Nuclear Spin Phenomena in Optically Active Semiconductor Quantum Dots

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Dissertation zur Erlangung des akademischen Grades des

Doktors der Naturwissenschaften (Dr. rer. nat.) an der Universität Konstanz

Fachbereich Physik

vorgelegt von Julia Hildmann

Nuclear Spin Phenomena in Optically Active Semiconductor

Quantum Dots

Referenten:

Prof. Dr. Guido Burkard Prof. Dr. Wolfgang Belzig

Tag der mündlichen Prüfung: 28-02-2014

Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-269285

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Abstract

The motivating background for this thesis lies in the recent development in the coherent optical control of electron spin states in semiconductor quantum dots. One possible and attractive application for these achievements is a physical implementation of a quantum computer based on spin qubits (quantum bit) in quantum dots. However, one of the obstacles in building such quantum computer is decoherence caused by the interaction of the electron spins with nuclear spins of the host material. In this thesis we theoretically investigate the effects caused by the hyperfine interaction in optically active semiconductor quantum dots.

In Chapter2we assume optically induced rotations of single electron spins in semiconductor quantum dots. The optical control of the electron spin states is considered to be conducted by means of Raman type optical transitions between electron spin states. We investigate the influence of nuclear spins on the performance of the single-qubit gates by incorporating the additional effect of the Overhauser field into the electron spin dynamics. To calculate the errors caused by the hyperfine interaction, we determine average fidelities of rotations around characteristic axes in the Bloch sphere in the presence of nuclear spins analytically with perturbation theory up to second order in the Overhauser field. By applying numerical averaging over the nuclear field distribution we find the average fidelity to the arbitrary orders of the hyperfine interaction.

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In Chapter3we investigate the dynamics of spin qubits interacting by means of an optical cavity. The qubits are represented by a single electron spin each confined to a quantum dot. Knowing that electron spins in III-V semiconductor quantum dots are affected by the decoherence due to the hyperfine interaction with nuclear spins, we find that the interaction between two qubits is consequently influenced by the Overhauser field. Assuming an unpolarizied nuclear ensemble, we investigate the fidelities for two-qubit gates depending on the Overhauser field. We include the hyperfine interaction perturbatively to second order in our analytical results, and to arbitrary precision numeri- cally.

Quantum dots can be used not only as physical systems for qubits, but also as sources for entangled photons. In chapter 4 we consider an emission of entangled photons by spontaneous decay of the biexciton state to the ground state. Due to electron-hole exchange interaction, the intermediate excitonic states have a finite fine structure splitting. The electron spins of the excitons also interact via hyperfine interaction with the nuclear spins. We study the temporal and fine structure splitting dependance of the coherence measures considering the effect of the nuclear magnetic field. We find that the hyperfine interaction contributes considerably to the reduction of the photon coherence.

One of the possibilities to minimize the decoherence effects of the hyperfine interaction is to polarize the nuclear spins to 100%. However, up to date it was not possible to reach such degree of the nuclear polarization in experiments. In a recent experiment by Chekhovichet al. it was shown that high nuclear spin polarization can be achieved in self-assembled quantum dots by exploiting an optically forbidden transition between a heavy hole and a trion state. A fully polarized state in this case was predicted by a classical rate equation, but could not be obtained experimentally. Therefore we theoretically investigate this problem with the help of a quantum master equation in Chapter 5, and we show that a fully polarized state cannot be achieved due to formation of a nuclear dark state. Moreover, we demonstrate that the maximal degree of polarization depends on the form of the electron wave function inside of the quantum dot.

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Zusammenfassung

Diese Doktorarbeit ist motiviert durch die raschen Fortschritten in der ko- härenten optischen Kontrolle von Elektronenspins in Halbleiter-Quanten- punkten. Von dem experimentellen Standpunkt her stellen die Halbleiter- Quantenpunkte ein geeignetes physikalisches System für die Realisierung von Qubits (quantenmechanische Bits) dar, die eine Grundlage für einen Quantenrechner bilden. Allerdings ist der Elektronenspin-Zustand wegen der Wechselwirkung mit den Kernspins der Gitteratome der Dekohärenz unter- worfen. Gegenstand dieser Arbeit sind theoretische Untersuchungen zu Ef- fekte in optisch aktiven Halbleiter-Quantenpunkten, die durch die Hyperfein- Wechselwirkung zwischen den Kernspins und dem Elektronenspin verursacht werden.

In Kapitel 2 werden optisch erzeugte Rotationen des Elektronspins in Halb- leiter-Quantenpunkten betrachtet. Es wird angenommen, dass die optische Kontrolle von den Elektronenspin-Zuständen durch optische Raman- Übergänge realisiert ist. Der Einfluss der Kernspins auf die Ausführung der Einzelqubit-Gatter wird durch Einbeziehung des Overhauser-Feldes in die Elektronenspin-Dynamik untersucht. Um die durch die Hyperfein-Wechsel- wirkung entstehenden Fehler zu berechnen, wird die durchschnittliche Güte (gemittelt über die statistische Verteilung für das Kernspinfeld) der Ro- tationen um charakteristische Achsen in der Bloch-Kugel mit Anrechnung von den Kernspin-Effekten bestimmt. Die durchschnittliche Güte wird bis zur zweiten Ordnung des Overhauser-Feldes mit Hilfe der Störungstheorie

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berechnet. Zusätzlich wird die durchschnittliche Güte in beliebiger Ord- nung in der Hyperfein-Wechselwirkung durch numerische Mitteln über die Verteilung der Kernspins erhalten.

In Kapitel3wird die Dynamik von zwei Spin-Qubits betrachtet, die mit Hilfe eines optischen Resonator wechselwirken. Da es bekannt ist, dass die Elektro- nenspins in III-V Halbleiter-Quantenpunkten durch die Dekohärenz aufgrund der Hyperfein-Wechselwirkung mit Kernspins beeinflusst sind, wird es untersucht, wie die Wechselwirkung zwischen zwei Qubits durch das Overhauser- Feld geändert wird. Unter der Annahme, dass die Kernspins nicht polarisiert sind, wird die Güte der Zweiqubit-Gatter in Abhängigkeit vom Overhauser- Feld berechnet. Die Ergebnisse für die durchschnittliche Güte werden bis zur zweiten Ordnung der Hyperfein-Wechselwirkung unter Verwendung der Störungstheorie und exakt durch numerische Mitteln erhalten.

