sonality: Edo Waks, Will Oliver, Anne Verhulst, David Fattal, Jonathan Goldman, ... Finally, I would like to thank Susan Clark for her enthusiasm, ideas, and work.
OPTICAL MANIPULATION OF ELECTRON SPINS BOUND TO NEUTRAL DONORS IN GAAS
A DISSERTATION SUBMITTED TO THE DEPARTMENT OF APPLIED PHYSICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
KaiMei C. Fu June 2007
c Copyright by KaiMei C. Fu 2007 ° All Rights Reserved
ii
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
(Yoshihisa Yamamoto) Principal Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
(Stephen E. Harris)
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
(David A.B. Miller)
Approved for the University Committee on Graduate Studies.
iii
iv
Abstract Donorbound electrons coupled to excitons provide a promising system for quantum information processing (QIP) applications. The bound electron spin forms a spinqubit system and exhibits strong radiative coupling to the bound exciton state with relatively little inhomogeneous broadening of the optical transitions. For any practical QIP system two issues must be addressed the dephasing time of the qubit and the ability to manipulate the qubit coherently. This work addresses both of these issues. The inhomogeneous dephasing time, T∗2 , of the donor electron spin is studied in a Raman scattering experiment. The observed nanosecond dephasing time is consistent with the nuclear hyperfine model. If this inhomogeneous dephasing is eliminated by spinecho techniques, the homogeneous decoherence time, T2 , should ultimately be limited by the measured millisecond T1 time. The optical manipulation of the electron spin can be achieved using twophoton stimulated Raman transitions. This control is demonstrated in two separate experiments. The first is a coherent population trapping experiment in which a superposition of the spin states is created with two continuouswave fields. In the second experiment, the spin state is controlled on a picosecond timescale using a single, offresonant, ultrafast optical pulse. This second technique could open the way for a new, general electron spin resonance technique in which spins can be probed and manipulated on timescales much shorter than the Larmor precession period.
v
Acknowledgements This work would not have been possible without the collaboration, help, and support of many people. I would first like to thank my advisor Yoshihisa Yamamoto for giving me the opportunity to do this research. I cannot imagine a better environment in which to do graduate research. I am thankful for the freedom I had to explore the physics I found interesting, as well as the guidance I received to keep me on a fruitful path. Additionally I would like to thank the other members of my reading committee: Stephen Harris, who first introduced me to Λtype systems and EIT, and David Miller. I am also indebted to Charles Santori. I had the opportunity to work closely with Charlie for my third and fourth year (for the acceptor and CPT work). Much of the knowledge I have gained about experimental optics I have learned from him. Even after he moved on to HP, our discussions and his advice continued to have a large impact on my research. I would like to thank the past and present members of the Yamamoto group. These individuals have given the group its unique, stimulating, and collaborative personality: Edo Waks, Will Oliver, Anne Verhulst, David Fattal, Jonathan Goldman, Cyrus Master, Na Young Kim, Gregor Weihs, Jocelyn Plant, Stephan G¨otzinger, Katsuya Nozawa, Patrik Recher, Kaoru Sanaka, Alex Pawlis, Hui Deng, Eleni Diamanti, Shinichi Koseki, Neil Na, David Press, Georgios Roumpos, Jason Pelc and Darin Slater. I would particularly like to thank Thaddeus Ladd for his never ending energy in the search for understanding (in both theory and experiment) and Wenzheng Yeo for the effort he took getting the T1 experiment running as part of his undergraduate vi
thesis. Finally, I would like to thank Susan Clark for her enthusiasm, ideas, and work on all of the experiments presented in Chapters 4, 5, and 6. This work was truly a joint effort and through her efforts and others I hope it will continue. I would also like to thank Suzanne Yelin and Renuka Medhavini for the many fruitful discussions on the acceptor bound exciton system. I would like to thank the Applied Physics department and the Yamamoto group administrative staff, Paula Perron, Claire Nicholas, Yurika Peterman, Rieko Sasaki and Mayumi Hakkaku, for all their help over the years. A special note of gratitude goes to the many sample growers who have made my research possible. I would like to thank Colin Stanley and M.C. Holland for the Glasgow MBE GaAs samples (most famous of which is b923), Bingyang Zhang for the Stanford MBE GaAs samples, Simon Watkins for the MOCVD GaAs and InP samples, M.L.W. Thewalt for the GaAs epitaxial layers, and H. Hiroyama for the ptype MBE GaAs samples. I would also like to thank Manual Cardona for the opportunity to study isotopically purified CdSe and ZnSe samples. Finally, I would like to thank my friends, my family, and my husband Michael Preiner for their support and love throughout the years.
vii
Contents Abstract
v
Acknowledgements
vi
1 Introduction
1
2 GaAs D0 D0 X system
4
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.2
Effective mass theory and GaAs shallow impurities . . . . . . . . . .
5
2.2.1
Effective mass theory . . . . . . . . . . . . . . . . . . . . . . .
5
2.2.2
EMT in action: Carbon doped GaAs . . . . . . . . . . . . . .
6
2.2.3
Neutral donors in a magnetic field . . . . . . . . . . . . . . . .
7
GaAs neutraldonorbound exciton . . . . . . . . . . . . . . . . . . .
8
2.3.1
GaAs D0 X at 0 T . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.3.2
GaAs D0 X in an external magnetic field . . . . . . . . . . . .
8
Spectroscopic study of the D0 D0 X Λ system . . . . . . . . . . . . . .
9
2.3
2.4
2.4.1
Magnetophotoluminescence spectroscopy . . . . . . . . . . . .
10
2.4.2
Photoluminescence excitation spectroscopy (PLE) . . . . . . .
10
2.4.3
Twolaser PLE . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.5
Comparison to other Λ systems . . . . . . . . . . . . . . . . . . . . .
13
2.6
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
3 CPT of D0 electron spins 3.1
18
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
18
3.2
The dark state and CPT . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2.1
The dark state . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2.2
Coherent population trapping . . . . . . . . . . . . . . . . . .
20
3.3
Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . .
22
3.4
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
4 D0 electron spin Raman study
29
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
4.2
Nuclear hyperfine interaction . . . . . . . . . . . . . . . . . . . . . . .
31
4.3
Raman theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.4
Raman linewidth measurement . . . . . . . . . . . . . . . . . . . . .
35
4.4.1
Experimental procedure . . . . . . . . . . . . . . . . . . . . .
36
4.4.2
Temperature dependence . . . . . . . . . . . . . . . . . . . . .
36
4.4.3
Magnetic field dependence . . . . . . . . . . . . . . . . . . . .
37
4.4.4
Density dependence . . . . . . . . . . . . . . . . . . . . . . . .
39
4.4.5
Two different donor gfactors . . . . . . . . . . . . . . . . . .
41
4.4.6
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Optical pumping of nuclei . . . . . . . . . . . . . . . . . . . . . . . .
42
4.5.1
Optical pumping of GaAs . . . . . . . . . . . . . . . . . . . .
44
4.5.2
Experimental procedure . . . . . . . . . . . . . . . . . . . . .
46
4.5.3
Results and discussion . . . . . . . . . . . . . . . . . . . . . .
46
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.5
4.6
5 Millisecond spinflip times of donor electrons
51
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
5.2
Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . .
52
5.3
Confirmation of optical pumping . . . . . . . . . . . . . . . . . . . .
55
5.4
Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
56
5.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
ix
6 Optical manipulation of the D0 spin
58
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
6.2
Ultrafast rotations: Theory
. . . . . . . . . . . . . . . . . . . . . . .
60
6.2.1
Adiabatic elimination of the excited state . . . . . . . . . . . .
61
6.2.2
Threelevel simulations . . . . . . . . . . . . . . . . . . . . . .
63
Ultrafast rotations: Experiment . . . . . . . . . . . . . . . . . . . . .
69
6.3.1
Observation of Raman population transfer . . . . . . . . . . .
69
6.3.2
Observation of 40 GHz electron spin precession . . . . . . . .
74
6.3.3
Ultrafast dephasing in the D0 D0 X system . . . . . . . . . . .
78
6.3.4
Yet another T∗2 measurement
. . . . . . . . . . . . . . . . . .
82
6.4
Spinecho using small angle pulses . . . . . . . . . . . . . . . . . . . .
83
6.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
6.3
7 Summary and Outlook
87
A GaAS A0 X study
90
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
A.2 Hydrogenic spectra of acceptors in GaAs . . . . . . . . . . . . . . . .
91
0
A.3 Lifetimes and selection rules of the A X complex . . . . . . . . . . . .
93
A.3.1 Energy levels and basis states . . . . . . . . . . . . . . . . . .
93
A.3.2 Selection rules, WignerWeisskopf theory . . . . . . . . . . . .
94
0
A.3.3 Polarization of A X spontaneous emission for selected excitation conditions . . . . . . . . . . . . . . . . . . . . . . . . . .
96
A.4 Timeresolved measurements . . . . . . . . . . . . . . . . . . . . . . .
98
A.5 Pumpprobe study of the A0 bound hole relaxation . . . . . . . . . .
99
A.5.1 Experimental technique . . . . . . . . . . . . . . . . . . . . . 100 A.5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 101 A.6 Basis states and dipole operator . . . . . . . . . . . . . . . . . . . . . 102 A.6.1 Basis states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 A.6.2 Dipole operator . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Bibliography
104 x
List of Tables 2.1
Comparison of essential D0 D0 X parameters to atomic systems. . . .
15
2.2
Parameters of other promising solidstate systems. . . . . . . . . . . .
16
4.1
Samples studied in the density dependence experiment. . . . . . . . .
41
A.1 Normalized dipole matrix elements for the Γ5 A0 transitions.
. . . .
95
A.2 Normalized dipole matrix elements for the Γ3 A0 transitions. . . . . .
95
xi
List of Figures 1.1
Threelevel Λ system. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2.1
EMT hydrogenic spectrum in the GaAs A0 XA0 system. . . . . . . .
7
2.2
Photoluminescence spectra of highpurity MBE GaAs.
. . . . . . . .
9
2.3
D0 D0 X Λ system studied in this work. . . . . . . . . . . . . . . . . .
10
2.4
Magnetophotoluminescence spectrum of the GaAs D0 D0 X system .
11
2.5
Twoelectron satellites (TES) in the D0 D0 X system. . . . . . . . . .
11
2.6
PL and PLE spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
2.7
Diagram of electron spin optical pumping in the D0 D0 X system. . . .
13
2.8
Onelaser and twolaser PLE scans. . . . . . . . . . . . . . . . . . . .
14
3.1
Energy diagrams for the GaAs D0 D0 X CPT experiment . . . . . . .
19
3.2
Theoretical CPT model for the population in the excited state 3i as a function of detuning ∆. . . . . . . . . . . . . . . . . . . . . . . . . .
22
0
3.3
Aboveband and resonant excitation photoluminescence spectra of D X 23
3.4
Coherent population trapping PLE scans . . . . . . . . . . . . . . . .
25
3.5
Dependence of CPT on coupling laser intensity. . . . . . . . . . . . .
27
4.1
Raman scattering energy diagrams. . . . . . . . . . . . . . . . . . . .
34
4.2
Raman spectra measured using a grating spectrometer. . . . . . . . .
34
4.3
Experimental setup for the Raman linewidth measurement. . . . . .
37
4.4
Temperature dependence of the Raman linewidth. . . . . . . . . . . .
38
4.5
Magnetic field dependence of the Raman linewidth. . . . . . . . . . .
38
4.6
ensity/sample dependence of the Raman linewidth. . . . . . . . . . .
39
xii
4.7
Raman splitting for two different GaAs donors, Si and S. . . . . . . .
42
4.8
Selection rules for σ − polarized light in bulk GaAs. . . . . . . . . . .
45
4.9
Experimental setup of the optical pumping experiment. . . . . . . .
47
4.10 Raman measurements of the Stokes and Antistokes splitting after optical pumping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.11 Relaxation of the Raman splitting and nuclear polarization after optical pumping.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
5.1
Energy level diagrams for the T1 experiment. . . . . . . . . . . . . . .
52
5.2
Pumpprobe T1 data at 9.9 T for a 30 µs delay. . . . . . . . . . . . .
53
5.3
Experimental setup for the T1 measurement. . . . . . . . . . . . . .
54
5.4
Onelaser and twolaser PLE scans demonstrating optical pumping of the D0 electron spin. . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
5.5
Loglog plot of the bound electron spin lifetime T1 dependence on Bfield. 57
6.1
Energy diagram for fastpulse Raman rotations. . . . . . . . . . . . .
6.2
Numerical simulation with 2 ps pulses including the relaxation operator
60
L(ρ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
6.3
Numerical simulation with 2 ps pulses excluding L(ρ). . . . . . . . . .
65
6.4
Numerical simulation with 100 fs pulses including the relaxation operator L(ρ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5
66
Numerical simulation with 100 fs pulses including the relaxation operator L(ρ) in the case where Ωp = 2Ωc . . . . . . . . . . . . . . . . . .
67
6.6
Λ system with two excited states. . . . . . . . . . . . . . . . . . . . .
67
6.7
Experimental setup for the singlepulse and twopulse experiment. .
70
6.8
Thermal relaxation experiment. . . . . . . . . . . . . . . . . . . . . .
71
6.9
Typical singlepulse experimental data. . . . . . . . . . . . . . . . . .
72
6.10 Polarization dependence of the singlepulse population transfer. . . .
73
6.11 Power dependence of the singlepulse electron spin population transfer. 74 6.12 Theoretical simulations for the twopulse experiment. . . . . . . . . .
75
6.13 Twopulse experiment. Population in state 2i is plotted as a function of the twopulse delay. . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
76
6.14 A fit of the twopulse experimental data to a twolevel model including Rabi rotations, decoherence, and Larmor precession. . . . . . . . . . .
77
6.15 Experimental data and threelevel simulations with (a) finite D0 X γ3 dephasing and (b) finite D0 γ2 dephasing. . . . . . . . . . . . . . . . .
