A Quantum Trajectory Analysis of Berry s Phase in a Dissipative Environment

Transcription

1 A Quantum Trajectory Analysis of Berry s Phase in a Dissipative Environment Martin Ams mams@physics.mq.edu.au August 13, 2003 Submitted in partial fulfilment of the requirements for the Honours Degree of Bachelor of Science (Physics) in the Division of Information and Communication Sciences at Macquarie University, 2001.

2 i Acknowledgements I think if any of us honestly reflects on who we are, how we got here, what we think we might do, and so forth, we discover a debt to others that spans written history. The work of some unknown person makes our lives easier everyday. I believe it s appropriate to acknowledge all of these unknown persons; but it is also necessary to acknowledge those people we know have directly shaped our lives and our work. This page is specifically designed to note my appreciation of those people who stand out most notably in my mind as contributing to the content of what you will find in this thesis. There is one man that urged me on by way of his un-tiring support and seemingly unlimited belief in me, to that man, all else pales. To my supervisor Dr J. D. Cresser, thank you. The many hours we spent chatting have led to fruitful developments. There are many that from behind the scenes have encouraged and supported my work, and I wish to thank them. I am indebted always to those people that do their best to improve on my best. Thanks goes to my parents, my family and the guys and girls from the honours room for keeping me sane. Gratitude and special thanks is due my partner Amie for giving me the strength and incentive that I needed to go on. Martin

3 Contents 1 Introduction 1 2 Berry Phase The Two-Level Atom The Density Operator The Bloch Sphere Atom-Field Interaction Unitary Transformation Parameter Space Pancharatnam Connection Adiabatic Theorem Observation of the Berry Phase The Quantum Trajectory Method Open Quantum Systems System-Reservoir Interactions Reduced Density Operator The Master Equation The Quantum Trajectory Method The Quantum Trajectory Selection of the Super Operators ii

4 CONTENTS iii Implementation of the QTM QTM and the Berry Phase Summary Collisional Damping Consequences The Jump Operator Bloch Equations The Quantum Trajectory Approach Effective Hamiltonian The Waiting-Time Distribution The Quantum Jump Quantum Trajectories Berry Phase in Collisional Damping Evolution of the State Vector No Collisions One Collision Summary Spontaneous Emission The Jump Operator Bloch Equations The Quantum Trajectory Approach Effective Hamiltonian The Waiting-Time Distribution The Quantum Jump Quantum Trajectories

5 CONTENTS iv 5.5 Berry Phase for Spontaneous Emission Evolution of the State Vector No Spontaneous Emissions One Spontaneous Emission Non-Hermitean Berry Phase Summary Conclusions & Further Research 93 A Computer Scripts For Matlab 103

6 Chapter 1 Introduction The evolution of the state of a quantum system is described by the Schrödinger equation Ĥ ψ = i h d dt ψ, where Ĥ is the Hamiltonian or energy operator of the system. The solution to this equation is particularly simple if ψ is an eigenstate of Ĥ with energy E: ψ(t) = ψ(0) e iet/ h. If this is the case, then ψ(t) is said to be a stationary state. By stationary state we mean that the state of the system will not change in time. In general, Ĥ will be a function of various parameters determined by the nature of the physical system it describes, such that its eigenstates and associated eigenvalues will also depend on these parameters. This leads to an important result embodied in the so-called adiabatic theorem [31] which effectively states that a system prepared in an eigenstate of its Hamiltonian will remain in the connected (instantaneous) eigenstate, as the parameters of the Hamiltonian are varied in time, provided the variation is carried out slowly enough. If the Hamiltonian Ĥ is returned to its original form, i.e. the parameters on which Ĥ depend have slowly returned to their initial values, then the system returns to its 1

7 CHAPTER 1. INTRODUCTION 2 original state multiplied by an appropriate phase factor. This means that the initial state of the system ψ(0) is accompanied by the familiar dynamical phase factor after a time t, i.e. ψ(t) = ψ(0) e i t 0 E(t )dt / h, a simple generalisation, taking account of the possible time dependence of the energy eigenvalue, of the more familiar time dependence of a stationary state given above. This phase difference simply reflects the time it took the system to complete the cycle. Figure 1.1: Sir Michael Berry. In addition to this dynamical phase, there was also an additional phase factor that was believed to be of no physical significance, and which could be removed by simply noting that the state of a system can be defined up to an arbitrary phase factor anyway. So it came as rather a shock to the physics community, when in 1984 Berry revealed that the adiabatic theorem was incomplete [6]. Berry had noticed that once a quantum system had adiabatically returned to its initial state, the multiplicative phase consisted of two components; the well known dynamical phase, and a left over part, the part that formerly was ignored, which depended

8 CHAPTER 1. INTRODUCTION 3 only upon the geometry of the path that the system traced out in parameter space. This geometric phase or Berry phase is independent of the time it takes the system to complete the circuit in parameter space. Berry showed that the geometric phase was related to the solid angle enclosed by the loop traced out in parameter space. Moreover, he was able to show that this phase factor had observable physical consequences, generally in circumstances involving interference phenomena. For most scientists, especially quantum physicists, the Berry phase was immediately recognised as providing a neat explanation of a wide variety of otherwise baffling observations. The Berry phase has its major underpinnings in quantum physics 1 but may also be applied to certain areas of classical physics. It so happens that the geometric phase provides an explanation of phenomena in any system whose parameters undergo a cyclic change. Below are listed some of the areas and applications of physics that the Berry phase has made an impact. Aharonov Bohm Effect [2, 6] Quantum Computing [32] Molecular Physics and the Born-Oppenheimer Approximation Quantum Hall Effect Neutron Spin Rotation [11] Gauge Field Theory Molecular Spectroscopy [19] Nuclear Magnetic Resonance [39] General Relativity 1 Note that the Berry phase is not a quantum entity but a property of wave systems.

