Many-body and spin-orbit aspects of the alternating current phenomena

The thesis reports on research in the general field of light interaction with matter. According to the topics addressed, it can be naturally divided into two parts: Part I, many-body aspects of the Rabi oscillations which a two-level systems undergoes under a strong resonant drive; and Part II, absorption of the ac field between the spectrum branches of two-dimensional fermions that are split by the combined action of Zeeman and spin-orbit (SO) fields. The focus of Part I is the following many-body effects that modify the conventional Rabi oscillations: Chapter 1, coupling of a two-level system to a single vibrational mode of the environment. Chapter 2, correlated Rabi oscillations in two electron-hole systems coupled by tunneling with strong electron-hole attraction. In Chapter 1, a new effect of Rabi-vibronic resonance is uncovered. If the frequency of the Rabi oscillations, R, is close to the frequency, !0, of the vibrational mode, the oscillations acquire a collective character. It is demonstrated that the actual frequency of the collective oscillations exhibits a bistable behavior as a function of ΩR ω0. The main finding in Chapter 2 is, that the Fourier spectrum of the Rabi oscillations in two coupled electron-hole systems undergoes a strong transformation with increasing ΩR. For ΩR smaller than the tunneling frequency, the spectrum is dominated by a low-frequency (<<ΩR) component and contains two additional weaker lines; conventional Rabi oscillations are restored only as ΩR exceeds the electron-hole attraction strength. The highlight of Part II is a finding that, while the spectrum of absorption between either Zeeman-split branches or SO-split branches is close to a -peak, in the presence of both, it transforms into a broad line with singular behavior at the edges. In particular, when the magnitudes of Zeeman and SO are equal, absorption of very low (much smaller than the splitting) frequencies become possible. The shape of the absorption spectrum is highly anisotropic with respect to the exciting field. This peculiar behavior of the absorption is also studied in wire geometry, where the interplay between two couplings (Zeeman and spin-orbit splitting) affects the shape of numerous absorption peaks.

MANY-BODY AND SPIN-ORBIT ASPECTS OF THE ALTERNATING CURRENT PHENOMENA by Rachel M. Glenn A dissertation submitted to the faculty of The University of Utah in partial ful llment of the requirements for the degree of Doctor of Philosophy in Physics Department of Physics and Astronomy The University of Utah August 2012 Copyright c Rachel M. Glenn 2012 All Rights Reserved The University of Utah Graduate School STATEMENT OF THESIS APPROVAL The dissertation of Rachel M. Glenn has been approved by the following supervisory committee members: Mikhail Raikh , Chair May 4, 2012 Date Approved Oleg Starykh , Member May 4, 2012 Date Approved Christoph Boehme , Member May 4, 2012 Date Approved Kyle Dawson , Member May 4, 2012 Date Approved Elena Cherkaev , Member May 4, 2012 Date Approved and by David Kieda , Chair of the Department of Physics and Astromony and by Charles A. Wight, Dean of The Graduate School. ABSTRACT The thesis reports on research in the general eld of light interaction with matter. According to the topics addressed, it can be naturally divided into two parts: Part I, many-body aspects of the Rabi oscillations which a two-level systems undergoes under a strong resonant drive; and Part II, absorption of the ac eld between the spectrum branches of two-dimensional fermions that are split by the combined action of Zeeman and spin-orbit (SO) elds. The focus of Part I is the following many-body e ects that modify the conventional Rabi oscillations: Chapter 1, coupling of a two-level system to a single vibrational mode of the environment. Chapter 2, correlated Rabi oscillations in two electron-hole systems coupled by tunneling with strong electron-hole attraction. In Chapter 1, a new e ect of Rabi-vibronic resonance is uncovered. If the frequency of the Rabi oscillations, R, is close to the frequency, !0, of the vibrational mode, the oscillations acquire a collective character. It is demonstrated that the actual frequency of the collective oscillations exhibits a bistable behavior as a function of R 􀀀 !0. The main nding in Chapter 2 is, that the Fourier spectrum of the Rabi oscillations in two coupled electron-hole systems undergoes a strong transformation with increasing R. For R smaller than the tunneling frequency, the spectrum is dominated by a low-frequency (<< R ) component and contains two additional weaker lines; conventional Rabi oscillations are restored only as R exceeds the electron-hole attraction strength. The highlight of Part II is a nding that, while the spectrum of absorption between either Zeeman-split branches or SO-split branches is close to a -peak, in the presence of both, it transforms into a broad line with singular behavior at the edges. In particular, when the magnitudes of Zeeman and SO are equal, absorption of very low (much smaller than the splitting) frequencies become possible. The shape of the absorption spectrum is highly anisotropic with respect to the exciting eld. This peculiar behavior of the absorption is also studied in wire geometry, where the interplay between two couplings (Zeeman and spin-orbit splitting) a ects the shape of numerous absorption peaks. I would like to dedicate this thesis to my husband, Ronald Glenn. Through his professorial demeanor I was able to achieve my goals. It is also dedicated to my mother, who taught me good study habits at an early age. CONTENTS ABSTRACT : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : viii ACKNOWLEDGMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : xii PART I MANY-BODY ASPECTS OF RABI OSCILLATIONS : : : : : : 1 1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2. RABI-VIBRONIC RESONANCE WITH LARGE NUMBER OF VIBRATIONAL QUANTA : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Modi ed Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.1 Oscillation frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.2 Vicinity of the resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.3 E ect of intrinsic anharmonicity of the oscillator . . . . . . . . . . . . . . . . . 22 2.4 Decay of the oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.1 Friction-dominated regime, 􀀀 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.2 The form of the decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4.3 Initial stage of the oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.4 Relaxation-dominated regime, 􀀀 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5 Number of vibrational quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7.2 Perturbative treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.7.3 Friction-dominated regime, 􀀀 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.7.4 Relaxation-dominated regime, 􀀀 . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.7.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3. MANY-BODY ASPECTS OF RABI OSCILLATIONS IN A QUANTUM DOT MOLECULE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 41 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Correlated dynamics of a photoexcited exciton in two coupled dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.1 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3 Exciton in two QDs with near-resonant light . . . . . . . . . . . . . . . . . . . . . . 51 3.3.1 Weak driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 Resonant driving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.5 Relation to the Rabi oscillations in spin pairs detected by PEDMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 PART II INTERPLAY OF SPIN-ORBIT COUPLING AND ZEEMAN SPLITTING IN ALTERNATING CURRENT ABSORPTION IN LOW-DIMENSIONAL SYSTEMS : : : : : : : : : : : : : : : : : : : : : : 74 4.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5. INTERPLAY OF SPIN-ORBIT COUPLING AND ZEEMAN SPLITTING IN THE ABSORPTION LINESHAPE OF 2D FERMIONS : : : : : : : : 77 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2 Optical conductivity with nite Z and SO . . . . . . . . . . . . . . . . . . . . . . 81 5.2.1 Hamiltonian and eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2.2 Optical conductivity at T = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2.2.1 SO Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2.2.2 Z SO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2.2.3 Z SO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2.3 Optical conductivity at nite temperature . . . . . . . . . . . . . . . . . . . . . . . 88 5.2.3.1 Temperature smearing of the chiral resonance . . . . . . . . . . . . . . . . . . 89 5.2.3.2 Temperature smearing of EDSR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.2.3.3 Z = SO at nite T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.3 ESR lineshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6. INTERPLAY OF SPIN-ORBIT COUPLING AND ZEEMAN SPLITTING IN THE ABSORPTION LINESHAPE OF 1D QUANTUM WIRE : 100 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 6.2 Hamiltonian and energy spectrum in dimensionless units . . . . . . . . . . . . . 106 6.3 Arbitrary con nement: general relations . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.3.1 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.4 Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 6.4.1 Analytical approach for absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.5 Rigid walls: derivation of the dispersion equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.6 Analysis of dispersion curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.6.1 Vicinity of Q2 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.7 wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 vi 6.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.9.1 Derivation of the constants 1, 2, 3, 4 . . . . . . . . . . . . . . . . . . . . . . . 131 6.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 vii LIST OF FIGURES 1.1 The broadening of the Rabi-peak in the frequency domain is illustrated. . . 8 1.2 Plot of the measured Rabi oscillations on pairs of electrostatically bound charge carriers, for weak spin-exchange coupling is shown in (a); while in (b), the Fourier transform spectra of Rabi oscillations obtained at di erent B1 eld strengths is shown, from Ref. [30]. . . . . . . . . . . . . . . . 9 2.1 Illustration describing main idea, showing the coupling between the en- vironment, which is modeled as an ensemble of oscillators, and the two-level system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 The system under consideration is illustrated schematically. . . . . . . . . . . . . 13 2.3 Illustration of the experimental realization of the system considered from Ref. [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Dimensionless frequency z, Eq. (2.17), versus the dimensionless deviation x from the R = !0 is plotted from Eq. (2.19). . . . . . . . . . . . . . . . . . . . 18 2.5 Dimensionless frequency z of oscillations of driven two-level system is plotted from Eq. (2.19) versus the dimensionless deviation x0 from the resonance for three positive dimensionless detunings. . . . . . . . . . . . 20 2.6 The same as in Fig. 2.5 for three negative detunings. . . . . . . . . . . . . . . . . . 21 2.7 The number of oscillation cycles, m, before collective motion stops falls o with dimensionless friction of the oscillator, =!0. . . . . . . . . . . . . . . . . . 25 2.8 Illustration depicting how population inversion evolves with time. . . . . . . . 27 2.9 Excitation level of the oscillator, N = 16N 2=3, where N is the number of vibrational quanta, is plotted from Eq. (2.19) vs dimensionless detuning and dimensionless deviation from the resonance. . . . . . . . . . . . . . . . . . . 29 2.10 Results of numerical study of resonantly driven two-level system coupled to a resonator with frequency tuned close to R from [18]. . . . . . . . . . . 30 2.11 Coupling-induced correction to the amplitude of the Rabi oscillations in the friction-dominated regime is plotted from Eq. (2.55). . . . . . . . . . . . 35 2.12 Coupling-induced correction to the amplitude of the Rabi oscillations in the relaxation-dominated regime is plotted from Eq. (2.59) . . . . . . . . . 37 3.1 Quantum dot molecule in the experiment Refs. [6]. . . . . . . . . . . . . . . . . . . . 42 3.2 A schematic illustration of the electrical detection of Rabi oscillations is shown from Ref. [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Optical detection of Rabi oscillations is shown from Ref. [11]. . . . . . . . . . . 45 3.4 A schematic illustration of single-exciton states in a quantum dot molecule. 47 3.5 Population of the states are plotted from Eqs. (3.12)-(3.15) versus the dimensionless time Ut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.6 Graphic solution to the cubic equation, Eq. (3.28). . . . . . . . . . . . . . . . . . . . 55 3.7 The function f(0; ; z) is plotted for four characteristic values of the di- mensionless detuning, , from Eq. (3.37). . . . . . . . . . . . . . . . . . . . . . . . 57 3.8 The function f(u; 0; z) is plotted for three values of the dimensionless interaction parameter, u, corresponding to perfect resonance, = 0, for an isolated dot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.9 The Fourier transform (top) and corresponding magnitudes (bottom) for the occupation of the vacuum state, Eq. (3.33), for = 0:1 and u = 0. 59 3.10 The same as Fig. 3.9 for = 0:5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.11 The same as Fig. 3.9 for = 1:5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.12 Both the peak positions and intensities are shown, and corresponds to Fig. 3.9-3.11, for u = 0 and three values of for . . . . . . . . . . . . . . . . . 64 3.13 Both the peak positions and intensities are shown for = 0 and three values of u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.14 Numerical results are shown corresponding to the Hamiltonian, Eq. (3.44), for spin-exchange coupling J=h = 1MHz, from Ref. [28]. . . . . . . . . . . . . 