79
6.16 Twopulse experimental visibility for long delays. . . . . . . . . . . .
82
A.1 A0 X A0 photoluminescence demonstrating the hydrogenic nature of the EMT acceptor. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
A.2 Energy levels of the A0 X and A0 complexes. . . . . . . . . . . . . . .
94
A.3 Streak camera data of the polarization visibility for special excitation conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
A.4 Streak camera measurement of the Γ5 lifetime. . . . . . . . . . . . . . 100 A.5 Photoluminescence spectra of the A0 A0 X transitions under applied tensile strain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 A.6 Pumpprobe experiment to determine holerelaxation rates of the A0 bound hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
xiv
Chapter 1 Introduction In classical computation, the smallest unit of information is the bit, which can either be in state 0i or 1i. In quantum computation, or more generally quantum information processing (QIP), the smallest unit of information is the qubit Ψqubit i, which can exist in any superposition state of 0i and 1i, Ψqubit i = a0i + b1i, with a2 + b2 = 1. A distinction can be made between two types of qubits: flying qubits which are generally considered suitable for communication, and stationary qubits which are considered suitable for quantum memory and local computation. Not surprisingly, longdistance quantum communication protocols require both, with flying qubits connecting entanglement nodes where quantum states are stored [1, 2, 3, 4, 5]. Additionally many quantum computation protocols require both types as well, because quantum interactions are often local in nature and require either the actual physical transport of the matter qubit [6] or some form of quantum bus that can interact selectively with distant qubits [7, 8]. The hardware for stationary qubits may be single, isolated spins or ensembles of spins. Additionally, communication between quantum nodes can occur using single photons [9] or macroscopic coherent light pulses [8]. In general, whether single particles or ensembles of particles 1
2
CHAPTER 1. INTRODUCTION
3i
}∆ E1
E2
2i 1i Figure 1.1: Λ system used for EIT and stimulated Raman transitions. The two longlived spin states, 1i and 2i, are connected through state 3i with applied fields E1 and E2 . The applied fields are detuned from resonance by ∆. provide the quantum hardware, the most exciting and versatile QIP systems invariably incorporate both quantum communication and quantum memory/computation. The final goal of a practical interface between the two motivates this study on the interaction of light with electrons spins in semiconductors. A threelevel Λtype system such as that shown in Fig. 1.1 provides a powerful interface between photonic fields and two lower, metastable spin states. In contrast to twolevel systems, in which the qubit’s excited state lifetime is limited by a fast radiative decay rate, in Λ systems the qubit information is stored in the longerlived lower spin states. Coupling between the two states is still possible through the strong optical transitions to the excited state 3i. The creation of the lower state coherence through the optical Raman transitions is at the basis of electromagnetically induced transparency (EIT) [10] which has been used in atomic ensembles and some solidstate systems to coherently store and release light [11, 12, 13]. Additionally, twophoton Raman transitions can be used to coherently drive population between the two lower states [14] as well as create arbitrary spin superposition states [15, 16]. Although to date the bulk of Λsystem QIP has been performed in atomic ensembles, solidstate systems could provide key advantages over atomic systems including onchip device fabrication, large oscillator strengths/densities in semiconductor systems, and the possibility of transferring the photonic information to the longlived
3
nuclear states in semiconductor and diamond systems [17, 18]. One promising solidstate candidate for EIT/Ramanbased QIP is the neutraldonorbound exciton (D0 X) system in semiconductors. Although still a new candidate for QIP, this thesis work demonstrates that many of the requirements for QIP with Λ systems can be achieved in the Si:GaAs system including the isolation of a Λ system (Chapter 2), coherent population trapping (Chapter 3), optical pumping of the electron spins (Chapter 5), and the ultrafast coherent optical manipulation of the electron spin state (Chapter 6). One of the primary advantages of the neutraldonor system over other solidstate systems is the homogeneity of the emitters, a characteristic not often found in solidstate systems. This homogeneity makes solidstate ensemble QIP possible with relatively high impurity densities without the aid of spectral holeburning/repumping techniques. The main limitation of the Si:GaAs system is the fast inhomogeneous dephasing rate due to the nuclear environment. Although in general the strong interaction between the GaAs lattice nuclei and the bound electron is detrimental for electronspinbased QIP, the nuclearelectron hyperfine interaction does provide a unique environment for the optical study of nuclear spin polarization in highpurity GaAs (Chapter 4). Additionally, theoretical proposals exist for harnessing the nuclear spins for extremely long quantum memory times [17]. For practical electronspinbased QIP, however, the nanosecond inhomogeneous dephasing rate will still need to be overcome with electron spin control and spinecho techniques. The final part of this work (Chapter 6) moves toward the goal of fast coherent control of the semiconductor electron spin. We theoretically propose and experimentally demonstrate the ultrafast coherent control of the GaAs neutral donor spin on a time scale much shorter than the electron Larmor period using terahertz detuned broadband optical pulses. Although rotation fidelities are still limited by a fast dephasing process in the particular donor system studied, we hope that the demonstration of this technique will prove useful for general ultrafast electron spin resonance applications in the solid state as well as for quantum information applications in other systems.
Chapter 2 GaAs D0D0X system 2.1
Introduction
A complete description of an electron in the vicinity of an ionized donor in a semiconductor must include not only the impurity atom and extra electron but also all of the lattice atoms. Fortunately, effective mass theory (EMT) greatly simplifies the problem by describing electrons and holes as the product of a Bloch wavefunction ψ(r) and a slowly varying EMT envelope function F (r) [19]. For neutral donors (D0 ), the EMT equation of motion is the hydrogenic one with the effects of the lattice potential condensed into two parameters: the effective electron mass m∗ and the dielectric constant ². It is the two D0 electron spin states which will form the lower two states 1i and 2i of the Λ system (Fig. 1.1) in this work. These two states are optically coupled to the more complicated donorbound exciton state (D0 X) which will serve as state 3i of our Λ system. In this chapter, we first introduce the GaAs D0 and D0 X levels at zero and finite magnetic field. Then, using magnetophotoluminescence spectroscopy (MPL) and photoluminescence excitation spectroscopy (PLE), we determine the selection rules and optical transition linewidths of the particular D0 D0 X Λ system that is extensively studied in this work. Finally, the GaAs D0 D0 X system is compared to other Λ systems (atomic and solidstate) proposed for quantum information processing. 4
2.2. EFFECTIVE MASS THEORY AND GAAS SHALLOW IMPURITIES
2.2
5
Effective mass theory and GaAs shallow impurities
2.2.1
Effective mass theory
The complete derivation of the wavefunction for a neutral donor in a semiconductor with only one conduction band minimum (as in GaAs) can be found in Ref. [19], from which we only reproduce the main results as an aid to the reader. The Hamiltonian for a conduction electron in the vicinity of an impurity is given by
h ¯2 2 H=− ∇ + V (r) + U (r), 2m0
in which m0 is the electron mass, V(r) is the periodic lattice potential, and U(r) is the Coulomb potential of the impurity ion (e.g. silicon in GaAs). For distances large compared to the lattice constant, this potential is given by U (r) = −e2 /(4π²r) in which ² is the static dielectric constant of the medium. The wavefunction of the electron, ψ, can be written as the superposition of the unperturbed crystal wavefunctions, ψn,k , ψ=
X n,k
An (k)ψn,k
in which k runs over all wavevectors inside the first Brillouin zone. If the impurity potential is sufficiently weak, the wavefunction ψ will closely resemble the Bloch wavefunction at the conduction band minimum and we can let An (k) = 0 for n 6= 0. The resulting equation for the coefficient A0 (k) is then the hydrogenic Schrodinger equation in momentum space. The equation of motion for its Fourier transform 1 X F (r) = √ A0 (k)eik·r V k
CHAPTER 2. GAAS D0 D0 X SYSTEM
6
is the hydrogen effective mass equation Ã
!
h ¯2 2 e2 F (r) = EF (r). − ∗∇ − 2m 4π²r
In GaAs, the effective electron mass is m∗ = 0.067 m0 [20] and the static dielectric constant is ² = 12.56 ²0 [21]. Thus, the EMT Bohr radius is a∗B = 99 ˚ A and the ground state binding energy is 5.8 meV. Due to the large dielectic constant and small electron effective mass, EMT theory works quite well in GaAs and experimental values of the binding energies of the shallow donors Te, Si, Sn, S and Ge all lie within 0.1 meV of the EMT value [22]. It is important to remember that the total wavefunction of the electron near a shallow donor impurity, ψ(r), is not given simply by the EMT envelope function F (r), but must also include the Bloch contribution u0,0 (r), ψ(r) = u0,0 (r)F (r).
2.2.2
(2.1)
EMT in action: Carbon doped GaAs
EMT can also be used to describe shallow acceptor levels in GaAs, although the situation is complicated somewhat by the degenerate valence band structure [19]. Due to their larger binding energies compared to donors, the photoluminescence spectrum of acceptors in GaAs provides a particularly striking example of the validity of EMT. By exciting with aboveband excitation (Fig. 2.1a) and also resonant excitation (Fig. 2.1b), transitions from the A0 X state to the 1s5s A0 states are observed in the photoluminescence spectra. As seen in Fig. 2.1c, the transition energies fit a hydrogenic spectra extremely well. From this fit the carbon A0 binding energy of 25.8±1.3 meV and the central cell correction (deviation from EMT for the 1s state) of 3.8 meV±1.0 meV is obtained. The binding energy is consistent with the value of 26.3 ± 0.1 meV derived from infrared absorption measurements [23]. Due to the availability of high quality ptype samples and our initial lack of access to high magnetic fields, we first studied the GaAs A0 X system for QIP applications (see Appendix A). However, the picosecond hole spin relaxation time of acceptors [24] is not suitable for practical QIP, and the remainder of this thesis focuses on donors.
2.2. EFFECTIVE MASS THEORY AND GAAS SHALLOW IMPURITIES
7
1.52 a)
1s
energy (eV)
A 0X
FE
1.51 1.51
3s 2s
1.5
1.515
b)
1.52
2s
3s 5s 4s
1s 1.485
£10 1.49
1.495
energy (eV)
1.49 c)
1
2
3
4
5
sorbital
Figure 2.1: a) A0 XA0 1s photoluminescence spectrum for aboveband excitation. Spectra are instrument resolution limited. Also present is the free exciton (FE) transition. b) Relaxation to the 2s5s A0 states observed under resonant A0 XA0 1s excitation. c) Hydrogenic fit of the A0 X transition energies to the A0 2s5s lines. The sample was grown by H. Hiroyama (NTT).
2.2.3
Neutral donors in a magnetic field
The neutral donor 1s state is twofold degenerate due to the spin of the electron. In a magnetic field, this degeneracy is lifted and the 1s state forms a twolevel system with an energy splitting of ∆E = ge µB B, in which ge is the electron gfactor, µB is the Bohr magneton, and B is the applied magnetic field. Due to the strong spinorbit interaction in GaAs, the gfactor of the bound electron, ge = −0.44 ± 0.02 [25], differs significantly from the vacuum electron gfactor, g0 = 2.00. These two spin states form the lower two states 1i and 2i of the Λ system in Fig. 2.3.
CHAPTER 2. GAAS D0 D0 X SYSTEM
8
2.3
GaAs neutraldonorbound exciton
The neutral donor complex (D0 ), composed of a positive ion and a bound electron, provides an attractive potential for an exciton (electronhole pair). The existence of this fourbody neutraldonorbound exciton (D0 X) consisting of two electrons in a spinsinglet state, a hole, and the positive ion, was first predicted by Lampert [26] and then experimentally observed in silicon by Haynes [27].
2.3.1
GaAs D0 X at 0 T
Fig. 2.2 is a photoluminescence (PL) spectrum of an ntype GaAs sample with the excitonic transitions labelled, including the neutraldonorbound exciton, the ionizeddonorbound exciton (impurity, one electron, one hole), and the neutralacceptorbound exciton. Several D0 X transitions (L = 03) can be observed, and though their origin is not entirely understood, the excited levels are believed to be due to the different rotational states of the bound hole [22, 28]. Although the D0 X complex predominantly relaxes to the 1s D0 state, there is a finite probability that the D0 X complex will relax to an excited (2s, 2p, etc.) D0 state. These twoelectron satellite transitions (TES) can also be observed in the zerofield PL spectrum (Fig. 2.2). As will be discussed in later sections, the TES transitions are valuable tools for monitoring the excited state D0 X population while performing resonant excitation on the main D0 X transitions.
2.3.2
GaAs D0 X in an external magnetic field
In the presence of a magnetic field, each D0 X level splits into four Zeeman levels, determined by the spin of the bound, J = 3/2, hole. This results is a complicated forest of D0 X peaks at large magnetic fields. Many of the transitions, however, have been identified by Karasyuk et al. [22] in a detailed magnetoPL study for the case in which the applied magnetic field is parallel to the [100] crystal axis. For all the experiments presented in this work, however, the GaAs crystal axis [110] is parallel to the Bfield. Throughout this work we will label the transitions
2.4. SPECTROSCOPIC STUDY OF THE D0 D0 X Λ SYSTEM
9
PL intensity
D0X, 1s L = 0,1
D0X, 1s L = 2,3
D0X, 2s,2 p L = 0,1
A0X
D0X, 2s,2 p L = 2,3
X D+X Band Edge
816
×10
817
818
819 λ (nm)
820
821
822
Figure 2.2: Photoluminescence spectra of highpurity MBE GaAs. The MBE epilayer was 10 microns thick with a background impurity concentration of n ∼ 5 × 1013 cm−3 . The sample was excited aboveband at 815 nm. Photoluminescence from free exciton transitions (X), neutralacceptorbound exciton transitions (A0 X), lowest energy and excited state neutraldonorbound exciton transitions (D0 X 1s), D0 X TES transitions (D0 X 2s,2p), and ionizeddonorbound exciton (D+ X) transitions are all observed. The sample temperature was 2 K and the instrumental resolution was 0.02 nm. The sample, b923, was grown by M.C. Holland and C. Stanley (University of Glasgow).
from the lowest energy D0 X level in this geometry with the letter A (Karasyuk et al.’s A1 state). A diagram of the Λ system studied most extensively in this work is shown in Fig. 2.3.