9 CHAPTER 1. INTRODUCTION 4 Foucault Pendulum [23] Polarisation of light beams [14, 15] Each individual application of the Berry phase may be subtle, but its overall effect has been enormous. Another significant feature of the Berry phase is that it can be observed most directly in an interference experiment such as, for instance, when an atom passes through an interferometer. If on one arm of the interferometer, the atom is subject to an adiabatic change of state, but not on the other arm, then when the two arms of the interferometer come together again, interference will be observed. This thesis however, explores another possibility where observing the Berry phase produces a significant effect. Following the approach of [4], we study a single quantum system (a two-level atom driven by a monochromatic laser field) which is prepared as a linear combination of the eigenstates of its Hamiltonian Ĥ. As the Hamiltonian is varied adiabatically, by slowly changing the phase of the field, a Berry phase connected with the evolution of each eigenstate is found to arise. It is then the difference between these phases which is observable in the measurable properties of the system. This situation commonly occurs in optical resonance and quantum optics. An important issue that cannot be overlooked in quantum optics is the fact that the system of interest is usually coupled to a much larger system, namely the environment, with which the system may exchange energy or matter. Because of a system s openness to the outside world, the dynamics of the system cannot be described by the Schrödinger equation, nor can its state be represented by a state vector. In general, the state of the system is specified by a density operator, and the dynamics by a so-called master equation for the reduced density operator.

10 CHAPTER 1. INTRODUCTION 5 The solution of the master equation is deterministic and evolves continuously in time, so that if it can be solved analytically, probabilities and expectation values of any observable may be readily obtained. However it is often the case that exact solutions cannot be found and recourse has to be made to numerical techniques. Amongst these techniques is one that also turns out to have a deep physical significance and that is the quantum trajectory method (QTM). The QTM relies on being able to replace the master equation by a Schrödinger equation in which there appears stochastic terms that describe the random interaction of the system with its environment. These random interactions are known as quantum jumps. Two of the most common types of unravellings of the master equation are the quantum state diffusion equations of Gisin and Percival [22] and the quantum jump or Monte Carlo wavefunction [17] unravellings used in quantum optics. This thesis is only concerned with the latter type of the afore-mentioned methods. The main idea of the quantum jump picture is that an open system may be described by a state vector but one which evolves by a sequence of sudden changes in state, quantum jumps, separated by periods of free evolution. The QTM offers a way of solving the master equation numerically using the interpretation of the quantum jump picture. It was Carmichael s interpretation of the QTM [13], that outlined the notion of quantum jumps being connected with the measurement process of the system properties. Carmichael implied that a quantum trajectory comprises the system s free evolution with quantum jumps taking place at random times throughout this evolution. An appropriately defined average of a large number of these quantum trajectories then yields the reduced density operator of the system. The overall aim of this project is to expose and discuss the effects of the Berry phase in the simplest model of optical resonance in which transitions between two

11 CHAPTER 1. INTRODUCTION 6 atomic levels are driven by resonant interaction with an intense laser field. If however, the atom is coupled to the environment, then the evolution of the atomic state is interrupted by dissipation due to collisions with other atoms or the atom in question experiencing spontaneous emission of a photon. This thesis will outline the application of the QTM to both the above mentioned processes, however, how the Berry phase phenomenon is affected by damping of the atomic evolution, and the explanation of its consequences thereof, remains the main focus of this project. The unravelling of this thesis into its main components is presented below. Chapter 2 introduces the representation of the two-level atom on a Bloch sphere. A parameter space is formulated and the derivation and observation of the Berry phase is presented. Chapter 3 exposes the reader to open systems and the density operator. The Lindblad form of the master equation is also featured. The development of the QTM and its properties are discussed in this chapter. Chapter 4 discusses the dissipation effect caused by colliding atoms. The master equation is derived from physical arguments and the simulation of the QTM is applied. Berry phase effects are grouped into a conclusion. Chapter 5 is almost a mirror image of the previous chapter, but in this chapter spontaneous emission is the cause of dissipation. The master equation is derived and the occurrence of a non-hermitean Berry phase is highlighted via the QTM. Chapter 6 presents my conclusions and a summary of the work completed. Suggestions for further research and investigation into the ideas presented in this thesis are made. The thesis is completed by a list of references and various appendices containing computer program scripts for the numerical package Matlab.

12 Chapter 2 Berry Phase 2.1 The Two-Level Atom Berry showed that the geometrical phase is most easily illustrated in two-state systems. So for the purposes of illustration and simplicity, a two-level atom has been specifically chosen for use in this project. A quantised atom has an infinite number of energy eigenstates but for many purposes it is sufficient to assume that the atom has only two states g and e with energies E g and E e respectively. e is the higher energy state known as the excited state while g is the lower state called the ground state. The energy difference between these two levels is E e E g = hω o where ω o is the transition frequency between the states (refer to figure 2.1). In the presence of an electromagnetic field the atom can make transitions from the ground state to the excited state and vice-versa by absorbing or releasing a quantum of energy hω o ; a photon. Although no such atom really exists, many coherent resonant interactions do involve only two levels of an atom. Thus it is frequently possible to assume, as a good first approximation, that the two-level idealisation is valid. The Hamiltonian describing a two-level atom is given by Ĥ = 1 2 hω o ( e e g g ). (2.1) 7

