SOFIA UNIVERSITY “ST. KLIMENT OHRIDSKI” Faculty of Physics

Design of quantum gates and entangled states for quantum information technologies

Elica Sotirova Kyoseva

Supervisor: Assoc. Prof. Nikolay V. Vitanov

Submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy

Sofia January 2009

To my mother. The candidate confirms that the work submitted is her own and that appropriate credit has been given where reference has been made to the work of others.

Copyright © 2009. This copy has been supplied on the understanding that it is copyright material and that no quotation from the thesis may be published without proper acknowledgement. Acknowledgements

First and foremost, I would like to express my gratitude to my PhD supervisor, Nikolay Vitanov, for his good-natured support and patient guidance over the past five years. From our first meeting he has been a continuous source of motivation and support for me. I have greatly benefitted from his encouragement, advice and expertise. I am very grateful for all opportunities he has given me over the years and for his good company. I would also like to thank the other members of the group, as well as many visitors with whom I have had useful discussions. A special thank you goes to Bruce Shore. His love of Physics is infectious and I have enjoyed all our discussions in Sofia and Kaiserslautern thoroughly. A milestone for me was my visit in the Quantum Information group at the University of Leeds and I would like to thank all its members for their warm welcome. I am extremely grateful to Almut Beige who supervised me during the nine months I spend there and who treated me as one of her own PhD students. I am very thankful for her great patience, her guidance, and for introducing me to the field of measurement-based quantum computing. I remain very impressed with her knowledge, broad background, and great attention to detail. I would like to say a heartfelt thank you to the leader of the group, Vlatko Vedral, for the motivating discussions I have had with him and his company. I am very grateful for his invitation to visit the Centre for Quantum Technologies in Singapore, which inspired me to continue my work in academia there. I would like to say thank you to Jonathan Busch for our work together. I want to mention also Libby Heaney, Bruno Sanguinetti and Michal Hajdusek for the good times we enjoyed in Leeds and elsewhere. I would like to thank all my family, for their love and constant support. The most special thanks goes to my mother Eli, for making me the person I am. She gave me the confidence to follow my heart and pursue my dreams. I felt she is behind me every step of the way, whichever path I took. This thesis is dedicated to her. A wholehearted thank you goes to my brother Vasil for taking care of his little sister throughout all her life; to my cousin Vesela, to my aunt Ani and to my grandparents Vasilka and Evtim. I love our family times together. Finally, I would like to give a loving thank you to Mark Williamson. You have been a

v vi constant source of happiness in my life for the last year and a half even when we were away from each other. I am extremely grateful for your love and kind support, without you this thesis wouldn’t have been completed now.

Sofia, January 2009 Contents

Acknowledgements v

1 Introduction 1

2 Theoretical background 11 2.1 Qubit dynamics ...... 13 2.2 Resonant excitation ...... 15 2.3 Optical Bloch equations ...... 17

3 Effect of dephasing on single-qubit gates 19 3.1 Motivation ...... 20 3.2 Analytic model ...... 20 3.3 Special cases ...... 23 3.4 Conclusions ...... 26

4 Controlled design of arbitrary quNit states 29 4.1 Motivation ...... 30 4.2 Definition of the problem ...... 30 4.3 General solution ...... 32 4.4 Types of population distribution ...... 36 4.5 Applications to exactly soluble models ...... 40 4.6 Multistate Landau-Zener model ...... 45 4.7 Conclusions ...... 48

5 Engineering of arbitrary unitaries by reflections 51 5.1 Motivation ...... 51 5.2 Quantum Householder Reflection (QHR) ...... 53 5.3 QHR decomposition of U(N) ...... 56 5.4 Quantum Fourier transform ...... 60

vii viii CONTENTS

5.5 Discussion and conclusions ...... 63

6 Physical realisation of coupled quantum reflections 65 6.1 Motivation ...... 66 6.2 The degenerate two-level system ...... 67 6.3 Exact analytical solution ...... 69 6.4 Quantum-state reflections ...... 74 6.5 Two degenerate upper states ...... 77 6.6 Discussion and conclusions ...... 82

7 Entangling distant qubits using classical interference 85 7.1 Motivation ...... 85 7.2 Experimental proposal ...... 86 7.3 Entangling scheme ...... 88 7.4 Evolution of the system ...... 91 7.5 Conclusions ...... 92

8 Summary and contributions 95

A Publications of the author 97

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|ψi = α |0i + β |1i , (1.1)

We say that a qubit is in a pure state if its state can be expressed as such superposition. It is not always possible however, to ascribe a state vector to a quantum system that is not isolated but is a part of a larger compound system. In this case, assume all we know is that the system is in a mixture (not a superposition) of states |ψi, not necessary orthogonal to each other and having a different expansion in the computational basis. Let pψ ≥ 0 be the probability that the system is in state |ψi. Then, the density operator must be written as X ρ = pψ|ψihψ|, (1.2) ψ which is clearly a Hermitian operator, ρ = ρ†. We can calculate that P2 P T r(ρ) = n=1 ρnn = ψ pψ = 1 which is consistent with the interpretation that pψ are probabilities for being in a state |ψi. Then, we say that the system is in a mixed state, i.e. a mixture of different pure states, which can not be represented as a state vector. For system in a pure state its density matrix is just ρ = |ψihψ|. For convenience, we assume that the qubit which represents our logical bit is in a pure state. Then, its state has the form (1.1) where the coefficients α and β are complex numbers. Remarkably, we can not examine a quantum state to determine the exact values of α or β provided we restrict to making measurements in the computational basis even if we have a very large number of copies of the state. The information we can retrieve by repeated measurements on identical copies of the state in different bases are the values of |α|2, |β|2 and their relative phase, |α||0i + eiφ|β||1i. The real numbers |α|2 and |β|2 are the probabilities of detecting 0 or 1 when measuring the state of the qubit. In fact, measurements in quantum computation represent probabilistic operations that convert a qubit into a classical bit M = {0, 1}. Since probabilities should sum up to one, we require that

|α|2 + |β|2 = 1. (1.3)

Geometrically we can interpret the qubit state vector as a unit vector in a two-dimensional complex vector space. Moreover, we can rewrite the state of the qubit given that (1.3) is 3

0 z

θ ψ

y x ϕ

1

Figure 1.1: Geometrical representation of the state vector |ψi of a single qubit on the Bloch sphere. fulfilled as ³ ´ iγ θ iφ θ |ψi = e cos 2 |0i + e sin 2 |1i , (1.4)

iγ θ with γ, θ and φ real numbers. The two coefficients α and β are expressed as α = e cos 2 , iγ iφ θ iγ β = e e sin 2 , with 0 ≤ ϕ < 2π and 0 ≤ θ < π. We can ignore the factor e and skip it from now on because it has no observable effects. Effectively, only the numbers θ and φ define the qubit state and they can be viewed as the coordinates of a point of a three-dimensional unit sphere. Hence, all possible pure states of a qubit are represented as points of a sphere. This gives rise to the Bloch sphere representation which provides a useful means of visualising the state of a single qubit as shown on Fig. 1.1. Physically, the qubit can be encoded in different systems. It can be represented as the two different polarizations of a photon; as the projections of an electron spin; as two states of an electron orbiting a single atom.

Single-qubit gates

During the course of quantum computation the single qubits are subject to various operations – quantum logic gates, which perform the desired state changes and determine the evolution of the qubit. For convenience, we will restrict attention to unitary quantum gates (which are reversible). Non-unitary (non-reversible) quantum operations can be simulated by unitary quantum gates if we allow the possibility of adding an ancilla and of discarding some output qubits. A single-qubit gate is mathematically represented by a two-dimensional unitary U which 4 CHAPTER 1. INTRODUCTION transforms a quantum state according to |ψi → U|ψi, or in the density matrix representation its action is expressed as ρ → UρU†. The quantum gate U is a linear transformation which is bijective and length-preserving. Any two-dimensional unitary represents a single-qubit logic gate and because of its unitarity it can always be inverted. Note, that there always exist a quantum logic gate U†, where U† is the adjoint of U, which undoes the state alteration produced by U. In the Bloch sphere representation such operations correspond to rotations and reflections of the sphere. On the other hand however, many classical logic operations are irreversible and hence applying them to the logical bits causes an irreversible loss of information. The requirement for unitary quantum evolution (= reversibility) restricts the number of logic operations that have both classical and quantum representations. For example, there is only one non-trivial single-bit gate – the not operation which changes the state of the logical bit from 0 → 1 and 1 → 0. To the contrary, there are an infinite number of single qubit gates which are not trivial. One of them is the quantum not gate which is the Pauli spin matrix X, " # 0 1 X = . (1.5) 1 0 X acts linearly on the state of the qubit and changes the role of |0i and |1i, " # " # α β X = . (1.6) β α In terms of the Bloch sphere this action is visualised as a rotation through an angle π about the x axis. Other important single qubit operations are the Pauli spin matrices Y, Z, the phase gate T, as well as the Hadamard gate H, " # " # " # " # 0 −i 1 0 1 0 1 1 Y = , Z = , T = , H = 1 , (1.7) i 0 0 −1 0 i 2 1 −1 which do not have classical alternatives. The Hadamard transformation turns |0i into 1 1 2 (|0i + |1i) and |1i into 2 (|0i − |1i). Note, that the H gate is in fact the two-dimensional quantum Fourier transform which is a basic ingredient in all quantum algorithms. Despite their infinite number, all single-qubit gates can be decomposed into a sequence of rotations about two of the axes of the Bloch sphere, along with a suitable phase factor. We define the rotation gates as the exponentials of the Pauli spin matrices

−iθX/2 −iθY/2 −iθZ/2 Rx(θ) = e , Ry(θ) = e , Rz(θ) = e , (1.8) which are unitary operators. Then, there exist real numbers α, β, γ and δ such that

iα U = e Rz(β)Ry(γ)Rz(δ). (1.9) 5

Such decomposition can be given for any two non-parallel axes of the Bloch sphere.

Quantum register and multi-qubit gates

Quantum computers operate on a register of many qubits, in the same way as classical com- puters process a classical register. There is a crucial difference between the two and it is contained in the amount of information which can be stored in each of them. Let us assume that both registers consist of n particles. Then, the classical register has 2n possible configu- rations but can store only one number. The quantum register however, has 2n basis states but can store a superposition of all n-binary numbers simultaneously. Hence, a processor that can use a register of qubits will in effect be able to perform calculations using all possible values of the input register simultaneously. Note, that in order to completely describe the state of an n-qubit register we require 2n complex numbers, while a classical n-bit register is fully described by n integers. This makes simulating quantum evolution on a classical computer an exponentially inefficient process. Consider now two-qubit or multi-qubit logic gates acting, respectively on two or more qubits from the quantum register. Since quantum computation is reversible, the number of input qubits must be equal to the number of output qubits for any unitary multi-qubit gate. It is extremely difficult though, to manipulate many qubits simultaneously and to realise multi- qubit quantum gates. This obstacle was overcome when it was shown that any computation on a register of qubits can be broken up into a series of two-qubit operations and single-qubit operations [6]. The prototype of a two-qubit gate is the controlled-not or the cnot gate. The cnot has two input qubits, known as the control qubit and the target qubit, and two output qubits. When applied the cnot gate changes the state of the target qubit accordingly to the state of the control qubit as follows: |0ic |0it → |0ic |0it , |0ic |1it → |0ic |1it , |1ic |0it → |1ic |1it and

|1ic |1it → |1ic |0it. In matrix form cnot is represented as   1 0 0 0      0 1 0 0  U =   . (1.10) cnot    0 0 0 1  0 0 1 0

Together with single-qubit operations, the cnot gate can be used to construct any multi- qubit gate, i.e. they form a universal set of gates. There are also other interesting two- qubit quantum gates, such as the swap gate which is analogous to the classical crossover transformation, it interchanges the states of the two qubits |ai|bi ← |bi|ai, and the controlled- 6 CHAPTER 1. INTRODUCTION

Z gates,     1 0 0 0 1 0 0 0          0 0 1 0   0 1 0 0  U =   , U =   . (1.11) swap   cZ    0 1 0 0   0 0 1 0  0 0 0 1 0 0 0 −1 Their action however, is not genuinely different and they can be constructed from the universal set of gates. The swap gate can be implemented by triple action of the cnot while the controlled-Z can be realised by one cnot and two Hadamard transformations. More generally, any controlled-U transformation, where U is a one-qubit gate, can be implemented on the qubit register using the universal set of gates.

Entanglement

Correlations between the individual qubits of the composite quantum register can occur during the course of computation. These correlations are classical if the state of the composite system can be approximated by a mixture of the subsystems’ states. For convenience, let us assume that the register consists of two qubits, labeled A and B, described by density matrices

ρA and ρB. Then, the state of the composite system is classically correlated if its density matrix (1.2) can be represented as X i i ρsep = piρA ⊗ ρB, (1.12) i where pi are the appropriate weights. These states are known as separable states. If no basis exist that allows ρ to be expressed in the form (1.12), i.e. X i i ρ 6= piρA ⊗ ρB, (1.13) i the state of the composite system is quantum correlated or entangled [7]. Important two-qubit entangled states are known as Bell states or EPR states after Einstein, Podolsky and Rosen who first pointed out their interesting properties. One of the Bell states is

|Belli = √1 (|0i |1i − |1i |0i ) , (1.14) 2 A B A B

th where we label the state | ii to be the state of the i qubit, i = A, B. Bell states are created by applying a cnot operation on the two qubits, provided that the target B is initially prepared in state |0i and the control qubit A – in a superposition state |0i + |1i, followed by an appropriately chosen single-qubit unitary. For convenience let us assume that the two qubits are encoded in spin 1/2 particles and |0i represents spin down ↓, while |1i represents spin up ↑. Suppose, the two qubits are 7 initially prepared in the state (1.14) after which they are separated and a spin-measurement is performed on one of them, let’s say qubit A. If the measurement yields spin down ↓, then a consequent measurement on qubit B will show a result spin up ↑ with probability equal to 1, even when there is no physical interaction between the qubits. Analogously, if the measurement outcome of qubit A is spin up ↑, then qubit B will be in state spin down ↓ upon measurement with certainty. Moreover, even if we perform a change of the computational basis (make a unitary transformation), measuring the two qubits prepared in state (1.14) will always show opposite spin orientations. In this sense the anti-symmetric Bell state is perfectly anti-correlated. Einstein, Podolsky and Rosen published a paper in 1935, famously known as the EPR pa- per, in which they provided two possible explanations for these measurement anti-correlations: (i) the measurement result of the first particle has a non-local effect on the physical state of the second particle. It is though the measurement result obtained from qubit A is instanta- neously transmitted to qubit B, which then changes its state, according to the information received. Or (ii) quantum mechanics is an incomplete theory and it provides only partial information about the state of a quantum system [8]. EPR believed that (ii) is true because (i) would violate special relativity in sense that information can not be transmitted at speeds faster than the speed of light. They also thought that there are some “elements of physical reality” - variables that characterise the state of a quantum system, which are not accounted for by quantum mechanics. Later, Bell proposed inequalities that could test whether (i) or (ii) is true [9]. Bell consid- ered a setup where two observers perform measurements on two spatially separated particles. He showed that for any theory including local hidden variables, i.e. describing a system with classical correlations, there is an upper bound on the maximal correlation of the measure- ment outcomes. Bell proved that to the contrary, a pair of particles in an entangled state with quantum correlations would violate his inequality. Clauser, Horne, Shimony and Holt (CHSH) framed Bell’s inequality in a more experimentally testable expression [10] and the first attempts to test Bell’s inequality using entangled pairs of photons were performed in the 1970s [11]. It is regarded by many that the first successful demonstration of the violation of the CHSH inequality was shown by Alain Aspect in 1981 [12] proving that “it is impossible to assign local physical reality” to systems with quantum correlations. Bell states have already been experimentally realised with trapped ions [13], photons [14], atoms [15] and quantum dots [16]. They are used for a very interesting quantum protocol called quantum teleportation which enables efficient communication of a state of a qubit, using just two bits of information sent from one spatial location to another via classical communication channel. Note, that the qubit is not physically teleported from one place to 8 CHAPTER 1. INTRODUCTION the other, rather, it is its state. The protocol for quantum teleportation includes two spatially separated parties, Alice and Bob, sharing a Bell state. Alice needs to communicate to Bob one qubit of information |ψi = α |0i + β |1i. Alice does not know the state of the qubit, i.e. the values of the coefficients α and β. Still, she can teleport the unknown qubit of information using only one cnot gate applied to her two qubits, followed by a Hadamard gate on the unknown qubit. Finally, Alice measures the two qubits in her possession and communicates the result to Bob via two classical bits of information, which encode the two possible states |00i, |01i, |10i or |11i. Once Bob has learned the measurement outcome he can fix up his qubit state, recovering |ψi, by applying at most one appropriate single-qubit gate. It is important to stress that the quantum teleportation protocol does not violate special relativity and the limit of the speed with which information can travel. Indeed, it is the use of a classical communication channel between the two parties that limits the speed of teleportation by the speed of light.

One – way quantum computation

A radical step forward in our understanding of how a quantum computer works was made by Raussendorf and Briegel in 2001 [17]. Before their proposal standard quantum computa- tion implied that the qubits are processed by single- and multi-qubit gates. Raussendorf and Briegel suggested a very different quantum computer model called a one - way quantum com- puter. The one-way quantum computer requires that qubits are initially prepared in a highly entangled cluster state. After that the computation is performed by applying processing mea- surements. These measurements fulfill two conditions: (i) they are single-qubit measurements with classical feedforward of their outcomes, and (ii) the choice of consequent measurements depend on the outcomes of earlier measurements. The actual order and choice of measure- ments determine the algorithm computed. Because of the crucial role of measurement, the one-way quantum computer is irreversible. The term cluster state does not refer to a single state, but rather to a family of quantum states. The idea is that to any graph we can define a cluster state by first associating to each vertex a corresponding qubit and then applying a graph-dependent preparation procedure. Cluster states are generated in d-dimensional lattices and were first introduced by Raussendorf and Briegel in the case when the graph is two-dimensional square lattice. While for pure states of bipartite systems there is only one class of entangled states - the Bell state, when the entanglement is shared between more qubits it turns out that there are several inequivalent classes of entanglement. Raussendorf and Briegel introduced their one-way quantum computer for multi-partite entangled cluster states, that are different from other multi-qubit entangled 9 states such as the Greenberger-Horne-Zeilinger (GHZ) state,

|00...0i + |11...1i , (1.15) or the W state, |10...0i + |01...0i + ... + |00...1i . (1.16)

Experimentally the one - way quantum computer has already been demonstrated with four qubits encoded into the polarization states of four photons [18] in 2005. A cluster state has also been created using neutral atoms trapped in the periodic potential of an optical lattice [19]. Other entangled multi-qubit states that have been successfully demonstrated include a six-atom GHZ state in 2005 by Wineland et al [13]. Quantum Information science has been mainly developed for qubits [20, 21]. All quan- tum algorithms and quantum information processing techniques so far have been based on qubits. A promising alternative to using qubits as building blocks of the quantum computer is to utilize for this purpose N-state quantum systems, known as quNits. The advantage is considerable. While in a qubit we can encode 2 continuous variables - the ratio between the population of the two states and the relative phase between their amplitudes, for a quNit in a pure state we can identify 2 (N − 1) independent parameters (N − 1 populations and N − 1 relative phases). Hence, for quantum computing based on quNits the manipulation of much fewer particles is required which significantly reduces complex multi-particle interactions as well as possible sources of decoherence. The presented thesis begins with a theoretical introduction to general quantum-mechanical description of a two-state system in an external pulsed field described in Chapter 2. The Schr¨odingerequation in the rotating-wave approximation as well as Bloch equation for a qubit are derived. In Chapter 3 the implementation of single-qubit operations is considered on a qubit when decoherence processes are present. Such processes are due to the coupling of the physical system in the laboratory to the environment and they cause irreversible loss of coherence. It is therefore crucial for the realisation of quantum computing schemes to be able to describe the evolution of the system in these cases and in Chapter 3 we present an exact analytical solution. It is given in terms of Γ functions and analyzed. Chapter 4 presents a natural generalisation of the qubit – a two-level quantum system which has an excited state and N times degenerate ground level. The degenerate sublevels represent a most natural candidate for representing a quNit. We derive the exact solution for the evolution of this system and provide generalisations of some of the well-known two-state models. Moreover, we show that the ground state quNit can be manipulated and various quNit operations can be performed to create arbitrary chosen superpositions and quantum gates. 10 CHAPTER 1. INTRODUCTION

In Chapter 5 the standard and generalized quantum Householder reflections (QHR) are introduced and their physical implementations are proposed. It is shown that QHRs can be utilized for the decomposition of general matrix ∈ U(N) group by means of standard and generalized QHR. Moreover, engineering of an arbitrary preselected superposition quNit states is presented. Chapter 6 describes how any statevector motion in the Hilbert space can be regarded as a succession of reflections rather than of two-dimensional rotations. We here show how such reflections (mirrors) can be designed for a system with two degenerate levels – each encoding a quNit – that allows the construction of propagators for angular momentum states. It is shown that if some conditions for the applied laser fields are fulfilled the propagators within each level of degenerate states represent products of coupled generalized Householder reflections, with orthogonal vectors. In Chapter 7 we present an entangling measurement-based scheme which in fact performs a control- operation on two distant qubits. We assume that the qubits are encoded in the electron spin state of two quantum dots placed inside two optical cavities and subject to laser driving. The successful creation of a maximally entangled Bell state is heralded via macro- scopic quantum jumps. This makes the scheme very robust against parameter fluctuations and spin-bath couplings, and enables the physical realisation of various schemes for distributed quantum computing, including loophole free Bell tests. h ai states, basis the vector state the tation system, coordinate a as serve to length | finite-dimensional a for Even a in system vector, quantum state any of state internal The an in field system pulsed quantum external a of presented. description is quantum-mechanical 2006, General Sofia thesis, Diploma Kyoseva, pulsed of a description with the interacting populations. following system field, oscillating a external of causes steady description radiation and quantum-mechanical general slow coherent its the a of from Hereafter, produces application different radiation significantly the incoherent is excitation an field While equilibrium laser coherent light. a polychromatic to to system response quantum a of response The ttso h osdrdqatmsse.TeHletsaeo ui o xml stwo- is example for qubit a of space Hilbert The dim system. dimensional, quantum considered the of states space Hilbert 0 i , | 1 i ..Ec hieo oriaesse ensa defines system coordinate a of choice Each ,... | H ψ ( h iesoaiyo h ibr pc seult h ubro distinct of number the to equal is space Hilbert the of dimensionality The . t ) H i hc sebde namliiesoa btatvco pc the – space vector abstract multidimensional a in embodied is which , .W nrdc opeesto rhgnlvcosin vectors orthogonal of set complete a introduce We 2. = | ψ ( t ) H ∈ i H | ψ hr r nifiienme fwy ocos h ai vectors basis the choose to ways of number infinite an are there ( t ) i sepesda ievrigchrn uepsto of superposition coherent time-varying a as expressed is = dim n X h =0 i hoeia background Theoretical H− | j i 1 11 = e − pure δ i ζ ij n . ( t ) c tt sdsrbdb time-dependent a by described is state n ( representation t ) | n i , o ie represen- given a For .

