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Relaxation and Spectra in Condensed Phases Relaxation and Spectra in Condensed Phases 1)V Davi( Rloberto ei(thmlan Submitted to the Department of Chemistry il partial fulfillment of the requirenments for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 1997 XMassachusetts Institute of Technology 1997. All rights reserved. Autihor ................ ....................... Department of Chemistry May 22. 1997 C ertified by ..... ................. Robert .Silbey Pr ffessor of Chemistry Thesis Supervisor Accepted by ..................................... .... ] . -......... Dietmar Seyferth Chairman, Department Committee on Graduate Students (4" TE-CHNO JUL 1 4 1997 Science L~BRARIES This doctoral thesis has been examined by a committee of the Department of Chemistry as follows: Professot :win Oppenheim......... ............ Y .; .. ......................... Chairman Professor Robert J. Silbey ......... •. Thesis Supervisor Si Professor Keith A. Nelson...... V ........... ....................... Relaxation and Spectra in Condensed Phases by David Robert Reichman Submitted to the Department of Chemistry on May 22, 1997, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this thesis, various problems in the quantum theory of relaxation are considered. We first discuss the dynamics of low temperature tunneling in condensed phases. With the aid of a varational unitary transformation, we are able to study the limits of both weak and strong coupling of the tunneling system to the environment. Next we discuss the physical implications of the standard Redfield equations when perturbation theory is carried out to fourth order. We then apply similar techniques to the study of the spin-boson problem. Here, various methods of aproximate resummation are used to calculate the mean position of a tunneling particle coupled to a fluctuating quantum environment. The next section deals with a theoretical attempt to explain recent hole burning experiments in a low temperature polymer glass. The last section deals with the nonperturbative theory of pure dephasing. Thesis Supervisor: Robert J. Silbey Title: Professor of Chemistry Acknowledgments First and foremost, I would like to thank my thesis advisor, Bob Silbey. Bob's keen insights, knowledge, and good humor were always helpful. I thank him most for the reminder "99 percent of the time that you think you'll fail, you'll be alright, the one time out of one hundred that you do fall flat on your face, only you will remember." I would like to thank professor Irwin Oppenheim. Irwin was always willing to answer my questions and listen to ideas, no matter how naive. I would also like to thank Irwin, as well as professor Bruce Tidor, for helping out with applications and recommendations. I would like to thank Professor Min Cho, Dr. Peter Neu and Dr. Alberto Suarez for helpful discussions and scientific collaborations. Some of these collaborative efforts appear in this thesis. I would also like to thank my "zoo" friends, Frank Brown, Cliff Liu, Welles Mor- gado, Joan Shea, and Sophia Yaliraki. I would separately like to thank Frank for time spent on scientific collaboration, and computer help. I would like to thank Cliff, Sophia, and Joan for help with this thesis. I would like to thank the DOD for financial support in the years 1993-1996. Lastly, I would like to thank my parents, brother and sister for their support and love. Contents 1 Introduction 2 A Study of the Coupling Dependence of Relaxation Rates in a Model of Low Temperature Tunneling Dynamics 2.1 Introduction ..................... 2.2 The Hamiltonian .................. 2.3 D ynam ics .... ...... ..... ...... 2.4 Discussion . ..................... 3 On the Relaxation of a Two Level System: Beyond the Weak- Coupling Approximation 30 3.1 Introduction ................................ 30 3.2 Review of the Problem . ............... 34 3.3 Population Relaxation . .... .......... 40 3.4 Application to the Spin-Boson Problem ...... ......... .. 43 3.5 Dephasing .................... ..... ...... 46 3.6 Variational Procedure . ........ ............... 51 3.7 Conclusion ...................... .. .. ..... 58 3.8 Appendix A ......... ..... 59 4 Cumulant Expansions and the Spin-Boson Problem 4.1 Introduction . ................ 4.2 Moment Expansion . ............ : : I 1 I I I I 4.3 Cumulant Expansions ................... .... 7 4.4 Results and Conclusions ................... ...... 84 5 Spectral diffusion on ultra-long time scales in low temperature glasses 96 5.1 Introduction ................... .... ....... 96 5.2 Hole-burning at ultra-long times and the Burin-Kagan theory . .. 99 5.3 The Kassner-Silbey approach for primary and secondary TLS 104 5.4 Stability analysis . .. .. ... 107 5.5 Comparison with experiment . .. 