13 CHAPTER 2. BERRY PHASE 8 Figure 2.1: The two-level atom. Solving the Schrödinger equation for the state of the system is usually the procedure followed once the Hamiltonian for the system is specified. However here, we need to deal with a more general description of the state of a system, i.e. the density operator. The next section deals solely with this concept. 2.2 The Density Operator Quantum mechanics makes statistical predictions on an ensemble of physical systems. A collection of systems all in the same physical state is known as a pure ensemble and the state of the ensemble, ψ, is known as a pure state. However, most physical systems have many degrees of freedom over which we have little or no knowledge or control. Thus we need a more general way of describing the state of the system suitable for these more realistic circumstances. Consider a mixture of independently prepared states ψ i with statistical weights P i. These states do not necessarily have to be orthonormal to each other [12]. The state representing this mixed ensemble is known as the density operator ˆρ, which is

14 CHAPTER 2. BERRY PHASE 9 defined as ˆρ = i P i ψ i ψ i, with i P i = 1 (2.2) where the sum extends over all states present in the mixture. The density operator is Hermitean (self-adjoint) and usually normalised so that it has unit trace, i.e. T r[ˆρ] = n φ n ˆρ φ n = 1 (2.3) which is independent of the choice of basis { φ n }. The diagonal elements of ˆρ, a ˆρ a, represent the probabilities or populations of observing the system in state a so that equation 2.3 is simply stating that ˆρ is normalised. The off-diagonal elements, a ˆρ a, are known as coherences which represent, amongst other things, the degree of randomness of the phase of state a with respect to the phase of state a. The expectation value of any operator ˆΩ is given by the trace of the product of ˆρ and ˆΩ: ˆΩ = T r[ˆρˆω]. (2.4) Since the expectation value of any operator can be obtained by use of equation 2.4, then the density matrix contains all physically significant information on the system. Note also that the trace is invariant under cyclic permutation of the arguments: T r[â ˆBĈ] = T r[ĉâ ˆB] = T r[ ˆBĈÂ]. (2.5) A pure state has the property that ˆρ 2 = ˆρ, iff P i = δ ik. This means the density operator for a pure state is ˆρ = ψ k ψ k. (2.6) The density operator is also time-dependent even in the Schrödinger picture and obeys the von Neumann equation of motion ˆρ = ˆρ t = ī [Ĥ, ˆρ] (2.7) h

15 CHAPTER 2. BERRY PHASE 10 where Ĥ is the Hamiltonian of the system. Equation 2.7 can be easily derived from the Schrödinger equation Ĥ ψ i = i h d dt ψ i. (2.8) 2.3 The Bloch Sphere Basic quantum mechanics tells us that an arbitrary state ψ of a two-state system with basis { 0, 1 }, can be written as ψ = a 1 + b 0, where probability is normalised; a 2 + b 2 = 1. As we are dealing solely with a two-state atom in this project, the basis states will be e and g. For the purposes of illustrating the Bloch sphere, we can conveniently rewrite the state ψ in the form ψ θ, φ = cos θ 2 e + eiφ sin θ g (2.9) 2 using the excited and ground states of the two-level atom. This equation represents the most general form of a two-state system including a relative phase difference φ between the two states. One also notices that this state ψ is normalised to unity via θ, φ θ, φ = cos 2 θ 2 + θ sin2 2 = 1. (2.10) The density operator, given in equation 2.2, can then be constructed to give ˆρ = θ, φ θ, φ = cos 2 θ 2 e e + sin2 θ 2 g g + cos θ 2 sin θ 2 e iφ e g + cos θ 2 sin θ 2 eiφ g e. (2.11) Using the { e, g } basis, the density operator can be represented in matrix form as ˆρ = cos 2 θ 2 cos θ 2 sin θ 2 e iφ cos θ 2 sin θ 2 eiφ sin 2 θ 2. (2.12)

16 CHAPTER 2. BERRY PHASE 11 Feynman was the first scientist to utilise the Bloch vector for a problem other than nuclear magnetic resonance [20]. Feynman generated the Bloch vector from the elements of the density matrix using the definitions u = ˆρ eg + ˆρ ge v = i (ˆρ eg ˆρ ge ) (2.13) w = ˆρ ee ˆρ gg. By applying these definitions to the matrix 2.12, one obtains the following results: u = sin θ cos φ v = sin θ sin φ (2.14) w = cos θ. Careful examination of equations 2.14 reveal that they can be pictorially represented as shown in figure 2.2 where u, v and w are the ˆx, ŷ and ẑ components of an arbitrary vector ˆn. This arbitrary vector ˆn is known as the Bloch vector. The single atom population difference, w, is known as the inversion. The value 2 hω 1 ow represents the internal energy of the atom, while u and v can be conveniently understood as being related to the dipole moment of the atom. More precisely, u and v are the components of the dipole moment in-phase and in-quadrature with an interacting electric field E. This idea will be introduced in the next section. One can also prove the relation u 2 + v 2 + w 2 = constant (2.15) which implies that the state of the atom remains normalised in time or that probabilities are conserved [1]. If the density matrix is in a pure state, given by equation 2.6, then u 2 + v 2 + w 2 = 1. In this situation, the Bloch vector is confined to a sphere

17 CHAPTER 2. BERRY PHASE 12 Figure 2.2: Bloch vector ˆn in u, v, w space. of radius one called the Bloch sphere. If θ = 0, equation 2.9 puts the state of the system in its excited state, i.e. the north pole of the Bloch sphere. Thus the ground state of the atom is equivalent to the south pole of the Bloch sphere, i.e. when θ = π. Because of these relations, the Bloch vector picture becomes a very useful interpretation of a two-level system. 2.4 Atom-Field Interaction The light interacting on resonance with a two-state atom is an important system, particularly in quantum optics. The two equations 2.7 and 2.8, form the starting point for studying the interaction of atoms with radiation within the context of the so-called semi-classical theory. In this approach, the radiation field is treated classically while the atom is described quantum mechanically. In this model the laser resonantly couples the ground state and the excited state of the atom, dominating the dynamics of the system, allowing the remaining states of the atom to be