67 3.15 Plot of the measured Rabi oscillations on pairs of electrostatically bound charge carriers, for weak spin-exchange coupling is shown in (a); while in (b), the Fourier transform spectra of Rabi oscillations obtained at di erent B1 eld strengths is shown, from Ref. [30]. . . . . . . . . . . . . . . . 68 3.16 Schematic illustration of the spin states in the electron-hole bound carrier pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1 Energy dispersion of two subbands in the presence of Zeeman splitting and spin-orbit coupling are plotted from Eq. (5.5). . . . . . . . . . . . . . . . . . . . 80 5.2 Two Fermi surfaces are plotted from Eq. (5.29). . . . . . . . . . . . . . . . . . . . . . 81 ix 5.3 Evolution of the Fermi surfaces near the point ky = 0 for di erent EF are plotted from Eq. (5.29). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.4 Smearing of the right edge of the chiral resonance at small Z !1 is plotted from Eq. (5.22). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.5 Optical conductivity at zero temperature, Eqs. (5.25), (5.26) vs. dimen- sionless frequency, != Z, are plotted in the units (e2= )(!1= Z), for three values of the ratio SO= Z. (a) SO= Z = 0:5, Zeeman splitting dominates. (b) Zeeman and SO splitting nearly \compensate" each other, SO= Z = 0:99. (c) SO= Z = 10, SO splitting dominates. . . 87 5.6 The optical conductivity, Eq. (5.32), in the case of spin-orbit coupling only, is plotted in the units e2=16 as function of dimensionless deviation !=!1, for three dimensionless temperatures, SO=T . . . . . . . . . . . . . . . 90 5.7 Evolution of optical conductivity with T in the regime Z = SO. . . . . . . 94 6.1 The three e ects that we consider are illustrated separately. . . . . . . . . . . . 101 6.2 Illustration of the conductance in a quantum wire from Ref. [1], with zero magnetic eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 6.3 Illustration of the conductance in a quantum wire from Ref. [11], with nonzero magnetic eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.4 Numerical plots of the dispersion relation and conductance in a quantum wire, from Ref. [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.5 Plot of the experimentally measured conductance in a quantum wire, from Ref. [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.6 Illustration of the spin hall e ect, which was rst demonstrated experi- mentally by Ref. [20] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.7 Energy dispersion plotted from Eq. (6.67) is shown. The inset illustrates how the dispersion behaves linearly with P, near the compensation point, P = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.8 The absorption lineshape is plotted from Eq. (6.43) for three characteristic temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 6.9 Energy dispersion for the intermediate value of , = 0:02, is shown from Eq. (6.67). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.10 Energy dispersion for the characteristic value of weak , = 0:02, is shown from Eq. (6.67). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.11 Energy dispersion for the strong value of , = 1:5, for lower energies, W < 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 x 6.12 Energy dispersion for the strong value of , = 1:5, for lower energies, W < 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.13 The probability density, P(y), and spin projections, y and z, are plotted for intermediate = 0:1, shown in red, orange, and blue, respectively. 126 6.14 Similar to Fig. 6.13 for strong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 xi ACKNOWLEDGMENTS I would like to express my deepest gratitude to my advisor, Dr. Raikh, for providing a welcoming atmosphere for studying physics. Dr. Raikh, with patience, gave me guidance and insight into theoretical physics. In the future, I hope to make Dr. Raikh proud of his invested time. I thank the members of my graduate committee for their guidance, Dr. Starykh especially, for introducing me to spin liquids and providing nancial assistance. I also thank Dr. Boehme for discussions which initiated part of the study in this thesis. I thank Dr. Dawson for his inquisitive nature that drove me to study harder and more in depth. I am very proud to have Dr. Cherkaev on my committee because the thesis involves challenging mathematical integrals. I would also like to thank a good friend, Jade, who taught me to believe in myself and helped me to believe that I deserve victory. Finally, I would like to thank my husband, Ronnie, and his son Jordyn. It was through Ronnie's professorial demeanor that I was able to achieve my Phd and Jordyn is light and cheerful personality helped to keep me motivated. PART I MANY-BODY ASPECTS OF RABI OSCILLATIONS 2 1.1 Preface In a seminal paper, Ref. [1], in 1937 I. Rabi pointed out that the level population of an ac-driven two-level system oscillates with frequency, R, proportional to the amplitude of the driving eld. As a two-level system, Rabi chose Zeeman split levels and an ac magnetic eld as the ac driving force. In notations more common in optics, his nding can be recapped as follows. Suppose the system of two levels with spatial wave functions 1 and 2 is illuminated by light with amplitude of electric eld, F, and frequency !L close to the energy di erence between the levels. Neglecting all other levels and spontaneous emission, we can describe the evolution of the system within the time-dependent Schr odinger equation, i @ @t = ^H + eFx 2 ei!Lt + e􀀀i!Lt : (1.1) The essence of the Rabi paper is that for weak-enough driving, the evolution of the level occupations proceeds with frequency much smaller than the frequency of the light. Formally, under this condition, the solution of the Schr odinger equation can be simpli ed as follows. The wavefunction, (t) can be presented as a sum of a product of slow and fast oscillating functions, which reads = a1(t) exp [􀀀iE1t] 1(x) + a2(t) exp [􀀀iE2t] 2(x): (1.2) Upon substituting this form into the Schr odinger equation, we neglect the fast terms coming either from ei!Lt or e􀀀i!Lt and get two coupled equations i @a1 @t = Ra2e􀀀i t; (1.3) i @a2 @t = Ra1ei t; (1.4) where = E2 􀀀 E1 􀀀 !L. The two coupled Eqs. (1.3) and (1.4) can be reduced to a single equation describing the slow amplitude, say a1, by rst di erentiating Eq. (1.3) and substituting @a2=@t and a2, which gives @2a1 @t2 + i @a1 @t + 2 Ra1 = 0: (1.5) Solution to the second order linear di erential equation for a1, Eq. (1.5), can easily be found. We present the solution in the form of the population, which reads ja1(t)j2 = 1 􀀀 4 2 R 4 2 R + 2 sin2 1 2 q 4 2 R + 2t : (1.6) The characteristic time scale in this system is the inverse dipole energy. In other words, the inverse Rabi frequency. Then the justi cation of the approximation made, which is 3 separation into the slow and fast parts or, in other words, the rotating wave approximation (RWA), is that the Rabi frequency is much smaller than the frequency of light. If this condition is met, we easily nd that if the level was occupied at time t = 0, then, under perfect resonance, its population oscillates as cos2 Rt. Away from resonance, the frequency of oscillations is higher, but the amplitude is smaller. Now we have to include the spontaneous emission, resulting from interaction of elec- tron with zero-point motion of a vacuum. The way to do it was paved by Felix Bloch, 9 years after Rabi. Using the optical language, to account for spontaneous emission we rewrite the system for amplitudes a1 and a2 as a system for ja1j2, ja2j2, a1a 2 and a 1 a2. In the density matrix language they are called 11, 22, 12, 21, respectively. Then the density matrix equations read @ 11 @t = 􀀀i R 2 ( 21 􀀀 12) + 􀀀 22; (1.7) @ 22 @t = 􀀀i R 2 ( 12 􀀀 21) 􀀀 􀀀 22; (1.8) @ 12 @t = i 12 + i R 2 ( 11 􀀀 22) 􀀀 􀀀 2 12; (1.9) @ 21 @t = 􀀀i 21 􀀀 i R 2 ( 11 􀀀 22) 􀀀 􀀀 2 21: (1.10) Note that the spontaneous emission is incorporated via the last terms in the four equa- tions. We see that the population of the upper state is decreased due to spontaneous emission, while the population of the lower state is increased. Adding the two, we con rm that the sum of the populations is zero, meaning the total probability is constant. Less trivial is to comment why the nondiagonal elements of the density matrix decay with the rate 􀀀=2, rather than 􀀀. In some way, spontaneous emission is similar to friction. Solving Eqs. (1.7)-(1.10), in small parameter 􀀀 for 11, we nd 11(t) = 2 R + 􀀀2 2 2 R + 􀀀2 + 2 R exp 􀀀 􀀀3􀀀 4 t cos( Rt) 2 2 R + 􀀀2 ; (1.11) that the Rabi oscillations always decay with spontaneous emission, despite the fact that the two-level system is constantly driven; whereas, a harmonic oscillator with friction will maintain oscillations, if it is under stationary driving. To observe Rabi oscillations, the driving should be strong enough so that R exceeds 􀀀. From Eqs. (1.7)-(1.10), we also see that the energy levels of the two-level system enter only into the equations for nondiagonal matrix elements. This fact explains how the environment a ects the Rabi oscillations via the mechanism additional to the spontaneous 4 emission. Namely, the environment can be viewed as objects which switch at random moments from \up" to \down" states in the course of their thermal motion, resulting in time-dependent shifts of the energy levels of the two-level system. These time-dependent shifts cause the damping of the Rabi oscillations after the characteristic time usually denoted with T2. By now Rabi oscillations were observed experimentally for a very wide variety of systems and in frequency ranges from optics !L 1:0 eV to nuclear !L 0:1meV. Obviously, in optical experiments, the RWA approximation is perfectly applicable, since the ratio R=!L is very small 10􀀀5. However, in magnetic-resonance experiments, this ratio equals B1=B0, where B0 is the dc and B1 is the ac magnetic eld, and can be much larger. For this reason, in the remainder of this Preface, we analyze quantitatively the accuracy of the RWA, i.e., the e ect of nite ratio R=!L on the Rabi oscillations. We start from the Schr odinger equation, Eq. (1.1), and substitute (r; t) = a1(t) 1(r)+ a2(t) 2(r), then multiplying by 1 ( 2) and integrating over the spatial coordinates, we get two coupled equations for a1 and a2, which read i @a1 @t = E1a1 + Ra2 ei!Lt + e􀀀i!Lt ; (1.12) i @a2 @t = E2a2 + Ra1 ei!Lt + e􀀀i!Lt : (1.13) The Bloch theorem dictates the following general form of the solution of the above system a1 = e􀀀is1t X p Cp exp(􀀀2ip!Lt); a2 = e􀀀is2t X n Dn exp(􀀀2in!Lt); (1.14) where the parameters s1 and s2 are related as s2 􀀀 s1 = !L: (1.15) Substituting Eqs. (1.14) into the system Eqs. (1.12) and (1.13) and equating the like terms yields the following in nite system of linear equations (s1 + 2p!L 􀀀 E1)Cp = R 􀀀 Dn + Dn+1 ; 􀀀 s2 + 2n!L 􀀀 E2 Dn = R 􀀀 Cp + Cp􀀀1 : (1.16) Consider the terms with p = n = 0 in Eq. (1.16), which read (s1 􀀀 E1)C0 = R(D0 + D1); 5 (s2 􀀀 E2)D0 = R(C0 + C􀀀1): (1.17) Keeping only the terms D0 and C0 immediately leads to the characteristic equation (s1 􀀀 E1)(s2 􀀀 E2) = 2 R; (1.18) which is the standard Rabi result. Neglecting D1 and C􀀀1 corresponds to RWA. To study the consequences of going beyond the RWA, we keep these terms, but neglect all other Cp and Dn. This results in the system which consists of Eqs. (1.17) plus two more equations 􀀀 s1 􀀀 2!L 􀀀 E1 C􀀀1 = RD0; (1.19) 􀀀 s2 + 2!L 􀀀 E2 D1 = RC0: (1.20) Expressing C􀀀1 and D1 from these two equations and substituting into Eqs. (1.17), we arrive at the modi ed characteristic equation s1 􀀀 E1 􀀀 2 R s2 + 2!L 􀀀 E2 s2 􀀀 E2 􀀀 2 R s1 􀀀 2!L 􀀀 E1 = 2 R: (1.21) The basic condition R !L allows one to replace s2 in the denominator of the rst bracket in Eq. (1.21) by E1 and s1 in the denominator of the second bracket in Eq. (1.21) by E2. After that, the second terms in the brackets coming from corrections to RWA can be simply viewed as shifts of the bare energy-level positions: E2 ! E2 􀀀 2 R 2!L , E1 ! E1 + 2 R 2!L . Overall, going beyond the RWA leads to the driving induced renormalization of the bare level separation E2 􀀀 E1 by a small amount 2 R !L without changing the Rabi frequency. This result was obtained by a di erent method in a seminal paper [3] by F. Bloch and Siegert. Another important consequence of a nite ratio R=!L is the existence of the mul- tiphoton resonances. Their existence was uncovered much later than the conventional Rabi oscillations [4]. It turns out that when the two-level system is illuminated by light with frequency !L (E2 􀀀 E1)=(2p + 1), where p is an integer, the populations of levels also undergo periodic oscillations with much smaller frequency p R=!(p􀀀1) L . Below we illustrate this phenomenon for the example of 3-photon oscillations corresponding to p = 1. It is convenient to rescale the energy levels from midway of the di erence E2􀀀E1 = E. Then E1 ! 􀀀E 2 and E2 ! E 2 . Adding and subtracting Eqs. (1.12) and (1.13) we get i @ @t (a1 + a2) = E 2 (a1 􀀀 a2) + R(a1 + a2)f(t); (1.22) 6 i @ @t (a1 􀀀 a2) = E 2 (a1 + a2) 􀀀 R(a1 􀀀 a2)f(t); (1.23) where f(t) = ei!Lt + e􀀀i!Lt. The coupled equations, Eqs. (1.22) and (1.23), and be cast in the form of a single second-order equation by rst solving for (a1􀀀a2) from Eq. (1.23) and then substituting it into Eq. (1.22). We get @2 @t2 + i R @f(t) @t + E2 4 + 2 Rf2(t) = 0; (1.24) where = a1 +a2. Again, the Bloch theorem dictates a solution of the form Eq. (1.14). For the 3-photon resonance, s is close to 1 2E, while E is close to 3!L. This suggests that, instead of the terms / exp(ist) and / exp[i(s􀀀!L)t] for conventional Rabi oscillations, the \resonating" terms for 3-photon Rabi oscillations are / exp(ist) and / exp[i(s 􀀀 3!L)t]. To provide the \minimal" (in terms of powers of R) coupling of these terms one has to keep two nonresonant terms, namely / exp[i(s 􀀀 !L)t] and / exp[i(s 􀀀 2!L)t]. Thus we search for (t) in the form (t) = aeist + bei(s􀀀!L)t + cei(s􀀀2!L)t + dei(s􀀀3!L)t: (1.25) Inserting Eq. (1.25) into Eq. (1.24) and equating like-terms gives E2 4 + 2 2 R 􀀀 s2 a 􀀀 !L Rb + 2 Rc = 0; (1.26) E2 4 + 2 2 R 􀀀 (s 􀀀 3!L)2 d + !l Rc + 2 Rb = 0; (1.27) E2 4 + 2 2 R 􀀀 (s 􀀀 2!L)2 c + !L R(b 􀀀 d) + 2 Ra = 0; (1.28) E2 4 + 2 2 R 􀀀 (s 􀀀 !L)2 b + !L R(a 􀀀 c) + 2 Rd = 0: (1.29) Expressing b and c from Eqs. (1.28), (1.29), and substituting them into Eqs. (1.26) and (1.27), we get the following two coupled equations relating a and d E2 4 + 5 2 2 R 􀀀 9 4 !2L 􀀀 3!Ls1 a = 􀀀 3 4 3 R !L d; (1.30) E2 4 + 5 2 2 R 􀀀 9 4 !2L + 3!Ls1 d = 3 4 3 R !L a; (1.31) where we introduced s1 = 3 2!L 􀀀 s and used the fact that s1 !L. The system Eqs. (1.30), (1.31) readily yields s21 = 1 16 6 R !4L + 1 !2L E2 4 􀀀 9!2L 4 + 5 2 2 R 2 : (1.32) Note that this expression has the conventional Rabi form with e ective Rabi frequency equal to 3 R=4!2L and Bloch-Siegert shift to equal to 5 2 R=6!L instead of 2 R!L for conventional Rabi oscillations. 