2.4
Spectroscopic study of the D0D0X Λ system
Due to the existence of several orbital angular momentum states of the D0 X complex, at high magnetic field, care must be taken in identifying D0 Zeeman pair transitions. A careful tracking of the D0 X photoluminescence lines as function of magnetic field (MPL) with a grating spectrometer is certainly useful. However, onelaser and
CHAPTER 2. GAAS D0 D0 X SYSTEM
10
D0X
3i
mh= 1 2
A, π A*, σ
D0,1s
me = 1 2 1i
2i
me= 1 2
Figure 2.3: D0 D0 X Λ system studied in this work. The two lower states correspond to the spin states of the bound D0 electron. The excited state is the L = 1, mL = 0, mh = −1/2 D0 X state. The polarizations for the A and A∗ transitions in the ~ are π and σ respectively. Voigt geometry (~c⊥B) twolaser high resolution photoluminescence excitation (PLE) spectroscopy must additionally be used to ensure transitions from two different D0 X states are not mistakenly identified as a D0 pair. In the next two sections the lowest state Zeeman pair is carefully identified at 7 T using MPL and PLE.
2.4.1
Magnetophotoluminescence spectroscopy
Fig. 2.4 shows a colormap of the PL spectra as a function of magnetic field. Two main effects of the Bfield on the red D0 X transitions can be observed. First there is a diamagnetic shift to higher energy due to the perturbation of the D0 X EMT wavefunction by the Bfield. Second, the somewhat broad, zerofield L=0,1 D0 X transitions split into many narrow, discrete levels. These are transitions between the discrete hole Zeeman levels of the D0 X complex and the electron Zeeman levels of the D0 complex. The MPL spectra are useful for tracking individual transitions, however, higher resolution spectra must be taken to positively identify D0 Zeeman pairs.
2.4.2
Photoluminescence excitation spectroscopy (PLE)
High resolution spectra can be obtained by performing photoluminescence excitation spectroscopy (PLE). In PLE, a narrow band laser (Coherent 89929 Ti:Sapphire) is
2.4. SPECTROSCOPIC STUDY OF THE D0 D0 X Λ SYSTEM
7
2p
2p0
11
2p+
2s
6
magnetic Field (T)
5
4
3
2
1
0 815
D0X 816
817
818
TES
819
820
821
822
823
824
825
wavelength (nm)
...
Figure 2.4: Magnetophotoluminescence spectra of the GaAs D0 D0 X system (sample b923). Diamagnetic shift and paramagnetic splitting of the main (red) D0 X transitions are clearly observed as the magnetic field is increased. Relaxation of the D0 X states to the excited 2s, 2p D0 states (TES transitions) is also observed.
L=2 L = 0,1
D0X TES
main D0
2s, 2p 1s
Figure 2.5: In PLE scans a narrowband laser is scanned over the main D0 X transitions as the PL is collected from the TES transitions. D0 X and D0 levels will be split in an external magnetic field. scanned over the main D0 D0 X transitions while the PL from the D0 X states to the 2s,2p D0 states is monitored (Fig. 2.5). The final spectral resolution is determined by the scanning resolution of the narrowband laser (10 MHz) instead of the spectrometer grating resolution.
CHAPTER 2. GAAS D0 D0 X SYSTEM
12
TES PL from D0X state A inte ns ity
817
(a)
A* A
817.1
817.2
817.3
π PLE σ PLE π PL σ PL
817.4
TES PL from D0X state B (b)
inte ns ity
B*
817
B
817.1
817.2 817.3 λ (nm)
817.4
Figure 2.6: PL and PLE spectra. PL spectra are labelled according to the collected polarization. PLE spectra are labelled according to the excitation polarization. A 1 GHz step size was used in the PLE scans. (a) TES PL from the lowest energy D0 X state was collected in the PLE scan. Zeeman pair transitions are labelled with A and A∗ . (b) TES PL from the first excited D0 X state is collected. Zeeman pair transitions are labelled with B and B ∗ . In Fig. 2.6 we plot the PLE scans in which PL is monitored from the lowest D0 X A state and the 2nd lowest D0 X B state. A comparison of the PLE to the PL spectra demonstrates that much higher resolution can be obtained with PLE and also that single PL lines may in fact correspond to several D0 D0 X transitions. All single PLE linewidths are less than 5 GHz, with transitions from the L=0 B states exhibiting linewidths as narrow as 1 GHz. These linewidths are only several times larger than the radiative relaxation rate of the D0 X complex [29]. D0 Zeeman pairs A, A∗ and B, B ∗ can be first identified in the PLE spectra using the known gfactor of the electron spin. To ensure that these are in fact Zeeman pairs, the splitting as a function of field can be mapped and also twolaser PLE scans can be performed.
2.4.3
Twolaser PLE
The verification that two D0 D0 X transitions split by approximately the D0 Zeeman energy are Zeeman pairs can be done with twolaser PLE spectroscopy. In this case, a
2.5. COMPARISON TO OTHER Λ SYSTEMS
13
D0X
D0,TES D0,1s me = 1 2
me= 1 2
me = 1 2
(a)
me= 1 2
(b)
Figure 2.7: (a) If only a single laser is resonant on the me = −1/2iD0 X transition, electrons will be pumped into the me = 1/2i state and the TES PL will be weak. (b) If a second laser is applied to the me = 1/2iD0 X transition, electrons will be repumped into the me = −1/2i state and the TES PL will be enhanced. laser (singlemode externalcavity diode laser) resonantly excites A (or A∗ ). A second laser (Ti:Sapphire) is then scanned over all of the D0 X transitions. When compared to the singlelaser PLE scan, some of the transitions will be weaker or the same in intensity, while others will have an intensity that is greatly enhanced. As shown in Fig. 2.7, in the singlelaser PLE scans, the PL is weak due to the optical pumping of the D0 electrons. If, however, a second laser is resonant on a D0 D0 X transition with a different ground state, the PL intensity will be greatly enhanced. By comparing the singlelaser and twolaser scans, the ground state level of the transition can be identified (Fig. 2.8). Thus we can confirm that the transitions labelled A and A∗ in Fig. 2.6a are indeed Zeeman pairs and form a Λ system. Coherent population trapping, which will be discussed in the next chapter, is not observed in these twolaser scans due to the low scan resolution.
2.5
Comparison to atomic systems and other solidstate systems
When evaluating a Λ system for a QIP application, the following should be considered. • Coherence time of the lower two levels (T2 ): T2 should be long compared to
CHAPTER 2. GAAS D0 D0 X SYSTEM
14
+
inte ns ity
(a)
diode , A +
817
+
+
817.1
+
+
817.2
inte ns ity
(b) 
817
817.1
817.3
817.4
diode , A*



817.2
λ (nm)
817.3

π 1laser PLE σ 1laser PLE π 2laser PLE σ 2laser PLE

817.4
Figure 2.8: Onelaser and twolaser PLE scans. The scanning laser was either π or σpolarized. All PL polarizations are collected. (a) The diode laser was fixed on the A transition. Enhanced lines in the twolaser scans correspond to transitions to the D0 me = +1/2i state. (b) The diode laser was fixed on the A∗ transition. Enhanced lines in the twolaser scans correspond to transitions to the D0 me = −1/2i state. the radiative lifetime of state 3i and the spin manipulation time. The coherence time is critical in that it determines the overall storage time and thus computation time in a quantum information processor. In an ensemble, due to inhomogeneous effects, the measured dephasing time T∗2 can be much shorter than the intrinsic, single particle T2 time. T2 can often be recovered in ensembles using spinecho techniques. • Oscillator strengths of the optical transitions.: Systems with larger oscillator strengths will have faster optical gate times. Additionally, the optical density of the system will be larger, ensuring complete single (and possibly multi) pulse storage in the medium. Systems with large oscillator strengths will exhibit short radiative lifetimes τrad . • High atomic/impurity density: This will contribute to a high optical density.
2.5. COMPARISON TO OTHER Λ SYSTEMS
T∗2
lower state T∗2 mechanism radiative lifetime, τrad density, cm−3
optical linewidth (inhomogeneous) fabrication potential temperature
15
D0 D0 X 110 ns nuclear environment 5 T). However at low field it was only possible to cleanly isolate the lowest energy A transition and all data presented is from this transition.
Intensity (arb. units)
5.3. CONFIRMATION OF OPTICAL PUMPING
55
fixed laser
*
A
A 816
816.1
816.2
816.3
816.4
816.5
Wavelength (nm)
Figure 5.4: Single laser PLE scans (solid lines) and double laser PLE scans (dashed lines) at 9.9 T. In the single laser PLE scans, a πpolarized (black) and σpolarized (grey) laser is scanned over the D0 X transitions. Photoluminescence (PL) is collected from an A TES transition and is proportional to population in the D0 X A state. Overall intensity is weak due to optical pumping. In the two laser PLE scans, a second laser is fixed resonantly on the A∗ transition as the first laser scans over the D0 X transitions. A large enhancement in PL intensity is observed for transitions originating from the  − 12 i state.
5.3
Confirmation of optical pumping
In order to ensure that population is indeed being pumped into the + 21 i electron state and not to some state outside our system, we performed singlelaser and twolaser photoluminescence excitation (PLE) scans which are described in more detail in Chapter 2. In the single laser PLE scan a continuous wave Ti:Sapphire laser was scanned over the D0 X transitions as emission from the mh = − 21 i state to its twoelectron satellites (TES) was detected (Fig. 5.1a. In this case, as the laser scanned over each transition, the photoluminescence was weak due to optical pumping. In a second scan, a diode laser resonantly excited the A∗ transition during the Ti:Sapphire scan. The diode laser repumped the trapped electrons back into the  − 21 i state allowing efficient population of the mh = − 12 i state (Fig. 5.4). An eightfold increase in the PLE intensity of the Atransition was observed. This result indicates that electrons were pumped between the two Zeeman states and not to a state outside the system.
56 CHAPTER 5. MILLISECOND SPINFLIP TIMES OF DONOR ELECTRONS
5.4
Results and discussion
The magnetic field dependence of the electron spin relaxation is shown in Fig. 5.5. At fields less than 4 T there appears to be a levelling off of T1 at several ms. This is related to the finite extinction ratio of the AOM (≈ 1000) and the small leakage field even when the beam is deflected away from the sample. Thus, for fields less than 4 T our data only gives a lower bound on T1 . As the magnetic field is increased, T1 decreases rapidly exhibiting a strong powerlaw dependence. This T1 dependence on Bfield is very different from previously reported T1 measurements in nGaAs by Colton et al. [64] Previously, T1 was measured in higher doped samples (n = 3 × 1015 cm−3 ) by timeresolved polarization photoluminescence measurements on free excitons. T1 was shown to increase with magnetic field reaching a maximum of 1.4 µs at the maximum attainable field of 5 T. This different dependence may be due to the larger doping density resulting in a greater interaction between neighboring donorbound electron spins. A further study on the effect of doping density on T1 is necessary to understand the discrepancy between the two experimental results. Theoretical calculations of the Bfield dependence of T1 due to the modulation of electron spinorbit coupling by phonons predict a B−4 dependence of T1 for neutral donors [56]. A fit to data for B > 5 T (Fig. 5.5) shows a strong power law dependence with an exponent of m = −3.8 ± 0.2. This indicates that the onephonon spinorbit relaxation process is the dominant process for GaAs neutral donors at high magnetic fields, gµB B > kB T . In our system the temperature T = 1.5 K, electron gfactor = 0.41 [42] and the lowfield to highfield transition occurs at B = 5.6 T. Our results are similar to previously reported quantum dot electronspin relaxation times. In InGaAs quantum dots, Kroutvar et al. [63] found an inverse powerlaw dependence of T1 on magnetic field. The observed power law exponent, m = −5, was theoretically predicted for disklike quantum dots [47, 48] with singlephonon spinorbit relaxation processes. The similarities between the donor electron and quantum dot systems indicate that in our sample electrons are wellisolated and noninteracting.
5.5. SUMMARY
57
10 maximum measurable T1
1
3.8± 0.2
6T arb. units
1
T (ms)
~B
T1 = 770 µs 0
0.5
1
1.5
time, ms
0.1 2
4 Bfield (T)
6
8
Figure 5.5: Loglog plot of the bound electron spin lifetime T1 dependence on Bfield. A strong power law dependence on Bfield is observed for B > 5 T. Recovery times measured at B ≤ 4 T are limited by the experimental apparatus and are only a lower bound on the electron T1 . Inset: Exponential fit of the pumpprobe data at 6 T.
5.5
Summary
We have observed long, millisecond T1 times of electrons bound to donors in bulk GaAs. This direct pumpprobe measurement is possible due to the homogeneity of the D0 and D0 X systems in highpurity GaAs. Our result represents a threeorder of magnitude increase over previously measured D0 electron T1 times. Both the long T1 ’s observed and the strong power law dependence of T1 on magnetic field indicate that donorbound electrons are noninteracting in sufficiently pure GaAs. If the D0 electron T2 time approaches the millisecond T1 time, the D0 D0 X system will be a very strong candidate for electronspin quantum information processing. The determination of the D0 electron T2 time will, however, require the coherent manipulation of the electron spins. The ultrafast control of the D0 electron spin is the subject of the next chapter.
