Coherent Exciton Phenomena in Quantum Dot Molecules

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A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University

In partial fulﬁllment of the requirements for the degree Doctor of Philosophy

Juan Enrique Rolon Soto November 2011

This dissertation titled Coherent Exciton Phenomena in Quantum Dot Molecules

by JUAN ENRIQUE ROLON SOTO

has been approved for the Department of Physics and Astronomy and the College of Arts and Sciences by

Sergio E. Ulloa Professor of Physics and Astronomy

Howard Dewald Dean, College of Arts and Sciences 3 Abstract

ROLON SOTO, JUAN ENRIQUE, Ph.D., November 2011, Physics and Astronomy Coherent Exciton Phenomena in Quantum Dot Molecules (144 pp.) Director of Dissertation: Sergio E. Ulloa We investigate different aspects of the coherent dynamics of excitons in quantum dot molecules. A theoretical model is developed in order to extract the Forster¨ energy transfer signatures in tunnel coupled quantum dot molecules in the presence of strong interdot tunneling. It is found that Forster¨ coupling can induce spectral doublets in the excitonic dressed spectrum, which is suitable for detection in level anticrossing spectroscopy. The coherent exciton dynamics is investigated both in the closed and open quantum system approach by means of the Lindblad master equation. An adiabatic elimination procedure using the projection operator formalism allows us to extract effective Hamiltonians to describe analytically all relevant anticrossing gaps of the dressed spectrum. It is found that a pair of two indirect excitons can be used as the computational basis of a qubit. An adiabatic control pulse is constructed in order to manipulate the indirect exciton qubit and characterize its coherent dynamics, as well as its decoherence due to spontaneous recombination. On the other hand, recent experiments have shown that indirect excitons in hole tunnel coupled quantum dot molecules exhibit indirect exciton oscillatory relaxation rates, as function of an applied electric field. To this end we developed a model for the experimental results, in which we incorporate relaxation due to exciton-acoustic phonon coupling. We characterize the scattering structure factor and found that it contains an electrically tunable phase relationship between the phonon wave and the hole wave function, which leads to interference effects and oscillatory relaxation rates. Approved: Sergio E. Ulloa Professor of Physics and Astronomy 4

To Anna my daughter, Diana my wife, Maria my mother and Max my brother 5 Acknowledgments

I would like to thank the Physics and Astronomy department for developing an encouraging atmosphere to pursue graduate studies in physics at Ohio University. In particular, to those colleagues who never hesitate to share informal conversations about physics and ordinary life matters, who never hesitate to bring out a laugh, or say hello. To the professors who hold graduate students in high regard and encourage them to fulfill themselves as professionals with critical thinking. In this special group, I would deeply thank Prof. Sergio E. Ulloa for being a very supportive advisor, always willing to share his time for discussions even in the busiest of days. I would like also to thank my advisor for creating an atmosphere of friendship and collaboration inside and outside the office, through our weekly group meeting discussions in diverse topics of condensed matter physics, and for its support for students to travel and share their research results with the rest of the world. I would also like to thank my friends and colleagues, Pedro L. Hernandez, Ginetom Diniz, Kushal Wijesundara, David Ruiz, Ahn T. Ngo, Greg Petersen, Mahmoud Asmar and Tejinder Kaur, for all those enjoyable times of traveling and comradeship. Special thanks to professors Dr. Nancy Sandler, Dr. Eric Stinaff, and Dr. Sasha Govorov, who have been excellent mentors inside and outside the classroom. And finally, I would also like to thank my family for their unconditional patience and support during my PhD studies. 6 Table of Contents

Page

Abstract...... 3

Dedication...... 4

Acknowledgments...... 5

List of Tables...... 8

List of Figures...... 9

1 Introduction...... 16

2 Forster¨ Energy Transfer Signatures in Optically Driven Quantum Dot Molecules. 20 2.1 Introduction...... 20 2.2 Model: FRET Coupling in Quantum Dot Molecules...... 21 2.3 Model: Quantum Dot Molecule Hamiltonian...... 24 2.4 Model: Eﬀective Exciton Hamiltonian...... 26 2.5 Model: Electrically Tunable Molecular Coupling...... 27 2.6 Model: Coherent Exciton Dynamics and Level Anticrossing Population Maps 30 2.7 Results: Competing Eﬀects of Tunneling and Forster¨ Energy Transfer in Monoexcitons...... 31 2.8 Results: FRET in Biexciton Optical Signatures...... 36 2.9 Conclusions...... 42

3 Coherent Control of Exciton Dynamics in Quantum Dot Molecules...... 43 3.1 Introduction...... 43 3.1.1 Coherent control...... 43 3.2 Effective Subspace Extraction From Level Anticrossing Spectra...... 45 3.2.1 Optical Signatures: Effective Three and Two Level Systems.... 45 3.2.2 Two Level System Optical Signatures: Indirect Exciton Qubits... 47 3.3 Effective Exciton Hamiltonian via Adiabatic Elimination...... 50 3.3.1 Adiabatic Elimination via Algebra of Projection Operators..... 51 3.3.2 Three Level System Subspaces: Effective Oscillator Strength.... 53 3.3.3 Spatially Indirect Protected Two Level Systems...... 55 3.4 Coherent Control of Indirect Excitonic Qubits...... 58 3.4.1 Excitonic Density Matrix Dynamics and Control Scheme...... 58 3.4.2 Indirect Exciton Qubit Initialization...... 61 3.4.3 Indirect Exciton Qubit Rotations...... 62 3.4.4 Indirect Exciton Qubit Readout...... 64 7

3.5 Additional dynamical eﬀects...... 65 3.5.1 Forster¨ Energy Transfer and Biexciton States...... 65 3.6 Dissipative Eﬀects...... 66 3.6.1 Numerical Results: Density Matrix...... 66 3.6.2 Analytical Results: Adiabatic Elimination...... 68 3.7 Discussion and Concluding Remarks...... 72 3.7.1 Stability of the Control Scheme...... 72 3.7.2 Conclusions...... 73

4 Modeling Electrical Control of Indirect Exciton Relaxation Rates in Quantum Dot Molecules...... 74 4.1 Introduction...... 74 4.2 Oscillatory Relaxation Rates of Indirect Excitons in InGaAs/GaAs QDMs: Experimental Evidence...... 75 4.3 Phonon Assisted Exciton Relaxation...... 77 4.3.1 Carrier Phonon Scattering Interaction Mechanisms...... 77 4.3.2 QDM Simpliﬁed Model: Electron and Hole Wave Functions.... 81 4.3.3 Uncorrelated Excitons...... 84 4.3.4 Phonon Induced Hole Scattering Rates...... 88 4.4 Radiative Recombination Rates...... 100 4.5 Discussion and Concluding Remarks...... 102

5 Conclusions and Outlook...... 104

References...... 110

Appendix A: Simulation Parameters...... 116

Appendix B: Adiabatic Elimination: Bloch-Feshbach Algebraic Method...... 120

Appendix C: Rabi oscillations and The Rotating Wave Approximation...... 130

Appendix D: Adiabatic Passage...... 133

Appendix E: Two Level Systems and Qubits...... 136

Appendix F: Articles and Conferences...... 142 8 List of Tables

Table Page

A.1 Constant parameters used in chapters2 and3...... 116 A.2 Parameters used in simulations corresponding to ﬁgures 2.5, 2.6 and 2.7.... 116 A.3 Parameters used in simulations corresponding to ﬁgures 2.8, 2.9...... 117 A.4 Parameters used for simulations in chapter3...... 118 A.5 Parameters used in simulations for chapter4...... 119

(Λ) B.1 Roots of the polynomial Π (z) at FR ...... 126 (Λ) B.2 Roots of the polynomial Π (z) at FI ...... 126 9 List of Figures

Figure Page

2.1 Excitation energy transfer between molecular complexes Q1 and Q2. A laser pulse pumps the excited state e1, upon de-excitation electrostatic energy is transfer to Q2; absorption on Q2 leads to the excited state e2. VF is the dipole- dipole Coulomb interaction responsible for this process...... 20 2.2 FRET process in a QDM (a) QDM schematics; two dissimilar disk shaped vertically stacked QDs. (b) Ground state excitons in each QD are represented by an electronic state (blue) and a hole state (red) in the conduction band (CB) and valence band (VB), respectively. A FRET interaction, VF, de-excites an exciton on QD1 and transfers its energy to create an exciton on QD2. (c) 10 The FRET process represented as a single transition between two excitons 10X and 01X. Here an exciton is denoted by e1e2 X, with h , e being the occupation 01 h1h2 i i numbers on the i-th QD...... 22 2.3 Level anticrossing spectroscopy schematics. (left panel) Vertically stacked QD layers are processed into Schottky photo diodes. Opaque shadow masks are patterned with apertures that isolate single QDMs for optical probing. (right panel) Full device schematics with realistic dimensions used in experiments. An applied bias voltage provides an axial electric field F that lifts the conduction and valence band edges. Upper inset by H.J. Krenner, et al., Phys. Rev. Lett. 94, 057402 (2005). ©(2011) American Physical Society. Main figure, courtesy of Prof. E.A. Stinaff, Ohio University...... 28 2.4 A QDM coupled by electron tunneling. (a) At zero bias the laser field pumps a spatially direct exciton, where both hole and electron are localized in the same QD; this correspond to a plateau on the F dependent eigenvalue spectrum. (b) The field tilts the band edges, inducing a resonance among the conduction bands of the two QDs; electron tunneling occurs, mixing the direct with the spatially indirect exciton with an anticrossing opening in the spectrum..... 29 2.5 (a) Single-exciton energy level diagram showing the relevant couplings with 00 the empty QDM state, 00X. (b) and (c) Show occupation maps of vacuum 00 10 01 state 00X in Eq. 2.13, exhibiting features as indicated, where direct, 10X, 01X, 01 10 and indirect excitons, 10X, 01X, have non-zero occupation; for vanishing (in b) and non-zero VF (c). Notice sizeable ∆F splitting at high fields, away from tunneling anticrossings. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 79, 245309 (2009). ©(2011) American Physical Society...... 32 10

10 2.6 LACS population maps of the monoexciton 10X for a system described by Eq. 2.13 and parameters given in appendix table A.2 (a) Shows FRET-induced weak satellite line (black arrows), parallel up to the tunneling anticrossing region. (b) Signature persists, even when taking all biexciton transitions into account (all 14 states). The respective population maps (c) and (d) show no satellites for VF = 0 (no FRET). (e) Acceptor population for two fixed values of F = −35 (black dashed line) and F = −70kV/cm (blue solid). (f) Acceptor population as in (e) but for full system in (b). Notice absence of features when VF = 0 in insets. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 79, 245309 (2009). ©(2011) American Physical Society...... 33 2.7 (a) Effective level diagram corresponding to truncated Hamiltonian Eq. 2.16. (b) Eigenvalue spectrum as function of energy detuning δ 10 , the spectrum X10 10 10 2 shows an anticrossing at δ10 = −VF and a level crossing at δ10 = VF −(Ω /VF) = VX. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 79, 245309 (2009). ©(2011) American Physical Society...... 35 11 2.8 FRET signatures on the biexciton state 02X according to parameters given in table A.3. (a) LACS population map showing splitting of indirect exciton line over large F range. Blue dashed line indicates the asymptotic position of the 02 direct biexciton 02X mixing with the acceptor level. Top right inset: map with no FRET. Bottom left inset: Magnification of the indirect line splitting. (b) Average population for three values of F, showing invariance of indirect split- off energy: satellite line in (a) tracks the brighter exciton line nearly parallel. (c) Reduced level diagram describing the system for high values of F. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 79, 245309 (2009). ©(2011) American Physical Society...... 39 11 2.9 (a) The trion-like state 02X has satellite after mixing with the biexciton 02 resonance 02X, as shown in (b). Notice inset shows no splitting when VF = 0. Comparison of (c) and (d) confirms mixing of FRET satellites between direct and trion-like biexciton states; FRET signature on direct biexciton state is weaker by an order of magnitude. Simulation parameters according to appendix table A.3. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 79, 245309 (2009). ©(2011) American Physical Society...... 41

3.1 Coherent control schematics. A coherent light source is pulsed using a control shaping function F(t). Light illuminates the sample which has a tunable spectrum. The optical response of the sample is sent to a detector. A feedback circuit is implemented in order to adjust the control parameters until the sample responds according to a predetermined ﬁnal state...... 44 11

00 3.2 Level anticrossing population map of the QDM vacuum state |00Xi for parameters given in appendix table A.4. As function of applied bias and pump laser energy the vacuum depopulates to other excitons at each resonance. The x-pattern dashed box indicates resonant excitation into molecular excitons. In 01 particular, for a state with mostly indirect character, 10X, at FI = 43.4kV/cm, Elaser = 1299.6meV, the system is governed by two level system (TLS) dynamics. In the central anticrossing enclosed in the diamond shaped box at FR = 2.3kV/cm, the system also behaves as a TLS, mixing only two indirect excitons...... 46 3.3 Average population of all neutral excitons for fixed laser excitation energy Elaser = 1271meV and parameters given in appendix table A.4. (a) and 10 01 (b) show the spectral peaks of excitons |01Xi and |01Xi at the LACS map coordinates (Elaser, F ' 7.5kV/cm), respectively. (c) and (d) show the spectral 01 10 peaks of excitons |10Xi and |10Xi at the LACS map coordinates (Elaser, F ' −3kV/cm), respectively. The population weights suggest a TLS behavior 00 between the vacuum |00Xi and each indirect exciton...... 48 3.4 (a) LACS population map inside the diamond dashed box at the central x- pattern in Fig. 3.2. (b) Eigenvalue spectrum as function of laser excitation energy for fixed field F = −3kV/cm (blue vertical line in (a)); two level 00 anticrossings occur between the vacuum |00X and each indirect exciton, corresponding to coherent oscillations at the intersection points with blue line in (a). (c) Eigenvalue spectrum as in (b) but at fixed field F = 2.4kV/cm (red vertical line in (a); the level anticrossings in (b) merge into a single double anticrossing mixing the two indirect states, corresponding to coherent oscillations between the two indirect excitons with minimal occupation of the vacuum...... 49 3.5 Effective Hamiltonian energy level diagrams. (a) Energy levels corresponding to the effective Hamiltonian in Eq. 3.5, showing indirect exciton effective couplings (solid lines) as well as the effective relaxation channels of indirect excitons (dashed red lines) (b) Energy levels according to the TLS Hamiltonian in Eq. 3.11, with an effective coupling connecting the two indirect excitons, given by tU = U + ξ...... 54 3.6 Dressed exciton eigenvalue spectrum as function of applied field F, for fixed excitation energy ~ω = 1299.6meV and parameters given in appendix table A.4. (a) and (c) show the anticrossings between the vacuum energy level and the indirect excitons, due to the effective coupling ΩI. (b) shows the anticrossing between the pair of indirect excitons due to the effective coupling U. In all cases the behavior corresponds to effective TLS which results in exciton population dynamics inside the relevant subspaces discussed above. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society...... 56 12

3.7 Dressed exciton eigenvalue spectrum at fixed excitation energy ~ω = 1299.6meV. The dashed boxes indicate the anticrossings associated to qubit 10 initialization (F = 43.4kV/cm) into the state |01Xi (red), as well as, qubit 01 rotation (F = 2.4kV/cm) into the final output state |10Xi (blue). Insets show the magnification of such anticrossings. The adiabatic reversal pulse state transfers 10 the population of the output state into |10Xi (green), which eventually relaxes into the vacuum. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society...... 59 3.8 (a) Electric field control pulse used for qubit initialization and rotation. (b) Density matrix numerical simulation of the exciton population dynamics corresponding to the control pulse in (a); it shows all steps involved in the coherent control of the indirect exciton qubit. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society.. 60 01 3.9 Initialization of the indirect exciton |10Xi by a 3π rotation. When driving the system at the anticrossing, corresponding to the coordinate (FI = 43.4kV/cm, ~ω = 1299.6meV), the initialization takes place after switching off the pulsed resonant excitation at a time t = 200ps. The initialization occurs with near 10 10 unity fidelity, F = h01X|ρ(ti)|01Xi ' 0.97, due to the almost perfect isolation of the subspace PΛ. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society...... 62 3.10 Two level system dynamics at FR exhibits Rabi oscillations that fully populate the indirect excitons. Here, the system evolves by its internal dynamics. The coherent oscillations allow for qubit rotations in the picosecond scale, with a π characteristic time 2U = 91.8ps. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society...... 63 3.11 Read out scheme of the output qubit state. In this step, the transported state passes trough the tunneling anticrossing at F ' +20kV/cm, transferring 10 adiabatically its population (blue line) to the direct state |10Xi (olive line), which in turn depopulates into the vacuum (dashed line) via spontaneous recombination. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society...... 65 3.12 Population relaxation dynamics for indirect excitons, from the numerical solutions of Eq. 4 with up to 14 exciton states included. (a) Population decay after switching off the laser light in the initialization regime at FI = 43.4kV/cm. Dashed line indicates the case where biexciton states are taken into account. Inset shows the corresponding depopulation into the vacuum. (b) QDM internal dynamics in absence of optical perturbations, shows decay of Rabi 10 01 oscillations of |01Xi and |10Xi (red line, blue line) in the qubit rotation regime at FR = 2.3kV/cm. Inset shows Rabi flops in the early stage of the dynamics. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society...... 67 13

3.13 Relaxation times for bare and dressed indirect excitons as function of applied electric ﬁeld F. Solid (dashed) line corresponds to relaxation time for the dressed (bare) indirect exciton eigenstates. Far from anticrossing value of electric ﬁeld FI = 2.38kV/cm, the dressed relaxation time approaches the bare ˜ (I) exciton relaxation time as molecular indirect eigenstates of HΓ become the 01 10 indirect states |10Xi, |01Xi. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society...... 71

4.1 (a) Lifetime measurements of spatially indirect excitons on QDMs samples with nominal interdot separation d ∼ 7nm. The lifetime as function of the indirect-direct exciton energy separation, oscillates with a local minimum at ∼ 6-8meV and a local maximum at ∼ 12meV (b) Level anticrossing PL 10 10 intensity of the spatially indirect exciton, 01X and direct exciton 10X. The region labeled with the X+ symbol indicates the energy window for which PL 10 arises from a positively charged trion. (c) 01X normalized PL intensity. K.C. Wijesundara, et al., Phys. Rev. B 84, 081404(R) (2011). ©(2011) American Physical Society...... 76 4.2 Simplified electron and hole energy level diagrams. Holes (red) hop between dots with tunneling strength th, whereas electrons (blue) are mostly confined in the bottom dot due to the large energy mismatch, ∆E0, of the top QD conduction band with respect to the bottom QD conduction band. Phonon assisted hole relaxation is indicated by Γph, whereas radiative relaxation, Γrad, occurs via direct state recombination in the bottom QD. The dashed lines indicate the electron hole pairs leading to exciton formation...... 80 4.3 Model schematics. (a) InAs QDM consisting of two QDs (bottom B and top T) with cylindrical symmetry separated by a distance d = d˜ + (hB + hT )/2 from their centers, having widths lB and lT , respectively; here d˜ is the sharp edges GaAs barrier width (not shown). (b) Parabolic potentials are matched to the square well confinement potential band edges, Ve(h). The lowest energy eigenstates correspond to the gaussian ground state of the harmonic oscillator problem...... 82 4.4 (a) Energy eigenvalues as function of applied electric field F for the single particle effective Hamiltonians, for holes as in Eq. 4.21 (solid red) and electrons as in Eq. 4.22 (dashed blue). (b) Mixing coefficients for bonding and anti- 2 2 2 2 bonding eigenstates of holes, a11 = a22, a12 = a21 (solid red) and electrons, 2 2 2 2 b11 = b22, b12 = b21 (dashed blue). See numerical values in appendix table A.5. 87 4.5 Structure factor as function of in-plane qk, axial qz phonon wave vectors, and different polar angles θ, for fixed applied electric field F = 0 and interdot distance d = 7nm. The structure factor exhibits prominent resonances mostly along the qzd axis, and these resonances shift towards higher values of qzd for increasing θ. In a sense for the range of energies used in the experiment of Fig. 4.1, only phonons emitted closely to the z-axis contribute the most to the relaxation rate. See numerical values in appendix table A.5...... 91 14

4.6 Structure factor contour plot as function of energy separation and phonon axial wave-vector, for fixed d = 7nm. The dotted purple line indicates the phonon wave vector dispersion as function of energy separation. The relaxation rates are strictly non-zero for allowed phonon wave vectors satisfying Eq. 4.47, in other words only at the intersection points between the purple line and the contour points. Resonances are prominent at qzd = π and qzd = 3π which correspond to an energy separation of ∆E = 2.6meV and ∆E = 6.2meV, respectively. See numerical values in appendix table A.5. K.C. Wijesundara, et al., Phys. Rev. B 84, 081404(R) (2011). ©(2011) American Physical Society.. 92 4.7 Phonon relaxation scattering channels as function of ∆E = ∆h(F). Left column panel: dependence on B QD lateral confinement length, dkB or lateral FWHM of the hole ground state wave function, fwhmkB. (a) LA-DP channel exhibits oscillations with decreasing amplitude as lateral confinement becomes weak (notice the developing of absolute minimum about δE ∼ 4.25meV). (b) and (c) TA and LA-PZ channels, respectively, exhibit no oscillations and weak dkB dependence. Right column panel: dependence on the B QD vertical confinement length, d⊥B. (d) Oscillations of LA-DP rate are prominent for strong confinements d⊥T ≤ 1.1nm, while vanishing completely for weaker confinements d⊥T ≥ 2.5nm. TA-PZ scattering channels do not exhibit qualitative changes. Other fixed parameters can be found in appendix table A.5...... 96 4.8 Phonon relaxation scattering channels as function of the field dependent energy separation, ∆E = ∆h(F), for different interdot separations. (a) LA-DP rate: the position of maxima/minima shift to lower energy separation with increasing d while accommodating a larger number resonances. For smaller separations there is a substantial broadening and suppression of oscillatory behavior. (b) and (c) TA and LA-PZ scattering rates: rates are suppressed and exhibit substantial broadening for smaller interdot separations. Inset text in (a) indicates fixed parameters. Other fixed parameters can be found in appendix table A.5...... 98 4.9 Phonon relaxation rates as function of the field dependent energy separation ∆E = ∆h(F). Green shows the total contribution arising from all scattering channels, LA-DP, LA-PZ, TA-PZ. Red shows the calculated radiative recombination rate. The resonance associated with the relaxation rate peak at 6.2 meV is due to maxima of the structure factor in Fig. 4.6 for a scaled phonon momentum wave vector qzd = 3π (green dot in Fig. 4.6). Other fixed parameters can be found in appendix table A.5...... 100

B.1 Eﬀective eigenvalue spectrum of H˜ (Λ)(z) (dashed red). Full Hamiltonian eigenvalues (solid black) (a) Eigenvalues for z = z3(FR) correctly reproduces the original spectrum (qubit rotation anticrossing), whereas in (b) z = z5(FR) does not. (c) Eigenvalues for z = z5(FI) correctly reproduces initialization anticrossing, whereas in (d) z = z3(FI) does not...... 127 15

B.2 Eﬀective eigenvalue spectrum of H˜ (I)(z). (a) Eigenvalue spectrum for diﬀerent solutions of Eq. B.31. Roots z3, z4 reproduce correctly the original spectrum (solid blue line). Substitution of the remaining roots, z3, z4 results in a false gap and anticrossing behavior of a purely indirect exciton subspace (dashed yellow green lines, respectively). Here, δ is a small arbitrary deviation from the root z3...... 129