7 Another interesting aspect of the Rabi oscillations pertains to their experimental observation in the situation when they are induced in a macroscopic sample, which can be viewed as an ensemble of two-level systems. Each two-level system has its own individual environment. For example, each Zeeman-split spin of a localized carrier in the organic material has its individual nuclear environment. This environment produces a random hyper ne eld which adds to the external eld. As a result, each two-level system possesses its individual detuning, , from the resonance. Modi cation of the Rabi oscillations due to the detuning is given by Eq. (1.6). The observed Rabi oscillations are, therefore, proportional to the oscillating part of Eq. (1.6) averaged over the distribution of . In this situation, valuable information about the e ect of the environment on the Rabi oscillations can be derived not from the time domain, but rather from the frequency domain, as it was suggested, e.g., in Ref. [24]. If one performs a Fourier transform of the measured Rabi oscillations, Eq. (1.6), then the integral averaging over reads I(s) = Z d 4 2 R 4 2 R + 2 s 􀀀 q 4 2 R + 2 f 􀀀 = 4 2 R jsj q s2 􀀀 4 2 R f q s2 􀀀 4 2 R ; (1.33) where f is the distribution function of the z-projection of a hyper ne eld. The remark- able property of I(s) is that it diverges as s approaches to R from above. This divergence suggests that the major contribution to I(s) comes from the two-level systems for which detuning is anomalously small, and is similar to the divergence of the density of states in a superconductor in the vicinity of the band-edges. It indicates that no matter how strong the disorder is, the Rabi oscillations can, in principle, be observed from analysis of I(s). This analysis should be performed by tting the measured I(s) by the function I(s) / (s 􀀀 2 R) p s 􀀀 2 R exp 􀀀 2 R 20 (s 􀀀 2 R) : (1.34) From the exponential fall-o in Eq. (1.34), one can infer the magnitude of the disorder, 0, which is the width of the distribution function f( ). Certainly, the realistic spectra of the Fourier transform of the Rabi oscillations do not have a sharp boundary at s = R, see Ref. [24]. The reason for this is the decay, exp(􀀀 t), of the Rabi oscillations with time due to inelastic processes. Due to this decay, the sharp boundary in I(s) will get smeared in the interval js􀀀2 Rj . It is important, however, that if the decoherence time, 􀀀1 is long enough, 20 = R, the tail of I(s) in the domain s < 2 R remains much narrower than the tail towards large s. Then the behavior of I(s) in the domain js 􀀀 2 Rj can be simply established, since the distribution function of can be replaced by f(0) in this domain. Then we have I(s) = f(0) Z 1 􀀀1 d s 􀀀 q 4 2 R + 2 2 + 2 : (1.35) Substituting = 2 R sinh u, we get I(s) = 1 2i Z du s + i R cosh u 􀀀 (s + i ) 􀀀 s 􀀀 i R cosh u 􀀀 (s 􀀀 i ) : (1.36) From Eq. (1.36), it can be seen that the main contribution of the integral comes from small u, i.e., from small detuning. Using this fact, we evaluate the integral, yielding I(x) = 2 R 1=2 "p x2 + 1 + x x2 + 1 #1=2 ; (1.37) where x = s 􀀀 2 R : (1.38) From Eq. (1.37), we see that the broadening remains asymmetric. In other words, the smearing of the right-edge, x > 0, is wider than the left-edge, x < 0. Graphical representation of Eq. (1.37) is shown in Fig. 1.1. When comparing our ndings, Fig. 1.1, to the experimentally observed Rabi oscillations, Fig. 1.2, we see the same characteristics: a narrower tail for small frequencies below the Rabi frequency than for frequencies above. x -5 0 5 10 15 0.0 0.2 0.4 0.6 0.8 1.0 Figure 1.1: The broadening of the Rabi frequency-peak in the frequency domain is illustrated from Eq. (1.37). We see that the smearing of the left-edge is narrower than for the right-edge. 9 Figure 1.2: Plot of the measured Rabi oscillations on pairs of electrostatically bound charge carriers, for weak spin-exchange coupling is shown in (a); while in (b), the Fourier transform spectra of Rabi oscillations obtained at di erent B1 eld strengths is shown, from Ref. [30]. The red line corresponds to the frequency R where both spins precess independently, while the purple line corresponds to 2 R where the spin pair precesses jointly. 10 1.2 References [1] I. I. Rabi, Phys. Rev. 51, 652 (1937). [2] F. Bloch and A. Siegert, Phys. Rev. 57, 522 (1940). [3] D. R. McCamey, K. J. van Schooten, W. J. Baker, S.-Y. Lee, S.-Y. Paik, J. M. Lupton, and C. Boehme, Phys. Rev. Lett. 104, 017601 (2010). [4] D. R. McCamey, K. J. van Schooten, W. J. Baker, S.-Y. Lee, S.-Y. Paik, J. M. Lupton, and C. Boehme, Phys. Rev. Lett. 104, 017601 (2010). CHAPTER 2 RABI-VIBRONIC RESONANCE WITH LARGE NUMBER OF VIBRATIONAL QUANTA 2.1 Introduction As mentioned in the Preface, coupling to the dynamic environment tends to damp the Rabi oscillations [1] of a resonantly driven two-level system via the random shifts of the levels E1 and E2. This mechanism is at work only at nite temperature. It is also assumed that the dynamics of the environment are completely independent of the dynamics of the two-level system. Such a common perception misses the following mechanism of \communication" between the two-level system and the environment, which persists even at zero temperature. Namely, the two-level system, undergoing Rabi oscillations, sets the environment into motion. Then the driven environment exercises feedback on the two-level system. This mechanism is schematically illustrated in Fig. 2.1 and is uncovered in the present thesis for the rst time. Clearly, it does not cause the decay of the Rabi oscillations. Rather, due to this mechanism, the Rabi oscillations acquire the collective character. To illuminate this mechanism, in the present chapter, we consider a particular form of environment for which this mechanism is most e cient. Usually, the environment is viewed as a medium with a continuous spectrum of modes. Less common is the situation when the environment possesses a single or several well-de ned frequencies. For concreteness, we will consider the situation depicted in Fig. 2.2 when the lower level of the two-level system is coupled to an oscillator (a mass M and a spring), which represents a single vibrational mode. Obviously, coupling to the oscillator has a strong e ect on the Rabi oscillations in the regime of the vacuum Rabi splitting [5] when the oscillator frequency !0 is close to the transition frequency !12. It is less obvious what e ect the coupling to the oscillator will have on the Rabi oscillations when !0 is much smaller than !12 and is comparable to the Rabi frequency R. One can argue on physical 12 Figure 2.1: Illustration describing the main idea. The environment is modeled as an en- semble of harmonic oscillators that are coupled to the two-level system. Namely, through coupling, the two-level system drives the environment and in turn, the environment drives the two-level system. 13 Figure 2.2: A schematic illustration of the system under consideration. The two-level system is driven by near-resonant light, !12 E2 􀀀 E1. The level E1 is linearly coupled to a classical oscillator with frequency !0. The Rabi oscillations are strongly modi ed when !0 is close to R, where R is the Rabi frequency. grounds that the e ect of coupling on the Rabi oscillations will be strong in the vicinity of the condition !0 R, which we dub Rabi-vibronic resonance. Indeed, consider the Hamiltonian H = ^X ^n1; (2.1) describing the linear coupling. Here ^X = p 1 2M!0 (by + b) is the operator of the oscillator displacement, by is a creation operator of the vibrational quantum, ^n1 is the occupation of the level E1, and = 􀀀 2M!3 0 1=2 , where is a dimensionless coupling constant. In de nition of ^X , , and thereafter we, set ~ = 1. In the course of the Rabi oscillations, the average ^n1 changes with time as n1(t) = 1 2 􀀀 1 + cos Rt : (2.2) Then at R !0, the second term in Eq. (2.2) gives rise to a resonant driving force acting on the oscillator. In turn, the strongly driven oscillator provides a resonant feedback [6, 7] on the two-level system. Thus, as R, which is proportional to the ac eld driving the two-level system, increases, we expect the Rabi oscillations to be strongly modi ed near the resonant condition. Among possible experimental realizations of the situation, Fig. 2.2 is a suspended carbon nanotube in an inhomogeneous electric eld, which creates a con nement for an exciton [8, 9, 10, 11], illustrated in Fig. 2.3. The localized exciton can be viewed as a two-level system. Bending modes have discrete frequencies due to nite nanotube length and can be viewed as oscillators with very low friction. While a typical transition frequency in such a system is [13, 12] !12 1015 Hz, the oscillator 14 Figure 2.3: Illustration of the experimental realization of the system considered from Ref. [11]. It is well known that nanotubes have inhomogeneities which play the role of the con ning potential for the electron and hole, frequency !12 1015Hz. Then the nanotube itself acts as a harmonic oscillator with frequency !0 109Hz. 15 frequency [10] is much smaller, !0 109 Hz. The resonant condition can be achieved by adjusting the illumination intensity. Another area in which the situation Fig. 2.2 is relevant, is the cavity QED, [14] where a two-level system is realized in the form of a superconducting qubit, while the oscillator is a LC circuit. The majority of experimental and theoretical studies in this eld are focused on the strong coupling in the domain !0 !12. However, in experiments from Refs. [15, 16], an ac driven superconducting qubit was coupled to a \slow" LC oscillator tuned to R. It was observed that the noise spectrum of the oscillator exhibits a Lorentzian peak [17] as a function of R 􀀀 !0. In theoretical papers [18, 19, 20] initiated by the experiment in Ref. [15], collective motion of the oscillator coupled to a qubit was studied within the density-matrix formalism, and both subsystems were treated quantum mechanically. In view of the complexity of this description, nal results were obtained numerically for particular values of a coupling parameter . A notable nding of Refs. [18, 19, 20] is that, in the vicinity of the condition !0 R, collective Rabi-vibronic motion becomes bistable. There are still several basic questions to be answered, among which are the following: (i) How does the frequency, s, of the collective oscillations depend on ? (ii) What is the width of the resonance, i.e., the domain 0 = R 􀀀 !0 of the Rabi frequencies where Rabi oscillations are modi ed due to coupling? (iii) How does the decay of the Rabi oscillations depends on the oscillator friction? The above questions are studied in the present paper. Our main nding is that the width of the Rabi-vibronic resonance is small for weak coupling, namely, 0 = 4=3!0 = !2=3 p !1=3 0 !0; (2.3) where !p = 2!0 is the polaronic shift. Equation (2.3) suggests that, while 0 is much smaller than !0, it is much bigger than !p. Most importantly, Eq. (2.3) guarantees that, in the resonant domain ( R􀀀!0) 0, the oscillator is highly excited and can be treated as classical. This allows the analytical description of the resonance. In this regard, the situation we consider, a two-level system coupled to a classical oscillator, is similar to the Rabi resonance considered in Refs. [21, 22], where the two-level system was driven by two classical elds: one with frequency close to !12 and one with frequency close to R. We will see that in the domain ( R􀀀!0) 0, the frequency s of collective oscillations di ers from R also by 0. Bistable behavior of the dependence s( R) emerges naturally within our approach; the frequency jump rate between two stable regimes is also 0. 16 In addition, in the present paper, we study how the Rabi-vibronic resonance depends on detuning of the driving frequency from !12, on intrinsic anharmonicity of the oscillator, and how the modi ed Rabi oscillations decay with time due to relaxation of the two-level system and due to friction in the oscillator. 2.2 Basic equations We rst assume that the displacement X(t) is a classical variable and will later check this assumption. The equation of motion for X(t) reads X + _X + !2 0X = 2M 􀀀 1 􀀀 w ; (2.4) where w = 1􀀀2n1 is the population inversion, and is the friction in the oscillator. The evolution of w with time is described by the system of optical Bloch equations. We write them for variables w(t), u(t), and v(t), where u(t) and v(t) are the real and imaginary parts of the nondiagonal elements of the density matrix, respectively, [23] w_ (t) = 􀀀 Rv 􀀀 􀀀 􀀀 1 + w ; (2.5) u_ (t) = 􀀀 􀀀 X(t) v 􀀀 􀀀 2 u; (2.6) v_(t) = 􀀀 X(t) u + Rw 􀀀 􀀀 2 v; (2.7) where 􀀀 is the relaxation rate of the excited state. Note that, while the oscillator is driven by w(t), it exercises a feedback on the two-level system via u(t) and v(t). We require that the level E1 at t = 0 be occupied while the level E2 is empty, i.e., w(0) = 􀀀1. We also assume that the dipole moment and dipole current are initially zero, leading to v(0) = 0 and u(0) = 0, respectively. From Eq. (2.5), we see that the initial conditions for v and w require that w_ (0) = 0. 2.3 Modi ed Rabi oscillations 2.3.1 Oscillation frequency The system Eqs. (2.5)-(2.7) can be reduced to two coupled equations by excluding v(t) and expressing u(t) in terms of w(t). Then one gets w + 3 2 􀀀w_ + 2 R + 􀀀2 2 w + 􀀀2 2 = 􀀀 R 􀀀 X(t) u(t); (2.8) u(t) = 􀀀 Z t 0 dt0 R e􀀀(t0􀀀t)=2􀀀 X(t0) 􀀀 h w_ (t0) + 􀀀 􀀀 1 + w i : (2.9) 17 We start from the simplest case, 􀀀 ! 0, ! 0, ! 