18 CHAPTER 2. BERRY PHASE 13 neglected. In order for this theory to be valid, the source of the field must be strong; as in the case of lasers. Given a huge number of photons, the quantisation of the field is hidden. With the quantum aspects of the field masked, the wave aspect will dominate. In this situation, the field can be described as a classical electromagnetic wave. The electric field will oscillate sinusoidally with some real frequency ω and amplitude E o, and the impact of the magnetic field will be neglected because it is small compared to that of the electric field [1]. In this project, the optical radiation field is perfectly monochromatic and almost exactly coincides in frequency with the transition frequency ω o of the atom. With these restrictions in mind, the semi-classical Hamiltonian of the system in the Coulomb gauge is Ĥ = 2 hω 1 oˆσ z ˆd Ê(t) (2.16) where the product of the dipole operator for the atom ˆd = dˆσ x, and the electric field operator Ê(t) = E o cos(ωt + φ(t)) in the Heisenberg picture, produces the term needed to describe the interaction between the two systems. The usual Pauli matrices ( ) ( ) ( i 1 0 ˆσ x =, ˆσ 1 0 y =, ˆσ i 0 z = 0 1 ( ) ( ) ˆσ + =, ˆσ 0 0 = 1 0 ), (2.17) in the { e, g } basis are assumed to be understood. Equation 2.7 is the basic dynamical equation which governs the evolution of the two-level atom variables. We can therefore substitute the Hamiltonian 2.16 into equation 2.7 and essentially solve for the matrix elements of the density operator. This substitution results in terms of the form cos(ωt + φ(t))e iωot from which we can

19 CHAPTER 2. BERRY PHASE 14 apply the expansion cos α = eiα + e iα 2. (2.18) We neglect the terms with oscillations at twice the optical frequency, e ±2i(ω+ωo)t, applying the rotating wave approximation (RWA). These fast oscillating terms correspond to processes which do not conserve energy for long interaction times, hence are left out of the problem altogether. To ensure all variables in the solution change slowly in time, we move all our definitions to a rotating frame of reference, i.e. we redefine the Bloch vector components 2.13 as u = ˆρ eg e iωt + ˆρ ge e iωt v = i(ˆρ eg e iωt ˆρ ge e iωt ) (2.19) w = ˆρ ee ˆρ gg. The rotating frame can be simply understood in the Bloch vector picture as transforming into a frame in which the Bloch vector processes about a static vector. As long as the applied field is near resonance, which is what has been assumed, all optical frequencies will have been removed from the problem. Finally, by noting the substitutions Ω = d E o 2 h = ω o ω, (2.20) one ends up with the optical Bloch equations: u = ( φ (t))v v = ( φ (t))u + 2Ωw (2.21) ẇ = 2Ωv.

20 CHAPTER 2. BERRY PHASE 15 The third Bloch equation from 2.21 explicitly shows that v is the component effective in coupling to the field to produce energy changes [1]. The general solution to equations 2.21, valid for any detuning and when φ(t) = 0, is a rotation of the constant length Bloch vector in u, v, w space on the Bloch sphere. The reason for this is because we are in a rotating frame of reference and only dealing with pure states. However, in the general case, the solution cannot be unravelled analytically. If the problem is confined to the Rabi case in which E o is constant, then a solution is possible with Rabi frequency Ω( ) = 2 + 4Ω 2. (2.22) The Rabi frequency gives the rate at which probability differences are coherently induced between the two atomic levels, ie. the frequency at which the inversion oscillates. By increasing the detuning, the Rabi frequency increases and the amplitude of the inversion decreases. This effect can be seen in figures 2.3 and Unitary Transformation Different quantum pictures are possible in terms of an arbitrary, possibly timedependent unitary operator Û. State vectors ψ and operators ˆ H can be defined in the new picture by ψ = Û ψ ˆ H = Û ĤÛ. (2.23) It is easily shown, via the Schrödinger equation presented in equation 2.8, that the equation for ψ can be written as [ ˆ H i hû dû ] ψ = i h d ψ. (2.24) dt dt

21 CHAPTER 2. BERRY PHASE = = 2 w = t Figure 2.3: Inversion for different values of when φ(t) = 0. Using the relevant information derived from equations 2.17, 2.18 and 2.20 we can rewrite the Hamiltonian equation 2.16 as ˆ H = 1 2 hω oˆσ z + hω(ˆσ + e i(ωt+φ(t)) + ˆσ e i(ωt+φ(t)) ). (2.25) By defining a unitary operator Û = e iωˆσzt/2 (2.26) and applying it to equation 2.24, one finds that the Hamiltonian 2.25 in the new picture is given by Ĥ φ = ˆ H i h Û dû dt

22 CHAPTER 2. BERRY PHASE 17 = w = = 4 0 v u Figure 2.4: Solution to equations 2.21 for different values of on the Bloch sphere with φ(t) = 0. = 1 2 h ˆσ z + hω(ˆσ + e iφ(t) + ˆσ e iφ(t) ) ( ) /2 Ωe iφ(t) = h Ωe iφ(t). (2.27) /2 Using the characteristic polynomial det(ĥφ E ± Î) = 0, where the identity Î is ( ) 1 0 Î =, (2.28) 0 1 one finds that the energy eigenvalues E ± are E ± = ± h(ω ) 1 2. (2.29) By substituting these eigenvalues into the eigenvalue equation Ĥ φ ± = E ± ±, (2.30)

