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Scattering from Spatially Localized Chaotic and 1 0 Disordered Systems 0 2 n L.E. Reichl and G. Akguc a Center for Studies in Statistical Mechanics and Complex Systems J The University of Texas at Austin 1 2 Austin, Texas 78712 ] February 8, 2008 D C n. Abstract i l A version of scattering theory that was developed many years ago to n treat nuclear scattering processes, has provided a powerful tool to study [ universalityinscatteringprocessesinvolvingopenquantumsystemswith 2 underlying classically chaotic dynamics. Recently, it has been used to v makerandommatrixtheorypredictionsconcerningthestatisticalproper- 0 ties of scattering resonances in mesoscopic electron waveguides and elec- 6 tromagneticwaveguides. Weprovideasimplederivationofthisscattering 0 theory and we compare its predictions to those obtained from an exactly 2 solvable scattering model; and we use it to studythe scattering of a par- 1 ticle wave from a random potential. This method may prove useful in 0 distinguishingtheeffectsofchaosfromtheeffectsofdisorderinrealscat- 0 / tering processes. n i l n : v 1 Introduction i X r Interest in the dynamical properties of open quantum systems at mesoscopic a and atomic scales has lead to a rebirth of a form of scattering theory which was originally developed to deal with very complicated nuclear collision processes. The originof the scattering theory that we consider here came fromthe recognition that a collision between two nucleons, one of which is very heavy, can lead to the creation of an unstable, but very long-lived, compound nucleus which eventually decays. In the late 1930’s, Kapur and Peierls [1] used this 1 fact to formulate a nonperturbative approachto scattering theory in which the compound nucleus was viewed as a stable object which was made unstable by weak coupling to the continuum. In the late 1940’s, Wigner and Eisenbud [2] developed an alternate version of the Kapur-Peierls theory which lead to the very widely used R-matrix approach to scattering theory [3]. The idea behind both these theories is to decompose configuration space into an internal region (reactionregion)andanexternalasymptotic scatteringregion. As we shallsee, this approach can be made clean and rigorous if there is an abrupt (in configuration space) transition between the scattering region and the asymptotic region. The internal regioncan be modeled in terms of a complete set of states with fixed boundary conditions on the surface of the internal region. The internal states can then be coupled to the external asymptotic states. Bloch [4] and Feshbach [5] both showed that a consistent theory requires that the coupling betweenthe internalandexternalregionsmustbe singular. Inthe 1960’s, Weidenmuller developed a phenomenological Hamiltonian approach to nuclear scattering based on this picture [6]. The Hamiltonian for the interior region was based on the shell model of the nucleus. The Hamiltonian for the exterior scattering regiondescribedthe asymptotic states. The strength of the coupling between the interior and exterior regions was an unknown input parameter. The Weidenmuller approach to scattering theory created a framework with which to study the manifestations of chaos in the scattering properties of nuclear systems, as well as mesoscopic and atomic systems. It is now well known that chaos manifests itself in bounded quantum systems by affecting the statistical distribution of spacings between energy levels [7]. For open quantum systems, in regimes where scattering resonances are not strongly overlapping, underlying chaos affects the statistical distribution of resonance spacings and