0, and search for a solution of the system Eqs. (2.4), (2.8), and (2.9), in the form w(t) = 􀀀cos st. Substituting this form into Eq. (2.4), we nd the displacement X(t) = cos st 2M 􀀀 !2 0 􀀀 s2 = X0 cos st: (2.10) Static displacement, =2M!2 0, can be neglected compared to the oscillating part. Substi- tuting X(t) into Eq. (2.9), we nd u(t) u(t) = 􀀀 2 (1 􀀀 cos 2st) 8M R 􀀀 !2 0 􀀀 s2 : (2.11) Substituting Eqs. (2.10) and (2.11) into the right-hand side of Eq. (2.8), and equating the terms / cos st in both sides, we nd a closed equation for s, 2 R 􀀀 s2 = !2 p!4 0 8 􀀀 !2 0 􀀀 s2 2 : (2.12) Thus coupling to the oscillator causes the shift of the oscillation frequency from R, as stated in the Introduction. Note that the term / cos 2st in u(t) will also give rise to the nonresonant contribution / cos 3st in w(t), causing a weak anharmonicity of the oscillations. Away from resonance, we can substitute s = R into the right-hand side of Eq. (2.12). Then Eq. (2.12) yields a correction to the Rabi frequency due to coupling to the oscillator s = R 􀀀 !2 p!0 64 􀀀 !0 􀀀 R 2 : (2.13) This expression is valid only if the correction on the right-hand side is much smaller than 􀀀 !0􀀀 R . Equating the correction to 􀀀 !0􀀀 R , we nd that the width of the resonance, 􀀀 !0 􀀀 R 0, is given by Eq. (2.3). Recall now our basic assumption that the oscillator is classical. We are now in position to verify this assumption. In the resonant domain, the amplitude X(t) can be estimated from Eq. (2.10) as X =M!0 0. Then for the ratio of the energy of oscillations to the vibrational quantum !0, we get the following estimate: MX2 0!2 0 !0 !0 !p 1=3 = 􀀀2=3 1: (2.14) Thus, for weak coupling, the classical treatment of the oscillator is justi ed. 18 2.3.2 Vicinity of the resonance To incorporate nite detuning into Eq. (2.12), it is convenient to rewrite Eq. (2.8) keeping all -dependent terms in the right-hand side, w + 2 Rw+ 2X(t) Z t 0 dt0X(t0)w_ (t0) = 􀀀 X(t)􀀀 w(t)+1 + Z t 0 dt0X(t0)w_ (t0): (2.15) The term / 2 in the right-hand side leads to a standard modi cation of the Rabi frequency to 􀀀 2 R + 2 1=2. The last term is proportional to sin2 st, and does not contain the rst harmonics. The term / cos st comes from the combination X(t) in the right-hand side. Emergence of this term, which is odd in detuning, is the result of the coupling of the vibronic mode only to the level E1. This term results in the following modi cation of Eq. (2.12): 2 R + 2 􀀀 s2 = !2 p!4 0 8 􀀀 !2 0 􀀀 s2 2 􀀀 !p!2 0 !2 0 􀀀 s2 : (2.16) Near the resonance 􀀀 R 􀀀!0 !0, this equation can be simpli ed. Upon introducing dimensionless variables, z = s 􀀀 !0 !2=3 p !1=3 0 ; (2.17) -0.3 0.0 0.3 0.6 -0.5 0.0 0.5 1.0 x z Figure 2.4: Red line: Dimensionless frequency z de ned by Eq. (2.17), versus the dimensionless deviation x from the R = !0 is plotted from Eq. (2.19). The unstable solution is shown with the dashed line. The blue line z(x) = x, corresponds to the absence of coupling to the oscillator. 19 x = x0 + 8 02; x0 = R 􀀀 !0 !2=3 p !1=3 0 ; (2.18) Eq. (2.16) assumes the form (z 􀀀 x)z2 + 0z = 􀀀 1 64 ; (2.19) where dimensionless detuning 0 is de ned as 0 = 4!1=3 p !2=3 0 : (2.20) Note that characteristic detuning !1=3 p !2=3 0 is much bigger than the width of the resonance 0, but much smaller than R. Figure 2.4 shows the solution of Eq. (2.19) for zero detuning. The blue line z = x corresponds to the Rabi oscillations without coupling. We see that bistability develops for x > 3 2􀀀8=3. At x = 3 2􀀀8=3, the frequency s experiences a jump by 2􀀀8=3 0. Two values of z corresponding to stable solutions de ne, via Eq. (2.17), two frequencies of the modi ed Rabi oscillations. They also de ne two corresponding amplitudes of the oscillator, X = 1 2 1=3z (2M!0)􀀀1=2 : (2.21) The last factor in Eq. (2.21) is the amplitude of a zero-point motion of the oscillator. As the dimensionless deviation x from resonance increases, the upper branch approaches z = x. For this branch, the frequency of the Rabi oscillations is close to R and the amplitude of the oscillator is small. For the lower branch, z is small, i.e., the frequency of the oscillations approaches !0 with increasing x. For this branch, the oscillator is highly excited. Figures 2.5 and 2.6 illustrate the e ect of detuning on the frequency of oscillations s. Note that there is a qualitative di erence between Fig. 2.4 for zero detuning, and Figs. 2.5, 2.6 for positive and negative detunings, respectively. For zero detuning, the domain of bistability exists only when R > !0, whereas for nite detuning, bistable regions emerge both to the left and the right from the resonance. As one changes the dimensionless deviation x0 from the resonance, from negative to positive, for 0 = 0, bistability corresponds to x0 > 3 2􀀀8=3. For nite positive detuning, 0 > 0, the rst domain of bistability occurs at x0 < 0, then disappears, and re-emerges at positive x0 greater than 3 2􀀀8=3. Conversely, nite negative detuning simply broadens the domain of bistability as compared to = 0. The bistable region starts for x0 < 0. A peculiar dependence of s on the deviation from resonance is also re ected in the amplitude of the oscillator. This e ect is discussed in Sec. 2.4. 20 -0.6 -0.3 0.0 0.3 0.6 0.9 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 ' ' ' x' z Figure 2.5: Dimensionless frequency z of oscillations of adriven two-level system is plotted from Eq. (2.19) versus the dimensionless deviation x0 from the resonance for three positive dimensionless detunings 0, de ned by Eq. (2.20). As detuning increases, the unstable branch shifts from positive z to negative z, and both stable values of z become positive (for positive x0) or negative (for negative x0). 21 -0.9 -0.6 -0.3 0.0 0.3 0.6 0.9 -1.0 -0.5 0.0 0.5 1.0 z ' ' ' x' Figure 2.6: The same as in Fig. 2.5 for three negative detunings. Note that negative 0 broadens the range of bistability. 22 2.3.3 E ect of intrinsic anharmonicity of the oscillator Suppose that in addition to the harmonic part of the oscillator energy, M!2 0X2=2, a weak intrinsic anharmonicity, X4=4, is present. Then Eq. (2.10) will assume the form 3 4 X3 a + 􀀀 !2 0 􀀀 s2 Xa = 2M : (2.22) The second relation, s2 = 2 R 􀀀 2X2 a=8, between the amplitude Xa and the frequency s, which follows from Eq. (2.8), remains unchanged. It is now more convenient to express s from this relation and substitute it into Eq. (2.22). This yields a cubic equation for Xa, 2!0 􀀀 !0 􀀀 R Xa + 6 + 2 8 X3 a = 2M : (2.23) If we now set = 0, then Eq. (2.23) will have multiple real solutions for Xa in the domain 􀀀 R 􀀀!0 < 􀀀3 2􀀀8=3 0, i.e., the same as determined from Eq. (2.19) with 0 = 0. We see from Eq. (2.23) that, depending on the sign of , intrinsic anharmonicity can either shift the threshold of bistability to the left (for positive ) or to the right (for negative ). Anharmonicity will also a ect the magnitude of the jump of the frequency s of the oscillations. This magnitude will get modi ed from 2􀀀8=3 0 to 2􀀀8=3 0 􀀀 1 + 6 = 2 􀀀2=3, i.e., the jump will become smaller for < 0. 2.4 Decay of the oscillations Up to now we disregarded both mechanisms of dissipation: nite relaxation 􀀀 and the friction in the oscillator, . Rabi oscillations will decay with the rate , which is determined by 􀀀, in the regime 􀀀 , or by friction in the regime 􀀀 . We will consider both cases separately. We emphasize that, as the oscillations decay, so does the coupling between the oscillator and the two-level system. Thus, the decay will be accompanied by the change of frequency back to R. We cannot capture this evolution of frequency with time analytically. To nd the decay rate only, we will adopt the approach based on the energy conservation. 2.4.1 Friction-dominated regime, 􀀀 Upon neglecting 􀀀 in Eq. (2.8) and setting = 0 we have w + 2 Rw = 􀀀 2X(t) Z t 0 dt0X(t0)w_ (t0): (2.24) Multiplying both sides by w_ and integrating from 0 to t, we arrive to the following conservation law: w_ 2 2 + 2 R 2 (w2 􀀀 1) = 􀀀 2 2 Z t 0 dt0X(t0)w_ (t0) 2 : (2.25) 23 The right-hand side describes the energy exchange between the two-level system and the oscillator. As a next step, we multiply the equation of motion of the oscillator Eq. (2.4) by _X and integrate from 0 to t. Then we arrive at the second conservation law, _X 2 2 + !2 0X2 2 + Z t 0 dt0 _X 2(t0) = 􀀀 2M Z t 0 dt0w(t0) _X (t0): (2.26) At long time t 􀀀1 we have _ w, w ! 0 and also _X , X ! 0. Then the left-hand side of Eq. (2.25) for w turns to 􀀀 2 R=2. Combining Eqs. (2.25) and (2.26), we arrive at the relation R = 2M Z 1 0 dt0 _X 2(t0): (2.27) This relation is convenient to nd the decay rate because it contains only _X 2, which is insensitive to the change of frequency in the course of decay. Substituting _X 2 / exp[􀀀2 t], we nd the following relation between and the amplitude of the oscillations at time . 1= : = (MX2 0 R): (2.28) In fact, Eq. (2.28) does not prove that decay is exponential. In the next subsection, we will see that it becomes exponential only for long times, t 1. Note that the second factor in the right-hand side can be rewritten as (MX2 0 2 R)= R. The oscillator is classical when this ratio is big. Thus, we conclude that , i.e., the Rabi oscillations in the region of resonance decay faster than the undriven oscillator. As a next step, we distinguish two cases of weak and strong friction. In the rst case, to nd X that should be substituted into Eq. (2.28) one can use Eq. (2.10), obtained without friction. Then one gets = !p R (!0 􀀀 s)2 ; (2.29) where s is determined from the cubic equation, Eq. (2.13). In the region of resonance, the di erence !0 􀀀 s is 0, which yields !0 !p 1=3 = 2=3 : (2.30) Weak friction requires that j!0 􀀀 sj 0 , i.e., !p. 24 In the region of strong friction, the di erence !0 􀀀 s should be replaced by . Then Eq. (2.29) contains in both sides. Upon solving this equation, we get !p 1=3 0; (2.31) for !p. Equations (2.30) and (2.31) match when !p. The validity of this expression is limited from above by the condition that the oscillator is classical. As we replace !0 􀀀 s by , the estimate for X is X =(M!0 ). Then the kinetic energy can be estimated as M!2 0 h M!0 i2 !1=3 p !4=3 0 2=3 : (2.32) The condition that it is bigger than !0 limits in Eq. (2.31) to (!0!p)1=2 = !0, and correspondingly limits to 0(!0=!p)1=6 = 0= 1=3. From Eqs. (2.30) and (2.31), we see that, upon increasing friction, the decay rate rst grows linearly with , and then sublinearly as 1=3. At the boundary of applicability of the classical description, we have = . For even bigger , classical treatment of the oscillator is not justi ed, but we expect that the oscillator will eventually decouple from the two-level system, and Rabi oscillations will proceed as they do in the absence of the oscillator. It is convenient to reformulate the above results in terms of number, m = !0= , of the oscillation cycles, after which the collective motion e ectively stops. See Fig. 2.8. From Eqs. (2.30) and (2.31), we have m = 8< : 2=3 !0 ; < 2!0; 1 2=3 !0 1=3 ; !0 > > 2!0: (2.33) In the crossover between weak and strong friction regimes, we have m 􀀀4=3. Equation (2.33) is illustrated in Fig. 2.7. For > !0, the assumption of classical motion of the oscillator is violated. The boundary value of m at !0 is still large, m 􀀀1. 2.4.2 The form of the decay For more quantitative analysis of the decay of oscillations, it is convenient to rewrite the energy conservation law Eq. (2.25) in terms of the displacement X(t). Expressing w(t) from Eq. (2.4) and substituting it into Eq. (2.25), we get h d dt 􀀀 X + _X+!2 0X i2 + 2 R 􀀀 X + _X +!2 0X 2 = 2 2 R 4M2 􀀀 2 Z t 0 dt0 h X ... X+!2 0X _X + X X i!2 : (2.34) 25 The rst two terms in the integrand can be presented in the form X ... X + !2 0X _X = d dt h X X 􀀀 X 2 2 + !2 0X2 2 i ; (2.35) so that the integral from these terms is equal to 1 2 (!2 0 􀀀 s2)X2(t). At the same time, the integral from the third term can be rewritten as 1 2 s2 R t 0 dt0X2(t0), and estimated at s2X2 0= . One can check using Eqs. (2.30) and (2.31) that both in the strong-friction and weak-friction regimes, the integral from the last term is bigger than the contribution from the rst two terms. Neglecting this contribution, we can cast Eq. (2.34) in the form of an integral equation for the slow decaying amplitude ~X of the oscillations. Since j!0􀀀sj in both regimes, the nonoscillating part of the left-hand side of Eq. (2.34) can be presented as 1 2 (!2 0 􀀀 s2)2(s2 + 2 R) ~X 2 2!2 0 ~X 2 4M2X2 0 ; (2.36) where we used the de nition Eq. (2.10) of the initial amplitude X0. Upon substituting Eq. (2.36) into Eq. (2.34) and introducing a dimensionless function F(t) = ~X(t)=X0, we arrive at the integral equation Figure 2.7: The number of oscillation cycles, m, before collective motion stops is plotted vs. the dimensionless friction of the oscillator, =!0. For > !0, the assumption that the oscillator motion is classical is violated. 26 F2(t) = 1 􀀀 2 hZ t 0 dt0F2(t0) i2 ; (2.37) where = MX2 0!0 is introduced according to Eq. (2.29). It is easy to check that this equation has a simple solution, F(t) = 1 cosh t : (2.38) We see that the decay of amplitude ~X (t) becomes exponential in the limit t & 1, as mentioned in the previous subsection. 2.4.3 Initial stage of the oscillations After the ac driving eld is switched on, the population inversion starts to oscillate with frequency R. After some number m0 of the oscillation cycles the frequency crosses over to s. See Fig. 2.8. The question of interest is how m0 depends on the coupling strength . We will estimate m0 using the fact that at the initial stage, the system Eqs. (2.4) and (2.24) can be solved perturbatively in a small parameter !p=!0. To nd the perturbative solution, we substitute the \bare" Rabi oscillations w(0) = 􀀀cos( Rt) into Eq. (2.4) and nd X(t) with initial conditions X(0) = 0, _X (0) = 0. The obtained X(t) together with w(0)(t) is then substituted into the right-hand side of Eq. (2.24). Solving this second-order di erential equation with a given right-hand side, we nd that the amplitude of oscillations becomes 1􀀀w(1)(t), i.e., it acquires a time-dependent correction with w(1)(t) given by the following expression w(1)(t) = 2!2 p!2 0 ( 2 + 4 2)2 t 2 􀀀 4 ( 2 + 2) e􀀀 t=2 sin t􀀀 2 􀀀 4 2 2 + 4 2 1 􀀀 e􀀀 t=2 cos t 2 ; (2.39) where = !0 􀀀 R. The derivation of Eq. (2.39) can be found in the Appendix. We can now estimate m0 as Rtc !0tc, where tc is the time after which w(1)(t) becomes 1. Consider rst the limit ! 0. We see that for t 1, w(1)(t) grows with time as !2 p!2 0t4, while for larger times, t 1, w(1)(t) grows as 􀀀 !p!0t= 2. If we determine tc from the small-t asymptote, we will get tc = 1 !0 ; (2.40) and correspondingly m0 = 1 . For this answer to be correct, this tc should belong to the small-t domain, i.e., tc should be smaller than 1. The latter condition, !0 , is met since it coincides with the condition that the oscillator is classical. Consider now the limit ! 