Chapter 6 Ultrafast optical manipulation of the D0 electron spin 6.1
Introduction
For any practical D0 D0 X QIP system, the singlequbit spin manipulation time for both quantum information processing and electron spinecho will need to be much shorter than T∗2 . Although it is possible to perform such fast, subnanosecond, rotations with classical microwave electron spin resonance (ESR) techniques, it is still technologically challenging. Additionally, due to the large microwave wavelength, microwave ESR is inherently nonlocal and will be difficult to implement in a network of closely spaced (micronscale) D0 ensembles. A promising alternative to microwave ESR is ESR based on twophoton Raman transitions. Optical ESR has advantages over microwave ESR in terms of speed and locality. In optical ESR, the area of operation is limited to the micron size laser spot. In atomic physics experiments, optical ESR is routinely done using two lasers on twophoton resonance as depicted in Fig. 1.1. A powerful twolaser technique is fractional stimulated Raman adiabatic passage (fSTIRAP), in which any coherent superposition of the lower two states can be created. During the twopulse fSTIRAP sequence, the Λ system remains in the dark state (see Sec. 3.2.1). Thus decoherence effects due to real population in state 3i are avoided [66, 67, 14]. The major drawback 58
6.1. INTRODUCTION
59
of fSTIRAP is that the initial twolevel superposition state must be known in order to begin in the dark state. Rotations from an unknown state are only possible if more than three levels are used [68, 69]. A twolaser alternative to fSTIRAP is to perform a nonadiabatic stimulated Raman transition with a large detuning ∆. During this technique the system does not remain in the dark state, however decoherence effects due to the excitedstate population are minimized by the large detuning or by the speed of the operation. This technique has been successfully implemented in atomic interferometric experiments [16]. Such twolaser transitions work, but there are some drawbacks. First, the relative phase, frequency and intensity of the two lasers must be carefully controlled. Second, in a system without good polarization selection rules (i.e. many solid state systems), the bandwidth of the laser pulses must be much smaller than the lower state splitting in order to guarantee that each laser only couples the desired transitions. In the D0 D0 X system, the fastest rotation time possible at 7 T would be greater than 100 ps and will increase with decreasing magnetic field. We propose and demonstrate the use of a single, broadband pulse to simultaneously connect the three Λ states and manipulate the D0 spin state [8]. Previously, femtosecond and picosecond pulses have been demonstrated as powerful tools to probe and manipulate electron spins in semiconductors. However, the goal of full SU(2) rotations using optical ESR is still elusive in these systems. To date, the use of ultrafast lasers for solid state ESR can be grouped into three categories: passive Faraday rotation/differential transmission measurements [70, 30, 71], resonant stimulated Raman transitions which due to excitedstate population inherently have a limited fidelity [71, 72, 30, 73], and rotations based on the optical Stark effect [70]. Although this last technique can theoretically be used to obtain rotations of an arbitrary angle, it has limited control over the rotation axis. In contrast to the above techniques, singlepulse stimulated Raman transitions using a fardetuned field can perform general qubit rotations (full SU(2)). In this chapter we first present the theory of stimulated Raman transitions with a single detuned pulse. In Sec. 6.3 we present the experimental demonstration of rotations with areas up to π/3 using picosecond pulses. Full control over the rotation axis is demonstrated and 40 GHz D0 spin precession is probed using this technique.
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
60
3
ωL
2
ωL (a)
1
(b) (a)
Figure 6.1: (a) Frequency domain picture of the optical pulse, showing two pairs of frequency components within the pulse spectrum needed to complete a stimulated Raman transition. (b) Energy level picture of two pairs of frequencies contained within the applied pulse that will induce transitions between the D0 electron spin states. We also discuss factors that may be currently limiting larger angle rotations in the D0 system. The inhomogeneous dephasing time T∗2 is measured in variabledelay twopulse experiment and an extension to a homogeneous T2 measurement using small angle pulses is proposed.
6.2
Rotations using a single, fardetuned, optical pulse: Theory
When a single, off resonant, broadband pulse is applied to the threelevel system, pairs of frequency components separated by the electronZeeman splitting ωL are essentially picked out by the system and a coherent superposition of levels 1i and 2i is obtained through the stimulated Raman transitions (Fig. 6.1). The Hamiltonian for the threelevel system is similar to the CPT Hamiltonian (Eq. 3.7) with the following exceptions: • A single field connects both the 1i3i and the 2i3i transitions, thus the Rabi frequencies of the two transitions are only different if the dipole matrix elements
6.2. ULTRAFAST ROTATIONS: THEORY
61
are different. • Since there is only one field, the two photon detuning will always be the electron Zeeman splitting ωL . • Since this is a pulsed experiment, the Rabi frequencies are time dependent. The Hamiltonian for the threelevel system in the rotating wave approximation and the interaction picture is thus given by
H3 =
0
0
0
ωL
−Ωp (t)/2
, −Ωc (t)/2
−Ω∗p (t)/2 −Ω∗c (t)/2
(6.1)
∆
in which ∆ is the detuning from the 1i3i resonance and is positive for red detuning. Throughout this chapter we have let h ¯ = 1.
6.2.1
Adiabatic elimination of the excited state
A simple way to see that the above Hamiltonian produces rotations about an axis perpendicular to the applied magnetic field is to reduce the threelevel system to a twolevel system. The equation of motion for the threelevel system evolves according to
dΨi = −iH3 (t)Ψ(t)i dt
in which Ψ(t)i = a1 (t)1i + a2 (t)2i + a3 (t)3i. The equations of motion for the three coefficients are a˙ 1 (t) =
i Ωp (t)a3 (t) 2
i a˙ 2 (t) = −iωL a2 (t) + Ωc (t)a3 (t) 2 i ∗ i a˙ 3 (t) = Ωp (t)a1 (t) + Ω∗c (t)a2 (t) − i∆a3 (t). 2 2 If we are far detuned such that a˙ 3 (t) ¿ i∆a3 (t), then a˙ 3 (t) can be eliminated and the system can be reduced to a twolevel system (see, for example, Ref. [74]) and the
62
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
effective twolevel Hamiltonian is given by
H2 = −
Ωp (t)2 4∆p Ωp (t)∗ Ωc (t) 4∆p
Ωp (t)Ωc (t)∗ 4∆p Ωc (t)2 − ωL 4∆p
.
(6.2)
In the special case where Ωp (t) = Ωc (t), Eq. 6.2 simplifies significantly. The singlepulse Hamiltonian becomes
H2 = −Ω(t)/2
1 0 0 1
−
0
Ω(t)/2
Ω∗ (t)/2
−ωL
in which we have defined an effective Rabi frequency Ω(t) =
,
Ωp (t)Ω∗c (t) . 2∆
(6.3)
In a real
system, different Rabi frequencies will result in a component of the rotation axis parallel to the magnetic field. In this case, largearea rotations can still be obtained with more than one pulse. Often, the two Rabi frequencies can be made equal by utilizing the polarization selection rules in a particular Λ system.
The twolevel Hamiltonian in Eq. 6.3 is composed of two parts, a term proportional to the identity matrix and a second, rotational term. To see that this second term causes rotations, the Hamiltonian can be rewritten in a frame rotating with the electron spin. The twolevel Hamiltonian becomes
H2 = −
0
Ω(t) i(φ−ωL t) e 2
Ω(t) −i(φ−ωL t) e 2
0
,
(6.4)
in which Ω(t) = Ω(t) exp(iφ). Using this Hamiltonian we can derive the equations of motion for the expectation value of the Pauli spin operators si (t) = hˆ si (t)i in which ~ The equation of motion for the i = x, y, z (see for example [75]). We define zˆkB. vector ~s = (sx , sy , sz ) is the simple vector precession equation, ~ × ~s ~s˙ = Π
(6.5)
6.2. ULTRAFAST ROTATIONS: THEORY
63
in which the rotation axis Π is given by
Ω(t) cos (φ − ωL t)
Π = Ω(t) sin(φ − ωL t) .
(6.6)
0 According to Eq. 6.6, the instantaneous rotation axis is determined by the quantity (φ − ωL t). Thus the rotation axis is determined by time t. If the pulse length ∆tpulse is much shorter than the Larmor period 2π/ωL , Eq. 6.5 will describe a rotation about the single axis (0, cos (φ − ωL tarrival ), sin(φ − ωL tarrival )) where tarrival is the pulse arrival time. Thus in order to have welldefined rotations using a single broadband pulse the following condition must hold, ∆tpulse ¿
2π . ωL
Another way of stating this condition is that the pulse bandwidth must be large compared to the lower state splitting. There is an additional restriction on the pulse length in any real system. The pulse length must also be much shorter than the shortest relaxation/decoherence time in the system. This includes upper state relaxation since the system does not remain in the dark state during this fast Raman process.
6.2.2
Threelevel simulations
The twolevel approximation derived in the previous section gives a simple model for understanding how a single pulse in the threelevel system can cause rotations. However, in a real system the adiabatic elimination of the excited state may not be valid and relaxation between the levels will occur. The complete threelevel system needs to be treated using a densitymatrix model. Full densitymatrix model The densitymatrix model is the same model that was used to describe the CPT results in Chapter 3. The threelevel density matrix evolves according to Schrodinger’s
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
64
equation,
∂ ρ = −i[H, ρ] + L(ρ) ∂t
(6.7)
in which H is given by Eq. 6.1. The relaxation operator L(ρ) is given by Eq. 3.8. The relaxation terms described in Eq. 3.8 are known from the literature or from previous experiments. In our threelevel simulations we use the following parameters: Γ21 = (1 ms)−1 , Γ32 = Γ31 = (1 ns)−1 , γ3 = 10 GHz, and γ2 = (1 ns)−1 . The results for the numerical simulation, in which we assume Ωp = Ωc , are shown in Figs. 6.2 and 6.4. In both simulations we use hyperbolic secant pulses which are close in shape to the modelocked laser pulses output by the Spectraphysics Tsunami Ti:Sapphire modelocked laser used in the experiments. In the experiment, pulses are characterized using frequency resolved optical gating (FROG). Additionally, in both simulations ωL = 2π × (40 GHz) which corresponds to the D0 Zeeman splitting in a 7 T field. In the first simulation the applied pulse has a fullwidthhalfmax (FWHM) of ∆tpulse = 2 ps. For these pulses the pulse length is approximately 1/10 of the Larmor period. In Fig. 6.2a, the population after the picosecond pulse is applied is plotted as a function of the pulse maximum of the single transition Rabi frequency Ωp (t). Rabi oscillations are observed with a π/2pulse fidelity F = hΨρΨi = 0.99, in which ρ is the density matrix for the obtained state and Ψi is the desired ideal state. The π pulse fidelity is slightly lower at F = 0.95. By comparing Fig. 6.2 to simulations in Fig. 6.3 which do not include any relaxation terms, it is clear that the finite fidelity is not due to relaxation (since π pulse fidelities are the same) but instead due to the ratio of the pulse length to the fast Larmor precession of the electron spin. As seen in Fig. 6.4, higher pulse fidelities can be obtained by using shorter, 100 femtosecond pulses, at the price of larger detunings and higher powers.
Unequal Rabi frequencies Ideally, the system should be engineered such that the Rabi frequencies of the two transitions are equal. This however, may be difficult in a real system. The problem with unequal Rabi frequencies is that they cause the rotation to be about an axis not
6.2. ULTRAFAST ROTATIONS: THEORY
65
population
Population after 2 ps pulse (relaxation included) 0.8 (a) 0.6 0.4 0.2
π π/2 0.2
0.4
0.6 Ωp (THz)
0.8
π/2 pulse population
1
ρ22
π pulse 1
F= 0.99
(b)
ρ11
1
0.5
(c)
ρ33
F= 0.95
0.5
0 5
0 time (ps)
5
0 5
0 time (ps)
5
Figure 6.2: Numerical simulation of the 3level densitymatrix model including the relaxation operator L(ρ). Detuning ∆ = 2π × (1 THz) and the pulse length ∆tpulse = 2 ps. The system begins in state 1i. (a) Population 10 ps after the fast pulse. (b) Population evolution for a π/2pulse. (c) Population evolution for a π pulse.
population
Population after 2 ps pulse (no relaxation included) 0.8 (a) 0.6 0.4 0.2
π π/2 0.2
0.4
0.6 Ωp (THz)
π/2 pulse population
1
(b)
ρ11
1
ρ22 π pulse
1
F > 0.999
0.5 0 5
0.8
(c)
ρ33
F= 0.95
0.5
0 time (ps)
5
0 5
0 time (ps)
5
Figure 6.3: Numerical simulation of the 3level densitymatrix model excluding L(ρ). Detuning ∆ = 2π(×1 THz) and the pulse length ∆tpulse = 2 ps. The system begins in state 1i. (a) Population 10 ps after the fast pulse. (b) Population evolution for a π/2pulse. (c) Population evolution for a π pulse.
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
66
population
Population after 100 fs pulse (relaxation included)
(a)
0.8 0.6 0.4 0.2
π
π/2 5
10 Ω (THz)
ρ
15
11
ρ
p
π/2 pulse population
1
F > 0.999
(b)
0.5 0 0.5
22
π pulse 1
(c)
ρ33
F = 0.999
0.5
0 time (ps)
0.5
0 0.5
0 time (ps)
0.5
Figure 6.4: Numerical simulation of the 3level densitymatrix model including the relaxation operator L(ρ). Detuning ∆ = 2π×(10 THz) and the pulse length ∆tpulse = 100 fs. The system begins in state 1i. (a) Population 1 ps after the fast pulse. (b) Population evolution for a π/2pulse. (c) Population evolution for a π pulse.
in the xˆˆ y plane. In Fig. 6.5 we plot the final state population for the case Ωp = 2Ωc for 100 fs pulses. Rabi oscillations are still be observed, however complete population transfer cannot be obtained with the application of only a single pulse. In this case certain rotations may require 2 or more pulses.
Multiple upper levels A pulse detuned by 1 THz from the lowest energy D0 D0 X transition will actually interact with all the excited D0 X levels. At 7 T, the splitting between the lowest two D0 X levels is only 17 GHz. As will be shown, the existence of these additional excited levels should not significantly effect the singlepulse qubit rotations. In the case of the fourlevel system shown in Fig. 6.6, the Hamiltonian in the
6.2. ULTRAFAST ROTATIONS: THEORY
67
Population after 100 fs pulse, Ωp = 2 Ωc 0.9 0.8 population
0.7
ρ
11
0.6
ρ22
0.5
ρ
33
0.4 0.3 0.2 0.1
5
10 15 Ωp (THz)
20
Figure 6.5: Numerical simulation of the 3level densitymatrix model including the relaxation operator L(ρ) in the case where Ωp = 2Ωc . Full rotations cannot be obtained with a single pulse if Ωp 6= Ωc . Detuning ∆ = 2π×(10 THz) and the pulse length ∆tpulse = 100 fs. The population of the three levels 1 ps after the fast pulse is plotted.