111 5.5.1 The limit Emax(T) > kBT 111 5.5.2 The limit Emax(T) < kBT 113 5.6 Discussion . ............ 114 5.7 Conclusions ............ 117 5.8 Appendix A ....... .. ................ ......• ° ° 119 5.9 Appendix B . ..... ...... 122 5.10 Appendix C ............ 123 6 On the Nonperturbative Theory of Pure Dephasing in Condensed Phases at Low Temperatures 130 6.1 Introduction. ............. .. ....... 130 6.2 Linear Electron-Phonon Coupling . ... ........ 132 6.3 Experimental Relevance .... .. ... ......... 135 6.4 Quadratic Electron-Phonon Coupling .. ........... 138 6.5 Conclusion . ...... .......... ...... 144 List of Figures 2-1 Energy level diagram of the tunneling part of the Hamiltonian (2.1). j indicates the tunneling matrix element in the upper (n= 1) vibrational manifold. 2j indicates the tunnel splitting of the upper vibrational manifold. Q is the energy difference of the two manifolds. I and - are the intramanifold relaxation rates caused by interaction with the phonon bath. Fol and Flo are the intermanifold relaxation rates caused by the interaction with the phonon bath. ............. 26 2-2 Behavior of R.es for values of A > 4.Solid lines represent Resc and dotted lines represent "quasiclassical" rate - exp(-3Q)± normalized to the same starting value. The upper group is for 3Q = 5 while the lower group is for 3Q = 6. J is taken to be 1 cm -1,w = 100cm - 1,and Q = 50cm - 1 ........... ....... ............. 27 3-1 Fr(w)/w 2 for "Ohmic-Lorentzian" density of states Fl(w) = A (W2+,\2)" The value of A, which has units of frequency, is taken to be 1 cm-' wo is taken to be 1 cm - '. The values of A (in units of cm-') are: (a) .6, (b) 1, (c) 1.47, (d) 2. For $- > 1.4679, (7)(4) becomes negative. 63 T2 3-2 Same plot for the super-Ohmic density of states FI(w) = e-w/wo. B is taken to be 1 cm-', wo has the value 1 cm -1,and the values of w, (in units of cm - ') are: (a) 0.1, (b) 0.15, (c) 0.281. ........... 64 3-3 Same plot as (Fig. 2) with the values of wu (in units of cm - ') (a) 0.7, (b) 1.006, (c) 2. The function (_)(4) is negative for 0.218 > wl/wo > 1.006. 65 2 4-1 Zero temperature plot of P(y) (y = Aefft) for a = 0.3 and ( = 6. The dotted line is the second order POP result, the dashed line is the second order non-crossing cumulant result, the dash-dotted line is the NIBA (second order COP) result, and the solid line is the fourth order non-crossing cumulant result. ................... ... 88 4-2 Zero temperature plot of P(y) (y = Aefft) for a = 0.6 and y = 6. The dash-dotted line is the NIBA (second order COP) result, the dashed line is the second order POP result, the solid line is the second order non-crossing cumulant result, and the open circles are the simulation result of Egger and Mak ................... ...... 89 4-3 Zero temperature plot of P(y) (y = Aefft) for a = 0.7 and ( = 6. The dash-dotted line is the NIBA (second order COP) result, the dotted line is the second order POP result, the solid line is the second order non-crossing cumulant result, and the open circles are the simulation result of Egger and Mak ........ ........... ...... 90 4-4 Zero temperature plot of P(y) (y = Aefft) for a!= 0.7 and ( = 6. The dotted line is the fourth order non-crossing cumulant result, the dashed line is the second order POP result, and the solid line is the second order non-crossing cumulant result. Note the change in the y-axis. 91 4-5 Relative magnitude of second and fourth cumulant effects in the POP for a = 0.7. The dashed line shows I f0 dy' fo dy"K2POP(y")I and the solid line shows If0 dy' fo dy" K4POP(y")|. KPOP(t) is defined in Eq. (38) ..... ....... .. ............. .. 92 5-1 Hole broadening in PMMA at 1 K (upper curve) and 0.5 K (lower curve) compared with the Kassner-Silbey theory for single TLS. The experimental data are from Ref.12. The two solid lines are An()(t), Eq. (5-62) for 1 (upper solid curve) and 0.5 K (lower solid curve) with G = 32 (ED =108 K) and Po(C) = 6 x 10-5. ............. 128 5-2 Same figure as above but now with the contribution of both the single and pair TLS (logarithmic and algebraic part of Eq.(5-61)). The pa- rameters Po(C) = 4 x 10- 5 and PoUo = 2.5 x 10-6 have been optimized to find best agreement with the 0.5 K data; G = 32 has been kept fixed. Comparison with the 1 K data clearly shows that the tempera- ture dependence as predicted by Eqs.(5-29) and (5-61) is too strong. 129 6-1 Plot of In alo(t) vs. t in units where h = 1. 3 = .1 and W = .3. The cutoff frequency of the bath is taken to be wa= 95. Dashed line is from (6.18) with a twenty mode bath. Dash-dot line is from (6.18) with 50 mode bath, solid line is the Skinner-Hsu result (6.19). The time domain for which (6.18) and (6.19) agree depends on the number of bath modes employed.
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