This functional is responsible for carrying all the physical effects produced by the environment on the evolution of the system. In fact, if there is no coupling between the system and the environment, the Influence Functional is equal to the identity, and the above expression reduces to the one corresponding to the free Schrödinger evolution for the isolated system. The Influence Functional is defined as F [x, x ] = dqdq dq ρ E (q, q ) DqDq exp(i(s SE [x, q] S SE [q, x ])), (8.16) where ρ E is the initial state of the environment and S SE [q, x] is the action of the environment (including the interaction term with the system). It is easy to see that if there is no interaction (or if the two systems trajectories are the same, i.e., x = x ), then the influence functional is equal to one. Calculating the Influence Functional for an environment formed by a set of independent oscillators coupled linearly to the system is a rather straightforward task. Assuming the initial state of the environment is thermal equilibrium at temperature T = 1/k B β, the result is F [x, x ] = exp( i t t t1 dt 1 dt 2 Y (t 1 )η(t 1 t 2 )X(t 2 ) t dt 1 dt 2 Y (t 1 )ν(t 1 t 2 )Y (t 2 )), (8.17) where X = x + x, Y = x x, and the two kernels ν(s) and η(s) are the so called noise and dissipation kernels that were defined above in (8.2). Thus, all the influence of the environment on the evolution of the system is encoded in the noise and dissipation kernels (two different environments that produce the same kernels would be equivalent as to the impact they have on the system). To obtain the above expression is a simple exercise in path integrals. However, the calculation can also be done by a more straightforward procedure that makes no reference to path integrals. Indeed, one can notice that the influence functional can always be expressed in operator language as F [x, x ] = T r E ( T (e i t dt 1V int [x (t 1 ),q(t 1 )] )ρ E T (e i t dt 1V int [x(t 1 ),q(t 1 )] )), where T ( T ) denotes the time ordered (antitime ordered) product of the corresponding Heisenberg operators, and V int is the interaction term between the system and the environment. If the interaction is bilinear and the initial state of the environment is thermal, one can easily realize that the result should be a Gaussian functional of both x and x. Therefore, one can just write down such most general Gaussian functionals in terms of unknown kernels. These kernels could be identified by using the above expression, taking functional derivatives with respect to x and x and evaluating the result when x = x. In this way, one realizes that the result is given by (8.17), where the noise and dissipation kernels are given by expectation of symmetric and antisymmetric two time correlation functions of the environment oscillators, exactly as in (8.2).

44 8 Quantum Brownian motion 43 Knowing the Influence Functional enables us to compute the exact expression for the evolution operator of the reduced density matrix. In fact, all we need is to perform the path integral in (8.15). If the system is linear we see that the integrand is Gaussian and, therefore, the integral can also be explicitly computed. To perform this integral is not so trivial because the integrand is not separable into a product of functions of x and x. However, the integral can be calculated simply by changing variables. First we should integrate over sum and difference coordinates X and Y. Then, we should change variables writing X = X c + X and Y = Y c + Ỹ where X c and Y c satisfy the equations obtained by varying the phase of the integrand and imposing the corresponding boundary conditions. In this way, we show that the result of the path integral is simply the integrand evaluated in the trajectories X c, Y c, multiplied by a time dependent function that can be determined by normalization. The only nontrivial part of this derivation is to realize that the trajectories X c and Y c can be chosen as the ones extremizing only the phase of the integrand, (and not the entire exponent that, as we saw, has a real part coming from the noise). For more details on this derivation the interested reader can look in [19]. Therefore, the final result is given in equation (8.9) where the coefficients b i and a jl are time dependent functions that are determined in the following way. Let the functions u 1 be two solutions of the 2 equation, ü(s) + Ω 2 u(s) + 2 s ds η(s s )u(s ) =, satisfying the boundary conditions u 1 () = u 2 (t) = 1 and u 1 (t) = u 2 () =. Then, the coefficients appearing in (8.9) are simply given by Summary b 1 = u 2 (t), b 3 = u 2 () 1 t t a jl = (1 + δ jl ) 1 ds ds u j (s)u k (s )ν(s s ). (8.18) The time dependence of the coefficients of the master equation can be investigated after specifying the spectral density and the temperature of the environment. For the case that is most interesting for studying decoherence, which is the underdamped (i.e., weakly coupled) harmonic oscillator, the time dependence of the exact coefficients is very similar to the one obtained by analyzing the coefficients appearing in the perturbative master equation. Indeed, the perturbative coefficients obtained above can be recovered by solving the equation for the functions u 1 perturbatively and replacing these equations inside (8.18) and (8.12). Thus, to get a qualitative idea about the behavior of the coefficients, we restrict ourselves to the analysis already made for the perturbative ones (see Figure 3). It will be useful to analyze decoherence not only using the reduced density matrix but also the Wigner function that is the phase space distribution function that can be obtained from the density matrix as + dz W (x, p) = 2π eipz/ ρ(x z/2, x + z/2). (8.19) It is simple to show that for the case of the harmonic oscillator, the evolution equation for the Wigner function can be obtained from the master equation and has the form of a Fokker Planck