D.1 Eigenvalues of the Hamiltonian in Eq. D.5 as function of a time dependent detuning ∆(t) (solid blue). The asymptotic dashed red lines indicate the bare atomic energies for the unperturbed atomic states. The red points indicate the values at which θ is measured...... 134

E.1 Bloch sphere. The poles of the sphere denote the coordinates of the bare computational basis states. The Bloch vector projections are indicated by the red dashed lines...... 138 E.2 Indirect exciton qubits. (a) Rabi oscillations during the qubit initialization 10 state. (b) Qubit is initialized on the 01X state (non-ideal ﬁdelity exaggerated on picture). (c) Qubit rotations as Rabi oscillations between the indirect excitons. 01 (d) The π rotated qubit 10X state is prepared for read out...... 140 16 1 Introduction

Semiconductor physics has made steady progress towards the development of astonishingly small electronic devices. The nanoscale is the length scale of all electronic and spintronic devices currently in development and perhaps to be commercialized within the next few years. Devices operating at such scales are revolutionary in the sense that they exploit the full quantization of its carriers degrees of freedom, such as crystal momentum, charge and spin. Among the most prominent examples of such devices are quantum dots (QDs). Semiconductor quantum dots represent important systems for 3-dimensional electron confinement with remarkable features, such as, atomic-like behavior with electronic state discrete spectrum and bulk (host) semiconductor properties like excitonic spectra.[1,2] An exciton is an optical interband transition that populates the conduction and valence band in the crystal bulk or within a nanostructure. In a quantum dot an exciton can be thought as being a Coulomb bound molecular system made of quantum confined electron and holes.[3] The term (mono)exciton applies when optically exciting a single neutral electron-hole pair, whereas a neutral biexciton is created when optically exciting two Coulomb bound electron hole pairs; charged exciton complexes can also exist in confined structures, when having an excess charge on the valence or conduction band; positive/negative singly charged excitons are called positive/negative trions, and so on. Excitons are well defined quasi-particles, having a lifetime of the order of hundreds of picoseconds up to a few nanoseconds. Exciton radiative recombination is what gives access to the electronic and excitonic level structure of many nanostructures. In particular, photoluminescence (PL) spectroscopy is a prominent technique for exciton characterization in quantum dots.[4] PL has demonstrated that exciton states can be harnessed as two level systems which might encode quantum bits (qubits).[5] Moreover, diverse ultrafast optical spectroscopy techniques have demonstrated excitonic coherent 17 control in the picosecond timescales. Upon the requirement of coupling two excitonic qubits, an interesting possibility is to use a self assembled quantum dot molecule (QDM).[6,7] A QDM is a vertically stacked pair of electrically/optically driven self assembled QDs coupled by charge tunneling (i.e. electron or hole tunneling).[8] An external electric field applied along the growth direction can be used to localize charges in either dot. When labeling the dots with an index, say the top QD as 1 and the bottom dot as 0, the indexes map into isospin degrees of freedom. Therefore, the electrically controlled interdot coupling effectively rotates the system isospin, giving rise to a quantum superposition (molecular states) of the individual excitons localized within the QDM.[9, 10] To date many other complex schemes of external coherent manipulation of qubits are in route to be implemented in coupled QDs, however all of them require a delicate control of the interdot couplings. In this fashion if such control is achieved, quantum dots and coupled quantum dot networks offer a robust and scalable architecture for quantum computation. A key question which naturally arises, is whether stable exciton molecular states can be engineered using information from excitonic optical signatures. An equally important question, is whether or not such states can sustain coherent oscillations when subject to a strong influence from the environment. The general aim of this dissertation is motivated by the premise that real functionality of QDMs requires a good understanding of the time evolution and coherent control of their excitonic states. The dissertation is organized as follows: In Chapter 2 we discuss the optical signatures of the Forster¨ resonant energy transfer mechanism (FRET) in coherently excited InGaAs/GaAs quantum dot molecules. Since FRET plays a fundamental role in the molecular dynamics of chemical and biochemical interactions, a key question is whether FRET could appear on the excitonic signatures of realistic QD molecules. Our computational results proved that FRET is a competing mechanism to charge tunneling, 18 with the possibility of enhancing its optical signatures by applying an external electric field. We construct a map of the excitonic level populations, which depends on the external applied electric field and the laser excitation energy. The constructed map in this fashion gives the level anticrossing molecular spectra. We demonstrate that FRET optical signatures appear as spectral doublets of spatially direct and indirect excitons, with an spectral weight modulated by the applied electric field and its degree of hybridization with other excitons present in the structure, such as biexcitons. We propose that these unique features can be readily identifiable in experiments. Chapter 3 is focused on a theoretical and computational scheme to understand the coherent control of exciton dynamics in InGaAs/GaAs QDMs. Our results indicate that there are specific regions within the excitonic landscape, where the level population dynamics can be controlled by applying time dependent electric pulses. We discuss the possibility of tailoring pulses to control exciton states with spatial indirect character. Our analysis shows that long lived exciton quantum bits can be constructed using indirect excitons as computational basis, and that coherent control can be implemented upon them by applying optical and electrical pulses. In our scheme, high fidelity qubit initialization and rotation are carried out faster than the spontaneous exciton recombination rate. Qubit readout is devised via adiabatic rapid passage from a delocalized indirect exciton state into a luminescent localized direct state. Adiabatic elimination of unwanted excitonic levels is carried out using an exact method based on projection operators. In this way, an effective Hamiltonian is obtained to represent the relevant exciton transitions and their associated molecular resonances (level anticrossings). Interestingly when projecting out all direct exciton states, we obtain an effective subspace containing only indirect excitons. These states are subsequently used to form a qubit computational basis. Coherent rotations of these states are performed and their fidelity analyzed in terms of the effects of resonant energy transfer processes, laser detuning, and charge tunneling. 19

In Chapter 4, a theoretical model is developed in order to understand experimentally measured oscillatory relaxation rates of indirect excitons in InGaAs/GaAs QDMs. Our model corresponds to a QDM with cylindrical symmetry under an applied axial electric field. The in-plane and vertical confinement potentials of the QDM are modeled as finite parabolic potentials matched to the valence and conduction band edges. Our simulations predict that the oscillatory nature of the observed rates is due to exciton-acoustic phonon scattering. In particular, when considering hole scattering rates by bulk acoustic phonons, the QDM structure factor contains a phase relationship between the hole wave function and the phonon wavelength. It is found that the aforementioned phase relationship leads to interference effects that manifest in the suppression or enhancement of the relaxation rates as function of the applied field. Moreover the relaxation rates periodicity is strongly dependent on the hole wave function confinement along the lateral and vertical directions, which in turn are determined by the entire QDM structural parameters such as the interdot distance, dot heights and lateral size. Our results also suggest that since the structure factor is the Fourier transform of the excitonic charge density distribution, the measured modulated phonon relaxation rates can provide additional insight into the geometry and composition of QD structures. 20 2 Forster¨ Energy Transfer Signatures in Optically

Driven Quantum Dot Molecules

2.1 Introduction

Excitation energy transfer within a molecular complex is a directed transfer of electrostatic energy in the absence of charge transfer. This process arises from non-radiative multi-pole Coulomb interactions among the conformational states of the molecule involving optically excited states. Furthermore, it produces excitation delocalization within the whole complex, which might be interpreted as exciton hopping between distant sites the molecule.[11] Excitation transfer kinetics can be modeled as a coherent process, or as an open quantum systems problem within the context of the equation of motion for the excitonic density matrix. Let us introduce the notion of excitation energy transfer among two molecules Q1 and Q2. Consider each molecule as a two level quantum system, with the levels being the ground state and the ﬁrst excited state of each molecule, g1, e1 and g2, e2, respectively, see Fig. 2.1.

Figure 2.1: Excitation energy transfer between molecular complexes Q1 and Q2. A laser pulse pumps the excited state e1, upon de-excitation electrostatic energy is transfer to Q2; absorption on Q2 leads to the excited state e2. VF is the dipole-dipole Coulomb interaction responsible for this process. 21

In addition, let us suppose that molecule Q1 (donor) is initially on its electronic excited state e1 after being pumped with an external laser pulse; whereas Q2 (acceptor) is on its ground state g2. The Coulomb interaction among the two molecules causes de-excitation of Q1 with electrostatic energy being transferred to Q2 causing its respective excitation. This processes is eﬀectively equivalent to the recombination of an exciton on site Q1 and its subsequent creation at site Q2, or an exciton hopping from Q1 to Q2. This processes is known on the literature as Forster¨ resonant energy transfer mechanism (FRET), based on the early works of T. Forster¨ and D.L. Dexter.[12, 13] FRET is involved in a great variety of physical chemistry phenomena, such as superradiance, long range delocalization and exciton transport. The most fascinating manifestation of these phenomena occurs in photosynthesis, which relies on the directed excitation transfer of the energy collected by chromophores and photosynthetic light harvesting antennas.[14]

2.2 Model: FRET Coupling in Quantum Dot Molecules

In a quantum dot molecule the roles of the donor and acceptor complexes are played by individual QDs.[15–18] Lets consider the two vertically and dissimilar disk shaped quantum dots, QD1 and QD2, separated by a distance |R~|, each having radius |~r1| and |~r2|, respectively, see Fig. 2.2. 22

Figure 2.2: FRET process in a QDM (a) QDM schematics; two dissimilar disk shaped vertically stacked QDs. (b) Ground state excitons in each QD are represented by an electronic state (blue) and a hole state (red) in the conduction band (CB) and valence band (VB), respectively. A FRET interaction, VF, de-excites an exciton on QD1 and transfers its energy to create an exciton on QD2. (c) The FRET process represented as a single transition between two excitons 10X and 01X. Here an exciton is denoted by e1e2 X, with 10 01 h1h2 hi, ei being the occupation numbers on the i-th QD.

The ﬁrst step on deriving the interdot coupling leading to FRET, is the calculation of the interband transition dipole moments of exciton levels in each dot. This is followed by the calculation of the Coulomb matrix element connecting exciton wave-functions at diﬀerent dots. The Coulomb matrix element can be approximated by a dipole-dipole interaction, VF, following the standard procedure of Forster¨ and Dexter.[12, 13, 17, 18] In the limit where the length-scale of exciton wave functions are small in comparison with |R~|, we can write,

2   e  3(h~r1i · R~)(h~r2i · R~) V = − h~r i · h~r i −  . (2.1) F 3  1 2 2  4π0rR R

Within the envelope wave function approximation, the single carrier wave functions are given by the products ψi(~r) = φi(~r)ui(~r). The envelope part, φi(~r), contains the slow variation of the wave function amplitude over the QD, and most of the important properties of single carrier states. ui(~r) is the Bloch wave function which possesses the lattice periodicity, and it is important for deriving the optical transition dipole moments, 23 and many-body interactions. Therefore, Z 3 ∗ h~r1i = d ~r1ψe,n(~r1)~r1ψh,n(~r1) (2.2) Z 3 ∗ h~r2i = d ~r2ψh,n(~r2)~r2ψe,n(~r2) , (2.3) where n is the label for single particle states. Using the results above, it is possible to separate the contributions arising from the envelope and Bloch parts of the wave function. This results in a simpliﬁed expression for the Forster¨ coupling,

e2 3 V − O O ~µ · ~µ − ~µ · R~ ~µ · R~ F = 3 1 2[ 1 2 2 ( 1 )( 2 )] (2.4) 4π0rR R where O1(2) are the overlap integrals of the envelope wave functions for QD1 and QD2, respectively Z 3 Oi = d ~rφe(~ri)φh(~ri) . (2.5)

The inter-band dipole matrix elements, connecting the electron and hole Bloch wave functions, ue(h)(~r), are given by Z 3 ~µi = e ue(~ri)~riuh(~ri)d ~ri . (2.6)

Using the axial symmetry of the problem, and assuming QDs with transition dipole moments along the in-plane direction, the expression for VF reduces to

µ µ V − 1 2 κ . F = 3 (2.7) 4π0rR

In the model, µ1 and µ2, are not calculated explicitly, since more accurate values are readily available from experiments.[19–21] The interaction strength of VF depends on the degree of exciton conﬁnement within each QD (via the electron-hole wave function overlap). It also depends on the relative orientation of their dipole moments, κ = ~µ1·~µ2 ; µ1µ2 assuming in-plane and parallel permanent dipoles, κ ∼ 1. Furthermore, in this 24

2π 2 approximation the FRET exciton hopping rate, KDA = V Θ, depends on the spectral ~ F overlap Θ between the donor emission and acceptor absorption.[22–24] However, the excitonic levels in self-assembled QDs are spectrally narrow, which typically makes FRET signatures diﬃcult to observe in experiments. On the other hand, the energy

10 01 mismatch, ∆X12 , between the bottom (10X) and top (01X) exciton levels, depends on growth conditions, so QDs might be grown to have near resonant exciton transitions.[25, 26] For

QDMs such that ∆X12 ≤ VF, the resonant energy transfer is nearly coherent “exciton

hopping;” on the other hand, when ∆X12 VF, the transfer is incoherent, requiring phonon assisted transitions via auxiliary excited levels.[27]

2.3 Model: Quantum Dot Molecule Hamiltonian

In what follows, we consider a quantum dot molecule formed by vertically stacking two self assembled InGaAs/GaAs quantum dots, a top QD (T) and a bottom QD (B) separated by a distance d. We assume QDs grown by the Stranski-Krastanow method on a GaAs host matrix, so the barrier material separating the dots is mostly GaAs with InGaAs/GaAs alloying on the QD boundaries.[28] Exciton transition energies in single QDs are critically affected by strain induced modifications to the single particle conduction and valence band offsets. In addition, the QDM has an intrinsic broken inversion symmetry, which leads to a difference between the hole confinement potential experienced at the base of the top dot from that experienced at the base of the bottom dot.[29] This difference suppresses interdot hole tunneling, which typically is measured to be an order of magnitude less than electron tunneling. Atomistic pseudopotential calculations of exciton wave functions on strained QDMs have shown that holes are mostly heavy-hole like, experiencing a higher interdot potential barrier. Thus, the interdot barrier experienced by the heavy-holes changes with the width of the interdot GaAs barrier, d, leading to strong modifications of the exciton transition energies. Our model 25 uses input numerical parameters from atomistic simulations carried out by Bester, et al.[29] In doing this, we explicitly incorporate the effects of changing d, since the exciton energies were obtained by configuration interaction fitted to tight binding parameters. The d-dependent input parameters include single particle energies, electron-hole Coulomb interactions, and electron and hole tunneling energies. We construct the basis for a two site Hamiltonian representing the QDM, as products

of electron and hole states, |eT i|hT i, |eT i|hBi, |eBi|hT i and |eBi|hBi, where the subindex indicates the QD in which the single particle is localized; all single particle states are assumed to be the ground state in each QD/band. In this basis, the non-interacting Hamiltonian is given by   E t t 0   eT hT h e     t E 0 t   h eT hB e  H0 =   . (2.8)  t 0 E t   e eBhT h      0 te th EeBhB where the diagonal matrix elements of H0 are the neutral exciton transition energies, given

eh in terms of the single particle energies, i, and the electron hole Coulomb interaction, U . Explicitly,

eh EeT hT = eT − hT + UTT

eh EeT hB = eT − hB + UTB

eh EeBhT = eB − hT + UBT

eh EeBhB = eB − hB + UBB , (2.9)

For simplicity we have only shown the exciton manifold corresponding to single neutral excitons. A similar construction can be made when including the biexciton manifold or charged exciton states. 26

2.4 Model: Eﬀective Exciton Hamiltonian

In our model exciton pumping is generated by a coherent external laser ﬁeld, which results in coherent Rabi oscillations among all diﬀerent exciton states, see appendix C.[30, 31] Now, lets us introduce a notation for exciton states derived from the optically perturbed Hamiltonian, H. This corresponds to constructing an optically pumped excitonic basis eBeT X, where h is the occupation number of holes in the B(T) QDs, and hBhT B(T) eB(T) the number of electrons, respectively.[10] The QDM is pumped by broadband laser pulse of frequency ω, which pumps nearby exciton states. We assume a rectangular pulse width long enough to capture several Rabi oscillations of the excitonic level populations.

When, eB(T), hB(T) = {0, 1, 2}, the resulting Hamiltonian basis contains a total of 14 neutral

00 10 01 exciton states. This corresponds to: the vacuum 00X; a pair of monoexcitons 10X,01 X; a 10 01 20 02 pair of spatially indirect monoexcitons, 01X,10 X; direct biexciton states 20X,02 X; indirect 20 02 11 biexciton states 02X,20 X; a delocalized biexciton 11X and a remaining set of trion-like 02 20 11 11 states, 11X,11 X,02 X,20 X.

Up to single occupancy of holes and electrons, eB(T), hB(T) = {0, 1}, the perturbed Hamiltonian corresponding to the monoexciton manifold in Eq. 2.8, is given by

Htot = HD + Htun + HVF + Hopt . (2.10)

The Hamiltonian in Eq. 2.10 includes the diagonal contribution of Eq. 2.8 plus the vacuum state |0i, the tunneling matrix elements of electrons and holes, Htun, the resonant energy transfer interaction, VF, and the laser excitation or optical perturbation, Hopt. Explicitly, X HD = |0ih0| + Ei|iihi| i

01 01 01 01 Htun = te(|01Xih10X| + H.c) + th(|01Xih10X| + H.c)

01 10 HVF = VF(|01Xih10X| + H.c)

−iωt 01 iωt 01 −iωt 10 iωt 10 Hopt = ΩT (e |0ih01X| + e |01Xih0|) + ΩB(e |0ih10X| + e |10Xih0|) . (2.11) 27

In the model, only spatially direct monoexcitons are coupled to the radiation ﬁeld by

~ 01 ~ 10 ΩT (t) = h0|~µT · E(t)|01Xi and ΩB(t) = h0|~µB · E(t)|10Xi, for QD(T) and QD(B) respectively. On the other hand, the spatially indirect excitons have typically a much weaker oscillator strength; it’s inclusion in the model is straightforward but will be ignored in what follows.

Here, µ~i are the interband transition dipole moments and E~(t) is the electric ﬁeld component of the radiation pulse amplitude. In order to simplify the equation of motion, we have performed the rotating wave approximation (RWA) and a unitary transformation that removes the time-dependent fast oscillatory terms of the matter-radiation interaction.[32, 33] The latter approximation is carried out by the following transformation,

∂Λ† H˜ = Λ†HΛ + i Λ ~ ∂t X Λ = e−iξkt|kihk| , (2.12) k

with the constraint ξi − ξ j = 0. The transformed Hamiltonian has diagonal matrix elements containing the detunings of the exciton transition energies with respect to the laser ˜ P excitation energy, HD = i δi|iihi|, with δi = Ei − ~ω; whereas the couplings to the radiation ﬁeld are striped of the e±iωt terms.

2.5 Model: Electrically Tunable Molecular Coupling

Level anticrossing spectroscopy (LACS) is an invaluable tool for probing the optical signatures of quantum coupling in QD aggregates. It consists of performing cryogenic resonant or non resonant photoluminescence spectroscopy while applying external perturbations which shift the exciton levels of the system.[9, 10, 34] In this manner, all the different possible resonances of connecting different transitions give a direct measure of the electronic and molecular structure of the system. In single QDs and QDMs, this is typically achieved by embedding them in an n-i Schottky junction. In this way the QD 28 system is subject to an external axial electric field controlled by a bias voltage[35] between the n− contact and the Schottky gate, see Fig. 2.3.

Figure 2.3: Level anticrossing spectroscopy schematics. (left panel) Vertically stacked QD layers are processed into Schottky photo diodes. Opaque shadow masks are patterned with apertures that isolate single QDMs for optical probing. (right panel) Full device schematics with realistic dimensions used in experiments. An applied bias voltage provides an axial electric field F that lifts the conduction and valence band edges. Upper inset by H.J. Krenner, et al., Phys. Rev. Lett. 94, 057402 (2005). ©(2011) American Physical Society. Main figure, courtesy of Prof. E.A. Stinaff, Ohio University.

The strongest signature of molecular coupling arises from interdot tunneling coupling.[9, 10, 36] This manifests as an anticrossing in the eigenvalue spectrum, and indicates the onset of carrier delocalization over the two QDs. Fig. 2.4 shows the hallmark of this delocalization, a QDM under optical excitation generates an exciton, or electron (blue) hole (red) pair; initially, emission would arise from recombination of a single neutral direct exciton as in Fig. 2.4(a); as we vary the electric ﬁeld, the conduction band levels of both QDs become resonant via the Stark shift, allowing electron tunneling, as shown in Fig. 2.4(b). In this regime, the QDM exciton states are superpositions of a spatially direct and indirect states and are therefore truly molecular in nature.[9, 10] 29

Figure 2.4: A QDM coupled by electron tunneling. (a) At zero bias the laser ﬁeld pumps a spatially direct exciton, where both hole and electron are localized in the same QD; this correspond to a plateau on the F dependent eigenvalue spectrum. (b) The ﬁeld tilts the band edges, inducing a resonance among the conduction bands of the two QDs; electron tunneling occurs, mixing the direct with the spatially indirect exciton with an anticrossing opening in the spectrum.

The effect of the external axial electric field is to induce on one hand a quantum confined Stark effect[37] (QCSE) shift of the exciton levels within each individual QD. On the other hand, it exerts a DC Stark shift of excitons with spatially indirect character, effectively separating electron and holes in opposite directions. In what follows we ignore QCSE since it will be rather small for the set of parameters used in our simulations. The

Stark shift on spatially indirect exciton shifts the energy by ∆S = eFd. With this in mind, we can ﬁnally write the interacting eﬀective QDM excitonic Hamiltonian as    δ Ω 0 0 Ω   0 T B     Ω δ01 t t V   T 01 e h F    ˜  10  H =  0 te δ + ∆S 0 th  , (2.13)  01     0 t 0 δ01 − ∆ t   h 10 S e     10  ΩB VF th te δ10 30

01 10 01 10 where the columns are associated with the states |0i, |01Xi, |01Xi, |10Xi, and |10Xi, where the third and fourth columns represent spatially indirect excitons.

2.6 Model: Coherent Exciton Dynamics and Level Anticrossing Population Maps

A level anticrossing map can be generated by calculating the average population of all exciton levels, as function of the laser excitation energy and applied electric ﬁeld. A

given coordinate on these maps, (F, ~ω), represents the long time dynamics of the QDM.

The contribution of a pumped exciton is given by an amplitude pi. Two or more excitons

can contribute with their own amplitude to the coordinate (F, ~ω), only when having non-vanishing components in a dressed state.[37–40] Using the maps corresponding to each exciton state, it is possible to reassemble the entire spectrum of the system. On the

00 other hand, mapping the population of the vacuum state 00X = |0i, one can obtain the complete LACS spectra, such that all coordinates (F, ~ω) on this map, represents an exciton that is being depopulated into the vacuum, or redistributing its weight among all remaining states.[41] The ﬁrst step of this procedure consist in the calculation of the eigenvalue spectrum of the Hamiltonian in Eq. 2.13 and the calculation of the unitary dynamics of the system.