Quantenpunkte können nicht nur als geeignete physikalische Systeme für Qubits verwendet werden, sondern sind auch Quellen für verschränkte Pho- tonen. In Kapitel 4 wird die Erzeugung von verschänkten Photonen durch spontane Emission beim Zerfall eines Biexzitons betrachtet. Wegen der Aus- tauschwechselwirkung zwischen dem Elektron und dem Loch sind die Zwis- chenzustände, die Exzitonen, nicht mehr energetisch entartet. Ausserdem wechselwirken die Elektronenspins aus den Exzitonen mit Kernspins. Die Photonenkohärenz wird als Funktion der Zeit und der Feinstrukturaufspal- tung unter Berücksichtigung des Einflusses vom Kernspinfeld untersucht. Die Ergebnisse weisen eine wesentliche Reduzierung der Photonenkohärenz wegen der Hyperfein-Wechselwirkung auf.

Eine der Möglichkeiten die Dekohärenzeffekte zu minimieren, die durch die Hyperfein-Wechselwirkung verursacht werden, ist die Kernspins bis zu 100%

zu polarisieren. Jedoch war es noch nicht möglich so eine hohe Kernspinpolar- isation in Quantenpunkten experimentell zu erreichen. Eine beachtlich hohe Kernspinpolarisation wurde im Experiment von Chekhovichet al. beobachtet, die durch spinverbotene optische Übergänge zwischen einem schweren Loch and dem Trion-Zustand erzielt wurde. Ein komplett polarisierter Kern- spinzustand wurde durch die Ratengleichung für diesen Experiment voraus- gesagt, aber konnten nicht erreicht werden. Deshalb wird diese Fragestellung theoretisch in Kapitel5mit Hilfe der quantenmechanischen Mastergleichung untersucht. Es wird gezeigen, dass ein komplett polarisierter Zustand wegen der Formation eines Dunkelzustands nicht erreicht werden kann. Ausserdem ergibt sich eine Abhängigkeit der Höhe der Sättigung der Kernspinpolarisa- tion von der Form der Elektronenwellenfunktion im Quantenpunkt.

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Acknowledgments

When I was still studying for my Bachelor degree, I took a seminar on quantum information. I was introduced to this fascinating world, where quantum physics is just a common part of the daily life. Most of all I was fascinated by those tiny miraculous objects, the quantum dots. Since then I had been wanting to conduct research projects involving quantum dots. Fortunately, this opportunity I was given during my PhD. I would like to thank Professor Guido Burkard for giving me the possibility to follow my own path in the physics and accepting me as a PhD student in his group. I do not know how far I would be able to go in my work without his insightful hints, guidance and fruitful discussions. I appreciate also the freedom and the independence spirit I was given for conducting my research.

In addition to science it was quite a pleasure to spend my PhD time in one of the German excellence universities and wonderful city of Konstanz. The Constance lake and the Swiss mountains make up a really lovely landscape for a cheerful live. As for not a native German I will always keep the carnival in Konstanz as the best in my memories. It was a fairy-tale experience.

I have spent a very good time in the theory of condensed matter group and would like to thank everyone for the interesting discussion and entertaining non-scientific activities: Adrian Auer (thank you for being a president), Dr.

Luca Chirolli (thank you for being so Italian and showing me, that superconducting qubits are actually fun), Erik Welander (you are a cool officemate and a true viking), Dr. Eleftheria Kavousanaki (thank you for all the pa- tience explaining me sometimes too basic things), Dr. Hugo Ribeiro (thank

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you for crazy ideas and sometimes too loud discussions), Dr. András Pályi (thank you for being mighty postdoc), Heng Wang (thank you for being a girl among all the men), Niklas Rohling (thank you for being the most toler- ant person in the world), Matthias Droth (thank you for showing the "Bison stube"), Franziska Maier (thank you for taking care of me in the beginning of my PhD), Mathias Diez (thank you for all the cakes), Marko Rančić (thank you for being the best drinking buddy in the world), Peter Machon (thank you for always being fun and an awesome person), Julien Rioux (thank you for the maple syrup), Dr. Philipp Struck (thank you for agreeing to buy the skimmed milk), Dr. Andor Kormányos (thank you for showing, that graphene is just the beginning), Marco Hachiya (thank you for being a cool Brasilian guy and a nice person) and all, all other physicists and people, who inspired and fascinated me.

I would like especially thank our computer guru Dr. Stefan Gerlach for all kind of support and our secretary Maria Rosner for her guidance through the bureaucratic jungles and beyond.

Finally, I would like to thank all my family and all friends for their love, support, and believing in me.

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1

Introduction

1.1 Quantum computation

Quantum computers are one alternative for future development beyond clas- sic computers. The main components of classical computers are composed of electrical circuits. Due to technological developments and trends the size of their components is constantly decreasing. At some point, it will reach a dimension where the quantum effects are unavoidable and need to be dealt with. Quantum computers were proposed by Feynman [1,2], Deutsch [3] and others as a natural way for simulating complex quantum systems, since their working principle is based upon quantum mechanical laws. As a consequence, the idea of quantum computers gave rise to a class of new computational algorithms, quantum algorithms, that make use of such quantum phenomena as entanglement [4]. Quantum algorithms can solve some specific problems much faster than classical computers [3, 5]. Of special importance are the quantum factorization algorithm [6] and the quantum algorithm for data search [7].

As in classical computers, the information in quantum computers is encoded in bits (binary digits). A quantum bit or qubit can be implemented in a two-level quantum system, whose basis states are denoted|0i and|1i. How- ever, in contrast to the classical bit, a qubit does not have to be in one of the basis states. Its arbitrary state can be written as a superposition state:

|ψi = α|0i+β|1i [8, 9]. The amplitudes α and β are in general complex and fulfill|α|²+|β|² = 1. The fundamental differences between classical and quantum computing lie in the specific properties of the qubit state |ψi. It cannot be copied, as it was demonstrated with the no-cloning theorem [10],

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it can be in any superposition, and two quantum states can be entangled.

There are some necessary requirements that should be fulfilled to build a quantum computer. The five most important criteria for physical realization of a quantum computer were defined by DiVincenzo [11].

