quantum computation with bose-einstein condensates · deutsch’s two bit problem [3]. this was the...

Quantum Computation with

Bose-Einstein Condensates

Diplomarbeit

von

Theresa Hecht

angefertigt unter der Leitungvon

Dr. Belen Paredesund

Prof. Dr. J. Ignacio Cirac

Technische Universitat Munchen

Max-Planck-Institut fur Quantenoptik

Munchen 2004

i

Abstract

We develop a scheme for quantum computation with atomic many-body states.The system we have in mind consists of a two-species interacting Bose-Einsteincondensate, which, under certain conditions, behaves like a robust two-levelsystem protected by an energy gap from higher excited levels. Using thesetwo states to encode the qubit, we show how to perform a universal set ofquantum gates by inducing energy shifts on the atomic levels, changing theRaman coupling between atomic states and allowing tunneling between pairsof condensates. Finally, we discuss the limitations of our scheme and find thatparticle losses are an important source of decoherence.

CONTENTS iii

Contents

1 Introduction 1

2 Brief introduction to quantum computation 5

2.1 Quantum bits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Quantum operations . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Two-species Bose-Einstein condensate 9

3.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.1 Quantum field model . . . . . . . . . . . . . . . . . . . . 93.1.2 Two-mode model . . . . . . . . . . . . . . . . . . . . . . 10

3.2 Analysis of the two-mode model . . . . . . . . . . . . . . . . . . 113.2.1 Mean field approximation . . . . . . . . . . . . . . . . . 123.2.2 Mean field ground state in spin representation . . . . . . 133.2.3 Beyond the mean field approximation . . . . . . . . . . . 153.2.4 Schrodinger-cat states . . . . . . . . . . . . . . . . . . . 16

3.3 Exact numerical calculations . . . . . . . . . . . . . . . . . . . . 173.3.1 Eigenstates . . . . . . . . . . . . . . . . . . . . . . . . . 173.3.2 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Scheme for quantum computation 23

4.1 Encoding a qubit . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 Quantum operations . . . . . . . . . . . . . . . . . . . . . . . . 25

4.2.1 Single-qubit gate via external potential . . . . . . . . . . 254.2.2 Single-qubit gate via adiabatic phase . . . . . . . . . . . 284.2.3 Two-qubit gate via tunneling . . . . . . . . . . . . . . . 314.2.4 Read-out process . . . . . . . . . . . . . . . . . . . . . . 40

4.3 Quantum gates at work . . . . . . . . . . . . . . . . . . . . . . . 404.3.1 Common quantum gates . . . . . . . . . . . . . . . . . . 404.3.2 Creation of the singlet Bell state . . . . . . . . . . . . . 41

5 Realizability and decoherence 43

5.1 Realization of the qubits . . . . . . . . . . . . . . . . . . . . . . 435.1.1 Basic experimental requirements . . . . . . . . . . . . . . 43

iv CONTENTS

5.1.2 Requirements for encoding a qubit . . . . . . . . . . . . 455.1.3 Multi-qubit systems . . . . . . . . . . . . . . . . . . . . . 485.1.4 Initializing the qubits by cooling . . . . . . . . . . . . . . 48

5.2 Realization and timescales of the quantum gates . . . . . . . . . 505.2.1 x-gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.2 Phase-gate . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2.3 Two-qubit gate . . . . . . . . . . . . . . . . . . . . . . . 52

5.3 Effects of decoherence . . . . . . . . . . . . . . . . . . . . . . . 525.3.1 Qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3.2 x-gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3.3 Phase-gate . . . . . . . . . . . . . . . . . . . . . . . . . . 535.3.4 Two-qubit gate . . . . . . . . . . . . . . . . . . . . . . . 56

5.4 Inelastic collisions . . . . . . . . . . . . . . . . . . . . . . . . . . 575.4.1 Two-body processes . . . . . . . . . . . . . . . . . . . . . 595.4.2 Three-body processes . . . . . . . . . . . . . . . . . . . . 60

5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Conclusion and Outlook 63

Bibliography

Acknowledgements

1

Chapter 1

Introduction

Quantum theory is already known for more than 100 years. The first ideassuggesting that quantum theory could allow for models of computation thatare more powerful than the ones used in classical computation came up in thebeginning of the 1980s.

Richard P. Feynman pointed out in 1982 [1] that the simulation of a quan-tum system of N particles on a classical computer cannot be done without anexponential slowdown in the efficiency of the simulation. However, Feynmanproposed that this slowdown could be avoided by using a computer exploit-ing the laws of quantum physics. In other words, a quantum system couldefficiently simulate a quantum system. Quantum computational models werealso constructed by Benioff [2] in 1982, but Deutsch argued in [3] that Benioff’smodel can be perfectly simulated by an ordinary computer.

Three years after the proposal of Feynman, in 1985, David Deutsch intro-duced a complete quantum model for computation and gave a description of auniversal quantum computer [3]. He also devised the first quantum algorithm,Deutsch’s two bit problem [3]. This was the first computational problem forwhich it was shown that a quantum mechanical system, i.e. a quantum com-puter, is superior to classical computers. Later on Deutsch and Jozsa [4] foundan extension of the algorithm yielding exponential speed-up compared to clas-sical computers.

For almost 10 years, there were no major breakthroughs and quantumcomputation remained a curiosity. This situation changed in 1994, when PeterShor introduced his quantum algorithm for factoring integers in polynomialtime, much more efficient than by any known classical algorithm. Shor’s algo-rithm was not only a surprise for complexity theorists for which factoring isan example of a hard problem for which no efficient solution was believed toexist. It is also a Holy Grail for eavsdropping secret services, since the securityof the most popular public key encryption systems relies on the assumed diffi-culty of factoring large numbers. Since then, a number of new algorithms werediscovered, the most prominent of which is Grover’s database search algorithm[5].

2 CHAPTER 1. INTRODUCTION

These perspectives for practical applications motivated the search for po-tential implementations of quantum computers. The basic requirement for aquantum computer is a set of quantum mechanical two-level systems whichcan be initialized, coupled and measured in a controlled way. Informationis stored in these two-level systems, named quantum bits, or qubits. Manyphysical systems seem to be promising candidates for implementing quantumcomputation, among those are single photons [6], quantum dots [7], atoms andions [8],[9], superconducting Josephson junctions [10], and nuclear magneticresonance samples [11]. For a review see also [12].

One promising system for implementing quantum computation are neutralatoms. In the last few years, a great progress has been done in the trappingand manipulation of neutral atoms. For instance, the atoms may be cooledto very low temperatures and atoms can be trapped and addressed individu-ally. In addition, the atoms can be initialized and manipulated to a precisequantum state. Therefore, over the last few years, several implementations ofneutral-atom computing, exploiting various trapping methods and entanglinginteractions, have been proposed [13],[14] most of them based on neutral atomsin optical lattices.

One of these proposals [9] is based on controlled collisions and on themanipulation of individual atoms in a perfectly loaded lattice. With suchan device it is possible to build a universal quantum computer [9] or quantumsimulator [15]. The controlled collisions have been demonstrated partly byMandel et al. [16],[17]. However, this proposal has some problems, like theloss of atoms, or most importantly, defects in the filling of the lattice, i.e. thenumber of atoms per lattice site cannot be controlled perfectly.

To avoid the problem of defects, we present an approach to quantum com-putation in which the qubits are encoded with atomic many-body states. Apromising candidate is a two-component Raman-coupled Bose-Einstein con-densate. As it was shown in reference [18], that under certain conditions, thissystem has an almost degenerate ground state which is separated from theexcited levels by an energy gap. The atomic ensemble then behaves like a two-level system that could be used to encode a qubit. Furthermore, the propertiesof these states are not very sensitive to the number of particles. We explorethe idea of using these two many-body states to encode a qubit and present ascheme for quantum computation.

The thesis is organized as follows:

In chapter 2 we give a brief introduction to the basic concepts of quantumcomputation. In chapter 3 we discuss the low-energy physics of a two-speciesinteracting Bose-Einstein condensate reviewing some work done in reference[18]. We discuss the low-energy physics of this system and show that undercertain conditions, the system can be regarded as a quantum two-level sys-tem.Chapter 4 contains the central part of this thesis. We present a schemefor quantum computation where these two many-body states encode the qubit.

3

Inducing energy shifts on the atomic levels, changing the Raman coupling be-tween atomic states and allowing tunneling between pairs of condensates, weshow how to realize a universal set of quantum gates. As an example we designa protocol for creating a maximally entangled state, the singlet Bell state. Inchapter 5 we discuss the feasibility of the scheme. We analyze the experimen-tal requirements for the preparation and initialization of the qubit systems aswell as for the realization of the quantum gates. In addition, we investigatethe effect of decoherence due to fluctuations of the number of particles on thepresented scheme.

4 CHAPTER 1. INTRODUCTION

5

Chapter 2

Brief introduction to quantum

computation

One of the new interdisciplinary fields taking advantage of the possibilities ofquantum theory is quantum computation. The aim is to process and transmitinformation exploiting the laws of quantum physics. We give here a briefintroduction to the basic concepts of quantum computing, such as quantumbits, quantum operations and universality. For this chapter [19] and [20] mayserve as general references.

2.1 Quantum bits

In classical computation, the basic unit of information is the bit, which canhave two possible states, 0 and 1, e.g. realized by a full or empty capacitor.Thinking about computation based on the laws of quantum mechanics, it isnatural to take as basic unit the corresponding quantum mechanical system,i.e. a system with two basis states, usually denoted | 0 〉 and | 1 〉. This can beany (effective) two-level system, like a spin, some atom or ion in its ground oran excited state, the polarization of a photon, or, as in this thesis, the groundand excited state of a many-body system. In correspondence to the classicalbit, such an two-level system is called a quantum bit, or qubit [21].

The outstanding property of a qubit is, that, in contrast to the classicalbit, it is not restricted to be in either the state 0 or 1–as a quantum system,it can be in any superposition of the state vectors | 0 〉 and | 1 〉,

|Φ1〉 = α | 0 〉+ β | 1 〉, (2.1)

with complex coefficients α and β, |α|2 + |β|2 = 1.

Talking of a single qubit, it can be useful to think of the qubit as a point(θ, φ) on a unit sphere called Bloch sphere. For this purpose, the qubit state

6 CHAPTER 2. INTRODUCTION TO QUANTUM COMPUTATION

|Φ1〉 (2.1) is rewritten in the form

|Φ1〉 = cosθ

2| 0 〉+ eiφ sin

θ

2| 1 〉, (2.2)

where α can always be taken real by properly choosing the unobservable globalphase. As illustrated in figure 2.1, |Φ1〉 can be represented by the unit vector(cosφ sin θ, sinφ sin θ, cos θ), called the Bloch vector.

Figure 2.1: Bloch sphere representation of a qubit.

Still, one qubit is not enough for arbitrary computations. Therefore, let usdiscuss some of the features of a N -qubit system. Quantum mechanics nowtakes place in the 2N -dimensional Hilbert space spanned by the product states|00..0〉, |01..0〉, .., |11..1〉.1 These states correspond to the classically possiblestates formed by N bits. In the same way as for one qubit, a many-qubitsystem can be in a superposition of all these states, e.g.

|ΦN〉 =1

2N/2(|00..0〉+ |01..0〉+ ..+ |11..1〉) . (2.3)

This superposition offers the possibility of using all classically possible statesas an input for quantum computation at only one operational step. This socalled quantum parallelism allows to run the computation on all 2N classicallypossible input states at the same time, which is one of the reasons for thecomputational power of a quantum computer.

Apart from this quantum parallelism, which also exists for one-qubit sys-tems, the possibility of superposition additionally offers a new resource (com-pared to classical computation) within composite systems: entanglement. Inquantum mechanics, composite systems usually cannot be described by giv-ing the states of all the subsystems separately, that is by a product state

1The mathematical structure behind the composition of quantum systems is the tensorproduct. Hence, a vector like |00..0〉 has to be read |0〉 ⊗ ..⊗ |0〉 = |0〉⊗n.

2.2. QUANTUM OPERATIONS 7

|ΦN〉 = |Φ11〉⊗ ..⊗|ΦN

1 〉. Some of the characteristics of one subsystem may de-pend on the state of the other subsystem, a phenomenon called entanglement.Entanglement is one of the characteristics of quantum theory which does notexist in classical physics. It seems to be a key resource in quantum informationand quantum computation [22]. A typical example for an entangled state isthe singlet Bell state

|ΦBell〉 =1√2

(|01〉−|10〉) . (2.4)

Obviously, it cannot be written as a product of two one-qubit states,

|ΦBell〉 6= [αA| 0 〉+ βA| 1 〉]⊗ [αB| 0 〉+ βB| 1 〉] . (2.5)

In contrary, (2.3) is not entangled, since it is a product state,

|ΦN〉 =1

2N/2(| 0 〉+ | 1 〉)⊗N . (2.6)

2.2 Quantum operations

Similar to the classical case where computation can be decomposed into a se-quence of elementary logical gates like AND or NOT, the evolution of quantumbits is described by the successive application of quantum gates. These arecomposed out of unitary transformations and are therefore reversible. As forclassical logic networks, there exist universal sets of quantum gates: any logicquantum gate, i.e. any unitary acting on arbitrary many quantum bits, canbe composed out of an entangling two-qubit gate, together with the arbitraryoperations on single qubits (single-qubit gates) [20].

Each unitary transformation corresponds to a rotation in the Hilbert space.The generators of rotations in the two-dimensional Hilbert space are the Paulimatrices

σx =

(0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

). (2.7)

Each of these matrices generates a rotation about the x, y, z-axis of the Blochsphere, respectively. Hence, each single-qubit gate can be described by a rota-tion of the Bloch vector about a normalized axis ~n by an angle θ,

R~n(θ) = e−i θ

2~n~σ = cos

θ

211− i sin θ

2(~n~σ). (2.8)

Furthermore, each rotation within a two-dimensional unit sphere can bedecomposed into successive rotations about two fixed non parallel axes [20].Therefore, for an arbitrary transformation of the state of one qubit, it is suffi-cient to possess two sets of single-qubit gates corresponding to rotations abouttwo different axes.

8 CHAPTER 2. INTRODUCTION TO QUANTUM COMPUTATION

Two examples of single qubit gates that directly follow from a rotationgenerated by one of the Pauli matrices (2.7), are the NOT gate (negating thestate of the qubit) and the phase-gate (where the state | 1 〉 acquires a relativephase φ). They are represented by

(0 11 0

),

(1 00 eiφ

), (2.9)

and result from a rotation about the x- and the z-axis, respectively.