23 CHAPTER 2. BERRY PHASE 18 the corresponding energy eigenstates 1 are found to be { + = e iµ + cos θ 2 e + sin θ } 2 eiφ(t) g, { = e iµ sin θ 2 e cos θ } 2 eiφ(t) g where sin θ 2 = E h [2E + (E h )] 1 2, cos θ 2 = hω [2E + (E h )] 1 2. (2.31) The phases µ ± are arbitrary functions of, Ω and φ(t). One notices that as the coupling strength diminishes, i.e. Ω 0, the terms sin θ 2 0 and cos θ 2 1. Hence the + state becomes the excited state and the ground state becomes the state. This shows that the vectors representing the + and states point in exactly the opposite directions. In fact, if we replace θ in the + eigenstate with π θ and φ by π + φ, then one readily obtains the eigenstate. The Hamiltonian 2.27 is now in a form for application of the adiabatic theorem and the derivation of a Berry phase [4], i.e. all rapid time-dependence has been removed. 2.6 Parameter Space An important feature used when obtaining the Berry phase is the idea of changing, slowly in time, the parameters that Ĥφ depends on. Therefore we need to define the parameter space of the transformed Hamiltonian Ĥφ. The most general 2 2 hermitean matrix coupling the two states of Ĥφ(R) can be brought into the following standard form [6] Ĥ φ (R) = 1 2 ( Z X + iy X iy Z ), (2.32) 1 these states differ from those found in Barnett s paper [4] because of a different choice of arbitrary phase

24 CHAPTER 2. BERRY PHASE 19 where R = Xî + Y ĵ + Zˆk represents a vector in the parameter space of Ĥφ. Comparison of this matrix to the matrix of Ĥ φ, equation 2.27, leads us to conclude that X = 2 hω cos φ(t) Y = 2 hω sin φ(t) (2.33) Z = h. Using X, Y and Z as the components of R, figure 2.5 illustrates the parameter space of Ĥ φ (R). One should note here that the angles θ and φ are the same angles as found in the + eigenstate of equations 2.31, and those found in equation 2.9. Figure 2.5: Parameter Space of Ĥφ(R).

25 CHAPTER 2. BERRY PHASE Pancharatnam Connection Berry was the first to discover the relation between the adiabatic phase acquired by the wave function under a slow variation of the Hamiltonian parameters and parallel transport of a quantum state around a loop in parameter space. Hence this unexpected geometrical phase γ m, is picked up by a quantum eigenstate evolving under a parameter-dependent Hamiltonian Ĥφ(R), where R is made to slowly trace out a loop Γ in parameter space. The Berry phase thus emerges when R completes its circuit, and when the quantum phase does not return to its initial value. This phenomenon can be illustrated by the following simple example making use of figure 2.6. The red line represents the state of the system and the blue vector Figure 2.6: Illustration of parallel transport. tangential to the sphere, its arbitrary phase. One can easily see that this blue vector

26 CHAPTER 2. BERRY PHASE 21 can point in any direction without altering the state of the system. The question now that needs to be asked is: Is there a unique way of comparing phases at different points on a curved surface to achieve parallel transport?. In general the answer is no - there are many methods that may be used. In fact, what is meant by parallel transport is a matter of definition. The method implemented here was demonstrated by Pancharatnam in 1955 [33] in the context of a cyclic change in the state of polarisation of light. Pancharatnam stated that for parallel transport to be achieved, the inner product of two vectors of infinitesimal separation must be real to first order, i.e. if we let θ, φ = e [cos iµ θ 2 e + sin θ ] 2 eiφ g, (2.34) where µ is an arbitrary phase for θ, φ, then a vector infinitesimally separated from θ, φ is given by: θ + δθ, φ + δφ = e i(µ+δµ) [ cos θ + δθ 2 e + sin θ + δθ ] e i(φ+δφ) g 2 One finds that the inner product of these two vectors gives. (2.35) θ, φ θ + δθ, φ + δφ = 1 i(sin 2 θ δφ + δµ) + 2nd order terms. (2.36) 2 By choosing δµ = sin 2 θ δφ, the first order change is removed so that 2 θ, φ θ + δθ, φ + δφ 1, (2.37) which is real to first order as required. This means that parallelism on a curved surface is defined as a zero phase change between the two states concerned. A physical analogy of the finding made by Pancharatnam may be achieved if you let your arm be the red line and your hand the point on the sphere. A ruler held in your hand will then represent the phase of the state. The method of Pancharatnam

27 CHAPTER 2. BERRY PHASE 22 applied in this case means that you may move your arm in any direction but must keep your wrist rigid at all times. If you raise your arm above your head and parallel transport the ruler around the purple loop shown in figure 2.6, you will notice that the direction of the ruler has changed even though you are back to where you started from, i.e. a rotation angle has developed because of the geometry of the sphere and not on how fast you travelled around the loop. Berry showed that in tracing such a curve, a particle acquires a geometric phase given by the product of its spin-state quantum number and the solid angle enclosed by the curve on the sphere. So for a two-level atom with quantum spin number 1/2, the Berry phase γ m is actually half the solid angle enclosed by the purple loop in figure Adiabatic Theorem Parallel transport is obtained in quantum states by adiabatic variation of the parameters R that the Hamiltonian Ĥφ depends on. The adiabatic theorem states that a system prepared in an eigenstate of its Hamiltonian will remain in the connected (instantaneous) eigenstate as the Hamiltonian is varied, provided the variation is carried out slowly enough. This means that if n(0) = l C e (R(0)) φ l, then the connected eigenstate is n(t) = l C e (R(t)) φ l where { φ l } is a basis independent of R, e.g. e and g. If during this variation, the Hamiltonian returns to its original form, then the system returns to the original eigenstate multiplied by an appropriate dynamical phase factor [4]. It was always believed that since the phase of the eigenstate was of no physical significance, and in any case could be removed by an appropriate unitary transformation, that it could be ignored. But to the surprise of everyone, Berry showed that if the parameters that the Hamiltonian depended on returned to their original