The first criterion: one needs a physical system, where many qubits can be well defined and the system itself is scalable. A well defined qubit means, that the qubit can be externally controlled and coupled to other qubits. The scalability is an important feature to build a quantum register. It is useful to have a quantum computer, only if it contains a large number of qubits.

The second criterion is the possibility to initialize the qubit in one of the basis states. Each computation starts from some known initial state and the quantum error correction codes require a large number of additional (ancil- lary) qubits initialized in a specific known state [9].

The third criterion suggests long relevant coherence times for the qubit, which should be much longer than the gate operation times. It is important to have a qubit that is coupled to the environment, so that its state can be controlled (the first criterion). However, quantum states interacting with the environment become completely mixed states after a certain time due to decoherence. In this case, the time provided for the quantum computation would be limited by the shortest decoherence time in the system. With quantum error correction the errors produced by decoherence can be fixed, if a certain number of gates can be performed within the decoherence times.

According to the accuracy threshold theorem, the error probability per gate should be smaller than a certain threshold to enable the error correction [12].

Depending on the error models and the codes, this threshold can range from 10⁻⁵ to10⁻³ [12,13]. Thus, for a quantum computer the gate time should be thousands to hundred thousands times shorter than the shortest decoherence time. For quantum computation based on surface codes, however, the error threshold exceeds 1% [14].

The fourth criterion is the possibility to implement a “universal” set of quantum gates. A generic quantum algorithm can be performed using only single- and two-qubit gates, among which only the controlled-NOT (CNOT) gate is required [15]. If these gates cannot be performed directly (e. g. by switching on a certain coupling mechanism), it should be possible to perform a gate using a set of gates giving the same outcome.

The fifth criterion requires the possibility to measure the qubit state. The most obvious reason is that one needs to know the result of the computation. The measurement of the qubits is also necessary for verification steps in quantum error correction codes. It is possible to measure a quantum state only with a certain probability, but complete knowledge of the state can be established by repeating the computation process as many times as neces-

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1.1. Quantum computation 3

e^- e^- e^-

Figure 1.1: Scheme for the physical implementation of a quantum register with quantum dots as in the Loss-DiVincenzo proposal. Electric gates are used for creating a confinement potential for electrons in quantum dots QD1, QD2 and QD3. Additional electric gates control the tunneling between two corresponding quantum dots, which defines the strength of the exchange coupling. If the barrier between two quantum dots is lowered (QD1 and QD2), the interaction between two electron spins is turned on.

sary.

One of the schemes for universal scalable computing was proposed by Loss and DiVincenzo [16]. The main idea of the proposal is to use the spin states of a localized electron as the basis states for quantum computation: |↑i ≡ |0i and|↓i ≡ |1i. It can be applied in general to electrons confined in any kind of structures: atoms, molecules, or defects [17]. The original proposal suggests the physical implementation of a quantum register in an array of electrostatically defined quantum dots with single electrons (see Fig.1.1). Such systems can be created by transverse confinement of electrons in a two-dimentional electron gas, which is formed at a heterostructure interface. Voltages applied to the metallic contacts on top of the heterostructure are used to create a confinement potential. Additional contacts are needed for a controllable coupling between two neighboring dots. Recent developments in fabrication and electron spin control in GaAs single and double quantum dots [18, 17, 19]

make such systems especially suitable for the physical realization of a quantum computer. The scalability of this proposal is based upon the possibility to control the interaction between two adjacent quantum dots independently, without affecting interaction to other qubits.

The initialization of the quantum register can be achieved at low temper- atures with an external magnetic field, such that |gµbB| kBT. Here g denotes the electron g-factor, µ_B is the Bohr’s magneton, and k_B is the Boltzmann constant. At the thermodynamical equilibrium, all qubits will be in the state |↑i. Some faster initialization schemes involve spin-injection from a ferromagnet or using spin-polarized current from a spin-filter device [9, 20]. Single-qubit operations (electron spin rotations) can be performed

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independently on each qubit using electron spin resonance if the electrons have different Zeeman splitting, such that an external oscillating magnetic field affects only a qubit with the matching spin splitting [21]. Zeeman energies specific for each qubit can be achieved by inducing magnetic field gradients [22, 23], using micromagnets [24] or engineering the g-factor [25].

All-electrical control is possible by using the spin-orbit interaction [26,27,28], when the electron spin state is controlled by electrical pulses.

Two-qubit operations are performd by controlling the electrical gate between two quantum dots. The interaction is switched “on”, when the central barrier is lowered for some switching timeτ_g and the electron wave functions overlap.

The interaction between two spins is described by the isotropic Heisenberg Hamiltonian [16, 29]:

H₁₂(t) = J₁₂(t)S₁ ·S₂, (1.1.1) with time-dependent exchange couplingJ₁₂(t), which depends on the central gate voltage (see Fig.1.1). The time evolution of two-spin state is described by the unitary evolution operator:

U12(τg) = Texp h

− i

~ Z τg

0

H12(t)dt i

= exp h

− i

~ Z τg

0

J12(t)dtS1·S2

i , (1.1.2) whereT is the time-ordering operator. If the switching time meets the condi- tionRτg

0 J₁₂(t)dt=π, the state of the two qubits is exchanged. Such an operation is called SWAP. If the interaction is turned on only for a timeτ_g/2, the interaction results in so-called square-root of SWAP: U₁₂(τ_g/2) =U₁₂(τ_g)^1/2. Together with the square-root of SWAP single-spin rotations around the z- axis can be combined into the CNOT gate [16]:

U_CNOT=eⁱ^π²^S¹^ze⁻ⁱ^π²^S^z²U₁₂(τ_g)^1/2e^iπS¹^zU₁₂(τ_g)^1/2. (1.1.3) Two-qubit gates based on electrical control and built from single spin rotations and pulsed exchange interaction have been recently successfully realized experimentally [30] in double quantum dots, as a further achievement besides the square-root SWAP gate demonstrated previously [31]. The qubit measurement can follow also by electrical means [32].