C h a p t e r H funit of 2 (2.2) (2.1) 12 CHAPTER 2. THEORETICAL BACKGROUND

where the phases ζn(t) are chosen a priori for mathematical convenience, and the complex valued functions of time cn(t) are probability amplitudes. The absolute square of cn(t) gives the probability Pn(t) that upon measurement the quantum system will be found in state |ni,

2 2 Pn(t) = |cn(t)| = |hn|ψ(t)i| . (2.3)

As the latter part of (2.3) equation shows the probability amplitude cn is regarded as the projection of the state vector onto the coordinate axis |ni. If we consider that the quantum system is closed and there are no loss-inducing processes, then all probabilities should sum up to 1, dimXH Pn(t) = 1, (2.4) n=1 for any time t. Probability conservation (2.4) amounts to the fact that the state vector maintains constant (unit) norm at all time, hψ |ψi = 1. We have complete freedom in choosing the basis of the Hilbert space but typically the basis vectors are chosen to be eigenstates of some operator acting in H. Here, we adopt the energy representation which assumes that |ni are stationary energy states. They are the eigenstates of the time-independent Hamiltonian operator H0,

0 0 H |ni = En|ni, (2.5)

0 0 where the label n identifies a stationary energy value En. The Hamiltonian H governs the evolution of the quantum system when there are no external fields applied. If the system is initially prepared in a stationary state (or a superposition) it will remain in it as long as it evolves under H0. However, quantum information processing requires that we can selectively manipulate the dynamics of the system and engineer desired unitary gates. Hence, we apply laser fields and create a time-dependent system-laser interaction. The changes in the state vector are described by the time-dependent Schr¨odingerequation,

∂ i~ |ψ(t)i = H(t) |ψ(t)i , (2.6) ∂t where ~ is the Planck’s constant h = 6.626 × 10−34J.s divided by 2π. The time-dependent Hamiltonian operator H(t) represents the the total energy of the system, the sum of kinetic, potential and interaction energies. We require that H(t) is a hermitian operator, H†(t) = H(t), which ensures that observable excitation energies, the eigenvalues of H(t), are real. The full Hamiltonian is expressed as H(t) = H0 + V(t), (2.7) 2.1. QUBIT DYNAMICS 13 where V(t) is the part which accounts for the interaction between the atom and the laser field. All excitation processes that we consider here are induced by laser pulses with infrared or optical region frequencies whose wavelengths are much larger than the atomic dimensions. Typically, the magnetic dipole interaction relates to the electric dipole interaction as 1/274 [22]. Thus, we assume that the interaction experienced by our system is that between a spatially uniform electric field evaluated at the center of mass of the atom E(t) and the atomic dipole moment d, V(t) = −d.E(r, t). (2.8)

The fundamental Schr¨odingerequation (2.6) underlies all nonrelativistic descriptions of microscopic temporal behaviour. It is the basis for the quantum-mechanical description of excitation. In particular, it provides the foundation of all discussions of coherent excitation and defines the dynamical behaviour of a quantum system.

2.1 Qubit dynamics

The idealized notion of a qubit provides the simplest application of the time-dependent Schr¨odingerequation. The state vector (2.2) reduces to

−iζ0(t) −iζ1(t) |ψ(t)i = e c0(t)|0i + e c1(t)|1i. (2.9) and for time-independent dynamics it fulfills ∂ i~ |ψ(t)i = H0 |ψ(t)i = E0e−iζ0(t)c (t)|0i + E0e−iζ1(t)c (t)|1i. (2.10) ∂t 0 0 1 1 Considering the orthogonality of the basis states, this partial differential equation reduces to a set of two decoupled first-order, linear, homogeneous, ordinary differential equations for the probability amplitudes

0 ˙ i~c˙n(t) = (En − ~ζn(t))cn(t), (n = 0, 1). (2.11)

Note, that the probability amplitudes do not change their absolute values in the course of evolution according to H0 but only accumulate a dynamical phase,

0 −i(E −~ζ˙n(t))t/~ cn(t) = e n cn(0). (2.12)

If we apply a laser field on the qubit this will cause state transitions and changes in the probability amplitudes. For all electric-dipole transitions amongst bound atomic states of an isolated atom the matrix representation of V(t) (2.8) usually has no diagonal elements. If the external field is static the diagonal elements, V00 and V11, would produce an energy shift in 14 CHAPTER 2. THEORETICAL BACKGROUND

the undisturbed energies of the atom. This energy shift however, can be incorporated in the definition of the probability amplitudes with a simple phase transformation. This allows us to assume that the diagonal elements of V(t) are zero and the only nonvanishing element of

the interaction Hamiltonian is V01 and V10. It is the fact the H(t) is hermitian that ensures ∗ that V(t) is also hermitian and hence, V01 = V10. We can write the Schr¨odingerequation for

the probability amplitudes cn(t) and accounting for the phases we obtain, " # " #" # 0 −i(ζ (t)−ζ (t)) d c0(t) E − ~ζ˙0(t) V01(t)e 1 0 c0(t) i~ = 0 . (2.13) dt ∗ i(ζ1(t)−ζ0(t)) 0 ˙ c1(t) V01(t)e E1 − ~ζ1(t) c1(t) Let the polarization axis of the applied E field be ˆe, then the interaction Hamiltonian equals V(t) = −d.E(t) = −DE(t), (2.14)

where D = d.ˆe is the projection of the electric dipole operator on the polarization direction ˆe of the electric field which amplitude is

E(t) = Ee−iωt + E∗eiωt, (2.15)

with ω being its frequency and φ its phase, E = |E| eiφ. Then, we can express the interaction Hamiltonian elements as

−iωt ∗ iωt V01 = D01(Ee + E e ), −iωt ∗ iωt V10 = D10(Ee + E e ), (2.16)

where Dij = hi| D |ji are the matrix elements of the dipole operator D. We have assumed that ∗ Dij = Dji, which is practically always valid in this context, and often these matrix elements can be taken real [23]. If the phase of the electric field remains constant, it is always possible to choose the phases

of the probability amplitudes ζn(t) to be equal to

ζ1(t) − ζ0(t) = ωt + φ,

˙ 0 ~ζ0(t) = E0 , (2.17) which is the the so-called Dirac picture. Substituting the appropriately chosen phases in the Hamiltonian we obtain, " # −i2ωt 0 D01 |E| (1 + e ) H(t) = , (2.18) i2ωt D01 |E| (1 + e ) ~∆

where we have defined the detuning ∆ = ω0 − ω, of the resonance frequency of the atomic

transition ω0 from the radiation frequency. 2.2. RESONANT EXCITATION 15

The real-valued amplitude of the off-diagonal elements of the Hamiltonian (2.18) is called the Rabi frequency Ω, 1 2 ~Ω = D01 |E| , (2.19) which has dimension of angular frequency. Together with the Bohr frequency ω0 and the interaction frequency ω, the Rabi frequency provides one of the three characteristic time scales for the coherent atomic excitation. It parameterizes the interaction strength between the atom and the external field and is therefore called coupling. With the understanding that we shall be interesting in near-resonant transitions

∆ ¿ ω, ω0, we can adopt the rotating-wave approximation (RWA). It amounts to neglecting the terms involving the exponentials e±i2ωt, i.e. oscillating with frequency ±2ω, in the Hamil- tonian (2.18). Finally, we obtain that the Hamiltonian in the rotating-wave approximation reads, " # 0 1 Ω(t) H(t) = ~ 2 , (2.20) 1 2 Ω(t) ∆(t) which underlies all of the results in this thesis. The two-state rotating-wave approximation is widely used throughout quantum physics and can be generalised to multi-state systems. It was first applied in magnetic resonance.

2.2 Resonant excitation

Coherent resonant excitation represents an important notion in quantum mechanics [22, 24]. Resonant pulses of specific pulse areas are widely used in a variety of fields in quantum physics, including quantum information processing [20, 21], nuclear magnetic resonance [25] and co- herent atomic excitation [22]. Resonant excitation allows to establish a full control over the quantum system, particularly in a two-state system, and realize any unitary transformation (qubit rotation) in it. Particularly important transformations include complete population transfer (transition probability P = 1), complete population return (P = 0) and Hadamard 1 transformation H (1.7) providing P = 2 . Consider a resonantly driven system, ω = ω0, with detuning zero and coupling - an arbitrary function of time, ∆ = 0, Ω(t). (2.21)

Substituting these interaction parameters into Eqn.(2.20) we obtain a set of two coupled differential equations for the probability amplitudes, d i c (t) = 1 Ω(t)c (t), dt 0 2 1 d i c (t) = 1 Ω(t)c (t). (2.22) dt 1 2 0 16 CHAPTER 2. THEORETICAL BACKGROUND

The solution of this system is easily obtained if we introduce a new variable R t 0 0 d t → A(t) = −∞ Ω(t )dt and change the derivative dt according to d dA(t) d d = = Ω(t) . (2.23) dt dt dA(t) dA The variable A(t) is the pulse area of the applied field. Written in terms of the new variable, the set of differential equations (2.22) decouples and we obtain

00 1 c0(A(t)) = − 4 c0(A(t)), (2.24)

0 which holds for c1(A(t)) as well and labels a derivative over A(t). The solution of this equation is a superposition of cos(A(t)) and sin(A(t)) weighted with the appropriate initial constants - the values of c0(−∞) and c1(−∞). If we assume that the system was initially prepared in state |ψ1i, then 1 c0(t) = cos 2 A(t), 1 c1(t) = −i sin 2 A(t). (2.25) The population of the excited state |1i oscillates as

2 1 1 P1(t) = sin 2 A(t) = 2 [1 − cos(A)], (2.26) which is shown on Fig.2.1. At all times the probability amplitudes are expressible as trigono- metric functions of A(t); when A(t) is some even-integer multiple of π (e.g., a 2π pulse), the probabilities repeat their initial values. Thus the response, for resonant excitation, does not depend on any details of the pulse shape.

1.0

0.8

0.6

0.4

0.2

Population of excited qubit state state qubit of excited Population 0 0 2 4 6 8 10 Pulse Area (units of π) Figure 2.1: Rabi oscillations of the population of the excited qubit state.

These oscillations of the population are known as Rabi oscillations and they often serve as a test for quantum behaviour. Note, that at times when the pulse area A(t) is an odd 2.3. OPTICAL BLOCH EQUATIONS 17 multiple of π the population is completely inverted. For pulse areas equal to even multiples of π, the system returns to its initial state. It is straightforward to calculate that in order to realise a Hadamard gate we require a pulse with area that is an odd multiple of π/2, i.e. A(t) = (2k + 1)π/2. The Rabi oscillations provide a useful tool for coherent control of quantum dynamics via pulses with precise pulse areas.

2.3 Optical Bloch equations

A more general description of the dynamics of a two-state quantum system is provided by the Bloch equations. They provide the description of the evolution of the density matrix elements. The density matrix for a qubit in a pure state |ψi = c0(t)|0i + c1(t)|1i is defined as the outer product of its state vector and its hermitian conjugate, " # " # 2 ∗ |c0(t)| c0(t)c (t) ρ11 ρ12 ρ = |ψihψ| = 1 = . (2.27) ∗ 2 c0(t)c1(t) |c1(t)| ρ21 ρ22

The diagonal elements of the density matrix ρ11 and ρ22 are the populations of the states

|0i and |1i. The off-diagonal elements ρ12 and ρ21 are called coherences and they represent the response of the system at the driving frequency. Our goal is to obtain equations which describe the evolution of the elements of ρ. ∗ ∗ ∗ ∗ ∗ The time derivatives ofρ ˙12 =c ˙0.c1 + c0.c˙1 =ρ ˙21 andρ ˙22 =c ˙1.c1 + c1.c1 = −ρ˙11 are expressed in terms of the probability amplitudes. Considering the RWA Hamiltonian from Eq. (2.20) we obtain that

d d ρ = i Ω(ρ − ρ ) + i∆ρ = ρ∗ , dt 12 2 11 22 12 dt 12 d d ρ = i Ω(ρ − ρ ) = − ρ . (2.28) dt 22 2 21 12 dt 11

The real and imaginary parts of the coherences are labeled with u(t) and v(t),

u(t) = 2<ρ12, (2.29)

v(t) = 2=ρ12, (2.30)

w(t) = ρ22 − ρ11, (2.31) and the population difference is w(t)

w(t) = ρ22 − ρ11. (2.32) 18 CHAPTER 2. THEORETICAL BACKGROUND

In terms of u(t), v(t) and w(t) equations (2.28) become

d u(t) = −∆v(t), (2.33a) dt d v(t) = ∆u(t) − Ωw(t), (2.33b) dt d w(t) = Ωv(t), (2.33c) dt which are the optical Bloch equations. They can be written in vector notation as

R˙ = R × (−Ωˆe1 − ∆ˆe2) (2.34) by taking u, v and w as the components of the Bloch vector and hence they define a point on the surface of the Bloch sphere [26]. eie.Mroe,aayi eut o h mltd n h hs hf ftedme Rabi damped the of is shift Γ phase function the gamma presented. and Euler are amplitude of result oscillations the terms analytic for in exact results states analytic An Moreover, qubit’s equation. the derived. Bloch between optical transfer and the population present by the is system for process the dephasing a of when evolution interesting case the several realistic describe of the field discuss realisation we external the Here, pulsed for by allows gates. driven situation single-qubit resonantly This is qubit shape. the hyperbolic-secant that a assume with We [27]. Ref. in published effect This coherence. of loss irreversible the as between an known coupling produces is the environment However, the with and qubit transfer. the system population of physical desired manipulation precise produce the and require fields we Trans- optics external computing. quantum of quantum language reversible the for in techniques lated processing ingredient information basic the quantum is all laboratory the of in operations single-qubit of realisation experimental The rcsigqatmifraini scuilta eoeec ffcstk lc namuch a on place evolution. take state effects desired For decoherence the that dot. than quantum time-scale crucial a larger is or of it fluctuations spin magnetic information laser the quantum to and processing can due spin processes electron dephasing the Such states, between atomic interaction place. hyperfine excited take from processes emission decoherence spontaneous when important be extremely dynamics is qubit it the laboratory the describe in to processing information quantum of realisation the nti hpe h ffc fdpaigo igeqbtgtsi eie.Terslsare results The derived. is gates single-qubit on dephasing of effect the Chapter this In ffc fdpaigo igeqbtgates single-qubit on dephasing of Effect decoherence n tmksteeouino h unu ytmnnuiay For non-unitary. system quantum the of evolution the makes it and 19

C h a p t e r 3 20 CHAPTER 3. EFFECT OF DEPHASING ON SINGLE-QUBIT GATES

3.1 Motivation

Crucial conditions for resonant coherent excitation are resonance (the frequency of the exter- nal field ω must be equal to the Bohr transition frequency ω0) and coherence. Deviations from resonance (∆ 6= 0) are detrimental and lead to rapid departure of the transition probability from the desired value. In this respect, an alternative to resonant excitation is provided by adiabatic excitation [28], which is robust against such variations. Even more crucial for resonant excitation is coherence. Incoherent excitation, as described by Einstein’s rate equations, allows only partial population transfer, e.g., at most 50% in a two-level system with equal degeneracies of the two levels [22]. Deviations from perfect coherence, which can be described by the quantum Liouville or Bloch equations, inevitably cause departure from the desired unitary transformation. Two general types of decoherence processes can be present: depopulation (e.g., due to spontaneous emission or ionisation) and dephasing (e.g., due to elastic collisions, field fluctuations, coupling to the environment, etc.). In this Chapter, we present an exact analytic solution for pulsed resonant excitation of a qubit in the presence of pure dephasing. Dephasing is recognised as one of the main obstacles in quantum information and the availability of precise analytic estimates of its effect can be very useful and important. We derive the exact solution of the Bloch equation for a hyperbolic-secant pulse shape and a constant dephasing rate. We provide examples of the general solution, which is expressed in terms of gamma functions, in various special cases of interest, e.g. for pulses with specific pulse areas. These results allow us to determine explicitly the deviations from the desired probabilities caused by the dephasing.

3.2 Analytic model

Dephasing processes can be incorporated into the quantum-mechanical description of resonant excitation by including a phenomenological dephasing rate Γ0 into the Bloch equation (2.33) from Chapter 2,       u(t) −Γ0 −∆(t) 0 u(t) d        v(t)  =  ∆(t) −Γ −Ω(t)   v(t)  . (3.1) dt    0    w(t) 0 Ω(t) 0 w(t)

This dephasing rate is the inverse of the transverse relaxation time T2,Γ0 = 1/T2 [22, 24]. We shall solve the Bloch equation (3.1) with the initial conditions

u(−∞) = v(−∞) = 0, w(−∞) = −1, (3.2) 3.2. ANALYTIC MODEL 21

which correspond to a system initially in state |1i: ρ11(−∞) = 1, ρ22(−∞) = 0. Our objective is to find the Bloch vector [u, v, w]T as t → +∞, and particularly, the population inversion w(∞), since the coherences vanish at infinity due to the dephasing. We suppose an exact resonance, a hyperbolic-secant pulse and a constant dephasing rate,

∆(t) = 0, (3.3)

Ω(t) = Ω0sech(t/T ), (3.4)

Γ0 = const. (3.5)

The dephasing rate Γ0 is a positive constant and T is the characteristic pulse width. The peak

Rabi frequency Ω0 will be assumed also positive. The pulse area of the sech pulse (3.4) is

Z +∞ A = Ω(t)dt = πΩ0T. (3.6) −∞

The exact solution in the coherent limit

For Γ0 = 0, the Bloch equation (3.1) is solved exactly [22, 24],

w(∞) = − cos A. (3.7)

Of particular interest are the cases when A is equal to an integer or half-integer multiple of π. There are three cases of special significance.

• The odd-π pulses with area

A = (2n + 1)π, (n = 0, 1, 2,...), (3.8)

invert the population, w(∞) = 1 (ρ11 = 0, ρ22 = 1). A special case is the π pulse with A = π.

• The even-π pulses with area

A = 2nπ, (n = 0, 1, 2,...), (3.9)

restore the population to the initial state, w(∞) = −1 (ρ11 = 1, ρ22 = 0). A special case is the 2π-pulse with A = 2π.

• The half-integer-π pulses with area

A = (2n + 1)π/2, (n = 0, 1, 2,...), (3.10)

1 create an equal superposition between states 1 and 2, w(∞) = 0 (ρ11 = ρ22 = 2 ). A special case is the half-π pulse with A = π/2. 22 CHAPTER 3. EFFECT OF DEPHASING ON SINGLE-QUBIT GATES

All these three cases are of great importance and such pulses are widely used in various applications in quantum physics, e.g. in nuclear magnetic resonance, coherent atomic excita- tion and quantum information. We shall therefore pay special attention to these cases in the analytic solution, which we shall derive below.

The exact solution with dephasing

Because of the resonance condition (3.3), it follows from Eq. (3.1) that the equation foru ˙ decouples (with the overdot denoting a time derivative),

u˙(t) = −Γ0u(t), (3.11) and can be solved independently,

u(t) = u(−∞)e−Γ0t = 0, (3.12) where we have used the initial conditions (3.2). We change the Bloch variable v(t) = −ix(t) and Eq. (3.1) is reduced to two coupled equations,

ix ˙(t) = −iΓ0x(t) + Ω(t)w(t), (3.13a) iw ˙ (t) = Ω(t)x(t). (3.13b)

These equations resemble the Schr¨odingerequation for the Rosen-Zener model [29] with irre- versible loss from one of the states [30, 31]. For convenience, the derivation is adopted to our case and given in the Appendix at the end of the Chapter. The exact solution for w(t) as t → +∞ reads ¡ ¢ 2 1 Γ 2 + γ w(∞) = − ¡ 1 ¢ ¡ 1 ¢, (3.14) Γ 2 + γ + α Γ 2 + γ − α where Γ(z) is the gamma function [32, 33, 34] and the dimensionless parameters α and γ are defined as Γ T α = Ω T, γ = 0 . (3.15) 0 2 Because of the dephasing, the coherences vanish,

u(∞) = v(∞) = 0. (3.16)

By using the reflection formula Γ(z)Γ(1 − z) = π/ sin(πz) [33], Eq. (3.14) can be written also as ¡ ¢ ¡ ¢ 2 1 1 Γ 2 + γ Γ 2 − γ + α w(∞) = − ¡ 1 ¢ cos π (α − γ) . (3.17) πΓ 2 + γ + α 3.3. SPECIAL CASES 23

For γ = 0, Eq. (3.17) reduces to the lossless solution (3.7). Equation (3.17) shows that 1 3 5 w(∞) vanishes whenever the cosine factor vanishes, i.e. for |α − γ| = 2 , 2 , 2 ,... Hence the values of the pulse area, for which an equal superposition between states |0i and |1i is created 1 (ρ11 = ρ22 = 2 ), are shifted from their half-π values (3.10),

A = (2n + 1 + Γ0T ) π/2. (3.18)

Several approximations to w(∞) are given below in some special cases.