The most famous example of a “universal” two-qubit gate is the controlled-NOT gate. It negates the state of the second qubit if, and only if, the firstqubit is in the state |1〉. However, any two-qubit gate producing entanglementout of an unentangled two-qubit state is universal [23],[24].

9

Chapter 3

Two-species Bose-Einstein

condensate

This chapter provides an overview of work done by Cirac, Lewenstein, Mølmerand Zoller [18] in which they investigate the low-energy physics of a trappedgas of bosonic atoms, with two internal degrees of freedom. The physical isdescribed in detail, followed by an analysis of its low energy physics, focusing onthe structure of the ground and the first excited state. As demonstrated in [18],in a certain regime of parameters, these two states consist of a superposition oftwo Bose-Einstein condensates, that is, they are Schrodinger-cat states. Thesetwo states, which are later used to encode a qubit, are the basic elements inthe scheme for quantum computation we present.

3.1 Hamiltonian

We consider Bose-Einstein condensation of a trapped gas of N atoms thatare confined by a quasi harmonic potential, as e.g. realized by a magnetic oroptical trap. Each atom possesses two internal degrees of freedom, denotedby |A〉 and |B〉, which could represent two hyperfine levels. The two internallevels are connected by a coherent Raman-like transition, |A〉 ↔ |B〉. Theinteraction between the atoms occurs via elastic collisions.

3.1.1 Quantum field model

The Hamiltonian of the system is given by

H0 = HAB +Hint +Hlas, (3.1)

where

HAB =∑

i=A,B

∫d3~x Ψ†

i (~x)

[− h2

2M∇2 +

1

2Mω2

i x2

]Ψi(~x), (3.2)

10 CHAPTER 3. TWO-SPECIES BOSE-EINSTEIN CONDENSATE

Hint =∑

i,j=A,B

∫d3~x Ψ†

i (~x)Ψ†j(~x)

h Uij

2Ψj(~x)Ψi(~x), (3.3)

Hlas = − hΩ2

∫d3~x

[Ψ†

B(~x)ΨA(~x)e−i∆t + Ψ†A(~x)ΨB(~x)ei∆t

]. (3.4)

The Hamiltonian HAB describes the system in the absence of interactions.The first term of HAB refers to the kinetic energy and the second term to thetrapping potential. The frequencies are denoted ωA, ωB for atoms in state |A〉and |B〉, respectively.

The term Hint describes the interactions due to collisions between theatoms. The interaction strength UAA (UBB) characterizes collisions betweenatoms in the same state |A〉, (|B〉), and UAB refers to interspecies collisions.The interaction strengths depend on the scattering length asc

i and the mass

M of the particles like Ui =4πhasc

i

M, i = AA,BB,AB. Throughout this thesis

repulsive interactions are considered, i.e. the scattering lengths (and there-fore the Ui) are assumed to be positive. Furthermore, it is assumed that allcollisions are purely elastic and conserve the internal state.

The Hamiltonian Hlas describes a Raman transition induced by a laser ora microwave field detuned by ∆ from the Raman resonance. This Josephson-like coupling induces a coherent transfer between particles in different internalstates at an effective Rabi frequency Ω > 0.

The operators Ψk(~x), k = A,B, are bosonic field operators that annihilatean atom in the internal state |k〉 at position ~x. They obey the standard bosoniccommutation relations

[Ψk(~x), Ψ†k(~x

′)] = δ(~x− ~x′) (3.5)

and [ΨA(~x), ΨB(~x)] = [Ψ†A(~x), Ψ†

B(~x)] = 0. (3.6)

3.1.2 Two-mode model

The Hamiltonian presented above cannot be solved analytically [18]. However,for low temperatures and a low density of particles [25], the system can bedescribed by a two-mode model, leading to much simpler expressions for theterms of the Hamiltonian (3.1). For this reason it is assumed that the bosonicgas is cooled down to the ground state of the trap, so that the motional degreesof freedom of the atoms are frozen. As a consequence, the spatial degrees offreedom can be described by a single mode function, namely the ground stateφ0(x) of the trap. Therefore, only the dynamics of the internal levels |A〉 and|B〉 are relevant and the field operators are expressed as

ΨA(~x) = φ0(~x) a and ΨB(~x) = φ0(~x) b , (3.7)

where a, b are bosonic annihilation operators that destroy a particle in theinternal state |A〉 and |B〉, respectively. They satisfy the standard bosoniccommutation relations [a, a†] = [b, b†] = 1 and [a, b] = [a†, b†] = 0.

3.2. ANALYSIS OF THE TWO-MODE MODEL 11

This two-mode model simplifies the multimode terms of the Hamiltonian(3.1) to

HAB = hωA a†a+ hωB b

†b, (3.8)

Hint =1

2h UAA a

†a†a a+1

2h UBB b†b†b b+ h UAB a†b†b a (3.9)

Hlas = −h λ ( a†b e−i∆t + b†a ei∆t ), (3.10)

where the coefficients Ui and λ, derived from those in (3.2, 3.3, 3.4) are givenby

Ui =∫d3~x Ui |φ0(~x)|4 i = AA,BB,AB (3.11)

and λ =∫d3~x

Ω

2|φ0(~x)|2 =

Ω

2. (3.12)

For the sake of simplicity it is assumed that the internal states |A〉 and |B〉are degenerate and that the Raman coupling is resonant, that is ∆ = 0. Inaddition, it is assumed that the trapping potential and the scattering lengthdo not depend on the internal state of the atoms.

Thus we set ωAA = ωBB ≡ ω, ascAA = asc

BB ≡ a0 and therefore UAA =UBB ≡ U0. For convenience, one can define asc

AB ≡ a1 and UAB ≡ U1. Sincethe number operator N = a†a+ b†b commutes with the Hamiltonian, the totalnumber of particles is conserved and the term HAB = hωN can be neglected.Under these conditions, the two-mode model Hamiltonian reads

H0 =U0

2(a†a†aa+ b†b†bb) + U1a

†b†ba− λ(a†b+ b†a), (3.13)

where we set h = 1. The Hamiltonian is invariant under the exchange A↔ B.This ideal symmetric situation is barely realizable in a real experiment, sinceatoms in different hyperfine levels experience different Zeeman shifts in themagnetic field, and therefore feel different trapping potentials. In general, ifωA 6= ωB, one can always compensate the potential difference, choosing thedetuning ∆ = ωA − ωB appropriately. For this reason it should be noted thatthese simplifications only have a technical character. In [18] it is mentionedthat by appropriately choosing the detuning ∆, it should also be possible tocompensate for other effects, like the displacement of the traps with respectto each other, due to the different Zeeman interactions, and to gravity.

3.2 Analysis of the two-mode model

Even the two-mode model Hamiltonian H0 (3.13) cannot be solved exactlyby analytical methods [18]. However, for studying the low-energy physics of


H0, a mean field approximation can be used. Within this approximation theground state of the system is studied. In particular, under certain conditionsthe ground state is a Schrodinger-cat state. The Hamiltonian is diagonalizedexactly numerically for N = 200 and the exact numerical results are comparedwith the mean field solutions. Over a wide range of parameters, the resultsare in good agreement.

3.2.1 Mean field approximation

A gas of cold bosonic atoms is a weakly interacting system. Therefore, in orderto obtain an analytical expression for the ground state, a reasonable approachis to use a mean field Ansatz, where it is assumed that in the ground state ofthe N -particle system, all the atoms are in the same single-particle state

|ψ1〉 = α|A〉+ β|B〉. (3.14)

α and β are the probability amplitudes of the states |A〉 and |B〉, respectively.These probability amplitudes α and β satisfy the normalization condition |α|2+|β|2 = 1. Thus, the mean field Ansatz for the ground state is given by thecondensate state

|ψN〉 = |ψ1〉⊗N =1√N !

[αa† + βb†]N |vac〉, (3.15)

where |vac〉 denotes the vacuum state.The variables α and β characterizing the mean field ground state (3.15)

are determined by minimizing the energy of the state |ψN〉 (3.15), i.e. theexpectation value 〈ψN |H0|ψN〉. Depending on the parameters of the system,the following solutions are obtained:

• (a) U1 < U0 : For the case where the A−A and B−B collisions dominatethe collisional interactions, one obtains α = β = 1√

2. Hence, in the

ground state each atom is in an equal superposition between the twointernal states, so that the wave function is given by

|ψ′N〉 =

1√2N N !

[a† + b†]N |vac〉. (3.16)

The corresponding energy is E0 = N2−N4

(U1−U0)− λN .

• (b) U1 > U0 : For the case where the interspecies collisions A − Bdominate the collisional interactions one has to distinguish between tworegimes. They are characterized by Λ ≥ 1 and Λ < 1, respectively, wherethe parameter

Λ =2λ

(N − 1)(U1 − U0)(3.17)

is determined by the relative strengths of the laser coupling and thecollisional interaction.


– (i) Λ ≥ 1: In this case α = β = 1√2

and we again find the mean field

wave function |ψ′N〉 (3.16) with energy E0 as in the case of U1 < U0.

– (ii) Λ < 1: In this case the solution is two-fold degenerate. The twostates are

|ψ+N〉 =

1√N !

[αa† + βb† ]⊗N | 0 〉, (3.18)

|ψ−N〉 =

1√N !

[ βa† + αb† ]⊗N | 0 〉, (3.19)

where α =[

12

(1 +√

1− Λ2)]1/2

, β =[

12

(1−√

1− Λ2)]1/2

.

The corresponding energy is E = N2

[U0(N−1)− λΛ ].

The previous discussion shows that in case (b), U1 > U0, the parameterΛ = 2λ

(N−1)(U1−U0)(3.17) determines the structure of the mean field ground

state. In principle, the scattering lengths and the interaction strengths U0

and U1 are tunable by Feshbach resonances as will be described in chapter 5.However, they are usually kept constant during an experiment. Hence, thecharacteristics of the system can be adjusted by tuning the laser strength λ,and thereby changing the value of Λ.

Throughout the thesis, we will consider the case (b), i.e. U1 > U0, for whichwe demonstrated that under certain conditions, two mean field solutions forthe ground state exist. In chapter 4 we will use a two level system based onthese two states to encode a qubit.

3.2.2 Mean field ground state in spin representation

In this section, the structure of the mean field ground states obtained in theprevious section is discussed in more detail. For this purpose, a spin represen-tation of the two internal states of the atoms is introduced,

|A〉 ←→ | ↑ 〉 and |B〉 ←→ | ↓ 〉, (3.20)

where the spin quantization is relative to the z-axis. In this notation, super-positions of the two internal states of an atom may be pictured as

|α| > |β| : | 〉 = α | ↑ 〉+ β | ↓ 〉 (3.21)

|α| < |β| : | 〉 = α | ↑ 〉+ β | ↓ 〉 (3.22)

α = β : |→ 〉 =1√2| ↑ 〉+ 1√

2| ↓ 〉. (3.23)

The rotation angle of the spin, relative to the z-axis, depends on the relativemagnitudes of the absolute values of the coefficients α and β, that is of therelative weight of the states | ↑ 〉 and | ↓ 〉.


Using this formalism, we can better understand the structure of the meanfield ground states by analyzing the Hamiltonian H0 (3.13) for the differentregimes of Λ in a qualitative way. In order to simplify the discussion, we writehere once more the Hamiltonian (3.13),

H0 =U0

2(a†a†aa+ b†b†bb) + U1a

†b†ba− λ(a†b+ b†a). (3.24)

Let us analyze the following cases:

• Λ = 0: Rewriting the two degenerate mean field ground states |Ψ±〉(3.18, 3.19) in the spin representation, we obtain

|ψ+N〉Λ=0 = | ↑ 〉⊗N = | ↑↑↑ .. ↑ 〉|ψ−

N〉Λ=0 = | ↓ 〉⊗N = | ↓↓↓ .. ↓ 〉

Looking at the Hamiltonian (3.24), it can be easily understood why thesestates are the ground states of the system. For Λ = 0, no laser couplingbetween the two internal states of the atoms occurs. Thus, the lastterm of H0 vanishes and only the collisional interactions determine thestructure of the ground state. Since we are in the regime U1 > U0, theinteraction energy is minimized when all atoms are either in the spin upstate or in the spin down state. Furthermore, since UAA = UBB, thesetwo states must be degenerate.

• Λ ≥ 1: The ground state |Ψ′〉 (3.16) is represented by

|ψ′N〉 = |→ 〉⊗N = | →→→ ..→ 〉. (3.25)

In this case, the Raman coupling between the internal states of the atomsis the dominant term of the Hamiltonian (3.24). The energy of this cou-pling term is minimized if each atom is in an equal superposition between| ↑ 〉 and | ↓ 〉, i.e. |Ψ1〉 = |→ 〉⊗N . This is exactly the configuration real-ized in the ground state.

L

U > U1 0

U < U1 0

N

N

N

N

N

N

1

Figure 3.1: Spin diagram


• 0 < Λ < 1 : Here, the two degenerate mean field ground states |Ψ±〉(3.18, 3.19) rewritten in the spin representation are given by

|ψ+N〉Λ = | 〉⊗N = [α | ↑ 〉+ β | ↓ 〉]⊗N

|ψ−N〉Λ = | 〉⊗N = [β | ↑ 〉+ α | ↓ 〉]⊗N .

The rotation angle of the spin, more precisely α and β, only depends on Λ(3.17). Qualitatively, this configuration of the spins can be understood inthe following way. The parameter Λ characterizes the ratio between thestrengths of the collisional interactions and the laser coupling betweenthe internal states of the atoms. Hence, in the intermediate regime 0 <Λ < 1, the interaction terms of H0 (3.24) compete with the Ramancoupling. The mean field ground state continuously interpolates betweenthe two extreme cases for Λ = 0 and Λ ≥ 1. Starting from Λ = 0 andincreasing the strength of the laser coupling, the spin of each single atomof the ground state is rotated from the positive | ↑ 〉 (or negative | ↓ 〉)z-direction towards the x-axis (|→ 〉).