0. For t 1, Eq. (2.39) yields w(1)(t) 􀀀 !p!0t2 2. This leads to the estimate 27 Figure 2.8: Illustration depicting how population inversion evolves with time. Initially, the ac driving light is switched on and the system oscillates with frequency R. After some number of oscillations m0, the frequency crosses over to s, Eq. (2.13), and the system exercises collective motion. Some time later, as the oscillations decay, so does the coupling between the two-level system and the oscillator. E ectively, the two-level system and oscillator decouple and frequency changes back to R. The number of oscillations cycles before collective motion stops is described by the parameter m. tc 􀀀 !p!0 􀀀1=2 and m0 􀀀1. Small-t expansion of Eq. (2.39) is valid if the product tc is small. With tc found above, this product can be rewritten in the form tc 2 !p!0 1=2 : (2.41) On the other hand, the oscillator can be treated as classical when the ratio in the right- hand side of Eq. (2.41) is small. Thus, taking the limit t 1 in Eq. (2.39) is justi ed, and the frequency of the Rabi oscillations crosses over from R to s after m0 􀀀1 cycles. Correspondingly, after 􀀀1 cycles, the oscillator will \forget" about the initial phase, imposed by the initial conditions, and will execute a forced harmonic motion with frequency s. In conclusion of this subsection, we note that for the entire scenario of the collective oscillations to be consistent, the time during which the collective motion is established must be shorter than the time during which these oscillations decay. The corresponding condition is m0 < m. It follows from Eq. (2.33) and m0 = 1 that this condition is satis ed 28 in the the entire interval < !0, namely, m is always bigger than 􀀀1, while m0 is always smaller than 􀀀1. 2.4.4 Relaxation-dominated regime, 􀀀 At nite relaxation rate of the two-level system Eq. (2.26) assumes the form w_ 2 2 + 1 2 2 R + 􀀀2 2 􀀀 w2 􀀀 1 + 􀀀2 2 (w + 1) + 3 2 􀀀 Z t 0 dt0 w_ 2 = 􀀀 2 Z t 0 dt0X(t0)w_ (t0) Z t0 0 dt00e􀀀(t00􀀀t0)=2X(t00) w_ (t00) + 􀀀(1 + w) : (2.42) Without coupling to the oscillator, the right-hand side is zero, and Eq. (2.42) describes the decay of the Rabi oscillations due to relaxation. Indeed, upon substituting w_ = R sin 􀀀 Rt0 exp(􀀀 t0) and taking the limit t ! 1, the last term in the left-hand side takes the value 3􀀀 2 R=8 , which leads to = 3􀀀=4. Naturally, this value of follows directly from Eqs. (2.5) and (2.7). Finite coupling to the oscillator would increase the decay rate only if at t ! 1, the integral in the right-hand side exceeds 2 R. Contribution of the second term in the square brackets to the integral can be estimated upon noticing that the product X(t00)w(t00) is a slow function. Assuming that X and w both decay as exp(􀀀 t00) and that 􀀀, the integral over t0 reduces to R 1 0 dt0t0 sin(2st0) exp(􀀀2 t0) = =s3 =!3 0. Then one gets the estimate !2 p 􀀀= 􀀀 !0 􀀀 s 2 for this contribution. Since !0 􀀀 s cannot be smaller than , this contribution cannot exceed !2 p, which is much smaller than 2 R. The contribution from the rst term in the square brackets also cannot exceed 2 R. This becomes apparent upon performing integration by parts, after which the contribution from the rst term assumes the form 􀀀􀀀 2 Z 1 0 dt0e􀀀􀀀t0 Z t0 0 dt00e􀀀t00=2X(t00)w_ (t00) !2 : (2.43) If X and w_ decay much faster than exp(􀀀􀀀t00=4), the inner integral saturates at times t0 􀀀. Then the contribution Eq. (2.43) can be estimated as 2 R!2 p= 􀀀 !0 􀀀 s 2, which is again much smaller than 2 R. We thus conclude that, while coupling to the oscillator modi es the frequency, the decay of the Rabi oscillations in the relaxation-dominated regime is always dominated by the relaxation rate. 2.5 Number of vibrational quanta We studied the behavior of the frequency of the Rabi oscillations near the the reso- nance R = !0. The number of vibrational quanta N is also sensitive to the deviation, 29 R 􀀀 !0, from the resonance and to the detuning . Since N = M!2 0X2=!0 (we set ~ = 1), it can be expressed using Eq. (2.21), as the following: N = 1 2 1=3 1 16z2 = !0 !p 1=3 1 16z2 ; (2.44) where z is the solution of Eq. (2.19). The dependence of N on dimensionless deviation x0 and dimensionless detuning 0 is plotted in Fig. 2.9. The values of N shown correspond only to stable regimes of oscillations. The line of bifurcation points separates the (x0; 0) domains with and without bistability in Fig. 2.9. In the domain of bistability, the higher and lower values of N coincide along the red line. Away from the red line, the high-N and the low-N values di er very strongly. High-N values correspond to the regime of oscillations with frequency close to !0, see Fig. 2.5, whereas low N values correspond to the frequency of oscillations close to R. Figure 2.9: Excitation level of the oscillator, N = 16N 2=3, where N is the number of vibrational quanta, is plotted from Eq. (2.19) vs dimensionless detuning and dimen- sionless deviation from the resonance, x0 = 􀀀 R 􀀀!0)=(!0!2 p 1=3. The thick black line of bifurcations separates the \inner" domain of parameters (blue domain where bistability is absent) and outer domain (yellow domain where bistability is present). In the outer domain, the ratio of N values corresponding to the two stable solutions grows rapidly away from the boundary. 30 At x0 < 0:5 and to the right from the bifurcation line, there is no bistability. The value of N is low in this domain around 0 = 0. As 0 increases, N grows for both signs of detuning, 0. However, for 0 > 0 (blue detuning), the growth of N is monotonical. At the same time, for 0 < 0 (red detuning), the bistability sets in at a certain critical 0. Upon further increase of j 0j, the low-N value does not grow, while the high-N value grows rapidly. It is instructive to compare the results shown in Fig. 2.9 to the results of Refs. [18, 19], shown if Fig. 2.10. The curves N( 0; R) were obtained in Refs. [18, 19] by numerical solution of the system of master equations for the density matrix describing both the two-level system and the oscillator. First, there is a qualitative agreement in the shape of the boundary of bistability. In Refs. [18, 19], only the low-N values are plotted. The prime observation made in Refs. [18, 19] was that there is a strong di erence between these low-N values for blue and red detunings, namely, for the blue detuning, N is much higher. Our analytical results in Fig. 2.9 agree qualitatively with this observation. Figure 2.10: Results of numerical study of a quantum system consisting of a qubit coupled to a harmonic oscillator from [18]. The data are presented for a particular set of parameters of the system, such as coupling strength, spontaneous emission rate, level spacing of the qubit, and the friction in the oscillator. Only the inner curve, corresponding to a one-photon resonance, should be compared to Fig. 2.9. In the domain of bistability, the number of the oscillator quanta shown corresponds to the lower stable state. Compared to results in Fig. 2.9, we can see a similar shape in the bifurcation line. Note also that, similarly to Fig. 2.9, the N-values for positive detuning are much higher than for negative detuning. 31 2.6 Concluding remarks The frequency R of the Rabi oscillations is proportional to the square root of the excitation power. This linearity has been demonstrated in many experiments. Even when Rabi oscillations are damped, the dependence s( R) can be extracted from the position of maximum in the Fourier transform [24] of the signal w(t). We predict that, for a two-level system coupled to a vibrational mode, the position of maximum of the Fourier transform will deviate from the linear behavior near the resonance R = !0. Both to the left and to the right from the resonance the position of maxim corresponds to s < R. The relative width of the resonant regions depends on the coupling to the vibrational mode as 4=3. We also predict that, in the vicinity of the resonance, the dependence s( R) exhibits a hysteretic behavior with two stable values of s corresponding to two stable regimes of the Rabi oscillations. The underlying physics of the Rabi-vibronic resonance is the following. Without coupling, population inversion w and displacement X satisfy the harmonic oscillator equations with frequencies R and !0, respectively. With coupling, the two-level system acts as a driving force / w on the oscillator, while the back action of the oscillator on w is peculiar. The structure of back-action force is wX2, as can be seen from Eqs. (2.8) and (2.9). This structure implies that back action is of a parametrical type, i.e., X2 adds to 2 R. Thus, at R !0, it appears that R is modulated with frequency 2 R. This, however, does not lead to a parametric instability. Instead, the oscillator motion gets synchronized with the Rabi oscillations. In this regard, there is a certain analogy to the synchronization of the Rabi oscillations to a sequence of pulses [25] applied to the detector with a repetition period chosen to be 2 = R. As it was pointed out in the Introduction, the situation when a two-level system undergoing the Rabi oscillations is coupled to the oscillator is actively studied in con- nection to the circuit QED [14]. The most common situation in circuit QED is when the oscillator frequency !0 is tuned to the transition frequency !12 of the two-level system. Among physical e ects predicted for this domain is that two or multiple qubits can get strongly coupled to each other via coupling to a common oscillator, [26, 27]. Rabi-vibronic resonance corresponds to the domain !0 !12. Still, the e ects similar to those discussed in Refs. [26, 27] (see also recent experiments Refs. [28, 29]) will take place under the conditions of the Rabi-vibronic resonance. In particular, we anticipate that Rabi oscillations in two driven two-level systems with R = !0 coupled to the same oscillator will get synchronized. 32 As a nal remark, classical treatment of the vibrational mode adopted in the present paper does not allow one to capture the quantum jumps [20] between the stable regimes of collective motion of the two-level system coupled to the oscillator. We also did not consider the e ect of thermal noise, which leads to the activated switching [30] between the steady regimes even within a classical description of the oscillator. 2.7 Appendix 2.7.1 Introduction In the above chapter, we realized that Rabi-vibronic resonance develops when the coupling of electronic level to the vibrational mode is weak. Weakness of the coupling suggests tring to treat it perturbatively, namely, to assume that in zeroth order the oscillations have their \bare" form, w(0)(t), then to search for the correction, w(1)(t), to this form proportional to the coupling strength. The hope is that near the resonance R = !0, the correction \blows up". It is clear a priori that this program will not work for nondecaying Rabi oscillations, since we demonstrated in the previous chapter that, under the conditions of Rabi-vibronic resonance, the frequency, s, of collective motion di ers from R, so that w(0)(t) is not a good zero-order approximation. However, one might hope that, in the presence of friction or spontaneous emission, when w(0)(t) decays with time, so that the frequency R is \smeared", the correction w(1)(t) will blow up at the resonance but remain nite and smaller than w(0)(t), so that perturbative treatment remains justi ed. This is the motivation for why in the following Appendix we calculate w(1)(t) at nite friction in the oscillator and nite spontaneous emission of the two-level system and examine this solution in di erent limits. The prime outcome is that, while w(1)(t) remains nite at the resonance, the perturbative treatment does not apply within the entire time domain. The reason is that condition w(1)(t) w(0)(t) is incompatible with the requirement that the oscillator is classical. Still it o ers an accurate description of collective motion during the initial stage of oscillations when R gradually evolves into s. 2.7.2 Perturbative treatment We search for a solution of optical Bloch Eqs. (2.5)-(2.7), within the lowest order in X(t) in the form v = v(0) + v(1) and w = w(0) + w(1). The zero-order solution, can easily be found using the initial conditions: X(0), v(0) = 0 and u(0) = 0. From Eq. (2.5), we see that the initial conditions for v(t) and w(t) require that w_ (0) = 0. In the limit 33 R 􀀀, the function u(t) is zero, while the expressions for w(0) and v(0) read w(0)(t) = 􀀀 􀀀2 􀀀2 + 2 R 2 􀀀 2 2 R 􀀀2 + 2 2 R e􀀀3􀀀t=4 cos Rt 􀀀e􀀀3􀀀t=4 cos Rt; (2.45) v(0)(t) = 􀀀 1 R w_ (0) + 􀀀 􀀀 1 + w(0) 􀀀w_ (0)(t) R ; (2.46) In the rst order, the prime e ect of coupling to the oscillator is that u(t) = u(1)(t) becomes nonzero. It can be found from Eq. (2.6), which reads u_ (1) + 􀀀 2 u(1) = X(t) v(0)(t): (2.47) Using the initial condition u(1)(0) = 0, we nd u(1)(t) = e􀀀􀀀t=2 Z t 0 dt0e􀀀t0=2X(t0) v(0)(t0); (2.48) where v(0)(t) is determined from Eq. (2.46). From Eq. (2.5), we see that v(1)(t) and w(1)(t) are related as v(1)(t) = 􀀀 1 R w_ (1) + 􀀀w(1) : (2.49) Substituting this relation into Eq. (2.7), we get the following equation for w(1)(t) w(1) + 3􀀀 2 w_ (1) + R 2 + 􀀀2 2 w(1) = RX(t)u(1)(t): (2.50) We see that the product X(t)u(1)(t) plays the role of a driving force in the damped- harmonic-oscillator equation for w(1)(t). Note that this driving force is quadratic in X(t). With initial conditions w(1)(0) = w_ (1)(0) = 0, the solution for w(1)(t) reads w(1) = e􀀀3􀀀t=4 wc(t) cos Rt + ws(t) sin Rt ; (2.51) where the functions wc(t) and ws(t) are de ned as wc(t) = 2 Z t 0 dt0e􀀀t0=4 h X(t0) sin Rt0 i Z t0 0 dt00e􀀀􀀀t00=4 h X(t00) sin Rt00 i ; (2.52) ws(t) = 􀀀 2 Z t 0 dt0e􀀀t0=4 h X(t0) cos Rt0 i Z t0 0 dt00e􀀀􀀀t00=4 h X(t00) sin Rt00 i : (2.53) In Eqs. (2.52) and (2.53), we have substituted the explicit form Eq. (2.48) of u(1)(t). The rst term in Eq. (2.51), proportional to the cosine, describes the correction to the amplitude of the Rabi oscillations due to coupling to the oscillator, while the second term, proportional to sine, describes the change of the phase. 