To justify unitarity we calculate the typical Rabi period, TR, associated to a molecular resonance, which depends on all couplings of the Hamiltonian. To work in the coherent dynamics regime, this period should be of the order of a few ps [30], so that

TR τX ∼ 1ns, the radiative exciton recombination time found in experiments.[42, 43] An additional requirement is the strong coupling limit for the resonant transfer processes,

VF ∆XTB , ~/τX, so that the dynamics of the system is coherent for times t, such that,

TR < t τX. Therefore if the time evolution is unitary, the propagator is given by U(t) = exp( −iHt ), so that the population of an exciton state |ii is given at time t by ~

2 Pi(t) = |hi|U(t)|0i| . (2.14) 31

If the initial condition is full occupation of the vacuum |0i (i.e an ‘empty’ QDM state), then the averaged population for long integration times is given by

Z t∞ pi = (1/t∞) Pi(t)dt , (2.15) 0

where t∞ is the pulse duration for a broadband square pulse, which is long enough to

accommodate several Rabi oscillations of the excitonic populations, TR t∞ < τX. Only a few Rabi oscillations are enough to compute well-converged long-time population averages. A lower bound for the initial condition is the time for which transient eﬀects have elapsed, and an upper bound for the time to end the simulation would correspond to a value such that damping starts becoming important.

2.7 Results: Competing Eﬀects of Tunneling and Forster¨ Energy Transfer in Monoexcitons

Let us consider the general signatures of charge tunneling and investigate what are the overall features of FRET according to the level diagram in Fig. 2.5(a). To this end we have chosen the QD structural parameters (see appendixA table A.2), such that excitons

01 10 01X and 10X are initially in near resonance (strongly coupled). 32

Figure 2.5: (a) Single-exciton energy level diagram showing the relevant couplings with 00 00 the empty QDM state, 00X. (b) and (c) Show occupation maps of vacuum state 00X in Eq. 10 01 01 2.13, exhibiting features as indicated, where direct, 10X, 01X, and indirect excitons, 10X, 10 01X, have non-zero occupation; for vanishing (in b) and non-zero VF (c). Notice sizeable ∆F splitting at high ﬁelds, away from tunneling anticrossings. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 79, 245309 (2009). ©(2011) American Physical Society.

Fig. 2.5(a) shows the level diagram corresponding to the eﬀective Hamiltonian in Eq. 2.13. With this Hamiltonian we can construct a level anticrossing population map for the

00 vacuum state 00X. Fig. 2.5(b) show the LACS results for a QDM with an interdot distance 10 01 of d = 8.4 nm, assuming negligible Forster¨ coupling among 10X and 01X exciton states

(VF = 0). With these assumptions, the electron and hole tunnelings, te and th, are obviously the dominant interdot couplings. On the other hand, coupling to the radiation

field, ΩT(B), has the effect of dressing all exciton states, effectively coupling the direct states very weakly via higher order tunneling and excitation processes. A molecular resonance occurs at (F ' ±20kV/cm), resulting in a clear tunneling induced anticrossing.

Here, the applied ﬁeld F Stark shifts the spatially indirect exciton states by ±∆S , tuning them into resonance with the direct states. For simplicity we have neglected QCSE eﬀects which also shift weakly the direct exciton states, being a non essential element in our discussion. Lets us consider now the case when the direct excitons are near resonant, thus

strongly coupled by VF. In this case FRET splits considerably the direct exciton spectral 33 lines by ∆F = 0.16meV, Fig. 2.5(c) for ~ω ' 1248 meV. On the other hand, the tunneling remains unchanged, since its anticrossing is dominated by te VF.

10 Figure 2.6: LACS population maps of the monoexciton 10X for a system described by Eq. 2.13 and parameters given in appendix table A.2 (a) Shows FRET-induced weak satellite line (black arrows), parallel up to the tunneling anticrossing region. (b) Signature persists, even when taking all biexciton transitions into account (all 14 states). The respective population maps (c) and (d) show no satellites for VF = 0 (no FRET). (e) Acceptor population for two ﬁxed values of F = −35 (black dashed line) and F = −70kV/cm (blue solid). (f) Acceptor population as in (e) but for full system in (b). Notice absence of features when VF = 0 in insets. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 79, 245309 (2009). ©(2011) American Physical Society. 34

A more detailed view of the FRET optical signature is seen in Fig. 2.6, which shows

10 the level anticrossing map of the 10X exciton state. The splitting of the direct state for ω ' 1248 meV appears as a plateau satellite far from the tunneling anticrossing, see Fig. 2.6. However, for F ' (−35, −20) kV/cm, the satellite curvature increases, following the

10 (diabatic) spectral line of 01X. Interestingly, the FRET mechanism competes strongly with electron tunneling, making the indirect exciton to “light up”, by receiving some of the population of the direct state. Eventually the FRET signature quenches at the anticrossing where electron tunneling dominates, which demonstrates that tunneling is detrimental to FRET at low values of F. We emphasize that a proper description of the dynamics of such system needs to take into account the entire set of exciton states, since the direct coupling terms in the Hamiltonian and the various higher order virtual processes make the decoupling of direct and indirect exciton subspaces not possible. For completeness, we show in the right column panel of Fig. 2.6 the results of our simulation when expanding the Hamiltonian to include the biexciton manifold, expanding up to 14 the elements of the excitonic basis; clearly there is no qualitative difference between the LACS maps arising from the long time averaged dynamics. Figures 2.6(e) and (f), compare the populations for fixed values of the axial electric field, F = −70 and −35kV/cm. The FRET satellite peak amplitude increases at stronger fields, suggesting that the FRET signature strength can be controlled by electrical means. In order to understand the changing behavior of the FRET satellite peak amplitudes, we can truncate the Hamiltonian Eq. 2.13 such that its off-diagonal matrix elements

10 01 00 connect only the two direct excitons |10Xi, |01Xi, and the vacuum, |00Xi state; the truncated

Hamiltonian HD is given by,      0 ΩT ΩB     01  HD =  Ω δ V  . (2.16)  T 01 F     10  ΩB VF δ10 35

Figure 2.7: (a) Eﬀective level diagram corresponding to truncated Hamiltonian Eq. 2.16. (b) Eigenvalue spectrum as function of energy detuning δ 10 , the spectrum shows an X10 10 10 2 anticrossing at δ10 = −VF and a level crossing at δ10 = VF − (Ω /VF) = VX. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 79, 245309 (2009). ©(2011) American Physical Society.

The level diagram corresponding to the Hamiltonian Eq. 2.16 is shown in Fig. 2.7(a); Fig. 2.7(b) shows the corresponding eigenvalue spectrum as function of the direct state

10 10 01 laser detuning, δ10. For ΩB ' ΩT = Ω and δ10 ' δ01, the spectrum shows an anticrossing at 2 2 10 10 (VF −Ω ) δ = −VF and a level crossing at δ = = VX, with a relative separation 10 10 VF Ω2 ∆F = 2VF − . As a consequence, if we tune the laser excitation energy to VF

ω = E10 + VF, the system would be strongly coupled by the energy transfer mechanism ~ 10X 10 VF, which is manifest on the strong population of the exciton state 10X. However, when

ω ≈ E10 − VX, the coupling is weaker and the population lower; with VX having positive ~ 10X or negative values depending on the relative magnitude of VF and Ω. Hence, the exciton spectral line splits into a doublet; one high peaked line separated and a weak narrower satellite spectral line, both separated from each other by ∆F (for |F| 1). This result implies that a better spectral resolution for the FRET signature is achieved for low excitation power, Ω VF and when ∆F ' 2VF. For the parameters used in our simulations, the splitting in Fig. 2.5(b) is ' 0.25 meV (& 2VF = 0.14meV), since the 36 actual gap is slightly magniﬁed by perturbative corrections arising from charge tunneling and power broadening (not included in this simple algebraic analysis but included in the numerical simulations), even for values of F ' −80kV/cm. A proof of the arguments given above arises when one calculates the time evolution of the population (probability density) of the direct exciton states. Upon diagonalization of the truncated Hamiltonian Eq. 2.16, the time-dependent populations are given by ! 2Ω2 | 10 |2 | 01 |2 − ψ10(t) = ψ01(t) = 2 (1 cos(ΩRt)) , (2.17) ΩR

10 where the Rabi frequency in the regime where, δ10 ' −VF, is given by q 10 2 2 ΩR = (δ10 + VF) + 8Ω , (2.18) √ with the width of the gap at the anticrossing in Fig. 2.7(b) given by 2 2Ω. Integration of the time-dependent probability density gives the average exciton population,

10 2 2 10 10 p10 = 2Ω /ΩR, which has a maximum at δ10 = −VF. We readily notice that p10 increases monotonically with laser power for diﬀerent values of the detuning. The dependence of

10 p10 on the laser detuning and excitation power, suggests a high degree of tunability of the FRET optical signatures via optical parameters only. In this sense, our results suggest that the identiﬁcation of FRET signatures in QDMs, using monoexciton PL experiments, is attainable for realistic parameters and experimental conditions.

2.8 Results: FRET in Biexciton Optical Signatures

Increasing the QDM laser excitation power results in the possibility of pumping additional exciton states outside the direct exciton manifold.[44] On the one hand, excited states of single electron-hole pairs exist a few meV above the lowest transition energy, and correspond to excited states of holes or electrons.[34, 45] On the other hand, longitudinal optical (LO) phonons might be resonant at ∼35 meV above the lowest transition; for the parameters considered in the model, and for a suﬃciently narrow laser line-width and 37 narrow exciton transitions due to cryogenic temperatures, T ≤4K, render those processes negligible under suitable experimental conditions.[46–49] More probable situations arise when considering relaxation and exciton dephasing due to longitudinal acoustic (LA) phonons and pumping of charged excitons and biexcitons.[43, 50, 51] As we would prove in Chapter 4 of this dissertation, indirect exciton states would result to be more prone to LA phonon exciton dephasing in QDMs, rather than their direct counterparts, which are mostly involved in FRET processes. Charged excitons on the other hand can be suﬃciently suppressed by controlling carrier capture from the back contacts of the

20 02 Schottky structure. However, biexciton transitions in quantum dot molecules, 20X, 02X and 11 11X, are more favorable due to the existence of more intermediate transition cascades 00 20 02 leading to their formation. Direct transitions such as 00X →20 X(02X) are usually forbidden by selection rules. The most probable pathways for pumping biexciton in single QDs involves the transitions

00 10 20 00X →10 X →20 X

00 01 02 00X →01 X →02 X

00 01 01 11 00X →01 X(01X) →11 X , (2.19)

where the parenthesis indicates an alternative route for the preceding transition. However, in QDMs electron and hole tunneling expands the number of pathways leading to biexcitons, which should involve indirect excitons as intermediate states. For example,

00 01 10 11 02 11 01 10 00X →01 X →01 X →02 X →02 X(11X) →01 X(10X) , (2.20)

01 10 which, among many other pathways, interferes with the FRET transition 01X →10 X. On the other hand, biexciton recombination times are typically faster than monoexciton

1 0 ' 0 recombination times[43], τXX 4 τX , which might cause biexciton transition spectral lines to broaden signiﬁcatively in comparison. Moreover, binding Coulomb interactions 38 typically produce a red shift with respect to the monoexciton transition, reducing their relative detuning from as low as 1.7 meV up to 3.5 meV.[52] For these reasons biexciton resonances cannot be ruled out from the dynamics involving monoexcitons as just given by the Hamiltonian in Eq. 2.13.

20 02 11 In optically driven QDMs the pumping of biexcitons, 20X, 02X and 11X, strongly aﬀects the time evolution of the monoexcitons, as the additional biexcitonic manifold redistributes the monoexciton probability densities.[53] However, as we have shown in the previous section, the monoexcitonic FRET satellite in the molecular doublet spectral line, are quite robust. The right panel of Fig. 2.6 shows the results of simulations using the full Hamiltonian with 14 basis states, including monoexciton and biexciton manifolds. The main biexcitonic eﬀects appear as a small attenuation of the central peak in Fig. 2.6(f), with the satellite peak keeping its relative amplitude. Interestingly, our simulation shows that FRET signatures manifest on biexcitonic complexes by themselves, such as the

11 20 02 11 spatially indirect biexcitons, 02X, 11X, 11X and 20X, as seen in Fig. 2.8. These spatially indirect states possess an eﬀective dipole moment (due to their spatially direct component), such that they would be connected by the FRET mechanism. Additionally, these neutral biexcitons can be thought of being composed by a trion localized in one QD together with a single charge (electron or hole) localized in the second QD. 39

11 Figure 2.8: FRET signatures on the biexciton state 02X according to parameters given in table A.3. (a) LACS population map showing splitting of indirect exciton line over large 02 F range. Blue dashed line indicates the asymptotic position of the direct biexciton 02X mixing with the acceptor level. Top right inset: map with no FRET. Bottom left inset: Magniﬁcation of the indirect line splitting. (b) Average population for three values of F, showing invariance of indirect split-oﬀ energy: satellite line in (a) tracks the brighter exciton line nearly parallel. (c) Reduced level diagram describing the system for high values of F. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 79, 245309 (2009). ©(2011) American Physical Society.

Lets focus our attention on the biexciton manifold spawning the transition cascade

10 20 02 01X ↔11 X ↔11 X, as shown on Fig. 2.8(c). Truncation of the full Hamiltonian restricted 40 to such states yields,      ∆S ΩT ΩB     11  HXXI =  Ω δ + ∆ V  . (2.21)  T 02 S F     20  ΩB VF δ11 + ∆S A crucial symmetry of this Hamiltonian makes its eigenvalues invariant against variations

in the applied electric ﬁeld F, since all three states possess the same Stark shift ∆S . This symmetry makes the eigenvalue problem of Eq. 2.21 formally equivalent to that of Eq. 2.16; in this case FRET optical signatures appear as a bright exciton spectral doublet with the satellite peak following the Stark shifted spectral line associated to the spatially indirect component of the trion-like biexciton. This is precisely seen to be demonstrated

11 by our simulated LACS population map for the 02X state in Fig. 2.8(a). For F ∈ (−15, 0) kV/cm, the split-feature signature is evident (see lower inset); as expected, the splitting

20 disappears at the tunneling anticrossings where 11X hybridizes with the spatially direct 02 biexciton 02X. Interestingly, the width of the splitting does not change appreciably as we increase the value of the electric field F, as seen Fig. 2.8(b), which shows different cuts along the laser energy axis within the region enclosed by the dashed box. Most remarkably, the amplitude of the FRET peak steadily increases as the field is swept away from the molecular anticrossing, which corresponds to a further projection of the exciton

11 20 dynamics into the relevant subspace spanned by the FRET coupled states 02X and 11X. These results indicate that in experiments one can enhance the FRET signal (obtaining a better contrast) by just increasing the ﬁeld. 41

11 Figure 2.9: (a) The trion-like state 02X has satellite after mixing with the biexciton 02 resonance 02X, as shown in (b). Notice inset shows no splitting when VF = 0. Comparison of (c) and (d) conﬁrms mixing of FRET satellites between direct and trion-like biexciton states; FRET signature on direct biexciton state is weaker by an order of magnitude. Simulation parameters according to appendix table A.3. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 79, 245309 (2009). ©(2011) American Physical Society.

In order to support more our arguments we constructed LACS population maps of both of the hybridized states, at ﬁxed F ' −10kV/cm, within the region shown in dashed box at the bottom of Fig. 2.8(a). These maps appear in Fig. 2.9, which shows almost

11 20 identical level anticrossing signatures around ~ω ∼ 1251, for both 02X and 11X states, with the sum of the population at the peak maxima, in Figs. 2.9 (c) and (d), plus the amplitude of the vacuum, adding up to near unity. 42

2.9 Conclusions

In this chapter we have presented theoretical results regarding detection of resonant energy transfer processes in self-assembled InGaAs quantum dot molecules. We have shown that level anticrossing spectroscopy experiments would be a suitable methodology for FRET detection. In this regard, FRET optical signatures can be detectable in spite of the strong effects of charge tunneling. As interdot tunnel mixing is controlled by the application of an external axial electric field, enhancement of FRET optical signatures can be achieved in the same way. Although the effects we discuss have yet to be identified in experiments, the main message of this work is to point out that under suitable conditions Forster¨ interaction may give rise to clearly identifiable signatures in PL or differential transmission spectroscopy. In fact, to date most workers in this field see the Forster¨ interaction as difficult to detect in self-assembled QDMs, partly because clear signatures in experiments have not been identified from a theoretical point of view. We emphasize, especially on Fig. 2.6, for example, that this interaction can give rise to clearly defined split-off satellites which become stronger as an electric field is applied across the molecule (‘away’ from the region where the electronic tunneling is dominant). This should be easily testable in experiments. For example, the quantum dot molecule can be explored via pump-probe differential transmission spectroscopy measurements of exciton populations. These could in principle exhibit signatures as those discussed above. For example, the observation of dressed exciton states in QDs using this technique has appeared recently.[38–40] An exciton state can be pumped into the first QD via pulsed broadband laser excitation, creating a collection of dressed states in the system. A second weak probe pulse can be sent into resonance with the excitonic transition in the second QD, which has a slightly different transition energy respect to the QD, measuring in this way the transient differential transmission, which reflects the exciton population in the second QD. 43 3 Coherent Control of Exciton Dynamics in Quantum Dot

Molecules

3.1 Introduction

3.1.1 Coherent control

At the frontier of all subdisciplines of physics, quantum mechanics is still a subject for revolutionary discoveries. At the core of many of these discoveries are experiments that require an unprecedented degree of coherent control of light-matter interactions and quantum dynamics of the system. Examples of such confluent advances emerge in the areas of atomic and molecular physics, quantum optics and quantum information theory. But its has been until very recently that such advances have been ported into condensed matter physics, thanks to the development of new materials and nanostructures, and novel quantum effects that probe their most fundamental properties.[54, 55] A generic protocol for quantum coherent control involves the use of coherent electromagnetic fields to encode information on natural and artificial atomic and molecular systems. Systems such as trapped atoms and quantum dots, are manipulated in such a way that their subsequent quantum dynamics leads to a desired final state. Typically, the electromagnetic fields can be treated in a quantum or semiclassical way, while the matter system must have a discrete level structure. Advances in quantum optics have lead to the possibility of shaping in real time the form of electromagnetic field pulses, such that they can be the control agents while the matter system acts as the controlled object, as seen in Fig. 3.1. 44

Figure 3.1: Coherent control schematics. A coherent light source is pulsed using a control shaping function F(t). Light illuminates the sample which has a tunable spectrum. The optical response of the sample is sent to a detector. A feedback circuit is implemented in order to adjust the control parameters until the sample responds according to a predetermined ﬁnal state.

Coherent control of the optical response of nanostructures has several advantages over their atomic counterparts, such as enabling control in situ of the discrete energy levels of the controlled object. As we have seen in the previous chapter, the energy levels of quantum dots and quantum dot molecules, can be tuned at will by means of confinement engineering or by means of external magnetic or electric fields. Moreover, control over the continuum level environment in these systems is equally tunable in situ, offering invaluable experimental set ups for coherence phenomena to be studied in different conditions. But what is coherent control? The general principle of coherent control can be stated, according to Shapiro and Brumer[56], as follows: “Coherently driving a state with phase coherence through multiple, coherent, indistinguishable, optical excitation routes to the same final state allow for the possibility of control”. 45

In what follows, we apply the principle of coherent control for qubits defined by monoexcitons with spatial indirect character. The control rule is imposed on the Stark shift of indirect excitons, by means of an external electric field pulse. We demonstrate coherent oscillations between the photon field (dressed vacuum state) and indirect excitons, as well as, coherent oscillations between the two indirect monoexcitons. In both cases, the desired optical excitation routes are obtained by means of rigorous adiabatic elimination of states that affect adversely the phase coherence of the excitonic dynamics of interest.

3.2 Eﬀective Subspace Extraction From Level Anticrossing Spectra

3.2.1 Optical Signatures: Eﬀective Three and Two Level Systems

As we have discussed in Chapter 2, a level anticrossing signature indicates the onset of important molecular couplings. These couplings are in some instances not derived explicitly from oﬀ diagonal matrix elements of the full Hamiltonian, Hi j, connecting allowed transitions i ↔ j, but rather from higher order processes possible in the system. 46

00 Figure 3.2: Level anticrossing population map of the QDM vacuum state |00Xi for parameters given in appendix table A.4. As function of applied bias and pump laser energy the vacuum depopulates to other excitons at each resonance. The x-pattern dashed box indicates resonant excitation into molecular excitons. In particular, for a state with mostly 01 indirect character, 10X, at FI = 43.4kV/cm, Elaser = 1299.6meV, the system is governed by two level system (TLS) dynamics. In the central anticrossing enclosed in the diamond shaped box at FR = 2.3kV/cm, the system also behaves as a TLS, mixing only two indirect excitons.

In our model, we have assumed that the indirect exciton states have a vanishingly

01 10 small oscillator strength, such that h0|H|10Xi = h0|H|01Xi = 0. On the other hand, careful examination of the level anticrossing map due to the full Hamiltonian Eq. 2.13, as shown in Fig. 3.2, reveals that indirect excitons possess a non-vanishing amplitude for a wide range of applied field electric field, in particular far away from the tunneling anticrossings, where the system is weakly coupled by carrier tunneling. This indicates the existence of an effective oscillator strength which couples the indirect excitons to the radiation field

00 (vacuum state |00Xi), via higher order perturbative transitions that incorporate direct 47 exciton optical couplings, charge tunneling and FRET. All spatially indirect excitons with eﬀective oscillator strength are created by coherent oscillations between the vacuum and an indirect state. In a sense, all these regions can be thought to belong to a two level system (TLS) subspace, which is energetically detuned from the fast recombining spatially direct states. In Fig. 3.3 we present the average population of both indirect and direct excitons for a ﬁxed value of the excitation energy, Elaser = 1271meV, which corresponds to the horizontal dashed cut line across the central diamond dashed box in Fig. 3.2.

3.2.2 Two Level System Optical Signatures: Indirect Exciton Qubits

In Fig. 3.3, the total average population of the system is mostly shared with equal weights on the indirect exciton states, as seen in the upper panels, the indirect exciton

01 2 10 2 population peaks have almost the same amplitude, |10X| ' |01X| ' 0.45, with minimal weight on the direct states. For example, as shown in Figs. 3.3(c) and (d), by tuning the laser at Elaser = 1271meV and the electric ﬁeld at F = −3kV/cm, the system is let into the

− superposition state |Ψ( )i ' √1 |00Xi + |01Xi , which suggest that the dynamics in this X 2 00 10 regime is well represented by a two level system. Interestingly, the central region of the x-pattern in Fig. 3.2 corresponds to a molecular anticrossing involving only indirect excitons, which indicates that coherent oscillations among indirect states can be induced by optical and tunneling processes. As we would demonstrate in the following sections, the properties of this indirect excitonic two level system is well mapped into the properties of a quantum bit (see appendixE). 48

Figure 3.3: Average population of all neutral excitons for ﬁxed laser excitation energy Elaser = 1271meV and parameters given in appendix table A.4. (a) and (b) show the spectral 10 01 peaks of excitons |01Xi and |01Xi at the LACS map coordinates (Elaser, F ' 7.5kV/cm), 01 10 respectively. (c) and (d) show the spectral peaks of excitons |10Xi and |10Xi at the LACS map coordinates (Elaser, F ' −3kV/cm), respectively. The population weights suggest a 00 TLS behavior between the vacuum |00Xi and each indirect exciton.

In order to prove our hypothesis we have calculated the eigenvalue spectrum (dressed spectrum[33, 37, 39]) of Eq. 2.13 as function of the laser excitation energy, within the x-pattern central anticrossing region of Fig. 3.2, for two ﬁxed values of the applied electric ﬁeld as shown in Fig. 3.4. 49

Figure 3.4: (a) LACS population map inside the diamond dashed box at the central x- pattern in Fig. 3.2. (b) Eigenvalue spectrum as function of laser excitation energy for fixed field F = −3kV/cm (blue vertical line in (a)); two level anticrossings occur between 00 the vacuum |00X and each indirect exciton, corresponding to coherent oscillations at the intersection points with blue line in (a). (c) Eigenvalue spectrum as in (b) but at fixed field F = 2.4kV/cm (red vertical line in (a); the level anticrossings in (b) merge into a single double anticrossing mixing the two indirect states, corresponding to coherent oscillations between the two indirect excitons with minimal occupation of the vacuum.