3.3 Special cases

Specific pulse areas

For α = n (nπ pulse), where n is an integer, Eq. (3.14) reduces to

nY−1 2γ − 1 − 2k w(∞) = − , (3.19) 2γ + 1 + 2k k=0 where we have used repeatedly the recurrence relation Γ(z + 1) = zΓ(z). It follows from Eq. 1 (3.19) that w(∞) = 0 for γ = 2 for any integer α = n = 1. For α = 1 (π pulse) we have 1 − 2γ w(∞) = . (3.20) 1 + 2γ

Hence the population inversion is a decreasing function of γ, which decreases to wε = 1 − ε for 1 1 − wε γε = . (3.21) 2 1 + wε

1 1 1 For wε = 0.9, 0.5 and 0 we find γε = 38 , 6 and 2 , i.e. the inversion decreases very rapidly as γ increases. Similar simple expressions can be derived from Eq. (3.19) for other cases of integer-π pulses. 1 For α = n + 2 (half-integer-π pulse) we find from Eq. (3.14) that ¡ ¢ γΓ2 1 + γ Yn γ − k w(∞) = − 2 . (3.22) Γ2 (1 + γ) γ + k k=1

1 The factor in front of the product gives w(∞) for α = 2 (π/2 pulse). In Fig. 3.1 the population inversion w(∞) is plotted against the dephasing rate for different pulse areas. 24 CHAPTER 3. EFFECT OF DEPHASING ON SINGLE-QUBIT GATES

1.0 )

∞ π ( 3 π w 0.5

3π/2 0 π/2

π 3π4 2π -0.5 π 3π/2 4 π π 2 π/2 Population Inversion Inversion Population -1.0 0.001 0.01 0.1 1 10 100 1000 Dephasing Rate Γ (units of 1/T)

Figure 3.1: The population inversion, Eq. (3.19) for integer-π pulses and Eq. (3.22) for half-integer-π pulses, against the dephasing rate for different values of the pulse area A = πα = πΩ0T , denoted on the respective curve.

Weak dephasing £ ¤ When γ ¿ 1 we use the relation Γ(a + γ) = Γ(a) 1 + ψ(a)γ + O(γ2) , where ψ(a) is the psi function, and the relation ψ(1/2) = − (c + 2 ln 2), where c = 0.5772156649 ... is the Euler-Mascheroni constant [33, 32]. We thus find from Eq. (3.17) that

© £ ¡ 1 ¢¤ 2 ª w(∞) ≈ − 1 − 2 c + 2 ln 2 + ψ 2 + α γ + O(γ ) × cos [π (α − γ)] . (3.23)

The first factor {· · · } describes the amplitude of the damped Rabi oscillations and the cos factor describes the phase of the oscillations. The maxima and the minima of these oscillations are shifted by πγ (if the small additional shift from the damped amplitude is neglected) from their coherent values (3.8) and (3.9), respectively. The factor {· · · } in Eq. (3.23) displays explicitly the damping of the amplitude and its departure from 1 as γ rises from zero. Since ¡ 1 ¢ ψ 2 + α is an increasing function of α [33], the damping effect is stronger for larger pulse areas, which is shown explicitly below. For α = γ + n, near the nth extremum, we find from Eq. (3.23) by using Eq. (6.3.4) of 1 [33] for ψ(n + 2 ) that " # Xn 1 w(∞) ≈ (−1)n+1 1 − 4γ + O(γ2) . (3.24) 2k − 1 k=1 3.3. SPECIAL CASES 25

1.0 0.01 0.1 0.2 0.5 0.5

1.0 0 2.0

-0.5 Population Inversion Population

-1.0 0 1 2 3 4 5 6 7 8 Pulse Area (units of π)

Figure 3.2: The population inversion (3.14) against the pulse area A = πα for different values of the dephasing rate, Γ0T = 0.01, 0.1, 0.2, 0.5, 1, 2 (denoted on the respective curve).

For n = 1 − 4, Eq. (3.24) gives

n = 1 : w(∞) ≈ 1 − 4γ + O(γ2), (3.25a) 16 2 n = 2 : w(∞) ≈ −1 + 3 γ + O(γ ), (3.25b) 92 2 n = 3 : w(∞) ≈ 1 − 15 γ + O(γ ), (3.25c) 704 2 n = 4 : w(∞) ≈ −1 + 105 γ + O(γ ). (3.25d)

The cases of n = 1 and 3 correspond to the first and second maxima (π and 3π pulses), and n = 2 and 4 to the first and second minima (2π and 4π pulses). Equations (3.24) and (3.25) show explicitly how the values of the population inversion for these nπ pulses depart from their values ±1 as γ departs from zero. These equations also demonstrate that the effect of dephasing is stronger for larger pulse areas (since the coefficient in front of γ increases with n), which is indeed seen in Fig. 3.1. In Fig. 3.2 the population inversion is plotted against the pulse area for several values of 1 the dephasing rate. As predicted, for γ = 2 (Γ0T = 1), the nodes of w are situated at pulse areas A = nπ, where in the absence of dephasing one finds the extrema. For γ = 1 (Γ0T = 2), the maxima are situated approximately at α = 2, 4, 6,..., where one finds the minima (even-π pulses) for γ = 0; likewise, the minima are situated approximately at α = 3, 5, 7,..., where one finds the maxima (odd-π pulses) for γ = 0. 26 CHAPTER 3. EFFECT OF DEPHASING ON SINGLE-QUBIT GATES

Strong dephasing

For γ À α, 1, the population inversion w(∞) has the asymptotics · ¸ α2 w(∞) ∼ − exp − + O(γ−3) , (3.26) γ which is obtained from Eq. (3.14) by using the Stirling asymptotic expansion [33], · ¸ √ 1 1 ³ ´ Γ(z) ∼ 2πzz− 2 exp −z + + O |z|−3 , (3.27) 12z (|arg z| < π, |z| À 1) .

The inversion w(∞) decreases in a Gaussian fashion against α. For large γ, w tends to its initial value of −1, rather than to the incoherent limit w = 0, which is a result of quantum overdamping [30]. This behaviour is indeed seen in Fig. 3.1, where we have verified that Eq. (3.26) describes very accurately the asymptotic decrease of w(∞) for large γ (not shown for simplicity).

Large pulse area

When α À γ, 1, the population inversion w(∞) has the following behaviour

1 2 ¡ 1 ¢ −2γ £ −2 ¤ w(∞) ∼ − π Γ 2 + γ α 1 + O(α ) cos π(α − γ), (3.28) which is derived from Eq. (3.17) by using Eq. (3.27). Equation (3.28) shows that as α increases, the oscillation amplitude vanishes as α−2γ and for sufficiently large pulse areas,

1 " #− πε 2γ A & A = π ¡ ¢ , (3.29) ε 2 1 Γ 2 + γ the population inversion decreases below ε (|w| . ε), i.e. the two-state system evolves towards 1 a completely incoherent superposition of states |0i and |1i (ρ11 = ρ22 = 2 , ρ12 = 0). For 9 instance, A0.1 ≈ 1.58 × 10 π, 415π, and 3.18π for γ = 0.1, 0.3, and 1, respectively.

3.4 Conclusions

In this Chapter we have presented an exact analytic solution for resonant excitation of a qubit induced by a pulse with a hyperbolic-secant shape in the presence of dephasing processes. The exact solution (3.14) is given in terms of gamma functions. Dephasing affects the Rabi oscillations in two ways: shifting the oscillation phase by approximately πΓ0T/2 and damping the oscillation amplitude: the larger the pulse area, the stronger the damping. The implication 3.4. CONCLUSIONS 27 is that one cannot reduce the dephasing-induced losses of efficiency by increasing the intensity of the field (e.g. replacing a π pulse by a 3π pulse) since this will actually increase the losses. Various special cases of pulses with specific areas have been considered and various limits have been derived in terms of elementary functions. The results provide explicit and simple estimates of the effect of dephasing on single-qubit gates, e.g. in the cases of π, 2π and π/2 pulses, which are of great importance and widely used in quantum information science.

Appendix - Exact solution

The first step in solving Eqs. (3.13) is to decouple them by differentiating the equation forw ˙ and replacing v andv ˙, found from Eqs. (3.13); this gives à ! Ω˙ w¨ − Γ + w˙ + Ω2w = 0, (3.30) Ω with an overdot denoting d/dt. We change the independent variable from t to 1 z(t) = 2 [tanh(t/T ) + 1]; hence z(−∞) = 0 and z(+∞) = 1. Then

00 ¡ 1 ¢ 0 2 z(1 − z)W + 2 + γ − z W + α W = 0, (3.31) where 0 ≡ d/dz, W [z(t)] = w(t) and α and γ are defined by Eqs. (3.15). This equation has the same form as the Gauss hypergeometric equation [33, 34],

z(1 − z)F 00 + [ν − (λ + µ + 1)z] F 0 − λµF = 0, (3.32) upon the identification 1 λ = α, µ = −α, ν = 2 + γ. (3.33) The complete solution of this equation, expressed by a superposition of two linearly inde- pendent solutions of Eq. (3.31), depends upon the value of ν. The case ν 6= 1, 2, 3,...According to Sec. 9.153.1 of [34], the solution of Eq. (3.31) can be expressed in terms of the Gauss hypergeometric function [33, 34] as

W (z) = A1F (λ, µ; ν; z) 1−ν +A2z F (λ + 1 − ν, µ + 1 − ν; 2 − ν; z). (3.34)

From here and using Eq. (3.13), it can be found that p · i z(1 − z) λµ V (z) = A F (λ + 1, µ + 1; ν + 1; z) α 1 ν ¸ −ν + A2(1 − ν)z F (λ + 1 − ν, µ + 1 − ν; 1 − ν; z) , (3.35) 28 CHAPTER 3. EFFECT OF DEPHASING ON SINGLE-QUBIT GATES with V [z(t)] = v(t), where Eqs. (15.2.1) and (15.2.4) of [33] have been used. The integration constants A1 and A2 can be determined from the initial conditions (3.2),

A1 = −1,A2 = 0. (3.36)

Hence w(∞) = W (1) = −F (λ, µ; ν; 1) or Γ(ν)Γ(ν − λ − µ) w(∞) = − , (3.37) Γ(ν − λ)Γ(ν − µ) where Eq. (15.1.20) of [33] has been used. Referring to Eqs. (3.33), one obtains Eq. (3.14). Equation (3.37) has been derived under the assumption that ν 6= 1, 2, 3,...; then the two terms in Eq. (3.34) are linearly independent. Suppose now that ν = n where n = 1, 2, 3,..., 1 3 5 that is γ = 2 , 2 , 2 ,...; Then the two terms in Eq. (3.34) are linearly dependent for ν = 1 while the second term is not defined for ν = 2, 3, 4,... The case ν = 1. According to Sec. 9.153.2 of [34], the solution of Eq. (3.31) for ν = 1 is

W (z) = A F (λ, µ; 1; z) 1 " # X∞ (λ) (µ) ψ + A F (λ, µ; 1; z) ln z + k k 1 zk , (3.38) 2 (k!)2 k=1 with (x)k = Γ(x+k)/Γ(x) and ψ1 = ψ(λ+k)−ψ(λ)+ψ(µ+k)−ψ(µ)−2ψ(k+1)+2ψ(1), ψ(x) being the psi-function [33]. Since the second term diverges for z = 0, the initial conditions (3.2) require Eqs. (3.36) to be satisfied and Eq. (3.37) applies again. The case ν = n+1 (n = 1, 2, 3,...) and λ, µ 6= 0, 1, 2, . . . , n−1. According to Sec. 9.153.3 of [34] and Eq. (15.5.19) of [33], the solution of Eq. (3.31) in this case is ·

W (z) = A1F (λ, µ; n + 1; z) + A2 F (λ, µ; n + 1; z) ln z ¸ X∞ (λ) (µ) ψ Xn (k − 1)!(−n) + k k n+1 zk − k z−k (3.39) (n + 1)kk! (1 − λ)k(1 − µ)k k=1 k=1 with ψn+1 = ψ(λ + k) − ψ(λ) + ψ(µ + k) − ψ(µ) − ψ(n + 1 + k) + ψ(n + 1) − ψ(1 + k) + ψ(1). Again, the second term diverges for z = 0 and the initial conditions (3.2) require Eqs. (3.36) to be satisfied and hence, Eq. (3.37) holds again. The case ν = n + 1 (n = 1, 2, 3,...) and λ or µ = 0, 1, 2, . . . , n − 1. Suppose that λ = m + 1 < n + 1. Then, according to Sec. 9.153.4 of [34], the solution is given by Eq. (3.34), in which the second hypergeometric function reduces to a polynomial in z−1. Since it diverges for z = 0, the initial conditions (3.2) require Eqs. (3.36) to be satisfied and Eq. (3.37) holds again. In conclusion, for the model (3.5), in all cases the final population inversion w(+∞) is given by Eq. (3.37). odsrb n nls h yaiso w-ee unu ytmwihhsan has which states system the quantum of two-level consisting a level of ground dynamics degenerate the represent- analyse for and quantum candidate describe physical for to natural manipulated most and The qu stored 1. reduces a Chapter be This ing in to information. out need pointed more that as store particles computing, and of encode number to the capability significantly greater their of because ing Qu eut r ulse nRf [35] Ref. in published are results manipulate to possible it qu makes which detunings, specific at resonance, generalized such that qu show We various for pulses. solutions, creating multistate for the tools of as applications and various example, [24] suggest Allen-Eberly We [29], models. Rosen-Zener the [39] [37], Demkov-Kunike Landau-Zener models: soluble [36], exactly Rabi popular the most addition, solution, the resonance In of extensions states. degenerate ground for solutions of analytical superposition (bright) a system are and two-state there a state of excited solution the the to only transformation involving Morris-Shore the using by reduced be can state, excited one to pled otolddsg fabtayqu arbitrary of design Controlled N N twtotppltn h psil os)uprsae vntasety h presented The transiently. even state, upper lossy) (possibly the populating without it t ersn eypoiigatraiet uisfrqatmifrainprocess- information quantum for qubits to alternative promising very a represent its N N taetedgnrt usae fa tmceeg ee.I steeoeimportant therefore is It level. energy atomic an of substates degenerate the are it − aksae opsdo rudsae.W s hsdcmoiint derive to decomposition this use We states. ground of composed states dark 1 | ψ N +1 i nti hpe eso htteslto fti system this of solution the that show we Chapter this In . π ple a cu vnwe h pe tt sfroff far is state upper the when even occur can -pulses N tsproiinsaeb eeaie resonant generalized statesby superposition it 29 | ψ 1 i , | ψ 2 i ..., , | ψ N i omn h qu the forming , only N tstates it h rudstate ground the

C h a p t e r N N t cou- it, 4 -times π - 30 CHAPTER 4. CONTROLLED DESIGN OF ARBITRARY QUNIT STATES

4.1 Motivation

The problem of a two-state quantum system driven by a time-dependent pulsed external is interesting both physically and mathematically. Physically, because the two-state system is the simplest nontrivial system with discrete energy states in quantum mechanics and because it represents the basic ingredient for quantum information processing – the qubit ; math- ematically, because the Schr¨odingerequation for two states poses interesting mathematical challenges some of which are exactly soluble. Furthermore, already in the two-state case, important nonclassical phenomena occur, for instance, the famous Rabi oscillations (2.1) of the population. Finally, in almost all cases (except for a few exactly soluble), the behaviour of a multistate quantum system can only be understood by reduction to one or more effective two-state systems, e.g., by adiabatic elimination of weakly coupled states or by using some intrinsic symmetries. In this Chapter, we present the extensions of some of the well-known two-state models to the case when one of the states is replaced by a quNit, as displayed in Figure 4.1. By using the Morris-Shore (MS) transformation [43] we show that the (N +1)-state problem can be reduced to an effective two-state problem involving a bright state and the upper, nondegenerate state. If known, the propagator for this subsystem can be used to find the solution for the full (N + 1)-state system. Such analytic solutions can be very useful in designing general unitary transformations within the quNit manifold, which is very important for quantum information processing. We point out that an analogous system with N = 3 has been considered by Unanyan et al. [44], and by Kis and Stenholm [45] for general N, who have derived the adiabatic solution for pulses generally delayed in time; these schemes extend the well-known technique of stimulated Raman adiabatic passage (STIRAP) (see [46, 47] for reviews). Here we derive several exact analytic solutions for pulses coincident in time. This can therefore be considered as an extension to arbitrary N states of an earlier work [48], which treated the case N = 2.

4.2 Definition of the problem

The quNit states are coupled via the upper state with pulsed interactions, each pair of which are on two-photon resonance as shown on Figure 4.1. The upper state |ψN+1i may be off single-photon resonance by some detuning ∆(t) that, however, must be the same for all fields. In the usual rotating-wave approximation (RWA) the Schr¨odingerequation of the system reads d i~ C(t) = H(t)C(t), (4.1) dt 4.2. DEFINITION OF THE PROBLEM 31 where the elements of the (N + 1)-dimensional vector C(t) are the probability amplitudes of the states and the Hamiltonian is given by   0 0 ··· 0 Ω (t)  1     0 0 ··· 0 Ω2 (t)  ~    ......  H(t) =  . . . . .  . (4.2) 2      0 0 ··· 0 ΩN (t) ∗ ∗ ∗ Ω1 (t)Ω2 (t) ··· ΩN (t) 2∆ (t)

The Rabi frequencies of the couplings between the ground states and the excited state

Ω1(t), ..., ΩN (t) are assumed complex for generality. Furthermore, the Rabi frequencies are assumed to be pulse-shaped functions with the same time dependence f(t), but possibly with different magnitudes and phases,

iβn Ωn (t) = χne f(t)(n = 1, 2, ..., N) , (4.3) and hence different pulse areas,

Z ∞ Z ∞ An = |Ωn(t)|dt = χn f(t)dt (n = 1, 2, ..., N) . (4.4) −∞ −∞

ψ N +1 ∆

Ω Ω Ω Ω 1 2 3 N

ψ ψ ψ ψ 1 2 3 N

Figure 4.1: The system studied in this Chapter. N degenerate (in RWA sense) states |ψ1i, |ψ2i,..., |ψN i, forming a quNit, are coupled simultaneously to an upper state |ψN+1i, possibly off single-photon resonance by a detuning ∆(t), with Rabi frequencies Ωn(t) (n = 1, 2,...,N).

Physical implementations

The linkage pattern described by the Hamiltonian (4.2) can be implemented experimentally in laser excitation of atoms or molecules. For example, the N = 3 case is readily implemented 32 CHAPTER 4. CONTROLLED DESIGN OF ARBITRARY QUNIT STATES

ψ ψ 4 5 J = 0 J = 0

ψ ψ ψ 2 3 4

ψ ψ ψ ψ J = 1 1 2 3 1

J = 1 J = 1

Figure 4.2: Examples of physical implementations of the linkage pattern of N quNit states coupled via one upper state, considered in the present chapter. Left: N = 3 quNit states.

Right: N = 4 quN it (in the RWA sense) states (dashed arrows indicate two additional possible linkages). in the J = 1 ↔ J = 0 system coupled by three laser fields with right circular, left circular and linear polarizations, as shown in Figure 4.2 (left). These coupling fields can be pro- duced from the same laser by standard optical tools (beam splitters, polarizers, etc.), which greatly facilitates implementation. Moreover, the use of pulses derived from the same laser ensures automatically the two-photon resonance conditions and the condition (4.3) for the same temporal profile of all pulses. The cases of N = 4 − 6 can be realized by adding an additional J = 1 level to the coupling scheme and appropriately polarized laser pulses, as shown in the right frame of Figure 4.2(right).

4.3 General solution

Morris-Shore (dark-bright) basis

The Hamiltonian (4.2) has N − 1 zero eigenvalues and two nonzero ones,

λn = 0 (n = 1,...,N − 1), (4.5a) 1 h p i λ (t) = ∆ ± ∆2 + Ω2(t) , (4.5b) ± 2 where v u uXN t 2 Ω(t) = |Ωn| (t) ≡ χf(t) (4.6) n=1 is the real root-mean-square (rms) Rabi frequency, where v u uXN t 2 χ = χn. (4.7) n=1 4.3. GENERAL SOLUTION 33

The set of orthonormalized eigenstates |ϕni (n = 1, 2,...,N − 1) corresponding to the zero eigenvalues can be chosen as 1 h iT iβ2 iβ1 |ϕ1i = χ2e , −χ1e , 0, 0, ··· , 0 , (4.8a) X2 1 h iT iβ1 −iβ3 iβ2 −iβ3 2 |ϕ2i = χ1e χ3e , χ2e χ3e , −X2 , 0, ··· , 0 , (4.8b) X2X3 1 h iT iβ1 −iβ4 iβ2 −iβ4 iβ3 −iβ4 2 |ϕ3i = χ1e χ4e , χ2e χ4e , χ3e χ4e , −X3 , 0, ··· , 0 , (4.8c) X3X4 ··· 1 h iT iβ1 −iβN iβ2 −iβN 2 |ϕN−1i = χ1e χN e , χ2e χN e , ··· , −XN−1, 0 , (4.8d) XN−1XN where v u uXn t 2 Xn = χk (n = 2, 3, ..., N) . (4.9) k=1

These eigenstates are dark states, i.e. they do not involve the excited state |ψN+1i and are uncoupled from |ψN+1i, as we can see on Figure 4.3. All dark states are time-independent and cannot produce excitation followed by fluorescence [49]. We emphasize that the choice (4.8) of dark states is not unique because any superposition of dark states is a dark state too; hence their choice is a matter of convenience. The Hilbert space is decomposed into two subspaces: an (N−1)-dimensional dark subspace comprising the dark states (4.8) and a two-dimensional subspace orthogonal to the dark subspace. It is convenient to use the Morris-Shore (MS) basis [43], which, in addition to the dark states, includes the excited state |ψN+1i ≡ |ϕN+1i and a bright ground state |ϕN i. The latter does not have a component of the excited state and is orthogonal to the dark states; these conditions determine it completely (up to an unimportant global phase), 1 h iT iβ1 iβ2 iβN |ϕN i = χ1e , χ2e , ··· , χN e , 0 . (4.10) XN We point out that the Morris-Shore basis is not the adiabatic basis because only the dark states are eigenstates of the Hamiltonian, but |ϕN i and |ϕN+1i are not.