2i 1i
∆4
{
∆3 {
{
4i 3i
ωL
Figure 6.6: Λ system with two excited states.
interaction picture and RWA is given by
0
0
Ω31 /2 Ω41 /2
0 −ωL Ω32 /2 Ω42 /2 H4lev = − ∗ Ω31 /2 Ω∗32 /2 −∆3 0
Ω∗41 /2 Ω∗42 /2
0
−∆4
,
(6.8)
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
68
in which we have left out the time dependence of the Rabi frequencies for simplicity. Analogous to Sec. 6.2.1, if the optical field is fardetuned from all transitions, the fourlevel system can be reduced to a twolevel system. The effective twolevel Hamiltonian becomes
Hef f = −
1 Ω31 2 4∆3 1 Ω Ω∗ 4∆3 32 31
+ +
1 Ω41 2 4∆4 1 Ω Ω∗ 4∆4 42 41
1 Ω Ω∗ + 4∆1 4 Ω41 Ω∗42 4∆3 31 32 1 Ω32 2 + 4∆1 4 Ω42 2 − ωL 4∆3
.
It is straightforward to generalize this fourlevel case to nlevels,
Hef f = −
Ωk1 2 1P k>2 ∆k 4 Ω∗k1 Ωk2 1P k>2 4 ∆k
P
Ωk1 Ω∗k2 k>2 ∆k Ωk2 2 1P k>2 ∆k + −ωL 4 1 4
.
(6.9)
Thus we find that the nlevel case is analogous to the threelevel case (Eq. 6.2) except the effective twolevel Rabi frequency is now given by Ωef f =
1 X Ωk1 Ω∗k2 , 2 k>2 ∆k
where k runs over all the excited levels. Additionally, rotations about an axis in the xˆˆ y plane will only occur if the sum of the individual Rabi frequencies to each lower state are equal. That is, if X Ωk1 2 k>2
∆k
=
X Ωk2 2 k>2
∆k
.
(6.10)
This condition can be met with careful control over the polarization of the optical pulse if good polarization selection rules exist for the individual transitions.
Fourlevel densitymatrix model simulations indeed show good pulse fidelities as long as the Eq. 6.10 holds and all relaxation/decoherence times are short compared to the pulse length. If, however, the optical pulse couples the electron spin states to any excited state with dephasing times short or comparable to the pulse length, high fidelity rotations will not be obtained.
6.3. ULTRAFAST ROTATIONS: EXPERIMENT
6.3
69
Rotations using a single, fardetuned, optical pulse: Experiment
In this section we present the experimental demonstration of ultrafast rotations using a single broadband pulse on the D0 D0 X system. In the singlepulse experiment we observe population transfer between the two electron spin states, however, Rabi oscillations are not observed for large pulse energies. To measure the coherence of the small pulse area rotations, a twopulse experiment is performed.
6.3.1
Singlepulse experiment: Observation of Raman population transfer
The population transfer due to the application of the Raman pulse was observed in a singlepulse experiment. First, the electron spins were optically pumped into the ground electron spin state me = +1/2i by applying a narrow frequency field to the A transition (See Fig. 5.1b). After a time ∆τ short compared to the T1 time, a broadband pulse was applied. At time ∆τ following the broadband pulse, the optical pumping field was again applied. At the initial application of the optical pumping field, the intensity of the photoluminescence of the A∗ transition is proportional to the population in the excited electronspin state me = −1/2i. Thus, the intensity of the A∗ transition was measured as a function of pulse energy to obtain the electronspin population transfer as a function of the broadband pulse energy. A diagram of the singlepulse experiment is shown in Fig. 6.7. The experimental clock was set by the repetition rate of a modelocked Ti:Sapphire laser (Spectraphysics Tsunami, picosecond mode) and pulse timing was controlled with a Tektronix DG2020 Data Generator. A continuouswave Ti:Sapphire laser (Coherent 89929) provided the optical pumping field which was turned on and off with an acoustooptic modulator (AOM). Typical pump powers were between 510 µW for a 50 µm spot size. The optical pumping field was applied for 610 µs. After a 2 µs delay a 23 ps broadband pulse from the modelocked Ti:Sapphire was applied. All data presented in this section was for a broadband pulse detuned from the lowest D0 D0 X transition by 1 THz. The
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
70
B
Figure 6.7: Experimental setup for the singlepulse and twopulse experiment. In the singlepulse experiment, the Michelson interferometer on the fast pulse leg is either replaced with a mirror or one arm is blocked. The experimental procedure is explained in the text.
single pulse was picked from the modelocked pulse train using a Conoptics electrooptic modulator (EOM). A typical EOM extinction ratio was 80:1. Leakage through the EOM was further suppressed with an AOM immediately following the EOM. The entire pulse sequence was repeated 2 µs after the broadband pulse.
Photoluminescence (PL) from the A∗ transition was collected throughout the pulse sequence. Contamination from the πpolarized laser scatter was suppressed by only collecting σpolarized PL. Additional filtering of the A∗ transition was attained using a diffraction grating (Acton SpectroPro750). The PL was detected with a PerkinElmer single photon counting module (SPCM). SPCM counts were binned as a function of time using an SRS SR430 multichannel scaler. The minimum time bin was 5 ns, but since typical pump down times were on the order 12 microseconds, bin sizes of 80 or 160 ns were used.
6.3. ULTRAFAST ROTATIONS: EXPERIMENT
400 350
71
B=7T T = 1.5 K 0.2
thermal equilibrium state
0.15
250
ρ22
counts
300
200
0.1
150
optically 0.05 pumped state
100 50 0 0
10
20 time (µs)
30
0 40
Figure 6.8: Thermal relaxation experiment. A field is applied resonant to the A transition while PL from the A∗ transition is monitored. During the 20 µs optical pumping pulse, population is pumped from ρ22 = 0.21 to 0.05. After the optical pumping pulse, the system is allowed to relax back to thermal equilibrium for 5 ms. In this particular experiment, data is collected for 17549 5 ms pulse sequences. Thermal relaxation traces In order to determine the relationship between the A∗ photoluminescence and the me = −1/2i electron spin population, a thermal relaxation experiment was performed. In this experiment the optical pumping field was applied for 20 microseconds and population was pumped into the me = 1/2i state. The system was then left in the dark for 5 ms while the population fully recovered to the thermal equilibrium value. The sequence was then repeated. The results of this experiment are shown in Fig. 6.8. The initial PL intensity when the pumping field was applied was proportional to the population in the me = −1/2i state. At 7 T and 1.5 K this corresponds to an excitedstate electron population of ρ22 = 0.21. The finite population left at the end of the optical pumping pulse indicates that some unwanted parasitic coupling occured between the two electron states during the optical pulse. Polarization dependence of the population transfer In the singlepulse experiment, the repetition rate can be much faster than the (5 ms)−1 thermal relaxation experiment repetition rate. This is because in the pulse
72
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
500
Counts
400 300 200 100 0 0
20 pulse
pulse
40 60 Time (µs)
80
100
Figure 6.9: Typical singlepulse experimental data. The fast pulse is applied between optical pumping pulses (it cannot be seen due to the spectral filtering of the collected PL). The initial height of the optical pumping curve determines the amount of electron population transfer by the fast pulse. Data is collected for 17549 pulse sequences. All optical pumping traces are added together and fit to an exponential. Population transfer is calibrated using the data from the thermal relaxation experiment.
experiment population transfer is due to the application of the broadband pulse and not due to the slow T1 process. A typical data set for the singlepulse experiment described above is shown in Fig. 6.9. After each application of the fast pulse, the excitedstate spin population ρ22 returns to some value greater than the optically pumped value. The value of ρ22 after the broadband pulse depends on the pulseenergy density as well as the pulse polarization. In Sec. 6.2 it was seen that the maximum population transfer (singlepulse π pulses) can only occur if the condition in Eq. 6.10 holds. The sum of the Rabi frequencies coupling a single ground state to the excited states is strongly dependent on polarization. Thus in order to ensure maximum rotations, the population transfer due to a single pulse was measured as a function of the pulse’s linear polarization. As seen in Fig. 6.10, there is a strong dependence of the Raman transfer efficiency on polarization with the most efficient transfer occurring at θ = 45◦ . Working at the peak of the polarization curve, however, does not necessarily ensure that the condition Eq. 6.10 holds. A verification of Eq. 6.10 would be the observation of π pulses.
6.3. ULTRAFAST ROTATIONS: EXPERIMENT
73
0.14 0.13
0.11
ρ
22
0.12
0.1 0.09 0.08 0.07 0
20
40 60 polarization angle
80
100
Figure 6.10: Polarization dependence of the singlepulse population transfer. A peak in the transfer efficiency occurs at a 45 degree linear polarization.
Singlepulse saturation curve
Population transfer into state me = −1/2i, or ρ22 was measured as a function of the average power in the modelocked pulse train. From the theoretical analysis of three and multilevel systems, oscillations in ρ22 are expected with a peaks occurring for (2n + 1)π pulses. A plot of the ρ22 power dependence in Fig. 6.11, however, shows a saturation of the population transfer at ρ22 = 0.5. A theoretical simulation for the population transfer for 2 ps pulses is included in Fig. 6.11. The simulation is identical to Fig. 6.3 with the following exceptions: First, the initial density matrix for the spin system corresponds to the experimentally observed optically pumped state instead of all the population beginning in state 1i. Second, the population transfer is plotted as a function of Ωp 2 (since experimental data is a function of power) normalized to fit the beginning power dependence in the experimental curve. The experimental and theoretical curves agree well at low powers which suggests that in this regime coherent rotations are occurring. At higher powers, however, the saturation of the experimental curve suggests some fast dephasing process prohibits large angle rotations. Possible dephasing mechanisms at higher powers will be discussed in Sec. 6.3.3.
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
74
1 theory experiment 0.8
ρ22
0.6
0.4
0.2
0
50 100 150 Averaged pulse power (µW)
Figure 6.11: Power dependence of the singlepulse electron spin population transfer (solid line). The nonlinear power dependence at low powers suggests coherent Raman rotations are occurring. The saturation of the population at 0.5 at high powers indicates that large rotations are currently limited by some dephasing mechanism. A theoretical curve (dotted line) calculated using the threelevel densitymatrix model is included for reference.
6.3.2
Twopulse experiment: Observation of 40 GHz electron spin precession
A second experiment, using two fast pulses separated by a variable delay τD , can be performed to verify that coherent rotations are occurring at low pulse powers. The experimental setup for the twopulse experiment is the same as the singlepulse experiment except that a Michelson interferometer is used to split the broadband pulse path into two paths with a variable delay (Fig. 6.7). The delay between the pulses is varied using a Klinger stepper stage which has a minimum step size of 100 nm. The powers of the two arms are set such that the observed photoluminescence intensity due to each are equal to within 5%. In Sec. 6.2 it was found that the pulse arrival time determines the axis of rotation. For example, if two pulses are separated by the Larmor period τL , the rotation axis of each pulse (which we will define as xˆ) will be identical . If two pulses are separated by half the Larmor period τL /2, the second rotation will be about the −ˆ xaxis and will effectively undo the first rotation. Any other delay will result in a rotation about
population
6.3. ULTRAFAST ROTATIONS: EXPERIMENT
0.4
(a) τD = τL
0.4 0.2
(b)
ρ22
τD = τL/2
ρ33
0.2
0.2
0 10
ρ22
0.4
75
0
10 20 ps
30
0 10
0
10 20 ps
30
(c) τD = τL/2
0 10
τD = τL
20
30
40
50
60
τD (ps)
Figure 6.12: Theoretical simulations for the twopulse experiment. (a) State 2i and state 3i populations during the application of two pulses separated by the τD = τL . (b) State 2i and state 3i populations during the application of two pulses separated by the τD = τL /2. (c) Final state population after the two pulses as a function of the twopulse separation. some axis in the xˆˆ y plane which includes a yˆ component (e.g. τL /4 corresponds to the yˆrotation axis). As shown in Fig. 6.12c, if the population after the second pulse is measured as a function of the twopulse delay, oscillations will be observed with a frequency equal to the Larmor precession frequency. The twopulse experiment was performed at both 5 T and 7 T magnetic fields. The results are shown in Fig. 6.13. As predicted, oscillations with a period equal to the Larmor period were observed in both cases. The observation of 40 GHz Larmor precession in these twopulse experiments demonstrates both the full rotation axis control and the ultrafast control speed of the single pulse Raman ESR technique. In Fig. 6.14, ρ22 is plotted for the twopulse experiment at 7 T in which the single arm average power is 60 µW. In contrast to the theoretical dependence plotted in Fig. 6.12c, in the experimental data ρ22 never goes to 0. Defining the visibility of the twopulse curve to be V = (ρ22,max − ρ22,min )/(ρ22,max + ρ22,min ), the twopulse experimental visibility is 0.52. This nonunity visibility cannot be accounted solely by the finite population in the optically pumped state (dashed line in Fig. 6.14)
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
76
7T
population (a.u.)
population (a.u.)
5T
10
20
30
pulse delay (ps)
40
10
50
20
30
pulse delay (ps)
40
50
(b)
(a)
Figure 6.13: Twopulse experiment. Population in state 2i is plotted as a function of the twopulse delay. The twopulse delay has an overall offset between 0 and 50 ps.(a) B = 5 T. The oscillation frequency of 30.6 GHz corresponds to a gfactor of 0.44. (b) B = 7 T. The oscillation frequency of 41.5 GHz corresponds to a gfactor of 0.42. A decrease in the magnitude of the electron gfactor with increasing magnetic field is expected [22].
and indicates that in addition to coherent population transfer, incoherent population transfer is also occurring. We can use a simple, twolevel model to estimate the rotation angle and the rotation fidelity of the Raman pulse. In the model we assume the following 2level Hamiltonian,
Hpulse = −
0
Ω/2
Ω/2
0
.