Fig. 3.4(a) shows the LACS average population map of the dressed indirect exciton

01 10 states |10Xi and |01Xi, which clearly exhibit a level anticrossing. The blue and red vertical dashed lines represent the ﬁxed applied electric ﬁeld values, F = −3kV/cm and F = 2.4kV/cm, at which the dressed spectrum was calculated as function of the laser

excitation energy. At F = −3kV/cm and Elaser = 1262.5meV a relatively well isolated 50

00 10 Rabi splitting opens between the vacuum |00Xi and the indirect state |01Xi, while at 00 01 Elaser = 1271meV, the anticrossing mixes |00Xi and |10Xi. In both cases, the Rabi splittings indicate the presence of coherent oscillations between the self-avoided states, which averaged over long times gives rise to the bright spectral lines intersecting the blue dashed line in Fig. 3.4(a). By tuning the ﬁeld at F = 2.4kV/cm, the two splittings in Fig. 3.4(a) are tuned into near resonance, generating a vacuum mediated molecular superposition of purely indirect states, as shown in Fig. 3.4(c).

3.3 Eﬀective Exciton Hamiltonian via Adiabatic Elimination

The molecular states due to the superposition of two indirect excitons, arise from higher order perturbative couplings involving an effective coupling to the vacuum and the interplay of interdot couplings, such as charge tunneling and FRET. The spectral signatures we have found suggest that the dynamics of such purely indirect molecular states should be described by an effective two level system Hamiltonian.[57] This reduced Hamiltonian is constrained to a subspace of the Hilbert space spanned by eigenvectors with eigenvalues that approximate the diagonalization solutions of the full Hamiltonian. The formalism allowing the derivation of an effective Hamiltonian describing a sector of a complex multilevel system is called adiabatic elimination.[32, 33, 58, 59] The applicability of this procedure depends on the existence of irrelevant states which are not initially populated nor coupled to the relevant effective states. In our problem, the irrelevant states can be adiabatically eliminated provided that (1) the relevant energy levels are spectrally isolated from the irrelevant levels, a situation realized by controllable detuning of energy levels, either by varying the excitation energy or the applied electric field; (2) all external fields should not populate the irrelevant states. In what follows, we implement an adiabatic elimination procedure by means of an algebraic method using projection operators, which permits the derivation of an effective Hamiltonian with 51 exciton eigenstates of purely indirect character. The direct excitonic sector of the Hamiltonian is adiabatically eliminated (“projected out”), and its dynamical effects become embedded in the matrix elements of the effective Hamiltonian.

3.3.1 Adiabatic Elimination via Algebra of Projection Operators

Let us consider the case of a closed quantum system conformed by the exciton multilevel Hamiltonian given by Eq. 2.13. The Hamiltonian can be separated in to two

parts, H = H0 + V, where H0 is the unperturbed diagonal part, and V the perturbative interaction that contains the different couplings. Let P be the relevant exciton subspace spanned by excitons whose dressed energy levels anticross at a resonance identified in the dressed spectrum or the LACS population map. We define projector operators P and Q = 1 − P that project onto and outside of the relevant subspace P, respectively. The effective Hamiltonian of reduced dimensionality is given by

H˜ (z) = PH0P + PR(z)P , (3.1)

where z = E ± i, such that E is the real part, and the imaginary part of the generalized

complex eigenvalue z. Here, PH0P is the unperturbed part of the Hamiltonian inside P. The second term in Eq. 3.1, PRP, is a Hamiltonian in the irrelevant subspace, Q ⊥ P, that

when added to PH0P allows us to calculate the shifts of the perturbed energy levels relative to the unperturbed levels. Here R(z) is called the level shift operator and is given by

−1 R(z) = V + VQ[z − QH0Q − QVQ] V . (3.2)

Since the Hamiltonian depends on its eigenvalues, z, the diagonalization problem becomes non-linear. Additionally, for complex values of z, the resulting effective Hamiltonian becomes non-Hermitian, allowing for the description of dissipative phenomena. Solutions to the non-linear eigenvalue equation arise from the roots of a polynomial equation in z; the problem is drastically simplified when considering only the unitary evolution of the 52 system, in this case the roots are real, z → E. Remarkably, the projector operator formalism permits us to obtain analytical expressions for the effective matrix elements describing particular optical signatures obtained from the full multilevel Hamiltonian. In particular, we are interested in analytical expressions for the values of the gaps or their associated Rabi frequencies, for the vacuum indirect exciton anticrossings, as well as the anticrossing mixing purely indirect states. From an operational point of view adapted to our problem, the adiabatic elimination procedure is carried out as follows[57]:

1. Starting with the full Hamiltonian with real eigenvalue spectrum, identify an anticrossing “suﬃciently” detuned from the irrelevant exciton states. In doing so, identify the bare exciton states which mix at the anticrossing. These exciton basis states are selected to construct the projection operator P.

2. With ﬁxed values of the excitation energy, Elaser, excitation power, Ω, and ﬁxed

values of interdot couplings te, th, VF, tune the external electric ﬁeld, F, such that the system is at the desired coordinate in the spectrum.

3. Calculate the eﬀective Hamiltonian H˜ (z) and its eigenvalue spectrum, λi(z) which is now only function of z. For each eigenvalue ﬁnd the solutions to the equation

λi(z) − z = 0, with z being a real number. There is a unique root z j for which the spectrum of H˜ as function of the applied ﬁeld F, entirely reproduces the anticrossing feature of the original Hamiltonian. In other words, in a level anticrossing (and in the absence of accidental degeneracies) there exists a unique self-consistent solution z(F), for each value of the applied electric ﬁeld F.

4. There is a boundary spectral region beyond which the eﬀective eigenvalues as function of the applied ﬁeld F, does not approximate the full spectrum. In this regime, the irrelevant subspace contains states which can not be adiabatically 53

eliminated, in other words those states would be in fact populated by the dynamics associated to the full Hamiltonian.

3.3.2 Three Level System Subspaces: Eﬀective Oscillator Strength

In order to obtain the relevant subspace associated to molecular states of purely indirect character, we project the ﬁve level Hamiltonian in Eq. 2.13 onto a three level system, such that

00 00 10 10 01 01 P = |00Xih00X| + |01Xih01X| + |10Xih10X| (3.3)

10 10 01 01 Q = |10Xih10X| + |01Xih01X| . (3.4)

Let PΛ be the subspace spanned by the vacuum and the indirect exciton levels; then a

projection of Hamiltonian (2.13) onto PΛ gives      ∆00 (z) Ω˜ I(z) Ω˜ I(z)   00    ˜ (Λ)  ˜  H (z) =  ΩI(z) ∆10 (z) + ∆S U(z)  . (3.5)  01     Ω˜ I(z) U(z) ∆01 (z) − ∆S  10 54

Figure 3.5: Effective Hamiltonian energy level diagrams. (a) Energy levels corresponding to the effective Hamiltonian in Eq. 3.5, showing indirect exciton effective couplings (solid lines) as well as the effective relaxation channels of indirect excitons (dashed red lines) (b) Energy levels according to the TLS Hamiltonian in Eq. 3.11, with an effective coupling connecting the two indirect excitons, given by tU = U + ξ.

The resulting three level effective Hamiltonian, H˜ (Λ)(z), as shown schematically in Fig. 3.5(a), yields an effective coupling, U, connecting the indirect excitons to each other and effective couplings, Ω˜ 01 and Ω˜ 10 , connecting these states to the radiation field. In what 10 01 follows, as well as in Eq. 3.5, we have assumed that the spatially direct excitons are in near resonance, δ10 = δ01 = ∆, with both QDs coupling with the same strength, Ω, to the 10 01 Λ radiation field; this results in Ω˜ 01 = Ω˜ 10 = Ω˜ I. The diagonal matrix elements of H˜ (z) 10 01 contain shifted detunings, ∆i, given by

∆10 01 (z) = δ10 01 + δI(z) , (3.6) 01(10) 01(10) 2 2 (z − ∆)(t + t ) + 2tethVF δ (z) = e h , (3.7) I 2 2 (z − ∆) − VF 00 while the shift of the zero of energy (we have set δ00 = 0) is given by 2Ω2 ∆00 (z) = − . (3.8) 00 ∆ + VF − z The energy shifts of the indirect exciton detunings are inversely proportional to the direct

exciton detunings ∆, and proportional to VF up to second order terms in the interdot 55 couplings, te, th. Since the interdot couplings are ﬁxed by the structure, their weights on the correction terms are solely modulated by ∆, in other words, well controllable by the laser excitation frequency. We readily notice that a detuning of a few meV far from resonant excitation into the direct states results in smaller energy shifts. Interestingly, the shift of the relative vacuum energy level is mediated only by the direct transition dipole

matrix element Ω, and is small for low excitation power and larger values of δ and VF. On the other hand, the indirect exciton eﬀective coupling to the radiation ﬁeld is given by

Ω(te + th) Ω˜ I(z) = − . (3.9) ∆ + VF − z

3.3.3 Spatially Indirect Protected Two Level Systems

Figures 3.6(a) and 3.6(c) show the gaps of the vacuum-indirect exciton anticrossings.

Both gaps have a width 2Ω˜ I = 65µeV, and they occur at electric ﬁeld values of F ' −38.6

and 43.4kV/cm, respectively. Ω˜ I is directly proportional to the direct transition dipole

matrix element Ω and tunneling amplitudes (te + th). This implies that molecular indirect excitons possess in fact an eﬀective oscillator strength when the quantum dots in the QDM are coupled by either electron or hole tunneling. This is important, since it enables resonant excitation of the indirect excitons states, even if they have zero oscillator

01 10 strength. This explains the “lighting up” of |10Xi and 01X in Fig. 3.2, even when optically driving the QDM for laser excitation energies far from the direct exciton transitions, for electric ﬁeld values |F| ≥ 20kV/cm. 56

Figure 3.6: Dressed exciton eigenvalue spectrum as function of applied field F, for fixed excitation energy ~ω = 1299.6meV and parameters given in appendix table A.4. (a) and (c) show the anticrossings between the vacuum energy level and the indirect excitons, due to the effective coupling ΩI. (b) shows the anticrossing between the pair of indirect excitons due to the effective coupling U. In all cases the behavior corresponds to effective TLS which results in exciton population dynamics inside the relevant subspaces discussed above. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society.

A remarkable phenomenon exhibited by the LACS optical signatures and the adiabatic elimination procedure is the molecular coupling between two purely indirect states, with strength given by

2 2 2(z − ∆)teth + (t + t )VF U(z) = e h . (3.10) 2 2 (z − ∆) − VF The width of the gap associated with this interaction is 2U ' 45µeV, for the QDM parameters used in our model. The coupling U dominates when the electric ﬁeld is tuned to FR = 2.3kV/cm, and is independent of the laser excitation which is proportional to the direct dipole matrix element Ω. This is a remarkable feature for optically controlled two level systems in the purely indirect subspace, since it permits the implementation of coherent rotation using the“natural frequency” of the system, with minimal disturbance from the radiation ﬁeld environment. This is shown in Eq. 3.10, where U arises predominantly from electron and hole tunneling, with minimal contribution from FRET. In other words, the molecular indirect subspace evolves mainly by its internal dynamics. 57

Most remarkably, an eﬀective qubit subspace[60, 61] generated by the indirect excitons

10 01 |01Xi and |10Xi would be protected against the external variations of the excitation power, allowing for ﬂexibility in experiments if excitation power where to operate as an additional control parameter. As we would discuss later on this chapter, the indirect subspace is also resilient against the eﬀects of spontaneous recombination of spatially direct excitons. It is possible to continue further the projection procedure to adiabatically eliminate the dressed vacuum state. In this case, the projector operator associated to the relevant

10 10 01 01 subspace PI is given by P = |01Xih01X| + |10Xih10X|. Thus, the eﬀective two level system Hamiltonian after projection is given by    H˜ Λ (z) U(z)  ˜ (I)  22  H (z) =   + ξ(z)(σX + I) . (3.11)  ˜ Λ  U(z) H33(z)

˜ (I) ˜ (I) ˜ (Λ) ˜ (I) ˜ (Λ) ˜ (Λ) The diagonal elements of H are such that H11 = H22 and H22 = H33 , where Hii are the diagonal matrix elements of Eq. 3.5, while the oﬀ-diagonal elements are given by U as in Eq. 3.10. In writing Eq. 3.11 we have separated a small perturbation, ξ(z)(σX + I), arising from optical pumping, with σX and I being the Pauli-x and identity matrices. The strength of this perturbation is measured by

Ω˜ 2(z) ξ(z) = I (3.12) z − ∆00 (z) 00 and accounts for the fact that the eﬀective two level system is not perfectly isolated from the environment, in this case the radiation ﬁeld acting as reservoir. Interestingly, the

correction term ξ(z) vanishes in the absence of laser excitation, Ω = Ω˜ I = 0. For a purely indirect exciton qubit subspace, the representation given by H˜ (I) deﬁned on the subspace

PI, provides a more transparent framework for the implementation of coherent qubit rotations, while the representation of H˜ (Λ) is suitable for describing the initialization

01 10 dynamics of the qubit states, |10Xi or |01Xi, via the coupling of indirect excitons to the radiation ﬁeld, Ω˜ I. 58

3.4 Coherent Control of Indirect Excitonic Qubits

3.4.1 Excitonic Density Matrix Dynamics and Control Scheme

In what follows, we demonstrate that is possible to use the pair of indirect exciton

01 01 states, |10Xi and |01Xi, as the computational basis of a qubit.[57, 60] The qubit is initialized with near unity fidelity through a π rotation within the anticrossing, Ω˜ I, between the vacuum and either indirect exciton at F = −38.6kV/cm or F = +43.4kV/cm, respectively, see Figs. 3.6(a), (c). An adiabatic electric field pulse[62, 63] transports the initialized state into the central anticrossing U at F = 2.3kV/cm, see Fig. 3.6(b), where coherent qubit rotations occur strictly among the indirect exciton states. Qubit readout proceeds with an adiabatic pulse that returns the rotated output state into the initial electric field value, transferring via adiabatic passage[64–66] its population into a direct state which depopulates into the vacuum. 59

Figure 3.7: Dressed exciton eigenvalue spectrum at fixed excitation energy ~ω = 1299.6meV. The dashed boxes indicate the anticrossings associated to qubit initialization 10 (F = 43.4kV/cm) into the state |01Xi (red), as well as, qubit rotation (F = 2.4kV/cm) into 01 the final output state |10Xi (blue). Insets show the magnification of such anticrossings. The 10 adiabatic reversal pulse state transfers the population of the output state into |10Xi (green), which eventually relaxes into the vacuum. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society. 60

Figure 3.8: (a) Electric ﬁeld control pulse used for qubit initialization and rotation. (b) Density matrix numerical simulation of the exciton population dynamics corresponding to the control pulse in (a); it shows all steps involved in the coherent control of the indirect exciton qubit. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society.

In Fig. 3.7 we present the regions inside the exciton dressed spectrum where the initialization and rotation operation occurs. In order to verify the procedure described above, we use the full exciton Hamiltonian Eq. 2.13, such that the exciton population dynamics is given by numerical solutions to the dissipative Lindblad master equation, which gives the temporal evolution of the density matrix of the system, ρ. We incorporate

10 the effects of spontaneous recombination Γx of the spatially direct excitons, |10Xi and 01 |01Xi, as shown in Fig. 2.5(a). The Lindblad master equation is given by dρ i = − [H(t), ρ] + L(ρ) . (3.13) dt ~ The first term on the right hand side is the Liouville-von Neumann commutator, which describes the coherent evolution of the excitation dynamics[67], with H(t) being the full Hamiltonian of the system in Eq. 2.13. Here the time dependence of the Hamiltonian arises from the parametric time dependence of the control pulse of an applied external electric field, F(t), or from a pulsed control laser used to drive the transition i ↔ j, with transition matrix elements Ωi j(t). The second term on the right hand side, L(ρ), is the 61

Lindblad superopeator and incorporates the dissipative dynamics[68, 69],

X Γi j L(ρ) = − ({P , ρ} − 2ρ P ) , (3.14) 2 j j j i j

where P j = | jih j|, and | ji is an exciton which relaxes into a state |ii, with rate Γi j. The dynamics requires the solution of N2 coupled diﬀerential equations for a N-dimensional Hilbert space. For our particular problem, the dissipator operator takes the following form,

3.4.2 Indirect Exciton Qubit Initialization

The indirect exciton couplings U and Ω˜ I dominate at three diﬀerent electric ﬁeld regimes; in consequence the subspace representation of Hamiltonian Eq. (3.5) is

decoupled in to three diﬀerent manifolds of the eigenvalue spectrum. Since Ω˜ I dominates at large values of electric ﬁeld and positive energy detuning, we can extract two projected

00 10 00 01 subspaces, which are spanned by the excitons {|00i, |01Xi} and {|00i, |10Xi}, respectively. For a laser excitation energy of ~ω = 1299.6meV, coherent Rabi oscillations are induced in each of these subspaces for ﬁeld values of F = −38.6 and 43.4kV/cm, corresponding to resonant excitation at each level anticrossing, shown in ﬁgures 3.6(a) and 3.6(c), respectively. This enables the implementation of π rotations[37, 60] each having a period of π ' 63.6ps. Therefore, ultrafast initialization of the system in either of the logical 2Ω˜ I 01 01 ˜ states, |10Xi or |10Xi, can be implemented. We should notice that ΩI is inversely proportional to the direct exciton detuning, ∆, with its value setting an upper bound to the

initialization elapsed time, Ω˜ I ΓX, otherwise the initialization ﬁdelity would be signiﬁcantly diminished by spontaneous recombination of the direct exciton states. Figure 62

3.9 shows the coherent Rabi oscillations during the initialization of the indirect exciton

01 ˜ |10Xi by a 3π rotation. While the laser is on, it drives the system at the ΩI anticrossing in

Fig. 3.6(c), corresponding to the coordinate (FI = 43.4kV/cm, ~ω = 1299.6meV) in Fig. 3.2. The initialization takes place after switching off the resonant excitation laser field at a time t = 200ps. Moreover, the initialization fidelity[60] is close to unity,

10 10 F = h01X|ρ(ti)|01Xi ' 0.97. A near unity ﬁdelity is due to the protection of the subspace

PΛ from perturbations due to the second term in Eq. 3.11.

01 Figure 3.9: Initialization of the indirect exciton |10Xi by a 3π rotation. When driving the system at the anticrossing, corresponding to the coordinate (FI = 43.4kV/cm, ~ω = 1299.6meV), the initialization takes place after switching oﬀ the pulsed resonant excitation 10 10 at a time t = 200ps. The initialization occurs with near unity ﬁdelity, F = h01X|ρ(ti)|01Xi ' 0.97, due to the almost perfect isolation of the subspace PΛ. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society.

3.4.3 Indirect Exciton Qubit Rotations

Controlled indirect exciton qubit rotations can be performed by applying an adiabatic

10 electric ﬁeld pulse (see appendixD, this pulse transports the initial prepared state, |01Xi, from the anticrossing at FI ' 43.4kV/cm into the central anticrossing at FR ' 2.3kV/cm, see dashed box at center of Fig. 3.7. At FR, the two level system dynamics exhibits Rabi oscillations that fully populate the indirect excitons. However, we remark that even when the laser is on, the optically mediated perturbations due to the term ξ(z)(σX + I) in Eq. 3.11 are controlled at will by varying the excitation power and laser detuning. The Rabi 63 oscillations allow for ultrafast qubit rotations in the picosecond scale, with a characteristic

π period of 2U = 91.8ps, they exhibit a near unity amplitude within the time frame shown in Fig. 3.10. On the other hand, the effects of spontaneous recombination of the direct states can not be neglected from the dynamics after several π rotations. As seen in Fig. 3.10, after 2ns there is a slight damping of the Rabi oscillations, with more dramatic effects seen in Fig. 3.12(b), which shows total damping after 150ns. Other weak effects that contribute to damping are due to higher order transitions occurring outside the qubit subspace, due to perturbative corrections to the matrix element U in Eq. 3.5, and due to the slight energy shift of the indirect excitons, δI. The fidelity of the π rotation determines the amount of population transferred into the output state, and the extent to which the final indirect

01 eigenstates state would follow the eigenvalue spectral line, |10Xi, upon reversing the electric ﬁeld adiabatic sweep. Also in the model we have neglected possible excitation of trapping states in the continuum of the wetting layers, or due to exciton coupling to acoustic phonon; these eﬀects rather contribute weakly to the dynamics for resonant optical excitation at cryogenic temperatures.[70, 71]

Figure 3.10: Two level system dynamics at FR exhibits Rabi oscillations that fully populate the indirect excitons. Here, the system evolves by its internal dynamics. The coherent oscillations allow for qubit rotations in the picosecond scale, with a characteristic time π 2U = 91.8ps. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society. 64

3.4.4 Indirect Exciton Qubit Readout

After a π rotation, when the system is at the central anticrossing at FR, the probability

10 density of the output indirect exciton, |01Xi, has to be measured optically. Since the indirect excitons have vanishing oscillator strength in our model, optical inference about the amplitude of the output state can be obtained via photoluminescence of an auxiliary

01 direct state. An adiabatic reversal of the electric ﬁeld pulse transports the output state |10Xi

back to the original electric ﬁeld value of FI = 43.4kV/cm. In this step, the transported state passes through the tunneling anticrossing at F ' +20kV/cm, transferring

10 adiabatically its population via coherent tunneling[64, 72, 73] in to the direct state |10Xi, which in turn depopulates into the vacuum via spontaneous recombination. The ﬁdelity of

10 10 the readout depends on the conditional adiabatic population passage from |01Xi into |10Xi, where this state acquires maximal occupation at t = 3.6ns, see Fig. 3.11. We notice that

00 partial occupation of the vacuum state |00Xi takes place before the direct state acquires maximal occupation, but this is because in the simulation we let the qubit rotate several times inside the U anticrossing (3π rotation) in order to demonstrate damping, delaying the onset of reversal pulse until t = 3.1ns. Otherwise, starting the reversal pulse just after the ﬁrst π rotation at t = 1.35ns, would enhance the maximal occupation of the direct state. 65

Figure 3.11: Read out scheme of the output qubit state. In this step, the transported state passes trough the tunneling anticrossing at F ' +20kV/cm, transferring adiabatically its 10 population (blue line) to the direct state |10Xi (olive line), which in turn depopulates into the vacuum (dashed line) via spontaneous recombination. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society.

3.5 Additional dynamical eﬀects

3.5.1 Forster¨ Energy Transfer and Biexciton States

As we have discussed in the previous chapter, near resonant direct exciton transitions enhances the Forster¨ energy transfer mechanism, playing an important role on the system dynamics. The FRET strength is VF ' 0.08meV, for an interdot separation of d ' 8.4nm

(see appendix table A.4), which is small with respect to electron tunneling te, but enough to generate a direct exciton spectral doublet and to redistribute the exciton probability in the steady state regime. On the other hand, the Hamiltonian matrix elements in Eq. (3.9) and (3.10) contain perturbative corrections which depend on VF, such that it amounts to an additive shift in the direct exciton detuning ∆. Therefore, any inﬂuence of VF is suppressed for ∆ VF, and exciton dephasing due to FRET is also suppressed. In our model, |∆| ' 51.5 meV, this assures that qubit subspace is indeed protected against the perturbative dynamics due to FRET. 66

As we have also discussed in Chapter 2, biexciton pumping under strong laser excitation power cannot be in principle ignored, since biexcitons are energetically detuned by at most a few meV. However, even a few meV of detuning from the relevant anticrossing in the spectrum is a large value in comparison to the gaps Ω˜ I and U (∼ µeV), which makes biexciton pumping a very weak perturbation. Moreover, all biexciton subspaces decouple from monoexciton manifolds after switching oﬀ the excitation power during the sequence of control pulses.