In the new, still stationary basis {|ϕni}n=1,2,...,N+1, the Schr¨odingerequation reads d i~ B(t) = He(t)B(t), (4.11) dt where the original amplitudes C(t) are connected to the MS amplitudes B(t) by the time- independent unitary matrix W composed by the basis vectors |ϕni,

W = [|ϕ1i , |ϕ2i ,..., |ϕN+1i] , (4.12) according to C(t) = WB(t). (4.13)

34 CHAPTER 4. CONTROLLED DESIGN OF ARBITRARY QUNIT STATES

ψ N +1

∆

Ω

ϕ ϕ 1 N

Figure 4.3: The system from Fig. 4.1 in the Morris-Shore basis. There are N − 1 uncoupled dark states |ϕ i, ϕ i,..., |ϕ i, and a pair of coupled states, a bright state |ϕ i and the 1 2 N−1 N upper state |ψN+1i, with the same detuning ∆(t) as in the original basis and a coupling given by the RMS Rabi frequency Ω(t), Eq. (4.6).

The transformed Hamiltonian reads He(t) = W†H(t)W, or explicitly,

  0 0 ··· 0 0 0     0 0 ··· 0 0 0    ......  e ~ ......  H(t) =   . (4.14) 2 0 0 ··· 0 0 0      0 0 ··· 0 0 Ω(t)  0 0 ··· 0 Ω(t) 2∆(t)

We point out that the Hamiltonian of Eq. (4.2) is a special case of the most general Hamiltonian for which the MS transformation [43] applies and which includes N quNit lower states and M degenerate upper states which we will consider in Chapter 6. Hamiltonians of the same type as (4.2) and related transformations leading to Eq. (4.14), have appeared in the literature also before the paper by Morris and Shore [43], mostly in simplified versions of constant and equal interactions (see e.g. [50] and references therein).

Solution in the Morris-Shore basis

As evident from the first N − 1 zero rows of He the dark states are decoupled from states |ϕN i and |ϕN+1i and the dark-state amplitudes remain unchanged, bn(t) = const (n = 1, 2,...,N − 1). Thus the (N + 1)-state problem reduces to a two-state one involving

|ϕN i and |ϕN+1i, " # " #" # d b 1 0 Ω b i N = N . (4.15) dt bN+1 2 Ω 2∆ bN+1 4.3. GENERAL SOLUTION 35

The propagator for this two-state system, defined by " # " # bN (tf ) (2) bN (ti) = UMS , (4.16) bN+1 (tf ) bN+1 (ti) is a two-dimensional unitary and can be expressed in terms of the Cayley-Klein parameters as " # a b U(2) = , (4.17) MS −b∗e−iδ a∗e−iδ R with |b|2 = 1 − |a|2 and δ = tf ∆(t0)dt0. The unimportant phase factor e−iδ originates from ti the chosen representation of the Hamiltonian (4.2), which facilitates the application of the MS transformation. In the interaction representation, where the diagonal elements are nullified and the detuning appear in phase factors multiplying the couplings, the factor e−iδ disappears. We can express the propagator for the (N + 1)-state system in the MS basis as   1 0 ··· 0 0 0     0 1 ··· 0 0 0    ......  ......  UMS =   . (4.18) 0 0 ··· 1 0 0      0 0 ··· 0 a b  0 0 ··· 0 −b∗e−iδ a∗e−iδ

The solution in the original basis

We can find the transition matrix in the original, diabatic basis by using the transformation

† U(∞, −∞) = WUMS(∞, −∞)W , (4.19) or explicitly,

 2 iβ iβ iβ  χ1 χ1χ2e 12 χ1χN e 1N χ1e 1 1 + (a − 1) χ2 (a − 1) χ2 ··· (a − 1) χ2 b χ  −iβ 2 iβ iβ   χ1χ2e 12 χ2 χ2χN e 2N χ2e 2   (a − 1) 2 1 + (a − 1) 2 ··· 2 b   χ χ χ χ   ......  U =  . . . . .  , (4.20)  −iβ −iβ 2 iβ   χ1χN e 1N χ2χN e 2N χN χN e N   (a − 1) χ2 (a − 1) χ2 ··· 1 + (a − 1) χ2 b χ  −iβ −iβ −iβ ∗ −iδ χ1e 1 ∗ −iδ χ2e 2 ∗ −iδ χN e N ∗ −iδ −b e χ −b e χ · · · −b e χ a e where βkm = βk − βm (k, m = 1, 2,...,N). The ith column of this matrix provides the probability amplitudes for initial conditions

ci(−∞) = 1, (4.21a)

cn(−∞) = 0 (n 6= i). (4.21b) 36 CHAPTER 4. CONTROLLED DESIGN OF ARBITRARY QUNIT STATES

The initial state |ψii can be one of the quNit states or the upper state. This general unitary matrix and combinations of such matrices can be used to design techniques for general or special quNit rotations. As evident from Eq. (4.20) for finding the populations for the initial condition (4.21) it h i (2) 2 2 is sufficient to know only the parameter a = UMS(∞, −∞) because |b| = 1 − |a| [48]. 11 For the sake of simplicity, here we are interested only in cases when the system starts in a single state and below we shall concentrate on the values of the parameter a. In the more general case when the system starts in a coherent superposition of states, Eq. (4.20) can be used again to derive the solution; then the other Cayley-Klein parameter b is also needed.

4.4 Types of population distribution

We identify two types of initial conditions: when the system starts in one of the quNit states

|ψii or in the excited state |ψN+1i, which we shall consider separately.

System initially in a ground state

When the system is initially in the ground state |ψii, Eq. (4.21), we find from the ith column of the propagator (4.20) that the populations in the end of the evolution are

¯ ¯ ¯ χ2 ¯2 P = ¯1 + (a − 1) i ¯ , (4.22a) i ¯ χ2 ¯ χ2χ2 P = |a − 1|2 i n (n 6= i, N + 1), (4.22b) n χ4 ³ ´ χ2 P = 1 − |a|2 i . (4.22c) N+1 χ2

Therefore the ratio of the populations of any two quNit states, different from the initial state

|ψii, reads

2 Pm χm = 2 (m, n 6= i, N + 1). (4.23) Pn χn

Hence these population ratios do not depend on the interaction details but only on the ratios of the corresponding peak Rabi frequencies. For equal Rabi frequencies,

χ1 = χ2 = ··· = χN , (4.24) 4.4. TYPES OF POPULATION DISTRIBUTION 37

Eqs. (4.22) reduce to ¯ ¯ ¯ a − 1¯2 P = ¯1 + ¯ , (4.25a) i ¯ N ¯ |a − 1|2 P = (n 6= i, N + 1), (4.25b) n N 2 1 − |a|2 P = . (4.25c) N+1 N

Thus the populations of all ground states except the initial state |ψii are equal.

Special values of a

Several values of the propagator parameter a are especially interesting. For a = 0, which indicates complete population transfer (CPT) in the MS two-state system, Eq. (4.22) gives ¯ ¯ ¯ χ2 ¯2 P = ¯1 − i ¯ , (4.26a) i ¯ χ2 ¯ χ2 χ2 P = n i (n 6= i, N + 1), (4.26b) n χ4 χ2 P = i . (4.26c) N+1 χ2

For a = 1, which corresponds to complete population return (CPR) in the MS two-state system, we obtain

Pi = 1, (4.27a)

Pn = 0 (n 6= i, N + 1), (4.27b)

PN+1 = 0. (4.27c)

For a = −1, which again corresponds to CPR in the MS two-state system, but with a sign flip in the amplitude, we find µ ¶ χ2 2 P = 1 − 2 i , (4.28a) i χ2 4χ2 χ2 P = n i (n 6= i, N + 1), (4.28b) n χ4 PN+1 = 0. (4.28c)

It is important to note that although both cases a = 1 and a = −1 lead to CPR in the MS two-state system, they produce very different population distributions in the full (N + 1)- state system. The case a = 1 leads to a trivial result (CPR in the full system), whereas 38 CHAPTER 4. CONTROLLED DESIGN OF ARBITRARY QUNIT STATES the case a = −1 is very interesting because it leads to a population redistribution amongst the ground states with zero population in the upper state; hence this case deserves a special attention.

The case a = −1

The case of a = −1 is particularly important because it allows to create a coherent superpo- sition of all quNit states, with no population in the upper state. All ground-state populations in this superposition will be equal, 1 P = P = ··· = P = , (4.29a) 1 2 N N PN+1 = 0. (4.29b) if ³√ ´ χi = N ± 1 χ0, (4.30a)

χn = χ0 (n 6= i), (4.30b) where χ χ = r . (4.31) 0 ³ √ ´ 2 N ± N

This result does not depend on other interaction details (pulse shape, pulse area, detuning) as long as a = −1. For example, for N = 4 quNit states, equal populations are obtained when χi = χn or χi = 3χn. We shall discuss later how the condition a = −1 can be obtained for several analytically soluble models.

Another important particular case is when the initial-state population Pi vanishes in the end. This occurs for X 2 2 χi = χn. (4.32) n6=i

For example, an equal superposition of all lower sublevels except |ψii,

Pi = PN+1 = 0, (4.33a) 1 P = (n 6= i, N + 1), (4.33b) n N − 1 is created for √ χi = χ0 N − 1, (4.34a)

χn = χ0 (n 6= i), (4.34b) where χ χ0 = p . (4.35) 2 (N − 1) 4.4. TYPES OF POPULATION DISTRIBUTION 39

System initially in the upper state

If the system is initially in the excited state |ψN+1i, at the end of the evolution the populations will be

³ ´ χ2 P = 1 − |a|2 n (n = 1, 2,...,N) , (4.36a) n χ2 2 PN+1 = |a| . (4.36b)

For a = ±1 at the end of the evolution the system undergoes CPR, as in the MS two-state system. For a = 0 (CPT in the MS two-state system) the whole population will be in the ground states leaving the excited state empty, PN+1 = 0. If all the couplings are equal, Eq. (4.24), the ground states will have equal populations,

1 P = (n = 1, 2,...,N) . (4.37) n N

Discussion

In this section we discussed some general features of the population redistribution in the (N + 1)-state system. There are three particularly interesting results. First, the ratios of the populations of the quNit states (except the one populated initially) depend only on the ratios of the corresponding Rabi frequencies; hence they can be controlled by changing the corresponding laser intensities alone. The populations values, though, depend on the other interaction details. Moreover, it can easily be seen that the relative phases of the quNit states can be controlled by the relative laser phases. Second, it is possible to create an equal superposition of all ground states, with zero population in the upper state. This is possible when the system starts in a ground state: then condition (4.30) is required, along with the CPR condition a = −1. Alternatively, an equal superposition can be created when the system starts in the upper state: then condition (4.24) is required, along with the CPT condition a = 0. Equal superpositions are important in some applications because they are states with maximal coherence (since the population inversions vanish). Third, it is possible, starting from a ground state, to create a superposition of all other ground states, whereas the initial ground state and the excited state are left unpopulated. This requires a = −1 and condition (4.32). This case has interesting physical implications, which will be discussed in the next section. 40 CHAPTER 4. CONTROLLED DESIGN OF ARBITRARY QUNIT STATES

Model Resonance Ω(t) = χf(t), ∆(t) = 0 1 a = cos 2 A Rabi Ω(t) = χ (|t| 5 T ), ∆(t) = ∆0 ³ p ´ ∆ ³ p ´ a = cos T χ2 + ∆2 − i√ sin T χ2 + ∆2 Ω2 + ∆2 Landau-Zener Ω(t) = χ,£ ∆(t) =¤Ct a = exp −πχ2/4C Rosen-Zener Ω(t) = χsech(t/T¡), ∆(¢t) = ∆0 2 1 Γ 2 + iδ a = ¡ 1 ¢ ¡ 1 ¢ Γ 2 + α + iδ Γ 2 − α + iδ Allen-Eberly Ω(t) = χsech(t/T ), ∆(t) = B tanh(t/T ) ³ p ´ cos π α2 − β2 a = cosh (πβ) Demkov-Kunike Ω(t) = χsech(¡t/T ), ∆(t)¢ = ∆¡0 + B tanh(t/T¢ ) Γ 1 + i(δ + β) Γ 1 + i(δ − β) a = ³ p 2 ´ ³2 p ´ 1 2 2 1 2 2 Γ 2 + α − β + iδ Γ 2 − α − β + iδ

Table 4.1: Values of the Cayley-Klein parameter a = [UMS(∞, −∞)]11 for several exactly 1 1 1 soluble models. Here Γ(z) is the gamma function and α = 2 χT , β = 2 BT , δ = 2 ∆0T , are scaled dimensionless parameters, which are assumed positive without loss of generality.

4.5 Applications to exactly soluble models h i (2) The values of the propagator parameter a = UMS(∞, −∞) for the most popular ana- 11 lytically exactly soluble models are listed in Table 4.1. Equation (4.20), supplied with these values, provides several exact multistate analytical solutions, which generalize the respective two-state solutions. Among these solutions, the resonance case is the simplest and most important one, which will receive a special attention below. It will be followed by a detailed discussion of the Rosen- Zener (RZ) model, which can be seen as an extension of the resonance solution to nonzero detuning for a special pulse shape (hyperbolic secant). Both the resonance and the RZ model allow for the parameter a to obtain the important values 0, ±1. The Rabi model can also be used to illustrate the interesting cases of population distribution associated with these values of a. However, its rectangular pulse shape is less realistic than the sech-shape of the pulse in 4.5. APPLICATIONS TO EXACTLY SOLUBLE MODELS 41 the RZ model. The Landau-Zener (LZ) and Allen-Eberly (AE) models are of level-crossing type, i.e. the detuning crosses resonance, ∆(0) = 0. For these models in the adiabatic limit the transition probability approaches unity, that is a → 0. The parameter a is always nonnegative, i.e. the most interesting value in the present context, a = −1, is unreachable. Nevertheless, because of the popularity and the importance of the LZ model we will discuss this solution in details in Sec. 4.6. The Demkov-Kunike (DK) model is a very versatile model, which combines and generalizes the RZ and AE models. Indeed, as seen in Table 4.1, the DK model reduces to the RZ model for B = 0 and to the AE model for ∆0 = 0. For the DK model, the parameter a can be equal to the most interesting value of −1 only when B = 0, i.e. only in the RZ limit. Therefore, we shall only consider the RZ model below.

Exact resonance

In the case of exact resonance, ∆ = 0, (4.38) the elements of the evolution matrix for the MS two-state system for any pulse shape of Ω(t) are (2.2) A a = cos , (4.39a) 2 A b = −i sin , (4.39b) 2 where A is the rms pulse area defined as Z ∞ A = Ω(t0)dt0. (4.40) −∞ In the important case of N = 3 the propagator of the system reads

 2  χ1 2 1 χ1χ2 iβ12 2 1 χ1χ3 iβ13 2 1 χ1 iβ1 1 1 − 2 χ2 sin 4 A −2 χ2 e sin 4 A −2 χ2 e sin 4 A −i χ e sin 2 A  2   χ1χ2 −iβ12 2 1 χ2 2 1 χ2χ3 iβ23 2 1 χ2 iβ2 1  (4) −2 χ2 e sin 4 A 1 − 2 χ2 sin 4 A −2 χ2 e sin 4 A −i χ e sin 2 A U =  χ2  .  χ1χ3 −iβ13 2 1 χ2χ3 −iβ23 2 1 3 2 1 χ3 iβ3 1  −2 χ2 e sin 4 A −2 χ2 e sin 4 A 1 − 2 χ2 sin 4 A −i χ e sin 2 A χ1 −iβ1 1 χ2 −iβ2 1 χ3 −iβ3 1 1 −i χ e sin 2 A −i χ e sin 2 A −i χ e sin 2 A cos 2 A (4.41) We have a = 0, ±1 for the following pulse areas,

a = 0 : A = (2l + 1) π, (4.42a) a = 1 : A = 4lπ, (4.42b) a = −1 : A = 2 (2l + 1) π. (4.42c) 42 CHAPTER 4. CONTROLLED DESIGN OF ARBITRARY QUNIT STATES where l = 0, 1, 2, .... The pulse areas for the three important cases discussed in Sec. 4.4 are easily calculated. An equal superposition of all N ground states is created when starting from the excited state and all individual pulse areas are equal to (see Sec. 4.4) (2l + 1) π An = √ (n = 1, 2,...,N) , (4.43) N where l = 0, 1, 2, .... An equal superposition of all N ground states is created also when starting from one ground state |ψii and the pulse areas are [see Eq. (4.30)] s √ N ± 1 Ai = 2 √ (2l + 1) π, (4.44a) N s 2 An = √ (2l + 1) π (n 6= i) , (4.44b) N ± N where l = 0, 1, 2, ...

The other interesting case when the system starts in one ground state |ψii and evolves into an equal superposition of all other ground states is realised for pulse areas [see Eq. (4.34)] √ A = 2 (2l + 1) π, (4.45a) i r 2 A = (2l + 1) π (n 6= i) , (4.45b) n N − 1 where l = 0, 1, 2, ....

Multistate Rosen-Zener model

Equation (4.20) and the value of the parameter a in Table 4.1 represent the multistate RZ solution in the degenerate two-level system. It is easy to show that ¡ ¢ 2 1 2 sin 2 πχT |a| = 1 − 2 ¡ 1 ¢, (4.46) cosh 2 π∆0T 1 1 where we have used the reflection formula Γ( 2 + z)Γ( 2 − z) = π/ cos πz. Hence in this model 1 |a| = 1 for α = 2 χT = l (l = 0, 1, 2, ...). The phase of a, however, depends on the detuning ∆0 [48]; we use this to an advantage to select values of ∆0 for which a = −1. For α = l we find [48] nY−1 2l + 1 − i∆ T a = (−1)n 0 (α = l), (4.47) 2l + 1 + i∆0T k=0 where the recurrence relation Γ(z +1) = zΓ(z) [33] has been used. Thus, the equation a = −1 reduces to an algebraic equation for ∆0, which has l real solutions [48]. The first few values 4.5. APPLICATIONS TO EXACTLY SOLUBLE MODELS 43

χT ∆0T 2 0 4 ±1.732 6 0 ±4.796 8 ±1.113 ±9.207 10 0 ±2.756 ±14.913 12 ±0.943 ±4.936 ±21.903 14 0 ±2.243 ±7.595 ±30.171 16 ±0.855 ±3.916 ±10.708 ±39.715 18 0 ±1.988 ±5.907 ±14.265 ±50.534 20 ±0.799 ±3.418 ±8.195 ±18.260 ±62.627 22 0 ±1.830 ±5.098 ±10.766 ±22.687 ±75.993 24 ±0.759 ±3.113 ±7.006 ±13.613 ±27.545 ±90.634 26 0 ±1.719 ±4.606 ±9.130 ±16.729 ±32.833 ±106.549 28 ±0.728 ±2.901 ±6.289 ±11.461 ±20.113 ±38.548 ±123.736 30 0 ±1.636 ±4.268 ±8.150 ±13.994 ±23.760 ±44.690 ±142.198

Table 4.2: Some approximate solutions of the equation a(∆0) = −1 for the RZ model, where a is given in Table 4.1, for various even integer values of χT .

of χ and ∆0 for which a = −1 are shown in Table 4.2. As the table shows, ∆0 = 0 is a 1 solution for odd α = 2 χT but not for even α, in agreement with the conclusions in Sec. 4.5. Moreover, the a = −1 solutions do not depend on the number of quNit states N. In the present context the RZ model is interesting for it shows that one can create super- positions within the ground-state manifold even when the excited state is off resonance by a considerable detuning (∆0 À 1/T ), for which the transition probability in the MS two-state system is virtually zero, i.e. |a| ≈ 1. This fact allows us, for specific detunings, to essentially contain the transient dynamics within the ground states; in contrast, in the resonance case 2 1 the excited state can get significant transient population, PN+1(t) = sin 2 A(t), although it vanishes in the end.