3.2.3 Beyond the mean field approximation

The mean field solutions discussed in the previous sections are the lowest-energy states fulfilling the mean field Ansatz |ψN〉 = |ψ1〉⊗N (3.15). However,they are not the exact ground states of the system. In this section, a betterapproximation for the ground state is given by making use of the structurefrom the Hamiltonian (3.13). H0 is symmetric in the internal states and thusinvariant under the symmetry operator TAB, which exchanges |A〉 with |B〉.This means that in the case of no degeneracy, the Hamiltonian H0 and thesymmetry operator TAB must share the same set of eigenstates. The eigenval-ues +1 and −1 correspond to states that are symmetric and antisymmetric,respectively, under exchange of the internal levels. Obviously, the two degen-erate mean field ground states |ψ±

N〉 (3.18, 3.19) do not satisfy this condition.Hence, a better approximation for the ground state in the regime Λ < 1, havinga lower energy, can be found by using the wave function

Λ < 1 : |Ψ±〉 ≡ 1

n±(|ψ+

N〉 ± |ψ−N〉), (3.26)

for the variational Ansatz. These superpositions of two degenerate mean fieldsolutions |ψ±

N〉 lead to two orthogonal quasi-degenerate states, which are noweigenstates of TAB. Therefore, the symmetry of the Hamiltonian is now re-flected in the ground state. From now on, we will not write the normalization

factor n± =√

2 (1± 〈ψ+N |ψ−

N〉) =√

2 (1± ΛN) explicitly. 1

1Keep in mind that because of n+ 6= n− we have |Ψ+〉 ± |Ψ−〉 6= |ψ±N 〉 (for Λ 6= 0).


The expectation value of the energy in the states (3.26) is found to be

E± =N2−N

4

2U0− Λ2(U1− U0)± ΛN(3U0 − U1)

1± ΛN. (3.27)

Thus, the energy difference ε separating the two states is given by

ε = E− − E+ = λNΛN−1 1− Λ2

1− Λ2N. (3.28)

Therefore |Ψ+〉 has lower energy than |Ψ−〉, and for Λ 1 the two states |Ψ±〉are quasi-degenerate.

The mean field ground state obtained for strong laser coupling is alreadysymmetric in |A〉 and |B〉, thus we define

Λ ≥ 1 : |Ψ′〉 ≡ |ψ′N〉. (3.29)

3.2.4 Schrodinger-cat states

In this section the structure of the two states |Ψ±〉 (3.26) is analyzed in moredetail. Rewriting them as

|Ψ+〉 = |ψ+1 〉⊗N + |ψ−

1 〉⊗N (3.30)

|Ψ−〉 = |ψ+1 〉⊗N − |ψ−

1 〉⊗N (3.31)

one can easily see that they consist of a superposition of two macroscopic statesin which all atoms are in either the single-particle state |ψ+

1 〉 = α |A〉+ β |B〉,or in the single-particle state |ψ−

1 〉 = β |A〉 + α |B〉. In literature, states ofthis structure are known as Schrodinger-cat states.2 They are characterized bytheir coherent inclusion of macroscopically distinguishable states. Therefore, inorder to be a “good” (that is a macroscopic or at least mesoscopic) Schrodinger-cat state, the two macroscopic states |ψ±

N〉 have to be as different as possible.This requires that their overlap ξ,

ξ = 〈ψ+N |ψ−

N〉 = N⊗〈ψ+1 |ψ−

1 〉⊗N = ΛN , (3.32)

is as small as possible, i.e. ξ 1, or equivalently that the “size of the cat”,which can be defined as 1

ξis as large as possible.

As seen in equation (3.26), it is precisely in this regime of “big” cat states,ΛN 1, that the two states |Ψ±〉 (3.26) are quasi degenerate.

2This expression is derived from the famous Gedankenexperiment of Erwin Schrodingerof 1935 [26], in which he illustrated the problem that arises when applying the quantumsuperposition principle to states of macroscopic systems. The two superposed macroscopicstates in the Gedankenexperiment where a dead and an alive cat.

3.3. EXACT NUMERICAL CALCULATIONS 17

3.3 Exact numerical calculations

In the previous section the low-energy physics of the two-mode HamiltonianH0 (3.13) was investigated by using a mean field approximation and takingadvantage of the symmetry of the Hamiltonian. Now, the obtained analyticalresults are compared with the exact results of a numerical diagonalization ofH0. For this reason the structure of the exact lowest-energy eigenstates of theHamiltonian is discussed as the system is driven across the phase transitionto the Schrodinger-cat phase. We show, that outside the transition region,the analytic expressions obtained in the previous section are very satisfyingapproximations. Through the analysis of the energy spectrum, an insight intothe structure of the higher excited states can also be gained.In all the examples of this section, the number of atoms is fixed at N = 200,and the ratio of the binary interaction strengths is taken U1

U0

= 3.

3.3.1 Eigenstates

For the numerical calculation of the eigensystem of the Hamiltonian, theHamiltonian is represented in the Fock basis |N − n〉A ⊗ |n〉B , n= 0..N ,where each state is characterized by the number of particles in the internalstates |A〉 and |B〉. In this basis H0 is a real symmetric tridiagonal matrixof dimension N+1 × N+1. Thus, for a fixed number of particles, it can bediagonalized by numerical methods.

Let us denote the eigenstates of the Hamiltonian by

|Ψi〉 =N∑

n=0

cin |N− n〉A ⊗ |n〉B, (3.33)

with real3 coefficients cin and the corresponding energies Ei (i = 0, 1, .., N andE0 ≤ E1 ≤ .. ≤ EN). For the discussion, it is convenient to also express theapproximate mean field states |Ψ+〉, |Ψ−〉 (3.26) and |Ψ′〉 (3.29) in the Fockbasis. Then the index i in (3.33) additionally runs over +,− and ′. Usingthis notation the validity of the approximated solutions is investigated and thestructure of the lowest eigenstates of H0 is discussed. The results are presentedin figures. 3.2-3.4.

The mean field calculations done in the previous section predict a phasetransition for Λ = 1 from a phase where the two lowest-energy states areSchrodinger-cat states (Λ < 1), to a phase with no macroscopic superposi-tions of the ground state (Λ > 1). Figure 3.2 depicts the B-atom numberdistribution for the ground state |Ψ0〉 as a function of Λ. The Schrodinger-catstructure is observed, as well as the phase transition with a transition point

3Since H = H+ and Hij ∈ R, ∀i, j ∈ 0..N, we can always choose the eigenvectors tobe real: Hv = λv ⇒ Hv∗ = λv∗ ⇒ w ≡ v + v∗; Hw = λw and wi ∈ R


of Λt ≈ 0.97 for N = 200. Therefore, apart from the shifted transition point,the structure of the lowest-energy states is well predicted by the states (3.26,3.29) calculated by use of the improved mean field approximation.

Of central importance in the derivation of the approximated states wasthe A-B symmetry of the Hamiltonian (3.13), i.e. the invariance of H0 underthe exchange of the internal states |A〉 ↔ |B〉. This symmetry of the exacteigentstate |Ψ0〉 and |Ψ1〉 is depicted in figure 3.3. As for the mean field solu-tions, the ground state is symmetric and the first excited state is antisymmetricunder exchange of |A〉 and |B〉.

Figure 3.4 demonstrates the overlap between the mean field states |Ψ±〉,|Ψ′〉 and the exact numerical eigenstates as a function of Λ. It is observed,that outside the transition region Λ ≈ Λt, the states are in good agreement, sothat the mean field states (3.26, 3.29) are good approximations for the exacteigenstates of the Hamiltonian (3.13),

Λt 1 : |Ψ±〉 ≈ |Ψ0

1〉 (3.34)

Λt 1 : |Ψ ′ 〉 ≈ |Ψ0〉. (3.35)

However, the mean field approximation is not valid in the vicinity of the transi-tion point. This can be understood by the fact that the state structure obtainedby the mean field approximation should be valid in the limit N →∞, whereasthe numerical calculations are computed for a finite number of particles. As aconsequence, the phase transition occurs already for some Λt < 1 and not atΛt = 1, as predicted by the mean field theory (3.26, 3.29).

3.3.2 Energy spectrum

Figure 3.5 presents the numerically calculated low-energy part of the spectrumas a function of Λ. For Λ below a critical value, the energies E0 and E1 mergeand the two lowest states in energy become quasi-degenerate, as predicted forthe transition to the Schrodinger-cat phase in the previous section.

In order to analyze the energy spectrum, as well as the structure of theexcited states, in more detail the limiting cases of Λ = 0 and Λ Λt areconsidered.

• For Λ = 0, the two lowest states in energy are degenerate. In addition,the cat states are well separated from the rest of the spectrum. This canbe understood from the structure of the two states. For Λ = 0, the systemis in an equal superposition of the two states with either all atoms in thestate spin up, or all in the state spin down, |Ψ±〉 = | ↑ 〉⊗N ± | ↓ 〉⊗N . Apossibility for an excitation of such a state is the flip of a spin, | ↑〉 ↔ | ↓〉.The energy one has to pay for that is ∆0 = (N−1)(U1−U0). In fact,this is exactly the same energy found by the numerical calculation of


0 100 200

n

L = 0.1

L = 0.5

L = 0.7

L = 0.95

Lt = 0.97

L = 2

N

N

N N

N N

N N

N N

n

L

00

200

2

100

1Lt

a)

b)

Figure 3.2: B-atom number distributions for the ground state (i.e., the coeffi-cients |c0n|2 from the decomposition of |Ψ0〉 in the Fock basis (3.33)) forN = 200(a) as a function of n and Λ, and (b) as a function of n and the values of Λ in-dicated. Λt = 0.97 indicates the transition point to the Schrodinger-cat phasefor Λ < Λt. The quantities plotted are dimensionless.


0 100 200

0

0 100 200

0

nn

a) b)

Figure 3.3: (a) B-atom number distribution for the ground state (i.e., thecoefficients c0n from the decomposition of |Ψ0〉 in the Fock basis (3.33)) as afunction of n for N = 200 and Λ = 0.5. (b) Same as (a), but for the firstexcited state (i.e., c1n). The quantities plotted are dimensionless.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.0

0.2

0.4

0.6

0.8

1.0

q

q0

q1

L

Figure 3.4: Overlap q = |〈Ψnum|Ψmean〉|2 of exact numerically calculated stateswith the approximated mean field states: overlap q0 of |Ψ0〉 with |Ψ+〉 (Λ < 1)and |Ψ′〉 (Λ ≥ 1), respectively, and the overlap q1 of |Ψ1〉 with |Ψ−〉 (Λ < 1) asa function of Λ for N = 200. For Λ ≥ 1, no mean field approximation existsfor the first excited state.


0.0 0.5 1.0 1.5 2.00

500

1000

1500

E-E

0(U

0)

L

E5-E

0

E4-E

0

E3-E

0

E2-E

0

E1-E

0

Figure 3.5: Low-energy part of the energy spectrum of the two-mode Hamil-tonian H0 (3.13) as a function of Λ. All energies are measured with respect tothe ground state energy.

E2 − E1.4 Thus, the second and third excited states emerge from the

ground and first excited state, respectively, by the flip of one spin,

|Ψ2

3〉 = | ↓ 〉| ↑ 〉⊗N−1± | ↑ 〉| ↓ 〉⊗N−1, (3.36)

The next two excited states (|Ψ4

5〉) emerge from a second spin flip andare energetically raised in the spectrum by (N−2)(U1−U0), etc. Thesespin-flip excitations also explain why the merging of the energy levelsfor Λ < Λt occurs within consecutive pairs of levels (see figure 3.5).Note that the A-B symmetry of the states |Ψ0〉 and |Ψ1〉 is due to theirSchrodinger-cat structure. Therefore, the states arising from the groundstate are all symmetric, and the states arising from the first excited stateare all antisymmetric under exchange of the internal levels.

• In the region where Λ Λt, the energy difference between consecutivelevels increases linearly with Λ. As in the case Λ = 0, the structure ofthe spectrum can be explained by assuming the excitations to be spinflips. As described in section 3.2, this regime is dominated by the Ramancoupling of the internal atoic states, and the ground state of the systemis found to be |→ 〉⊗N , where the spin of each atom is pointing towardsthe x direction. By flipping one spin of this state, | →〉 ↔ | ←〉, thefirst excited state is reached, |←〉|→〉⊗N−1. This state has an increased

4For Λ small (Λ ≤ Λt) the energy gap E2−E1 can be approximated by ∆ ≈ ∆0

√1− Λ2.


energy of ε = E1−E0 ≈ Λ (N−1)(U1−U0) = 2λ with respect to theground state. The next excited states are constructed analogously andit is observed that the system as a whole behaves like a ferromagnet.

3.4 Summary

In this chapter Bose-Einstein condensation of a trapped gas of bosonic atomswas described, where the two internal states of the atoms are coupled by aRaman laser field. The low-energy physics of the system was analyzed numer-ically and by a mean field approximation improved by symmetry arguments.Hereby, it was focused on the case where the collisional interaction betweenatoms in different internal states dominates the one between atoms in the sameinternal state (U1 > U0). It was demonstrated for Λ < Λt that a phase tran-sition to a Schrodinger-cat phase occurs, where Λ is a measure at the relativestrength for the coupling and for the collisional interactions. In this regimethe two lowest-energy states are formed by a superposition of two condensatestates, that is, they are Schrodinger-cat states. They are quasi-degenerate andwell separated from the rest of the spectrum. The approximated results arein good agreement with the exact numerical solutions. Therefore analyticalexpressions are available for the low-energy physics of the system.

23

Chapter 4

Scheme for quantum

computation

In this chapter we present a scheme for quantum computation, using the idealtwo-species Bose-Einstein condensate, as analyzed in the previous chapter. Aswe have seen, under certain conditions the system is confined to a subspace oftwo states. In the first section of this chapter we discuss how these two statescan be used to encode a qubit: the basic element of any quantum computationscheme. In section two we discuss how to manipulate the state of this qubit ina controlled manner We show how to realize one-qubit gates as well as a two-qubit entangling gate. Together, these form a universal set of quantum gates.As an example, we describe the time evolution of the system that results fromusing the gates to create a maximally entangled singlet state.

If not mentioned explicitely, we assume throughout the chapter that we arein the regime of the Schrodinger-cat phase, that is Λ < Λt and U1 > U0.

4.1 Encoding a qubit

The basic element of quantum computation is a two-level system: a qubit. Inorder to be able to encode a qubit, we must identify a physical system thatacts as an effective two-level system.

We saw in the previous chapter that, under certain conditions (U1 >U0, Λ < Λt), the two lowest-energy states of an interacting two-componentBose-Einstein condensate are quasi-degenerate and well separated in energyfrom the rest of the spectrum. Therefore, as long as perturbations are muchsmaller than this energy gap, the subspace spanned by these two state is pro-tected against perturbations. Thus, within the regime Λ < Λt the Hilbert-space of the system can be projected onto this two-dimensional subspace, de-noted by Υ0 = span|Ψ0〉, |Ψ1〉.

24 CHAPTER 4. SCHEME FOR QUANTUM COMPUTATION

N N

-

N N

+

1

0of a QUBITencode

Figure 4.1: Encoding a qubit.