resonance widths [8]. If the Hamiltonian of the interior regionis chosento be a random matrix Hamiltonian, then predictions can be made as to the statistical properties of resonance spacings and resonance widths. These predictions can 2 then be compared to nuclear scattering resonances, or to resonances in electron waveguides [[9]], [13],and [[11]] and electro-magnetic resonators [[1]] and [14] whose cavities have been constructed to yield classically chaotic behavior. The agreement between random matrix theory predictions and numerical and laboratory experiments is quite good. The Weidenmuller Hamiltonian is phenomenological and does not have information about the strength of the interface coupling. The random matrix theories treat the interface coupling as an input parameter. In a real dynamical system, however, one needs a way to rigorously determine the coupling at the interface of the internal and external regions. A very useful way to fix the coupling at the interface was provided by Pavlov [12]. The idea of Pavlov was that the surface coupling could be fixed by the requirement that the total Hamiltonian (including interior and exterior regions) be Hermitian. In this paper, we use a simple textbook scattering problemto illustrate and clarify many issues concerning this approach to scattering, and we then use it to study the effect of disorder on the scattering process. In Section (2), we describe the “textbook” scattering problem and obtain exact expressions for the reaction function, the scattering function, and the Wigner delay times. In Section (3), we develop the Hamiltonian for interior and exterior regions of the scattering system. In Section (4) we derive the Hermiticitycondition. InSection(5),wederivethereactionfunction. InSection (6), we derive the scattering function andlocate resonanceenergies forthe case of a smooth scattering potential. In Section (7), we use this theory to study the scatteringofaparticlewavefromarandomscatteringpotential. Finally,in Section (8), we make some concluding remarks. 3 2 Exact Solution Wewillfirstconsiderthescatteringofaparticleofmass,m,duetothepotential shown in Fig. (1). The quantum particle enters from the right with energy E and is reflected back to the left by an infinitely hard wall located at x = 0. A barrier of height, V , is located 0 < x < a. The Schroedinger equation, which 0 describes propagation of a particle wave, Ψ(x,t), for all times, t, is given by ∂Ψ(x,t) ¯h2 ∂2Ψ(x,t) i¯h = +V(x)Ψ(x,t), (2.1) ∂t −2m ∂x2 where ¯h is Planck’s constant and the potential, V(x) = for ( < x 0), ∞ −∞ ≤ V(x) = V for (0 < x a), and V(x) = 0 for (a < x < ). Since the potential 0 ≤ ∞ is infinite for <x<0, no wavefunction can exist in that region of space. −∞ The Schroedinger equation for energy eigenstates ΦI (x), for Region I (0 < E x<a) in Fig. (1), is ¯h2 d2ΦI (x) E +V Φ (x)=EΦI (x), (2.1) −2m dx2 0 E E Energy eigenstates in Region I have the form ΦI (x)=Asin(k′x) (2.2) E wherek′ = 2m(E V )andAisanormalizationconstant. TheSchroedinger h¯2 − 0 equation forqenergy eigenstates ΦII(x), for Region II (a < x < ) in Fig. 1, E −∞ is ¯h2 d2ΦII(x) E =EΦII(x), (2.3) −2m dx2 E Energy eigenstates in Region II have the form ΦII(x)=B(e−ikx S(E)eikx), (2.4) E − where k = 2mE. The first term describes the incoming part of the energy h¯2 eigenstate aqnd the second term describes the outgoing part of that state. The constant, B, is a normalization constant and S(E) is the scattering function. 