45 8 Quantum Brownian motion 44 equation Ẇ = {H ren (t), W } P B + γ(t) p (pw ) + D(t) 2 ppw f(t) 2 pxw. (8.2) 8.3 Death of the Schrödinger cat: The quantum to classical transition We will analyze here the decoherence process in a simple example: the linear quantum Brownian motion model whose exact master equation is given in Eq. (8.3) or, in the position representation, in Eq. (8.11). For this we will first set up an initial state that is delocalized in position (or momentum) space, namely a superposition of coherent states with opposite phases. This is what it is generally meant to be an ideal implementation of a Schrödinger cat state, namely a superposition of macroscopically distinguishable quasi-classical states. In position space, coherent states are represented by Gaussian wave packets. Thus, we will consider a state of the form where Ψ(x, t = ) = Ψ 1 (x) + Ψ 2 (x), (8.21) Ψ 1,2 (x) = N exp ( (x L ) 2 ) 2δ 2 exp (±ip x), (8.22) N 2 N 2 πδ 2 = 1 [ 2π 2 δ exp ( L 2 )] 1 δ 2 δ2 P 2, (8.23) and examine its temporal evolution, paying special attention to the fate of interference effects. Note that we assumed (just for simplicity) that the two wave packets are symmetrically located in phase space. The above expression allows us to study two extreme cases: the coherent states are separated in position or in momentum. In both cases, as a consequence of quantum interference, the Wigner function oscillates and becomes negative in some regions of phase space (and therefore cannot be interpreted as a probability distribution). When the coherent states are separated in position (momentum), the fringes are aligned along the p (x) axis. To evolve this initial state, we should solve the master equation (8.11). Rather than doing this, one can use the explicit form of the evolution operator (8.9) and obtain the exact form of the reduced density matrix or the Wigner function at any time. We will adopt this strategy but will use the master equation (8.11) and the equation for the Wigner function (8.2) as a guide to interpret our results and to obtain simple estimates for the most important effects that take place as a result of the interaction between the system and the environment. The exact evolution of the above initial state is such that the Wigner function can be written always as the sum of two Gaussian peaks and an interference term, W (x, p, t) = W 1 (x, p, t) + W 2 (x, p, t) + W int (x, p, t), (8.24) where W 1,2 (x, p, t) = N 2 π δ 2 δ 1 exp ( (x x c) 2 δ 2 1 ) exp ( δ 2 2 (p p c β(x x c )) 2), W int (x, p, t) = 2 N 2 δ 2 δ2 2 (p βx) 2 ) π δ 1 cos (2κ p p + 2(κ x βκ p )x). (8.25)

46 8 Quantum Brownian motion 45 Figure 3: Wigner function for a quantum state which is a superposition of two Gaussian wavepackets separated in position. The interference fringes are alligned along the p axes.