The states |Ψ0〉 and |Ψ1〉 are entangled many-body Schrodinger-cat states.However, we will not investigate these states themselves, but, as illustrated infigure 4.1, use them to encode the two states | 0 〉 and | 1 〉 of a qubit.

Following the usual notation of quantum computation, we denote the twoqubit states by

| 0 〉 ≡ |Ψ0〉 and | 1 〉 ≡ |Ψ1〉. (4.1)

Projecting the Hamiltonian H0 (3.13) onto the subspace Υ0 of the twoqubit states, we derive the effective Hamiltonian

H0 = P H0 P = E0 |Ψ0〉〈Ψ0|+ E1 |Ψ1〉〈Ψ1|, (4.2)

where E0 and E1 are the exact eigenvalues of |Ψ0〉 and |Ψ1〉. P = |Ψ0〉〈Ψ0| +|Ψ1〉〈Ψ1| is the projection operator that projects onto Υ0, fulfilling P 2 = P .Operators projected onto the subspace Υ0 are written as O = POP .

Expanding the Hamiltonian (4.2) in the Pauli basis 11, σi, i = x, y, z(where the Pauli matrices are defined in the |Ψ0〉, |Ψ1〉 basis), H0 is givenby

H0 =E1+E0

211− E1−E0

2σz. (4.3)

Now, by substituting the exact qubit states |Ψ0〉 and |Ψ1〉, by their approxi-mations |Ψ+〉 and |Ψ−〉 (3.31), respectively, we obtain an analytical expressionfor the effective Hamiltonian,

H0 =E++E−

211− ε

2σz, (4.4)

where ε = λNΛN−1 1−Λ2

1−Λ2N is the energy splitting between the two qubit states.For Λ < Λt, the two states are quasi-degenerate so that we neglect the termproportional to σz. Therefore, as it evolves under the action of the resultingtrivial Hamiltonian, the qubit simply acquires an unobservable global phasethat we neglect.


4.2 Quantum operations

In this section we first show how to realize single-qubit operations, rotating thequbit state about the x- and the z-axis of the Bloch sphere. For this purpose weinduce energy shifts on the atomic levels by adding a state-dependent potentialto the Hamiltonian and adiabatically change the Raman coupling betweenatomic states. In addition, we show that by allowing tunneling between pairsof condensate systems, the states of the corresponding qubits can be entangled.Together, these operations form a universal set of quantum gates allowingarbitrary computation.

4.2.1 Single-qubit gate via external potential

The first one-qubit operation we present generates an x-rotation in the Bloch-sphere.

State dependent potential

In order to control the time evolution of the qubit, we assume that the trap-ping potential depends on the internal states |A〉 and |B〉 of the atoms. Thisadditional potential raises the energy of one of the internal states with respectto the other, so that the degeneracy of the internal states is lifted. Experimen-tally this can be realized by applying an external magnetic or electric field,so that the Zeemann effect or Stark effect, respectively, causes the degeneratehyperfine levels to split and thus they feel different trapping potentials. Fig-ure 4.2 illustrates this situation with a raised potential for atoms in the state|B〉 = | ↓ 〉.

B = 0

V

B 0¹

a) b)

Figure 4.2: Magnetic field dependency of the trapping potential for the twodifferent internal states of the atoms.

This state dependent perturbation is reflected in the addition of a potential


V to the Hamiltonian H0 (3.13) of the unperturbed system,

H = H0 + V, (4.5)

where V = µ b†b raises (or lowers) the potential for atoms in state |B〉. Theperturbation strength µ depends on the strength of the applied external mag-netic field.

Provided the perturbation does not couple the qubit subspace Υ0 to higherexcited levels, the system still acts as a two-level system, and we may projectthe perturbation term V onto Υ0. Approximating the exact eigenstates by theman-field solutions |Ψ±〉 (3.26), the projection can be expressed analytically:

V = µ b†b =Ωx

2σx, (4.6)

where Ωx = µN

√1− Λ2

1− Λ2N. (4.7)

Since V is proportional to σx, the state dependent potential generates a rota-tion of the qubit about the x-axis of the Bloch sphere with frequency Ωx. Thisrotation is described by the time evolution operator

U(t) =

(cos Ωx

2t −i sin Ωx

2t

−i sin Ωx

2t cos Ωx

2t

). (4.8)

Quantum gate

Choosing an appropriate rotation angle φ = Ωxtφ (by appropriately choosingthe duration t of the perturbation), a set of quantum gates based on therotation about the x-axis can be implemented,

Rx(φ) = U(tφ) =

(cos φ

2−i sin φ

2

−i sin φ2

cos φ2

). (4.9)

Examples are (0 11 0

)and

(1 ii 1

), (4.10)

where the first gate (φ = π) known as the NOT-gate, negates the state of thequbit, and the second one (φ = 3

2π) creates superpositions of | 0 〉 and | 1 〉.

Note that the timescale of the gates is determined by the rotation frequencyΩx (4.7). Since Ωx is proportional to N , for a macroscopic number of particlesthe gates can be made very fast.

In order to understand why the state dependent perturbation generates anx-rotation, let us analyze as an example the effect of the additional potentialon the time evolution of a qubit initially in the state | 0 〉 = | ↑ 〉⊗N + | ↓ 〉⊗N

(Λ = 0). During the evolution, the state with the highest energy, i.e. | ↓ 〉, will


acquire an additional phase as compared with the state of lowest energy, i.e.| ↑ 〉. Therefore, the initial state | 0 〉 evolves as

| 0 〉 = | ↑ 〉⊗N+ | ↓ 〉⊗N −→ | ↑ 〉⊗N+ e−iφt | ↓ 〉⊗N −→ | 1 〉 = | ↑ 〉⊗N− | ↓ 〉⊗N ...(4.11)

where the negation of the initial state (| 0 〉 → | 1 〉) for φt = π corresponds tothe NOT gate. In the basis | 0 〉, | 1 〉 the evolution (4.11) is equivalent, upto a global phase, to the x-rotation described by U(t) (4.8).

Validity of the projection

For the projection of the HamiltonianH = H0+V (4.5) onto the qubit subspaceΥ0 to be valid, the perturbation V must only weakly couple the subspace Υ0

to higher excited levels. This is satisfied for perturbations much smaller thanthe energy gap separating Υ0 from the rest of the spectrum. In section 3.3.2we found that in the regime Λ Λt, the gap scales like N−1. Thus, thetransition probability P|0〉→|2〉 for the excitation |0〉→|2〉, computed accordingto Fermi’s golden rule, has to fulfill

P Fermi|0〉→|2〉 =

|〈0 |V | 2〉|2(E2−E0)2/4

≈ µ2 Λ2

(1−Λ2)(U1−U0)2

N

(N−1)2 1, (4.12)

where the matrix element 〈0 |V |2〉 is computed numerically. Therefore, we canalways find a perturbation strenth µ such that the projection is valid.

Fidelity

Given the condition (4.12), we expect that the time evolution of the qubitunder the perturbation V is well described by the time evolution operator U(t)(4.8), obtained by a projection onto the mean field states |Ψ±〉 (3.26). We nowcompare this approximated time evolution with the exact solutions obtainedby numerical diagonalization of the two-mode model Hamiltonian H = H0 +V(4.5). Figure 4.3 shows the exact numerically calculated time evolution of theinitial state |Ψ0〉 under the Hamiltonian (4.5) for a particle number of N = 200.In order to judge the coincidence of this evolution with the evolution predictedby the time evolution operator U(t) (4.8) representing the quantum gate, wecompute the fidelity, defined as the overlap between two states.1 It takes thevalue 1 for states that are equal to each other and 0 for orthogonal states. In ourcase, the fidelity is given by the overlap between the expected wave-function|Ψideal(t)〉, described by the approximated time evolution operator U(t) (4.8)defining the quantum gate, and the wave-function |Ψexact(t)〉 obtained by exactnumerical calculations considering the full two-mode model,

F (t) = |〈Ψideal(t)|Ψexact(t)〉|2. (4.13)

1To be exact, it is defined as the square of the absolute value of the overlap.


Figure 4.3 shows also the fidelity as a function of time. It has almost themaximum value of 1. Thus, as expected, the time evolution of the qubit, thatis the quantum operations based on the spin-dependent perturbation V , iswell-described by the time evolution operator U(t) (4.8).

0.0 0.5 1.0 1.5 2.0 2.5 3.00.0

0.2

0.4

0.6

0.8

1.0

p1

p

t x 104

(1/U0)

p0

F

Figure 4.3: Exact time evolution of the initial state |Ψ0〉 under the state de-pendent potential. The occupation probabilities p0 and p1 of the qubit states|Ψ0〉 and |Ψ1〉, respectively, and the fidelity F (t) for N = 200, Λ = 0.1 andµU0

= 100 as a function of time. Times tπ and t 3

2π, where the gates (4.10) are

realized are emphasized in the plot.

Conclusion

To conclude we have shown that a set of quantum gates, based on rotationabout the x-axis, e.g. a NOT gate, can be implemented by changing the po-tential of one of the internal states |A〉 or |B〉. The quantum operations arewell-described by the analytical expression (4.8).

4.2.2 Single-qubit gate via adiabatic phase

Any arbitrary unitary operation on a single qubit can be decomposed intosuccessive rotations about two non-parallel axes (see chapter 2). As a secondone-qubit operation, we show how to perform a rotation about the z-axis, thatis a phase-gate.


Implementing a phase-gate

For the two states |Ψ0〉 and |Ψ1〉 to acquire a relative phase, they must beseparated in energy. In the Λ Λt regime, we know that this is not the casesince the two states are quasi-degenerate, i.e. the energy difference betweenthem is much smaller than any timescale of the system, thus does not lead toan appreciable relative phase. However, in section 3.3.2, when analyzing theenergy spectrum of the system, we saw that there exists a regime of Λ, namelyΛ ≥ Λt, in which these two states are separated in energy (see figure 3.5).

To construct a phase-gate, let us first assume that the system is in thisΛ > Λt regime, and analyze the time evolution of the eigenstates |Ψ0

1〉. Sincewe are considering free evolution, even in this regime of Λ, where the twolowest-energy states are not well separated from the rest of the spectrum, weare allowed to project the Hamiltonian onto the subspace Υ0 of these twoeigenstates. Therefore, the free Hamiltonian is given by (4.3),

H0 = −ε(Λ)

2σz, (4.14)

where ε = E1−E0 denotes the exact energy gap separating the two states at agiven value of Λ.2 The time evolution operator resulting from this Hamiltonianis

U0(t) = ei

2

∫ε(Λ)σzdt =

(1 00 eiφ

). (4.15)

Thus, in the regime Λ > Λt, the first excited |Ψ1〉 acquires a relative phaseφ = − ∫ ε(Λ)dt, corresponding to a rotation about the z-axis by an angle φ.

To exploit the time evolution (4.15) for implementing a quantum gate, thesystem has to be transferred from the initial regime Λ = Λ0 Λt (where|Ψ0

1〉 are quasi-degenerate Schrodinger-cat states), to the regime Λ ≥ Λt. Thisis realized by adiabatically increasing the parameter Λ. Experimentally thiscorresponds to appropriately tuning the laser strength λ which is proportionalto Λ. Under the condition of adiabaticity, we can assume, as we will explainlater, that if the system starts in the ground state it remains in the groundstate during the process. The system is kept in the Λ > Λt regime until thestate |Ψ1〉 has acquired the desired relative phase, and is then transferred backto the initial regime. Following this scheme, a phase-gate is realized, given by

Rz(φ) =

(1 00 eiφ

), (4.16)

where φ is now the phase accumulated during the process, φ = − ∫ ε(Λ(t))dt.For instance, a qubit initially in an equal superposition of the basis states,

2Note that ε = E1 − E0 is the exact numerical expression, whereas ε = E−E+ denotesthe mean field energy gap in the regime Λ < Λt.


evolves as

| 0 〉Λ0+ | 1 〉Λ0

−→ | 0 〉Λ + eiφ(t)| 1 〉Λ −→ | 0 〉Λ0+ eiφ| 1 〉Λ0

.3 (4.17)

Condition for adiabaticity

For this phase-gate to work, we have to ensure that the system remains withinthe two-level subspace during the whole evolution. This can be achieved byobeying the adiabatic theorem [27]. It states that, if the Hamiltonian of a sys-tem is changed sufficiently slowly, i.e. adiabatically, a system starting in theground state remains in the ground state (now corresponding to the Hamilto-nian of the new parameters), etc. Hence, the system can be transferred fromone regime of parameters to another one, H → H ′, without inducing transitionbetween the states. Thus, even when “perturbing” the system adiabatically,the projection (4.14) is still valid.

The condition for adiabaticity is given by

|t〈j|∂H(t)∂t|i〉t|

Ej − Ei

Ej − Ei

h, (4.18)

where |i〉t, |j〉t are eigenstates of the Hamiltonian at time t. Physically, the lefthand side is a measure of the change of the eigenvectors of the Hamiltonianwith time. It has to be much smaller than the frequency of the transition|i〉 → |j〉, described by the right hand side.

In implementing the phase-gate following the scheme presented above,equation (4.18) becomes a condition on Λ. Since the equation depends onthe energy gap Ej − Ei, it is reasonable to distinguish between three regionsof Λ in the energy spectrum (see figure (3.5)).4 Firstly, the Λ < Λt regime,with a large energy gap between the qubit subspace Υ0 and the higher excitedlevels, which only slowly varies with Λ. Secondly, the regime Λ → Λt witha small, quickly varying energy gap. Partly evaluated numerically, condition(4.18) can be estimated within these two regions by

Λ < Λt : | Λ | 2 (1−Λ2)(U1 − U0)√N − 1,

Λ→ Λt : | Λ | f(N) (U1 − U0),(4.19)

where the function f(N) slightly decreases with N, f(4) = 0.6, f(10) =0.5, f(1000) ≈ 0.01. We note that the first inequality becomes less restrictivefor increasing number of particles, whereas in the second case, Λ only smoothlydecreases with N . In the third regime, Λ > Λt, numerical calculation shows

3Do not confuse (4.17) with the example (4.11) given for the x-rotation: an x-rotation inthe | 0 〉, | 1 〉 basis corresponds to an z-rotation in the | ↑ 〉⊗N , | ↓ 〉⊗N basis.

4Note that since in our case [H0(t), TA,B ] = 0, with TA,B the symmetry operator thatexchanges |A〉 and |B〉, only states with the same symmetry as the initial eigenstate can beaccessed. Thus no transition | 0 〉 ↔ | 1 〉 can occur.


that the transition amplitude for excitation is negligible compared to the othercases. Thus, for large N , the timescale of the gate is primarily limited by theadiabatic evolution near Λt.