Figure 4.4 displays the populations against the detuning ∆0 for a hyperbolic-secant pulse with χT = 18 for couplings chosen to satisfy Eqs. (4.30) (left frame) and Eqs. (4.34) (right frame). In both cases we have |a| = 1 [see Eq. (4.46)], which leaves the excited state unpopulated in the end. For several special values of the detuning ∆0, as predicted in Table 4.2, we have a = −1. For these values, an equal superposition of all quNit states including the initially populated state |ψ1i is created in the left figure, and an equal superposition of all quNit states except |ψ1i is created in the right frame. Figure 4.5 shows the final populations versus the rms pulse area A = πχT for N = 3 44 CHAPTER 4. CONTROLLED DESIGN OF ARBITRARY QUNIT STATES

1.0 1.0

P 1 P 0.8 0.8 1

0.6 0.6

0.4 0.4 Populations Populations P ,P 2 3

P ,P 0.2 2 3 0.2

0 0 0.1 1 10 100 0.1 1 10 100 Detuning (units of 1/T) Detuning (units of 1/T)

Figure 4.4: Populations vs the detuning ∆0 for N = 3 lower states and χT = 18. The coupling strengths χn are given by Eqs. (4.30) in the left frame and Eqs. (4.34) in the right frame. The system is initially in state |ψ1i. quNit lower states for couplings chosen to satisfy Eqs. (4.30) (left frame) while and to satisfy Eqs. (4.34) in the right frame. As follows from Table 4.2, an equal superposition of all quNit states is created for rms pulse area A = 18π; this is indeed seen in the left frame of Figure 4.5, while the right frame shows that an equal superposition is created of all quNit states except the initially populated state |ψ1i. A closer examination (not shown) reveals that for these other values of the rms pulse area the created superposition has almost, but not exactly, equal components. Figure 4.6 displays the time evolution of the populations for N = 3 quNit lower states and rms pulse area of 18π, and detunings ∆ = 0 and ∆T = 50.534, left and right frame. For these pairs of areas and detunings, Figures 4.4 and 4.5 have already demonstrated that an equal superposition of all quNit states is created. Figures 4.6 show that the evolution towards such a superposition can be dramatically different on and off resonance. Indeed, for ∆ = 0 the nondegenerate upper state receives considerable transient population, which would lead to significant losses if this state can decay on the time scale of the pulsed interaction. In strong contrast, off resonance this undesired population is greatly reduced, and still the desired equal superposition of the quNit states emerges in the end. We have verified numerically that for larger detunings this transient population continues to decrease, e.g. for ∆T = 142.198 and ΩT = 30 it is less than 1%. To conclude this section we point out that one can create any desired superposition, with arbitrary unequal populations, in very much the same manner, on or off resonance, by 4.6. MULTISTATE LANDAU-ZENER MODEL 45

1.0 1.0

P P 1 1 0.8 0.8

0.6 0.6

0.4 0.4 Populations Populations

P ,P 2 3 P ,P 0.2 0.2 2 3

0 0 0 10 20 30 0 10 20 30

RMS Pulse Area (units of π) RMS Pulse Area (units of π)

Figure 4.5: Final populations versus the rms pulse area χ for N = 3 degenerate lower states and detuning ∆T = 50.534. The coupling strengths χn are given by Eqs. (4.30) in the left frame and Eqs. (4.34) in the right frame. The system is initially in state |ψ1i. appropriately choosing the individual couplings, while still maintaining particular values of the overall rms pulse area. Tuning on resonance gives the advantage of smaller pulse area required, whereas tuning off resonance (with larger pulse area) provides the advantage of greatly reducing the transient population of the possibly lossy common upper state.

4.6 Multistate Landau-Zener model ¡¡ ¢¢ As seen in Table 4.1 the propagator parameter a for the LZ model, a = exp −πχ2/4C , cannot be equal to 0 or 1 or −1, but may approach 0 or 1 arbitrarily closely. However, it is always positive and cannot approach the value of −1; hence the LZ model is unsuitable for unitary operations within the quNit manifold, in contrast to the resonance and RZ mod- els discussed above. Still, the present multistate LZ solution represents an interesting and important addition to the available LZ solutions (see [51] and references therein).

The Demkov-Osherov model

The present multistate LZ model complements the Demkov-Osherov (DO) model [52], wherein a slanted energy crosses N parallel nondegenerate energies. In the DO model, the exact proba- bilities Pn→m have the same form — products of LZ probabilities for transition or no-transition applied at the relevant crossings — as what would be obtained by naive multiplication of LZ 46 CHAPTER 4. CONTROLLED DESIGN OF ARBITRARY QUNIT STATES

1.0 1.0 P P 1 1 0.8 0.8

0.6 0.6

P ,P 0.4 2 3 0.4 Populations Populations

P 4 P ,P 0.2 0.2 2 3

P 4 0 0 -6 -2 2 6 -6 -2 2 6 Time (units of T) Time (units of T)

Figure 4.6: Populations versus time for N = 3 lower states and rms Rabi frequency χT = 18. The coupling strengths χn are given by Eqs. (4.30). The detuning is ∆ = 0 in the upper frame and ∆T = 50.534 in the lower frame. The system is initially in state |ψ1i.

probabilities while moving across the grid of crossings from |ψni to |ψmi, without accounting for phases and interferences. For example, if the states |ψni (n = 1, 2,...,N) are labeled such that their energies increase with the index n, and if the slope of the slanted energy of state

|ψN+1i is positive, the transition probabilities in the DO model are

Pn→m = pnqn+1qn+2 ··· qm−1pm (n < m), (4.48a)

Pn→m = 0 (n > m), (4.48b)

Pn→n = qn, (4.48c)

Pn→N+1 = pnqn+1qn+2 ··· qN , (4.48d)

PN+1→n = q1q2 ··· qn−1pn, (4.48e)

PN+1→N+1 = q1q2 ··· qN , (4.48f)

¡ ¢ 2 where qn = exp −πχn/2C is the no-transition probability and pn = 1 − qn is the transition probability between states |ψN+1i and |ψni at the crossing of their energies.

The degenerate case

The present multistate LZ solution provides the transition probabilities for the special case when all parallel energies are degenerate, which cannot be obtained from the DO model. 4.6. MULTISTATE LANDAU-ZENER MODEL 47

The propagator The elements of the transition matrix for our (N + 1) -state degenerate LZ problem are readily found from Eq. (4.20) to be ¡ ¢ χnχm −Λ Um,n = − χ2 1 − e (4.49a) 2 ¡ ¢ χn −Λ Un,n = 1 − χ2 1 − e , (4.49b)

χn Un,N+1 = χ b, (4.49c)

χn ∗ −iδ UN+1,n = − χ b e , (4.49d) −Λ −iδ UN+1,N+1 = e e , (4.49e) where m, n = 1,...,N; m 6= n, and Λ = πχ2/4C and |b|2 = 1 − e−2Λ.

System initially in the excited state When the system begins initially in the excited state |ψN+1i, with the tilted energy, the system ends in a coherent superposition of all states with populations

2 ¡ ¢ χn −2Λ Pn = χ2 1 − e (n = 1,...,N) , (4.50a) −2Λ PN+1 = e . (4.50b)

In the adiabatic limit Λ À 1 the population is distributed among the quNit states according to their couplings, whereas the initially populated state |ψN+1i is almost depleted, PN+1 ≈ 0.

For equal couplings, all quNit states’ populations will be equal, Pn ≈ 1/N. In the opposite, diabatic limit Λ ¿ 1 the population remains in state |ψN+1i with almost no population in the quNit states.

System initially in a degenerate state When the system is initially in an arbitrary quNit state |ψii, at the end of the evolution the populations are

h 2 ¡ ¢i2 χi −Λ Pi = 1 − χ2 1 − e , (4.51a) 2 2 ¡ ¢2 χnχi −Λ Pn = χ4 1 − e (n = 1,...,N; n 6= i) , (4.51b) 2 ¡ ¢ χi −2Λ PN+1 = χ2 1 − e . (4.51c)

In the adiabatic limit Λ À 1 and for equal couplings, the populations will be

¡ 1 ¢2 Pi ≈ 1 − N , (4.52a) 1 Pn ≈ N 2 (n = 1,...,N; n 6= i) , (4.52b) 1 PN+1 ≈ N . (4.52c)

Obviously, Eqs. (4.50) and (4.51) cannot be reduced to the DO solution (4.48), which implies that the non-degeneracy assumption in the DO model is essential. 48 CHAPTER 4. CONTROLLED DESIGN OF ARBITRARY QUNIT STATES

1.0

0.8

0.6 P i

Populations 0.4 P N+1

0.2 P n

0 0 2 4 6 Landau-Zener Parameter Λ Figure 4.7: Populations for the degenerate LZ model vs the LZ parameter Λ = πχ2/4C for N = 3 quNit states and equal couplings. The system is supposed to start in one of the quNit states |ψii. The arrows on the right point the adiabatic values (4.52).

Figure 4.7 shows the transition probability for the multistate LZ model plotted against the LZ parameter Λ = πχ2/4C. As Λ increases the populations approach their steady adiabatic values (4.52). Different coherent superpositions can be created by choosing appropriate values for the couplings χn.

4.7 Conclusions

In this Chapter we have described a procedure for deriving analytical solutions for a multistate system composed of N degenerate lower states, forming a quNit, coupled via a nondegener- ate upper state with pulsed interactions of the same temporal dependence but possibly with different peak amplitudes. The multistate resonance and Rosen-Zener solutions have been discussed in some detail because they allow one to find special values of parameters, termed generalised π pulses, for which arbitrary unitary single-qubit gates can be realised in the quNit’s subspace. The RZ solution is particularly useful because it allows to prescribe ap- propriately detuned pulsed fields for which the dynamics can be essentially contained within the quNit space, without populating the upper state even transiently, thus avoiding possible losses from this state via spontaneous emission, ionization, etc and realising arbitrary unitary 4.7. CONCLUSIONS 49 transformations. We have analyzed in some detail also the multistate Landau-Zener model, which comple- ments the Demkov-Osherov model in the case of degenerate energies. The presented analytical solutions and general properties have a significant potential for manipulation of quNits with various applications in quantum information processing.

oiinsae racniumo tts uhtcnqe r eldvlpd particularly developed, well are techniques super- Such or single states. another, to of state continuum energy a initial or bound popu- state, one of position from transfer partial, for or scenarios involves complete traditionally lation, dynamics quantum of control Coherent (QHR) reflection Householder quantum the of M implementation physical simple a describe We eoato eryrsnn usdetra ed.W loitouetegnrlzdQHR generalized by the introduce induced also are We couplings fields. The external M pulsed state. resonant (excited) nearly or ancillary resonant an to simultaneously coupled hwta h otgeneral qu most the the on operators transformations two that unitary these show preselected use arbitrary We realising in state. blocks excited building the as with resonance from detuned appropriately are . Motivation 5.1 [53]. Ref. in published for QHR single a requires provide factorizations the QHR these of mathematically, parametrizations Viewed gate. phase one-dimensional a and either by h unu ore rnfr QT ya most at by (QFT) transform Fourier quantum the ( ( v v = ) ; ϕ = ) I N − I + − 2 ¡ tnadQR n an and QHRs standard 1 | niern fabtayuiaisby unitaries arbitrary of Engineering e v i i| ϕ v − | 1 nteqatmsse osdrdi hpe ossigo qu a of consisting 4 Chapter in considered system quantum the in ¢ | v i| U v ( | N N hc a epoue ntesame the in produced be can which , U ru.A neape epooearcp o constructing for recipe a propose we example, an As group. ) ,adol w Hsfor QHRs two only and 2, = ( N rnfraincnb atrzd(n hrb produced) thereby (and factorized be can transformation ) N dmninlpaegt,or gate, phase -dimensional 51 N neato tp.Freape QFT example, For steps. interaction N n .Teerslsare results These 4. and 3 = N pdsse hntefields the when system -pod N reflections − eeaie QHRs generalized 1

C h a p t e r N t We it. 5 N N it 52 CHAPTER 5. ENGINEERING OF ARBITRARY UNITARIES BY REFLECTIONS for two-state and three-state systems, e.g. π pulses [22], adiabatic passage using one or more level crossings [47], or stimulated Raman adiabatic passage (STIRAP) and its extensions [46]. Essentially all these techniques start from a single initial state; such a state can be prepared experimentally, e.g. by optical pumping. In the same time, in contemporary quantum physics implementations of specific propaga- tors are often demanded. For example, the two most significant quantum algorithms proposed by Shor [3] and Grover [4] lean heavily on the quantum Fourier transform [20, 21]. Another example is quantum state engineering when a system starts in a coherent superposition of states; then one must construct the entire propagator, while the above mentioned techniques provide only some transition probabilities. The implementation of such propagators is well understood and used for qubits, upon which the theory of quantum information is primarily built [20, 21]. On the other hand, we are interested in designing quantum gates for quNits due to the advantages they offer for storing quantum information. If fewer particles are required for quantum computing decoherence processes can be reduced [20, 21]. Furthermore, there are indications that using quNits can improve error thresholds in fault tolerant computation. Physical realizations of quNit operations in the existing proposals [54], however, are dif- ficult to implement. These implementations use sequences of U(2) operations, i.e. transfor- mations acting at each instance of time upon only two of the N states of the quNit. The general U(N) transformation of a quNit requires O(N 2) such U(2) operations [54]; hence the complexity increases rapidly with the quNit dimension N, which makes quNit manipulations challenging, even for qutrits (N = 3). In this Chapter, we show that a general U(N) transformation can be implemented phys- ically in a quantum system with only N interaction steps. For this purpose we introduce a compact quantum implementation, in a single interaction step, of the Householder reflection [55]. The latter is a powerful and numerically very robust unitary transformation, which has many applications in classical data analysis, e.g., in solving systems of linear algebraic equations, finding eigenvalues of high-dimensional matrices, least-square optimization, QR decomposition, etc. [56]. The Householder transformation, acting upon an arbitrary N- dimensional matrix, produces an upper (or lower) triangular matrix by N − 1 operations. When the initial matrix is unitary, the resulting final matrix is diagonal, i.e. a phase gate or a unit matrix. We use this property to decompose an arbitrary U(N) matrix into Householder matrices and hence, design a recipe for physical realization of a general U(N) transformation. The quantum Householder reflection (QHR) consists of a single interaction step involving N simultaneous pulsed fields. In contrast to the existing U(2) realizations of quNit transfor- mations, here each Householder reflection acts simultaneously upon many states: N states in 5.2. QUANTUM HOUSEHOLDER REFLECTION (QHR) 53 the first step, N − 1 states in the second, etc. This allows us to greatly reduce the number of physical steps, from O(N 2) in U(2) realizations to only O(N) in our proposal. We introduce two types of QHRs: standard QHR and generalized QHR; the latter involves an additional phase factor. The physical realizations of both use simultaneous pulses of precise areas in a system with an N-pod linkage pattern, the difference being that the standard QHR operates on exact resonance, whereas the generalized QHR requires specific detunings. Any unitary matrix can be decomposed into N − 1 standard QHRs and a phase gate, or into N generalized QHRs, without a phase gate. This advantage of the generalized-QHR implementation derives from the additional phase in each step, which delivers N additional phases in the end, thereby making the phase gate unnecessary.

5.2 Quantum Householder Reflection (QHR)

Standart QHR

An N-dimensional quantum Householder reflection (QHR) is defined as the operator

M(v) = I − 2 |vi hv| , (5.1) where |vi is an N-dimensional normalized complex column-vector and I is the identity oper- ator. The QHR (5.1) is hermitian and unitary, M(v) = M(v)† = M(v)−1, which means that M(v) is involutary, M2(v) = I (like the Pauli matrices); in addition, det M(v) = −1. If the vector |vi is real, M(v) has a simple geometric interpretation: reflection with respect to an (N − 1)-dimensional plane with a normal vector |vi; in the complex case the interpretation is more involved. In general, the Householder vector |vi is complex, which implies that it contains 2(N − 1) real parameters (taking into account the normalization condition and the unimportant global phase).

Generalized QHR

We define the generalized QHR as ¡ ¢ M(v; ϕ) = I + eiϕ − 1 |vi hv| , (5.2) where |vi is again an N-dimensional normalized complex column-vector and ϕ is an arbitrary phase. The standard QHR (5.1) is a special case of the generalized QHR (5.2) for ϕ = π: M(v; π) ≡ M(v). The generalized QHR is unitary,

M(v; ϕ)−1 = M(v; ϕ)† = M(v; −ϕ), (5.3) and its determinant is det M = eiϕ. 54 CHAPTER 5. ENGINEERING OF ARBITRARY UNITARIES BY REFLECTIONS

Physical implementations

Coherently driven N-pod system

The standard and generalized QHRs have simple physical realizations in the system described in Chapter 4. The N degenerate [in the rotating-wave approximation (RWA) sense [22]] ground states |ni (n = 1, 2,...,N), which represent the quNit, are coupled coherently and simultaneously by N external fields to an ancillary excited state |ei ≡ |N + 1i, as shown in Fig. 4.1. Such an N-pod system can be formed, e.g., by coupling the magnetic sublevels of several J = 1 levels to a single J = 0 level by polarized laser pulses as shown on Fig. 4.2; for a qutrit only one J = 1 level suffices. The propagator UN+1(t, t0) of this system obeys the Schr¨odingerequation, d i~ U (t, t ) = H(t)U (t, t ) , (5.4) dt N+1 0 N+1 0 with the RWA Hamiltonian [22]

  0 0 ··· 0 Ω (t)  1     0 0 ··· 0 Ω2 (t)  ~    ......  H(t) =  . . . . .  , (5.5) 2      0 0 ··· 0 ΩN (t)  ∗ ∗ ∗ Ω1 (t)Ω2 (t) ··· ΩN (t) 2∆ (t)

and the initial condition UN+1 (t0, t0) = I. The Rabi frequencies again have the same time dependence, described by the envelope function f (t), but we assume them complex for gen- erality,

iβn Ωn(t) = χnf (t) e (n = 1, 2,...,N). (5.6)

We derived the exact solution for the propagator in Chapter 4 and it is given by Eq. (4.20).

Standart QHR: exact resonance

In the case of exact resonance (∆ = 0) the Cayley-Klein parameter a = −1 for any pulse shape f(t) if the pulse area A is equal to (see Sec. 2.2)

A = 2 (2k + 1) π (k = 0, 1, 2,...) . (5.7) 5.2. QUANTUM HOUSEHOLDER REFLECTION (QHR) 55

Then, sin (A/2) = 0, and the last row and column of the propagator (4.20) vanish, except for the diagonal element, which becomes −1 (δ = 0); the propagator (4.20) reduces to   p q 0    .   Uπ .  U =   . (5.8) N+1    x y 0  0 ··· 0 −1

Here Uπ is an N-dimensional unitary matrix (with det Uπ = −1), which represents the propagator within the N-state degenerate manifold; it has exactly the QHR form (5.1), Uπ = M(v; π) = M(v). The components of the N-dimensional QHR vector |vi are the normalized Rabi frequencies, with the accompanying phases, 1 h iT |vi = χ eiβ1 , χ eiβ2 , . . . , χ eiβN . (5.9) χ 1 2 N Hence the propagator Uπ for the ground level quNit from the N-pod system driven by the Hamiltonian (5.5), with ∆ = 0 and rms pulse area (5.7), represents indeed a physical re- alization of QHR in a single interaction step. Any QHR vector (5.9) can be produced by appropriately selecting the peak couplings χn and the phases βn, while obeying Eq. (5.7) (e.g., by adjusting the pulse duration).

Generalized QHR

The unitary propagator (4.20) for a = eiϕ (|b| = 0) reduces to   p q 0    .   Uϕ .  U =   , (5.10) N+1    x y 0  0 ··· 0 e−iϕe−iδ where, as is easily verified, we have Uϕ = M(v; ϕ), and hence, the propagator Uϕ represents a physical realization of the generalized QHR (5.2). The vector |vi is again given by Eq. (5.9). The condition a = eiϕ for ϕ 6= 0, π can only be realized off resonance (∆ 6= 0). There is a beautiful off-resonance solution to the Schr¨odingerequation – the Rosen-Zener (RZ) model – which we shall use here to exemplify the generalized QHR. The Rozen-Zener (RZ) model [29] can be seen as an extension of the resonance solution (4.39) to nonzero detuning for a hyperbolic-secant pulse shape,

f (t) = sech (t/T ) , (5.11a)

∆ (t) = ∆0. (5.11b) 56 CHAPTER 5. ENGINEERING OF ARBITRARY UNITARIES BY REFLECTIONS

The solution for the Cayley-Klein parameter a is derived in the Appendix of Chapter 3 and it reads, ¡ ¢ 2 1 1 Γ 2 + 2 i∆0T a = ¡ 1 1 1 ¢ ¡ 1 1 1 ¢, (5.12) Γ 2 + 2 χT + 2 i∆0T Γ 2 − 2 χT + 2 i∆0T where Γ(z) is Euler’s gamma function. Using the reflection formula ¡ 1 ¢ ¡ 1 ¢ Γ 2 + z Γ 2 − z = π/ cos πz, we find ¡ ¢ 2 1 2 sin 2 πχT |a| = 1 − 2 ¡ 1 ¢. (5.13) cosh 2 π∆0T Hence in this model, |a| = 1 for χT = 2l (l = 0, 1, 2,...); then the last row and the last column of the propagator (4.20) vanish, except the diagonal element. The phase ϕ of a = eiϕ depends on the detuning ∆0 and for an arbitrary integer l we find from Eq. (5.12)

Yl−1 ∆ T + i (2k + 1) a = eiϕ = 0 , (5.14) ∆0T − i (2k + 1) k=0 and hence Yl−1 ϕ = 2 arg [∆0T + i (2k + 1)] . (5.15) k=0

This can be seen as an algebraic equation for ∆0, which has l real solutions. For example, for l = 1 [which corresponds to rms pulse area A = 2π], we have ∆0T = cot (ϕ/2). Hence the generalized-QHR phase ϕ can be produced by an appropriate choice of the detuning ∆0. The use of nonresonant interaction, besides providing an additional phase parameter, has another important advantage over resonant pulses: lower transient population of the intermediate state. This can be crucial if the lifetime of this state is short compared to the interaction duration. Equation (5.15) provides the opportunity to control this transient population, which is proportional to ∆−2, by using large peak Rabi frequency (implying larger l) and find the largest solution for ∆. It is important that the standard QHR can also be realized off resonance, by selecting a detuning ∆0 for which ϕ = π.