1.2 Optically Active Quantum Dots

Besides nano-structures that enable all-electrical control of spin qubits, there are quantum dots that allow optical spin manipulation. Quantum dots repre- sent structures in which mobile carriers of spin in semiconductors (electrons

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1.2. Optically Active Quantum Dots 5

Figure 1.2: Scheme for electronic band ordering in semiconductor quantum dots in the vicinity of theΓpoint (zinkblende lattice structure). Energies are represented as a function of wavevectork perpendicular to the strain axis on an arbitrary scale.

and holes) are confined along three spacial directions. This situation is com- parable to a particle trapped in a box-like potential, and in analogy to the particle, the energy levels of the confined carriers are quantized. The energy levels can be occupied by two electrons or holes with opposite spin corresponding to the Pauli’s exclusion principle and filled like the orbital atomic levels sequentially from the ground state.

In the case of quantum dots formed by III-V semiconductor compounds with zinkblende lattice symmetry such as gallium arsenide (GaAs) or indium arsenide (InAs), the conduction and valence bands of confined states have certain similarities to the electronic band structure of the bulk materials (see Fig.1.2). The lowest conduction band and the highest valence band are separated by the band gap energy Eg and the split-off band differs in energy from the light and heavy hole bands (the energy difference from the light hole band is denoted with ∆_so in Fig.1.2). Because of the confinement the heavy hole and light hole bands in quantum dots are split. If we consider a certain confinement direction and assume the confinement to be given by a rectangular potential well, the quantized energies for a particle are~²k²/(2m), wherek is the wavevector and m is the mass of the particle. Since the heavy and light holes (what actually gave the names to these states) have different effective masses m_hh and m_lh respectively, they have different confinement energies.

At the Γ point (k=0) the heavy and light holes are split by a confinement

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energy [33]:

|∆hh−lh| ∼ 1

m_lh − 1 m_hh

, (1.2.1)

where the effective masses are taken along the confinement direction. The light hole band has a higher energy than the heavy hole band due to the confinement. However, strain can change the valence band structure and overcompensate the heavy-light hole splitting caused by confinement. This is the case for self-assembled quantum dots, which have the valence sub-band ordering shown in Fig. 1.2.

Quantum dots are sometimes called artificial atoms, since their properties resemble the properties of atoms. They have localized energy eigenstates and the energy separation between levels increases with the confinement potential. Because of the similar physics, the same techniques can be applied for optical control and pumping of the quantum dot states. These techniques are versatile tools for coherent control, initialization and readout (also non- demolition measurement) of the spin states in quantum dots [34, 35,36] and therefore are important for physical realization of quantum computing with spins in quantum dot.

In analogy to atomic states the optical transitions in a quantum dot are possible (or allowed) only between states fulfilling certain symmetry conditions. These conditions are usually comprised to the optical selection rules.

We consider optical transitions between quantum dot eigenstates in the form

|l, j, m_zi and |l⁰, j⁰, m⁰_zi, where l and l⁰ are the quantum numbers for the orbital angular momentum, j an j⁰ describe the total angular momentum states, and m_z and m⁰_z are the projections of the total angular momentum.

The conduction band states in a quantum dot haves-type orbital wave function and can be considered to have l = 0. The valence band states from the heavy hole, light hole and split-off subbands have p-type orbital wave function and therefore havel = 1(see Fig.1.2). In the presence of spin-orbit coupling the optical selection rules for quantum dot states are [33]:

l−l⁰ =±1, j −j⁰ = 0,±1 and mz−m⁰_z = 0,±1. (1.2.2) Optical transitions, where m_z − m⁰_z = ±1 are circularly or σ polarized, since they change the angular momentum (see Fig. 1.3). Transitions, where mz−m⁰_z = 0, are linearly or π polarized.

The optical selection rules enable optical spin-selective initialization of electrons and holes in quantum dots. As shown in Fig.1.3, the optical transitions involving circularly polarized light are allowed between heavy hole, light hole and split-off states and the conduction electron. Creation of an exciton state (consisting of an electron and a hole) with a circularly polarized photons

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1.2. Optically Active Quantum Dots 7

Figure 1.3: Optical allowed transitions in a self-assembled quantum dot: circularly polarized transitions from valence states to conduction states. Rela- tive strength of optical transitions from different valence subbands is shown by the thickness of the arrows.

is possible corresponding to the arrows in the figure (angular momentum changes by ±~). Light frequency is usually chosen in such a way, that split- off states are not excited. If the heavy-light hole degeneracy is lifted by strain, it is also possible to excite only heavy or light holes by tuning the frequency of the photons.

1.2.1 Interaction with electromagnetic radiation of dif- ferent types quantum dots

In general, all kinds of quantum dots interact with electromagnetic radiation, but there are only certain types of quantum dots, that can be called optically active. Their coupling to light must be strong enough such that it is possible to use it for scientific or technological purpose. For example, it does not apply for the electrostatically defined (or electrically gated) quantum dots described in the Section 1.1 and assumed for the Loss-DiVincenzo proposal.

The potential in these quantum dots confines only one type of carriers, in most cases electrons. The confinement potential for holes is very weak and a stable exciton state cannot be formed. However, it does not mean that these quantum dots cannot couple to the electromagnetic radiation field. The concepts for interaction between distant qubits by coupling them to a cavity field can still be applied. Rather than using the spin qubit, the charge qubit is more suitable in this case. As the strength of the optical transition is associated with its dipole moment, the difference in dipole moments of single and two electron states in double quantum dots allows for a coupling to a

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Figure 1.4: Scheme for an optical excitation in a self-assembled InAs/GaAs quantum dot: e. g. σ₊ polarized light creates an exciton with a hole spin-up (in blue) and an electron spin-down (in orange).

superconducting cavity [37, 38,39,40]. Therefore, although electrostatically defined quantum dots are not optically active, they still can be coupled to electromagnetic radiation, namely to microwave radiation in superconducting cavities. Some concepts and techniques for interaction between optically active quantum dots and an optical cavity can also be used and applied for electrostatically defined quantum dots coupled to a superconducting cavity [41].

Besides electrically gated quantum dots, thin quantum wells of GaAs between two barriers of AlGaAs can form another type of dots with high optical quality [33]. These quantum dots are formed naturally due to the atom-thick fluctuations in the quantum well layer. The carriers are trapped in the mono- layer fluctuations of roughly 100 nm size. Both electrons and holes can be confined simultaneously, however the confinement potential is quite weak and only two pairs of carriers can be kept inside of such quantum dots. The properties of the interface fluctuation quantum dots cannot be controlled with the growth techniques and quantum dots with specific characteristics should be selected from many samples.