Fidelity

If the conditions (4.19) for Λ are obeyed, we expect the time evolution ofthe qubit to be well described by the time evolution operator U0(t) (4.15),i.e. no transitions to higher levels occur. We now present, as an examplefor the implementation of the adiabatic phase-gate, the time evolution of thequbit state | 0 〉 + | 1 〉 for the acquired relative phase φ = π. Calculationsare done by an exact diagonalization of the full two-mode model HamiltonianH0(Λ) (3.13). To compare the results with the evolution predicted by the timeevolution operator (4.15) describing the phase-gate, also the fidelity (4.13)is calculated. The evolution under the adiabatic phase-gate is presented infigure 4.4: For Λ ≈ Λt the quasi-degeneracy of the two lowest-energy states islifted and the first excited state acquires a relative phase. Furthermore, theoccupation probability of the qubit states as well as the fidelity are plotted asa function of time. Even for a factor five used to obey the conditions (4.19),the fidelity does not decrease significantly.

Summary

We conclude that by adiabatically transferring the system back and forth fromΛ0 < Λt to a regime where the two qubit states are separated in energy, anadiabatic phase-gate can be implemented. The adiabatic conditions are givenby (4.19), and the transformation of the state of the qubit can be calculatedfrom (4.16).

4.2.3 Two-qubit gate via tunneling

We have presented two sets of one-qubit operations that together allow the ar-bitrary manipulation of the state of a single qubit. A universal set of quantumgates additionally requires a non-trivial two-qubit gate to allow entanglementcreation. In this section, we present a quantum gate that can generate maxi-mally entangled states of two qubits. The interaction is mediated via secondorder tunneling processes between two adjacent traps.

Two-qubit problem

Consider two systems as described in chapter 3, characterized by the same pa-rameters U1, U0, Λ and by the same number of particles, N . The experimental


0.0 0.5 1.00.0

0.5

1.0

0.0 0.5 1.0

0.0 0.5 1.00.0

0.5

1.0

0

10

20

p

t (1/U0)

F

p0

p1

F

t (1/U0)

0

-p/2

L

L

t (1/U0)

-p

e e=

E1-

E0

(U0)

a)

b)

c)

Figure 4.4: Adiabatic phase-gate for phase φ = π, with Λ0 = 0.1 and N = 200for the initial qubit state |Ψ0〉 + |Ψ1〉. Λ is chosen to obey the conditions(4.19) by a factor of five and is taken constant within the two regimes Λ <0.85, Λ ≥ 0.85. Computation is done by numerical diagonalization of theHamiltonian. (a) Λ and the corresponding energy splitting ε, (b) the relativephase φ(t) = − ∫ ε(Λ(t))dt and (c) the exact occupation probabilities p0 andp1 of the states |Ψ0〉Λ and |Ψ1〉Λ, respectively, as well as the fidelity F (t) (4.13)of the exact evolution due to (4.15), as a function of time.


realization of such an equal filling of two traps will be discussed in chapter 5.Each system represents a qubit.

Controlled interactions between two qubits can be achieved by enablingtunneling between the systems. This can be realized e.g. by lowering the trap-ping potential of an optical lattice [16] or by bringing two microtraps confiningthe atoms close to each other [28]. The two-mode Hamiltonian describing thishas the form

HT = H01 +H02 + T (4.20)

Here, H0i is the Hamiltonian (3.13) describing the dynamics of each singletrap, i = 1, 2, and T is the tunneling term

T = −J (a†1a2 + a†2a1 + b†1b2 + b†2b1), (4.21)

which accounts for the tunneling of atoms from one trap to the other, conserv-ing their internal state. The subscripts 1, 2 identify the trap.

Because of the tunneling between the two systems the number of particlesin each trap is no longer conserved, thus the qubits are in general not welldefined. However, the ratio of tunneling to collisional interactions J/U0, canbe chosen so small that the dynamics of the system are dominated by secondorder tunneling processes, which do not change the number of particles of thesubsystems.

We show now that for J sufficiently small, these second order processes cre-ate entanglement between the qubits in the different traps, i.e. that a universaltwo-qubit gate can be implemented.

Energy spectrum

In order to analyze the tunneling process and to find an appropriate projectionof the two-mode Hamiltonian HT , let us first discuss the energy spectrum ofthe unperturbed system (J = 0) of two traps, with total number of particlesK = 2N , as shown in figure 4.5.

Since no tunneling is allowed, the spectrum is composed of the unperturbedspectra of the two traps for the different possible number distributions N1 +N2 = K. As we have seen in section 3.3.2, the lowest-energy subspace Υ0 ofa single system is, for Λ Λt, well separated from the rest of the spectrum.Therefore, the low-energy part of the spectrum of two traps is composed ofthese one-trap subspaces Υ0

Ni= |Ψ0〉, |Ψ1〉Ni

, where Ni is the number ofparticles in trap i = 1, 2. For Λ = 0 it is easy to verify that the subspace ofequal filling, Υ = Υ0

N ⊗ Υ0N , is lowest in energy. The first excited subspace

Υ1 is characterized by unequal trap filling N + 1 and N − 1. It is separatedfrom Υ by the interaction energy U0. Since the problem is symmetric in thetraps 1 and 2, Υ1 consists of 8 states, Υ1 = |Ψ0〉, |Ψ1〉N±1⊗|Ψ0〉, |Ψ1〉N∓1.The second excited subspace in turn is raised by 3U0 with respect to Υ1. It isdefined by the one-qubit states of N ± 2 atoms, etc.


U0

N, N

N-1, N+1 N+1, N-1

N+2, N-2

3U0

E

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

1

2

3

4

5

6

7

8

9

10

E-E

0(U

0)

L

a)

b)

Figure 4.5: Low-energy part of the spectrum of the two-trap system (a) as afunction of Λ with respect to the ground state, (b) schematically for Λ = 0.


First order tunneling processes lead to transitions from the two-qubit sub-space Υ to excited states, Υ −→ Υ1. Clearly, such a change in the numberof particles per trap is not desirable. However, choosing the ratio J/U0 suffi-ciently small, transitions to excited levels become negligible and the dynamicsof the system takes place within the qubit subspace Υ, driven by second orderprocesses via Υ1.

Projection

We can introduce an effective Hamiltonian to account for the dynamics in atruncated Hilbert space, by projecting the system onto the subspaces Υ andΥ1 (see figure 4.5).5 Second order processes via Υ1 are then automaticallyincluded. In order to derive a simple analytical expression for the effectiveHamiltonian, we again approximate the states |Ψ0

1〉 by the corresponding meanfield solutions |Ψ±〉 (3.26), and use Λ ≡ Λ(N) = Λ(N ± 1), ΛN ≡ ΛN±1 andε = 0.

The effective Hamiltonian resulting from the full two-mode Hamiltonian(4.20) can be written in the form:

HT = PHTP =

W V 0V 0 V0 V W

, (4.22)

where P is the operator projecting onto Υ⊕Υ1. The 3× 3 block structure ofHT results from the different couplings of the three subspaces Υ1

N+1,N−1, ΥN,N

(the qubit subspace) and Υ1N−1,N+1 (see figure 4.5). Each of these subspaces

is spanned by the four states |Ψ+Ψ+〉, |Ψ+Ψ−〉, |Ψ−Ψ+〉 and |Ψ−Ψ−〉, whichdepend on the number distribution of the corresponding subspace. Hence, HT

is a 12× 12 matrix.

The matrices on the diagonal determine the energy of the unperturbedsubspaces. Setting the energy of the states in Υ to zero (recall that we neglectthe energy splitting ε within the subspaces), W is given by

W = Γ11, (4.23)

where Γ = EΥ1 − EΥ is the energy difference separating Υ1 from Υ (Γ = U0

for Λ = 0). Since T does not couple the states within the subspaces, W is a4× 4 diagonal matrix.

5The dimension of the Hilbert space of the double system grows very quickly with in-creasing number of particles, dim=

∑2N

m=0(m + 1)(2N −m + 1). Thus, not only for a nice

description of the processes, but also to allow the simulation of the time evolution for ar-bitrary number of particles, an appropriate projection of the Hamiltonian HT (4.20) has tobe found.


The matrix V represents the tunneling of one particle, i.e. the couplingbetween Υ and Υ1,

V = −J2

√N(N + 1)

1 + Λ 0 0 1− Λ0 1 + Λ 1− Λ 00 1− Λ 1 + Λ 0

1− Λ 0 0 1 + Λ

. (4.24)

T does not couple Υ1N+1,N−1 and Υ1

N−1,N+1.

In order to judge the validity of this projection, figure 4.6 shows both thetime evolution due to the full two-mode HamiltonianHT (4.20), and that underthe effective Hamiltonian HT (4.22) for N = 6 atoms each qubit. Even forthis small number of particles, the results are in good agreement. Therefore,the projection leads to a satisfying description of the system.

0 10 20 30 40 50 60 70 80 90 1000.0

0.2

0.4

0.6

0.8

1.0

pexact

00

p00

p

t (1/U0)

pexact

11

p11

Figure 4.6: Time evolution of the two-qubit system for N = 6, Λ = 0.1 andJ/U0 = 0.032 starting from the ground state |Ψ0Ψ0〉. Occupation probabilitiesp00 and p11 of the states |Ψ0Ψ0〉 and |Ψ1Ψ1〉, respectively, as a function oftime. Superscript indicates exact diagonalisation of the full two-mode modelHamilttonian HT (4.20), without superscript indicates the time evolution dueto the effective Hamiltonian HT (4.22).

Time evolution

We use the effective Hamiltonian HT (4.22) for further investigation of thetime evolution of the two-qubit system. Figure 4.7 shows the time evolutionaccording to UT = e−iHT t for N = 200, starting from the ground state of the


0 20 40 60 80 1000.0

0.5

1.0

0 20 40 60 80 1000.0

0.5

1.0

0 20 40 60 80 1000.0

0.5

1.0

p

t (1/U0)

Pou

t

t (1/U0)

p01

+p10

p00

p11

pt (1/U

0)

p11p

01+p

10

p00

a)

b)

c)

Figure 4.7: Time evolution of the system for N = 200, Λ = 0.1 and A = 0.1(J/U0 = 8.8 × 10−4) of the initial state |Ψ0Ψ0〉. pij, i, j = 0, 1 denotes theoccupation probability of the state |ΨiΨj〉 of the qubit subspace Υ. (a) Time

evolution under UT = e−iHT t, (b) probability Pout(t) (4.25) and (c) dynamicsdue to the slow second order processes described by U(t) (4.27) as a functionof time.

total system |Ψ0Ψ0〉. The evolution of the qubit states is governed by twodifferent timescales:

• The fast oscillation superimposed on the slow dynamics are due to firstorder tunneling processes. Since we are in a regime of weak tunneling,these processes can be understood as far off-resonant Rabi-like oscillationbetween the two subspaces, Υ ↔ Υ1. The total occupation probabilityPout of the excited subspace Υ1 is shown in figure 4.7. It can be extractedfrom UT to be

Pout ≈ A sin2 ω

2t, (4.25)

where the amplitude A and the oscillation frequency ω are given by

A = 4J2N(N + 1)

Γ2 + 8J2N(N + 1)and ω =

√Γ2 + 8J2N(N + 1). (4.26)


• The dominant contribution to the time evolution of the qubit states isdue to slow second order transitions via the excited subspace Υ1. Straightforward algebra shows that for Λ = 0, the time evolution operator de-scribing this dynamics is given by

U(t) =

cos Ω2t 0 0 i sin Ω

2t

0 cos Ω2t i sin Ω

2t 0

0 i sin Ω2t cos Ω

2t 0

i sin Ω2t 0 0 cos Ω

2t

, (4.27)

with the frequency Ω = ω−Γ2

. Numerical calculation shows that for 0 ≤Λ Λt, the time evolution is well described by an operator of the samestructure as (4.27) with a frequency

Ω =ω − Γ

2

(1− Λ2

). (4.28)

The resulting time evolution of the two qubits is plotted in figure 4.7. Itis in good agreement with the slow dynamics depicted in figure 4.7.

Quantum gate

The two-qubit operations induced by tunneling can now be identified usingthe approximated time evolution operator,

Σ(φ) = U(tφ) =

cos φ2

0 0 i sin φ2

0 cos φ2

i sin φ2

0

0 i sin φ2

cos φ2

0

i sin φ2

0 0 cos φ2

, (4.29)

where φ is defined as φ = Ωtφ. For particular times tφ, maximally entangledstates can be produced.

To illustrate this, we investigate the time evolution of the ground state |00〉,which leads to the maximally entangled state |00〉+ i|11〉. The correspondingφ is given by φ = π

2, so that the gate is

Σ ≡ U(tπ

2) =

1√2

1 0 0 i0 1 i 00 i 1 0i 0 0 1

. (4.30)

Figure (4.8) shows this gate at work. In order to judge the fidelity of the

final state, we consider the time evolution under both UT = e−iHT t and U(t)(4.27). The latter does not account for first order processes. One can observethat the fidelity is reduced by these processes and may thus be approximated


0 5 10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

p01

+p10

p11

p

t (1/U0)

F

p00

Figure 4.8: Time evolution of the initial state |Ψ0Ψ0〉 due to UT = e−iHT t forN = 200, Λ = 0.1 and J/U0 = 8.6 × 10−4. The occupation probabilities pij,i, j = 0, 1 of the state |ΨiΨj〉 of the qubit subspace Υ and the fidelity F as afunction of time.

by F (t) ≈ 1 − Pout(t) (see (4.26)). In order to suppress this undesirableeffect, the probability amplitude A (4.26) can be adjusted to a small value byappropriately tuning J ,

J2 =U2

0

4N(N + 1)

A

1− 2A. (4.31)

Additionally, the parameters can be chosen such that Pout(t) is minimal at thefinal time tφ of the gate, i.e. sin2(ω

2tφ) = 0. Nevertheless, in order to make

A small, J cannot be increased too much. otherwise higher excited states arepopulated. The parameters used in figure 4.8 obey these conditions.

A high fidelity implies a small occupation probability of levels outside Υ,i.e. weak tunneling. This leads to a slowdown of the dynamics of the twoqubits. The rotation frequency Ω expressed in terms of A is given by

Ω =Γ

2

(1√

1− 2A− 1

)(1− Λ2). (4.32)

As can be seen, there is no direct dependence on the number of particles, sothat the dominant timescale of the evolution of the two qubits only dependson A and on the energy difference Γ between Υ and Υ1. Difficulties arisingfrom this will be discussed in chapter 5.