5.3 QHR decomposition of U(N)

Standard-QHR decomposition

We shall show that QHR is a very efficient tool for constructing a general U(N) quNit gate. In particular, we shall show that any N-dimensional unitary matrix U (U−1 = U†) can be expressed as a product of N − 1 standard QHRs M(vn)(n = 1, 2, ..., N − 1) and a phase gate

Φ (φ1, φ2, . . . , φN ),

U = M(v1)M(v2) ··· M(vN−1)Φ (φ1, φ2, . . . , φN ) , (5.16) 5.3. QHR DECOMPOSITION OF U(N) 57 where

iφ1 iφ2 iφN Φ (φ1, φ2, . . . , φN ) = diag(e , e ,..., e ). (5.17)

We shall prove this assertion by explicitly constructing the decomposition (5.16). The standard QHRs M(vn) involve vectors |vni, which we construct as follows. First we define the normalized vector |v1i as iφ |u1i − e 1 |e1i |v1i = p , (5.18) −iφ 2 [1 − < (u11e 1 )] where the vector |uni denotes the nth column of U = {ukn}, φ1 = arg u11, and T |e1i = [1, 0, ..., 0] . We find

iφ1 M(v1) |u1i = e |e1i , (5.19a)

−iφ1 M(v1) |uni = |uni + 2e u1n |v1i , (5.19b)

he1| M(v1) |uni = 0 (n = 2, 3,...,N) . (5.19c)

Hence the action of M(v1) upon U nullifies the first row and the first column except for the first element,   eiφ1 0 ··· 0      0 p q  M(v )U =   , (5.20) 1  .   . UN−1  0 x y where UN−1 is a U(N − 1) matrix. We repeat the same procedure on M(v1)U and construct the vector |v2i, 0 iφ2 |u2i − e |e2i |v2i = p , (5.21) 0 −iφ2 2 [1 − < (u22e )] 0 where the vector |u2i is the second column of M(v1)U, φ2 = arg [M(v1)U]22, and T |e2i = [0, 1, 0,..., 0] . The corresponding QHR M(v2), applied to M(v1)U, has the following effects: (i) nullifies the second row and the second column of M(v1)U except for the diagonal element, which becomes eiφ2 , and (ii) does not change the first row and the first column. By repeating the same procedure N − 1 times, we construct N − 1 consecutive Householder reflections, which nullify all off-diagonal elements, to produce a diagonal matrix comprising N phase factors,

M(vN−1) ··· M(v1)U = Φ(φ1, φ2, . . . , φN ), (5.22) which completes the proof of Eq. (5.16) since M(v) = M(v)−1. If U is a SU(N) matrix then PN det Φ = ±1, meaning n=1 φn = 0 or π.

We note that the choice of the QHRs M(vn) is not unique; for example, the first QHR

M(v1) can be constructed from the first row of U, instead of the first column. Furthermore, 58 CHAPTER 5. ENGINEERING OF ARBITRARY UNITARIES BY REFLECTIONS the final diagonal matrix (5.17) occurs due to the unitarity of U, which leads to Eq. (5.19c); a QHR sequence produces a triangular matrix in general. The QHR decomposition (5.16) of the U(N) group into N − 1 Householder matrices (5.1) and a phase gate provides a simple and efficient physical realization of a general transformation of a quNit by only N − 1 interaction steps and a phase gate; this is a significant advantage compared to O(N 2) operations in existing recipes. Each QHR vector is N-dimensional, but the nonzero elements decrease from N in |v1i to just 2 in |vN−1i, and so does the number of fields required for each QHR, see Eq. (5.9). The decomposition (5.16) is also of mathematical interest because it provides a very nat- ural parametrization of the U(N) group. Indeed, a QHR vector with n nonzero elements con- tains 2(n − 1) real parameters (because of the normalization and the irrelevant global phase). PN 2 The phase gate (5.17) contains N phases. Hence Eq. (5.16) involves n=2 2(n−1)+N = N real parameters, as should be the case for a general U(N) matrix.

Generalized-QHR decomposition

We shall show how any unitary matrix U can be expressed as a product of N generalized

QHRs M (vn; ϕn)(n = 1, 2,...,N) defined by Eq. (5.2), without a phase gate, that is

YN U = M(vn; ϕn). (5.23) n=1

We first define the normalized vector

|u1i − |e1i |v1i = p , (5.24) 2 [1 −

M(v1; −ϕ1) |u1i = |e1i , (5.25a)

he1| M(v1; −ϕ1) |uni = 0 (n = 2, 3,...,N ). (5.25b)

Therefore, the action of M(v1; −ϕ1) upon U nullifies the first row and the first column except for the first element, which is turned into unity,   1 0 ··· 0      0 p q  M(v ; −ϕ )U =   , (5.26) 1 1  .   . UN−1  0 x y 5.3. QHR DECOMPOSITION OF U(N) 59

where UN−1 is a U(N − 1) matrix. We repeat the same procedure on UN−1 and construct the vector |u0 i − |e i |v i = p 2 2 ,, (5.27) 2 0 2 [1 −

The action of M(v2; −ϕ2) upon M(v1; −ϕ1)U has the following effects: (i) nullifies the second row and the second column of M(v1; −ϕ1)U except for the diagonal element which is turned into unity, and (ii) does not change the first row and the first column of M(v1; −ϕ1)U. By repeating the same procedure N times, we construct N consecutive generalized Householder reflections, which nullify all off-diagonal elements to produce the identity matrix, Y1 M(vn; −ϕn)U = I. (5.28) n=N By recalling Eq. (5.3) we obtain Eq. (5.23) immediately. Note that the last QHR

M(vN ; ϕN ) = Φ(0,..., 0, ϕN ) is actually a one-dimensional phase gate. Therefore the use of generalized QHRs replaces the N-dimensional phase gate needed in the standard-QHR implementation (5.16) by a one-dimensional phase gate Φ(0,..., 0, ϕN ).

We point out that again, as for the standard QHRs M(vn), the choice of any of the generalized

QHRs M(vn; ϕn) is not unique because it can be constructed from the respective row, rather than the column, of the corresponding matrix.

Examples

Qubit

As an example of the QHR decomposition we first consider the qubit, which is the conventional system for quantum information processing. The conventional realization of a general U(2) transformation involves three interactions: two phase gates and one rotation R (ϑ) [20, 21],

U = Φ (α1, α2) R (ϑ) Φ (0, α3) . (5.29)

Already for a qubit, the QHR implementations (5.16) and (5.23) are superior to Eq. (5.29) because they only require one QHR and one phase gate,

U = M(v)Φ(φ1, φ2), (5.30a)

U = M(v; ϕ1)Φ (0, ϕ2) . (5.30b)

Qutrit

As a second example we consider a qutrit – a three-state quantum system. The most general transformation of a qutrit belongs to the U(3) group, which can be parametrized by nine real 60 CHAPTER 5. ENGINEERING OF ARBITRARY UNITARIES BY REFLECTIONS parameters; respectively, the SU(3) group is described by eight real parameters. A SU(2) factorization of SU(3) reads [57]

U = R23 (α1, β1, γ1) R12 (α2, β2, α2) R23 (α3, β3, γ3) , (5.31) where Rmn are SU(2) subgroups of SU(3), with the SU(2) submatix occupying the mth and nth rows and columns of Rmn. Hence this implementation (5.31) of SU(3) requires three SU(2) gates, each involving three qubit gates (5.29), i.e. nine qubit gates in total (which can be reduced to seven by combining adjacent phase gates). With the present QHR implementation (5.30) of SU(2) the number of operations can be reduced to six. Already for SU(3) or U(3), the present QHR implementations (5.16) and (5.23) are con- siderably more efficient because they require only two QHRs and a phase gate,

U = M(v1)M(v2)Φ(φ1, φ2, φ3), (5.32a)

U = M(v1; ϕ1)M(v2; ϕ2)Φ(0, 0, ϕ3). (5.32b) As an example, the arbitrarily chosen SU(3) gate   0.864e−2πi/3 0.282e15πi/19 0.416e−7πi/8    0.140πi 7πi/11 0.808πi  U =  0.382e 0.902e 0.203e  (5.33) 0.327e−0.789πi 0.328e4πi/5 0.886e0.035πi (keeping 3 significant digits) can be realized with two standard QHRs and a phase gate, with

iπ/3 0.140πi −0.789πi T |v1i = [0.260e , 0.734e , 0.628e ] , (5.34a) −0.134πi 0.710πi T |v2i = [0, 0.651e , 0.759e ] , (5.34b) © ª Φ = diag e−0.667πi, e0.866πi, e−0.199πi . (5.34c)

Alternatively, the same SU(3) gate (5.33) can be realized by two generalized QHRs and a phase gate (5.32b), with ϕ1 = −0.693π, ϕ2 = 0.653π, ϕ3 = 0.04π, and

£ 0.307πi −0.707πi 0.364πi¤T |v1i = 0.955e , 0.226e , 0.193e , (5.35a) £ 0.347πi −0.383πi¤T |v2i = 0, 0.987e , 0.161e . (5.35b)

5.4 Quantum Fourier transform

The quantum Fourier transform (QFT) is a key ingredient in quantum factoring, quantum search, generalized phase estimation, the hidden subgroup problem, and many other quantum algorithms [20, 21]. The QFT is defined as the unitary operator with the following action on an orthonormal set of states |ni (n = 1, 2 ...,N): XN F 1 2πi(n−1)(k−1)/N UN |ni = √ e |ki . (5.36) N k=1 5.4. QUANTUM FOURIER TRANSFORM 61

Qubit

For a qubit, UF is the Hadamard gate [see Eq. (1.7)], " # F 1 1 1 U2 = √ , (5.37) 2 1 −1

F which can be written as a single QHR, U2 = M(v), with · q q ¸ 1 √ √ T |vi = − 2 − 2, 2 + 2 . (5.38) 2 Here the standard and generalized QHRs coincide.

Qutrit

For a qutrit the QFT matrix reads   1 1 1 1   UF = √  1 e2πi/3 e−2πi/3  . (5.39) 3 3   1 e−2πi/3 e2πi/3 The standard-QHR decomposition reads

F U3 = M(v1)M(v2)Φ(0, π/4, −3π/4) (5.40a) 1q h √ iT |v i = 1 + √1 1 − 3, 1, 1 , (5.40b) 1 2 3 s √ 1 + 2 h √ iT |v2i = √ 0, 1 − 2, −i . (5.40c) 2 2 The generalized-QHR decomposition reads

F U3 = M(v1; π)M(v2; π/2), (5.41a) 1q h √ iT |v i = 1 + √1 1 − 3, 1, 1 , (5.41b) 1 2 3 1 T |v2i = √ [0, 1, −1] . (5.41c) 2

Here the first QHR M(v1; π) = M(v1) is the same for the standard- and generalized-QHR implementations.

Quartit

For a quartit (N = 4) the QFT matrix reads   1 1 1 1     1  1 i −1 −i  UF =   . (5.42) 4 2    1 −1 1 −1  1 −i −1 i 62 CHAPTER 5. ENGINEERING OF ARBITRARY UNITARIES BY REFLECTIONS

1.0

0.8

0.6 N = 3 0.4 N = 4 Deviation

N = 2 0.2

0 -10 -5 0 5 10 Time (units of T) ¯ ¡ ¢ ¯ PN ¯ F ¯ Figure 5.1: Deviation j,k=1 ¯(UN )jk − UN jk¯ of the propagator UN (t) from the QFT ma- F trix UN versus time for N = 2, 3, 4, for generalized-QHR implementations. The pulses for N = 2 are centered at time τ = −5T , whereas for N = 3 and 4 at times τ = −5T and τ = 5T . We have assumed sech pulse shapes (5.11a) and rms pulse area A = 2π (χ = 2). The individual couplings χn are given by the components of the generalized-QHR vectors (5.38) for N = 2, (5.41b) and (5.41c) for N = 3, (5.44b) and (5.44c) for N = 4, each multiplied by χ. All phases βn are zero. The detunings are ∆ = 0 for the first steps and ∆ = 1/T for the second.

The standard-QHR decomposition reads

F U4 = M(v1)M(v2)Φ(0, π/4, 0, −3π/4), (5.43a) 1 |v i = [−1, 1, 1, 1]T , (5.43b) 1 2 s √ 1 + 2 h √ iT |v2i = √ 0, 1 − 2, 0, −i . (5.43c) 2 2

The generalized-QHR decomposition reads

F U4 = M(v1; π)M(v2; π/2), (5.44a) 1 T |v1i = 2 [−1, 1, 1, 1] , (5.44b) |v i = √1 [0, 1, 0, −1]T . (5.44c) 2 2

Again, the first QHR M(v1; π) = M(v1) is the same for the standard- and generalized-QHR implementations. Interestingly, the QFT for N = 4 is decomposed with only two QHRs, rather than three, without phase gates.

Figure 5.1 shows the time evolution of the propagator UN (t) towards the respective QFT F matrix UN , for N = 2, 3, 4, for realizations with generalized QHRs. As time progresses, the 5.5. DISCUSSION AND CONCLUSIONS 63

F deviation of UN (t) from UN vanishes steadily in all cases. As predicted, QFT is realized with just a single QHR for N = 2 and with just two QHRs for N = 3 and 4.

5.5 Discussion and conclusions

We have proposed a simple physical implementation of the quantum Householder reflection in a coherently driven N-pod system consisting of a quNit coupled to an auxiliary state. We have shown that the most general U(N) transformation of a quNit can be constructed by at most N − 1 standard QHRs and an N-dimensional phase gate, or by N − 1 generalized QHRs (each having an extra phase parameter compared to the standard QHR) and a one- dimensional phase gate, i.e. by only N physical operations. This significant improvement over the existing setups [involving O(N 2) operations] can be crucial in making quantum state engineering and operations with quNits experimentally feasible. The Householder gate is superior already for a qubit because the general U(2) gate needs just two gates, a QHR and a phase gate, compared to three gates in existing implementations. For a qutrit, the QHR realization of U(3) requires only three gates, compared to at least seven hitherto. The QHR implementation of the U(N) gate is particularly important for qutrits because of the straightforward physical implementation in a J = 1 ↔ J = 0 transition (Fig. 4.2); the results, of course, apply to any N, and can be accomplished, for instance, by using more J = 1 levels. We have given examples for QHR implementations of quantum Fourier transforms. The QHR realization of QFT for a qubit requires a single interaction step, compared to two steps hitherto. The QHR realization is particularly efficient for a quartit (N = 4), where the QFT is synthesized with only two QHR gates [as for a qutrit (N = 3)], much fewer than O(42) in the existing SU(2) proposals. The components of the Householder vectors are the amplitudes of the respective couplings. It is important that all QHR phases are relative phases of the external control fields, e.g. relative laser phases, which are much easier to control than dynamic and geometric phases. The generalized QHR requires off-resonant pulsed interactions, appropriately detuned from resonance. The standard QHR can be realized both on and off resonance. The off- resonance implementation has the advantage that only negligible transient population is placed into the (possibly decaying) ancillary excited state; however, it requires a specific value of the detuning. In the existing SU(2) proposals, each interaction step involves a single SU(2) (or Givens) rotation. The difference between the Givens rotation and the Householder reflection is that, when applied to an arbitrary matrix, the Givens rotation nullifies a single matrix element; 64 CHAPTER 5. ENGINEERING OF ARBITRARY UNITARIES BY REFLECTIONS the Householder reflection nullifies an entire row (or column). When the matrix is unitary, a single Householder reflection nullifies one column and one row simultaneously. Hence the Householder reflection is N times faster than the Givens SU(2) transformation. In atoms and ions qubits are encoded usually in degenerate ground sublevels, and the coupling between them is accomplished by off-resonant interactions, via an intermediate state, which is eliminated adiabatically to produce an effective Raman coupling. In doing so, the phase relation between the two Raman fields is lost. In our proposal we use resonant, or nearly- resonant, fields; no adiabatic elimination is performed and the phase relation is preserved in the resulting QHR propagator. Therefore, already for N = 2, the QHR contains an additional phase parameter compared to previous realizations, which reduces the number of steps for U(2) operations from 3 to 2. Hence, even for a qubit there is a clear improvement. It is also significant that resonant interactions, which we use, require less interaction energy than off-resonant interactions; this may be crucial in the case of weak couplings. We conclude this Chapter by emphasizing that the wide-spread use of the Householder re- flection in classical data analysis promises that the proposed quantum implementation has the potential to become a powerful tool for quantum state engineering and quantum information processing. egv eea xmlso mlmnain nra tm rmlcls h eut from results The molecules. 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C h a p t e r 6 66CHAPTER 6. PHYSICAL REALISATION OF COUPLED QUANTUM REFLECTIONS

6.1 Motivation

Manipulation of discrete quantum states has long held interest, most recently for application to quantum information processing [20, 21, 59]. In the simplest realizations one deals with a qubit and through pulsed resonant coherent excitation produces a specified superposition of the two states, starting from a single state. More recently interest has shifted to the realisation of arbitrary single-qubit gates, and to producing transitions between superposition states. Here, we are interested in the more general tasks which involve control and manipulation of quNits. The goal of quantum-state manipulation is to realign the statevector |ψ(t)i from some given Hilbert-space direction |ψ(ti)i at an initial time ti to some prescribed state |ψ(tf )i at a

final time tf , each of these statevectors being defined by the set of complex-valued probability amplitudes Cn(t) associated with a set of quantum states |ψni. Expressed in matrix form, the goal is to obtain a propagator matrix U(tf , ti) that transforms the vector C(t) of components

Cn(t),

C(tf ) = U(tf , ti)C(ti). (6.1)

Here we consider, as a particular generalization of the two-state system, a two- level system involving two sets of degenerate sublevels: a less energetic set of N states

|ψni (n = 1, 2,...,N) with common energy EN – the lower set representing a quNit, and a more energetic set of M states |φmi (m = 1, 2,...,M) that share energy EM – the upper set representing an M-state quNit. The Hilbert space is of dimension N +M. For definiteness we here assume that N = M. We assume that interactions produced by laser fields can induce direct transitions between lower and upper quNits, but not directly within either manifold of states (i.e. we allow electric-dipole transitions only). Because the statevector has unit length at all times, any such change can be regarded as a rotation, and any allowable motion in several dimensions can be decomposed into a succession of two-dimensional rotations [54], analogous to the three Euler angles that define an arbitrary rotation in Euclidean space. However, length-preserving changes in multi-dimensional space can also be produced by a succession of reflections and such unitary operations can be imple- mented very efficiently using coherent excitation techniques as we already showed in Chapter 4 and Chapter 5. Moreover, the use of quantum Householder reflections (QHR) permits efficient quantum-state engineering of transitions between arbitrary superpositions [60]. The QHR implementation proposed in Chapter 5 requires a particular multistate linkage pattern, in which a low-lying quNit states link, via radiative interaction, with a single upper state – a generalization of the tripod linkage to an Npod. Such a linkage pattern occurs with a lower level having angular momentum J = 1 (three sublevels) excited to an upper level 6.2. THE DEGENERATE TWO-LEVEL SYSTEM 67 having J = 0 (a single sublevel) but the pattern is difficult to realise for more than 3 lower states. It is therefore desirable to extend the QHR technique to more general linkage patterns. The present manuscript describes a procedure that allows implementation of the QHR when there are multiple states in the upper set. Such an extension makes possible the appli- cation of QHR to arbitrary angular momentum states, as occur with free atoms and molecules. As we show, the propagator for such situations is not a single QHR, but a product of QHRs, moreover with orthogonal vectors. The expression for the resulting propagator has a clear geometric interpretation as the effect of a succession of reflections, i.e. coupled mirrors. The key to this extension of the QHR is a transformation of the underlying basis states, the so-called Morris-Shore (MS) transformation [43, 61, 62]. This replaces the original system, with its multiple linkages between states, by a set of independent nondegenerate two-state systems [see Chapter 4]. In this Chapter we present the solution of the degenerate two-level problem in a simple closed form involving sums of projectors of MS states. These expressions are useful for deriving analytical solutions, generalizations of known two-state solutions, for systems having degenerate levels. For certain conditions, wherein the transition probabilities between states from different sets vanish, the propagator for states of the lower set is given by a product of QHRs with orthogonal vectors, each vector being a bright state; a similar property applies to the upper set. We discuss properties of this novel geometric object, the coupled mirrors. To illustrate the procedure we develop a useful explicit analytic formalism for two upper states and present some examples.