Another type of optically active quantum dots can be produced by epitax- ial deposition of semiconductor materials layer by layer. If a semiconductor with a slightly smaller lattice constant is grown on a surface of another

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1.2. Optically Active Quantum Dots 9 semiconductor, after some certain thickness the elastic forces will break the deposed layer into the islands. In the case of In(Ga)As grown on GaAs the self-assembled quantum dots (islands) are randomly distributed and have di- ameters of 20-30 nm and heights of 5-10 nm [33]. A possible mechanism for an optical excitation in such a quantum dot is represented in Fig.1.4. Quan- tum dots of different sizes have varying electronic energies, therefore due to the size distribution there is a certain variety for the emission wavelengths.

When a quantum dot with desired optical properties is found, it can be easily incorporated into an optical cavity. A clever technique employs the fabrication of two-dimentional slab photonic microcavity directly around the chosen quantum dot [42]. The integration of a quantum dot into a cavity enables the all-optical control, initialization and measurement of spin qubits [43] and at the same time provides optically mediated interaction with another, spatially separated quantum dot [44].

1.2.2 Interaction between quantum dots and a cavity

The coupling of a quantum dot to electromagnetic radiation field can be am- plified by inserting it into a cavity. A cavity represents a structure, where the electromagnetic field is spatially confined, and the light irradiates the emitter many times while it is reflected from the cavity walls. Photons escape from a cavity with a rate κ called cavity loss rate. Because of this photon decay, the cavity mode has a finite width∆ω_c, which together with the cavity mode frequencyωc defines the cavity quality factorQ=ωc/∆ωc.

For the experimental realization of a considerable coupling between an emitter and a cavity, the cavity mode must spectrally match the emission wave- length of the emitter. The spatial matching requires that the emitter (e. g.

quantum dot) is placed at the maximum of the electric field of the cavity mode. The polarization of the quantum dot transition must also be as close as possible to the polarization of the cavity mode [33].

We assume that the cavity is on resonance with the optical transition of a quantum dot and there are only 0 or 1 photon in the cavity mode. Then the ground state of the quantum dot |gi (e. g no confined electrons) couples to the excited state |ei (e. g. exciton) by means of the light-matter interaction and the coupling strength is given by ~g =|he,0|E·d|g,1i| [45]. The operator E denotes the the electrical field at the position of the quantum dot and d is the dipole moment of the transition |ei → |fi. Depending on the coupling strength, different coupling regimes can be distinguished for the cavity quantum electrodynamics (cavity QED) [46].

In the weak coupling regime the coupling strength is less or on the same range as the relevant relaxation rates: g .κ, γ, where γ is the spontaneous

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emission rate for the quantum dot (typically around 1 ns). In this regime, the spontaneous emission of the emitter is modified, but the decay of the pop- ulation is still described by an exponential function. The decay rate can be enhanced, because of the specifically designed electromagnetic field around the emitter. According to Fermi’s golden rule, spontaneous relaxation rate of an emitter is proportional to the density of final states. The density of final states in a cavity (not necessarily the number of states) is higher than the free space density of states. Therefore the spontaneous emission rate of an emitter in a resonant cavity is larger compared to the spontaneous emission rate in free space. This phenomenon is known as Purcell effect [47] and was observed for quantum dots experimentally [33, 45]. Control of the emission rate of quantum dots is an important tool for improving their properties as single photon sources. The incorporation of a quantum dot into a cavity in the weak coupling regime is also used for an efficient extraction and collection of the quantum dot emission.

In the strong coupling regime the coupling strength is considerably larger than the relaxation rates: g κ, γ. The lifetime of the photons in the cavity is sufficient long that they can be reabsorbed by the emitter and emitted again. The strong coupling regime is manifested by the formation of

“dressed states” and consequently by the energy shifts of the photon and emitter states. This effect is observed in the vacuum Rabi splitting of a single cavity mode into two peaks corresponding to the energies of the dressed photon states. Since the linewidth of the cavity mode is given by ¹₂(κ+γ) and the splitting of the dressed states by 2g, it can indeed be observed by spectroscopic means only for large g compared to κ and γ. In the time domain, photon emission exhibits damped oscillations instead of a strictly exponential decay.

The strong coupling between a quantum dot and a cavity was realized in experiments [48, 49] and represents a useful mechanism for quantum information schemes with optically active quantum dots. This regime allows to have an interface between a quantum dot state and a photon state and to make use of schemes, where the state of a stationary qubit is transferred to a flying qubit. This allows quantum communication and information exchange [50]. If in the Loss-DiVincenzo proposal the two-qubit interaction can be implemented only between two neighboring qubits, coupling between distant qubits can be achieved for qubits interacting with a single cavity mode in the strong coupling regime [41, 51].

To describe an emitter (labeled withi) coupled to a cavity mode we use the Jaynes-Cummings Hamiltonian [33]:

H_JC= ¹₂~ω_q,iσ^z_i +~ω_ca^†a+~g_i a σ_i⁺+a^†σ⁻_i

, (1.2.3)

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1.2. Optically Active Quantum Dots 11 where the pseudo-spin operators are σ_i^z = (|ei_ihe|_i − |gi_ihg|_i), σ⁺_i =|ei_ihg|_i and σ⁻_i = |gi_ihe|_i. The energy difference between the ground and excited states of the emitter is given by~ω_q,iand the frequency of the cavity mode is denoted by ω_c. The cavity photon creation/annihilation operators are a/a^† and the interaction energy is ~g_i. The coupled system is in the dispersive regime when the cavity mode is not in resonance with the emitter transition;

the detuning of the cavity mode is given by∆_i =ω_q,i−ω_c6= 0. In this case, the interaction can be treated perturbatively (g_i/∆_i 1) and the effect of the coupling can be best seen by applying a Schrieffer-Wolff transformation [52, 53,54, 55].

In the following, we briefly outline general concepts of the Schrieffer-Wolff transformation for a Hamiltonian of the form:

H =H₀+V

=H0+Vd+Vod, (1.2.4)

where H₀ is the diagonal interaction free part of H and the interaction V is decomposed into a diagonal partV_d and an off-diagonal partV_od. In general an exact diagonalization of H is a difficult procedure, however, it is possible to find easily a transformation that perturbatively diagonalizes H [54, 55].