3.6 Dissipative Eﬀects

3.6.1 Numerical Results: Density Matrix

Through all the discussion above we have considered neutral indirect excitons with vanishing oscillator strength for our chosen QDM structural parameters; in other words, the indirect exciton intrinsic recombination rate was set to ΓI = 0. However, since the QDM is a coupled multilevel system, all eigenstates should be affected by relaxation from at least one of its components. In our case, the eigenstates are exciton molecular states whose direct components relax into the vacuum. Ideally, at initialization and rotation of the exciton qubit, the computational basis has “mostly” spatially indirect character, but non-zero spatially direct character. The effects of direct exciton spontaneous recombination on the indirect exciton states is due to higher order transitions mediated by charge tunneling, taking place outside the qubit subspace (Q manifold). This causes the molecular indirect exciton states to acquire a finite effective oscillator strength and a lifetime, which is ultimately dependent on the spatially direct exciton recombination. 67

Figure 3.12: Population relaxation dynamics for indirect excitons, from the numerical solutions of Eq. 4 with up to 14 exciton states included. (a) Population decay after switching oﬀ the laser light in the initialization regime at FI = 43.4kV/cm. Dashed line indicates the case where biexciton states are taken into account. Inset shows the corresponding depopulation into the vacuum. (b) QDM internal dynamics in absence of 10 01 optical perturbations, shows decay of Rabi oscillations of |01Xi and |10Xi (red line, blue line) in the qubit rotation regime at FR = 2.3kV/cm. Inset shows Rabi ﬂops in the early stage of the dynamics. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society.

Figure 3.12(a) shows the numerical solution to the density matrix population

01 dynamics of the input state |10Xi after switching oﬀ laser excitation power, Ω(τ = 200ps) = 0, see Fig. 3.9; after this time the exciton relaxes into the vacuum via

˜ −1 higher order transitions (see inset in Fig. 3.12a) acquiring a lifetime ΓI ' 218ns. For completeness, we also show the depopulation of the state when the biexcitonic degrees of freedom are taken into account, indicated by the dashed line. The dashed red curve in Fig. 3.12(a) shows that biexcitons do not aﬀect the relaxation time when Ω = 0. On the other 68 hand, Figure 3.12(b) shows the input state dynamics when the system has been transported adiabatically into the rotation anticrossing. Since the time evolution is due to the qubit subspace internal dynamics, both indirect excitons (blue and red solid lines) enter a Rabi oscillation regime, while damping is due to depopulation into the vacuum,

˜ −1 with characteristic lifetime ΓI ' 25.2ns. Notice that no population has been transferred into the direct excitons, and that the coherent oscillation period (' 91.8ps), is orders of magnitude shorter than the relaxation time. Remarkably, our results show that despite assuming an inﬁnite lifetime for the qubit computational basis (bare indirect excitons), the interdot couplings mediate a strong relaxation channel for the indirect excitons at any value of the applied electric ﬁeld F. In fact, experiments have shown that it is possible to electrically modulate exciton lifetimes (around ∼ 2ns to ∼ 10ns) near a tunneling induced molecular anticrossing, even in the presence of acoustic phonon induced relaxation.[74–76]

3.6.2 Analytical Results: Adiabatic Elimination

10 Relaxation of the state |01Xi inside the initialization subspace PΛ, as well as damping of Rabi oscillations inside the qubit subspace PI, results in a finite lifetime of the input qubit state during initialization and rotation. In considering only intrinsic spontaneous recombination of spatially direct excitons, the effective relaxation rates of the molecular states can be appreciably different, since they depend on all interdot couplings, and thus can be controlled at will by varying the laser detuning, excitation power and applied electric field. Let us consider the problem of finding analytical expressions for the effective relaxation rate of indirect excitons inside the subspace PI. Since VF |∆|, we can ignore the effects of resonant energy transfer in the following discussion. A formulation of the 69 relaxation dynamics can be done with the non-Hermitian Hamiltonian

iΓ H = H − X (|10Xih10X| + |01Xih01X|) . (3.16) Γ 2 10 10 01 01

We can project this Hamiltonian onto the subspace PI, such that one obtains an eﬀective ˜ (I) two-level system Hamiltonian, HΓ (z). This Hamiltonian is also non-Hermitian, with

matrix elements that depend on the relaxation rates, ΓX, due to spatially direct exciton recombination. Explicitly

˜ (I) ˜ (I) ˜ (I) ˜ (I) HΓ (z) = H0 + HRe (z) + iHIm(z) . (3.17)

˜ (I) This Hamiltonian is equivalent to Eq. 3.11 for ΓX = VF = 0. The diagonal part of HΓ (z) is given by,      δ10 + ∆S 0  H˜ (I)(z) =  01  , (3.18) 0    0 δ10 − ∆S  01 with a real perturbative part given by   1  α(z) υ(z)  H˜ (I)(z) =   , (3.19) Re 2 − 2   ΓX + 4(z ∆)  υ(z) α(z) 

and imaginary part given by   1  β γ  H˜ (I)(z) =   , (3.20) Im 2 − 2   ΓX + 4(z ∆)  γ β 

where 2 2 2 2 α(z) = 4(te + th)(z − ∆), β = −2(te + th)ΓX (3.21)

υ(z) = 8teth(z − ∆), γ = −4tethΓX . ˜ (I) The Hamiltonian, HIm, contains two terms describing relaxation processes, a term proportional to β and an off-diagonal term proportional to γ. Here, β accounts for the intrinsic relaxation rate of dressed indirect excitons, which is different from zero when we 70 have either non-zero electron or hole tunneling. This term is the leading contribution to the relaxation rate, and is non-zero for all the electric field sweep range, within and outside the projected qubit subspace PI. On the other hand, γ describes dephasing due to sequential tunneling of an electron and a hole, and is interpreted as an interference term due to the simultaneous paths followed by the tunneled charges. The dephasing term is much smaller than β as long as te th, and its effect is mostly dominant inside the qubit subspace PI.

The relaxation rates of the dressed indirect exciton states inside PI, are obtained by diagonalizing the Hamiltonian (3.17). This rate is given by

˜ ˜ (I) ˜ ˜ ΓD(z) = Im[Diag(HΓ )] j j = ΓI(z) + ΓΦ(z) , (3.22)

for each indirect exciton | ji, where

1  !4 2 2 2 2 2 2 2  4  δ10 − δ01 8te th(32te th − (δ10 − δ01 + 2∆S ) (ΓX − 4(z − ∆) )) Γ˜ (z) =  01 10 + ∆ + 01 10  Φ  S 2 2 2  2 (ΓX + 4(z − ∆) ) × sin θ(z) .

with  !2   16teth   (∆ − z) ΓX  −1  δ10 −δ01 +2∆S  tan  01 10  θ(z) =   .  !2  2   (Γ2 + 4(z − ∆)2)2 − 8teth (Γ2 − 4(z − ∆)2)  X δ10 −δ01 +2∆S X  01 10 whereas 2 2 2(t + t )ΓX Γ˜ (z) = e h . (3.23) I 2 2 ΓX + 4(z − ∆) 71

Figure 3.13: Relaxation times for bare and dressed indirect excitons as function of applied electric ﬁeld F. Solid (dashed) line corresponds to relaxation time for the dressed (bare) indirect exciton eigenstates. Far from anticrossing value of electric ﬁeld FI = 2.38kV/cm, the dressed relaxation time approaches the bare exciton relaxation time as molecular ˜ (I) 01 10 indirect eigenstates of HΓ become the indirect states |10Xi, |01Xi. Juan E. Rolon and Sergio E. Ulloa, Phys. Rev. B 82, 115307 (2010). ©(2011) American Physical Society.

˜ −1 Figure 3.13 shows the electric ﬁeld dependence of the relaxation times, ΓI (z) ˜ −1 (dashed line) and ΓD (z) (solid line). For values of F far from the purely indirect exciton ˜ −1 ˜ (I) anticrossing shown Fig. 3.6, the relaxation time, ΓD (z), of eigenstates of HΓ (z), ˜ −1 asymptotically approaches the relaxation time of the dressed indirect excitons, ΓI (z), while on this regime the dephasing time, Γ˜ Φ, is signiﬁcatively suppressed. At

FR = 2.3kV/cm, the eigenstate relaxation time has a minimum of ∼ 22.2 ns. Remarkably, ˜ −1 ΓI (z) gives the lifetime of the qubit computational basis states for a broad range of values of F, ranging from the initialization regime at FI = 43.4kV/cm, up to the qubit rotation regime at FR = 2.3kV/cm, acquiring value of 214ns and 24.5ns, respectively. This is 72 physically sound and it is an important result since it matches the numerical solutions of the Lindblad master equation shown on Fig. 3.12 with that those obtained from the analytical expressions.

3.7 Discussion and Concluding Remarks

3.7.1 Stability of the Control Scheme

The qubit control scheme we have just described requires a purely indirect-exciton anticrossing to be spectrally isolated from other exciton manifolds and molecular resonances taking place out of the qubit subspace. We can choose diﬀerent QDM structural parameters, such that coherent rotations occur in an isolated region within a very narrow spectral window (≈ µeV) suﬃciently detuned from other transitions. Since

10 01 such anticrossings occur even if δ10 , δ01, the qubit subspace is protected, with this protection being manifest in the almost ideal coherence of the Rabi oscillations within PI. In the case of pumping strong excitation power, other nearby states can also be populated, such as those associated with excited electron or hole levels. These additional excited states can appear in the vicinity of the qubit window, but their effects can be naturally incorporated in the theoretical description. This added manifold would change the value of the field at which initialization is performed, but for the same reasons exposed above, it would not to affect the fidelity of the qubit rotations. On the other hand, strong distortion of the control anticrossings in Fig. 3.6, as result of molecular mixing with nearby states, would certainly affect the rotation and initialization fidelity. We can understand those undesired effects from the fact that adiabatic elimination of such states would not be possible; in other words, the two-level system approximation would not be valid. Different non-ideal behavior could also arise from charge tunneling into the Schottky contacts, providing a competing relaxation rate that limit the control time scales. In this case, charged excitons would not be easily eliminated adiabatically. However, this non-ideal 73 behavior can be suppressed by optimization of the QDM structural parameters tailoring the exciton spectrum and optical pumping conditions.[25, 26]

3.7.2 Conclusions

In summary, we have shown that the optical signatures of a QDM can be used to identify regions of the exciton spectrum where qubit subspaces can be defined. In our case, interplay of optical pumping and interdot charge tunneling resulted in signatures identifiable with a purely indirect exciton qubit subspace. In our model, the QDM was treated as an open quantum system, with relaxation rates arising from spontaneous decay of spatially direct excitons. We found however, that indirect exciton qubits are protected to an extent from spontaneous recombination, due to suppression of the optical interaction between the qubit and the laser field acting as a reservoir. Remarkably, we were able to initialize the qubit with near unity fidelity via tunneling mediated virtual transitions, which provided an effective oscillator strength for indirect excitons. In our control scheme, realized with an adiabatic electric field pulse, the input state was shelved for times longer than the spontaneous recombination time. The rotated qubit final state was also shelved and transported adiabatically into a tunneling induced anticrossing. This adiabatic passage transferred all the indirect exciton population into a direct state, which subsequently relaxed into the vacuum, providing a mechanism for optical read-out of the qubit final state. The high degree of controllability of indirect excitons was achieved via controlled suppression of undesired transitions into states outside the qubit subspace. Our scheme has an advantage over qubits defined via spatially direct excitons, because it enables control over the duration of the indirect exciton relaxation times, either using purely electrical or purely optical means, or a combination of both.[17, 18, 77, 78] Moreover, it opens the possibility of implementing spatially indirect neutral excitons as ancillary qubits in scalable quantum computation using quantum dot molecules. 74 4 Modeling Electrical Control of Indirect Exciton

Relaxation Rates in Quantum Dot Molecules

4.1 Introduction

As we have discussed in previous chapters, coherent control of exciton states in quantum dots and molecules offers an excellent platform for ultrafast switching devices.[79–81] A crucial paradigm inherent to excitons is the tunability of their recombination lifetimes. In one hand, spatially direct states typically possess recombination lifetimes[2, 43] ranging from tens of picoseconds (ps) up to a nanosecond (≈ 1ns), which makes them impractical for implementing reliable switching operations. On the other hand, spatially indirect excitons have typically long lifetimes ranging from a few to hundreds of nanoseconds[74] due to their small electron-hole wave function overlap, which also results in a large built in dipole moment ed, where d is the effective separation between the electron and the hole layers, which is roughly the interdot separation in a quantum dot molecule. In a layered nanostructure the indirect exciton energy can be electrically controlled by its Stark shift ∆S = eFzd, where Fz is an applied electric field perpendicular to the growth layers. This energy tunability precisely enables control of the exciton radiative recombination rates and lifetimes. However, in quantum wells and dots, hot photo-excited excitons thermalize via emission of longitudinal acoustic (LA) phonons, a relaxation channel that is much more efficient in confined nanostructures in comparison to bulk semiconductors.[82–84] Since indirect excitons couple weakly to the light field, it is expected that phonon induced exciton relaxation would be a much more important mechanism for indirect exciton dephasing. Seminal work by Bockelmann and Bastard [85, 86], pointed out the importance of phonon mediated relaxation between discrete electronic states in quantum dots, predicted to take place over time scales comparable to the spontaneous recombination lifetime of direct excitons. In particular, for 75 interlevel energy separations of a few meV, acoustic phonons have been confirmed as the dominant source of carrier relaxation and exciton decoherence, a typical situation realized in single quantum dots and quantum dot molecules. Therefore, QD and QDMs based devices require the characterization and controllability of phonon induced relaxation rates, if optimal performance is desired. Recent theoretical and experimental evidence have suggested that acoustic phonon induced exciton relaxation can indeed be controlled by electrical means.[75, 76, 87, 88] In quantum dot molecules, an applied external electric field is capable of tuning both the energy separation of electron and hole levels, as well as their spatial wave function extension (delocalization) over the quantum dots comprising the molecule, resulting in the controllability of the scattering matrix elements and structure factor of the exciton wave functions.

4.2 Oscillatory Relaxation Rates of Indirect Excitons in InGaAs/GaAs QDMs: Experimental Evidence

Recent experiments carried out by K.C. Wijesundara et. al, in Prof. E.A. Stinaff’s laboratory at Ohio University, have observed oscillatory lifetimes and PL intensity modulations of indirect excitons in InAs/GaAs quantum dot molecules as function of an applied external electric field, see Fig. 4.1.[88] In these experiments, the indirect exciton lifetimes were tuned from 0.3 up to 2.0 ns, as the spatially indirect and direct exciton relative energy separation, ∆E, was varied from 0.8 up to 19 meV. The observed behavior in Fig. 4.1(a) can be interpreted as a variable exciton non-radiative relaxation rate which is slow at minimum exciton energy separation while oscillatory with increasing applied field, a behavior partially explained by fast efficient photoluminescence spike in Fig. 4.1(c) and its subsequent fluctuations. At intermediate energy separations, radiative recombination seems to stabilize as indicated by the PL plateau between 3 ≤ ∆E ≤ 9meV; 76 however in the same energy range the indirect exciton lifetime decreases reaching a minimum then increasing again.

Figure 4.1: (a) Lifetime measurements of spatially indirect excitons on QDMs samples with nominal interdot separation d ∼ 7nm. The lifetime as function of the indirect-direct exciton energy separation, oscillates with a local minimum at ∼ 6-8meV and a local maximum at 10 ∼ 12meV (b) Level anticrossing PL intensity of the spatially indirect exciton, 01X and 10 + direct exciton 10X. The region labeled with the X symbol indicates the energy window 10 for which PL arises from a positively charged trion. (c) 01X normalized PL intensity. K.C. Wijesundara, et al., Phys. Rev. B 84, 081404(R) (2011). ©(2011) American Physical Society.

The observed behavior on Fig. 4.1 suggests the strong activation of phonon mediated non-radiative relaxation channels, via variations of exciton energy and carrier wave 77 function confinement within the QD, which traduces in scattering matrix modulation as carriers delocalize over the QDM. On the other hand for ∆E ≥ 12meV, charging effects promote the formation of a positive trion X+, see Fig. 4.1(b), such that the second peak of the lifetime is probably promoted by hole capture from the Schottky contacts. For this particular setting, the electron is mostly localized in the bottom QD, while it is the heavy-hole component which suffers delocalization, leading to the small anticrossing observed at ∆E = 0.8meV in Fig. 4.1(b).

4.3 Phonon Assisted Exciton Relaxation

In what follows, we develop a model for acoustic phonon induced relaxation of indirect excitons in quantum dot molecules. We consider predominantly the effect of interaction with longitudinal acoustic (LA) bulk phonons inside the QD material. We leave aside scattering due to longitudinal optical (LO) phonons on the basis of energy-momentum conservation, although an important effect in other situations, for the particular QDM samples considered here the exciton transition energies are highly detuned with respect to the LO phonon energies. We neglect interface effects due to the different mass compositions of the ions in the lattice, such that confinement energies of phonons are assumed much smaller than carrier confinement energies. Moreover, for strongly confined carriers in QDs this approximation is well justified, since the GaAs host matrix which acts as tall barriers along the growth z-direction, posses nearly equal lattice properties as the InAs QD wells. In this sense, LA phonon modes are not affected by interface effects and propagate through the QDM as if they were bulk-like.

4.3.1 Carrier Phonon Scattering Interaction Mechanisms

Let us consider acoustic phonons with linear dispersion Eµ(q) = ~cµq, where µ is a subindex indicating the phonon polarization µ = {LA, TA, }, such that q = |~q|, where ~q = (q cos φ sin θ, q sin φ sin θ, q cos θ) is the phonon momentum wave vector represented 78 in spherical coordinates. The calculation of the scattering probability of an electron or hole, requires knowledge of the carrier-phonon interaction Hamiltonian, whose perturbative effect promotes carrier-phonon assisted transitions. This Hamiltonian can be written as follows X ˆ e(h) ˆ i~q·~r ˆ† −i~q·~r Hh(e)−ph = Mν (~q)(be + b e ) , (4.1) ν~q e(h) where Mν (~q) are the scattering matrix elements corresponding to the electron (hole) relaxation channel ν, and bˆ, bˆ† are the phonon annihilation and creation operators. Here ν = µη, with η = {DP = Deformation Potential, PZ = Piezoelectric} indicates the nature of the scattering potential. Scattering due to deformation potential (DP) is expected to dominate charge relaxation in vertically coupled self assembled QDs, while piezoelectric (PZ) scattering[89] has been demonstrated to be more relevant in spin relaxation[90, 91] for laterally coupled QDs [92, 93]; moreover, PZ scattering is also weaker for small interdot barrier thickness.[76] However, since the tunneling energy in our system is controlled by an external electric field, we shall expect PZ scattering to have a moderate contribution specially near the hole tunneling anticrossing. In calculating the time decay of optical excitation, one can consider two types of processes, virtual and real transitions[83]. In the first case, the initial and final states correspond to the same exciton level, and as such these processes do not change exciton level population, resulting in pure dephasing involved in long time decay of optical polarization and coherence. On the other hand, real transitions change the exciton state occupation, a process which leads to exciton population decay. All these processes are adequately described by Fermi’s golden rule,

2π X 1 1 τ−1 = |hΨ |W |Ψ i|2δ(E − E ± E ) × [n (E , T) + ± ] , (4.2) i→ f f ν i f i ~q B ~q 2 2 ~ ν~q 79 where nB is the Bose-Einstein distribution which accounts for thermal population of phonons at the lattice temperature T, and Wν is the carrier phonon interaction given by

±i~q·~r Wν = Mν(~q)e . (4.3)

The minus and plus sign in Eq. 4.2 corresponds to emission and phonon absorption, respectively. The sum extends over all phonon momenta ~q and over all polarization and scattering mechanisms described above. In our case, we consider emission of single phonons, thus phonon absorption and multi-phonon eﬀects are neglected; in this setting, we work at zero temperature. The most important contribution to possible non-trivial

−~q·~r features in the relaxation rates arises from the transition matrix elements hΨ f |e |Ψii.

01 When calculating the relaxation rate of the indirect exciton, |10Xi, into the lower energy 01 direct exciton state, |01Xi, it is important to consider that the electron is mostly localized in the bottom dot.[94] Therefore, the scattering rate is eﬀectively the rate associated for relaxation of the (higher energy) anti-bonding hole state into the (lower energy) bonding state. 80

Figure 4.2: Simpliﬁed electron and hole energy level diagrams. Holes (red) hop between dots with tunneling strength th, whereas electrons (blue) are mostly conﬁned in the bottom dot due to the large energy mismatch, ∆E0, of the top QD conduction band with respect to the bottom QD conduction band. Phonon assisted hole relaxation is indicated by Γph, whereas radiative relaxation, Γrad, occurs via direct state recombination in the bottom QD. The dashed lines indicate the electron hole pairs leading to exciton formation.

Figure 4.2 shows schematically the exciton energy level diagram, interdot couplings, and relaxation channels for this case. Here, te, th are the electron and hole tunneling strengths, and although te th, the electron tunneling rate is much slower due to the large relative detuning, ∆E0(F), of the electron levels in separate QDs. On the other hand, the hole levels become resonant as we tune the electric field, as indicated by the hole tunneling anticrossing in Fig. 4.1(b). Therefore, the two competing relaxation rates considered here are radiative recombination, Γrad, and hole-phonon scattering rate, Γph, calculated according to Eq. 4.55 and Eq. 4.31, respectively, as we describe in the following section. In the following discussion we shall consider two regimes, the first is concerned with indirect-direct exciton energy separations of the order of the tunneling splitting near the anticrossing (near ∆E = 0.8meV) in the PL LACS map of Fig. 4.1(b), while the second 81 regime would be concerned with higher energy separations away from this anticrossing. Since the energy separation between the initial (indirect exciton) and final state (direct exciton) fixes the energy (frequency and wavelength) of the emitted phonons, these two regimes would correspond to distinct scales of the emitted phonon wavelengths, such that for a fixed value of q, the distinct scattering channels would result in distinct contributions in these two distinct regimes.

4.3.2 QDM Simpliﬁed Model: Electron and Hole Wave Functions

Let us consider a simplified effective model for InGaAs/GaAs quantum dot molecules with cylindrical symmetry under an applied axial electric field. The quantum dot vertical and lateral confinement potentials are modeled as narrow quantum wells along the growth and lateral directions matched to parabolic potentials at the valence and conduction band edges. Since the lateral size has a large experimental uncertainty, we have decided to interpret its size in terms of the wave function confinement length of the ground state single particle solutions wavefunctions. 82

Figure 4.3: Model schematics. (a) InAs QDM consisting of two QDs (bottom B and top T) with cylindrical symmetry separated by a distance d = d˜ + (hB + hT )/2 from their centers, having widths lB and lT , respectively; here d˜ is the sharp edges GaAs barrier width (not shown). (b) Parabolic potentials are matched to the square well conﬁnement potential band edges, Ve(h). The lowest energy eigenstates correspond to the gaussian ground state of the harmonic oscillator problem.