6.2 The degenerate two-level system

Multiple states as two degenerate levels

An important extension to the two-state quantum system is the linear chain of N states, each state being linked only to its nearest neighbor. The simplest of this is the three-state chain, involving two independent pulses, the mathematics of which has been extensively reported [63], as have lengthier chains [64]. Here we consider an extension in which each of the states of the two-state system is replaced by a degenerate set of states that together form an energy level. Such situations occur commonly when one deals with angular-momentum eigenstates, as happens with the electronic structure of atoms in free space. For a given angular momentum J there are 2J + 1 magnetic sublevels that, in the absence of an external electric or magnetic field, all have the same energy. We shall consider the possibility that there be no nonzero elements of the Hamiltonian matrix linking any states of the same energy; nonzero couplings occur only between states 68CHAPTER 6. PHYSICAL REALISATION OF COUPLED QUANTUM REFLECTIONS

ϕ ϕ ϕ 1 2 M

∆(t )

ψ ψ ψ ψ 1 2 3 N

Figure 6.1: Schematic linkage pattern for multistate system consisting of two coupled sets of degenerate levels.

from the lower set and the states in the upper set. We make no restriction on the number of states that connect with any single state. This generalizes the usual situation of angular momentum states excited by electric-dipole radiation, when any state can link to no more than three other states (the selection rules ∆m = −1, 0, +1). Figure 6.1 illustrates the linkage pattern we consider. We shall assume that we can control the magnitude and the phase of each interaction (e.g. each electric field envelope), and that every Rabi frequency has the same time dependence, which we write as

Ωmn(t) = 2Vmnf(t), (6.2)

where Vmn is a complex-valued constant and f(t) a real-valued pulse-shaped function of time bounded by unity. We assume that the duration of the pulse is shorter than any decoherence time, so that the dynamics is governed by the Schr¨odingerequation. Such is the situation for excitations of free atoms and molecules by picosecond or femtosecond pulses [65]. We shall also assume that each transition has exactly the same detuning ∆. In the simplest situations, those associated with angular-momentum states, three distinct fields can be distinguished by their polarization, while sharing a common carrier frequency. By utilizing three independent polarization directions of a laser pulse we can ensure three linkages with common time dependence. More general linkage patterns, still within the model described here, are also possible [62]. 6.3. EXACT ANALYTICAL SOLUTION 69

6.3 Exact analytical solution

The RWA Hamiltonian of the system shown in Fig. 6.1 has the block structure " # OVf(t) H(t) = . (6.3) V†f(t) D(t)

Here O denotes the N-dimensional square null matrix; the zeros signify that the lower states do not interact with each other and that they all have the same energy, which we take as the zero-point of our energy scale. The matrix V is an (N × M)-dimensional matrix whose † elements Vnm are the magnitudes of the couplings between the lower and the upper states; V is its hermitian conjugate. Lastly, D(t) is an M-dimensional square diagonal matrix whose elements are all equal to the shared detuning ∆(t),

D(t) = ∆(t)I. (6.4)

The diagonal nature of D(t) indicates the absence of interaction of the upper states amongst themselves. We allow the detuning ∆(t) to vary with time, bearing in mind use of known analytic solutions to the two-state model with frequency-swept detuning. It proves use- ful to write the matrix of interactions V as a row vector of N-dimensional column vectors T |Vni = [V1n,V2n,...,VNn] (n = 1, 2,...,M),   V V ··· V  11 12 1M     V21 V22 ··· V2M  V =   = [|V i , |V i ,..., |V i]. (6.5)  .. .  1 2 M  ······ . .  VN1 VN2 ··· VNM

The Morris-Shore transformation

A significant property of the linkage pattern shown in Fig. 6.1 is that a transformation of Hilbert-space coordinates – the Morris-Shore (MS) transformation – reduces the dynamics to that of a set of independent two-state systems, together with decoupled states [43]. When

N > M, as is the case with the system from Fig. 6.1, the N0 = N − M additional states are a part of the lower-level manifold. These are the dark states which do not take part in the evolution of the system and do not produce fluorescence. The states that can produce excitation (and thence fluorescence) are the bright states of the system and their number is the lesser of N and M, in this case N0. Figure 6.2 shows the new linkage pattern in the MS basis. 70CHAPTER 6. PHYSICAL REALISATION OF COUPLED QUANTUM REFLECTIONS

β β β 1 2 M ∆(t )

λ λ λ 1 2 M

α α α γ 1 2 M 1 γ N0

Figure 6.2: Linkages of Fig. 6.1 transformed by the Morris-Shore transformation into a set of M independent nondegenerate two-state systems, with couplings λn, and a set of N0 = N −M decoupled (dark) states.

Because, by assumption, all elements of the interaction matrix share a common time dependence, the required MS transformation is achieved by a constant unitary transformation

" # AO S = . (6.6) OB

Here A is a unitary N-dimensional square matrix (AA† = A†A = I) which transforms the lower set of states, and B is a unitary M-dimensional square matrix (BB† = B†B = I) which transforms the upper set. This transformation casts the dynamics into the MS basis, with new

M N0 M MS bright lower states {|αmi}m=1, dark lower states {|γki}k=1, and upper states {|βmi}m=1. The transformed MS Hamiltonian has the form

" # O Vef(t) He(t) = SH(t)S† = , (6.7) Ve†f(t) D(t) where

Ve = AVB†. (6.8)

Because the matrices VeVe† = AVV†A† and Ve†Ve = BV†VB† must be diagonal (possibly after removing null rows and rearranging the basis states), the matrices A and B are defined by the conditions that they diagonalize VV† and V†V, respectively. V†V has M generally nonzero 2 † eigenvalues λn (n = 1, 2,...,M). The N-dimensional matrix VV has the same M eigenvalues † as V V and additional N0 = N − M zero eigenvalues. From the vector form (6.5) of V we 6.3. EXACT ANALYTICAL SOLUTION 71 obtain XM † VV = |Vni hVn| , (6.9a) n=1  hV |V i hV |V i · · · hV |V i  1 1 1 2 1 M     hV2|V1i hV2|V2i · · · hV2|VM i  V†V =   . (6.9b)  . . .. .   . . . .  hVM |V1i hVM |V2i · · · hVM |VM i

† V V is the Gram matrix [73] for the set of vectors |Vni,(n = 1, 2,...,M). Thus if all these vectors are linearly independent then det V†V 6= 0 and all eigenvalues of V†V are nonzero [73]; however, this assumption of independence is unnecessary. The MS Hamiltonian (6.7) has the explicit form   O O      0 0 ··· 0 λ1f(t) 0 ··· 0       0 0 ··· 0 0 λ2f(t) ··· 0   ......   ......   ......    He(t) =  0 0 ··· 0 0 0 ··· λ f(t)  . (6.10)  M   O   λ1f(t) 0 ··· 0 ∆ 0 ··· 0       0 λ2f(t) ··· 0 0 ∆ ··· 0     ......   ......  0 0 ··· λM f(t) 0 0 ··· ∆

The structure of He(t) shows that in the MS basis the dynamics decomposes into sets of N0 decoupled single states and M independent two-state systems |αni ↔ |βni (n = 1, 2,...,M), each composed of a lower state |αni and an upper state |βni, and driven by the two-state RWA Hamiltonians " # 0 λnf(t) ehn = , (n = 1, 2,...,M). (6.11) λnf(t) ∆

Each of these two-state Hamiltonians has the same detuning ∆, but they differ in the cou- † plings λn. Each of the new lower states |αni is an eigenstate of VV that corresponds to a 2 † † specific eigenvalue λn of VV . Similarly the new upper state |βni is the eigenstate of V V, 2 corresponding to the same eigenvalue λn. The square root of this common eigenvalue, λn, represents the coupling (half the Rabi frequency) in the respective independent MS two-state † system |αni ↔ |βni. The N0 zero eigenvalues of VV correspond to decoupled (dark) states in the lower set (we assume throughout that N = M; therefore, dark states, if any, are in 72CHAPTER 6. PHYSICAL REALISATION OF COUPLED QUANTUM REFLECTIONS the lower set). The dark states are decoupled from the dynamical evolution because they are driven by one-dimensional null Hamiltonians.

The propagator in the original basis

The eigenvectors of VV† and V†V form the transformation matrices A and B,   hγ |  1   .   .      hβ1|      hγN0 |  . A =   , B =  .  . (6.12)    .   hα1|    hβ |  .  M  .  hαM | They obey the completeness relations

XM XN0 |αnihαn| + |γkihγk| = I, (6.13a) n=1 k=1 XM |βnihβn| = I. (6.13b) n=1

The propagator eun(tf , ti) for the independent MS two-state system |αni ↔ |βni (n = 1, 2,...,M) is unitary and therefore can be parameterized in terms of the complex- valued Cayley-Klein parameters an and bn, " # an bn eun(t , ti) = , (6.14) f ∗ −iδ ∗ −iδ −bne ane R 2 2 tf 0 0 where |an| + |bn| = 1 and δ = ∆(t )dt . ti By taking into account the MS propagators (6.14) for the two-state MS systems, the ordering of the states (6.12), and the MS Hamiltonian (6.10), we write the propagator of the full system in the MS basis as   I O      a1 0 ··· 0 b1 0 ··· 0       0 a2 ··· 0 0 b2 ··· 0   ......   ......   ......    Ue =  0 0 ··· a 0 0 ··· b  . (6.15)  M M   O   −b∗e−iδ 0 ··· 0 a∗e−iδ 0 ··· 0   1 1   ∗ −iδ ∗ −iδ   0 −b2e ··· 0 0 a2e ··· 0     ......   ......  ∗ −iδ ∗ −iδ 0 0 · · · −bM e 0 0 ··· aM e 6.3. EXACT ANALYTICAL SOLUTION 73

It is straightforward to show that the propagator in the original basis U = S†USe reads " # U U U = N NM , (6.16a) UMN UM where

XM XN0 UN = an|αnihαn| + |γkihγk|, (6.16b) n=1 k=1 XM UNM = bn|αnihβn|, (6.16c) n=1 XM −iδ ∗ UMN = −e bn|βnihαn|, (6.16d) n=1 XM −iδ ∗ UM = e an|βnihβn|. (6.16e) n=1

The propagator UN connects states within the lower set, UM connects states within the upper set, and UNM and UMN mix states from the lower and upper sets. By using the completeness relations (6.13a) and (6.13b) we find

XM UN = I + (an − 1)|αnihαn|, (6.17a) n=1 " # XM −iδ ∗ UM = e I + (an − 1)|βnihβn| . (6.17b) n=1

Hence the propagator UN does not depend on the decoupled states |γki (k = 1, 2,...,N0). This has to be expected because, owing to their degeneracy, the choice of the decoupled states is not unique: any superposition of them is also a zero-eigenvalue eigenstate of VV†. Because the dynamics in the original basis must not depend on such a leeway of choice, the full propagator U must not depend on the decoupled states at all, and this is indeed the case.

Analytical solutions for degenerate levels

Equation (6.16a) expresses the dynamics of the degenerate two-level system in terms of the dynamics of the M independent non-degenerate two-state systems, each with the same de- tuning ∆(t) and pulse shape f(t) of the couplings but with different coupling strengths λn. Therefore, Eq. (6.16a) allows one to generalize any analytical non-degenerate two-state solu- tion to a pair of degenerate levels. Recently, such a generalization of the Landau-Zener model to two degenerate levels has been presented [74]. This generalization displays several inter- esting properties, for instance, not all transition probabilities between degenerate states are 74CHAPTER 6. PHYSICAL REALISATION OF COUPLED QUANTUM REFLECTIONS defined for infinite time duration. Other analytical solutions involving two degenerate levels have been derived for five chainwise-coupled states in M or W linkage configurations [61]. We present below another interesting aspect of the solution (6.16): its geometrical nature.

6.4 Quantum-state reflections

Coupled reflections

Of particular significance is the special case when the Cayley-Klein parameters bn are all equal to zero,

bn = 0 (n = 1, 2,...,M); (6.18) then all transition probabilities in the MS basis, as well as these in the original basis from the lower set to the upper set, vanish, i.e. UNM = UMN = 0. Equation (6.18) implies that

|an| = 1, or

iφn an = e (n = 1, 2,...,M), (6.19) for all MS two-state systems, where φn are arbitrary phases. After substituting Eqs. (6.18) and (6.19) in Eqs. (6.16), the propagator in the original basis reads " # U O U = N , (6.20a) OUM XM iφn UN = I + (e − 1)|αnihαn|, (6.20b) n=1 " # XM −iδ −iφn UM = e I + (e − 1)|βnihβn| , (6.20c) n=1 where the propagator UN operates in the lower set of states and UM acts in the upper set.

Taking into account that the coupled lower states |αni (n = 1, 2,...,M) are orthonormal basis vectors, i.e. hαn|αki = δnk, as are the upper MS states |βni (n = 1, 2,...,M), we rewrite UN and UM as the products

YM UN = M(αn, φn), (6.21a) n=1 YM −iδ UM = e M(βn, −φn), (6.21b) n=1 where M(ν; φ) = I + (eiφ − 1)|νihν|, (6.22) 6.4. QUANTUM-STATE REFLECTIONS 75 are the generalized quantum Householder reflection (QHR) operators introduced in Chapter 5.

The orthogonality of the QHR vectors |αni in Eq. (6.21a) automatically ensures the commutation of the QHRs,

[M(αn, φn), M(αm, φm)] = 0. (6.23)

Therefore their ordering in the product in Eq. (6.21a) is unimportant. The same argument applies to Eq. (6.21b). The importance of QHR derives from the fact that any N-dimensional unitary matrix can be decomposed into a set of at most N generalized QHRs as we proved in Chapter 5. The resulting ease with which pulse sequences can be designed to realise the QHR therefore enables one to synthesize any desired unitary transformation of a quNit state, for example a quantum Fourier transform [Chapter 5] or transition between any two pure or mixed quNit states [60].

Special case: orthogonal interaction vectors

2 In the special case when the interaction vectors |Vni are orthogonal, hVm|Vni = |Vn| δmn, the MS eigenvalues and the MS states simplify greatly. Then the matrix V†V of Eq. (6.9b) 2 2 becomes diagonal, and hence its eigenvalues are λn = |Vn| (n = 1, 2,...,M). Moreover, the eigenstates of V†V – the MS states in the upper set – coincide with the original states,

|βni ≡ |φni, (n = 1, 2,...,M). (6.24)

The coupled MS states in the lower set – the eigenstates of VV† – are readily found from Eq. (6.9a), 1 |αni = |Vni ≡ |Vˆni. (6.25) |Vn|

The propagator UN in the lower set is a product of QHRs, with the normalized interaction vectors |Vˆni serving as QHR vectors, whereas the propagator UM in the upper set is a phase gate,

YM UN = M(Vˆn, φn), (6.26a) n=1 XM −iδ −iφn UM = e e |φnihφn|. (6.26b) n=1

The advantage of having orthogonal interaction vectors |Vni is that they serve as QHR vectors. The implication is that in order to construct a pre-selected coupled-QHR transfor- mation (6.26a), the required couplings are directly obtained from Eq. (6.25). Otherwise, in 76CHAPTER 6. PHYSICAL REALISATION OF COUPLED QUANTUM REFLECTIONS

the general case of non-orthogonal interaction vectors, a set of pre-selected QHR vectors |αni (n = 1, 2,...,M), defined as the eigenvectors of VV†, demand numerical derivation of the required couplings from Eq. (6.9a).

Realizations

Off-resonant hyperbolic-secant pulses

The condition (6.19) can be realised with the Rosen-Zener model [see Section 4.5], which assumes constant detuning and hyperbolic-secant time dependence f(t) for the couplings, with pulse duration T ,

f(t) = sech(t/T ), (6.27a) ∆(t) = const. (6.27b)

The independent MS two-state systems share the same detuning ∆ and pulse shape f(t), but have different MS couplings λn. For the Rosen-Zener model the Cayley-Klein parameters an (n = 1, 2, ..., M) read ¡ ¢ 2 1 1 Γ 2 + 2 i∆T an = ¡ 1 1 ¢ ¡ 1 1 ¢, (6.28) Γ 2 + λnT + 2 i∆T Γ 2 − λnT + 2 i∆T We can rewrite this expression in the form

2 2 sin (πλnT ) |an| = 1 − 2 ¡ 1 ¢. (6.29) cosh 2 π∆T

It follows that the condition |an| = 1 is satisfied when λnT = l (l = 0, 1, 2,...). The phase φn

iφn of an = e depends on the detuning ∆, but not on the corresponding coupling λn, and for an arbitrary integer l we find [Chapter 4]

Yl−1 φn = 2 arg [∆T + i (2k + 1)] . (6.30) k=0

Hence the QHR phase φn can be produced by an appropriate choice of the detuning ∆. This result shows that even though the couplings for the MS two-state systems are not the same, the phases φn of the Cayley-Klein parameters an coincide, φn ≡ φ. This feature is unique for the sech pulse. For other non-resonant pulses, e.g. Gaussian [69], the phase φn would depend also on the coupling and therefore will be generally different for each MS two-state system.

Resonant pulses

For exact resonance (∆ = 0), the Cayley-Klein parameter an reads an = cos(An/2), where the R ∞ pulse area is An = 2λn −∞ f(t)dt. When the pulse area is An = 2(2l + 1)π (l = 0, 1, 2,...), 6.5. TWO DEGENERATE UPPER STATES 77

the phase φn is equal to π; hence we obtain a physical realization for the standard QHR (5.1). This result is not resticted to the sech pulse (6.27a) but it is valid for any pulse shape with such an area as noted in Chapters 4 and 5. When the pulse areas are multiples of 4π, the phases

φn vanish, an = 1, and the corresponding QHRs reduce to the identity. Resonant pulses therefore do not produce variable QHR phases φn, which can be used as free parameters.

Far-off-resonant pulses

Far-off-resonant pulses provide the opportunity for easy adjustment of the phase φn, albeit only approximately. Then the condition (6.19) is fulfilled automatically because of the small- ness of the transition probabilities in each MS system. Specifically, if the common detuning

∆ exceeds sufficiently much the largest MS coupling λn, then all transition probabilities will be negligibly small, |bn| ¿ 1. By adiabatic elimination of each upper MS state one finds

2 Z ∞ λn 2 φn ≈ f (t)dt. (6.31) ∆ −∞ p For a sech pulse the integral is equal to 2T , and for a Gaussian to π/2. Because each MS coupling λn is generally different, each phase φn will also be different. Any desired phase φn, or a set of such phases, can easily be produced by choosing the original couplings Vmn, and hence the MS couplings λn, appropriately.

6.5 Two degenerate upper states

Above, we described the dynamics of the degenerate two-level quantum system in the general case when the lower and upper levels had arbitrary degeneracies, N and M respectively. In this section, we will illustrate these results with a specific example: when the upper set consists of just two degenerate states, i.e. M = 2. This case is interesting because of the possible implementations in different real physical systems, several examples of which will be presented below. Moreover, this special case allows for an elegant analytical treatment.

General case

We retain the notation for the lower states |ψni (n = 1, 2, ..., N), and we denote the two upper states |φ0i and |φ00i. The interaction matrix (6.5) reads   V 0 V 00  1 1   0 00   V2 V2  £¯ ® ¯ ®¤ V =   ≡ ¯V 0 , ¯V 00 , (6.32)  . .   . .  0 00 VN VN 78CHAPTER 6. PHYSICAL REALISATION OF COUPLED QUANTUM REFLECTIONS where |V 0i and |V 00i are N-dimensional interaction vectors comprising the couplings between the lower states and the corresponding upper state. The product V†V reads " # |V 0|2 hV 0|V 00i V†V = . (6.33) hV 00|V 0i |V 00|2

With the introduction of parameters θ and σ through the definitions

2|hV 0|V 00i| = tan 2θ (0 < θ < π/2), (6.34a) |V 00|2 − |V 0|2 arghV 0|V 00i = σ, (6.34b) we write the eigenvalues λm and the associated eigenvectors |βmi (m = 1, 2) within the upper set as |V 0|2 + |V 00|2 |V 0|2 − |V 00|2 λ2 = ± , (6.35a) 1,2 2 2 cos 2θ " # " # cos θ eiσ sin θ |β1i = , |β2i = . (6.35b) −e−iσ sin θ cos θ

The next step is to find the MS states |α1i and |α2i within the lower set of states. They are the eigenstates of the N-dimensional matrix,

VV† = |V 0ihV 0| + |V 00ihV 00|, (6.36) which correspond to the (nonzero) eigenvalues (6.35a). We construct them as superpositions of the interaction vectors |V 0i and |V 00i, and find after simple algebra ¯ 1 ¡ 0® −iσ 00 ¢ |α1i = cos θ ¯V − e sin θ|V i , (6.37a) λ1 1 ¡ iσ 0 00 ¢ |α2i = e sin θ|V i + cos θ|V i . (6.37b) λ2

The propagators

After we have found the explicit form of the MS states (6.35b) and (6.37), we obtain the exact form of the propagators UN and UM ,

UN = M(α1, φ1)M(α2, φ2) (6.38a)

iφ1 iφ2 = I + (e − 1)|α1ihα1| + (e − 1)|α2ihα2|, (6.38b) h i −iδ −iφ1 −iφ2 UM = e e |β1ihβ1| + e |β2ihβ2| , (6.38c) with φ1 and φ2 being the phases of the Cayley-Klein parameters for the MS two-state prop- agators (6.14). 6.5. TWO DEGENERATE UPPER STATES 79

It is easy to verify that the bright states (6.37a) and (6.37b) are eigenstates of the propa- iφ iφ gator UN with eigenvalues e 1 and e 2 , respectively. Physically this means that if the quNit starts in one of these states it will end up in this same state, acquiring only a phase factor. This occurs because of the conditions (6.18) and the independence of the different MS two- state systems. If the QHR phases are equal, φ1 = φ2, then any superposition of |α1i and

|α2i is also an eigenvector of UN . The other eigenstates of the propagator are all degenerate, with unit eigenvalue, and they are orthogonal to |α i and |α i. For a qutrit (N = 3) there is 1 2 only one such eigenvector (up to an unimportant global phase factor) and it is proportional to |α1i × |α2i. For higher-dimensional quNits, any vector in a hyperplane orthogonal to |α1i and |α2i is an eigenvector of UN .

−1/ 2 1/ 2 ∆(t ) J = 1/ 2 V+′ V+′′ ′ ′′ V0 V0

V−′ V−′′ J = 3/ 2 −3/ 2 −1/ 2 1/ 2 3/ 2 Figure 6.3: Linkage pattern for the four degenerate magnetic sublevels of J = 3/2, the lower set, shown coupled by arbitrary polarization of electric-dipole radiation to the two sublevels of J = 1/2, the upper set. The states are labeled by their magnetic quantum number.

In the special case when the vectors |V 0i and |V 00i are orthogonal, hV 0|V 00i = 0, the expressions simplify. Then θ = 0 and σ = 0, and hence

0 00 λ1 = |V |, λ2 = |V |, (6.39a) 0 00 |α1i = |Vˆ i, |α2i = |Vˆ i, (6.39b)

|β1i = |φ1i, |β2i = |φ2i. (6.39c)

Then the propagator UN in the lower set is a product of QHRs, in which the interaction 0 00 vectors |V i and |V i serve as QHR vectors, while the propagator UM in the upper set is a phase gate,

0 00 UN = M(Vˆ , φ1)M(Vˆ , φ2), (6.40a) " # −iφ1 −iδ e 0 UM = e . (6.40b) 0 e−iφ2 80CHAPTER 6. PHYSICAL REALISATION OF COUPLED QUANTUM REFLECTIONS

1.0

P -3/2 P -1/2 0.8 P 1/2 P 3/2 0.6

0.4 Populations

0.2

0 -4 -2 0 2 4 Time (in units 1/T)

Figure 6.4: Time evolution of the numerically calculated populations of the magnetic sublevels of a J = 3/2 level coherently coupled to a J = 1/2 level by three polarized (σ+, σ− and π) −1 pulsed laser fields, with sech shape f(t) = sech(t/T ), V− = V0 = V+ = 8.5T and detuning −1 ∆ = 80T . Then θ = π/4, σ = π, λ1T = 106.3, λ2T = 38.2, φ1 = 2.65772, φ2 = 0.954776. The arrows on the right indicate the values derived by the QHR theory.