4.2.4 Read-out process

Finally, the result of a computation must be read out. This can be done bymeasuring in the rotated basis of the condensate states (3.18,3.19):

|ψ+N〉, |ψ−

N〉 (4.33)

Transferring the system for the read-out process to Λ = 0, this basis simplifiesto

| ↑ 〉⊗N , | ↓ 〉⊗N, (4.34)

where all atoms are either in the internal state |A〉 or |B〉.

4.3 Quantum gates at work

In the previous sections we presented a scheme for quantum computation witha two-species Bose-Einstein condensate. We demonstrated that the two lowest-energy states of this system can encode a qubit, and showed how to implementa universal set of quantum gates. In conclusion, we show how to construct someof the standard quantum gates, namely the Hadamard gate and the controlled-NOT gate. In addition, we investigate the quantum gates required to createthe singlet Bell state.

4.3.1 Common quantum gates

Single-qubit gates

In the previous section we showed how to perform single-qubit rotations aboutthe x- and the z-axis:

Rx(φ) =

(cos φ

2−i sin φ

2

−i sin φ2

cos φ2

)and Rz(φ) =

(1 00 eiφ

)(4.35)

From these operations, the Hadamard gate can be constructed as

H =

(1 11 −1

)= Rx

(π

2

)Rz

(π

2

)Rx

(π

2

), (4.36)

up to a global phase.

Two-qubit gate

The most common universal two-qubit gate is the controlled-NOT gate,

CNOT =

1 0 0 00 1 0 00 0 0 10 0 1 0

. (4.37)

4.3. QUANTUM GATES AT WORK 41

One can prove that the two-qubit gate (4.30)

Σ =1√2

1 0 0 i0 1 i 00 i 1 0i 0 0 1

, (4.38)

is equivalent to CNOT, up to local unitaries [29]. One possible constructionof the CNOT gate out of the quantum operations we posses is given by

CNOT =[Rx(−

π

2)Rz(−

π

2)]⊗ 11 · Σ ·

[Rx(−

π

2)Rz(−

π

2)Rx(−

π

2)]⊗Rx(−

π

2).

(4.39)

4.3.2 Creation of the singlet Bell state

In order to put the quantum gates to work, we present a simple quantumcircuit in terms of the quantum gates available in our scheme. Starting from|00〉, we show how to create the singlet Bell state |01〉 − |10〉. The sequence ofquantum gates is given by

|01〉 − |10〉 = Rz(−π

2)⊗ 11 · Σ · 11⊗Rx(π) |00〉. (4.40)

The implementation of the circuit is shown in figure 4.9.For the sake of completeness we show the sequences for creating the triplet

Bell states:

|00〉+ |11〉 = Rz(π2

)⊗ 11 · Σ |00〉|01〉+ |10〉 = Rz(

π2

)⊗ 11 · Σ · 11⊗Rx(π) |00〉|00〉 − |11〉 = Rz(−π

2)⊗ 11 · Σ |00〉

(4.41)


Figure 4.9: Time evolution under the quantum circuit (4.40) creating thesinglet Bell state. pij, i, j = 0, 1 denotes the occupation probability of thestate |ΨiΨj〉 of the qubit subspace Υ. F is the fidelity of the time evolution.Note the different timescales of the quantum gates.

43

Chapter 5

Realizability and decoherence

In the preceding chapter, a scheme for quantum computation with a two-species Bose-Einstein condensate was presented, where the qubit is encodedby the two lowest-energy states of the system. Furthermore, it was shown howto realize a universal set of quantum gates.

In this chapter, we discuss the feasibility of the presented scheme. Firstly,the experimental requirements for that qubits can be realized and initializedare discussed. Secondly, the effect of decoherence on the presented schemeis investigated. For this purpose, the timescale of the quantum operations isanalyzed and compared with the decoherence time of the system.

In order to give some numbers, we consider the cases of N = 4, N = 10and N = 105 atoms, respectively. We choose the two hyperfine states |F =1,mF =−1〉 and |2, 2〉 of the electronic ground state of 87Rb to represent thetwo internal states of the atoms: Rb is the atomic species for which BEC wasrealized for the first time [30], it has well known scattering properties and thetwo states can be condensed simultaneously [31].

5.1 Realization of the qubits

In this section we discuss the requirements for satisfying the conditions imposedon the physical system in chapter 3, where the two-species Bose-Einstein con-densate was presented. We show how to choose and prepare the system inorder to realize and initialize the quantum bits.

5.1.1 Basic experimental requirements

We consider Bose-Einstein condensation of a trapped gas of N atoms that areconfined by a three dimensional quasi-harmonic and isotropic potential. Sucha potential can be experimentally realized by a magnetic or optical trap inIoffe-Pritchard geometry [32],[33]. The trapping frequency is denoted ω. Forcalculating explicit examples of different numbers of particles, we consider the

44 CHAPTER 5. REALIZABILITY AND DECOHERENCE

N 4 10 105

ω 5 kHz 2 kHz 100 HzU0 142.6 Hz 36.1 Hz 0.40 Hzλ 7.13 Hz 5.42 Hz 667 HzTmax 20 nK 16 nK 1.9 µK

Table 5.1: Examples for N = 4, N = 10 and N = 105 for the hyperfine states|1,−1〉 and |2, 2〉 of 87Rb for the trapping frequency ω: scattering strengthU0 (5.2), λ for Λ = 0.1 and the temperature Tmax required for cooling tothe ground state at Λ = 3. The used parameters are a0 = a1 = 5.45 nm,M = 1.44× 10−25kg and a1

a0

= 43.

appropriate size of the trap by ωN=4 =5 kHz, ωN=10 =2 kHz and ωN=105 = 100Hz.

Internal state structure of the atoms

The atomic species has to be chosen such that the atoms possess two internaldegrees of freedom, denoted by |A〉 and |B〉. It is essential that |A〉 and |B〉can be coupled to each other by a coherent transition.

|A〉 and |B〉 could be two hyperfine levels of the atoms connected by aRaman transition [34]. Such a two-photon transition has to obey the selectionrule |∆mF | ≤ 2, where mF is the z-component of the total spin, representedby the quantum number F . For the 87Rb isotope, as can be seen from thelevel scheme of the D2 line of figure 5.1, this could be the two hyperfine states|F = 1,mF = 1〉 and |2, 2〉. The hyperfine states |1,−1〉 and |2, 2〉 of 87Rbweuse for our numeric examples, do not satisfy the selection rules. However, theycan, in principle, be coupled by a multi photon transition, also indicated infigure 5.1.

Requirements on the scattering properties

The interaction between the atoms occurs via elastic collisions, characterizedby the s-wave scattering length asc. In the discussion of chapter 3, the scat-tering length for collisions of atoms in the same internal state, A–A, B–B, areassumed to be equal, a0 = asc

AA = ascBB. This condition is e.g. almost fulfilled

for the two hyperfine states |1,−1〉 and |2, 2〉 of 87Rb with asc0 = 5.45 ± 0.26

nm [35]. The scattering length for interspecies collisions A–B is denoted a1.

The collisional interaction strengths U0 and U1 are directly related to thescattering lengths. The first one characterizes A–A as well as B–B processes,whereas U1 describes A–B collisions. For calculating the interaction strengths,we assume the spatial mode function φ0(~x) (3.7) to be a normalized gaussian

5.1. REALIZATION OF THE QUBITS 45

F’ = 2

F’ = 1

s+

p

0 1 2-1

0 1-1

-2

0 1 2-1

0 1-1

-2

F = 2

F = 1

0 1 2-1

0 1-1

-2

0 1 2-1

0 1-1

-2

F’ = 2

F’ = 1

F = 2

F = 1

s+

s-

s-

p

a) b)

mF’

mF

mF’

mF

Figure 5.1: Level scheme of the D2 line of the rubidium isotope 87Rb with thenuclear spin quantum number I = 3/2. mF denotes the z-component of thetotal spin, represented by the quantum number F =I ± 1/2. (a) Raman tran-sition coupling the hyperfine states |F =1,mF =1〉 and |2, 2〉. (b) Multiphotontransition coupling the hyperfine states |1,−1〉 and |2, 2〉.

function,

φ0(~x) =√x−3

0 π− 3

2 e− ~x

2

2 x2

0 , (5.1)

with the characteristic length x0 = ( hMω

)1/2 of the harmonic trap. Therefore,the collisional interaction strength Ui (3.11), i = 1, 2, becomes

Ui =4πhasc

i

M

∫d3~x |φ0(~x)|4 =

√2

πωasc

i

x0

=

√2M

πhasc

i ω3/2. (5.2)

For the example of 87Rb, the interaction strength for elastic collisions betweenparticles in the hyperfine states |1,−1〉 or |2, 2〉 yields UN=4

0 = 142.6 Hz,UN=10

0 = 36.1 Hz and UN=105

0 = 0.40 Hz.

5.1.2 Requirements for encoding a qubit

The analysis of the condensate in section 3.2 showed that for the conditionsU1 > U0 and Λ Λt, the two lowest-energy states of the system are quasi-degenerate Schrodinger-cat states that are well separated from the rest of thespectrum. Therefore, these are the two states used to encode a qubit. Wediscuss now how to satisfy the necessary conditions.


Feshbach resonance

To satisfy the condition U1 > U0 or U1

U0

> 1, equation (5.2) is used to show thedependence on the scattering length,

U1

U0

=a1

a0

> 1. (5.3)

For most atomic species, the desired value of a1

a0

> 1 is not satisfied by theintrinsic parameters of the atoms. A mechanism to influence and control thescattering properties of an atomic ensemble is the Feshbach resonance [36].

A Feshbach resonance occurs an elastic binary collision when the energyof a bound state in a closed channel is close to the energy of the incomingatoms (open channel) (see figure 5.2). If there exists some coupling mechanismbetween open and closed channel, the atoms can temporarily occupy the boundstate, thereby changing the elastic scattering properties of the system. Ingeneral, the energies of the involved states depend on external parameters,like a magnetic or an optical field. Hence, by varying this parameter, thesystem can be tuned into resonance and therefore the scattering length canbe varied over a wide range of negative and positive values, as it was realizedusing magnetic [37],[38] and optical [39] fields.

Therefore, in principle, we are able to tune the scattering length a1, charac-terizing collisions between atoms in different internal levels, such that a1/a0 >1.1 Nevertheless, in the vicinity of a Feshbach resonance, inelastic processesdrastically increase and limit the range of a1

a0

. We will discuss the related prob-lems in section 5.4. It is not known yet if there exists an easily accessibleFeshbach resonance for the channel (|1,−1〉 + |2, 2〉) of 87Rb, the example weuse for calculations.

Weak laser coupling

In order to encode a qubit, we also need the condition Λ Λt, where pa-rameter Λ is defined as Λ = 2λ

(N−1)(U1−U0)(3.17). It is determined by the rela-

tive strengths of the laser coupling and the collisional interaction. Λt (whereΛt ≤ 1) denotes the transition point to the Schrodinger-cat phase. In the rangeof Λ Λt, the two lowest states in energy of the system can encode a qubit.As we saw, the interaction strengths U1 and U0 are in principle tunable, butusually kept constant during an experiment. The value of Λ can be chosen andvaried easily by adjusting the laser strength λ of the Raman process. Valuesfor the examples of LL = 0.1 are given in table 5.2. Note that in this regime,Λ Λt, the two Schrodinger-cat states are well described by the mean field

1Since a Feshbach resonance is a one-channel process will not change more than onescattering length at the same time. Therefore we choose the scattering processes of atomsin different internal states to be altered.


Atomicseperation

Ebound

Eth

Energy

closed channel

open channel

b)a)

asc

BBres.

Figure 5.2: Feshbach resonance: (a) Schematic plot of the potential energycurves for two different channels illustrating the formation of a Feshbach reso-nance in an elastic collisional process. Eth is the energy of the entrance channel,and Ebound is the energy of the bound state in the closed channel giving riseto the Feshbach resonance. (b) Schematic plot of the dependency of the scat-tering length asc on an external parameter (here a magnetic field B), near theFeshbach resonance at Bres..


solutions |Ψ±〉 (3.26). A physical system satisfying these conditions can beused to encode a qubit.

5.1.3 Multi-qubit systems

Obviously, for quantum computation processing, it is not enough to possessone qubit. Therefore, several identical copies of the qubit-system have tobe prepared. In doing so a difficult task is to obtain the same number ofparticles in each trap. One possibility to achieve commensurate filling is toprepare all the atoms in one trap first, and then deform the trapping potentialadiabatically into a multi-well potential [40] (see figure 5.3). Since the energy ofthe whole system is minimized for equal filling and the process is adiabatic, wecan assume that the condensates in each well have the same number of particlesN . For small N , one can also think of loading a Bose-Einstein condensate intoan optical lattice with commensurate filling [16],[41].

E

x

l

Figure 5.3: Preparation of several identical qubits. Adiabatic deformation ofan initial single-well trap with 2 × N atoms into a double-well trap with Natoms in each well.

5.1.4 Initializing the qubits by cooling

The single qubits are encoded by the two lowest-energy states of a two-speciesBose-Einstein condensate. Therefore, the system has to be cooled down tocondensation. In order to initialize the qubit in the state |0〉, the system hasto be prepared in the ground state |Ψ0〉 for Λ Λt, which is an even morerestrictive requirement. In this regime, the two lowest-energy states are quasi-degenerate. Thus, direct cooling of the system to the absolute ground statewould be a difficult task. The idea is therefore to first cool the system to atemperature T close to zero for Λ > Λt, such that the thermal energy is lowerthan the energy gap between the ground and the first excited state. Note thatthis is only possible in this regime of Λ, since only there the first-excited stateenergy is high enough so that practically all of the atoms can be cooled downto the ground state. Then, we decrease Λ adiabatically to the desired value.According to the adiabatic theorem (see section 4.2.2), the system remainsin the ground state, which now becomes the Schrodinger-cat state, the qubitstate | 0 〉.


We give the temperature needed to cool the system to the ground state ofthe regime Λ > Λt and present a method how to cool a two-component systemdown to condensation.