47 8 Quantum Brownian motion 46 All the coefficients appearing in these expressions are somewhat complicated functions of time that are determined by the coefficients that appear in the propagator (8.9) and the initial state (in the same way, they also depend on temperature and on the spectral density of the environment). The initial state is such that δ1 2 = δ 2 2 = δ2, κ x = P = p c, κ p = L = x c and A int =. From the form of the exact solution, it is clear what the qualitative behavior of the quantum state is. The two Gaussian peaks follow the two classical trajectories (which get distorted by the interaction with the environment) and change their width along their evolution. On top of this, the interference fringes change their wavelength and also rotate somewhat following the rotation of the two wavepackets. The effect of decoherence is clearly manifested in the damping of the interference fringes that, in the above formulae, is produced by the exponential term exp( A int ). Thus, we will look carefully at this term, which can be seen to be the fringe visibility factor defined as exp ( A int ) = 1 W int (x, p) peak. (8.26) 2 1/2 (W 1 (x, p) peak W 2 (x, p) peak ) A close analysis of the definition of A int shows that it vanishes initially and is always bounded from above, i.e., A int L 2 δ 2 + δ2 P 2 = A int max. (8.27) The value of A int cannot grow to infinity as a consequence of the fact that the two Gaussian initial states have a finite overlap that is proportional to exp ( A int max ). To understand qualitatively and quantitatively the time dependence of the fringe visibility factor, it is interesting to obtain an evolution equation for A int. Using its definition, we know that (Ẇ1 A int = Ẇint peak 1 + Ẇ2 W int 2 W 1 W 2 ) peak. (8.28) This, after using the form of the Wigner function together with the evolution equation, can be transformed into A int = 4D(t)κp 2 4f(t)κ p (κ x βκ p ). (8.29) This equation enables us to obtain a clear picture of the time evolution of the fringe visibility function. Thus, we can see that the first term on the right hand side is always positive and corresponds to the effect of normal diffusion. The normal diffusion will tend to wash out interference. The initial rate at which A int grows is determined by the diffusion coefficient and by the initial wavelength of the fringes in the momentum direction (remember that initially we have κ p = L /. As time goes by, we see that the effect of this term will be less important as the effective wavelength of the fringes grows (making κ p decrease). Various simple estimates of the temporal behavior of the fringe visibility factor can be obtained from this equation. The most naive one is to neglect the time dependence of the diffusion coefficient and assume that the fringes always stay more or less frozen, as in the initial state. In such a case, we have A int 4L 2 Dt/ 2. Thus, if we use the asymptotic expression of the diffusion coefficient, we obtain (at high temperatures) A int γt4l 2 /λ2 DB where λ DB is the thermal de Broglie wavelength. Consequently, we find that decoherence takes place at a rate t dec = γ 1 (λ DB/L ) 2, (8.3)

48 8 Quantum Brownian motion 47 which is the relaxation rate multiplied by a factor that could be very large in the macroscopic domain (for typical macroscopic parameters, i.e., room temperature, cm scale distances and masses on the order of a gram, the factor 4L 2 /λ2 DB can be as large as 14 ). By analyzing the temporal behavior of A int obtained using the exact solution, we can check that this naive estimate is an excellent approximation in many important situations. However, it may fail in other important cases. Here, we want to stress a message that we believe is very important: It may be rather dangerous to draw conclusions that are too general from the theoretical analysis of simple models of decoherence (like the one of linear QBM). The reason is that simple estimates like the one corresponding to the decoherence timescale (8.3) are just that: simple estimates that apply to specific situations. They do not apply in other circumstances, some of which we will describe here (and in the next section). For example, the above simple estimate of the decoherence timescale fails in the simple case of ultrafast decoherence. For, in the high temperature approximation of the master equation we neglected (among other things) initial transients occurring in the timescale fixed by the cutoff. Nothing (not even decoherence) can happen faster than the cutoff timescale since only after such timescale the diffusion coefficient reaches a sizable value. Thus, studying the initial time behavior of the normal diffusion coefficient one realizes that for very short times, A int always grows quadratically (and not linearly). In fact, we have A int 4Mγ k B T L 2 2 Λt 2. From this expression one sees that in this case A int is smaller than the one obtained under the assumption of a constant diffusion coefficient (at least for times t Λ 1 ). In this case, the decoherence timescale may be longer than the one corresponding to the high temperature approximation, t dec = 2L Mγ Λk B T. (8.31) On the other hand, the above estimate for A int also fails to take into account the fact that A int does not grow forever because it finally saturates to the value fixed by equation (8.27). Saturation is achieved in a timescale that can be estimated to be t sat γ 1 ( Ω/k BT ). At approximately this time the saturation of A int takes place (it is clear that this is a very short time, much shorter than any dynamical timescale).