In Fig. 4.3(a) we show a schematic view of the QDM considered in this model, here the interdot separation, d, is the distance between the two middle points in each QD, whereas the in plane QD lateral sizes are given by lT , lB and the corresponding QD heights by hT , hB for the bottom (B) and top (T) QDs. In Fig. 4.3(b) we show the parabolic potentials ﬁtted to the band edges, with the matching being done for both conduction and valence bands. The single carrier ground state wave function is given by the lowest energy

e(h) solution φ0 to the parabolic potential problem, which in this case corresponds to the envelope part of the total wave-function.[16, 18, 95] 83

The Schrodinger¨ equation corresponding for the envelope wave functions is as follows,  2  − ~ ∇2  He(h)φe(h)(~r) =  ∗ + Ve(h)(~r) φe(h)(~r) = Ee(h)φe(h)(~r) , (4.4) 2me(h) where the parabolic potential Ve(h) is given by

1 1 V(~r) = m∗ ω2 r2 + m∗ ω2 z2 (4.5) 2 e(h) e(h)k k 2 e(h) e(h)z 2 2 2 r = rk + z . (4.6)

The eigenfunctions of He(h) in Eq. 4.4 are separable in all coordinates and well known. For

each coordinate x j they can be typically written as

φe(h)(~r) = ψe(h)x(x)ψe(h)y(y)ψe(h)z(z) (4.7)

1 ! 2 1 2 x − x 2d2 ψn(x) = Hn( )e x , (4.8) n 1 n!2 dxπ 2 dx

2 ~ where d = ∗ , is the squared wave function conﬁnement length along the x-direction. x m ωx For ground state excitons, we assume wave-function solutions to the single particle problem,

1   2 !  1  x2 y2 z2 φ (~r) =   exp − − − (4.9) e(h)  3  2 2 2 π 2 dxdydz 2dx 2dy 2dz   1 2  2 2   1   rk z  =   exp − −  , (4.10)  3   2 2   2 2   2d 2d  π dk d⊥ k ⊥

where dk = dx = dy and d⊥ = dz. Let us focus our attention on the hole ground state wave functions. The ground state simpliﬁed orbitals are centered at the middle of each quantum dot, with the origin placed at midpoint of the B QD, such that

φB(~r) = φh(~r) (4.11)

φT (~r) = φh(~r − dkˆ) , (4.12) 84 here d is the separation of the two orbitals which corresponds to the eﬀective interdot

˜ hB hT separation d = d + 2 + 2 . Therefore, the ground state solutions for holes conﬁned at each QD are given by, 2 −rk exp ( 2d2 ) B(T) kB(T) φ (rk) = √ (4.13) dkB(T) π

−z2 −(z−d)2 exp ( 2 ) exp ( 2 ) B 2d⊥B T 2d⊥T φ (z) = q √ φ (z) = q √ (4.14) d⊥B π d⊥T π where we have separated explicitly the in-plane and axial components of the wave function. In order to estimate the length scales of the wave function conﬁnement lengths in connection to the in-plane and vertical QD quantum well potentials, we notice that the

1 2 well depth, Vi, should match the parabolic potential, 2 ci jr , at the band edge,

1 lw j 2 2 ci j( 2 ) = Vi, with lw j being the width of the quantum well along the j-th direction and ∗ 2 2 √ ~ i = e, h and j = x, y, z. Since ci j = mi ωi j and d j = ∗ , this results in mi ci j s 2l2 2 ~ wk dk = ∗ (4.15) 8mi Vk s 2 2 2 ~ lw⊥ d⊥ = ∗ , (4.16) 8mi V⊥ 2 2 2 where rk = x + y , and dkB(T) ≤ l(B)T , d⊥B(T) ≤ hB(T) are the characteristic conﬁnement lengths of the wave functions along the vertical and lateral directions, and bounded by the QDs physical dimensions.

4.3.3 Uncorrelated Excitons

The exciton wave functions are constructed by direct products of bonding and anti-bonding states of holes and electrons,

           T     T   a11 a12   ψ  b11 b12 ψe  XeBeT =    h  ⊗     (4.17) hBhT            B     B a21 a22 ψh a21 a22 ψe 85 where the hole wavefunctions are written as

B(T) B(T) B(T) ψe(h) = φ (rk)φe(h) (z) (4.18)

with the coeﬃcients ai j, bi j obtained by diagonalization of the single particle Hamiltonians for holes and electrons, Hˆ h, Hˆ e,      h −th  Hˆ =   , (4.19) h   −th h + ∆S (F)      e −te  Hˆ =   , (4.20) e   −te e + ∆E0 − ∆S (F) where h, e are the hole, electron conﬁnement energies. The eigenvalues of each Hamiltonian are given by,

1 q E± = + ∆ (F) ± ∆2 (F) + 4t2 , (4.21) h h 2 S S h

1 q E± = + (∆E − ∆ (F)) ± (∆E − ∆ (F))2 + 4t2 , (4.22) e e 2 0 S 0 S e

and without loss of generality, we can shift the zero energy, such that both h, e = 0, when calculating the eigenvalue spectrum as function of the field. On the other hand, the mixing coefficients that define the bonding and anti-bonding normalized states of the hole are given by the eigenvector coefficients of Eq. 4.19. Explicitly,         a11 1 (∆S (F) + ∆Eh)/th   =   , (4.23)   h     A+   a12  2          a21 1 (∆S (F) − ∆Eh)/th   =   , (4.24)   h     A−   a22  2  where q 2 2 ∆Eh = ∆S (F) + 4th , (4.25) 86 is the energy separation between the bonding and anti-bonding states of the hole, under applied bias, which is in turn the energy separation, ∆E, between the spatially indirect and direct exciton states, as the electron is primarily localized in the B QD, and where

− 1 h 2 2 2 A± = 4 + (∆Eh ± ∆S (F)) /th , (4.26)

are normalization constants. For electrons we obtain formally equivalent expressions,         b11 1 ((∆E0 − ∆S (F)) + ∆Ee)/te   =   , (4.27)   e     A+   b12  2      b  ((∆E − ∆ (F)) − ∆E )/t   21 1  0 S e e   = e   , (4.28)   A−   b22  2  where q 2 2 ∆Ee = (∆E0 − ∆S ) + 4th , (4.29) − 1 e 2 2 2 A± = 4 + (∆Ee ± (∆E0 − ∆S )) /te . (4.30)

In ﬁgure 4.4(a) we present the energies associated to hole and electron bonding and anti-bonding states. Two molecular anticrossings are evident from the spectrum, a hole molecular anticrossing with gap ∆h = 2th ' 0.8meV occurs at zero relative electric ﬁeld,

F = 0, or at ∆E = ∆Eh = 0.8meV, while the corresponding to the electron occurs highly detuned at electric fields beyond experimental accessibility at about F ' 85kV/cm. On the other hand, 4.4(b) shows the localization or mixing eigenvector coefficients associated to the molecular energies, we can readily see that the hole delocalizes over a very narrow window of applied field (or Stark shift energy) near F = 0, and for moderately higher fields the hole localizes in the top QD, which causes the formation of the indirect exciton

10 state 01X. Interestingly, despite assuming a signiﬁcantly higher value for the electron tunneling te ' 20meV, while ﬁxing the initial electron level detuning in the B QD at 87

∆E0 ' 60meV, the occupation of the electron in the top QD is ' 10% at F = 0, as

2 measured by b21, suggesting that indeed the electron is almost completely localized in the B QD.

Figure 4.4: (a) Energy eigenvalues as function of applied electric field F for the single particle effective Hamiltonians, for holes as in Eq. 4.21 (solid red) and electrons as in Eq. 4.22 (dashed blue). (b) Mixing coefficients for bonding and anti-bonding eigenstates of 2 2 2 2 2 2 2 2 holes, a11 = a22, a12 = a21 (solid red) and electrons, b11 = b22, b12 = b21 (dashed blue). See numerical values in appendix table A.5. 88

4.3.4 Phonon Induced Hole Scattering Rates

We consider acoustic phonon assisted transitions from an initial anti-bonding state of

+ T B − T B the hole Ψh = a11ψh + a12ψh to the ﬁnal bonding state Ψh = a21ψh + a22ψh . The relaxation rate is calculated using the Fermi golden rule in Eq. 4.2,

2π X h 2 −i~q·~r 2 γph = |Mν (~q)| |hΨ−|e |Ψ+i| δ(∆Eh − ~cs|~q|) . (4.31) ~ ν~q

3 R P → P Ωc 3 The relaxation rate calculation involves the sum (integral) ν~q ν (2π3) d ~q, which is carried out in spherical coordinates ~q = (q cos φ sin θ, q sin φ sin θ, q cos θ), where Ωc is the crystal volume, and (θ, φ) are the (polar,axial) angles, respectively. Since we consider a QDM with axial symmetry, any contribution from integrations over φ amounts to a constant factor of 2π. The scattering rates depend upon the direction of the phonon wave vector through the scattering matrix elements, Mν, and the structure factor

−i~q·~r 2 |hΨ−|e |Ψ+i| . On the one hand, the scattering matrix elements for different scattering channels depend differently upon the magnitude and individual components of the phonon wave vector, these differences shall result in qualitative differences on relaxation arising from distinct channels. On the other hand, it is the structure factor which contains the phase relationship between the hole wave function and the phonon wavelength leading to possible interference effects. In order to compute the structure factor we calculated the following transition matrix elements,

−i~q·~r B −i~q·~r B T −i~q·~r B B −i~q·~r T hΨ−|e |Ψ+i = a11a21hψh |e |ψh i + a22a11hψh |e |ψh i + a21a12hψh |e |ψh i

T −i~q·~r T + a22a12hψh |e |ψh i

= IBB + ITB + IBT + ITT . (4.32)

In a sense, the terms in Eq. 4.32 represent the Fourier transform of the probability density

of the hole on each QD, IBB + I + TT, plus the Fourier transform of the gaussian orbitals 89 overlap, ITB + IBT . Upon integration over the spatial coordinates one obtains,

2 2 d q d2 q2 − kB k − ⊥B z IBB = a11a21e 4 e 4 (4.33)

2 2 d q d2 q2 kT k ⊥T z − − −iqzd ITT = a22a12e 4 e 4 e (4.34)

2 2 D q D2 q2 − kBT k − ⊥BT z −i qzd Iover = IBT + ITB = 2(a22a11 + a21a12) fBT e 2 e 4 e 2 , (4.35)

where qk is the in-plane component of the phonon wave vector, DkBT and D⊥BT are the eﬀective perpendicular and in-plane conﬁnement lengths weighted over the whole

molecule, and fBT is a factor that decays exponentially with the interdot distance, explicitly

2 2 2 qk = qx + qy (4.36)

dkBdkT DkBT = 2 2 (4.37) dkB + dkT d⊥Bd⊥T D⊥BT = 2 2 (4.38) d⊥B + d⊥T s ! d d d d d2 kB kT ⊥B ⊥T − fBT = 2 2 2 2 exp 2 . (4.39) dkB + dkT d⊥B + d⊥T 4D⊥BT

Once we have integrated over all spatial coordinates, we can write the structure factor by taking the squared module of the transition matrix elements in Eq. 4.32,

−i~q·~r 2 2 2 2 ∗ ∗ ∗ |hΨ−|e |Ψ+i| = |IBB| + |ITT | + |IBT | + 2<(IBBITT ) + 2<(IBBIBT ) + 2<(ITT IBT ) , (4.40) 90

Simpliﬁcation of each one of the terms above leads to the following expressions,

2 q 2 2 2 2 2 2 − (d sin θ+d⊥ cos θ) |IBB| = |a11a21| e 2 kB B (4.41)

2 q 2 2 2 2 2 2 − (d sin θ+d⊥ cos θ) |ITT | = |a12a22| e 2 kT T (4.42)

D2 sin2 θ+D2 2 kBT ⊥BT 2 2 2 2 −q ( 2 cos θ) |IBT | = 4|a11a22 + a21a12| fBT e (4.43) − 1 2 2 2 2 2 2 2 ∗ 4 q (dkB+dkT ) sin θ+(d⊥B+d⊥T ) cos θ 2<(IBBITT ) = 2a11a21a12a22e cos (qd cos θ) (4.44)

 2 2 2   dk d +D  − 1 q2( B +D ) sin2 θ+( ⊥B ⊥BT ) cos2 θ ∗ 2  2 kBT 2  2<(IBBIBT ) = 4a11a12(a11a22 + a21a12) fBT e ! qd × cos cos θ (4.45) 2  2 2 2   dk d +D  − 1 q2( T +D ) sin2 θ+( ⊥T ⊥BT ) cos2 θ ∗ 2  2 kBT 2  2<(ITT IBT ) = 4a12a22(a11a22 + a21a12) fBT e ! qd × cos cos θ (4.46) 2 91

Figure 4.5: Structure factor as function of in-plane qk, axial qz phonon wave vectors, and different polar angles θ, for fixed applied electric field F = 0 and interdot distance d = 7nm. The structure factor exhibits prominent resonances mostly along the qzd axis, and these resonances shift towards higher values of qzd for increasing θ. In a sense for the range of energies used in the experiment of Fig. 4.1, only phonons emitted closely to the z-axis contribute the most to the relaxation rate. See numerical values in appendix table A.5.

According to Eqs. 4.35-40, the structure has a strong dependence upon the magnitude q 2 2 of the phonon wave vector, q = qk + qz , and the polar angle θ. When calculating the phonon relaxation rate in Eq. 4.31, the energy-momentum conservation constraint, (via the Dirac delta function) selects all angles and phonon wave vectors satisfying the following conditions,

∆Eh q = qk = q sin θ qz = q cos θ . (4.47) ~cs 92

The condition above indicates that the frequency, ωq = csq, of the emitted phonon is controlled by the field dependent tunneling energy, ∆Eh, (relative energy separation of the indirect and direct states, ∆E), which is minimal at the hole tunneling anticrossing. In this sense, phonons would be emitted with the largest wavelength at low relative exciton energy separations. Notice that the sound velocities, which are fixed by the QD alloying composition and strain effects, determine the energy scale and produce a shift of the emitted phonon frequencies and the structure factor resonances, accordingly.

Figure 4.6: Structure factor contour plot as function of energy separation and phonon axial wave-vector, for ﬁxed d = 7nm. The dotted purple line indicates the phonon wave vector dispersion as function of energy separation. The relaxation rates are strictly non-zero for allowed phonon wave vectors satisfying Eq. 4.47, in other words only at the intersection points between the purple line and the contour points. Resonances are prominent at qzd = π and qzd = 3π which correspond to an energy separation of ∆E = 2.6meV and ∆E = 6.2meV, respectively. See numerical values in appendix table A.5. K.C. Wijesundara, et al., Phys. Rev. B 84, 081404(R) (2011). ©(2011) American Physical Society. 93

It is clear from the results mentioned above that the interdot distance and the phonon sound velocities play a critical role in fixing the phase relationship between the hole wave function and the phonon plane wave. Remarkably, for small tunneling energies it is the plane wave component along the z-direction which determines most of the oscillatory behavior of the structure factor. This result mean that phonons would be emitted prominently close to the z-direction, as the sweeping of the different phonon momenta qz takes place by the variation of the relative energy separation ∆E. As the structure factor is a non-monotonic function, it results in oscillations of the relaxation rates, see Fig. 3.12.A clear phase relationship is demonstrated by both Figs. 4.5 and 4.6, with maxima occurring at odd multiples of π. In these figures, the structure factor is shown as function of the axial and in-plane phonon wave vectors, qz and qk, evaluated at different polar angles θ. For

θ = 0, the structure factor exhibits three resonances within a range 0 ≤ qzd ≤ 6π, which corresponds to the indirect-direct exciton energy separation, 0.8 ≤ ∆E ≤ 18meV, accessible to the experiment. It is in this regime where the resonances occur almost

|h | −i~q·~r| i|2 ∼ 2 qzd 1−cos (qzd) exactly at odd multiples of π, which implies that Ψ− e Ψ+ sin ( 2 ) = 2 , 2πd and has maxima for qzd = ( ) ∼ (2 j + 1)π or d ∼ ( j + 1/2)λq. For non-zero polar angles, λq

θ = 0.5 and θ = 1, the resonances start shifting to higher axial phonon momenta qz, while

most of the allowed in plane phonon momenta, qk, start to lack the proper Fourier ∼ π components necessary to allow constructive interference. Finally, at θ 2 , all resonances are shifted out of the experimental range of accessible energy detuning ∆E, which

corresponds to qzd 6π. In order to complete the calculation of the relaxation rate in Eq. 4.31, we can

h 2 compute separately the scattering matrix elements, |Mν (~q)| . For zinc-blende structures the scattering matrix elements arising from diﬀerent polarizations, and distinct carrier-phonon 94 couplings[76, 87, 96] are given by,

2 h 2 ~Dh |Mν=LA−DP(~q)| = q (4.48) 2ρcLAΩc 2 2 2 2 32e ~π e (3qxqyqz) |Mh ~q |2 14,v ν=LA−PZ( ) = 2 7 (4.49) ρcLAΩcq 2 2 2 2 2 2 2 2 2 2 32e ~π e qxqy + qyqz + qz qx (3qxqyqz) |Mh ~q |2 14,v | − | ν=TA−PZ( ) = 2 5 7 (4.50) ρcLAΩc q q

Finally, we can write the total relaxation rate as

γph = γν=LA−DP + γν=LA−PZ + γν=TA−PZ . (4.51) where

D2q3 Z π LA−DP h |h | −i~q·~r| i|2 γph = 2 dθ sin θ Ψ− e Ψ+ (4.52) 4π~cLAd 0 18πe2e2 Z π LA−PZ 14,v 5 2 −i~q·~r 2 γ = q dθ sin θ cos θ|hΨ−|e |Ψ i| (4.53) ph 2 2 + ~cLA d 0 8πe2e2 Z π γTA−PZ = 14,v q dθ(sin5 θ + 8 sin3 θ cos2 θ − 9 sin5 θ cos2 θ) ph 2 2 4~cTA d 0 −i~q·~r 2 × |hΨ−|e |Ψ+i| , (4.54)

here Dh is the acoustic deformation potential shift for holes, ρ is the density of the crystal,

Ωc is the crystal volume, e14 is the piezoelectric modulus, is the relative dielectric

constant for InAs, while cLA and cTA are the sound velocities for longitudinal and transversal polarizations. An interesting result derived from the equations above is that the diﬀerent allowed phonon assisted transitions are fully controlled via the tunable Stark shift q 2 2 of the hole energy level, i.e ∆E = ∆Eh = ∆S (F) + 4th. This electric ﬁeld controllability

includes the tunability of the diﬀerent coeﬃcients ai j(th, F), the scattering matrix elements

h 2 −i~q·~r 2 |Mν (q)| and the structure factor, |hΨ−|e |Ψ+i| . We shall discuss now the eﬀects of varying the wave function conﬁnement lengths,

such as dkB(T) and d⊥B(T), and the eﬀects of varying the interdot separation d. Figures 95

4.7(left column panel) show the relaxation rates dependence on the lateral conﬁnement length of the bottom QD, dkB, which alternatively could be pictured as the full width at half maximum (FWHM) of the ground state wave function of a localized hole in the B

QD, fwhmkB ' 2.3548 × dkB. Fig. 4.7 (right column panel) show the dependence on the B QD vertical conﬁnement length. Fig. 4.7(a) shows the deformation potential (LA-DP)

scattering as function of dkB, here each of the peak amplitudes shift weakly as the conﬁnement length increases, clearly the minima are deepen as well, with the second

minima reaching an almost exact anti-phase regime for dkB ≥ 3.5nm (olive color curve), strongly indicating that the scattering rate is suppressed for weak lateral conﬁnements. In contrast, the piezoelectric scattering curves in Fig. 4.7(b) and (c) shift slightly with

variations in dkB, but show no qualitative changes on their monotonic decay behavior as function of ∆E. 96

Figure 4.7: Phonon relaxation scattering channels as function of ∆E = ∆h(F). Left column panel: dependence on B QD lateral confinement length, dkB or lateral FWHM of the hole ground state wave function, fwhmkB. (a) LA-DP channel exhibits oscillations with decreasing amplitude as lateral confinement becomes weak (notice the developing of absolute minimum about δE ∼ 4.25meV). (b) and (c) TA and LA-PZ channels, respectively, exhibit no oscillations and weak dkB dependence. Right column panel: dependence on the B QD vertical confinement length, d⊥B. (d) Oscillations of LA-DP rate are prominent for strong confinements d⊥T ≤ 1.1nm, while vanishing completely for weaker confinements d⊥T ≥ 2.5nm. TA-PZ scattering channels do not exhibit qualitative changes. Other fixed parameters can be found in appendix table A.5. 97

We should notice that LA-PZ scattering is one to two orders of magnitude smaller than LA-DP and TA-PZ scattering in all cases. Moreover, the relaxation rates dependence on the B QD vertical confinement length, d⊥B, as shown in Figs. 4.7(d,e,f). do not exhibit appreciably features for piezoelectric scattering. However, it is the deformation potential channel which is affected the most, and this is reflected in the extension of the shoulders of the relaxation curves at higher relative energy separations for short confinement lengths. At the same time enhancing the oscillatory behavior (black solid curve).

Interestingly, for larger values of d⊥T , the oscillations are blurred away and the overall relaxation rate is suppressed at lower energy separations (not shown). We should point out that although the qualitative behaviors of lateral and vertical conﬁnement lengths are quite distinct, the relative positions of the maxima and minima of the LA-DP relaxation curves depend only weakly on substantial variations of these parameters, with the single peaks in PZ relaxation curves shifting even less. The scattering rates, on the other hand should exhibit a more dramatic behavior as function of the interdot separation d. As seen in Fig. 4.8, all scattering relaxation channels shift their maxima and amplitude with changing interdot separation d. This is in accordance to the results discussed previously regarding the structure factor, for which we

found that the product qzd ﬁxes the periodicity of the oscillations observed in the relaxation rates. Interestingly, in stark contrast to the piezoelectric scattering dependence on the lateral conﬁnement, TA-PZ and LA-PZ scattering exhibit an appreciable broadening and suppression for smaller interdot separations. 98

Figure 4.8: Phonon relaxation scattering channels as function of the field dependent energy separation, ∆E = ∆h(F), for different interdot separations. (a) LA-DP rate: the position of maxima/minima shift to lower energy separation with increasing d while accommodating a larger number resonances. For smaller separations there is a substantial broadening and suppression of oscillatory behavior. (b) and (c) TA and LA-PZ scattering rates: rates are suppressed and exhibit substantial broadening for smaller interdot separations. Inset text in (a) indicates fixed parameters. Other fixed parameters can be found in appendix table A.5.

Combining the results above and using realistic parameters extracted from the experiment, we can obtain model dependent fits to the oscillatory relaxation rates which yield the observed features in Fig. 4.1. The results of our best fit are shown in Fig. 4.9. Here, the total relaxation rate is bounded by ∼ 1.25ns−1 corresponding to the amplitude of the first maximum. The different peaks for all relaxation channels have the same order of magnitude at low ∆E. This is in sharp contrast with previous investigations for electron charge relaxation in weakly confined GaAs QDs [76, 87]. We attribute our results to the 99 large difference between the hole and electron effective mass and deformation potential for valence band holes, both being orders of magnitude less than the corresponding electronic case. On the other hand, in agreement with previous investigations mentioned above, we clearly see that LA-DP scattering dominates and exhibits distinct maxima positions, unlike TA-PZ scattering, since for fixed phonon wave vector q one has distinct energetically allowed transitions due to the difference in sound velocities, and due to the difference in functional dependence on q for these two relaxation channels. Clearly, LA-PZ scattering does not contribute appreciably to the total relaxation rate. It is evident as well, that TA-PZ scattering contributes for low energy separations ∆E (longer phonon wavelengths) and it is rapidly suppressed for ∆ ≥ 5meV. TA-PZ effects are important at low phonon emission energies or weak lateral confinements. Interestingly, our results indicate that hole localization suppresses PZ effects for excitons with prominent indirect character, supporting the idea that the QDs under investigation have both vertical and lateral strong hole confinement energies. 100

Figure 4.9: Phonon relaxation rates as function of the ﬁeld dependent energy separation ∆E = ∆h(F). Green shows the total contribution arising from all scattering channels, LA- DP, LA-PZ, TA-PZ. Red shows the calculated radiative recombination rate. The resonance associated with the relaxation rate peak at 6.2 meV is due to maxima of the structure factor in Fig. 4.6 for a scaled phonon momentum wave vector qzd = 3π (green dot in Fig. 4.6). Other ﬁxed parameters can be found in appendix table A.5.