Examples

Following are examples of linkage patterns amongst angular-momentum states, which allow application of the QHR theory.

Two levels, J = 3/2 ↔ J = 1/2

Figure 6.3 shows linkage patterns possible with arbitrary polarization between the four mag- netic sublevels of J = 3/2, the lower set, and the two of J = 1/2, the upper set. The interaction matrix has the elements (with the Clebsch-Gordan coefficients included),  √  3V 0  +   √  1 − 2V0 V+  V = √  √  , (6.41) 6  V− − 2V0  √  0 3V− where the subscripts +, − and 0 refer to right circular (σ+), left circular (σ−) and linear (π) polarizations.

6.5. TWO DEGENERATE UPPER STATES 81

−1 0 1 = ∆( t ) J 1

V+′ V+′′

′ ′′ V− V− J = 2 −2 −1 0 1 2 Figure 6.5: Linkage pattern for five states in an M configuration.

Figure 6.4 shows an example of time evolution of the populations of the magnetic sub- levels in the J = 3/2 level, starting with all population in state |ψ−3/2i. The conditions

iφn an = e (n = 1, 2) are realised approximately, by using large detuning from the upper J = 1/2 level. In the end of the interaction, the numerically calculated populations are seen to approach the values predicted by the QHR theory (the arrows on the right).

Two levels, J = 2 ↔ J = 1

The linkage pattern for electric-dipole couplings between sublevels of J = 2, the lower set, and J = 1, the upper set will, for polarization expressed as a combination of left- and right- circular polarization, appear as two uncoupled systems: three states form a Λ-linkage, while five form an M-linkage, as shown in Fig. 6.5. The interaction matrix is √  6V 0  +  1   V = √ V− V+ . (6.42) 10  √  0 6V−

2 2 Then tan 2θ = |V+V−|/[5(|V−| − |V+| )] and σ = arg V+ − arg V−.

Three levels, J = 0 ↔ J = 1 ↔ J = 0

The MS transformation can be applied not only to a pair of degenerate levels but also to a ladder of degenerate levels, as long as there is only a single detuning. Figure 6.6 illustrates an example in an angular momentum basis between the magnetic sublevels of three levels with angular momenta J = 0, 1, 0. The interaction matrix has the form   V 0 V 00  + +   0 00  V = V0 V0  , (6.43) 0 00 V− V−. and the formalism of this section applies. In this case all fields can be changed independently. 82CHAPTER 6. PHYSICAL REALISATION OF COUPLED QUANTUM REFLECTIONS

0 ∆(t ) J = 0

V+′ V−′ ′ V0

J =1 −1 0 1 ′′ V0 V−′′ V+′′

∆(t ) J = 0 0

Figure 6.6: Linkage pattern for three-level ladder involving a degenerate middle level. The two ends of the chain, magnetic sublevels with J = 0 have the same detuning from the three intermediate sublevels of J = 1. The various linkages are invoked by adjusting the direction of the polarization with respect to the quantization axis. The sublevels with J = 1 form the lower set, while those with J = 0 form the upper set.

6.6 Discussion and conclusions

We have here extended the previous applications of QHR in Hilbert space [see Section 5] to allow more general linkage patterns between the quantum states, with particular attention to degenerate sublevels that occur with angular momentum states. The extension relies on the use of the Morris-Shore transformation to reduce the original multi-linkage Hamiltonian to a set of independent two-state systems, thereby allowing the utilization of various known two-state analytic solutions. Such solutions, when expressed in terms of Cayley-Klein param- eters, readily lead to conditions upon the pulse areas and the time-varying detunings of the excitation pulses. We have found that the propagator within the lower (or upper) set of degenerate states represents coupled quantum-state mirrors. The realisation of these coupled QHRs within the lower quNit requires certain conditions on the interaction parameters, specifically that all transition probabilities between the lower and upper sets must vanish. We have proposed three physical realizations of this condition: with resonant, near-resonant (hyperbolic-secant), and far-off-resonant pulses. Resonant and near-resonant pulses provide exactly zero transition probabilities, but offer less flexibility in the QHR phases φn; moreover they require a carefully chosen pulse area in each MS two-state system, which leads to a number of conditions on 6.6. DISCUSSION AND CONCLUSIONS 83 the interaction parameters. Far-off-resonant pulses fulfill the zero-probability conditions only approximately, but offer much more flexibility. Then the only restriction is for sufficiently large detuning, without specific constraints on pulse areas, because the zero-probability conditions are fulfilled simultaneously in all MS two-state systems. For angular-momentum states, there are six independent interaction parameters: three polarization amplitudes, two relative phases between different polarizations, and the common detuning. Therefore the constructed QHR has six free parameters (with the far-off-resonance realization). For a more general linkage, the number of independent parameters can be much larger. We conclude by pointing out that the confinement of the statevector evolution to the lower set of states, and the availability of simple and efficient tools for its engineering, such as coupled QHRs, can be an essential ingredient for decoherence-free quantum computing [75].

aeoeain n xml fmaueetbsdqatmcmuigaelna optics quantum linear desired are a by computing only quantum one measurement-based initial of the example when from One useful differs is or This operation. entangled fidelity. gate high highly very is a state with state final obtained, well-defined the is a onto outcome measurement projected certain indepen- is a system is Whenever the performance parameters. its experimental that the is of computing dent quantum measurement-based of strength The Motivation 7.1 are results presented The qubits. distant to applied [76]. be to Ref. qubits resonator, in atomic optical published of an shuttling of the out enables require otherwise and computing would in quantum periods that measurement-based proposals long other for by a of scheme interrupted realisations This i.e. the fluorescence jumps, intense emission. quantum of photon macroscopic no periods by with long heralded of is signal preparation and telegraph anti-symmetric state techniques entangled random The reservoir maximally using the (1.14). cavities. onto them state optical them between Bell to project interaction coupled to effective measurements are an performing which in create by dots encoded to laser quantum are is distant common qubits goal two the of Our of that help states assume the We spin with electron interference. achieved the classical be and qubits can fibers non-interacting and This optical distant driving, two entangle engineering. to reservoir possible employing is it by that show we Chapter this In nagigdsatqbt sn classical using qubits distant Entangling 85 interference

C h a p t e r 7 CHAPTER 7. ENTANGLING DISTANT QUBITS USING CLASSICAL 86 INTERFERENCE

Quantum dot 1 Optical fiber

Measurement Laser driving box

Quantum dot 2

Figure 7.1: Experimental setup to entangle two distant quantum dots via the observation of macroscopic quantum jumps. schemes based on the detection of single photons [77]. Further examples are the processing of atomic qubits via the detection of single or no photons [78] and the manipulation of the electron-spin states of quantum dots via charge detection [79]. However, the scalability of these approaches depends strongly on the respective measurement efficiency. The entangling scheme by Metz et al. [80] alleviates the detection problem via the obser- vation of macroscopic quantum jumps [81]. This means, the interactions in the system are engineered such that it emits a random telegraph signal of long periods of intense fluorescence (light periods) interrupted by long periods of no photon emission (dark periods). The success- ful state preparation is heralded by a macroscopic dark period. Ref. [80] describes a scheme that prepares two laser driven atoms inside an optical cavity in a maximally entangled state. The same authors have shown that electron shelving techniques allow even for the build up of large cluster states [82]. However, this requires the shuttling of atomic qubits in and out of an optical cavity, which is time consuming and susceptible to decoherence.

7.2 Experimental proposal

In this Chapter, we propose a scheme for distributed entanglement preparation with inherent scalability. We require neither the transport of qubits from one interaction zone to another nor the detection of single photons. This is achieved via reservoir engineering based on an in- terference effect that has already been observed experimentally [83, 84]. For example, in their famous two-atom double-slit experiment, Eichmann et al. [83] detected the spontaneously emitted photons from two laser-driven ions on a distant photographic plate. The pattern formed resembles that of a classical double-slit experiment. In fact, the ions behave like clas- sical dipole emitters sending out electromagnetic waves [85]. Here we use optical fibers to create an analogous coupling of two distant cavities to common modes of the free radiation field. 7.2. EXPERIMENTAL PROPOSAL 87

Fiber 1 Lens

ϕ Fiber 2 Detector Phase plate

Figure 7.2: Measurement box in Fig. 7.1. Illustration of the interference of cavity fiber photons.

Qubits placed in these distant cavities experience only one common resonator mode, a so- called bus mode [86], with a tunable spontaneous decay rate. Consequently, it is now possible to generalize ideas for the generation of scalable entanglement in atom-cavity systems to solid state systems. As in previous quantum dot schemes (see e.g. Refs. [87]), we encode information in electron-spin states. Each dot is driven by a laser field and placed inside an optical cavity. This is feasible with current technology [88]. The light coming from the fibers is constantly monitored by a distant measurement box (c.f. Fig. 7.1). The detected fluorescence signal exhibits macroscopic quantum jumps such that a dark period indicates the shelving of the qubits in a maximally entangled state. Transitions from one fluorescence period into another are now caused by spin-bath couplings, parameter fluc- tuations, or the spontaneous emission of photons into free space. These jumps play a vital role in the proposed state preparation and make it relatively robust against experimental im- perfections. We require only that the cavities experience the same system-bath interaction. The quantum dots do not have to be identical. In the following, spontaneous emission due to the interaction between the cavity field modes and the radiation field propagating through the optical fibers 1 and 2 is taken into account by the dipole Hamiltonian X Hdip = e Di · E(ri) . (7.1) i=1,2

Here Di is an effective dipole moment and proportional to the sum of the annihilation and † the creation operator ci + ci for a photon in cavity i and E(ri) describes the radiation field, † where cavity i couples to fiber i. Suppose ak is the creation operator for a photon with wave vector k and frequency ωk and both cavities have the same fiber coupling constants gk, the same frequency ωc. Then Hcond in Eq. (7.1) equals, in the rotating wave approximation, X h i ik·r1 i(k·r2+ϕ) † Hcond = ~gk e c1 + e c2 ak + H.c. (7.2) k CHAPTER 7. ENTANGLING DISTANT QUBITS USING CLASSICAL 88 INTERFERENCE for the measurement box depicted in Fig. 7.2. Since the fiber photons are constantly moni- tored, a quantum jump description [89, 85] based on Eq. (7.2) can be used to show that the conditional Hamiltonian for the no-photon time evolution of the system equals

0 i † Hcond = Hcond − 2 ~κc c(ϕ) c(ϕ) . (7.3)

0 Here Hcond is the no-photon time evolution Hamiltonian in the absence of cavity decay, c(ϕ) is the annihilation operator

h i √ ikc·r1 i(kc·r2+ϕ) c(ϕ) ≡ e c1 + e c2 / 2 , (7.4)

κc is the spontaneous decay rate of the common cavity mode c(ϕ), and kc is the fiber photon wave vector. In case of a cavity photon emission, the state of the system changes from |ψi into c(ϕ)|ψi/kc(ϕ)|ψik. Notice that the two cavities possess two common cavity modes (one with annihilation operator c(ϕ) and one with c(ϕ + π)) but only one decay channel. Photons in the c(ϕ + π) mode cannot leak out through the cavity mirrors and have a negligible decay rate. Their electromagnetic field components interfere destructively, thereby making the detection of c(ϕ + π) photons in the fiber impossible. Field components created by c(ϕ) photons, to the contrary, interfere constructively and can be detected easily. Consequently, the two cavities do not emit photons independently but are effectively coupled via the radiation field in the fibers. The role of the lens in Fig. 7.2 is to maximize this coupling by focussing the light from the different cavities onto the same spot on the detector, thereby safely erasing all information about the origin of the arriving photons.

7.3 Entangling scheme

Let us now have a look at a variation of the above setup, in which the measurement box in Fig. 7.2 is replaced by the measurement box shown in Fig. 7.3. The new box contains a second photon detector and two parallel polarising beam splitters. Depending on their orientation, some parts of the light field in the fibers are now reflected towards detector b. In analogy to Eq. (7.2), the interaction of the cavity photons with the environment is given by the Hamiltonian

X h i ik·r1 i(k·r2+ϕ) † Hdip = ~gk;a e c1 + e c2 ak k h i ik·r1 ik·r2 † +~gk;b e c1 + e c2 bk + H.c. (7.5) 7.3. ENTANGLING SCHEME 89

PBS Lens Fiber 1

ϕ Fiber 2 Detector a

Lens

Detector b

Figure 7.3: Measurement box in Fig. 7.1. Box to entangle two qubits using, in addition, two polarising beam splitters (PBS’s), a second photon detector and two lenses.

† Here the bk are the creation operators for the photons arriving at detector b and the gk ;a and the gk;b are fiber coupling constants. If we assume, for example,

ϕ = π and eikc·r1 = eikc·r2 , (7.6) then the no-photon time evolution of the system is described by the conditional Hamiltonian X 0 i † Hcond = Hcond − 2 ~κx cxcx (7.7) x=a,b with the annihilation operators √ √ ca ≡ (c1 − c2)/ 2 and cb ≡ (c1 + c2)/ 2 . (7.8)

Here κa and κb are the decay rates of the common cavity a and b-modes. The ratio of both rates can be adjusted to any size by simply changing the orientation of the polarizing beam splitters. In the following we assume that κa is much larger than κb and all other relevant parameters. As we shall see below, the cavity a-mode is then so overdamped that it plays effectively no role in the time evolution of the system. Consequently, the two cavities operate like a single cavity with the tunable spontaneous decay rate κb! Suppose now, each cavity is a part of a quantum dot-cavity system [88]. The internal level configuration of each quantum dot is shown in Fig. 7.4 and should be as in a recent experiment by Atat¨ure et al. [90] on spin-state preparation with near-unity fidelity. In the ground states |0i = |↓i and |1i = |↑i, the dot contains one spin up or one spin down electron with angular momentum projection mz = −1/2 and mz = +1/2. In the excited states |2i = |↓↑⇓i and CHAPTER 7. ENTANGLING DISTANT QUBITS USING CLASSICAL 90 INTERFERENCE

↓↑ ⇓ = 3 ↓↑ ⇑ = 2 ∆(i ) 3 ∆(i ) 2 Ω(i ) (i ) eff geff

(i ) Ω(i ) g 1 Ω(i ) Ω(i ) 0 1 0 ξ (i ) ↑ = 1 ↓ = 0 ξ (i )

Figure 7.4: Level configuration and effective level scheme of a single quantum dot.

|3i = |↓↑⇑i, the dot contains two electrons in a singlet state and a heavy hole with spin projections mz = −3/2 and mz = +3/2. The 1–2 dipole transition of dot i is driven by a (i) (i) circularly polarised laser with Rabi frequency Ω1 and detuning ∆2 . Additional laser fields (i) (i) drive the quadrupole transitions 0–2 and 1–3 with Rabi frequency Ω0 and Ω and detuning (i) (i) (i) (i) ∆2 and ∆3 . The 0–3 transition couples with detuning ∆3 and coupling constant g to the quantised mode of cavity i. Introducing the rotating wave approximation and choosing the 0 appropriate interaction picture, the Hamiltonian Hcond in Eq. (7.7) becomes time independent and equals

X h X 0 (i) † 1 (i) Hcond = ~g |0iiih3| ci + 2 ~Ωj |jiiih2| i=1,2 j=0,1 1 (i) 1 (i) + 2 ~Ω |1iiih3| + 2 ~ξ |0iiih1| + H.c. X ³ i ´ i + ~ ∆(i) − Γ(i) |ji hj| . (7.9) j 2 j ii j=2,3

(i) (i) Here the Γj are the decay rates of the states |2i and |3i. The ξ -dependent terms take uncontrolled spin-bath interactions into account which mix the states |0i and |1i [91]. Without (i) (i) (i) restrictions, we can assume that Ωj ,Ω , and g are real by including their phases in the definition of |2i, |3i and the cavity photon states. In the following, we show that the population of the cavity a-mode and spontaneous decay of electron-hole pair states remains negligible when

(i) (i) (i) (i) (i) Ωj , Ω , g ∼ κb ¿ ∆2 , ∆3 , κa . (7.10)

In this parameter regime, the a-mode and the states |2i and |3i can be eliminated adiabatically 7.4. EVOLUTION OF THE SYSTEM 91 and the Hamiltonian (7.3) simplifies to X h1 ¡ ¢ H = − ~ Ω(i) − ξ(i) |0i h1| cond 2 eff ii i=1,2 (i) † (i) † +~geff |0iiih1| cb + H.c. + ~∆eff;cav |0iiih0| cbcb X i i + ~∆(i) |ji hj| − ~κ c†c , (7.11) eff;j ii 2 b b b j=0,1 √ (i) (i) (i) (i) (i) (i) (i) (i) (i) (i)2 (i) with Ωeff ≡ Ω0 Ω1 /2∆2 , geff ≡ g Ω /2 2∆3 , ∆eff;0 ≡ Ω1 /4∆2 , (i) (i)2 (i) (i)2 (i) (i) (i)2 (i) ∆eff;1 ≡ Ω1 /4∆2 + Ω /4∆3 , and ∆eff;cav ≡ g /2∆3 . The quantum dots indeed behave as if they were placed into only one cavity with a tunable decay rate κb! To resemble the situation in Ref. [80] even more closely, we further assume that both qubits experience the same effective interactions. This requires

∆(1) Ω(1) 2 Ω(1) 2 ∆(1) Ω(1) 2 g(1) 2 2 = 0 = 1 , 3 = = (7.12) (2) (2) 2 (2) 2 (2) Ω(2) 2 g(2) 2 ∆2 Ω0 Ω1 ∆3 and changes the conditional Hamiltonian (7.11) into

~ £¡ (1) (1) † (1) (2)¢ Hcond = − √ 2Ω + 4g c − ξ − ξ 2 2 eff eff b ¡ ¢ ¡ ¢ × |00ihs01| + |s01ih11| + ∆ξ |00iha01| − |a01ih11| ¤ ¡ (1) (1) (1) † ¢¡ +H.c. + ~ ∆eff;0 − ∆eff;1 + ∆eff;cav cbcb |00ih00| ¢ i −|11ih11| − ~κ c†c (7.13) 2 b b b with ∆ξ ≡ ξ(1) − ξ(2), √ |a01i ≡ (|01i − |10i)/ 2, (7.14a) √ |s01i ≡ (|01i + |10i)/ 2. (7.14b)

7.4 Evolution of the system

For ∆ξ = 0, there are no transitions between the symmetric and antisymmetric subspace. Once in a symmetric state, the system emits photons towards detector b. However, when the qubits are in the only antisymmetric qubit state |a01i, no photons arrive at this detector. The detector signal hence reveals information about the state of the quantum dots. The overall effect of this is the continuous projection of the qubits either onto the symmetric subspace or onto |a01i. In the case of small deviations ∆ξ 6= 0, macroscopic quantum jumps occur from one subspace into the other. The system exhibits long periods of intense photon emissions (light CHAPTER 7. ENTANGLING DISTANT QUBITS USING CLASSICAL 92 INTERFERENCE b

(a) κ 60

40

20 I in units of 0 0 5 10 15 t in units of T dark

(b) 1

F 0.5

0 0 5 10 15 t in units of T dark

Figure 7.5: (a) Possible trajectory of the photon density I(t) at detector b obtained (1) (1) 1 (1) 1 (1,1) from a quantum jump simulation with Ωeff = ∆eff;0 = 2 ∆eff;1 = 4 κeff and with (1) (2) ξ = −ξ = 0.005 κb. (b) Logarithmic plot of the corresponding fidelity F of the max- imally entangled state |a01i. periods) interrupted by long periods of no emission (dark periods) [81], as shown in Fig. 7.5(a).

The population in |a01i is very close to unity during a dark period (c.f. Fig. 7.5). A dark period hence prepares the qubits with a very high fidelity in a maximally entangled state.

Identifying a successful state preparation is easy when the mean dark period length, Tdark, is long compared to the mean time between photon emissions within a light period, Tem. Due 2 to the constant projection of the qubits, Tdark and Tlight scale as 1/∆ξ (c.f. Fig. 7.6) and can be very long. Transitions between light and dark periods are also caused by parameter fluctuations violating condition (7.12) and the spontaneous emission of photons, which are not collected by the fibers. The effect of these errors on the fidelity of the prepared state has already been studied in Ref. [92] for an analogous setup. The analysis there suggests that spontaneous emission from excited states can be tolerated, even if the system is operated in the vicinity of the bad-cavity limit. Moreover, random variations of the coupling constants up to 30 % do not affect the fidelity of the prepared state. They only reduce the occurrence of relatively long dark periods.

7.5 Conclusions

We have shown that it is possible to entangle distant quantum dots with electron-spin qubits via the observation of a macroscopic fluorescence signal. There are two conditions for the 7.5. CONCLUSIONS 93

750

T /T dark em T /T light em 500

250

0 0.02 0.04 0.06 0.08 0.1 0.12 ∆ξ in units of κ b

Figure 7.6: Quadratic dependence of the mean length of the light and dark periods, Tdark and (1) (1) 1 (1) 1 (1,1) Tlight, on ∆ξ obtained from a quantum jump simulation with Ωeff = ∆eff;0 = 2 ∆eff;1 = 4 κeff and with ξ(1) = −ξ(2). scheme to work. Eq. (7.6) can be realised by driving the cavity with a common laser field and placing detector a into an interference maxima. Eq. (7.12) can be adjusted by maximizing the mean length of the light and dark periods and the intensity of the emitted light within a light period. Notice that the quantum dots do not have to have the same frequencies and coupling constants. Moreover, the generalisation of the scheme to the build up of large cluster states is straightforward [82] and opens new perspectives for scalable solid state quantum computing.

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