28 CHAPTER 2. BERRY PHASE 23 values, the phase acquired by the eigenstate contained, in addition to a dynamical component e i t 0 E(t )dt / h, a further contribution determined solely by the geometrical form of the path taken through parameter space. As the Hamiltonian is varied, each eigenstate of the Hamiltonian may acquire a Berry phase thus leading to readily observable physical consequences. The Berry phase is an element in an approximate solution of the Schrödinger equation presented in equation 2.8. At any instant in time, the Hamiltonian has a basis of orthonormal and degenerate eigenstates satisfying the eigenvalue equation Ĥ(t) n(t) = E n (t) n(t). (2.38) By use of the completeness relation, we know that ψ(t) = n n(t) n(t) ψ(t). (2.39) Therefore the wavefunction ψ(t) maybe expanded in the form ψ(t) = n { a n (t) n(t) exp ī t } E n (t )dt h 0 (2.40) and then substituted into the Schrödinger equation to give the equations of motion for the amplitudes a k : ȧ k = n { a n exp ī t } [E n (t ) E k (t )]dt k ṅ. (2.41) h 0 The ket vector ṅ is not the derivative of the state, but rather a concept of the evolution of the connected state over a time interval δt due to the slow varying parameters R, i.e. the ket vector is defined as n; R(t + δt) n; R(t) ṅ = lim δt δt. (2.42) To find the amplitudes a k defined in equation 2.41, we require the terms k ṅ. Differentiating equation 2.38 with respect to time, and taking the inner product

29 CHAPTER 2. BERRY PHASE 24 with k, gives (for n k) k ṅ = k ( Ĥ/ t) n (E k E n ). (2.43) If the system is initially prepared in the state m, then a n (0) = δ nm. The adiabatic theorem will hold true if a k 0 for k n, i.e. k ( Ĥ/ t) n << 1 h E k E m 2. (2.44) In this limit only a m is non-zero: ȧ m = a m m ṁ. (2.45) If we let a m = e iγm, then the Berry phase γ m obeys the equation of motion γ m = i m ṁ. (2.46) So one notices that the ability to change γ m is related to making a phase change in m, and that the Berry phase is uniquely defined only for a closed loop [4]. then If we define m(t) = e iγm(t) m(t), (2.47) m(t) = [iγ m (t) m(t) + ṁ(t) ] e iγm(t). (2.48) Taking the inner product of the latter equation with m(t), and using the relation 2.46, one finds that which leads to m(t) m(t) = 0 (2.49) m(t) m(t + δt) = m(t) m(t) = 1 (2.50) to first order. Equation 2.50 is equivalent to the following form θ, φ θ + δθ, φ + δφ = 1, (2.51)

30 CHAPTER 2. BERRY PHASE 25 where δθ and δφ are changes in θ and φ occurring in a time δt. Thus we see that adiabatic change fulfils the requirements of parallel transport given in equation One may simply rewrite equation 2.42 as ṅ = n; R + Ω n; R Ω + φ n; R φ. (2.52) The simplest non-trivial Berry phase is generated if the laser phase φ(t) is varied while the detuning and coupling strength Ω are kept fixed [4]. This means that equation 2.52 can be simplified to ṅ = φ ṅ; R φ. (2.53) Using equation 2.53 in conjunction with equation 2.46, one finds that, for the eigenstates ± of the Hamiltonian Ĥφ, γ satisfies γ ± = sin 2 θ 2 φ. (2.54) This result may be formally integrated to give γ ± = sin 2 θ [φ(t) φ(0)]. (2.55) 2 In the case where the loop in parameter space is complete, [φ(t) φ(0)] = 2π. This accounts for the loop Γ shown in figure 2.5. Bearing this in mind and using the relevant expansion for sin θ 2 in equations 2.31, one simplifies the Berry phase form to just γ ± = π { 1 ± h } 2E +. (2.56) Equation 2.56 clearly shows that the Berry phase is independent of the form of φ(t). As long as φ(t) varies slow enough and completes a full loop in parameter space, the Berry phase will be exactly the same for all laser phases φ(t) chosen.

31 CHAPTER 2. BERRY PHASE Observation of the Berry Phase The effects of the Berry phase may be seen by exploring the following example. Consider the two-level atom initially in its ground state g, i.e. ψ(0) = g = sin θ 2 + cos θ (2.57) 2 using a superposition of the eigenstates ±. The ± states will evolve freely under the Hamiltonian Ĥφ and each will acquire its own Berry phase as φ(t) is varied adiabatically. Hence, the time-evolved wavefunction becomes ψ(t) = sin θ 2 e ie +t/ h e i sin2 θ 2 δφ(t) + (t) cos θ 2 e ie t/ h e i sin2 θ 2 δφ(t) (t) (2.58) where δφ(t) = φ(t) φ(0). The wavefunction is then used to calculate the probability of being in the ground state at time t, i.e. [ g ψ(t) 2 = 1 + Ω2 cos {2Λt } ] 2Λ 2 2Λ δφ(t) 1 (2.59) using Λ = E + / h = (Ω ) 1/2. This ground state probability exhibits the familiar Rabi oscillations at a frequency of 2Λ [4]. Noting that probability is conserved g ψ(t) 2 + e ψ(t) 2 = 1, (2.60) the inversion w = ˆρ ee ˆρ gg can be shown to be [ w = Ω2 cos {2Λt } ] Λ 2 2Λ δφ(t) 1 1. (2.61) The inversion here not only oscillates at the frequency 2Λ, but also depends on the geometrical phase factor produced by varying φ(t). Thus, this is a clear illustration of Berry s phase in the simplest model of optical resonance. To further confirm this result, the inversion was calculated using the computer program Matlab. Firstly φ(t) had to be calculated to vary slowly enough and complete a full circuit in parameter space. This was achieved by choosing any function of