4 When V =0, then S(E)=1 and the incoming wave is reflected from the wall 0 at x = 0 with an overall phase shift of π since we can rewrite the minus sign as eiπ. Thus, S(E), contains informationabout the phase shift of the scattered wave due to the potential barrier, V , in the region, 0<x<a. 0 We can determine how the two parts of the energy eigenstate are connected by the condition that the wavefunction and the slope of the wavefunctionmust becontinuousatx=a. Thesetwoconditionscanbecombinedintoastatement thatthe”logarithmicderivative”ofthewavefunctionmustbecontinuousatthe interface. Thus we must have 1 a dΦI a dΦII E = E (2.5) R(E)≡ΦI (a) dx ΦII(a) dx E (cid:12)a E (cid:12)a (cid:12) (cid:12) at the interface. The function, R(E), is(cid:12)(cid:12)called the reactio(cid:12)(cid:12)n function and it will play an important role in the following sections. The S-matrix is then given by (1+ikaR(E)) S(E)=e−2ika . (2.6) (1 ikaR(E)) − where the reaction function, tan(k′a) R(E)= . (2.7) k′a Since the scattering function has unit magnitude we can also write it S(E)=eiθ(E) =e−2ikaeiθ′(E), (2.8) whereθ(E)isthephaseshiftofthescatteredwaverelativetothecaseforwhich V =0. It is straightforwardto show that 0 2kaR(E) θ′ tan(θ′)= and tan( )=kaR(E). (2.9) 1 k2a2R(E)2 2 − The time delay of the scattered particle due to its interaction with the barrier, V , is given by 0 dθ τ ¯h . (2.10) ≡ dE k=k0 (cid:12) This time delay is called the Wigner dela(cid:12)y time [[18]]. 5 For the scattering system we consider here, the phase angle, θ′(E) is given by β θ′(E)=2arctan tan(β′) , (2.11) β′ where β =ka and β′ =k′a. The Wigner(cid:2)delay time(cid:3)is given by dθ(E) τ(E)=h¯ dE 2ma2 1 β′ β′ 2ma2 = tan(β′)+βsec(β′)2 . (2.12) ¯h2 β′2+β2tan(β′)2 β −β − ¯hβ (cid:18) (cid:20)(cid:18) (cid:19) (cid:21)(cid:19) In Fig. (2.a) we show the phase angle, θ(E), as a function of energy, and in Fig. (2.b), we show the Wigner delay time for parameter values, ¯h=1, m=1, a=1,andV =10. ThepeaksinFig. (2.b)occuratvaluesoftheenergywhere 0 the particle wave resonates with the barrier region. The fact that the Wigner delay time goes negative for low energies means that when V =0, the particle 0 6 can be reflected earlier by the barrier, V , than it would be if V =0. 0 0 In Fig. (3), we show that exact energy eigenstates as a function of position for a range of energies. At resonant energies, the energy eigenfunctions have enhanced probability in the scattering region. These resonances are also often calledquasi-boundstatesbecausetheyareassociatedwithcomplexenergypoles of the scattering function. In Fig. (4), we plot those complex energy poles of S(E)whichgiverisetothefirstthreeresonancepeaksintheWignerdelaytime. Itisimportanttonote thatforscatteringsystemswithhigherspacedimension, complex energy poles can have a profound effect on the scattering properties. Forexample,intwodimensionalelectronwaveguidestheycancompletelyblock transmission in some channels [[15]], [[16]]. 3 Scattering Hamiltonian Most of the interesting information regarding scattering events is contained in theenergydependenceofthescatteringphaseshiftsandinthelocationofquasi- bound state poles. However, we generally cannot find those quantities exactly 6 as we did in the previous section. There are a number of techniques for finding them if the scattering potential is weak, but not if the scattering potential is strong. In subsequent sections, we describe an approach to scattering theory whichcandealwithstronginteractions,providedtheyareconfinedtoalocalized regionofspace. AswementionedinSect. (1),thisapproachtoscatteringtheory was originally developed to deal with nuclear scattering processes,but recently has provided an important tool for studying universal properties of scattering processes induced by underlying chaos. We shall use this alternate approach to scattering theory to study the system considered in Sect. (2), and a