(6.11)
During the pulse rotation the densitymatrix evolves according to Eq. 6.7. Decoherence during the rotation is modelled by the following simple decoherence matrix
Lpulse =
0
−γρ12
−γρ21
0
.
in which γ is a constant dephasing term applied during the pulse application. Between
6.3. ULTRAFAST ROTATIONS: EXPERIMENT
77
0.8 data two level model
0.7 0.6
ρ22
0.5 0.4 0.3 0.2 0.1 0
optically pumped population
10
20 30 40 twopulse delay (ps)
50
Figure 6.14: A fit of the twopulse experimental data to a twolevel model including Rabi rotations, decoherence, and Larmor precession. Data is taken at 7 T.
the twopulses, the system precesses according to the Larmor Hamiltonian
HLarmor =
0
0
0 ωL
for a time τD . Both the applied pulse time ∆t and the decoherence parameter γ are varied to fit the experimental Larmor precession curve in Fig. 6.14. The simulation begins with the twolevel system in the observed experimental optically pumped state. The fit from this twolevel model gives a singlepulse rotation angle of θ = Ω∆t = 1.14 (slightly greater than π/3) and a pulse fidelity of F = 0.91. The pulse fidelity F is defined as hΨρΨi in which Ψ is the wavefunction obtained for a perfect rotation of area θ = 1.14. The twopulse rotation area is thus θ = 2.28 and the twopulse fidelity is F = 0.73. Although the population saturation curve in Fig. 6.11 indicates that single pulse areas greater than π/2 are not yet achieved in this system, the above analysis show that significant pulse areas as still possible. The question that remains is what is limiting the larger pulse areas in the D0 D0 X system?
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
78
6.3.3
Ultrafast dephasing in the D0 D0 X system
In Sec. 6.2, a threelevel theoretical model using the known GaAs D0 D0 X relaxation parameters predicted that π pulses with high fidelities (0.95) could be obtained with 2 ps pulses. Experimentally, however, π pulses were not observed. Using only a threelevel densitymatrix model, the observed saturation curve in Fig. 6.11 can be reproduced if some fast dephasing/decoherence process occurs during the pulse duration. Although this fast decoherence process is most likely due to interactions with additional levels not included in the threelevel model, the timescale of the interaction can be estimated using a threelevel model and additional dephasing terms. Specific dephasing mechanisms will be discussed, but in general they can be divided into two groups: 1) dephasing of the D0 X state and 2) dephasing of the D0 states. Dephasing of the D0 X states Dephasing of the D0 X states can be modelled by replacing the relaxation operator in Eq. 3.8 with
L(ρ) = −
0
0
0
0
γ3 (I)ρ31 γ3 (I)ρ32
γ3 (I)ρ13
γ3 (I)ρ23
(6.12)
0
in which we have only kept the fast excitedstate dephasing term γ3 . Additionally, we have allowed γ3 to depend on the pulse intensity I. This power dependence is important since a constant dephasing term large enough to cause the ρ22 population saturation will have a significantly lower twopulse visibility at lower powers than that experimentally obtained. Additional relaxation terms included in Eq. 3.8 are so small compared to the picosecond pulse duration that they do not need to be included. In Fig. 6.15a, we plot the experimental data and the threelevel simulation which includes γ3 relaxation. γ3 in this simulation is proportional to power and varies between 0 and 5 THz for the applied powers. At 60 µW, a dephasing of 1.4 THz reproduces the experimental twopulse visibility in Fig. 6.14. In the twopulse simulations, the dephasing term is only on during the pulse duration.
6.3. ULTRAFAST ROTATIONS: EXPERIMENT
0
6
4
V = 0.52
0.2
3
0.4
V = 0.27
0.2
1 20
30 τ D
100 200 power (µW )
40
6
0.6
5
0.4 0.2 0 10
4 20
0.2
30 τD
2
ρ
22
0.6
0 10
0.3
3
ρ22
5 0.3
7
0.4 γ (THz)
0.4
0 0
0.5
7
8
(b)
40
50
γ2 (THz)
(a)
0.1
D dephasing
8
ρ22
D0X dephasing 0.5
79
3 2
0.1
1
50
0 300
0 0
100 200 power (µW)
0 300
Figure 6.15: (a) Experimental data (black dots) and threelevel simulation (black dashed line) that includes fast D0 X γ3 dephasing. γ3 (red line) is linearly dependent on pulse power. Due to the fast γ3 , population at the end of the pulse is left in the excited D0 X state. The simulation splits this excitedstate population evenly between the lower two ground states. Reasonable agreement between the data and model cannot be obtained for a constant γ3 . Inset: 2pulse simulation at 60 µW using the threelevel model. The model gives the experimentally observed visibility of 0.52. (b) Experimental data (black dots) and threelevel simulation (black dashed line) that includes fast D0 γ2 dephasing. γ2 (red line) is linearly dependent on pulse power. Additionally, a reasonable agreement between the data and the model can also be obtained for a constant γ2 = 5 − 7 THz for this dephasing model. Inset: 2pulse simulation at 60 µW. The model gives a visibility of 0.27, which is significantly lower than the 0.52 experimental value. This threelevel model indicates that if fidelities are limited by the excitedstate dephasing, dephasing times on the order of 100 fs1 ps are necessary. There are several possible dephasing mechanisms that could be occurring, all of which will require further theoretical and/or experimental work to confirm. • Bound exciton interactions: Theoretically we have mainly modelled threelevel systems with a single D0 X excited level. However, photoluminescence spectra and photoluminescence excitation (PLE) spectra show more than 14 transitions between 817 nm and 817.4 nm. The Raman pulses are detuned from the lowest
80
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
level D0 X lines by 1 THz or 2.3 nm. Thus Raman pulses are interacting with all the D0 X levels, including unbound levels at energies higher than the free exciton transitions at 816.5816.9 nm. PLE spectra show that excited D0 X transitions (see Fig. 2.6) between 817817.4 nm have transition linewidths less than 10 GHz. However, these linewidths are for continuous wave excitation which has a relatively low excitation intensity compared to pulsed excitation. It is possible that a high virtual D0 X population leads to strong excitonexciton interactions and significant dephasing. This theory is also qualitatively consistent with the γ3 power dependence. A theoretical estimate of the D0 XD0 X interaction strength should be performed. • Short lived D0 X states: Additionally, no spectroscopic data is currently available for unbound D0 X states (states with higher energy than the free exciton energy). If these states are very short lived, coupling to these states could also be modelled by a fast γ3 term in the threelevel model. • D0 Xfree exciton interactions: In addition to exciting virtual D0 X states, the ultrafast optical pulse also excites virtual free exciton states. The pulse is detuned from these states by 1.21.4 THz. In a semiconductor, free carriers are known to have strong interactions with bound carriers. For example, the strong interaction between free and bound carriers is the cause of efficient spinexchange averaging between electronic states which allows the optical pumping of semiconductor nuclei with aboveband photoexcitation [76]. Additionally, the strong interaction between virtual excitons and quantum dot electrons has been proposed as an efficient way to couple charged quantum dots for quantum computation [77]. A theoretical estimate of the spinexchange time for virtual free and bound excitons should be performed. • Superradiant effects: A final possible dephasing mechanism is the significant shortening of the excitedstate lifetime due to superradiant effects (see for example Chapter 8 in Ref. [75]). Spontaneous emission (S.E.) rates would need to increase by a factor of 1000 to reach THz dephasing times. Preliminary calculations based on the sample geometry and donor concentration indicate such
6.3. ULTRAFAST ROTATIONS: EXPERIMENT
81
high S.E. enhancements are not likely in our lightly doped sample. However, lifetime measurements with resonant, coherent excitation could be performed to confirm this.
Dephasing of the D0 states Dephasing of the lower D0 states could also account for the saturation effect observed in Fig. 6.11. To model this lower state dephasing, Eq. 6.12 can be replaced with
0
L(ρ) = γ2 (I)ρ21 0
γ2 (I)ρ12 0
0
0
0
0
(6.13)
in which γ2 is allowed to depend on the pulse intensity I. In Fig. 6.15b, we plot the experimental saturation curve and the results of the threelevel model with fast γ2 dephasing which is linearly dependent on I. The model suggests dephasing rates of 12 THz are necessary to see saturation effects. A comparison between the data and the γ2 dephasing model are not quite as good as in the γ3 dephasing model. We also note that a reasonable agreement between the γ2 model and the single pulse experimental data can also be obtained for a constant, fast, dephasing rate γ2 of 37 THz. If the singlepulse parameters are used to calculate the twopulse visibility at 60 µW (inset in Fig. 6.15b), a visibility of 0.27 is obtained. This visibility is significantly lower than the experimental value and it is not possible to fit both the single pulse and twopulse data satisfactorily using only the threelevel model with γ2 dephasing. One significant difference between γ2 and γ3 dephasing is that in the former no real excitation of the D0 X states occurs. Thus an experiment that measures timeresolved excitedstate luminescence could distinguish between the two models. If γ2 dephasing does turn out to be a factor, one possible dephasing mechanism would be the D0 free exciton exchange interaction which is similar to the D0 Xfree exciton interaction discussed above.
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
82
Visibilities as a function of pulse delay counts
0.5
3 ns
0.4 Visibility
0
20 40 short delay, ps
0.3
0.1
counts
0.2
0
0
0
0 ns 20 40 short delay, ps
1
2 delay (ns)
3
4
Figure 6.16: Twopulse experimental visibility for long delays. The decay of the visibility is a measure of T∗2 .
6.3.4
Decay of the 2pulse fringe visibility: Yet another T∗2 measurement
Since the visibility obtained in the twopulse experiment depends on the coherence between the lower two (D0 ) Λ states, a measurement of the twopulse visibility for long delays should yield the inhomogeneous electron spin dephasing time, T∗2 . To perform this experiment a twofoot optical rail was added to one of the Michelson interferometer arms in the twopulse experiment (Fig. 6.7). The twopulse experiment described in Sec. 6.3.2 was performed at delays of 03.5 ns. The complete data set of delays was taken three times. Data was taken for increasing delays for sets 1 and 3 and for decreasing delays for set 2. The average power in each arm was 60 µW. Additionally, T∗2 data was taken at higher (120 µW) and lower (30 µW) powers and no power dependence of the visibility decay was observed. In Fig. 6.16 the twopulse experiment visibility is plotted as a function of delay. Not surprisingly, the visibility decays to one half its initial value in 2.53 ns. This is consistent with the values obtained for T∗2 in both the coherent population trapping (Chapter 3) and the Raman (Chapter 4) experiments.
6.4. SPINECHO USING SMALL ANGLE PULSES
6.4
83
Spinecho using small angle pulses
In the last section it was shown that it is possible to use the 2pulse experiment and the 2pulse visibility to estimate T∗2 (Fig. 6.16). In theory, using three broadband pulses, it should be possible to measure T2 even without π/2 and π pulses. (A good treatment on the standard spinecho sequence can be found Chapter 9 of Ref. [75]). In this section we derive the expectation value of the spin projection on the x ˆaxis (hσx i) for a twolevel system after two small rotations separated by time τ1 and find that a partial rephasing of the spins occurs at a time τ2 = τ1 after the application of the second pulse. We begin with the system in the ground state,
Ψi =
1 0
.
(6.14)
Next we apply an instantaneous rotation with area θ1 about the xaxis. The resulting rotated state immediately following the θ1 pulse is
Ψi =
cos( θ21 ) i sin( θ21 )
.
(6.15)
The twolevel system is left to freely precess for a time τ1 . At the end of this precession the wavefunction is given by
Ψi =
cos( θ21 ) i sin( θ21 )e−iωL τ1
,
(6.16)
in which ωL is the Larmor precession frequency. During this precession the xspin projection is given by hˆ σx i. hˆ σx i = hΨˆ σx Ψi = sin θ1 sin(ωL τ1 ). If we assume an inhomogeneous Bfield with a Gaussian distribution, the ensemble
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
84
averaged spinprojection, hˆ σx i, is given by hˆ σx i = in which ωL (B) =
gµB B h ¯
Z ∞ ∞
dB sin(θ1 ) sin(ωL (B)τ1 )g(B),
and g(B) =
√ 1 2πσB
2
0) ). This can be solved anaexp(− (B−B 2σ 2 B
lytically and the final averaged xˆspin projection is hˆ σx i = sin(θ1 ) sin(ωL τ1 )e−
(∆ωL τ1 )2 2
in which the Larmor frequency ωL is given by ωL = frequencies ∆ωL is given by ∆ωL =
gµB σB . h ¯
gµB B0 h ¯
, and the spread in Larmor
This result is the standard free induction
decay (FID) curve in which the Larmor precession is damped by an exponential with a time constant proportional to the inverse spread of the inhomogeneous magnetic field. Additionally, the overall initial magnitude of the signal is given by sin(θ1 ) as this is the initial tipping angle into the xˆˆ y plane. This FID curve is what was measured in the twopulse visibility experiment in Sec. 6.3.4. At time τ1 , a second rotation about the xˆaxis of area θ2 is applied and the spin is allowed to precess freely for a time τ2 . The state after the pulse and the free precession is given by
Ψi =
cos( θ21 ) cos( θ22 ) − sin( θ21 ) sin( θ22 ) exp(−iωL τ1 ) (i cos( θ21 ) cos( θ22 ) + i sin( θ21 ) sin( θ22 ) exp(−iωL τ1 )) exp(−iωL τ2 ).
.