Required temperature

In order to cool all atoms down to the ground state of the regime Λ > Λt,the thermal energy has to be smaller than the energy separation ε to the firstexcited level,

kBT < ε(Λ), (5.4)

with the Bolzmann constant kB = 1.38 · 10−23 JK

. As discussed in section 3.3.2,for Λ 1 (Λ > 3) the system behaves like a ferromagnet. The energy gapin this regime can be approximated by ε(λ) = 2λ = (N−1)(U1−U0)Λ, thuscondition (5.4) becomes

T < (N − 1)(U1 − U0) Λ1

kB

. (5.5)

The preparation of the ground state is favoured for a high number of particlesand strong collisional interactions. For (Λ = 3), condition becomes forN = 105

T < 1 µK whereas for a small number of particles (N = 4, 10) the cooling is anexperimentally more challenging task, since T < 20 nK, 16 nK, respectively.Note that U0 depends on N , thus T (N = 4) > T (N = 10) is due to the chosenvalues ωN=4 = 5 kHz and ωN=10 = 2 kHz.

Sympathetic cooling

Bose-Einstein condensation was first reported in a cloud of atoms in a singlespin state of rubidium [30] and later in single spin states of sodium [42] andlithium [43]. To reach the necessary ultralow temperatures below the transi-tion temperature of Bose-Einstein condensation, these experiments used lasercooling and trapping, followed by magnetic trapping and evaporative cooling.

The creation of two different condensates of neutral atoms in the sametrap was first demonstrated by Myatt et al. [31], with the hyperfine states|1,−1〉 and |2, 2〉 of 87Rb. They used the technique of sympathetic cooling,known from cooling of trapped ions [44], but at higher temperatures. Theycooled the cloud of atoms in the |1,−1〉 state by lossy evaporative cooling.The atoms in state |2, 2〉 were only cooled by thermal contact with the |1,−1〉atoms. For this method to be effective requires the elastic collision rate formomentum transfer between the two components to be large, and the inelasticcollision rate K2 that converts either component to an untrapped species tobe small. They measured K2 between the two components to be surprisinglysmall, K2 = 2.2 ± 0.9 × 10−14 cm3/s. This stability of the double condensatewas explained [35],[45] with the almost equal scattering lengths of the twohyperfine states, asc

AA = ascBB.


N 4 10 105

tx (ms) 1.8 1.5 1.2× 10−4

tad (ms) 12 33 12t2 (ms) 2.9 11.4 1018

Table 5.2: Timescales tx, tad, t2 of the x-gate, phase-gate and two-qubit gate,respectively, for the example of 87Rb with Λ = 0.1, a1

a0

= 43, φx = π

2, φ2 = π

4

and κ = 1.23. tad results from numerical integration of the adiabatic conditions(4.18).

N 5 10 105

µ 213 Hz 102 Hz 126 Hzλmax 71.3 Hz 54.2 Hz 6.7 kHzJ 5.64 Hz 0.61 Hz 7× 10−7 Hz

Table 5.3: Experimental parameters for realizing the quantum gates: µ refersto the strength of the state dependent potential, λmax to the maximal laserstrength in the adiabatic process and J to the tunneling strength. Parametersfor calculation see table 5.1.

5.2 Realization and timescales of the quantum

gates

In this section the quantum gates presented in chapter 4.2 are investigatedin more detail. It is assumed that we have a physical system satisfying allconditions as discussed above, so that the quantum bits are initialized in thestate |00..0〉, ready for quantum computation processing.

In order to successfully apply the quantum-gate operations, the dynamicsof the system, i.e. the dynamics due to the gates, have to work on a much fastertimescale than the decoherence of the physical system. This is expressed bythe condition

tG τ, (5.6)

where tG is the typical timescale of a quantum gate and τ the decoherencetime of the physical system (e.g. for a gate generating a rotation of frequencyΩ by an angle φ, the required time is given by t = φ

Ω). Therefore, we discuss

how to realize the quantum gates and analyze their timescales. Decoherencewill be treated in the next sections. The examples of very small numbers ofparticles, N = 4, N = 10, and a big condensate, N = 105, are calculated withthe parameters Λ = 0.1 and U1

U0

= a1

a0

= 43. The results are summarized in table

5.2.

5.2. REALIZATION AND TIMESCALES OF THE QUANTUM GATES 51

5.2.1 x-gate

The first single qubit operation we presented (see section 4.2.1) is a rotationabout the x-axis of the Bloch sphere, Rx(φ) (4.9). For this purpose a potentialdepending on the internal state |A〉, |B〉 of the atoms was added to the Hamil-tonian. Experimentally, this can be realized by applying an external magneticor electric field, so that due to the Zeemann or Stark effect, respectively, thedegenerate hyperfine levels split up (see figure 4.2).

To ensure that the perturbation does not couple the qubit states to higherexcited levels, the probability P Fermi (4.12) for occupying the higher excitedstates has to be much smaller than one. This leads to the condition (4.12) forthe perturbation strength µ,

µ

U0

=1

η

[1− Λ2

Λ2

(N − 1)2

N

] 1

2

, (5.7)

where the factor η ≥ 10 ensures P Fermi 1. Possible values for µ (η = 10)are summarized in table 5.2.

The timescale of the x-gate results from the rotation frequency Ωx (4.7) byeliminating µ with equation (5.7):

tx =φ

Ωx

≈ η

2

Λ

1−Λ2

φ

U1 − U0

1

(N − 1)√N. (5.8)

tx decreases very quickly with N . But even for a particle number of N = 4 itis the fastest of the three gates (see table 5.2).

5.2.2 Phase-gate

The second one-qubit operation we presented (see 4.2.2) is an adiabatic phase-gate, performing an z-rotation, Rz(φ) (4.16). The gate requires the adiabatictransfer of the system from 0 ≤ Λ0 Λt to Λ ≈ Λt , forth and back.

For this purpose the laser strength λ has to be tunable from zero to Λ =2λ

(N−1)(U1−U0)≈ 1 (see table 5.2). Note that in the range Λ ≈ Λt, the laser

intensity has to be controlled with a very high precision.

Obviously, the timescale of the phase-gate is not determined by the rotationangle φ of the gate, but by the adiabaticity of the process. Hence, we expecta slow gate. The required time tad for applying the phase-gate is obtainedby numerical integration of the adiabatic condition for Λ (4.19). tad slightlydecreases with N and is indirectly proportional to U1−U0. For instance, N = 4leads to tad = 12 ms, N = 10 to tad = 33 ms and for N = 105, the timescale isreduced to tad ≈ 12 ms. Thus, for small numbers of particles the phase-gateis the slowest of the three gates (see table 5.2).


5.2.3 Two-qubit gate

In section 4.2.3, we presented a universal two-qubit gate, realized by allowingtunneling between two adjacent traps. This can be implemented e.g. by low-ering the potential of an optical lattice [16], or by bringing the traps confiningthe atoms close to each other, e.g. as it can be realized with microtraps [28].

The interaction between the particles in the two traps is mediated viasecond order tunneling processes. In order to keep the probability amplitudeA for first order processes small, we consider weak tunneling. A and thetunneling parameter J are correlated as (4.31)

J2

U20

=1

4N(N + 1)

A

1− 2A, (5.9)

where we used that for Λ Λt, the energy difference between the qubitsubspace Υ and the next excited levels, Γ, can be approximated by Γ ≈ U0.Number examples for A = 0.1 are given in table 5.2.

Using (5.9), the frequency of the two-qubit gate can be expressed in termsof A, Ω = κ

4(1−Λ2)U0 (4.32), where κ = 1√

1−2A− 1. Therefore, the timescale

of the two-qubit gate is given by

t2 =4 φ

κ(1− Λ2)U0

. (5.10)

Note that by the replacement of J according to (5.9), t2 depends only via theinteraction strength U0 on the number of particles. Therefore, decreasing thenumber of particles slightly shortens t2. Another possibility to speed up thegate is to increase A, the amplitude of the probability Pout (4.25) for that theexcited subspace Υ2 gets occupied. A maximal value of A = 0.4 correspondsto κ = 1.23.2 The resulting conditions for the example of low and high numberof particles are listed in table 5.2.

Comparing the timescales of the three quantum gates (table 5.2) one findsthat the slowest dynamics is due to the adiabatic phase-gate. Also the possi-bilities of creating a fast two-qubit gate are limited.

5.3 Effects of decoherence

Quantum systems can never be perfectly isolated from the environment. Theresulting uncontrolled interaction with the environment leads to a loss of co-herence, a process known as decoherence. Some sources of decoherence are

2Note that the exact value of A has to be chosen such that Pout is minimal for tφ (seesection 4.2.3). Nevertheless, by choosing such a high A the probability for transitions tohigher subspaces becomes significant. Obviously these is a trade-off between excitation andspeed up of the gate.

5.3. EFFECTS OF DECOHERENCE 53

thermal fluctuations, trap losses due to inelastic collisions, scattering processeswith background thermal atoms or spontaneous light scattering.

In this section, the effect of number fluctuations onto the presented schemeof quantum computation is investigated. A reduction of the number of particlesleads to the problem that the calculated parameters for applying the quantumgates do not match to the real number of particles in the experiment. Theresulting problems onto the time evolution under the three quantum gates arediscussed and it is show that already the loss of one particle may reduce thefidelity of the time evolution in a unacceptable way.

5.3.1 Qubit

The effect of particle fluctuations onto the parameter Λ (3.17), defining thestructure of the qubit states, is found to be ∂Λ

∂N= − Λ

N−1. For small fluctuations,

this can be linearized to∆Λ

Λ= − ∆N

N − 1. (5.11)

The loss of a small fraction of particles leads to a small relative change of Λ.Thus, Λ and therefore the structure of the qubit states, is quite stable underparticle fluctuation. That means that the exact knowledge of the number ofparticles is not necessary for defining the qubit.

In the following, we investigate the effect of number fluctuations onto thetime evolution of the qubit states under the different quantum gate operations.Note that the laser strength λ is an experimental parameter that does notdepend on N .

5.3.2 x-gate

The dynamics induced by the x-gate is almost stable under small variationsof the number of particles: the rotation frequency Ωx (4.7) and the rotationangle φx =Ωx t change as

∆Ωx

Ωx

=∆φx

φx

=∆N

N+

Λ2

1− Λ2

∆N

N − 1, (5.12)

where the last term, resulting from the variation of Λ, can be neglected forΛ Λt. Starting from | 0 〉, the fidelity of the gate is reduced from 1 toFx(tφ) = cos2(∆φ

2). Thus, even for a loss of 10% of the initial number of

particles, the fidelity for φ = π is still Fx(tπ) = 0.97 (see figure 5.4).

5.3.3 Phase-gate

The evolution of the qubit under the phase-gate is extremely sensitive to num-ber fluctuations. The reason for this lies in the dependence of the energy


0.0 0.4 0.8 1.2 1.60.0

0.2

0.4

0.6

0.8

1.0p

fluc

0

pfluc

1

p0

p1

F

p

t x 104

(1/U0)

Figure 5.4: Time evolution of the initial qubit state |0〉 under the x-rotationRx(π) for a fluctuation of ∆N

N= 0.1. The parameters are N = 200, Λ = 0.1

and µ/U0 = 100 The occupation probabilities p0 and p1 of the two qubit states|0〉 and |1〉, respectively as well as the fidelity F (t) (4.13) are plotted as afunction of time. Superscript indicates the fluctuation, no superscript refersto the ideal case.

5.3. EFFECTS OF DECOHERENCE 55

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

F

t (1/U0)

0.0 0.5 1.0 1.5 2.0 2.5

Ffluct

F

0

-pF

t (1/U0)

-2p

0.0 0.5 1.0 1.5 2.0 2.50

5

10

15

20e

fluct

e

e(U

0)

t (1/U0)

a)

b)

c)

t

Figure 5.5: Phase-gate for a fluctuation of ∆NN

= 0.1 for N = 100. Superscriptindicates the fluctuation, no superscript refers to the ideal case. (a) Energysplitting ε = E1− E0, (b) accumulated relative phase and (c) fidelity as afunction of time.


spectrum on the number of particles (see figure 5.5). Numerical calculationshows that the loss of even a small fraction of particles leads to an intolera-ble decrease of the fidelity. Figure 5.5 shows that for a number of particlesN = 100, already the loss of one particle can lead to a decrease of the fidelityto F < 0.5. Therefore, for small number of particles, N < 100, not even asingle particle may be lost.

These results imply that the initial number of atoms must be known exactly.Slightly modifying the phase-gate, this experimental problem can be evaded.By further increasing Λ in the adiabatic process, Λ Λt, a value of Λ (Λ ≈3) can be found such that for each number fluctuation ∆N

N 1, the phase

error is a multiple of 2π, and hence not observable. Since the condition foradiabaticity for Λ > Λt restricts Λ only very weakly, this process does not affectthe timescale of the phase-gate significantly. However, this method only solvesthe problem of the uncertainty of the initial number of particles. During theadiabatic phase-accumulating process (ε(Λ) > 0), no fluctuations are allowed- independent on the number of particles.

5.3.4 Two-qubit gate

The rotation frequency Ω = ω−Γ2

of the two-qubit gate is found to be stableunder small fluctuations of N , ∆Ω

Ω≈ ∆N

N. However, the world is not that

simple: when calculating Ω in section 4.2.3, we assumed that the two systemsbetween which tunneling occurs have exactly the same number of particles atall times, N1(t) = N2(t). Discussing the effect of fluctuations, this conditionimplies that the loss of particles has to occur in both traps at the same time.Obviously, for a statistical process this is not realistic.

Therefore, we investigate the problem arising from an unequal trap filling.So let us assume that the two traps containing the qubit systems are not equallyfilled, N1 6= N2. Figure 5.6 shows the energy spectrum for this situation.Compared to equal filling, the qubit subspace Υ defined by two different qubits

N, N

N+1, N-1

N , N1 2

N +1, N -11 2

N -1, N +11 2

N-1, N+1

E

~~

Figure 5.6: Schematic energy spectrum for the two-trap system for Λ = 0.

5.4. INELASTIC COLLISIONS 57

0 5 10 15 20 25 30 35 40 45 500.0

0.2

0.4

0.6

0.8

1.0p

fluct

00

pfluct

11

p00

p11

p

t (1/U0)

Figure 5.7: Time evolution of the initial qubit state |00〉 under the two-qubitgate for equal filling with N = 200 atoms each qubit and for unequal filling,N1 = 200, N2 = 199, denoted by superscript. The occupation probabilities p00

and p11 of the two-qubit states |00〉 and |11〉 are plotted as a function of time.The parameters are Λ = 0.1 and J/U0 = 8.8× 10−4.

is not only raised in energy. Also the energy differences of the two subspaces towhich Υ is coupled by first order tunneling processes depend on the difference∆N = |N1−N2|, as well as on the direction of tunneling, so that the symmetricsituation characterizing the two-qubit gate is broken.

Thus, independent on the total number of particles, already a particledifference of ∆N = 1 between the two traps leads to dynamics completelydifferent from the one predicted by the two-qubit gate assuming equal filling(see figure 5.7). This effect can only be avoided if there are no losses beforeand during the application of the two-qubit gate.