49 9 The Monte Carlo Wave Function (MCWF) Quantum Jumps approach 48 9 The Monte Carlo Wave Function (MCWF) Quantum Jumps approach 9.1 Quantum Jumps: Introduction This section is partly based on the review of Ref. [22]. Quantum mechanics is a statistical theory which makes probabilistic predictions of the behaviour of ensembles (an ideally infinite number of identically prepared quantum systems) using density operators. This description was completely sufficient for the first 6 years of the existence of quantum mechanics because it was generally regarded as completely impossible to observe and manipulate single quantum systems. For example, Schrödinger, in 1952 wrote... we never experiment with just one electron or atom or (small) molecule. In thought-experiments we sometimes assume that we do; this invariably entails ridiculous consequences. {...} In the first place it is fair to state that we are not experimenting with single particles, any more than we can raise Ichthyosauria in the zoo. This (rather extreme) opinion was challenged by a remarkable idea of Dehmelt which he first made public in He considered the problem of high precision spectroscopy, where one wants to measure the transition frequency of an optical transition as accurately as possible, e.g., by observing the resonance fluorescence from that transition as part (say) of an optical frequency standard. However, the accuracy of such a measurement is fundamentally limited by the spectral width of the observed transition. The spectral width is due to spontaneous emission from the upper level of the transition which leads to a finite lifetime τ of the upper level. Basic Fourier considerations then imply a spectral width of the scattered photons of the order of τ 1. To obtain a precise value of the transition frequency, it would therefore be advantageous to excite a metastable transition which scatters only a few photons within the measurement time. On the other hand one then has the problem of detecting these few photons and this turns out to be practically impossible by direct observation. So obviously one has arrived at a major dilemma here. Dehmelt s proposal however suggests a solution to these problems, provided one would be able to observe and manipulate single ions or atoms which became possible with the invention of single ion traps. We illustrate Dehmelts idea in its original simplified rate equation picture. It runs as follows. Instead of observing the photons emitted on the metastable two-level system directly, he proposed to use an optical double resonance scheme as depicted in Fig. 4. One laser drives the metastable 2 transition while a second strong laser saturates the strong 1; the lifetime of the upper level 1 is for example 1 8 s while that of level 2 is of the order of 1s. If the initial state of the system is the lower state then the strong laser will start to excite the system to the rapidly decaying level 1, which will then lead to the emission of a photon after a time which is usually very short (of the order of the lifetime of level 1). This emission restores the system back to the lower level ; the strong laser can start to excite the system again to level 1 which will emit a photon on the strong transition again. This procedure repeats until at some random time the laser on the weak transition manages to excite the system into its metastable state 2 where it remains shelved for a long time, until it jumps back to the ground state either by spontaneous emission or by stimulated emission due to the laser on the 2-transition. During the time the electron rests in the metastable state 2, no photons will be scattered on the strong transition

50 9 The Monte Carlo Wave Function (MCWF) Quantum Jumps approach 49 Figure 4: The V system. Two upper levels 1 and 2 couple to a common ground state. The transition frequencies are assumed to be largely different so that each of the two lasers driving the system couples to only one of the transitions. The 1 transition is assumed to be strong while the 2 transition is weak.

51 9 The Monte Carlo Wave Function (MCWF) Quantum Jumps approach 5 Figure 5: A few periods of bright and dark sequences in the fluorescence intensity I from a three-level system. The bright periods last on average T B, and the dark periods T D.