34 The remaining task is to solve the equation, Eq. (2.4), for the oscillator and substitute the found X(t) into Eqs. (2.52) and (2.53). In doing so, one should keep in mind that, in addition to forced oscillations, caused by the r.h.s. of Eq. (2.4), the oscillator also undergoes the free oscillations with frequency !0. The relation between the amplitudes of the free and forced oscillations is governed by the initial conditions. As a result, the products X(t) cos Rt and X(t) sin Rt in the integrands of Eqs. (2.52), (2.53) will contain a slow nonoscillating term and a slow term oscillating with frequency = R􀀀!0. The integrands of wc(t) and ws(t) contain two combinations of the type X(t) cos Rt. As a result, w(1)(t) will contain a nonoscillating term, terms oscillating with frequency , and terms oscillating with frequency 2 . Concerning the magnitude of w(1)(t), note that, away from resonance, X(t) is proportional to 1 . This leads to 1 2 factor in w(1)(t). Less trivial is that two additional integrations over t0 and t00 in Eqs. (2.52), (2.53) give rise to another factor 1 2 . So overall, we have w(1) / 1 4 . The physics underlying the additional 1 2 factor is the nonlinear feedback exercised by the driven oscillator on the two-level system. 2.7.3 Friction-dominated regime, 􀀀 In the limit 􀀀 , the r.h.s. of Eq. (2.4) is proportional to cos Rt. Then the solution is straightforward. It contains an inessential constant term =(2M!2 0) and the oscillating part X(t) = 2M R( 2 + 4 2) sin Rt 􀀀 2 cos Rt 􀀀 e􀀀 t=2 sin !0t 􀀀 2 cos !0t : (2.54) In deriving Eq. (2.54), we used the fact that , R and !0 R. We also neglected the friction-induced correction, 2=(8!0), to !0. In the limit 􀀀 , the relaxation exponents in the integrand of Eq. (2.52) can be neglected. After that, the double integral simply reduces to 1=2[ R t 0 dt0X(t0) sin Rt0]2. Performing integration and keeping only resonant terms we obtain, wc(t) = !2 p 2 R 2( 2 + 4 2)2 " t 2 􀀀 4 2 + 4 2 e􀀀 t=2 sin t 􀀀 2 􀀀 4 2 2 + 4 2 1 􀀀 e􀀀 t=2 cos t #2 : (2.55) Equation (2.55) coincides with Eq. (2.39) of the main text. In the main text, it was used to determine the number of oscillation periods, m0, before collective motion sets in. As we expected, wc(t), carries the "knowledge" about the Rabi-vibronic resonance, via the resonant prefactor proportional to ( 2 + 4 2)􀀀2, where is the deviation from 35 resonance. However, the main message sent by Eq. (2.55) is that its growth with time is not contained by friction or detuning, as one could hope. For times exceeding the time, 1 , of oscillator damping, the correction wc(t) grows with time quadratically. Note also that this growth is preceded by several oscillations, as illustrated in Fig. 2.11. More speci cally, for t 1, the magnitude of wc(t) increases as !2 p 2 Rt4, for = 0, while for longer times, t 1, the term t=2 dominates and the growth of wc(t) is / 1 (!p Rt)2. However, perturbation theory no longer applies, after time 1 , as was discussed in Section 1.4.3. 0 5 10 15 20 0.00 0.01 0.02 0.03 0.04 dt wc d g=1.9 d g=2.3 d g=3.1 Figure 2.11: Coupling-induced modi cation of the amplitude of the Rabi oscillations in the friction-dominated regime versus the dimensionless time, t, is plotted from Eq. (2.55), in the units of !2 p 2 R=(2 4), for three dimensionless detunings from resonance; = 1:9 (red); = 2:3 (black); = 3:1 (green). We see that for small detuning, = = 1:9, wc(t) grows rapidly, meaning that for small, detuning perturbation theory only applies for very short times. For large detuning, = = 3:1, when the growth is much slower, wc(t) exhibits several oscillations, while it is still small. 36 2.7.4 Relaxation-dominated regime, 􀀀 In this section, we consider decoherence of the excited state, assuming no detuning. In this regime, the damped oscillator, Eq. (2.4), is a driven harmonic oscillator, where driving force, w(0), is described by Eq. (2.45). The solution of X(t) can easily be found as X(t) = 4 M!0 􀀀 16 2 + 9􀀀2 " 3􀀀 4 sin !0t + cos !0t 􀀀e􀀀3􀀀t=4 3􀀀 4 sin Rt+ cos Rt # ; (2.56) where the static displacement causes a small correction =(2M 2 R) to X(t) and it is neglected. Comparing Eqs. (2.56) and (2.54), it can be seen that X(t) assumes the same form as in the previous section with the substitution 􀀀 = 2 =3. Recall that the solution of w(1)(t) is found by Eqs. (2.52), (2.53). Returning, we see that both equations contain the integrand e􀀀􀀀t00=4X(t00) sin Rt00. Employing the RWA when expanding the product X(t00); sin Rt00, gives X(t00) sin Rt00 = 2 M!0 􀀀 16 2 + 9􀀀2 sin t0 + 3􀀀 4 cos t0 􀀀 e􀀀3􀀀t0=4 # : (2.57) Concentrating on the rst integral over t00 in wc(t), we multiply Eq. (2.57) by e􀀀􀀀t00=4 and integrate, noting that, Y (t0) = R t0 0 dt00e􀀀􀀀t00=4X(t00) sin Rt00, Y (t0) = 2 M!0 􀀀 9􀀀2 + 16 2 8 􀀀 􀀀2 + 16 2 e􀀀t0=4 sin t0 + 3􀀀2 + 16 2 􀀀2 + 16 2 1 􀀀 e􀀀􀀀t0=4 cos t0 + 3 4 e􀀀􀀀t0 􀀀 1 : (2.58) Although the cosine and sine-terms in Y (t0) grow exponentially, w(1)(t) will decay expo- nentially due to damping factor, e􀀀3􀀀t=4, multiplying wc(t) and ws(t). The second integral over t0, in wc(t), Eq.(2.52), contains the integrand X(t0)e􀀀t0=4 sin Rt0Y (t0). Introducing dimensionless variables, z = =􀀀 and = t, we covert the product of trigonometrical functions in the integrand into a sum, and then evaluate wc(t) wc(t) = 􀀀 16!2 p 2 R 􀀀4(16z2 + 1)(16z2 + 9) " 8z 􀀀 4z 16z2 + 1 e =4zsin + 1 4 16z2 􀀀 3 16z2 + 1 e =4zcos 􀀀 12z 16z2 + 9 e􀀀3 =4zsin + 3 4 16z2 + 5 16z2 + 9 e􀀀3 =4zcos + 1 2 16z2 􀀀 3 16z2 + 9 sin2 + 1 16z 80z2 + 9 16z2 + 9 sin 2 􀀀 h( ; z) # ; 37 (2.59) where h( ; z) = 1 4 (16z2 􀀀 3)(16z2 + 9) 16z2 + 1 + 3 8 16z2 + 9 16z2 + 1 e􀀀 =2z + 3 8 e􀀀3 =2z: (2.60) Obviously, the rst three terms in the square brackets of Eq. (2.59) dominate for all times. For small times, t 􀀀 4 , wc( ) grows linearly, as described by the term =8z, while for times, t > 􀀀 4 , the second and third terms dominate, and wc(t) becomes exponential. On the other hand, the correction w(1)(t) is proportional to e􀀀3 =4zwc( ) and does not grow exponentially. For this reason, we plot e􀀀3 =4zwc( ) in Fig. 2.12. We see that for small z = 0:03, the amplitude of e􀀀3 =4zwc( ) has a maximum height. With increasing detuning further z = 0:1, the initial amplitude of e􀀀3 =4zwc( ) decreases. Upon further increasing z, the function e􀀀3 =4zwc( ) simply approaches zero. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 t e-3 t 4 zwc z=0.03 z=0.10 z=0.30 Figure 2.12: Coupling-induced modi cation of the amplitude of the Rabi oscillations in the relaxation-dominated regime is plotted from the function e􀀀3 =4zwc( ), Eq. (2.59), in the units of 16!2 p 2 R=􀀀4 for di erent dimensionless detunings z = 0:03 (red), z = 0:1 (black), z = 0:3 (green). With increasing the detuning, the correction, e􀀀3 =4zwc( ), approaches zero. 38 Overall, as in the friction-dominated regime, the prefactor in Eq. (2.59) exhibits resonant behavior as a function of dimensionless deviation z = =􀀀 from the Rabi-vibronic resonance. Unlike the friction-dominated regime, the correction e􀀀3 =4zwc( ) remains nite at all times. However, it falls o with time slower than the zero-order result w(0)(t), and, thus, the perturbation treatment fails at large times. 2.7.5 Analysis As was stated in the Introduction, perturbation theory does not apply for the entire domain. The time domain where perturbation theory applies is where both the assump- tions made coincide. First consider the assumption that the oscillator is classical. This requires that the kinetic energy be greater than ~!0, 1 2 M _X 2 !2 0M 2 X2(t) ~!0: (2.61) In the friction-dominated regime, this condition can be written as !p !2 0 2 + 4 2 !0; (2.62) Condition, Eq. (2.62), has the same form in the relaxation-dominated regime, where one should replace by 􀀀. On the other hand, as it follows from Eq. (2.55), the condition that w(1)(t) 1 at all times smaller than 1= reads !2 p 2 R ( 2 + 4 2)2 1: (2.63) We see that these two conditions are incompatible at all -values and -values in both regimes. 2.8 References [1] I. I. Rabi, Phys. Rev. 51, 652 (1937). 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CHAPTER 3 MANY-BODY ASPECTS OF RABI OSCILLATIONS IN A QUANTUM DOT MOLECULE 3.1 Introduction An obvious reason why optical properties of semiconductor quantum dots are inten- sively studied both theoretically and experimentally is that quantum dots are considered to be promising candidates for the basic device unit of quantum information processing. In plain words, a dot with one electron and hole level can be viewed as a qubit, in the sense that the presence of an exciton in a dot can be viewed as an \on" state of the qubit, while absence of an exciton can be viewed as an \o " state. Compared to other candidates for the qubits, such as nuclear spins [1] or electron spins [2], a qubit based on an exciton in a semiconductor quantum dot can operate at a much higher (terahertz) rate. This is because the resonant frequency of the exciton lies within the optical range. First, the principal requirement for exciton(s) contained within a quantum dot to serve as an e cient quantum bit is that they have to possess a long coherence time. Second, they should allow coherent optical manipulation, the simplest example being coupling and entangling of quantum states in a pair of dots. In simple terms, coherent optical manipulation in a pair of two quantum dots relies on the fact that creation of an exciton in one dot a ects the resonant energy of the other. As a result, the quantum state of one dot controls the outcome of measurements on another dot. This correlation, plus the fact that exciton(s) in both dots are coherent and are characterized by a single many-body function, was argued [3] to be su cient for two dots to operate as a quantum gate. A natural characteristic of the degree of coherence of an optical excitation in a dot is its ability to exhibit Rabi oscillations. In fact, Rabi oscillations of an exciton in a single dot is what is meant by coherent control. In this regard, experimental demonstrations of 42 Rabi oscillations in a single dot reported in Refs. [4, 5] was vital. In the language of the quantum computation community, observation of Rabi oscillations in a single quantum dot constitutes proof of the exciton qubit rotation. Equally important was a decade-old experimental demonstration [6] that excited excitons in two close-spaced quantum dots can be e ciently coupled to each other. The evidence of coupling was inferred from measuring the evolution of the photoluminescence emission lines as a function of the interdot separation, shown if Fig. 3.1. In a later paper, Ref. [7], it was demonstrated that this coupling can be facilitated by applying an external electric eld which aligns the single particle levels in the neighboring dots. In early experiments [4, 5] on coherent control, Rabi oscillations were registered either by means of transient nonlinear optical spectroscopy [4] performed on a single dot, or by excitation of a dot by a sequence of pulses and recording photoluminescence as a function of temporal pulse delay between two pulses [5]. A remarkable progress in registration of coherence of excitations in a single quantum dot was reported in Ref. [8]. Instead of optical registration of the Rabi oscillations in Refs. [4, 5], the authors had put forward Figure 3.1: Quantum dot molecule in the experiment Refs. [6].(A) The assumed shape of the two vertically aligned quantum dots. (B) Transmission electron microscopy (TEM) image of the two layers of quantum dots. (C) Assuming the shape shown in (A), the authors numerically calculate how the symmetric-antisymmetric splitting of the levels depends on the vertical separation of the dots. 43 an elegant idea: in the course of the Rabi oscillations, the exciton population of a dot changes with time as cos Rt, where R is the Rabi frequency. If the duration of the excitation pulse is = R (a -pulse), an exciton is present in the dot with probability 1. Then electric eld applied to the dot will tear the electron and hole apart, as illustrated in Fig. 3.2, and both electron and hole will contribute to the current in the circuit. As a result, after many pulses, the measured current, I, will be simply equal to fe, where f is the frequency of the repetition of -pulses. Rabi oscillations manifest themselves in the dependence of the current on the amplitude, A, (in the frequency units) of an excitation pulse I = ef cos2 A= R. If A is equal to n R, where n is an integer, then the current is maximal. See Fig. 3.2d. If A = (n + 1 2) R, the current is zero, since an exciton is not excited during the pulse duration. With the help of this novel technique, the authors of Ref. [8] were able to probe Rabi oscillations electrically in a remarkably weakly disturbing fashion. In all present-day experiments, the quantum dots used were self-assembled. They emerged in the course of interruption of molecular-beam growth of GaAs followed by growing of several layers of InGaAs [9]. The strain caused by lattice mismatch of two materials results in formation of a layer lens-shaped InGaAs dots. Repetition of this interruption yields double layers of these quantum dots used in experiments on double dots. Pioneering experimental papers [4, 5, 3, 7, 8] were followed by a stream of subsequent studies. We cannot quote all of them. The most prominent advances [10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] can be divided into two groups. The rst group of papers, Refs. [10, 11, 12, 13, 14, 15, 16], reported the measurements of characteristics of excitons in two coupled dots (quantum dot molecule) by means of photoluminescence. Quantum dot molecules are realized experimentally by extending the self-assembly techniques of Ref. [9] to include strained layers. Then InGaAs dots turn out to grow \one above the other", as illustrated in Fig. 3.1a. Thus, two vertically-aligned dots serve as a quantum dot molecule. In the second group, Refs. [11, 17, 18, 19, 20, 21] studied the quantum dynamics of an exciton in a single dot driven by excitation pulses into the regime of Rabi oscillations. One of the techniques to register Rabi oscillations optically is illustrated in Fig. 3.3 and uses the resonant pulses initiating Rabi oscillations between the ground and the rst excited state of an electron in the dot. In the course of these oscillations, the electron relaxes nonradiatively to its excitonic ground state. Then the photoluminescence from this state is measured as a function of pulse duration and laser power. Since the probability that, 44 Figure 3.2: A schematic illustration of the electrical detection of Rabi oscillations is shown from Ref. [8]. (a) Two level system, an excition in a quantum dot. (b) An applied electric eld bends the energy pro le of the two-level system, allowing both the electron and hole to tunnel out and contribute to the current. (c) Experimental setup to selectively excite a single dot. (d) The measured photocurrent; the fact that the period of oscillations of the measured photocurrent is a function of excitation amplitude proves they are able to measure Rabi oscillations electrically. 