We conclude that decoherence due to fluctuations of the number of particlesstrongly restricts the number of gates one can apply without decreasing thefidelity unacceptably. The phase-gate and the two-qubit gate are found tobe very sensitive to fluctuations, they work only within no losses. Thus, therequired lifetime τ tG is the time during which no particle is lost. Whereasa small uncertainty in the initial number of particles only causes small errors.

5.4 Inelastic collisions

A fundamental source of decoherence of a Bose-Einstein condensate is due totrap losses by density-dependent inelastic two-body collisions (mostly spin-exchange) and three-body processes. Unlike other loss mechanisms such as


collisions with the background gas or spontaneous light scattering, these traplosses are intrinsic loss mechanisms that cannot be eliminated by simply engi-neering a better trapping environment. For small losses the rate equation forN can be linearized to

∆N

∆t= −K2〈n〉N −K3〈n2〉N, (5.13)

where 〈n〉 = N∫|φ0(~x)|4d3~x (5.14)

〈n2〉 = N 2∫|φ0(~x)|6d3~x (5.15)

and n(~x) = N φ0(~x)∗φ0(~x). (5.16)

K2 and K3 are the inelastic rate coefficients for two- and three-body processes,respectively. The inelastic rate coefficientsK0

2 andK03 for 87Rb (the superscript

indicates the absence of a Feshbach resonance) are summarized for the differentscattering channels in table 5.4. Note that K0

2 of 87Rb is very small comparedto other species [35].

As discussed in the previous section, in order to fulfill the condition a1/a0 >1 (5.3), a Feshbach resonance is used to alter the scattering length a1 that char-acterizes elastic collisions between atoms in different internal states. However,as was shown e.g. by Roberts et al. [46], inelastic processes are highly enhancedin the vicinity of a Feshbach resonance. Therefore, the dominant trap loss ofthe presented system will be due to inelastic interspecies collisions. Thus, wefocus on the corresponding scattering channel |A〉+ |B〉.

In the previous analysis of the effect of number fluctuations on the quantumgates, we derived the very severe condition that during quantum computationprocessing, the number of particles in the traps may not change, ∆N = 0.Therefore, we define the decoherence time τ of the system as the time inwhich one particle is lost

τ ≡ ∆t(∆N = 1). (5.17)

For successfully applying a quantum gate, this decoherence time has to bemuch larger than the typical timescale tG of the gate,

τ tG. (5.18)

For analyzing this condition in detail, we discuss the loss mechanisms due toinelastic two- and three-body processes separately and compare the resultingdecoherence times (5.17) with the tG for the different quantum gates, summa-rized in table 5.2. Analyzing the rate equation (5.13), τ is found to scale atleast with 1

N2 . In addition, the decay rates are enhanced in the vicinity of theFeshbach resonance. Therefore, condition (5.18) is expected to be fulfillable

5.4. INELASTIC COLLISIONS 59

Entrance Channel K th2 (cm3/s) Kexp

2 (cm3/s) Kth3 (cm6/s) Kexp

3 (cm6/s)|1,−1〉+ |1,−1〉 3.1× 10−18 a 2.3× 10−28 c 9× 10−29 d

|2, 2〉+ |2, 2〉 2× 10−16 a 2.3× 10−28 c 2.2× 10−28 e

|1,−1〉+ |2, 2〉 2× 10−14 a 2.2× 10−14 b

Table 5.4: Theoretical and experimental results for the inelastic collision ratesK0

2 (two-body processes) and K03 (three-body processes) of 87Rb. a[47] b[31]

c[48] d[49] e[50]

only for a low particle number and a weak Feshbach resonance, a1/a0 ≈ 1.The decoherence times for N = 4, 10 and N = 105 are summarized in table5.4.1. For the discussion, recall table 5.2 summarizing the timescales of thequantum gates.

5.4.1 Two-body processes

Only considering inelastic two-body collisions, the rate equation (5.13) writes

∆N

∆t= −K2N

2∫|φ0(~x)|4d3~x. (5.19)

As far as we know, there are no proposals about the scaling of the inelasticcollisional rate K2 in the vicinity of a Feshbach resonance. Therefore, we firstlyevaluate (5.21) without considering the effect of the Feshbach resonance. Sec-ondly, we assume from experimental data for 85Rb [46] a maximal dependenceof K2 ∝ a4

sc (like for three-body losses, see below).In order to obtain the order of magnitude of N for which quantum gates

can be applied, we evaluate condition (5.18) for the two-qubit gate (5.10),

τ t2 =4 φ

κ(1− Λ2)U0

. (5.20)

Together with (5.19) this leads to

N2 4πha0

M

κ

4φ

1

K2

. (5.21)

Note that since U0 (5.2) and the two-body rate-equation (5.19) have the samedependency on the mode function φ0(~x), the condition does not depend on thedensity of particles or on the trapping frequency.

Even only considering the non-resonant rate K02 = 2.2× 10−14 cm3

sof 87Rb,

the condition (5.21) demands for φ2 = π2

and a1/a0 = 4/3 a very small numberof particles, N 30. Evaluating the rate equation (5.19), one can see that the


N 4 10 105

τ 02 (ms) 174 110 10× 10−5

τ2 (ms) 55 35 3× 10−5

τ3 (ms) 50 50 4× 10−7

τ (ms) 26 21 4× 10−7

Table 5.5: Decoherence times τ = ∆t(∆N = 1) (5.17) for N = 4, 10 and 105

due to inelastic two- (τ2) and three-body (τ3) processes and considering bothloss mechanisms (τ). The superscript 0 denotes that no Feshbach resonance isconsidered.

decoherence time τ 02 is longer than the timescales of the single gates. Assuming

now the effect of the Feshbach resonance to be described by the scaling K2 ∝a4

sc, the decoherence time is reduced by a factor of (a1/a0)4. Together with

tx, tad that are proportional to 1a1/a0−1

(see section 5.2), the condition τ tGis optimized for a1/a0 = 4/3. Using this value, the decoherence time for N = 4computes to τ2 = 55 ms, which is still longer than the timescales of all threegates. For a particle number of N = 10, the decoherence time τ = 35 ms is ofthe order of the timescale of the adiabatic phase-gate.

5.4.2 Three-body processes

Only considering particle losses due to inelastic three-body processes, (5.13)writes

∆N

∆t= −K3N

3∫|φ0(~x)|6d3~x. (5.22)

In the vicinity of a Feshbach resonance, the three-body recombination rate K3

scales as the scattering length to the fourth power, K3 = K03 (a1/a0)

4 [48], [46].Therefore, the decoherence time is given by

τ3 = x60(3√π)3 1

K03 (a1/a0)4

1

N3(5.23)

where x0 = ( hMω

)1/2 is the characteristic length of the harmonic trap. For allthree different quantum operations tG is proportional to 1

U0

∝ x30 (see section

5.2). Thus, unlike for the case of two-body processes discussed above, the trap-ping frequency ω ∝ x−2

0 influences the relation between τ3 and the timescalesof the gates; small frequencies are favorable.

For the non-resonant rate of 87Rb K03 = 2.2× 10−28 cm6

s(table 5.4) and the

ratio a1/a0 = 4/3, the decoherence times for N = 4 and N = 10 are bothcomputed to be τ3 = 50 ms. For N = 10 atoms, this is only little more thanthe timescale of the adiabatic phase-gate.

5.5. SUMMARY 61

Accounting for both inelastic two- and three-body processes, the decoher-ence time τ computes from the rate equation (5.13) to be τ = 62.2 ms forN = 4 particles, and τ = 20.5 ms for N = 10 particles per qubit. Therefore,we conclude that for particle numbers of about N = 4, a small amount of x-rotations as well as two-qubit gates can be applied. Choosing the parameterscarefully, it is even possible to apply the adiabatic phase-gate once, so thattogether with the other operations, a Bell state can be produced. For N = 10particles per qubit, it is still possible to create maximally entangled two-qubitstates.

5.5 Summary

The ideal atomic species for realization of the scheme we presented, possessestwo internal states that can be trapped and condensed simultaneously. Fur-thermore they can be coherently coupled to each other and have the samescattering lengths. The scattering between atoms in different internal states iseither stronger than between equal atoms or there exists a Feshbach resonancefor this channel. The inelastic rates are extremely small.

The most profound problem of using the system for quantum computa-tion or at least for creating an arbitrary two-qubit state, is that the quantumgates are very sensitive to number fluctuations, no particle may be lost. Com-paring the resulting decoherence time by only considering the dominant loss-mechanisms due to inelastic two- and three-body collisions with the timescalesof the quantum gates, it results that only a very small number of quantumgates can be applied. Therefore, without suppressing the loss of particles, thesystem cannot be used for quantum computation. In fact, only very recentlyit was proposed how to reduce the three-body recombination rate by using asequence of laser pulses [51]. However, we obtain that for small numbers ofparticles and a weak Feshbach resonance, maximally entangled states betweentwo qubits can be produced.

63

Chapter 6

Conclusion and Outlook

The goal of this thesis is to explore an approach for quantum computation, inwhich qubits are encoded in atomic many-body states not sensitive to defectsin the number of particles.

For this purpose, we focused on a two-component interacting Bose-Einsteincondensate. We reviewed and analyzed the results of [18], where it was shownthat under certain conditions, the ground state of the condensate is quasi-degenerate and well separated from the excited levels. The atomic ensemblethen behaves as a two-level system, where the properties of the two statesare not very sensitive to a change in the number of particles. Using thesetwo many-body states for encoding the qubit, we further demonstrated howto realize a universal set of quantum operations. We showed that one-qubitoperations can be performed by exploiting the Zeeman effects or Stark ef-fects induced by a magnetic or electric field, respectively, and by adiabaticallychanging the Raman coupling between atomic states. A universal two-qubitgate can be performed by allowing tunneling between two neighboring qubitsystems.

Finally, we discussed the feasibility of the presented scheme. We analyzedthe experimental requirements, both for the preparation and the initializationof the qubit systems as well as the realization of the quantum gates. In addi-tion, we investigated the limitations of the scheme resulting from decoherence.Our attention hereby was focused on the most important source of decoher-ence, the loss of particles due to inelastic two- and three-body processes. Asexpected, the properties of the many-body qubit states are found to be robustunder changes in the number of particles. Thus, in order for the qubits are welldefined, it is not necessary to know the exact number of particles. In addition,the loss of a small fraction of particles does not lead to a loss of information.

However, we found that this situation changes when applying the quan-tum gates. In particular, coherent evolution of the states of the qubits is onlypossible if (i) the number of particles does not change during the adiabaticprocess of the phase-gate, and (ii) two qubits interacting via tunneling have

64 CHAPTER 6. CONCLUSION AND OUTLOOK

to be characterized by the same number of particles. Thus, a successful appli-cation of these gates requires that not even a single particle is lost. Therefore,the challenging problem to be solved consists in finding an optimal system forrealization, above all an atomic species with extremely small inelastic collisionrates and the required scattering properties. In that case it turns out that forsmall numbers of particles per qubit a limited amount of gates can be applied.

Thus, the presented scheme cannot be used for complex quantum compu-tational processes unless a way to suppress the loss of particles is found. Infact, only very recently it was proposed to suppress the inelastic three-bodyrecombination rate using a sequence of laser pulses [51]. However, even if thesuppression of losses should turn out to be unrealistic in practice, by applyingonly a small number of gates, interesting multi-particle states can be created.Using the example of 87Rb, we showed that it is possible to prepare two qubitsof four atoms each in any Bell state. Two qubits of ten atoms each can bemaximally entangled.

In fact, it should be possible to create even more complex states. Eventhough decoherence imposes severe restrictions onto the number of gates thatcan be applied subsequently, the number of quantum operations acting simul-taneously are in principle not limited. Therefore, by applying the two-qubitgate onto pairs of qubits in parallel, and then entangling these pairs in a sec-ond step, a large number of qubits can be entangled in only two operationalsteps. Together with the fast rotation of single qubits about the x-axis, a vari-ety of entangled multi-qubit states could be created. Expanding the presentedscheme, one could imagine the creation of entangled multi-qubit states in onlyone operational step by arranging the traps in an appropriate way and allow-ing tunneling between more than two qubits. For the example of a three-qubitsystem, the arrangement could be a linear or trigonal one. Already for a num-ber of four qubits, a multitude of configurations opens up.

In this thesis, we explored a specific approach for quantum computationbased on qubits encoded in condensate states. However, one can partly gener-alize the problems arising from decoherence to any approach based on a similarsomehow simple way of encoding qubits in condensate states and using secondorder tunneling processes for entanglement creation. It remains to be seen ifthere can be found a clever way to encode qubits in condensate states suchthat not only the states of the qubits themselves are robust under defects, butalso the computational processes.

BIBLIOGRAPHY 65

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ACKNOWLEDGEMENTS 71

Acknowledgments

Many people have contributed to make this work a success. I would like toexpress my thanks to everyone for their support and encouragement. Especialthanks to:

Prof. Dr. J. Ignacio Cirac, for giving me the opportunity to do mydiploma works as a member in his group at the Max-Planck Institute forQuantum Optics.

Dr. Belen Paredes, for giving me the chance to deal with this excitingtopic and for her continuous support in all aspects. I am very gratefulfor the valuable advice and the good atmosphere.

Dr. Juan Jose Garcıa Ripoll, for fruitful discussions concerning coldatoms.

Dr. Michael Wolf, for his lecture and the constructive discussionsabout quantum computation - and for the word hence.

Thomas Volz, for the good advice and help at any time with experi-mental questions.

Klemens Hammerer, Valentin Murg and Christian Schon, foralways been helpful and full of ideas concerning quantum optics.

Markus Popp and Henning Christ, for answers about cold atomsand for making me laugh.

Toby Cubitt, Norbert Schuch and Dr. Karl Vollbrecht, for theirknowledge about quantum information.

Simon Nigg, for a very pleasant company in the office.

all “my” PhD students and PostDocs, for all the fun I had withthem. Indeed, I will miss calling them for lunch.

Felix Hofbauer, Niels Syassen, Mike Reimer, Heinz Thieme andmost of the people already mentioned, for proofreading this thesis.

Finally, I would like to thank my parents for their generous support duringall my studies.

72 ACKNOWLEDGEMENTS

STATEMENT 73

Statement

With the submission of this diploma thesis I testify that I have written itindependently and did not use other sources than the cited references.

Munchen, 21. Mai 2004 Theresa Hecht

quantum computation with bose-einstein condensates · deutsch’s two bit problem [3]. this was the...

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