We assume thatHis diagonalised by a unitary transformationU = e^S, where S is an antihermitian operator. We can write

H →e^SHe^−S =

∞

X

j=0

1

j![S, H]^(j), (1.2.5) where we have defined the recursive relation:

[S, H]⁽⁰⁾ =H, [S, H]⁽¹⁾ = [S, H], [S, H]^(j)=h

S,[S, H]^(j−1)i .

(1.2.6)

The idea of the Schrieffer-Wolff transformation is to look for S as a pertur- bative series :

S =

∞

X

k=1

S^(k), (1.2.7)

where S^(k) ≡ O λ^k

and it was assumed that V ≡ O(λ). By imposing that S^(k)must cancel the non-diagonal part of H up to order k, we find equations

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for the successive terms of the series defined in Eq. (1.2.7):

S⁽¹⁾, H₀

=−V_od S⁽²⁾, H₀

=−

S⁽¹⁾, V_d S⁽³⁾, H₀

=−

S⁽²⁾, V_d

−1 3

S⁽¹⁾,

S⁽¹⁾, V_od . . .

(1.2.8)

An elegant way to find S^(k) can be obtained by using a superoperator L₀ given by

L₀O = [H₀, O]. (1.2.9)

With this definition the equation defining, e. g. S⁽¹⁾, becomes

L₀S⁽¹⁾ =V_od. (1.2.10)

A formal solution for S⁽¹⁾ is then given by

S⁽¹⁾ =L⁻¹₀ V_od. (1.2.11) The right hand side of Eq. (1.2.11) can be expressed as a solution of the following integral:

S⁽¹⁾ =−i

~ limε→0

Z ∞ 0

dt e^−εte^iL⁰^t/^~V_od, (1.2.12) whereε >0is a convergence parameter. We obtain by expendingexp (iL₀t/~) into a series and using the definition ofL₀:

S⁽¹⁾ =−i

~ limε→0

Z ∞ 0

dt e^−εt

∞

X

j=0

(it/~)^j

j! [H₀, V_od]^(j). (1.2.13) In analogy to Eq. (1.2.5), we identify the expansion of V_od in the interaction picture,

V_od^I =e^iH⁰^t/^~V_ode^−iH⁰^t/^~ =

∞

X

j=0

(it/~)^j

j! [H₀, V_od]^(j). (1.2.14) Combining Eq. (1.2.12) and Eq. (1.2.14), we can write

S⁽¹⁾ =−i

~lim

ε→0

Z ∞ 0

dt e^−εte^iH⁰^t/^~V_ode^−iH⁰^t/^~, (1.2.15)

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1.2. Optically Active Quantum Dots 13 which expresses S⁽¹⁾ as the integral of V_od in the interaction picture. Simi- larly S^(k) can be found by computing the integral of the right hand side of Eq. (1.2.8) in the interaction picture.

We use this method to find an effective Hamiltonian describing the interaction between an emitter and a cavity. The HamiltonianH_JC defined in Eq. (1.2.3) can be divided into a diagonal part,

H₀ = ~

2ω_q,iσ^z_i +~ω_ca^†a, (1.2.16) and an off-diagonal interaction,

V_od=~g_i aσ_i⁺+a^†σ_i⁻

. (1.2.17)

In the case of the Jaynes-Cummings Hamiltonian, we haveV_d = 0. It implies according to Eq. (1.2.8) that expandingS_i up to orderg_i/∆_igives an effective Hamiltonian, that is correct up to order(g_i/∆_i)². The interaction picture of the Hamiltonian Eq. (1.2.17) is given by:

V_odÎ =~g_ie²ⁱ^ω^q,i^σⁱ^z^teîω^câ^†ât aσ⁺_i +a^†σ_i⁻

e⁻²ⁱ^ω^q,i^σ^zⁱ^te^−iω^c^a^†^at (1.2.18) where we used the fact that σ_i^z commutes with both a and a^†. To simplify the above equation we first find

σ_I,i⁺ =e²ⁱ^ω^q,i^σ^zⁱ^tσ⁺_i e⁻²ⁱ^ω^q,i^σⁱ^z^t. (1.2.19) Applying the Euler formula for the Pauli matrices yields,

σ⁺_I,i = h

cos ωq,i

2 t

1+isin ωq,i

2 t

σ_i^z i

σ⁺_i h

cos ωq,i

2 t

1−isin ωq,i

2 t

σ_i^z i

. (1.2.20) After an explicit calculation we have:

σ_I,i⁺ =

cos²ωq,i

2 t

σ_i⁺+icosωq,i

2 t

sinωq,i

2 t

σ^z_i, σ_i⁺

+isin²ωq,i

2 t

σ_i^zσ⁺_i σ_i^z. (1.2.21) By using the commutation relation [σ^z, σ⁺] = 2σ⁺, the relation σ^zσ⁺σ^z =

−σ⁺, and trigonometric relationssin 2ϑ= 2 sinϑcosϑ andcos 2ϑ= cos²ϑ− sin²ϑ, we obtain

σ_I,i⁺ =e^iω^q,i^tσ_i⁺. (1.2.22)

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With a similar calculation it can be shown that

σ_I,i⁻ = e^−iω^q,i^tσ⁻_i . (1.2.23) We perform now an identical calculation for the cavity operators:

a_I=eîω^câ^†âta e^−iω^câ^†ât. (1.2.24) To evaluate Eq. (1.2.24), we use the expansion defined in Eq. 1.2.5, but with S → iω_ct a^†a and H → a. The series can be evaluated exactly by applying the commutation relation

a^†a, a

=−a and we have

a_I=e^−iω^c^ta. (1.2.25)

Analogically we obtain

a^†_I =e^iω^c^ta^†. (1.2.26) By placing Eqs. (1.2.22), (1.2.23), (1.2.25), and (1.2.26) into Eq. 1.2.18 we finally find

V_od^I =~g_i e^i∆ⁱ^taσ_i⁺+e^−i∆ⁱ^ta^†σ⁻_i

(1.2.27) It is now straighforward to evaluateS_i:

S_i =−ig_ilim

ε→0

Z ∞ 0

dt e^{−(ε−i∆}ⁱ^)taσ_i⁺+e^−(ε+i∆ⁱ^)ta^†σ_i⁻

=−ig_ilim

ε→0

1

ε−i∆_iaσ⁺_i + 1

ε+i∆_ia^†σ_i⁻

= g_i

∆_i aσ_i⁺−a^†σ_i⁻ .