Solving for hˆ σx i for a single spin, we find that the expectation value is the sum of three terms hˆ σx i = cos(θ1 ) sin(θ2 ) cos(ωL τ2 ) θ2 + sin(θ1 ) cos2 ( ) sin(ωL (τ1 + τ2 )) 2 θ 2 + sin(θ1 ) sin2 ( ) sin(ωL (τ1 − τ2 )). 2
(6.17)
We note that in the case where θ2 = π only the third term in Eq. 6.17 remains (and a complete echo of the original signal will appear at τ1 = τ2 in the inhomogeneous
6.5. SUMMARY
85
case). It is straightforward to integrate hˆ σx i over a Gaussian distribution of magnetic fields. The result is again three terms (∆ωL τ2 )2
hˆ σx i = − cos θ1 sin θ2 cos(ωL τ2 )e− 2 (∆ωL (τ1 +τ2 ))2 θ2 2 − sin(θ1 ) cos2 ( ) sin(ωL (τ1 + τ2 ))e− 2 (∆ωL (τ1 −τ2 ))2 θ2 2 + sin(θ1 ) sin2 ( ) sin(ωL (τ1 − τ2 ))e− . 2
(6.18)
The first term in Eq. 6.18 is the FID signal due to the second pulse. The third term is the echo term and it is maximum when τ1 = τ2 . We see that this echo is greatest if θ2 is a π pulse and θ1 is a π/2 pulse. Even without these large pulse capabilities, however, the echo pulse should be able to be detected in the D0 D0 X system. The decay of this echo will be due to the intrinsic decoherence in the system and thus a T2 measurement should be possible by rotating the spin back on to the zˆaxis using a third pulse.
6.5
Summary
In this chapter the ultrafast control of the D0 electron spins using fardetuned, broadband pulses has been proposed. Theoretically we have shown that complete SU(2) control of the electron spin state should be possible with this optical Raman technique in a threelevel system. Experimentally, we were able to achieve appreciable pulse rotation angles, with single pulse rotations exceeding π/3 and twopulse rotations exceeding 2π/3. Additionally, the twopulse experiment demonstrated complete control over the rotation axis using the pulse arrival timing. To our knowledge this work presents the first experimental demonstration of coherent ultrafast spinmanipulation using single, far detuned optical pulses in the solid state. Although fidelities currently seem limited by a fast, unknown dephasing process in the D0 D0 X system, experimentally obtained pulse areas and fidelities should be sufficient to perform a homogeneous T2 measurement in the GaAs D0 system.
86
CHAPTER 6. OPTICAL MANIPULATION OF THE D0 SPIN
Additionally, if dephasing effects are due to bandedge effects such as freeexcitonD0 X interactions, this dephasing may not be a factor for the application of this ESR technique in deeper impurities and quantum dot systems.
Chapter 7 Summary and Outlook This chapter provides a summary of the work performed in this thesis as well as an outlook for the future of donorbound exciton systems in the field of quantum information processing. This work almost exclusively focused on the GaAs D0 D0 X ensemble system in highpurity MBE grown GaAs epilayers. In this system we sought to address three main questions: 1. Can the D0 D0 X system provide a Λtype system? 2. What are the coherence properties of the D0 spin state? 3. Can the D0 spin be optically controlled for QIP applications and to lengthen the D0 spin dephasing time through spinecho techniques? To answer the first question, magnetophotoluminescence spectroscopy and high resolution photoluminescence excitation spectroscopy were used to isolate a D0 D0 X Λ system and to demonstrate optical pumping between the D0 electron spin states. We found that the optical transition linewidths were extremely narrow, 110 GHz, for a bulk semiconductor system. We first attempted to control the D0 spin in a coherent population trapping (CPT) experiment. Using the optical Raman transitions, CPT using continuouswave fields 87
88
CHAPTER 7. SUMMARY AND OUTLOOK
was demonstrated for the first time in a semiconductor. However, the effect was limited by the short nanosecond dephasing time of the electron spins. The CPT result led to a study designed to answer the second question. The bulk dephasing properties of the system were studied in a Raman linewidth experiment. We found that in samples with a low donor density, the dependence of the Raman linewidth on magnetic field and temperature was consistent with nuclearhyperfine theory. Additionally, we were able to use the Raman lines to study the nuclearspin environment inside the D0 wavefunction in a nuclearspin optical pumping experiment. Unfortunately, the Raman experiments showed that the nanosecond bulk dephasing time observed in the CPT experiment remained short under a variety of experimental conditions. For any practical QIP device based on the D0 D0 X system, any operation on the D0 spin state will need to be performed on a timescale fast compared to the bulk dephasing time T∗2 . Additionally, spinecho techniques utilizing subnanosecond electron spin control will be necessary for extending the processing time to its fundamental limit (possibly the millisecond T1 time measured in this system). The extension of the spin dephasing time was the motivation for the final part of the work which addressed the third question above. We thus proposed and implemented the ultrafast spin control using single, fardetuned, picosecond pulses. Theoretically we were able to show that this technique should provide complete SU(2) control over the electron spin state. Using this technique, we were able to experimentally demonstrate reasonable rotations with areas exceeding π/3, the probing of the electron spin on a timescale much faster than the electron Larmor period, and the control of the rotation axis using the pulse arrival timing. Although there are still some unanswered questions on the experimental side, we hope that the experimental demonstration of this fast Raman ESR technique will prove useful in the general study of solidstate spin dynamics. I conclude by noting that this body of work only realizes a small part of the potential of D0 D0 X systems, as it focuses almost entirely on ensembles in GaAs. While there is still much work to be done in ensembles of donors in GaAs, in the long run, one of the most attractive features of the D0 D0 X system is its ubiquitousness
89
in all semiconductors. Different materials offer different advantages. For example, the bright, nuclearspin free host matrices of isotopically purified IIVI materials may eventually prove to have long enough bulk T∗2 times to eliminate the need for spinecho techniques. Additionally, the deeper impurity binding energies could lead to a reduction in the bandedge dephasing effects observed in the fastpulse experiments. Another potentially promising system is the P:Si system that exhibits a strong impurity nucleuselectron hyperfine interaction [78, 79]. This strong hyperfine interaction could be utilized for longlived quantum memory if the weak optical properties of silicon are enhanced with a semiconductor microcavity [80]. In addition, to date, single D0 D0 X emitters have not been isolated from the bulk. The isolation of nearly identical D0 X emitters opens up the D0 D0 X system to the rich QIP research based on single emitters. Thus I hope that the work presented in this thesis will be just one part of a much broader effort of D0 D0 X quantum information processing.
Appendix A Selection rules and relaxation rates of acceptorbound excitons in GaAs In this appendix we present a spectroscopic study of the GaAs neutral acceptor (A0 ) coupled to the neutralacceptorbound exciton (A0 X). Using timeresolved spectroscopy we confirm that the A0 X system has strong optical transitions, with a radiative lifetime of 350 ps. Pumpprobe experiments indicate that population relaxation times between the heavy and light hole levels is fast faster than the 30 ps experimental resolution. Although A0 X systems may be appropriate for some QIP applications such as single photon sources [81], the picosecond relaxation time makes them unsuitable for EITtype applications and other QIP applications requiring long processing times.
A.1
Introduction
Neutralacceptorbound excitons in GaAs exhibit three sharp photoluminescence lines near 1.5124 eV at liquid helium temperatures. Although at one time controversial [82, 83, 84, 85, 86], the origin of the triplet is now established and is attributed to holehole jj coupling and the crystal field potential [87]. Despite the excitation of millions of 90
A.2. HYDROGENIC SPECTRA OF ACCEPTORS IN GAAS
91
impurity centers, the bulk luminescence linewidths can be as narrow as 0.02 meV [87] and can be used to study neutral acceptor energy levels, acceptorbound exciton selection rules and radiative relaxation rates, and neutral acceptor hole spin relaxation rates. In this appendix we first present data on the A0 X twohole transitions and show how this data may be used to obtain information on the type of acceptor, the acceptor ionization energy, and the centralcell potential. In Sec. A.3.2 a full theoretical model of the acceptorbound exciton is derived. Using a generalized WignerWeisskopf (WW) theory we obtain the surprising result, not noted in previously published work [88, 89, 29], that the Γ3 and Γ5 levels have different radiative lifetimes. Additionally, the Γ5 level is composed of two sets of states with radiative lifetimes that differ by a factor of 4. In Sec A.4 we present timeresolved and polarizationresolved spectra of the Γ3 and Γ5 photoluminescence after a short excitation pulse. The initial polarization of the photoluminescence after resonant excitation is in agreement with the general WW model. Additionally, we estimate that the fast Γ5 lifetime is 350±10 ps. In Sec. A.5 we use the A0 A0 X transitions to study the A0 hole spin relaxation between the light (mh = ±1/2i) and heavy (mh = ±3/2i) hole states for a sample under uniaxial strain. Pumpprobe measurements indicate a fast, > 30 GHz relaxation rate between the light and heavy hole states.
A.2
Hydrogenic spectra of acceptors in GaAs
At liquid helium temperatures acceptors in GaAs can be described by effective mass theory (EMT). The wave function of the bound hole is given by the product of its Bloch function ψB (~r) and hydrogenic envelope function F (~r) [19]. In the EMT hydrogenic wavefunction, the mass is the effective hole mass and the charge of the acceptor impurity is screened by the dielectric constant of the material. For acceptors in GaAs, the acceptor ionization energy can range from 2635 meV [23] and the corresponding effective Bohr radius is < 20 ˚ A. There are several ways to determine ionization energies and interlevel transitions of neutral acceptors (A0 ) including photoconductivity
92
APPENDIX A. GAAS A0 X STUDY
measurements, infrared absorption measurements [90], and conduction band to acceptor photoluminescence measurements [91]. Additionally, the same information can be obtained using the sharp photoluminescence lines of the A0 X transitions [92]. An acceptorbound exciton is formed when an electronhole pair (exciton) becomes bound to the neutral acceptor. The A0 X complex is thus composed of two holes, an electron, and the acceptor impurity. Due to the exciton confinement potential, the A0 X A0 1s transitions are narrow ( 2, the observed spectrum fits a hydrogenic one very well (Fig. A.1c). From this fit we obtain a carbon ionization energy of 25.8±1.3 meV and a central cell correction of 3.8±1.0 meV. This value is in agreement with the carbon ionization energy 26.3±0.1 meV previously derived from infrared absorption measurements [23]. In addition to providing information on the energy levels of the A0 state, A0 X THT lines are valuable tools for monitoring A0 X population during resonant excitation experiments as will be shown in Sec. A.5.
A.3. LIFETIMES AND SELECTION RULES OF THE A0 X COMPLEX
93
1.52 a)
1s
energy (eV)
A 0X
FE
1.51 1.51
3s 2s
1.5
1.515
b)
1.52
2s
3s 5s 4s
1s 1.485
£10 1.49
1.495
energy (eV)
1.49 c)
1
2
3
4
5
sorbital
Figure A.1: a) A0 X A0 1s PL for aboveband excitation. Spectra is instrument resolution limited. Also observed is the free exciton (FE) line. b) THT 2s5s states observed under resonant Γ3 excitation. c) Hydrogenic fit of the 2s5s THT lines.
A.3
Lifetimes and selection rules of the A0X complex
In this section we briefly review the energy levels and eigenstates of the A0 X complex in the absence of strain or a magnetic field. We then derive the selection rules for the A0 XA0 optical transitions. Using a generalized WignerWeisskopf method, we find that the A0 X states are composed of 3 sets of spontaneous emission (SE) eigenstates with different lifetimes. Additionally, with the knowledge of the SE eigenstates we are able to predict the polarization of the emitted photoluminescence in certain excitation conditions.
A.3.1
Energy levels and basis states
The triplet splitting of the A0 X complex arises from the holehole coupling and the crystal field potential[85, 86, 87]. As shown in Fig. A.2, the A0 X complex consists of two Jh = 3/2 holes and a J = 1/2 electron. The two holes couple to form the singlet J=0 and triplet J=2 states. This work concentrates on the J=2 states, as at
APPENDIX A. GAAS A0 X STUDY
94
spherical symmetry, noninteracting
tetrahedral symmetry
holes interacting
2
A0X
2
12 6 10 4
A0
4
4
4
Figure A.2: Energy levels of the A0 X and A0 complexes. Degeneracies of the levels are labelled at the lower right of each line. The final observed levels are labelled according to their group theoretic representation [96]. liquid He temperatures emission from the higher level J=0 state is very weak. In the tetrahedral crystal field, the fivefold degenerate J=2 state is further split into a 3fold Γ5 symmetric state and 2fold Γ3 symmetric state [85]. Although the final coupling of the twohole states and the Γ6 electron can reduce the symmetry further, experimentally no such splitting is observed.