45 Figure 3.3: Optical detection of Rabi oscillations is shown from Ref. [11]. (a) Optical detection of Rabi oscillations requires the third level. (b) Image of a quantum dot shows it is well isolated from other quantum dots. (c) Intensity of the transition j2 >! j0 > versus the excitation amplitude for a given pulse duration, p. Oscillating character, photoluminescence, is due to the fact that for a given pulse duration, it can contain either integer or half-integer Rabi cycles depending on the amplitude. (d) Illustration of the qualitative tendency of damping of the oscillations for di erent pulse widths as a function of input pulse area. 46 after the pulse, the electron resides in the rst excited state is a periodic function of both, the luminescence signal exhibits Rabi oscillations, as illustrated in Fig. 3.3. The most recent progress in the eld is the experimental demonstration [22] of the Rabi oscillations of two excitons in a quantum dot molecule. The oscillations were detected using the pump-probe technique. The above brief review motivates the research reported in the present chapter. We will study the exciton Rabi oscillations in the system of two tunnel-coupled quantum dots with account of interaction-induced correlation between electron and hole. As we consider two quantum dots, the following new physical e ects come into play: (a) due to a nite transparency of barriers, separating the dots, an electron photoexcited, say, in the left dot can, after some time, be found in the right dot. The same applies to the hole. (b) the photo-excited electron and hole in the dot attract each other. In an optical experiment, the electron and hole are created simultaneously. The major consequence of this intradot attraction of electron and hole is that, as a result of this attraction, the dynamics of the exciton acquires a correlated character. The prime objective of the present chapter is to trace how this correlation manifests itself in the shape of the Rabi oscillations, which arise when the two-dot system is driven by intense resonant light. We will show that the shape of the Rabi oscillations is strongly a ected by the interaction e ects. Previous theoretical studies [24, 25, 26] were conducted by solving the system of equations for the density matrix numerically, and could not reveal the e ect of interaction on the Rabi oscillations in the quantum dot molecule systematically. 3.2 Correlated dynamics of a photoexcited exciton in two coupled dots The system being considered is depicted in Fig. 3.4. It consists of two coupled quantum dots (QDs) each containing one electron and one hole levels. Possible one-exciton states of the system are labeled a1, a2, b1, and b2, where a1 corresponds to an exciton in the left QD, a2 corresponds to an exciton in the right QD, while b1 and b2 correspond to two possible indirect exciton states. Before including interactions and studying Rabi oscillations in the system, we will consider the simplest auxiliary problem: suppose that at time t = 0 an exciton is excited in the left QD. What is the exciton evolution with time? This evolution is due to tunneling of carriers between the dots. We will also assume that no electric eld is applied, so that, 47 Figure 3.4: A schematic illustration of single-exciton states in a quantum dot molecule. A resonant light shone on the molecule can create a direct exciton in either dot (states a1, a2). Due to tunneling, the exciton dynamics involves indirect excitons (states b1, b2). Note that the initial vacuum state, c, is not shown. without tunneling, electron and hole levels in the dots are perfectly aligned. Denote with e and h the tunnel splittings of the electron and hole levels, respectively. Exciton dynamics are described by 4 probabilities ja1(t)j2, ja2(t)j2, jb1(t)j2, and jb2(t)j2, where a1(t), a2(t), b1(t), b2(t) are the amplitudes to nd the exciton in the states a1, a2, b1, and b2 after time t, respectively. The probabilities are related by the normalization condition ja1(t)j2 + ja2(t)j2 + jb1(t)j2 + jb2(t)j2 = 1: (3.1) Without electron-hole interactions, the dynamics reduce to independent beatings of elec- tron and hole between the dots, so that ja1(t)j2 = Ne(t)Nh(t); jb1(t)j2 = Ne(t) 1 􀀀 Nh(t) ; ja2(t)j2 = 1 􀀀 Ne(t) 1 􀀀 Nh(t) ; jb2(t)j2 = 1 􀀀 Ne(t) Nh(t); (3.2) where Ne(t) = cos2 et and Nh(t) = cos2 ht. Note that, with regard to observables, ja1(t)j2 describes the luminescence from QD1, while ja2(t)j2 describes the luminescence from QD2. 48 As a next step, we incorporate electron-hole interaction in the simplest form. Namely, we assume that, at time moments when electron and hole are located in the same dot, they attract each other with Coulomb energy, U. Now, in order to nd the time evolution of an exciton, one has to solve the Schr odinger equation i @ @t 0 BB@ a1 b1 b2 a2 1 CCA = ^H 0 BB@ a1 b1 b2 a2 1 CCA ; (3.3) for the four-component wave function (a1(t); b1(t); b2(t); a2(t)). In Eq. (3.3), we have set ~ = 1. In the matrix representation the Hamiltonian, ^H , has the form ^H = 0 BB@ U + !10 e h 0 e !10 0 h h 0 !10 e 0 h e U + !10 1 CCA ; (3.4) where the interaction U is incorporated into the energies of the states a1 and a2. To get the time evolution of the system, we will now solve Eqs. (3.3) and (3.4). 3.2.1 Eigenvalues and eigenvectors A standard way to nd the time evolution of the exciton is to nd the eigenvalues of the Hamiltonian Eq. (3.4) and choose the linear combination of the eigenvectors satisfying the initial conditions. The form of the eigenvalues, ^H i = ( i + !01) i, is quite simple 1;2 = U 2 1 2 U2 + 4( e + h)2 1=2 = U 2 1; 3;4 = U 2 1 2 U2 + 4( e 􀀀 h)2 1=2 = U 2 2; (3.5) where 1 and 2 are de ned as 1 = 1 2 p U2 + 4( e + h)2; 2 = 1 2 p U2 + 4( e 􀀀 h)2: (3.6) Using Eq. (3.5), we nd from the Schr odinger equation the following normalized eigen- vectors, Xi: X1 = 1 p 2(1 + 2 1) 0 BB@ 1 1 1 1 1 CCA ; X2 = 1 p 2(1 + 2 1) 0 BB@ 1 􀀀1 􀀀1 1 1 CCA ; (3.7) 49 X3 = 1 p 2(1 + 2 2) 0 BB@ 􀀀1 􀀀 2 2 1 1 CCA ; X4 = 1 p 2(1 + 2 2) 0 BB@ 􀀀 2 1 􀀀1 2 1 CCA ; (3.8) where the constants 1 and 2 are de ned as follows, 1 = 2( e + h) U + p U2 + 4( e + h)2 ; (3.9) 2 = 􀀀2( e 􀀀 h) U + p U2 + 4( e 􀀀 h)2 : (3.10) If U is much smaller than e, h, the eigenvalues 1; 2 are equal to e h, respectively. It is easy to check that this corresponds to independent beating of electron (with frequency e) and hole (with frequency h) between the dots. In the realistic limit U e; h, the eigenvalues can be simpli ed as 1;2 U 2 U 2 + ( e + h)2 U ; 3;4 U 2 U 2 + ( e 􀀀 h)2 U : (3.11) We see that eigenvalues 1 and 3 are large and approximately equal to U, while two other eigenvalues are much smaller. The underlying physical meaning is the following: without interactions, all four eigenstates (within e, h) had approximately the same energies. With interactions, the states are almost degenerate pairwise: a1 and a2 have close energies which exceed the energies of b1 and b2 by the interaction energy, U. Below, we study how the interaction modi es the dynamics of the exciton, which, without interaction, is described by Eqs. (3.2). Using the eigenvectors, Eqs. (3.7), (3.8), and the initial conditions, a1(0) = 1, b1(0) = b2(0) = a2(0) = 0, we get ja1j2(t) = cos2 1 2 􀀀 1 + 2 t cos2 1 2 􀀀 1 􀀀 2 t + 1 4 cos(2 1) sin 1t+cos(2 2) sin 2t 2 ; (3.12) ja2j2(t) = sin2 1 2 􀀀 1 + 2 t sin2 1 2 􀀀 1 􀀀 2 t + 1 4 cos(2 1) sin 1t􀀀cos(2 2) sin 2t 2 ; (3.13) jb1j2(t) = 1 4 sin(2 1) sin 1t 􀀀 sin(2 2) sin 2t 2 ; (3.14) jb2j2(t) = 1 4 sin(2 1) sin 1t + sin(2 2) sin 2t 2 ; (3.15) where the constants 1 and 2 are de ned as 1 = arctan 1; 2 = arctan 2: (3.16) Equations (3.12)-(3.15) solve the auxiliary problem of dynamics of one exciton in two coupled quantum dots. It is instructive to show that, in the limit U ! 0, we recover 50 the noninteracting result Eq. (3.2). Indeed, in the limit U ! 0, we have 1 = 1 and 2 = 􀀀1, then ja1(t)j2 = cos2 e t cos2 h t, as stated earlier. Also for jb1j2, it can easily be seen that for U = 0, jb1j2 = sin2 e t cos2 h t. Also, notice that for e = h, 2 = 0 and 2 = U=2, resulting in jb1j2 = jb2j2, and ja1j2 and ja2j2 di er by the phase factor of =2. Consider now the opposite limit U e; h. Then the beating between the two frequencies, 1 and 2, emerging from the last terms in Eqs. (3.12), (3.13) gives rise to slow frequency 2 e h=U, much slower than intrinsic frequencies e and h. On the other hand, expanding Eqs. (3.14) and (3.15), we see that sin2-terms oscillate with the fast frequency U=2 e; h. Both slow and fast frequencies contain the interaction energy, U, so that their emergence is a many-body e ect. Concerning the general picture of the exciton dynamics, it is illustrated in Fig. 3.5. We see that jb1j2, jb2j2 move with the fast frequency, while ja1j2 and ja2j2 move with both: the fast frequency is \riding" on the slow frequency. Electron and hole initially in a1, stay in a1 for a long time. Occasionally one particle tunnels to the states b1 or b2 and then the other particle tunnels, and both particles are then in the state a2 where they stay for a long time. Another prominent feature of the exciton dynamic is that probability to nd electron and hole in the same dot is much higher than probability to nd them in di erent dots. To illustrate this, we Figure 3.5: Population of the states are plotted from Eqs. (3.12)-(3.15) versus the dimensionless time Ut, for the values e=U = 0:1, and h=U = 0:2. (a) The occupation of the states a1 and a2 evolve with a fast frequency U=2 riding on a slow frequency 2 e h=U. (b) Conversely, the intermediate states, b1 and b2, evolve with a fast frequency modulated by the slow frequency. 51 notice that in the large-U limit, the occupation of the states can be simpli ed to ja1j2 = cos2 2 e h U t ; jb1j2 = 2 e U 2 sin2 U 2 t ; ja2j2 = sin2 2 e h U t ; jb2j2 = 2 h U 2 sin2 U 2 t ; (3.17) so that jb1j2, jb2j2 are 1. To conclude this section, we would like to make the following remark. We assumed for simplicity that electron and hole attract each other only in the states a1 and a2, when they reside in the same dot. In reality, they also attract each other in the states b1 and b2, but this attraction is weaker than in a1 and a2. The above consideration applies to the realistic situation if we identify U with the di erence of attractive interactions in the states a1 and b1. Concerning the numbers, the tunnel splittings e and h are certainly strongly dependent on the vertical separation between the dots, which varies within interval 2-10 nm. It can be as high as several meV and as low as 0.1meV [6, 7, 13, 15]. In the latest study [23] which involved the photocurrent techniques, the value e was reported to be 1:7meV for the vertical separation of the dots being 10nm. To study the interplay of the Rabi oscillations and e ects of tunneling, to which we turn in the next section, it is important to note that the actual Rabi frequency lies within the same interval. Indeed, according to Ref. [11] the typical Rabi frequency in an experiment on a self-assembled quantum dot was several meV. 3.3 Exciton in two QDs with near-resonant light We now turn to the prime question: how many-body e ects between two quantum dots modify the Rabi oscillations. Assume that a quantum dot molecule is being driven continuously by the near-resonant light. It is convenient to characterize these oscillations by their Fourier spectrum. For a single-dot, the light will initiate the Rabi oscillations with a single frequency and the magnitude depending on detuning from the resonance. Then the Fourier transform of oscillations would represent a single peak. At perfect resonance, the peak position will shift with light intensity, I, as I1=2 / R. In reality, the decoherence processes, such as spontaneous emission, will broaden the peak. To account for these processes, it requires one to use a description based on the density matrix. However, as long as we are interested only in peak positions and the integral magnitude of the peaks, the description based on the Sch odinger equation is su cient. 52 The dynamics of the quantum-dot molecule can be modeled using ve states. Four of them are shown in Fig. 3.4, while the fth state is simply the vacuum state, c. Using this model, the Schr odinger equation can be written in the rotating wave approximation as i @ @t 0 BBBB@ a1 b1 b2 c a2 1 CCCCA = 0 BBBB@ !10 + U e h Re􀀀i!Lt 0 e !10 0 0 h h 0 !10 0 e Rei!Lt 0 0 0 Rei!Lt 0 h e Re􀀀i!Lt !10 + U 1 CCCCA 0 BBBB@ a1 b1 b2 c a2 1 CCCCA : (3.18) The eigenvalues, , of Eq. (3.18), are determined in a standard way by using the following substitutions: a1 = A1 exp[􀀀i( + !10)t]; b1 = B1 exp[􀀀i( + !10)t]; a2 = A2 exp[􀀀i( + !10)t]; b2 = B2 exp[􀀀i( + !10)t]; c = C exp[􀀀i( + !10 􀀀 !L)t]: (3.19) The matrix equation, Eq. (3.18), written as a set of equations, now reads ( 􀀀 U)A1 = eB1 + hB2 + RC; B1 = eA1 + hA2; ( 􀀀 U)A2 = hB1 + eB2 + RC; B2 = hA1 + eA2; ( + 􀀀 U)C = R(A1 + A2); (3.20) where = U +!10 􀀀!L, and represents detuning of the driving light. In this regime, the eigenvalues can be separated into symmetric and antisymmetric modes. This physically obvious fact is revealed by reducing the system, Eq. (3.20), to two coupled equations by excluding the terms, B1, B2, and C. Then one gets ( 􀀀 U) 􀀀 1 e 2 + h 2 􀀀 2 R 􀀀 U + A1 = 2 e h + 2 R 􀀀 U + A2; (3.21) ( 􀀀 U) 􀀀 1 e 2 + h 2 􀀀 2 R 􀀀 U + A2 = 2 e h + 2 R 􀀀 U + A1: (3.22) From Eqs. (3.21) and (3.22), we see that for antisymmetric modes, A1 = 􀀀A2, the Rabi frequency vanishes, while for symmetric modes, the modi ed Rabi frequency is determined from the roots of the cubic equation. Using A1 = A2, Eqs. (3.21) and (3.22) give the following cubic equation ( + 􀀀 U) h ( 􀀀 U) 􀀀 ( e + h)2 i = 2 2 R : (3.23) Our main nding in this chapter, the nontrivial evolution of the spectrum of the Rabi oscillations with U, is encoded in the roots of the cubic equation Eq. (3.23). 