© 2016 Aiyin Liu A COMPUTATIONAL ELECTROMAGNETIC FRAMEWORK FOR WAVEGUIDE QUANTUM ELECTRODYNAMICS BY AIYIN LIU THESIS Submitted in partial fulfillment of the requirements for the degree of Master of Science in Electrical and Computer Engineering in the Graduate College of the University of Illinois at Urbana-Champaign, 2016 Urbana, Illinois Adviser: Professor Weng C. Chew ABSTRACT In this thesis, a framework for cavity and waveguide quantum electrodynamic (CQED, and WQED, respectively) problems is presented. The essence of this framework is to use the dyadic Green's function (DGF) to obtain the local density of states (LDOS) and radiative shift of a two-level system (TLS) embedded in an arbitrary electromagnetic continuum. The dressed states of the system are also found using a method of direct diagonalization of operators. The physics and essential features of the dressed states are related to the DGF. The dynamics of the system are solved in connection with the resolvent formalism of quantum scattering theory. The work presented here represent the first attempts in this important topic of study. This thesis accomplishes the full solution of the problem of a two-level atom coupled to arbitrary lossless electromagnetic environments. The cases of transmission line, hollow rectangular waveguide and free space are considered. At the same time, an attempt to gather much of the required background for this highly interdisciplinary topic is made, in particular, the quantization of an arbitrary lossless waveguide is presented in detail, which to the best of our knowledge, has not appeared in the literature. More detailed investigations will follow in a Ph.D. dissertation. ii TABLE OF CONTENTS CHAPTER 1 GREEN'S FUNCTION ................. 1 1.1 Inhomogeneous transmission line: Classical .......1 1.2 Inhomogeneous transmission line: Quantization .... 14 1.3 General waveguide structure: Classical .......... 19 1.4 General waveguide structure: Quantization ....... 26 CHAPTER 2 FANO DIAGONALIZATION . 28 2.1 Statement of the problem ................... 29 2.2 Two-level atom single continuum .............. 32 2.3 Two-level atom multiple continua .............. 36 2.4 New state below the continuum ............... 44 CHAPTER 3 RESOLVENT ....................... 47 3.1 Introducing the resolvent .................... 47 3.2 Representations of the resolvent ............... 50 3.3 Interpretations of the resolvent ............... 53 3.4 Applications of the resolvent ................. 58 3.5 The resolvent in the continuum limit ........... 65 REFERENCES . 76 iii CHAPTER 1 GREEN'S FUNCTION 1.1 Inhomogeneous transmission line: Classical Consider an inhomogeneous transmission line at x ∈ [0;L] whose per unit length capacitance and inductance are functions of space, c(x) and l(x), re- spectively. The equations of motion governing the voltage, V (x) and current, I(x), on this transmission line are the telegraphers' equations: ⎪⎧ ⎪ @xV = − l(x) @t I ⎨ (1.1) ⎪ ⎩⎪@x I = − c(x) @tV It is convenient to normalize the problem using the average values of l(x) and c(x). ⎧ ⎪ ⎪ 1 L l(x) ⎪ l = dx l(x) ; and l (x) = ⎪ 0 ∫0 r ⎨ L l0 ⎪ ⎪ 1 L c(x) ⎪c = dx c(x) ; and c (x) = ⎪ 0 ∫0 r ⎩ L c0 The second-order equations are in the form of wave equations. ⎪⎧ ⎪ −1 2 ⎪@x lr @xV − l0c0 cr @t V = 0 ⎨ (1.2) ⎪ ⎪ @ c−1 @ I − l c l @2 I = 0 ⎩⎪ x r x 0 0 r t For time harmonic solutions the dispersion relation is !2 = v2k2, where v = 1 2 (l0c0)− ~ is the phase velocity of the transmission line. Plugging in this 1 relation and transforming to the frequency domain turns Equation (1.2) into: ⎪⎧ ⎪ −1 2 ⎪@x lr @xV + cr k V = 0 ⎨ (1.3) ⎪ ⎪ −1 + 2 = ⎩⎪ @x cr @x I lr k I 0 These are the source free equations governing the eigenfunctions of the trans- mission line. Obtaining these eigenfunctions will allow us to quantize the problem in terms of modes along the lines of canonical quantization. We make several observations on the properties of these equations and their so- lutions before proceding to the quantization of this transmission line. 1.1.1 Continuity of materials and fields From Equation (1.3) it is apparent that V (x) and I(x) are everywhere contin- uous. To obtain the continuity condition on their first order derivatives, con- sider the equation for voltage in Equation (1.3) and assume a delta-function source is exciting the transmission line. Then, −1 2 ′ @x lr @xV + cr k V = a δ(x − x ) Integrating the above equation over a vanishing segment x ∈ [x′ ; x′ ] around − + x′ gives: −1 −1 lr @xV T ′ − lr @xV T ′ = a x=x+ x=x− Hence, in regions free of delta-function sources, we must have the following continuity conditions on the first-order derivatives: ⎪⎧ ⎪ −1 ⎪lr @xV is continuous; KCL ⎨ ⎪ ⎪ −1 ⎩⎪cr @x I is continuous; KVL These conditions reflect the local Kirchoff equations where the transmission line material changes. Sudden jumps are allowed in lr(x) and cr(x) in our discussions. 