4.4 Radiative Recombination Rates

In order to model the total rate of indirect exciton relaxation and lifetime we need also to calculate its radiative relaxation rate, Γrad. To be consistent in the modeling, we used the relative separation of the indirect-direct exciton transition energies, and the two orbital Hamiltonian in the envelope wave function approximation. In the model, the 101

indirect exciton recombination depends on an eﬀective oscillator strength due to direct exciton recombination[3, 97, 98], and results from both the strong hole tunneling and weak electron tunneling mixing the indirect and direct excitons. This mixing is controlled

by an “overlap” parameter OI and the energy detuning, ∆E, extracted from ﬁtting the eigenvalue spectrum of Eq. 4.19 and Eq. 4.20 to the observed PL spectrum of Fig. 4.1(c). Explicitly, 1 3 2 2 2 E O µ 10X I B 01 , Γrad = 3 (4.55) 3π0c ~

where µB ∼ 6.2eÅ is the inter-band transition dipole moment in the bottom QD,

E10 = 1290.35meV+∆Eh is the transition energy of the indirect exciton, is the dielectric 01X 2 2 constant for InAs and OI = θeh(a12 + b12) is the eﬀective electron hole “overlap” which characterizes the mixing of indirect and direct exciton states through the wave function

coeﬃcients, a12, b12 and the intra-dot electron-hole overlap, θeh. For large energy

0 detunings of the electron level, ∆Ee ∆Eh, the coeﬃcient that characterizes the electron

2 delocalization over the two dots, b12, is negligibly small in the regime of interest. We should remark that the radiative relaxation considered here considers the process of radiative relaxation of the mostly indirect QDM molecular state via the recombination of its direct component, which dominates when the applied electric ﬁeld tunes the energy

near the hole tunneling anticrossing. As expected in the context of our model, Γrad is monotonically decreasing as function of ∆E, as seen clearly in the red solid line of Fig. 4.9. This is compatible with a model in which the indirect exciton recombination quenches both due to decreasing oscillator strengths, (quench of inter-dot recombination) and due to the fact that it loses its mixing with the intra-dot recombining direct exciton as the energy detuning increases. 102

4.5 Discussion and Concluding Remarks

We have demonstrated theoretically that phonon induced oscillatory relaxation rates in hole tunneling QDMs are possible. In analogy to GaAs electron tunneling coupled QDMs, we found that the dominant scattering mechanism is due to deformation potential, in spite of the relatively distinct length scales, confinement and deformation potential parameters in self assembled InAs QDMs. The structure factor characteristics provided a clear picture of the different scattering resonances, which in turn resulted in strong dependence of the indirect exciton lifetimes on the structure parameters of the QDM such as the lateral in-plane confinement potentials, but especially on the interdot distance in the molecule. A key mechanism for the observed oscillatory behavior, arises from the strict phase relationship between the phonon wave vector and the localized hole molecular wave functions. Electric field tunability of the indirect exciton energy provides a way to scan different relaxation rates reaching different maxima or minima, resulting in this way, in the controllability of the indirect exciton relaxation rates or lifetimes. An interesting application of the results found here, is the use of experimentally measured oscillatory relaxation rates, in order to obtain QDM structural parameters that are difficult to measure in-situ, such as the lateral sizes of the QDs.[25, 99, 100] This can be done by extracting the inverse Fourier transform of the measured structure factor, and use it to map the spatial distribution of the electron and hole probability densities in the QDM. Another possible application is the introduction of oscillatory relaxation rates in the model proposed in chapter 3, which concerned with the stability of indirect exciton qubit rotations in a more realistic setting, in which acoustic phonons would play a prominent role on the decoherence processes, especially at finite temperatures, leading to damping of indirect exciton Rabi rotations. In this case, an important question would be how the control scheme needs to be modified in order to accommodate phonon relaxation channels without losing fidelity on qubit rotations, or if the controllability of phonon relaxation 103 leads to an advantageous resource instead to purely detrimental effects. These questions suggests for future investigations which may lead to new interesting results. 104 5 Conclusions and Outlook

Recent progress in nanofabrication and ultrafast spectroscopy has brought near the possibility of implementing integrated arrays of coupled quantum dots in the solid state realization of a quantum computer. Quantum dots offer unique advantageous features due to the high degree of controllability of their discrete energy levels and their coupling to the environment, which can be done by changing their geometry and material composition, or by external manipulation using electrostatic pulses and coherent optical excitation. Electrostatic and optical carrier injection enables the generation of confined excitons, offering an additional active control element. Both, individual carriers and excitonic degrees of freedom can be engineered into an interconnected array of two levels systems that can be harnessed into several realizations of quantum bits (qubits). These realizations include spin memories and ancillary qubits encoded in the charge and spin of electrons, holes, excitons and nuclear spins, as well as photon flying and entangled qubits, all of them coexisting on the same device. Fundamental challenges are still ahead and a great deal of research effort is currently devoted to the fundamental understanding and control of possible quantum coupling mechanisms in QD arrays, as well as the challenging problem of suppressing charge and spin decoherence in these systems. This dissertation was conceived and motivated by these premises. It has the purpose of presenting our theoretical investigations and additional contributions in regard to the use of Forster¨ coupled quantum dots and coherent control of indirect excitons acting as possible qubits, while devoting attention to possible decoherence mechanisms such as radiative and acoustic phonon induced relaxation. Following this direction we investigated possible quantum coupling mechanisms in quantum dot molecules and presented theoretical results for detection of resonant energy transfer processes in self-assembled InAs/GaAs QDMs. We have demonstrated that level anticrossing spectroscopy experiments would be a suitable methodology for FRET 105

detection. In this regard, FRET optical signatures can be detectable in spite of the strong effects of charge tunneling. In level anticrossing spectroscopy, tunnel coupling is controlled by the application of an external electric field, and our proposal is that FRET interdot coupling can be controlled in the same way, and if given suitable experimental conditions, FRET could give rise to identifiable optical signatures. To date Forster¨ interaction mechanisms have been difficult to detect experimentally in self-assembled InAs/GaAS QDMs. In this sense, our simulations might offer an additional guide for experimental set ups devoted to the identification of FRET optical signatures in these systems. A hallmark of our results is the clearly defined split-off exciton spectral doublets which become stronger as an electric field is applied across the molecule (‘away’ from the region where the electronic tunneling is dominant). This signature can be tested in experiments using pump-probe differential transmission spectroscopy or exciton photoluminescence. When developing our numerically simulated level anticrossing spectra, we found ourselves in the need to develop analytical tools in order to identify distinct effective molecular couplings inducing specific optical signatures, as well as regions of the exciton spectrum where qubit subspaces can be defined. In our case, interplay of optical pumping and interdot charge tunneling resulted in signatures identifiable with a purely indirect exciton qubit subspace. In order to make our approach more realistic, the model QDM was treated as an open quantum system, with relaxation rates arising from spontaneous decay of spatially direct excitons. Our results showed that indirect exciton qubits are protected to an extent from spontaneous recombination, due to suppression of the optical interaction between the qubit and the laser field acting as a reservoir. This subspace protection enabled us to formulate an scheme for qubit initialization with near unit fidelity. Our procedure involves tunneling mediated virtual transitions, which provides an effective oscillator strength for indirect excitons. A coherent control pulse was realized with an 106 electric field adiabatic pulse. In our control procedure, the input state was shelved for times longer than the spontaneous recombination time. The state of the rotated qubit was also shelved using the same pulse and transported adiabatically into a tunneling induced anticrossing. This adiabatic passage enabled coherent population transfer of the indirect exciton into a direct state, such that relaxation of this direct state provides a way of reading-out the rotated indirect exciton qubit state. Thus we found that indirect excitons offer a high degree of controllability using electrical bias pulses and their decoherence is suppressed via adiabatic elimination of undesired transitions. Therefore, indirect excitons offer a great advantage over qubits defined via spatially direct excitons, and this opens the possibility of implementing spatially indirect excitons as ancillary qubits in scalable quantum computation in QDMs. On the other hand, all relaxation and dephasing mechanisms considered in Ch. 2 and Ch. 3 were effectively due to spontaneous recombination and due to the non-idealities of the control pulse. However, recent experiments carried out by K.C. Wijesundara et. al, in Prof. E.A. Stinaff’s laboratory at Ohio University, demonstrated oscillatory lifetimes and PL intensity modulations of indirect excitons in biased InAs/GaAs quantum dot molecules. This presented us with an interesting problem concerning the origin of such oscillatory lifetimes, which they correctly attributed to non-radiative relaxation channels. As proposed in previous works concerning charge and exciton dephasing in quantum dots, longitudinal acoustic phonons may indeed play an important role in the non-radiative thermalization of confined carriers. In this regard, we decided to test such hypothesis for the experimental samples under investigation, by developing an effective model for the structure factor due to phonon scattering with bonding and anti-bonding states of holes. Scattering rates due to standard phonon relaxation channels in semiconductors were performed, such as deformation potential and piezoelectric scattering. The results presented in Ch. 4 were in general agreement with previous investigations concerning 107

GaAs QDs containing weakly confined electrons. A key mechanism for the observed oscillatory behavior was identified as arising from the strict phase relationship between the phonon wave vector and the localized hole molecular wave functions. Our results indicated that oscillatory deformation potential scattering rates are the origin of the lifetime oscillatory behavior seen in experiments, specially at moderately high phonon energies, when holes tend to create molecular states with highly indirect character. On the other hand, piezoelectric scattering rates on the same order of magnitude as deformation potential rates, contributed substantially by emission of long wave length phonons near the regime where holes tended to create excitons with highly mixed direct and spatially indirect character. An interesting application of these results is the use of experimentally measured oscillatory relaxation rates, in order to obtain QDM structural parameters that are difficult to measure in-situ, such as the lateral sizes of the QDs. Ideally one might devise an algorithm to perform an inverse Fourier transform of the measured structure factor, and use it to map the spatial distribution of the electron and hole probability densities distributed across a quantum dot molecule. As a final remark, it should be pointed out that the field of semiconductor non-linear optics, coherent spectroscopy and coherent control is advancing very rapidly. At the present time there is an intense amount of research focused on building a qubit register using solid-state spins. For spins confined in QDs, two pressing problems are: (1) the controllability of spin dephasing due to coupling to acoustic phonons and (2) spin dephasing due to the electron or hole hyperfine interaction with the QD host nuclear environment. A possible direction along the first problem, might consist on investigations of possible non-Markovian effects an the rapid switching between coherent and incoherent regimes during the qubit manipulation dynamics. These effects are the result of the strong coupling of the system to a phonon bath, as suggested by the oscillatory behavior of 108 acoustic phonon induced relaxation rates discussed in chapter4. Reconsideration of exciton qubits protection conditions should be addressed for realistic non-zero effective temperatures. In this case, the temperature dependence should account for the role of system-bath correlation and fluctuations. A model might be formulated in the context of the time-convolutionless (TCL) evolution equation for the non-Markovian excitonic density matrix, which includes the Lindblad regime as the weak system-bath coupling regime. A new control scheme must be investigated, and possess the ability to couple and decuple the system from the phonon bath. In regard to the second problem, which constitutes a research field of its own, a formulation of an all optical coherent control of electron or hole spins confined in QDMs is highly sought. Any realistic model along these lines must include realistic decoherence channels, such as the carrier coupling to the nuclear spin ensemble of the host semiconductor. An interesting problem consist in to investigate the entanglement of two remote spins localized in separated QDs, via the implementation of adiabatic electrical and optical pulses. In place of the neutral charge indirect excitons presented in3, we might use instead trion states confined at each QD, such that the spin qubit sector would be comprised of the lowest energy spin states. To this end, we might apply an adiabatic elimination procedure to extract an effective Hamiltonian for the qubit sector. We might include as well noise effects, which would be expressed in terms of realistic couplings to unwanted states, such as levels in the wetting layer continuum or due to the coupling to the nuclear spin ensemble of the host semiconductor. In this case the dynamics should capture non-Markovian effects due to spin bath memory effects. This effects have been shown to be of importance for spin-spin and spin-bath coupled systems, in the limit of a small number of spins in the bath. To this end, we might employ the Nakajima-Zwanzig master equation, in which the effective two level system is coupled to the bath via a Jaynes-Cummings type of coupling. To this extent, questions shall be formulated in regard 109 to the controllability of such interactions, as well as the fidelity of single qubit rotations or entanglement operations. 110 References

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[100] A. D. Giddings, J. G. Keizer, M. Hara, G. J. Hamhuis, H. Yuasa, H. Fukuzawa, and P. M. Koenraad, Phys. Rev. B 83, 205308 (2011).

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[106] J. Mller and J. Rau, Physics Letters B 386, 274 (1996). 116 Appendix A: Simulation Parameters

Name Symbol Value

Interband transition moments µT(B) 6.2eÅ

Exciton recombination time τX 1ns

Biexciton recombination time τXX 0.5ns Table A.1: Constant parameters used in chapters2 and3.

Name Symbol Value

Interdot distance d 8.4nm

Forster¨ coupling strength VF 0.08meV

Electron tunneling strength te 2.0meV

Hole tunneling strength th 0.1meV Radiation ﬁeld coupling Ω 0.4meV

T QD direct exciton bare energy E01 1248.1meV 01X

B QD direct exciton bare energy E10 E01 +10µeV 10X 01X

Indirect exciton eBhT E10 1263.11meV 01X

Indirect exciton eBhT E01 1267.12meV 10X Table A.2: Parameters used in simulations corresponding to ﬁgures 2.5, 2.6 and 2.7. 117

Name Symbol Value

Interdot distance d 8.25nm

Forster¨ coupling strength VF 0.09meV

Electron tunneling strength te 2.1meV

Hole tunneling strength th 0.12meV Radiation ﬁeld coupling Ω 1.4meV

T QD direct exciton bare energy E01 1248.62meV 01X

B QD direct exciton bare energy E10 1248.79meV 10X

Indirect exciton eBhT E10 1263.49meV 01X

Indirect exciton eBhT E01 1267.60meV 10X

T QD direct biexciton bare energy E02 2492.85meV 02X

Trion like biexciton energy E20 E02 + 17meV 11X 02X

Trion like biexciton energy E11 E20 - 20µeV 02X 11X Table A.3: Parameters used in simulations corresponding to ﬁgures 2.8, 2.9.

. 118

Name Symbol Value

Interdot distance d 8.4nm

Forster¨ coupling strength VF 0.08meV

Electron tunneling strength te 2.0meV

Hole tunneling strength th 0.1meV Radiation ﬁeld coupling Ω 0.75meV

T QD direct exciton bare energy E01 1248.1meV 01X

B QD direct exciton bare energy E10 E01 +10µeV 10X 01X

Indirect exciton eBhT E10 1263.11meV 01X

Indirect exciton eBhT E01 1267.12meV 10X

T QD direct biexciton bare energy E02 2491.69meV 02X

Trion like biexciton energy E20 2508.77meV 11X

Trion like biexciton energy E11 2508.91meV 02X Table A.4: Parameters used for simulations in chapter3. 119

Name Symbol Value

Crystal density InAs ρ 5600 kg/m3

Dielectric constant 12.4

Piezoelectric modulus InAs e14 12.65

Deformation potential Dh 950 meV

3 3 LA phonon speed cLA 7×10 m/s

3 3 TA phonon speed cTA 3.2 ×10 m/s

Hole tunneling strength th 0.43meV

Intradot e-h overlap θeh 1

Electron level detuning ∆E0 60meV

Electron tunneling strength te 20meV

Excitation energy E10 1290.35meV 01X Interdot distance d 7.0nm

Lateral conﬁnement length B QD dkB 1.5nm

Lateral conﬁnement length T QD dkTB 3.5nm

Vertical conﬁnement length B QD d⊥B 2.5nm

Vertical conﬁnement length T QD d⊥T 1.0nm Table A.5: Parameters used in simulations for chapter4. 120 Appendix B: Adiabatic Elimination:Bloch-Feshbach

Algebraic Method

In the calculation of the dynamics of a quantum system one is often confronted with the problem of finding effective interactions relevant to a particular set of states. Such states define the so called model space, while the irrelevant states, which might or not include a continuum, define the orthogonal space. The basic idea of the appendix is to illustrate a powerful partitioning technique, or algebraic adiabatic elimination, which permits the construction of effective operators which act only within the model space, but which generate the same physics as do the operators acting on the full Hilbert space. For example, in open quantum systems, the power of this method is illustrated when trying to find a Hamiltonian describing interactions among the model space eigenstates, such that localized effective Hamiltonian might be non-Hermitian, thus permitting the description of decoherence processes.[101, 102] This technique was originally introduced by Feshbach[58], and independently developed by Bloch[103] during the late 1950s and early 1960s, with subsequent development towards projection operator algebraic methods in perturbation theories[104, 105], and non-perturbative theories such as the renormalization group[106].

B.1 Green’s Functions

Let H = H0 + V be the Hamiltonian of the system, such that H0 and V are its unperturbed and perturbing parts respectively. The evolution operator for a system evolving from the initial time t0 towards a ﬁnal time t is given by U(t, t0), such that

dU i = (H + V)U(t, t0) , (B.1) ~ dt 0

such that U(t0, t0) = 1 . (B.2) 121

For time independent H0 and V, the solutions to the above equation are

Z t 0 0 1 0 U(t, t ) = U0(t, t ) + dt1U0(t, t1)VU(t1, t ) (B.3) i~ t0 0 i 0 U0(t, t ) = exp[− H0(t − t )] (B.4) ~

By imposing boundary conditions on the propagators it is possible to deﬁne the following Green’s functions

0 0 0 K±(t, t ) = ±U(t, t )θ[±(t − t )] , (B.5)

dθ such that θ(τ) is the Heaviside function. Remembering that dτ = δ(τ), with δ being the Dirac delta function, the Green’s functions obey the following evolution equations, ! d i − H K (t, t0) = i δ(t − t0) . (B.6) ~dt ± ~

It is possible to write the Green’s functions in the energy domain by taking the Fourier

transform of K±,

Z +∞ 1 iEτ/~ G±(E) = dτe K±(τ) . (B.7) i~ −∞

−iHτ/ Since K±(τ) = e ~θ(±τ),

Z ∞ 1 i(E−H±iν)τ/~ G±(E) = lim dτe (B.8) ν→0+ i~ 0 1 = lim (B.9) ν→0+ E − H ± iν

where ν is a positive inﬁnitesimal. Finally, by inverse Fourier transform we can write the entire propagator of the system as

Z +∞ Z 1 −iEτ/~ 1 −izτ/~ U(τ) = dEe [G−(E) − G+(E)] = dze G(z) (B.10) 2πi −∞ 2πi C+∪C−

where C = C+ ∪ C− is the union of the semicircle contours in the upper/lower half complex plane, bounded below and above by the real axis E, respectively. 122

The Green’s functions in energy domain can be obtained from a general operator G(z) of the complex variable z, such that

1 G(z) = (B.11) z − H

G±(E) = lim G(E ± iν) . (B.12) ν→0+

G(z) is called the resolvent operator of the Hamiltonian H, it is a generalized Green’s

function in the complex domain, such that G±(E) are the limits of the resolvent when z approaches the real axis from above(+)/below(-) the complex plane[33].

B.2 Projection Operators

Let P be the relevant model subspace onto which we would like to constrain the dynamics of the system and Q be its orthogonal complement. If H is the full Hamiltonian,

then it posses a subset of relevant eigenvectors {|ψ1i, |ψ2i, ...|ψni}, from which we can construct the eﬀective propagator of the system. If these states are orthogonal, there exists a projector operator, P, onto the subspace P,

P = |ψ1ihψ1| + |ψ2ihψ2| + ... + |ψnihψn| , (B.13)

with an associated complementary projector, Q = 1 − P, which projects onto the orthogonal (irrelevant) subspace Q, both satisfying the following relations,

P = P† P2 = P (B.14)

Q = Q† Q2 = Q . (B.15)

Since P ⊥ Q, we also have PQ = QP = 0 . (B.16) 123

If the model states in P are eigenvectors of an unperturbed Hamiltonian H0, such that such states lead to the most important terms of an expansion of G(z) in powers of V, then

[P, H0] = [Q, H0] = 0 (B.17)

PH0Q = QH0P = 0 . (B.18)

B.3 Eﬀective Green’s Functions and Hamiltonians

By calculating the projection of the Green’s function G(z) in the relevant subspace, P, we can extract from it an eﬀective Hamiltonian containing the relevant interactions within P. We can rearrange Eq. B.11 such that (z − H0 − V) G(z) = 1. If we multiply this expression from both sides by P, then if we insert the identity P + Q = 1 in between

(z − H0 − V) and G(z), and make use of B.18, the following expressions are obtained after several algebraic manipulations, " # Q P = P z − H0 − V − V V PG(z)P , (B.19) z − QH0Q − QVQ where Q R(z) = V + V V , (B.20) z − QH0Q − QVQ is an operator that allows to determine the shifts of the perturbed energy levels respect to the unperturbed ones. Substitution of Eq. B.20 in Eq. B.19 yields the desired result,

P PG(z)P = , (B.21) z − PH0P − PR(z)P which is the projected Green’s function onto the relevant model space P. This function

P yields the eﬀective interaction Hamiltonian He f f , since PG(z)P = , where z−He f f (z)

He f f (z) = PH0P + PR(z)P . (B.22)

The matrix elements of G(z) are analytic functions in the entire complex plane except on the real E axis, such that its poles yields the eigenvalues of H. Therefore, the poles of 124

PG(z)P should yield the eigenvalues of He f f , which should coincide with the original eigenvalues associated with eigenstates spanning P. The problem of ﬁnding the eﬀective Hamiltonian reduces to compute the solutions to the equation

det[z − He f f (z)] = 0 . (B.23)

B.4 Eﬀective Hamiltonians and their Eigenvalue Spectra

We start with the full Hamiltonian H, in the RWA approximation as given by Eq. 2.13,    δ Ω 0 0 Ω   0 T B     Ω δ01 t t V   T 01 e h F     10  H =  0 te δ + ∆S 0 th  . (B.24)  01     0 t 0 δ01 − ∆ t   h 10 S e     10  ΩB VF th te δ10 In order to obtain the eﬀective Hamiltonian H˜ (Λ)(z) in Eq. 3.5 we need to identify the

unperturbed and perturbed parts of the Hamiltonian, H0 and V, respectively. In our case, this results in      δ 0 0 0 0   0 Ω 0 0 Ω   0   T B       0 δ01 0 0 0   Ω 0 t t V   01   T e h F       10    H0 =  0 0 δ + ∆S 0 0  V =  0 te 0 0 th  . (B.25)  01         0 0 0 δ01 − ∆ 0   0 t 0 0 t   10 S   h e       10    0 0 0 0 δ10 ΩB VF th te 0

Next, we construct the projection operators P an Q represented in the basis,

00 01 10 10 01 {|00Xi, |01Xi, |01Xi, |10Xi, |10Xi} . (B.26) 125

Explicitly,      1 0 0 0 0   0 0 0 0 0           0 0 0 0 0   0 1 0 0 0          P =  0 0 1 0 0  Q =  0 0 0 0 0  . (B.27)          0 0 0 1 0   0 0 0 0 0           0 0 0 0 0   0 0 0 0 1 

The numerical or symbolic computation of He f f is facilitated by calculating separately the following intermediate terms,

A1 = PH0PA2 = QH0QA3 = QVQ

(5) −1 A4 = [zI − A2 − A3] A5 = V + VQA4VA6 = PA5P , (B.28)

where I(5) is the 5 × 5 identity matrix which results in,

He f f (z) = A1(z) + A6(z) (B.29)

Formally, the resulting matrix He f f is still a 5 × 5 matrix with null rows and columns intersecting the positions of the nonzero elements of Q, discarding these null elements results in the reduced matrix, H˜ (Λ)(z), whose dimension m corresponds to the relevant subspace of interest. Now, when calculating Eq. B.23, we obtain the following polynomial

Π(Λ)(z) = det[zI(3) − H˜ (Λ)(z)] . (B.30)

Since our Hamiltonian is real and has a strictly discrete spectrum, we shall discard all complex roots for Π(z). Since in our problem the spectrum depends on the value of the applied electric field, it is possible to either express the roots as functions of the applied field. Depending on the complexity of the z-dependence of effective Hamiltonian matrix elements, we might reconstruct the entire original spectrum with all roots zi(F) which would correspond to the original eigenvalue problem. However, this is not algebraically easy task. Instead, we focus on a particular value of the applied electric field, FA, for 126

which an isolated anticrossing occurs in a well predeﬁned place in the spectrum, such that

there is a unique root zi(FA), such that the eigenvalue spectrum of He f f (zi(FA) would reproduce the original spectrum within a spectral region adequately described by the eﬀective Hamiltonian. For the simulation parameters listed on table A.2, the roots of Eq. B.30 at

F = FR = 2.3kV/cm and F = FI = 43.4kV/cm are respectively given by,

z1(FR) z2(FR) z3(FR) z4(FR) z5(FR) Units -51.8271 -51.7364 -34.3548 -34.2116 0.0219 meV (Λ) Table B.1: Roots of the polynomial Π (z) at FR

z1(FI) z2(FI) z3(FI) z4(FI) z5(FI) Units -69.1980 -51.6452 -51.3045 -0.0107 0.0504 meV (Λ) Table B.2: Roots of the polynomial Π (z) at FI

At FR = 2.3kV/cm two roots, z3(FR), z4(FR), match the two eigenvalues

λ3(FR),λ4(FR), corresponding to the molecular indirect exciton eigenstates of the full Hamiltonian. Therefore, within the purely indirect exciton qubit subspace, either choice among these two roots in Eq. B.22 reproduces the original eigenvalue spectrum in an electric ﬁeld interval for which the only relevant eigenstates mix just two indirect

excitons, as seen in Fig. B.1(a). If instead we substitute z5(FR) in Eq. B.22, the eﬀective eigenvalue spectrum results in an incorrect anticrossing behavior as seen in Fig. B.1(b).