A.3.2
Selection rules, WignerWeisskopf theory
With the knowledge of the basis states of the Γ3 , and Γ5 twohole states [85] and the dipole matrix operator µ ˆ [97], it is possible to derive the transition matrix elements for the A0 X complex. The basis states and dipole matrix operator can be found in the Sec. A.6.2. The ground and excited states are connected by the dipole operator ~µν (photon energy ν) such that hg, i, ω~µν e, ji = ~µhg, i, ωˆ µν e, ji = δων ~µMij , in which e denotes the excited state, j denotes the degenerate level within the excited state, g denotes the ground state, i denotes the degenerate level within the ground
A.3. LIFETIMES AND SELECTION RULES OF THE A0 X COMPLEX
mh mh mh mh
3 2 1 2
= = = − 12 = − 32
Γxy ↑ + iy) 0 − 12 (x − iy) − √13 z
1  2√ (x 3
Γxz ↓ − iy) 0 1 2 (x + iy) √1 z 3
1 √ (x 2 3
Γyz ↓ − iy) 0 1 2 (x + iy) − √13 z
Γxy ↓ − √13 z 1 2 (x + iy) 0 1 √ (x − iy) 2 3
1 √ (x 2 3
Γxz ↑ − √13 z 1 2 (x − iy) 0 1 √ (x + iy) 2 3
95
Γyz ↑ − √13 z − 12 (x − iy) 0 1 − 2√ (x + iy) 3
Table A.1: Normalized dipole matrix elements for the Γ5 A0 transitions. A0 transitions are labelled according to the single hole spin. Γ5 states are given in Sec. A.6.1. (x, y, z) denote photon polarizations. A0 X electron states are denoted by ↑ and ↓ . mh mh mh mh
= 32 = 12 = − 12 = − 32
Γa ↑  2√1 3 (x + iy) 0 1 (x − iy) 2 √1 z 3
Γa ↓ − √13 z 1 (x + iy) 2 0 1 √ − 2 3 (x − iy)
Γb ↑ 1 (x − iy) 2 − √13 z 1 √ (x + iy) 2 3 0
Γb ↓ 0 1 √ (x − iy) 2 3 √1 z 3 1 (x + iy) 2
Table A.2: Normalized dipole matrix elements for the Γ3 A0 transitions. Γ3 states are given in the Sec. A.6.1. state, and ω denotes the photon energy. The complete normalized dipole matrix operator Mij for the Γ3 and Γ5 states are given in Tables A.1 and A.2. The combined A0 X photon system can be described by the following wavefunction X
Ψi =
j
aj (t)e, ji +
XX i
bi,ω (t)eiωt g, i, ωi,
ω
in which aj (bj ) is the probability amplitude for being in the excited (ground) state. Following WignerWeisskopf theory [98], the probability amplitudes for the ground states evolve according to X X d aj = −ig Mij∗ bi,ω e−iωt dt ω i X d bi,ω = −ig Mij aj eiωt dt j
(A.1) (A.2)
where g is the vacuum Rabi coupling strength proportional to ~µ. By formally solving for bi,ω and assuming all population begins in the ground state, the solution for aj
APPENDIX A. GAAS A0 X STUDY
96
becomes d γ0 X X ∗ aj = − Mij Mik ak dt 2 i k d γ0 ~a = − (M † M )~a dt 2
(A.3) (A.4)
in which γ0 is the spontaneous emission lifetime in the case that M † M = 1. Thus the eigenvalues of M † M give us the spontaneous decay rate of the M † M eigenvectors. For the Γ5 states we find 2 distinct eigenvalues of M † M of 4/3 and 1/3, indicating that the Γ5 state is characterized by two lifetimes. For the Γ3 states we find that M † M is diagonal with a single eigenvalue of 1. Thus, although previous theoretical and experimental work have found a single decay constant for the A0 X states [89, 29], a full analysis of the A0 X system including symmetry considerations indicates three relaxation times.
A.3.3
Polarization of A0 X spontaneous emission for selected excitation conditions
The spontaneous emission matrix Mij can be used to calculate which ground and excited states are coupled by certain excitation polarizations and to calculate the polarization of the spontaneous emission (SE) after such an excitation. We examine two cases in which a strong SE polarization is predicted. Spontaneous emission polarization after xpolarized excitation: Γ5 In the absence of relaxation mechanisms, when xpolarized light is incident, the excited states evolve according to d ~a = Ω2 Mx† Mx~a dt in which Ω is the Rabi frequency and Mx contains only the xpolarized elements of M . The resulting Mx† Mx is block diagonal and composed of two 3 × 3 matrixes. Focusing on the upper diagonal block we find only one nonzero eigenvalue and thus
A.3. LIFETIMES AND SELECTION RULES OF THE A0 X COMPLEX
97
only one of the three degenerate excited states couples to xpolarized excitation. By symmetry, the same holds true for the lower diagonal block. Similarly, we can find the eigenvectors and eigenvalues of Mx Mx† which determine the coupled ground states. The resulting coupled excited and ground states are, √ Ã ! 1 1 3 3 1 xy xz yz √ (Γ ↑i − Γ ↓i − Γ ↓i) ←→  i+  i 2 2 2 2 3 √ Ã ! 1 1 3 3 1 xy xz yz √ (Γ ↓i + Γ ↑i − Γ ↑i) ←→ − i +  i . 2 2 2 2 3 Experimentally, for xpolarized excitation pulses that are short compared to A0 X relaxation rates, the excited states should correspond to the excited states above. The polarization of the excited state spontaneous emission can be found by decomposing the excited states into the SE eigenstates. The visibility of the polarization as a function of time is Ix − Iy Ix + Iy 24e−2γ0 t + 12e−γ0 t = . 40e−2γ0 t + 4e−γ0 t + 1
Vxy (t) =
The polarization visibility is a somewhat complicated function dependent of time, especially if intralevel and interlevel relaxation are included. However, there is a strong polarization visibility at t=0 of Vxy = 0.8 which is readily observed in experiments as shown in Sec. A.4. A similar calculation for ±45◦ polarized light shows only a weak polarization visibility.
Spontaneous emission polarization after +45◦ polarized excitation: Γ3 When a similar calculation is carried out for 45◦ polarization excitation of the Γ3 states, the situation is further complicated by the existence of two Rabi frequencies, √ Ω± = Ω (2 ± 3). The ground and excited states couple in the following manner, 6 µ
3 1 1 Ω+ : √  i + i− i 2 2 2
¶
´ 1 ³ ←→ √ Γa ↑i − iΓb ↑i 2
APPENDIX A. GAAS A0 X STUDY
98 ¶
µ
´ 1 1 ³ 1 3 Ω+ : √  i − i− i ←→ √ Γa ↑i + iΓb ↑i 2 2 2 2 µ ¶ ´ 1 1 3 1 ³ a Ω− : √  i + i− i ←→ √ Γ ↓i + iΓb ↓i 2 2 2 2 ¶ µ ´ 1 1 3 1 ³ Ω− : √  i − i− i ←→ √ Γa ↓i − iΓb ↓i . 2 2 2 2
In this more complicated case it is still possible to estimate the polarization of the spontaneous emission immediately after the excitation pulse in the low excitation limit, Ω± t ¿ 1. Since this state has a single radiative lifetime, the visibility for all time in the absence of other relaxation mechanisms is given by √ V±45 =
3 Ω+ − Ω−  3 = . 2 Ω+ + Ω−  4
A similar calculation for (x, y)polarized light shows no spontaneous emission polarization.
A.4
Timeresolved PL excitation spectroscopy of the A0X states
To in order to observe the polarization dependence of the A0 X PL after a short excitation pulse, it is necessary to do timeresolved measurements. In these experiments a short 30 ps pulse with polarization x or +45◦ was applied resonant on either the Γ3 or Γ5 transitions. PL from the THT lines, which are well separated in energy from the main lines, was detected as a function of time with a streak camera. In Fig. A.3 the PL intensities for the two cases discussed in Sec. A.3.2 are plotted. As WW theory predicts, a strong dependence of the PL polarization on the initial pump polarization was observed. In the Γ5 case, the polarization initial visibility was 0.8 as predicted. In the Γ3 case the initial visibility was 0.6 which is slightly lower than the predicted value. This may be because the excitation power was too strong to be in the weak excitation limit. It is difficult to determine the radiative relaxation rate for the A0 X system, since a
A.5. PUMPPROBE STUDY OF THE A0 BOUND HOLE RELAXATION
(a)
99
intensity
Γ5 x polarization Γ5 y polarization
0
0.5
1 time (ns)
(b)
1.5
Γ +45 polarization
intensity
3
Γ3 _ 45 polarization
0
0.5
1 time (ns)
1.5
Figure A.3: (a) Time dependence of x and ypolarized PL from Γ5 after Γ5 was resonantly excited with xpolarized light. A strong polarization anisotropy was observed immediately following the pump. (b) Time dependence of the +45◦ and −45◦ polarized PL from Γ3 after Γ3 was resonantly excited with +45◦ polarized light. Again, a strong polarization anisotropy was observed. full description of the system must include the three radiative relaxation rates, intralevel relaxation rates, and interlevel relaxation rates. However, if Γ5 is resonantly excited, the majority of the PL will due to the fast Γ5 radiative relaxation rate. In Fig. A.4 we fit the PL after Γ5 excitation to an exponential and find a relaxation of 350 ± 10 ps. We note that this is less than half of the previously measured value of 1.0 ± 0.1 ns [29].
A.5
Pumpprobe study of the A0 bound hole relaxation
If the degeneracy of the ground state can be lifted, pumpprobe experiments on the A0 A0 X transitions can be used to gain information on the spin state of the A0 hole. One simple way to split the A0 states is to apply strain to the sample. In this section we discuss a pumpprobe experiment on a strained GaAs sample. We find a lower
APPENDIX A. GAAS A0 X STUDY
100
PL Γ5
PL Γ3 + Γ5
PL Γ3
fit
(b)
intensity
Log(intensity)
(a)
0
0.5 1 1.5 time (ns)
0
0.5 1 time (ns)
1.5
Figure A.4: (a) PL from the Γ3 and Γ5 levels after resonant excitation of Γ5 with xpolarized light. (b) A fit to the total PL indicates a 350 ps radiative relaxation time for the fast Γ5 rate. bound on the relaxation rate between the heavyhole and lighthole states of 30 GHz.
A.5.1
Experimental technique
The GaAs sample was sandwiched between two plates with 1 mm apertures. A micrometer controlled springloaded pin pressed into the center of the back of the sample. Photoluminescence was collected from the front MBE epilayer. The effect of applied strain is twofold. First, the A0 states split into the heavy ±3/2i and light hole ±1/2i states. The splitting is linear with applied stress 0.67 meV/(kg× mm−2 )) [99] and for tensile strain the light hole (lh) band lies above the heavy hole (hh) band. Second, the A0 X Γ3 state exhibits a nonlinear splitting as a function of applied stress [85, 87]. We worked in a regime such that the Γ3 splitting was much less than the A0 splitting. A high resolution photoluminescence excitation scan of the A0 X transitions under strain can be found in Fig. A.5. In the pumpprobe experiment a short 30 ps pulse was applied resonant with the Γ5 lh transition. This pulse pumped the lh state to the Γ5 state where it then relaxed to both the lh and hh states. A weak, 30 ps probe pulse was then applied to
A.5. PUMPPROBE STUDY OF THE A0 BOUND HOLE RELAXATION
intensity (arb. units)
pump
lh
819.9
101
probe hh
820
wavelength (nm)
Figure A.5: Photoluminescence spectra of the A0 A0 X transitions under applied tensile strain. In pumpprobe experiments the pump pulse was resonant on the Γ5 lh transition and the probe pulse was resonant on the Γ3 hh doublet. the Γ3 (doublet)hh transition to probe the population in the hh state. If the hhlh relaxation, T1 , is long compared to the Γ5 radiative relaxation τr , an increase in the hh population is expected at times long compared to τr and short compared to T1 . If however, T1 is short compared to τr a decrease will be observed at times < τr . Additionally, if T1 is shorter than the 30 ps pulse duration, the population in the hh state can approach 0 for strong pump pulses.
A.5.2
Results and Discussion
The result of the pump probe experiment is shown in Fig. A.6. To remove the effects of the photoluminescence from the pump, we plot the combined pumpprobe emission minus the emission when only the pump pulse is applied. This is then normalized by the emission when only the probe pulse is applied. To avoid contamination from laser scatter the PL intensity is measured in the window 150650 ps after the probe pulse. What is observed experimentally is a sharp decrease in the probe emission at short delays. This sharp decrease indicates a lower state relaxation time shorter than the pump time. The dashed and solid lines give the theoretical predictions for a three state system with an upper state radiative lifetime τr of 800 ps and a lower
APPENDIX A. GAAS A0 X STUDY
102
1
30 ps, theory 1 ps, theory experiment
ratio
0.8 0.6 0.4 0.2 0 0
2
4 6 pumpprobe delay (ns)
8
Figure A.6: Pumpprobe experiment to determine holerelaxation rates of the A0 bound hole. The small intensity of the PL due to the probe for short pumpprobe delays indicates a fast hole relaxation rate. state lifetime T1 of 30 ps and 1 ps respectively. At long times a slight increase in the efficiency of the probe in the pumpprobe case is observed (ratio > 1). This can be attributed to a small increase in sample temperature due to absorption of the strong pump pulse. The magnitude of the increase indicates a 0.8 K increase in temperature over the 2.5 K measured temperature. The short holerelaxation time in this small strain limit (kB T > ∆Estrain ) is in agreement with the 6 ps lifetimes measured using the Hanle effect in a similar system, holes bound in donorbound exciton complexes in unstrained bulk GaAs [24].
A.6
Basis states and dipole operator
In this section we give the explicit expressions for the A0 X Γ5 and Γ3 basis states and the dipole matrix operator.
A.6.1
Basis states
Basis states are obtained from Mathieu et al. [85]. We denote these states in terms of the hole (h) and electron (e) creation and annihilation operators. For identical
A.6. BASIS STATES AND DIPOLE OPERATOR
103
√ fermions we use the condensed notation h†a h†b 0i ≡ 1/ 2(a, bi − b, ai). Γ5 basis states ¶ √ µ † † † † = 1/ 2 h 3 h 1 + h− 3 h− 1 e†± 1 0i 2 2 2 2 2 µ ¶ √ † † † † † Γxz 5 i = 1/ 2 h 3 h− 1 − h− 3 h 1 e± 1 0i 2 2 2 2 2 ¶ √ µ † † yz † † † Γ5 i = 1/ 2 h 3 h− 1 + h− 3 h 1 e± 1 0i
Γxy 5 i
2
2
2
2
2
Γ3 basis states ¶ √ µ Γa3 i = 1/ 2 h†3 h†1 − h†− 3 h†− 1 e†± 1 0i 2 2 2 2 2 µ ¶ √ Γb3 i = 1/ 2 h†3 h†− 3 + h†1 h†− 1 e†± 1 0i 2
A.6.2
2
2
2
2
Dipole operator
The dipole matrix operator is given by ´ √ a†x + ia†y ³ √ h 3 e− 1 + (1/ 3)h 1 e 1 2 2 2 2 2 † † ³ ´ √ a − ia − x √ y h− 3 e 1 + (1/ 3)h− 1 e− 1 2 2 2 2 2 ´ ³ 2 + √ a†z h 1 e− 1 + h− 1 e 1 2 2 2 2 6
µ ˆ =
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