(1.2.28)

The transformed Jaynes-Cummings Hamiltonian is then given up to second order in g_i/∆_i by:

H_JC^eff = H_JC+ [S_i, H_JC] + ¹₂[S_i,[S_i, H_JC]] +O g_i³

∆³_i

= ¹₂~ω_q,iσ_i^z+~ω_ca^†a+ ~g²_i

∆_i a^†a+ ¹₂

σ_i^z. (1.2.29) The last term in the last line represents the ac Stark shift and the Lamb shift. The ac Stark shift is proportional to ±g_i²a^†a/∆_i and its sign depends on the state of the qubit. It induces a shift of the cavity resonance and can be used to measure the state of the qubit. At the same time the ac Stark shift also produces a shift of the energy levels of the emitter depending on the number of photons. It can be seen as an effective magnetic field B_Stark

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1.2. Optically Active Quantum Dots 15 for spin states. If two circularly polarized modes are taken into account for the Schrieffer-Wolff transformation (1.2.29), the effective Zeeman splitting is defined by the difference between the intensity of the right (+) and left (−) circularly polarized light:

gµ_BB_Stark= ~g_i²

∆_i

a^†₊a₊−a^†₋a₋

. (1.2.30)

This way the ac Stark effect provides the possibility for effectively applying magnetic pulses using circularly polarized light pulses. These tipping pulses were used in the experiment by Berezovskyet al. [56] for manipulating single electron spin states in a quantum dot.

We can extend our analysis by adding another emitter (qubit) coupled to the same cavity mode. The new Hamiltonian reads

H_JC=~ω_ca^†a+ X

i=1,2 1

2~ω_q,iσ^z_i +~g_i a σ⁺_i +a^†σ_i⁻

. (1.2.31)

The couplings can be assumed equal for simplicity: g₁ = g₂ = g. The effective Hamiltonian can again be obtained by applying the Schrieffer-Wolff transformation defined by S =S₁+S₂ with S_i (i= 1,2) given by Eq. (??).

The resulting Hamiltonian to leading order is [41, 51, 57]:

H_JC^eff = ~

ω_c+2g²

∆₁σ₁^z+2g²

∆₂σ₂^z

a^†a+~ X

i=1,2

ω_q,i 2 + g²

∆_i

σ^z_i +~

2 g²

∆1

+ g²

∆2

σ⁺₁σ⁻₂ +σ⁻₁σ₂⁺

. (1.2.32)

The last term of the Hamiltonian (1.2.32) represents processes, when a photon is emitted into the cavity mode by one emitter and absorbed by the other.

It describes an effective coupling mechanisms between two qubits, which is mediated by a common cavity mode. This type of interaction can be used for constructing two-qubit gates for spin qubits in quantum dots and coupled to an optical cavity mode by means of stimulated Raman transitions [51].

1.2.3 Coupling of spins in a cavity

We assume a quantum dot placed into an optical cavity and an external magnetic field in the x direction (the configuration with a magnetic field perpendicular to the growth direction z is known as Voigt geometry). The Hamiltonian describing the spins in the quantum dot consists of two parts:

H =H₀+H_int, (1.2.33)

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with (~= 1) H₀ =ω_Z^e

2 (|↑i_xh↑|_x− |↓i_xh↓|_x) +ω_v(|⇑i_zh⇑|_z+|⇓i_zh⇓|_z) +ω_ca^†_ca_c+ω_La^†_La_L,

(1.2.34)

where the states |↑/↓i_x ≡ |m^x_e =±1/2i denote the electron spin states along the external magnetic field and the energyω_Z^e is the Zeeman splitting.

The states |⇑/⇓i_z ≡ |m^z_hh=±3/2i are the heavy hole states of the valence band. We denote the energy of the valence band states ωv. Due to the confinement (and/or strain) the energies of the heavy-hole up and down states are non-degenerate. Here, we can set the splitting between heavy hole up and heavy hole down states to zero, since it is negligible compared to the Zeeman splittingω_Z^e. The annihilation and creation operators with indices c and L describe the cavity and the laser field respectively, the corresponding frequencies are ωc and ωL. The interaction part is given by

H_int=g

a^†₊|⇓i_zh↓|_z+a^†₋|↑i_zh⇑|_z+h.c.

(1.2.35) and describes the allowed optical transitions between conduction and valence band states. The circularly polarised optical field is composed of the x- polarised cavity mode and the y-polarised laser light:

a±= (a_c±ia_L)/√

2. (1.2.36)

The electron spin eigenstates along thez direction can be expressed in terms of the eigenstates in the xdirection:

|↑/↓i_z = (|↓i_x± |↑i_x)/√

2. (1.2.37)

We make a further replacement:

|vi= (|⇓i_z +|⇑i_z)/2, (1.2.38) to obtain the Hamiltonian H₀:

H₀ = ω_Z^e

2 (|↑i_xh↑|_x− |↓i_xh↓|_x) +ω_v|vi hv|+ω_ca^†_ca_c+ω_La^†_La_L, (1.2.39) and the interaction Hamiltonian

H_int =g

a^†_c|vi h↑|_x−ia^†_L|vi h↓|_x+|↑i_xhv| a_c+i|↓i_xhv|a_L

, (1.2.40)

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1.2. Optically Active Quantum Dots 17 in the new representation. An effective Hamiltonian without valence band states can be obtained by performing a canonical Schrieffer-Wolff transformation [51, 58]. By using Eq. (1.2.15), we have

S⁽¹⁾ =−iglim

ε→0

Z ∞ 0

dt e^−εt e^−i∆^c^ta^†_c|vi h↑|_x+e^i∆^c^t|↑i_xhv| ac

−glim

ε→0

Z ∞ 0

dt e^−εt

e^−i∆^L^ta^†_L|vi h↓|_x−e^i∆^L^t|↓i_xhv|aL

,

(1.2.41)

where ∆_c is the cavity detuning given by

∆_c= ω^e_Z

2 −ω_v−ω_c, (1.2.42)

and the laser detuning ∆_L is defined as

∆_L=−ω^e_Z

2 −ω_v−ω_L. (1.2.43)