32 CHAPTER 2. BERRY PHASE Omega = 3 Matlab Theory w Omega = 10 t w Omega = 30 t w t Figure 2.7: Inversion adjustment to ensure adiabatic variation of Ĥφ. t starting at 0 and ending at 2π, and also by suitably fixing Ω sufficiently large. The equations 2.21 were then solved for u, v and w using the Runge Kutta 4 method [30]. To check whether the numerical inversion using Matlab corresponded to what the theory predicted, equation 2.61 was plotted on the same axes as the calculated Matlab inversion. To make the plots overlap each other, the amplitude of φ(t) and the coupling strength Ω, had to be adjusted accordingly. Figure 2.7 portrays a simple example of this adjustment. The execution of Rabi oscillations by the inversion is also clearly seen in figure 2.7. Once the inversions coincided with each other, it was safe to assume φ(t) was

33 CHAPTER 2. BERRY PHASE 28 φ(t) Variables w A sin(et) + 2πt/b e = A(t c (t b) d ) + 2πt/b c = 4, d = A(t(t a)(t c)(t d)(t b)) + 2πt/b a = b/2, c = b/3, d = b/ A(t c (t b) d ) + 2πt/b c = 1, d = Table 2.1: Value of the inversion after a time b = 0.3 with = 2, Ω = 2821 and A = 1. The system was initially in its ground state g. varying adiabatically. With φ(t) sorted out, the inversion for off-resonance interaction ( 0) was calculated for various phases φ(t). The results of these calculations are shown in table 2.1. Plots comparing the inversion for various phases φ(t) were indistinguishable by eye, therefore a table was used to present the final inversion value after a certain time interval. One notices, from table 2.1, that the inversion is invariant to φ(t) when a phase φ(t) is actually present. This confirms equation Also shown in table 2.1 is that the inversion for a non-zero φ(t) does not correspond to the inversion when φ(t) = 0, i.e. a phase shift has been added to the inversion when φ(t) is present. Applying an interference type experiment, which was introduced in the previous chapter, leads us to conclude that the phase shift mentioned above is due to the geometrical Berry phase only. Also mentioned in the previous chapter was the aim of this project, i.e. what happens to the Berry phase when we introduce the simple two-level atom into the real world. Before we can attempt to answer this question, we have to define the real world, its equations of motion, viz. the master equation, and introduce a method of solving these equations, i.e. the QTM. Once the features of the QTM are in place, the two-level atom will be free to experience collisions with other atoms, or transfer energy into the real world resulting in spontaneous emission.

34 Chapter 3 The Quantum Trajectory Method 3.1 Open Quantum Systems From a classical point of view, an open system is one that can exchange heat, energy or matter with the surrounding environment. In the general case, a quantummechanical system will show not only the thermal fluctuations and dissipation due to interactions with the environment that are observed classically, but also fluctuations due to the non-commutative nature of quantum theory [35]. This then results in a wide range of new phenomena. The simplest model of an open system is one in which the system is subject to an externally applied time-dependent interaction. An example of this has already been outlined in the case of an atom coupled to a classical electromagnetic field. In this model of an open system, the external influence is not included in the dynamics, i.e. it just continues to grind away in its predetermined fashion. A more physically correct description of an open system requires a quantum dynamical approach to the externally applied influence. This amounts to considering the external influence as a new, enlarged quantum system. Thus the problem now involves the description of coupled quantum systems where one is still only interested in the properties of the original system. The ability to extract this information requires a generalisation of the notion of the state of a quantum system. Hence an 29

35 CHAPTER 3. THE QUANTUM TRAJECTORY METHOD 30 open system must be characterised in terms of its density matrix 1 rather than the usual Schrödinger equation approach System-Reservoir Interactions The density operator formalism is found necessary to describe the situation in which two physical systems are interacting with each other and it is the properties of just one of these systems that is of particular interest. A situation that may occur in quantum optics is one where the system of interest is coupled to another system usually referred to as a reservoir or bath which has a continuous range of energy levels and degrees of freedom. This state of affairs can be arranged but is usually unavoidable as no system can ever be completely isolated from its environment. It is the environment then that acts as the reservoir for the system. The reservoir is so large that the energy and matter that are exchanged between itself and the smaller system of interest do not affect the state of the system. This so happens because the source of the interaction, the environment, is assumed to be very large so that any effect due to the system would be absorbed into its many degrees of freedom. This tells us then, that the variation in time of the open system of interest exhibits irreversible behaviour. Therefore the description of the system of interest, in terms of the density operator concept, is a valid one Reduced Density Operator When considering an open quantum system one is usually faced with the problem of describing a comparable small set of active degrees of freedom, the system, which interacts with a macroscopic environment. If the measurements of interest only concern observables exclusively defined in the state space of the system, one can reduce the whole description to the determination of the reduced statistical operator. 1 strictly speaking, a reduced density matrix is required