Similarly, z5(FI) is the correct root who reproduces correctly the anticrossing at FI, as seen in Fig. B.1(c). We need to remark that in order to assert the correct values of the roots 127

z, it is not necessarily to know a priori the spectrum of the full Hamiltonian, which might in principle contain many degrees of freedom, for which straight diagonalization would be computationally intensive. In solving equation Eq. B.23 the solutions parameterized by F are exact, therefore all real solutions zi(F) reconstructs the entire level anticrossing (dressed) spectrum; within a speciﬁc range of F all the eigenvalues of both the eﬀective and the full Hamiltonian match unambiguously the solutions of Eq. B.23.

Figure B.1: Eﬀective eigenvalue spectrum of H˜ (Λ)(z) (dashed red). Full Hamiltonian eigenvalues (solid black) (a) Eigenvalues for z = z3(FR) correctly reproduces the original spectrum (qubit rotation anticrossing), whereas in (b) z = z5(FR) does not. (c) Eigenvalues for z = z5(FI) correctly reproduces initialization anticrossing, whereas in (d) z = z3(FI) does not. 128

Let us focus now on the purely indirect exciton subspace whose dynamics is associated to the Hamiltonian H˜ (I)(z) in Eq. 3.11. In this case the solutions to the equation

Π(I(z) = det[zI(2) − H˜ (I)(z)] = 0 , (B.31)

result in the same set of roots of roots given in B.2. This is not surprising since the Bloch-Feshbach projection operator equations are exact and independent of the rank of resulting eﬀective Hamiltonian, and should reproduce the same spectrum for the same particular Hilbert subspace of interest. 129

Figure B.2: Eﬀective eigenvalue spectrum of H˜ (I)(z). (a) Eigenvalue spectrum for diﬀerent solutions of Eq. B.31. Roots z3, z4 reproduce correctly the original spectrum (solid blue line). Substitution of the remaining roots, z3, z4 results in a false gap and anticrossing behavior of a purely indirect exciton subspace (dashed yellow green lines, respectively). Here, δ is a small arbitrary deviation from the root z3.

In summary, we have shown that the adiabatic elimination procedure and effective Hamiltonian extraction is greatly simplified by projection operator algebraic methods. For our case, we were able to extract analytical expressions for effective Hamiltonians of reduced dimensionality, as well as their effective eigenvalue spectra via solving for the zeros of a simple polynomial expression. 130 Appendix C: Rabi oscillations and The Rotating Wave

Approximation

C.1 Two Level System Bloch Equations

Let us consider an atomic-like (excitonic-like) system with two lowest energy levels

E0 and E1. Let us suppose that the system is being driven by an external coherent light

ﬁeld, V(t) = e~r · E~(t), linearly polarized along the x-axis, such that E~(t) = (E0 cos ωt, 0, 0), where E0 is the amplitude of the electric ﬁeld envelope and ω the driving frequency. Here we have used already the semiclassical electrical dipole approximation, therefore the light-matter coupling takes the form,

exE V(t) = 0 eiωt + e−iωt , (C.1) 2

with matrix elements given by

E µi j V (t) = − 0 eiωt + e−iωt , (C.2) i j 2

with µi j = −ehi|x| ji being the transition dipole matrix elements, and |ii, | ji the atomic (excitonic) eigenstates. The Bloch equations for the driven two level system in consideration are given by,

dc0 = − i c (t)V + c (t)V e−iω0t (C.3) dt ~ 0 00 1 01 dc1 = − i c (t)V eiω0t + c (t)V , (C.4) dt ~ 0 10 1 11

where ω = E1−E0 , is the natural transition frequency. Inserting the matrix elements in Eq. 0 ~ C.2 in Eqs. C.3 and C.4 results in

dc0 = iΩ ei(ω−ω0)t + e−i(ω+ω0)t c (t) (C.5) dt 2 1 dc1 = iΩ e−i(ω−ω0)t + ei(ω+ω0)t c (t) , (C.6) dt 2 0

where Ω = | µ01E0 | is the Rabi frequency of the driven system. ~ 131

C.2 Rotating Wave Approximation

Let us deﬁne the detuning frequency as δω = ω − ω0. When driving the system close

to resonance, δω ω0 and furthermore δω ω + ω0. Moreover, since absorption or emission of light is dominated by resonant transitions, they should dominate over non-resonant processes associated with the fast oscillating terms in Eqs. C.5 and C.6 that involves the sum of the driving and natural frequencies, in other words terms that oscillate

with ±(ω + ω0). One can identify such processes by expressing the electric-dipole coupling as follows, Ω V(t) = ~ σ e−iωt + σ eiωt + σ e−iωt + σ eiωt , (C.7) 2 + − − + where σ+ = |1ih0| and σ− = |0ih1|. The first terms describes absorption of light followed by the atomic excitation, |0i → |1i, whereas the second term describes emission followed by the atomic de-excitation, |1i → |0i. For near resonant driving we can neglect the last two terms of Eq. C.7, which implies ignoring the fast oscillatory off-resonant terms (counter-rotating terms) in Eqs. C.5 and C.6. In doing this, we are performing the so called rotating-wave approximation (RWA). The term derives from resonant excitation, for which the fictitious magnetic field associated to V(t) retains only the component rotating in the same direction of the pseudo-spin generated by the two level system[33].

C.3 Rabi Oscillations

Let us consider the case of resonant excitation δω = 0 under these RWA approximation. Under this conditions Eqs. C.5 and C.6 reduce to dc 0 = iΩ c (t) (C.8) dt 2 1 dc 1 = iΩ c (t) , (C.9) dt 2 0

such that if we apply the initial conditions c0(0) = 1 and c1(0) = 0, one obtains the

Ωt Ωt solutions c0(t) = cos 2 and c1(t) = i sin 2 . This results in the oscillatory behavior of the 132

atomic level populations, ! ! Ωt Ωt |c (t)|2 = cos2 |c (t)|2 = sin2 . (C.10) 0 2 1 2

For the general case of oﬀ-resonant excitation, the population dynamics of the upper level is modiﬁed as follows, 2 ˜ ! 2 Ω 2 Ωt |c1(t)| = sin , (C.11) Ω˜ 2 2 √ where Ω˜ = Ω2 + δω2 is the generalized Rabi frequency. Thus, a non-zero detuning introduces a frequency blue-shift and an amplitude damping as function of the increasing detuning. 133 Appendix D: Adiabatic Passage

D.1 Pseudo-Spin Representation

Let us consider a two level system interacting with an external driving ﬁeld. After performing the RWA approximation, the Hamiltonian can be written as follows,

H = −~Ω~ · σ~ , (D.1) 2 R where Ω~ R = (Ω, 0, ∆(t)) is the generalized Rabi rotation axis and ∆(t) is a time dependent detuning. The eigenvalues of the system are given by

p E (t) = ±~Ω˜ (t) = ±~ Ω2 + ∆(t)2 , (D.2) ± 2 2 with associated instantaneous eigenvectors given by, ! ! θ(t) θ(t) |±i = cos |±i + i sin |∓i , (D.3) 2 z 2 z

with |+iz = |1i and |−iz = |0i, and ! Ω ∆(t) Ω θ(t) = tan−1 cos θ(t) = sin θ(t) = . (D.4) ∆(t) Ω˜ (t) Ω˜ (t)

In other words, the Hamiltonian in Eq. D.1 can be rewritten as follows,

Ω˜ (t) H = −~ σ (t) , (D.5) 2 n where σn(t) = ~n(t) · σ~ and ~n(t) = (sin θ(t), 0, cos θ(t)) . (D.6)

D.2 Adiabatic Theorem

The adiabatic theorem states that if the initial condition of the dynamics is to start at t = 0 in an eigenstate of a slowly varying time dependent Hamiltonian, the system remains on the same local eigenstate at t , 0 up to a phase. 134

Figure D.1: Eigenvalues of the Hamiltonian in Eq. D.5 as function of a time dependent detuning ∆(t) (solid blue). The asymptotic dashed red lines indicate the bare atomic energies for the unperturbed atomic states. The red points indicate the values at which θ is measured.

Let us consider Fig. D.1, if we start the system in the lowest state |0i, and if ∆(t) is varied slowly, thus the system adiabatically follows the lower eigenvalue band (solid blue), so that the systems evolves from |0i at θ = −π, then passes through the state

|+i = √1 |0i − i|1i to finally reach |1i at θ = 0. In other words, the adiabatic condition θ=π/2 2 dθ ˜ is satisfied if dt Ω(t). It should be pointed out that the local eigenstates of the system are dressed states resulting from the interaction with the external electromagnetic field. Let us consider explicitly the dressed state |+i,  r r  1  ∆ ∆  |+i = √  1 + |1i + i 1 − |0i , (D.7) 2 Ω˜ Ω˜ 135

therefore for ∆ Ω, |+i → |0i, while for ∆ Ω, |+i → |1i. In terms of the Rabi frequencies of the system, an adiabatic passage (adiabatic following) occurs provided that

1 dΩ˜ | | Ω , (D.8) Ω˜ dt where Ω is the smallest gap of the anticrossing between the two bands. Thus, the general condition for adiabatic passage, from the lower (higher) energy state into the higher (lower) state, is that the driving perturbation must vary slowly in comparison to the frequency associated to energy splitting of such states. For the case of the electric ﬁeld sweep presented in Fig. 3.8 section 3.4.1, the adiabatic condition is imposed on the velocity of the time dependent energy detuning of the indirect excitons. For a time dependent energy detuning, we can write Eq. D.8,

1 d∆ | | Ω . (D.9) Ω2 + ∆2(t) dt

d∆ The detuning speed dt in the qubit initialization procedure, is such that no exciton population is transferred into states outside the qubit subspace while transporting the initialized state into the central anticrossing; while in the readout procedure, the rotated

01 10 state 10X does not populate other excitons except the radiative state 10X. 136 Appendix E: Two Level Systems and Qubits

E.1 Deﬁnitions

A two level system (TLS) is among the simplest and most fundamental systems in

1 quantum mechanics. Natural examples are spin 2 systems and Hilbert spaces spanned by the two lowest energy orthogonal states of atomic-like systems. Since two-dimensional Hilbert space vectors are isomorphic to SU(2) isospin states, the algebraic properties of angular momentum are inherited to states and operators acting on a TLS. A quantum bit (qubit) is a two level system with a robust representation, such that it should be able to be initialized, and maintain a superposition state for enough time in order to be controlled and measured. On one hand, a qubit should be as much as possible isolated from its environment in order to preserve its quantum properties. On the other hand, it should be also accessible to environment in order enable its control by use of external ﬁelds. A qubit is the smallest unit of information that a quantum computer is able to store and operate on. Contrary to classical bits, qubits can also exists on superposition or linear combinations of classical bit states 0 and 1, this means that a general state of a qubit takes the form

ψ = c0|0i + c1|1i , (E.1)

2 2 directly, instead we perform ensemble measurements in order to obtain |c0| and |c1| . A quantum register of size N is a collection of N qubits. For example, an arbitrary state of a two qubit register is given by the state vector,

ψ = c00|00i + c01|01i + c10|10i + c11|11i . (E.2) 137

N An N-qubit register would be speciﬁed by 2 state amplitudes ci jk.., and its clear that the amount of quantum information available increases exponentially with register size. Before any measurement takes place, all information is inaccessible and arbitrary measurements can destroy large portions of such information. Ideally all qubits in a register can be manipulated, such that a control scheme can be harnessed in order for them to interact with each other coherently with minimal disturbance. When dealing with operations deﬁned on qubits it is useful to have at hand the fundamentals of the SU(2) pseudo-spin algebra. Let us consider the pseudo-spin vector

~ ~ S = 2 σ~ , such that the components of σ~ are the Pauli matrices σx, σy and σz,

       0 1   0 −i   1 0        σx =   σy =   σz =   . (E.3)  1 0   i 0   0 −1 

σx±σy Diﬀerent pseudo-spin projections are obtained by the ladder operators σ± = 2 . In the basis of pseudo-spin along the z-direction, {|+iz, |−iz}, the following set of properties are also standard,

[σi, σ j] = 2ii jkσk [σz, σ±] = ±2σ± (E.4)

2 [σ+, σ−] = σz σi = 1 (E.5)

Tr(σi) = 0 det σi = −1 (E.6)

Moreover, any operator A in a two-dimensional Hilbert space can be represented as linear combinations of the SU(2) algebra generators σi,

1 A = a 1ˆ + ~a · σ~ , (E.7) 2 0 where a0 = A, ai = Tr(Aσi), for i = 1, 2, 3. 138

E.2 Bloch Representation

Alternatively, the state of a two level system can be speciﬁed by the density matrix. According to Eq. E.7 the density operator can be written as,

1 ρ = 1ˆ + R~ · σ~ , (E.8) 2 where R~ = Tr(ρσ~ ) = hσ~ i is the Bloch vector. Therefore, there is univocal correspondence between the state of TLS or qubit and the coordinate speciﬁed by the Bloch vector on a unit sphere, called the Bloch sphere, see Fig. E.1.

Figure E.1: Bloch sphere. The poles of the sphere denote the coordinates of the bare computational basis states. The Bloch vector projections are indicated by the red dashed lines.

The Bloch representation of the qubit state in Eq. E.1 results from specifying the matrix elements of E.8. Since ρi j = hcic ji, this results in    h|c |2i h|c c∗|i   0 0 1  ρ =   (E.9)  ∗ 2  h|c0c1|i h|c1| i 139 which implies,

2 2 Rx = 2Rehc0c1i Ry = 2Imhc0c1i Rz = |c1| − |c0| (E.10)

In spherical coordinates R~ = (sin θ cos ϕ, sin θ sin ϕ, cos θ), therefore

θ c = sin (E.11) 0 2 θ c = eiϕ cos . (E.12) 1 2

Single qubit operations, which evolve the system between the north and south poles, or between two arbitrary points on the Bloch sphere, can be achieved either by letting the system evolve via its natural or externally driven Rabi oscillations (seeC) or via rapid adiabatic passages (seeD). A clear mapping exists between the Bloch sphere representation of a qubit and the dressed level diagram presented in Fig. D.1. The understanding of this mapping is facilitated by the pseudo-spin representation of both the two level system Hamiltonian and the density matrix of the system.

E.3 Indirect Exciton Qubits

Let us discuss the indirect exciton qubits in the context of the Bloch sphere picture. For the case of indirect exciton qubit initialization, Fig. 3.9 in section 3.4.2, the vacuum

00 10 state |00Xi is located at the north pole of the Bloch sphere while, |01Xi, is located at the south pole. Initially, the Bloch vector is rotating about the y-axis (the initial axis and the azimuthal angle are initially defined arbitrarily) going back and forth between the poles, see Fig. E.2(a). After switching on the adiabatic ramp, the Bloch vector is fixed and tipped along an axis slightly tilted from the z-axis along the south pole, not completely aligned with the vertical axis due to small effects of population relaxation, see Fig. E.2(b). In this case, the system follows the local eigenstate whose eigenvalue is marked by red labeled exciton in Fig. 3.7, section 3.4.1. At the central anticrossing for F = FR, a new Bloch

10 sphere is needed to describe qubit rotations, see Fig. E.2(c). In this case, the state |01Xi is 140

located at the north pole. Notice that the initial condition is specified by the position near the south pole in Fig. E.2(b); however, an accumulated phase factor (not shown) defines completely a new starting horizontal axis. At the end of the qubit rotation operations in Fig. 3.10 section 3.4.3, and just after the onset of the reversed ramp, the Bloch vector is fixed and tipped along an axis slightly tilted from the north pole, as seen in Fig. E.2(d). At this stage the state of the qubit is prepared for measurement.

Figure E.2: Indirect exciton qubits. (a) Rabi oscillations during the qubit initialization 10 state. (b) Qubit is initialized on the 01X state (non-ideal ﬁdelity exaggerated on picture). (c) Qubit rotations as Rabi oscillations between the indirect excitons. (d) The π rotated qubit 01 10X state is prepared for read out. 141

During the reversed ramp in Fig. 3.11 section 3.4.4, the system enters the tunneling anticrossing at F ' 20kV/cm; on this stage, the Bloch sphere picture is not adequate, since

01 10 there are three excitons in the neighborhood of the anticrossing, namely 10X (blue), 10X 01 (green) and 01X (black), which is an spectator state since it does not mix appreciably with the remaining states. The tunneling creates the upper and lower branch of dressed states

01 10 which are coherent superpositions of 10X (blue) and 10X (green). The new initial condition on this subspace, is that the system starts at the eigenstate corresponding to the upper branch. We notice, that upon passing through the anticrossing, the system remains in the local eigenstate corresponding to the upper branch, which at larger ﬁelds asymptotically

10 01 10 becomes 10X (green). Therefore, 10X (blue) depopulates entirely into 10X (green) which rapidly relaxes into the vacuum, without populating any other state. 142 Appendix F: Articles and Conferences

In what follows I present the list of articles we have published, conferences attended and talks given during the course of my PhD studies.

F.1 Articles

• Competing eﬀects of hyperﬁne and spin-orbit interactions in two-electron spin qubits, E. Cota, J.E. Rolon and S.E. Ulloa (in preparation)

• Tunable exciton relaxation in vertically coupled semiconductor InAs quantum dots, K.C. Wijensundara, J.E. Rolon, S.E. Ulloa, A.S. Bracker, D. Gammon and E.A. Stinaﬀ, Phys. Rev. B 84, 081404 (2011)

• Coherent control of indirect excitonic qubits in optically driven quantum dot molecules, J.E. Rolon and S.E. Ulloa, Phys. Rev. B 82, 115307 (2010)

• F¨orster energy-transfer signatures in optically driven quantum dot molecules, J. E. Rolon and S. E. Ulloa, Phys. Rev. B 79, 245309 (2009)

• F¨orster signatures and qubits in optically driven quantum dot molecules, Physica E, 40, 1481 (2008)

F.2 Conferences

• Charge dynamics and phonon induced oscillatory relaxation rates of indirect excitons in quantum dot molecules, Juan E. Rolon, Kushal C. Wijesundara , Eric A. Stinaﬀ and Sergio E. Ulloa, American Physical Society, APS March Meeting 2011, Dallas, Texas, March 21-25, (2011)

• Resilient neutral exciton qubits in self-assembled quantum dot molecules, Juan E. Rolon and Sergio E. Ulloa, American Physical Society, APS March Meeting 2010, Portland, Oregon, March 15-19, (2010) 143

• Coherent control of indirect excitonic qubits in optically driven quantum dot molecules, Juan E. Rolon and Sergio E. Ulloa, International Conference on Electronic and Optical Coherence in Low Dimensional Semiconductors and Atomic Gases, Turunc Marmaris, Turkey, September 20-29, (2009)

• Qubit extraction and manipulation in optically-driven self-assembled quantum dot molecules, Juan E. Rolon and Sergio E. Ulloa, American Physical Society, APS March Meeting 2009, Pittsburgh, Pennsylvania, March 16-20, (2009)

• F¨orster optical signatures in quantum dot molecule photoluminescence, Juan E. Rolon and Sergio E. Ulloa, American Physical Society, APS March Meeting 2008, New Orleans, Louisiana, March 10-14, (2008)

• F¨orster Optical Signatures on the Dressed Spectrum of Quantum Dot Molecules, Juan E. Rolon and Sergio E. Ulloa, NSS5/SP-STM2 International Conference of Nanoscale Spectroscopy and Nanotechnology 5, Athens, Ohio, July 15-19 (2008)

• F¨orster Signatures and Qubits in Optically Driven Quantum Dot Molecules, Juan E. Rolon and Sergio E. Ulloa, The 17th International Conference on Electronic Properties of Two-Dimensional Systems (EP2DS-17) and the 13th International Conference on Modulated Semiconductor structures (MSS-13), Genoa, Italy, July 15 - 20, (2007)

• Qubit identiﬁcation and entanglement in tunneling and F¨orster coupled quantum dots, Juan E. Rolon and Sergio E. Ulloa, American Physical Society, APS March Meeting 2007, Denver, Colorado, March 5-9, (2007)

• Tunneling, dipole interactions and coherent Rabi oscillations in quantum dot molecules, Juan E. Rolon, Jose M. Villas-Boasˆ and Sergio E. Ulloa, American 144

Physical Society, APS March Meeting 2006, Baltimore, Maryland, March 13-17, (2006) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

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