After integration of Eq. (1.2.41), we obtain:

S⁽¹⁾ = g

∆_c a^†_c|vi h↑|_x− |↑i_xhv|a_c

−i g

∆_L

a^†_L|vi h↓|_x+|↓i_xhv|a_L . (1.2.44) The effective interaction Hamiltonian to leading order in g²/∆_c and g²/∆_c is given by [51, 58]:

H˜int = 1

2[Hint, S⁽¹⁾]

= g²

∆c

a^†_ca_c|↑i_xh↑|_x+ g²

∆L

a^†_La_L|↓i_xh↓|_x + ig²

2 1

∆_L + 1

∆_c

a^†_c|↓i_x|↑i_xaL−a^†_L|↑i_xh↓|_xac

.

(1.2.45)

The first two term in the second line of Eq. (1.2.45) describe the ac Stark shifts induced by the cavity and the laser field respectively. We use the semi- classical description for the laser field and replacega_LbyΩ_Lexp(−iω_Lt). The effective Hamiltonian describing the interaction between spins in a quantum dot with the index i, an optical cavity and laser light is given by [51, 58]:

H˜_i =ω_ca^†_ca_c+ ω_Z,i^e

2 + g²

∆ⁱ_ca^†_ca_c

|↑iⁱ_xh↑|ⁱ_x+ Ω²_L

∆ⁱ_L,i − ω_Z,i^e 2

!

|↓iⁱ_xh↓|ⁱ_x +igⁱ_eff

a^†_c|↓iⁱ_xh↑|ⁱ_xe^−iω^Lⁱ^t−h.c.

.

(1.2.46)

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The effective two-photon coupling for the quantum dot with index iis given by

˜

g_effⁱ = gΩⁱ_L,i 2

1

∆ⁱ_L + 1

∆ⁱ_c

, (1.2.47)

and the two-photon resonance condition is satisfied, when ∆_c = ∆_L. A further Schrieffer-Wolff transformation can be used to eliminate the cavity degrees of freedom in the Hamiltonian (1.2.46) and to obtain an effective interaction Hamiltonian describing the coupling between two quantum dots.

The generator of the second transformation is S˜_i⁽¹⁾ = igⁱ_eff

∆_i (a^†_c|↓iⁱ_xh↑|ⁱ_xe^−iωⁱ^L^t+h.c.), (1.2.48) where∆i = ∆ⁱ_c−∆ⁱ_L is the two-photon detuning. The resulting Hamiltonian in the interaction picture defined by

H_0,Z= ω^e_Z

2 (|↑i_xh↑|_x− |↓i_xh↓|_x), (1.2.49) is given by:

H˜_int⁽²⁾ =X

i6=j

gⁱ_effg^j_eff 2∆i

(σ_↑↓ⁱ σ^j_↓↑e^−i(∆ⁱ^−∆^j^)t+σ^j_↑↓σ_↓↑ⁱ e^i(∆ⁱ^−∆^j^)t), (1.2.50) where the projection operators are denoted by σ_↑↓ⁱ = |↑iⁱ_xh↓|ⁱ_x and σⁱ_↓↑ =

|↓iⁱ_xh↑|ⁱ_x. By choosing the laser frequencies ω_Lⁱ and ω_L^j, so that ∆_i−∆_j = 0, the Raman coupling to a common cavity mode enables long-distance controllable interaction between two electrons spins. If we set∆_i−∆_j = 0, we obtain a transverse spin-spin interaction:

H_int,ij⁽²⁾ = ˜g_ij

2 (σⁱ_yσ_y^j+σ_zⁱσ_z^j), (1.2.51) where σy and σz are the Pauli matrices. The effective coupling strength between two electron spins is given by

˜

g_ij = gⁱ_{ef f}g^j_{ef f}

2∆_i . (1.2.52)

1.3 Relaxation and decoherence

One important feature for a quantum system to be operated as a qubit is that it should possess sufficiently long coherence and lifetimes. In the beginning

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1.3. Relaxation and decoherence 19

Figure 1.5: (a) Schematic representation of a relaxation process on the Bloch sphere for an initialized spin up state. The spin is flipped to the ground state by emitting energy to the environment. A typical mechanism in quantum dots is a spin-orbit mediated spin flip with emission of a phonon. (b) Time evolution of the spin expectation value along z axis. Forz kBT it expo- nentially decays to zero, where k_B is the Boltzmann constant and T is the phonon bath temperature.

of the era of developing quantum computing with spin qubits in quantum dots this was one of the main challenges. There are three important time scales for describing coherence of a quantum system: the relaxation time T₁, the decoherence time T₂ and the inhomogeneous dephasing time T₂^∗. We can make an example for a spin qubit with states |↑i and |↓i, which are eigenstates of the Pauli operator σ_z and are split by the energy _z. The relaxation from |↑i to |↓i, caused by the interaction with the environment (e. g. with the lattice), is described by the relaxation timeT1, called spin-flip time (see Fig. 1.5). The relaxation process involves energy dissipation to the environment. This time scale gives the decay time of hσ_zi(t), with the system being initialized in state |↑i [59]. The decoherence time T2 describes the decay of a superposition state |ψi = α|↑i +β|↓i, more precisely loss of the relative phase between |↑i and |↓i (see Fig. 1.6). This process does not involve energy exchange with the environment. The time T2 is called decoherence time, because it gives the decay time for the coherence terms of the density matrix. The density matrix of the superposition relaxes afterT₂

Nuclear Spin Phenomena in Optically Active Semiconductor Quantum Dots

Nuclear Spin Phenomena in Optically Active Semiconductor

Quantum Dots

Prof. Dr. Guido Burkard Prof. Dr. Wolfgang Belzig

Abstract

Zusammenfassung

Acknowledgments

Contents

1

Introduction

1.1 Quantum computation

1.2 Optically Active Quantum Dots

1.2.1 Interaction with electromagnetic radiation of dif- ferent types quantum dots

1.2.2 Interaction between quantum dots and a cavity

1.2.3 Coupling of spins in a cavity

1.3 Relaxation and decoherence