53 What is left to do is to determine the eigenvectors and use the initial conditions, that the state c is initially occupied, c(0) = 1, while all other states are unoccupied, a1(0) = a2(0) = b1(0) = b2(0) = 0. Before we proceed further with studying how many-body e ects modify the Rabi oscillations, we will rst incorporate R perturbativly to study the dynamics of the weakly driven system. 3.3.1 Weak driving As R ! 0, the system initially in the vacuum state c will remain in this state. More interesting is how the population of the states evolves with time at weak driving R U; e; h, when the change of the population of the ground state is small and proportional to 2 R. If U, e, h are all zero, then the known result for c2(t) has the form jc(t)j2 = 1 􀀀 p 2 2 p R 2 2 R + 2 sin2􀀀q 2 + 2 2 Rt ; (3.24) so that for small R, the magnitude of the correction is 2 R= 2, while the oscillation frequency is simply equal to detuning between the incident light frequency and the spacing between the ground and excited states. To contrast this result to the case of nite U, e, h we notice that at small R, the solutions of Eq. (3.23) are = U 􀀀 and also 1 = 1, 2 = 2, which are given by Eq. (3.5). Using these solutions and satisfying the initial conditions, we get jc(t)j2 = 1 􀀀 4 2 R (U 􀀀 ) + 􀀀 e + h)2 " 1 􀀀 1 􀀀 2 􀀀 1 cos [( 2 􀀀 ) t] 2 1 􀀀 U + 2 􀀀 1 􀀀 2 cos [( 1 􀀀 ) t] 2 1 􀀀 U # : (3.25) Equation (3.25) has a transparent physical meaning: weak driving of the system causes relatively weak / 2 R oscillating correction to the occupation of the ground state, while corresponding frequencies are the \internal" frequencies of the system. In a way, this is analogous to a harmonic oscillator under weak nonresonant drive. A nontrivial question is, what are the relative amplitudes of the two internal frequencies in the oscillating correction in Eq. (3.25). It can be readily seen from Eq. (3.25) that, when U is comparable to ( e+ h), the amplitudes are also comparable to each other. However, for U ( e+ h), the \fast" frequency, being close to U, has a much smaller amplitude than the \slow" frequency, ( e + h)2=U. The ratio of amplitudes is then ( e + h)2=U2. At this point, an important remark is in order. Equation (3.23) has three roots, and, in principle, there should be three oscillating corrections in Eq. (3.25). However, the 54 third correction having frequency 􀀀 U has an amplitude / 4 R, i.e., much smaller than the oscillating terms kept in Eq. (3.25). This situation will change in the case of resonant driving to which we now turn. 3.3.2 Resonant driving It is clear that, when R exceeds all internal frequencies of the system, i.e., the spacings between the energies of the states a1 and b1, then jcj2 will assume the conventional \Rabi" form c2(t) = cos2( p 2 Rt). The really interesting issue is the evolution of the frequencies and amplitudes in Eq. (3.25) as R becomes comparable to internal frequencies. To assess this regime, we rewrite the cubic equation, Eq. (3.23), by using the dimensionless variable, = s + 1 3 􀀀 2U 􀀀 ; (3.26) where s = 1 12 (2U 􀀀 )2 + 2 2 R + e + h 2 + 1 4 2 1=2 : (3.27) Then the cubic equation, Eq. (3.23), assumes the form 3 􀀀 + f = 0: (3.28) The dimensionless eigenvalues are now to be determined by a single dimensionless pa- rameter f de ned as f = 1 108(2U 􀀀 )3 􀀀 1 6 􀀀 2U 􀀀 h 4 2 R 􀀀 􀀀 e + h 2 + 1 2 2 i 􀀀 1 2 􀀀 e + h 2 1 12 􀀀 2U 􀀀 2 + 2 2 R + e + h 2 + 1 4 2 3=2 : (3.29) Graphical solution of Eq. (3.28) is illustrated in Fig. 3.6. Since all three roots, i, must be real, the parameter f must reside within the interval 􀀀 2 3 p 3 ; 2 3 p 3 for any set of parameters, U; R; e + h, and . It is seen from Eq. (3.29) that weak interaction corresponds to small f 1, while the limiting value f 2 3 p 3 corresponds to U ! 1. As we will see below, the nontrivial evolution of the spectrum of the Rabi oscillations takes place at intermediate f. At intermediate f, the roots of Eq. (3.28) are given by 1 = 􀀀sgn(f) 2 p 3 cos 3 ; 2 = 􀀀sgn(f) 2 p 3 cos 3 + 2 3 ; 55 h1 h -1 1 - 2 3 3 1 3 2 3 3 h2 - 1 3 h3 f Figure 3.6: Graphic solution of the cubic equation, Eq. (3.28). Note that at f = 0 we have 1 = 1, 2 = 0, and 3 = 􀀀1, while at the maximum f = +2=3 p 3 we have two degenerate eigenvalues, 1 = 2 = p1 3 , and a third distant, 3 = 􀀀p2 3 . 3 = 􀀀sgn(f) 2 p 3 cos 3 􀀀 2 3 : (3.30) where the phase is determined as = arctan 1 f r 4 27 􀀀 f2 ! : (3.31) The eigenvectors of the system Eq. (3.20) corresponding to the roots can be conveniently cast in the form Xi = 0 BBB@ 3s i􀀀(U􀀀2 ) 6 R 1 3s i􀀀(U􀀀2 ) 6 R 1 CCCA : (3.32) We are now in position to calculate the population of the state c. Using the eigenvectors, Eq. (3.32), and eigenvalues, Eq. (3.30), the general expression of jcj2 can be written as jcj2 = C2 1+C2 2+C2 3+2C1C2 cos ( 1􀀀 2)st +2C1C3 cos ( 1􀀀 3)st +2C2C3 cos ( 2􀀀 3)st ; (3.33) 56 where the three constants C1, C2, and C3, are determined from the initial conditions. The value of these three constants is easily solved for, which gives C1 = 3s 1 + (2U 􀀀 ) 3s 2 􀀀 (U 􀀀 2 ) 3s 3 􀀀 (U 􀀀 2 ) 27s2 􀀀 U 􀀀 (3 2 1 􀀀 1) ; C2 = 3s 2 + (2U 􀀀 ) 3s 1 􀀀 (U 􀀀 2 ) 3s 3 􀀀 (U 􀀀 2 ) 27s2 􀀀 U 􀀀 (3 2 2 􀀀 1) ; C3 = 3s 3 + (2U 􀀀 ) 3s 1 􀀀 (U 􀀀 2 ) 3s 2 􀀀 (U 􀀀 2 ) 27s2 􀀀 U 􀀀 (3 2 3 􀀀 1) : (3.34) Equations (3.30), (3.31), and (3.34) constitute the analytical solution of the problem. We see that the Fourier spectrum of the Rabi oscillations consists of three peaks corresponding to the frequencies ( 1 􀀀 2)s, ( 1 􀀀 3)s, and ( 2 􀀀 3)s (the third equal to the di erence between the rst two) with heights given by 2C1C2, 2C1C3, and 2C2C3, respectively. Before proceeding to the analysis of the spectrum, we would like to point out that our study of the Rabi oscillations in a quantum dot molecule is conceptually related to the study reported in Refs. [27, 28, 29], of the Rabi oscillations in the system of two localized spins driven by the ac magnetic eld. In Refs. [27, 28, 29] the spins were coupled by \density-density" type interaction, which was either weak exchange or weak dipole-dipole interaction, which, in our theory, is \mimicked" by U being present only in the states a1, a2. On the other hand, there is no analog of the tunnel splittings, e, h in the two-spin model. 3.4 Analysis Below, we will measure R, , and U in the natural units, e + h, which is the net tunnel splitting: z = p 2 R ( e + h) ; = 2( e + h) ; u = U ( e + h) : (3.35) Then parameter s introduced above takes the form s(u; ; z) = ( e + h) 1 3 (u 􀀀 )2 + 􀀀 1 + z2 + 2 1=2 : (3.36) In terms of new variables the expression for parameter f takes the form f(u; ; z) = (u 􀀀 ) h 2 27 (u 􀀀 4 )(u + 2 ) 􀀀 1 3(2z2 􀀀 1) i 􀀀 1 3 (u 􀀀 )2 + 􀀀 1 + z2 + 2 3=2 : (3.37) What makes the Rabi oscillations in a quantum dot molecule nontrivial is the fact that f exhibits a nonmonotonic dependence on z within a certain domain of u and , as can 57 be seen in Fig. 3.7. Besides the obvious extremum at z = 0, the position of the nontrivial extremum z = zc can be found analytically zc = r u2 􀀀 2 u 􀀀 1 2 9 􀀀 7 ; (3.38) which corresponds to the value of f fc = 􀀀16 27 (u 􀀀 )3 + (3 􀀀 2)(u 􀀀 ) 􀀀 4 3 (u 􀀀 )2 􀀀 9 2( 􀀀 1) 3=2 : (3.39) Presence of extrema in Fig. 3.7, 3.8 translates into a nontrivial transformation of the spectrum from weak R, where it is determined by Eq. (3.25), to large R, where the conventional Rabi oscillations are restored. To illuminate this nontriviality, we start from the noninteracting case. Setting u = 0, we plot f(0; ; z) from Eq. (3.37) for four characteristic values of . See Fig. 3.7. For these -values, we plot from Eq. (3.33) the corresponding Fourier spectrum and the peak intensities as a function of z. See Fig. 3.9, 3.10, 3.11. Figure 3.7: The function f(0; ; z) is plotted for four characteristic values of the dimensionless detuning, , from Eq. (3.37). For small = 0:1 the function f remains small; as increases, the z = 0 value of f decreases to the limiting value 􀀀 2 3 p 3 , which is achieved at = 1=2. Further increasing leads to the growth of f(0; ; 0) until it reaches the upper limiting value of 2 3 p 3 . 58 Figure 3.8: The function f(u; 0; z) is plotted for three values of the dimensionless interaction parameter, u, corresponding to perfect resonance, = 0, for an isolated dot. As u increases, the zero-z value of f increases monotonically and approaches the limiting value of 2 3 p 3 . The behavior of the spectrum at small z can be, essentially, understood from the weak driving limit. See Eq. (3.25). As follows from Eq. (3.25), the dimensionless positions of the main peaks are ( 1 4u2+1)1=2+ 1 2u􀀀2 and ( 1 4u2+1)1=2􀀀 1 2u+2 . These peaks have the magnitudes proportional to z2. The third peak at dimensionless frequency (u2+4)1=2 is not captured by Eq. (3.25), since its magnitude is proportional to z4. Results of the exact calculation for u = 0 and = 0:1, see Fig. 3.9, are in accord with this prediction. The exceptional behavior at small z corresponds to = 1=2. We see that positions of two peaks are degenerate at z = 0, and, correspondingly, the third peak starts from zero frequency. Moreover, the positions of degenerate peaks go apart linearly with z. The precursor of this peculiar behavior can be again traced to Eq. (3.25). Namely, the denominator of Eq. (3.25) turns to zero at = U 2 r U2 4 + ( e + h)2: (3.40) For u = 0, Eq. (3.40) de nes = 1=2. The physical origin of this anomaly is that, without interaction, the point of zero detuning shifts from = 0 to = 1=2 due to tunneling. For this reason, the position of the lowest peak in Fig. 3.10 is linear with 59 Figure 3.9: The Fourier transform (top) and corresponding magnitudes (bottom) for the occupation of the vacuum state, Eq. (3.33), for = 0:1 and u = 0. The behavior at small z is described by Eq. (3.25), while at large z 1, the system resumes the conventional Rabi oscillations. For intermediate z, we see that there is a strong redistribution of the magnitudes. 60 Figure 3.10: The same as Fig. 3.9 for = 0:5. Note that at small z, the system exhibits \true" resonance, i.e., Rabi oscillations for weak driving, described by Eq. (3.25). 61 Figure 3.11: The same as Fig. 3.9 for = 1:5. At small z, we see two degenerate frequencies, which can be predicted from Fig. 3.7. In Fig. 3.7, for small z, we have f 0, which corresponds to the relation 1 􀀀 2 2 􀀀 3, as seen in Fig. 3.6. 62 z and has a magnitude close to 0:5, corresponding to fully developed Rabi oscillations. Based on the exact expressions Eq. (3.34), we specify the behavior of intensities of all three peaks at small z. Note that at = 1=2, the value of f(0; 1=2; 0) is equal to the limiting value 􀀀 2 3 p 3 . In the vicinity of this value the asymptotic expressions for -values are 1 = 1 3 2 p 3 􀀀 2 3 p 3 + f ; 2 = 􀀀 1 p 3 1 + 31=4 s 2 3 p 3 + f ! ; 3 = 􀀀 1 p 3 1 􀀀 31=4 s 2 3 p 3 + f ! : (3.41) On the other hand, at small z, the di erence between f(0; 1=2; z) and 􀀀 2 3 p 3 is proportional to z2. This suggests that 2 and 3 have corrections linear in z, which is re ected in the positions of spectral lines in Fig. 3.10. We now turn to Eq. (3.34) and nd that at = 1=2, we have (C2 􀀀 1=2) = (1=2 􀀀 C3) / z, while C1 / z2. The di erence of signs in the corrections (C2 􀀀 1=2) and (C3 􀀀 1=2) comes from the last brackets in the numerators in the expressions for C2 and C3, and the fact that linear in z corrections to 2 and 3 have opposite signs. Therefore, the magnitude, 2C2C3, of the low-frequency peak decreases quadratically from z = 0, where it is equal to 0:5, while the magnitudes of two other peaks grow with z as z2. Since f(0; 3=2; z) behaves quadratically with z, Eq. (3.43) suggests that the two degenerate spectral lines go apart as z2 at small z. Concerning the peak magnitudes, the lower of two degenerate peaks grows slowly, as z4, at small z, while the upper peak, as well as the double-frequency peak, behave as z2. This could be expected on the basis of weak-driving expansion Eq. (3.25), which suggests that only two peaks exhibit z2-behavior. Another peculiarity in the small-z behavior of the spectral lines takes place at = 3=2, where the two peaks are again degenerate, while the third peak is at double frequency. The origin of this degeneracy is that f(0; 3=2; 0) = 0, as follows from Eq. (3.37). At small nite f, the corresponding -values can be simpli ed to 1 = 1 􀀀 f 2 ; 2 = f; 3 = 􀀀1 􀀀 f 2 : (3.42) Since f(0; 3=2; z) behaves quadratically with z, Eq. (3.43) suggests that the two degener- ate spectral lines go apart as z2 at small z. Concerning the peak magnitudes, the lower of two degenerate peaks grows slowly, as z4, at small z, while the upper peak, as well as the double-frequency peak, behave as z2. This could be expected on the basis of weak-driving expansion Eq. (3.25), which suggests that only two peaks exhibit z2-behavior. Another 63 peculiarity in the small-z behavior of the spectral lines takes place at = 3=2, where the two peaks are again degenerate, while the third peak is at double frequency. The origin of this degeneracy is that f(0; 3=2; 0) = 0, as follows from Eq. (3.37). At small nite f the corresponding -values can be simpli ed to 1 = 1 􀀀 f 2 ; 2 = f; 3 = 􀀀1 􀀀 f 2 : (3.43) Analysis of the spectral lines for strong driving z 1 is straightforward. This is because parameter f falls o with increasing z as 1=z, see Eq. (3.37). This means that the expressions Eq. (3.43) for -values apply for large z. Taking into account that parameter s grows linearly with z, we conclude that the position of the upper spectral line is 2z, which corresponds to the conventional Rabi oscillations. The positions of two satellites are close to z. Their deviations from z have equal values and opposite signs. A nontrivial feature is that these deviations, 3sf=2, saturate with increasing z at 2 3 (u􀀀 )( e+ h). Concerning the magnitudes of the peaks, it can be easily seen from Eq. (3.34) that at large z, the coe cients C1 and C3 approach 1=2. This suggests that the main spectral line approaches the conventional magnitude 2C1C3 = 0:5. It is less trivial to extract from the expression for C2 that it behaves as 1=z2 at large z, which corresponds to ( e+ h)2=2 2 R in the dimensional units. This fall o of the magnitude of the satellites is illustrated by all numerical curves in Figs. 3.9-3.11. The most interesting behavior of the spectral lines takes place at intermediate z, namely around z = 2, i.e., R p 2( e + h). As can be seen in Figs. 3.9-3.11, a strong redistribution of magnitudes takes place in this domain. While for smaller z the green line s( 2 􀀀 3) has a maximal magnitude, while the yellow \Rabi" line is suppressed