2 1.1.2 Operators and boundary conditions We abstract Equation (1.2) into a set of operator equations by defining: ⎪⎧ ⎪ −1 ⎪LV = −@x lr @x ⎨ (1.4) ⎪ ⎪ L = −@ c−1 @ ⎩⎪ I x r x To facilitate the study of scattering problems in later chapters, we would prefer to work with traveling waves in the x direction. These waves have a definite wave number but are complex. Hence, it is convenient to consider energy inner products1 of the form [1]: L 1 ∗ ∗ ⟨V1 ; LV V2⟩ = S dxV1 −@x @x V2 (1.5) 0 lr The Hermitian adjoint of the operator LV is written as LV and defined through the inner product: ∗ ∗ ⟨V1 ; LV V2⟩ = bLV V1 ;V2g (1.6) Starting from the definition in Equation (1.5) and doing integration by parts twice, we obtain a Lagrange identity [2]: L 1 ∗ ∗ ⟨V1 ; LV V2⟩ = S dxV1 −@x @x V2 0 lr L L 1 1 L 1 ∗ ∗ ∗ = −V1 @xV2W + V2 @xV1 W + S dx −@x @x V1¡ V2 0 ∗ lr 0 lr 0 lr 1 1 L ∗ ∗ ∗ ∗ ∗ ⟨V1 ; LV V2⟩ = a{LV V1} ;V2f + V2 @xV1 − V1 @xV2 (1.7) lr lr 0 If we restrict ourselves to a everywhere lossless transmission line that is pe- riodic at the end points, x = 0 and x = L. Then, lr(0) = lr(L) and l(x) ∈ R, ∗ so LV = LV . According to Equation (1.7) the operator LV is Hermitian with 1The complex conjugation on one of the functions is reminiscent of complex power in the frequency domain, hence the name energy inner product. 3 the boundary condition:2 ∗ ∗ Im V2 @xV1T − V2 @xV1T = 0: (1.8) x=L x=0 A number of boundary conditions can make the operator LV Hermitian. The Dirichlet and Neumann boundary conditions are often used for closed problems where the transmission line is terminated by open or short circuits at the end points. These boundary conditions give real eigenfunctions. To allow traveling waves the suitable boundary condition that satisfy Equation 3 (1.8) is the periodic boundary condition on V (x) and @xV (x) = V ′(x). Now we summarize our consideration of the continuity and boundary conditions of the V (x) and I(x) fields. Hermitian and symmetry properties The operators in Equation (1.4) are associated with a lossless inho- mogeneous transmission line that is periodic at x = 0 and x = L. The a operator LV is Hermitian on the domain: 2 ′ ′ D(LV ) = V; LV V ∈ L [0;L] ∶ V (L) = V (0);V (L) = V (0) (1.9) Note also that LV is symmetric on the domain defined by Equation (1.9), which is to say: ⟨V2; LV V1⟩ = ⟨V1; LV V2⟩ (1.10) Similar results hold for LI . aHere V ′ denotes the first-order x derivative. The Hermitian property is very important for both the classical and quan- 2This is the analogy of self adjoint boundary conditions for linear operators on the space of real functions of a single variable x [2]. 3Equation (1.8) can also be satisfied by the so-called twisted boundary condition. V (L) = eiφV (0) Here, the same condition should also be applied to the first derivative of the function. This is a generalized periodic boundary condition. The phase φ may be induced by a magnetic field through the Ahoronov Bohm effect for charged particles [3], or effectively for microwave photons [4]. Hence, it is now possible to implement such a phase into a transmission line. This phase factor can be used to tune the dispersion relation. 4 tum mechanical problem. It ensures that LV and LI have real eigenvalues. Their eigenfunctions form complete orthonormal sets that span a Hilbert space. Notably, they are capable of expanding functions that are outside of the domain defined in Equation (1.9). 1.1.3 Eigenvalues and eigenfunctions Equation (1.3) can be rewritten as two generalized eigenvalue problems for the LV and LI operators, respectively: ⎧ ⎪ 2 ⎪LV V = cr k V ⎨ (1.11) ⎪ L I = l k2 I ⎩⎪ I r These two operators share the same eigenvalue and there eigenfunctions are related4 by Equation (1.1): ¾ 2 2 1 @xV (x; k ) l0 I(x; k ) = ; Z0 = (1.12) Z0 iSkSlr c0 Here Z0 is the average characteristic impedance of the transmission line. It suffices to consider LV . Since LV is Hermitian, we know it possesses only real eigenvalues. Using this fact, consider: ∗ ∗ ∗ 2 2 ∗ ⟨V1 ; LV V2⟩ − ⟨V2 ; LV V1⟩ = (k2 − k1)⟨V1 ; crV2⟩ = 0 The choices are degeneracy of eigenvalues or orthogonality of eigenfunctions. It is convenient to normalize the eigenfunctions with a weight of cr(x). 4This can be easily shown, given that V (x; k2) satisfy the generalized eigenvalue equa- 2 2 2 2 tion LV V (x; k ) = cr k V (x; k ), we plug in the corresponding I(x; k ), computed from 2 Equation (1.12), into LI I(x; k ).