36 CHAPTER 3. THE QUANTUM TRAJECTORY METHOD 31 Suppose we let two systems interact with each other; system S and reservoir R, giving the density operator for the combined state of S and R as ˆρ SR. The expectation value of a system observable ˆΩ A is then given by ˆΩ A = T r SR [ˆρ SR ˆΩA ], (3.1) where equation 2.4 has been employed. However, the presence of the reservoir variables limits the usefulness of this expression, hence it is more convenient to deal with the reduced density operator ˆρ S defined by ˆρ S = T r R [ˆρ SR ] (3.2) using a partial trace over reservoir states [34]. Thus ˆρ S is the density operator of the system variables only. The expectation value of equation 3.1 can then be rewritten in terms of system variables alone: ˆΩ A = T r S [ˆρ S ˆΩA ]. (3.3) The Master Equation The equation of motion of the reduced density operator described in equation 3.2 is often called the master equation. Master equations were first introduced into quantum statistical physics by Pauli (1928) [12]. The purpose of the introduction of a reduced distribution is to simplify the dynamical description by invoking various approximation schemes. In the derivation of the master equation [13] for the reduced density operator of the system, it is necessary to eliminate all reservoir variables from the equation of motion. In doing so, what is quite generally found is a result of the form ˆρ = ī h [ĤS, ˆρ S ] + T 0 T r R [ ˆK(t t )ˆρ SR (t )]dt (3.4) where there appears the usual so-called coherent evolution associated with ĤS, and a term with its origin in the system-reservoir interaction. This latter term contains

37 CHAPTER 3. THE QUANTUM TRAJECTORY METHOD 32 memory effects by virtue of the appearance of an integral over the past histories of the system and reservoir dynamics. But under fairly commonly occurring circumstances, this term can be greatly simplified by applying three major approximations to get to its final differential form. Born Approximation: interactions between the system S and reservoir R are sufficiently weak during the correlation time of the reservoir so that the effect of the system on the reservoir is negligible. ie. ρ SR ρ S (t )ρ R (0). Markov Approximation: ρ S (t τ) ρ S (t) since one only needs to know the system s evolution over the correlation time of the reservoir. Hence, the future state of the system is determined by the current state of the system only and not by the state s past history. ie. all memory terms of earlier behaviours of the system are removed. Long-Time Approximation: due to short correlations of the reservoir variables, one can ignore any small transient effects on the system during the correlation time of the reservoir. For example, an integral may be insensitive to its upper limit, thus one can replace the upper limit in the integral by t. Under very general conditions it can be shown that the master equation for a system S coupled to a reservoir R takes the Lindblad form ˆρ = ī h [ĤS, ˆρ] + i [ L i ˆρL i 1 ] 2 (L il i ˆρ + ˆρL il i ) (3.5) for some system operators L i. The dissipative part of the density operator equation consists of the terms i [ L i ˆρL i 1 ] 2 (L il i ˆρ + ˆρL il i ). (3.6) The Lindblad operators L i and L i are determined by the nature of the system and the system-reservoir interaction. If these terms were not present, equation 3.5 would

38 CHAPTER 3. THE QUANTUM TRAJECTORY METHOD 33 solely describe the evolution of the system without any damping effects as shown already in equation 2.7. It is convenient to write the master equation in the short-hand form ˆρ = Lˆρ (3.7) where L is a super operator 2 hiding away the specific details of the system s dynamics. When L is independent of time, equation 3.7 has the formal solution ˆρ(t) = e Lt ˆρ(0) (3.8) in which e Lt is defined by its power series expansion [34]. 3.2 The Quantum Trajectory Method In recent years there has been considerable interest in stochastic wavefunction methods for the description of the dynamics of open systems in quantum optics. These methods represent a generalisation of the conventional formulation of the dynamics in terms of equations of motion for the reduced density operator. The basic idea underlying the stochastic formulation is to describe the state of the open system by means of ensembles of pure states whose time-evolution is governed by a stochastic process in Hilbert space. The density operator ˆρ can be reconstructed from these pure states in essentially the way given in equation 2.2. As mentioned in the first chapter, there are a great variety of these approaches making it impossible to obtain a unique stochastic representation of the reduced state vector only on the basis of the master equation. In this project the quantum trajectory method (QTM) has been introduced as the theoretical framework to describe such a behaviour of a single open quantum system. 2 or Liouvillian operator since it operates on an operator

39 CHAPTER 3. THE QUANTUM TRAJECTORY METHOD 34 The QTM leads to a numerical solution of the master equation for the system of interest and is heavily based upon the concept of a quantum jump or event. One can also take the point of view that the QTM represents an alternative, and sometimes more efficient, technique to determine ˆρ S rather than proceeding with direct integration of the master equation. The general picture for the related reduced-system dynamics is based on showing that the coupling to the environment leads to interruptions of the coherent motion by quantum jumps that occur at random times. Carmichael [13] understands this as unravelling the quantum master equation for the density operator of the reduced system into stochastic trajectories for state vectors. The coherent motion is described by a Schrödinger-like wave equation whereas the stochastic interruptions of the coherent motion, the quantum jumps, give rise to instantaneous changes of state. Both of these processes are built into the master equation. An appropriately defined average of a large set of these quantum trajectories will then regain the reduced density operator for the system. Clearly, the resulting numerical efficiency depends essentially on the number of trajectories one needs to achieve good statistics The Quantum Trajectory All quantum-trajectory techniques make use of the Lindblad form of dissipation (equation 3.5). The quantum jumps are introduced via a super operator J which is, in most cases, directly implied by the Lindblad structure of the master equation. Essentially, J has the effect of mapping ˆρ at some instant into a new state J ˆρ. At the very least, J must be such that J ˆρ is hermitean and 0 < T r[j ˆρ] <. We will define J later, but for the present we will derive some general results, following [13] and [34].