similar system with a random scattering potential. In this way, we can compare its predictions to the exact results which were obtained in Sect. (2). Let xˆ denote the position operator and let x denote its eigenstates, so | i that xˆx = xx . The position eigenstates satisfy a completeness relation, | i | i ∞ dx x x =ˆ1, where ˆ1 is the unit operator,and they are delta normalized, −∞ | ih | R∞ dx xx′ = δ(x x′). Since the potential energy is infinite for x < 0, all −∞ h | i − sRtates will be zero in that region. We can divide the completeness relation into threeparts,Nˆ = 0 dx x x,Qˆ = adx x x,andPˆ = ∞dx x x,sothat −∞ | ih | 0 | ih | a | ih | Nˆ+Qˆ+Pˆ =ˆ1. HRowever,theoperatoRrNˆ actingonanystaRtegiveszerobecause thepotentialenergyisinfiniteinthatregion. Therefore,wecanremoveNˆ from the completeness relation without changing our final results. Thus, from now on we will write the completeness relation as Qˆ+Pˆ =ˆ1. The operators,Qˆ and Pˆ , are projection operators. They have the property that Qˆ = Qˆ2, Pˆ = Pˆ2, and QˆPˆ = PˆQˆ = 0. This is easily checked by explicit calculation. The operator Qˆ projects any state or operator onto the interval, 0 x a, while the operator Pˆ projects any state or operator onto the interval, ≤ ≤ a x . Inother words,if the state, Ψ hasspatialdependence, Ψ(x) xΨ , ≤ ≤∞ | i ≡h | i over the interval (0 < x < ), then the state xQˆ Ψ = Ψ(x) for (0 < x < a) ∞ h | | i and the state xPˆ Ψ =Ψ(x) for (a<x< ). h | | i ∞ 7 Inside the Region I, (0<x<a) in Fig.(1), we define a Hamiltonian, 1 Hˆ Qˆ pˆ2+V Qˆ, (3.1) QQ 0 ≡ 2m (cid:0) (cid:1) wherepîsthemomentumoperatorandmisthemassoftheparticle. TheHamil- tonian, Hˆ , is Hermitian and therefore it will have a complete, orthonormal QQ set of eigenstates which we denote as Qˆ φ . We can write the eigenvalue prob- j | i lem in the region, 0 < x < a, as Hˆ Qˆ φ = λ Qˆ φ , where λ is the jth QQ j j j j | i | i energyeigenvalue of Hˆ andj =1,2,... . Because there is aninfinitely hard QQ ∞ wall at x=0, the eigenstates φ (x) xQˆ φ must be zero at x=0. We have j j ≡h | | i some freedom in choosing the boundary condition at x=a and we will do that later. Thecompletenessofthestates,Qˆ φ ,allowsustowritethecompleteness j | i relation, jQˆ|φjihφj|Qˆ =Qˆ. Orthonormality requires that hφj|Qˆ|φj′i=δj,j′. The HPamiltonian for the region, a<x< , can be written ∞ 1 Hˆ Pˆ pˆ2 Pˆ. (3.2) PP ≡ 2m (cid:0) (cid:1) Its eigenvalues are continuous and have range, (0 E ). The eigenvector of ≤ ≤∞ Hˆ , with eigenvalue, E, will be denoted Pˆ E . The eigenvalue equation then PP | i reads, Hˆ Pˆ E =EPˆ E . PP | i | i Any state, Ψ , can be decomposed into its contributions from the two dis- | i jointregionsofconfigurationspaceas Ψ =Qˆ Ψ +Pˆ Ψ . WecanexpandQˆ Ψ | i | i | i | i in terms of the complete set of states, Qˆ φ and we obtain, j | i Ψ = β Qˆ φ +Pˆ Ψ , (3.3) j j | i | i | i j X where β = φ Qˆ Ψ . The function xQˆ Ψ must be equal to the function, j j h | | i h | | i xPˆ Ψ , at the interface, x = a. In addition, the slopes of these two functions h | | i must be equal at the interface. We couple the two regions of configuration space, at their interface, with the singular operator, Vˆ = Cδ(xˆ a)pˆ. The coupling constant, C, will be − determined later. Then a ∞ d Hˆ =QˆVˆPˆ =C dx dx x δ(x x )δ(x a) x , (3.4) QP 1 0 1 1 0 0 0 | i − − dx h | Z0 Za 0 8 and ∞ a d Hˆ =PˆVˆQˆ =C dx dx x δ(x x )δ(x a) x . (3.5) PQ 0 1 0 0 1 1 1 | i − − dx h | Za Z0 1 Itisusefultorememberthat adxδ(x a)= 1, adxδ(x x )=1if0<x <a, 0 − 2 0 − 0 0 a and 0dx δ(x−x0)=0 if aR<x0. Note also thRat R a ¯h2 d2 Hˆ = dx x − +V x (3.6) QQ | i 2m dx2 0 h | Z0 (cid:18) (cid:19) and ∞ ¯h2 d2 Hˆ = dx x − x (3.7) PP | i 2m dx2 h | Za (cid:18) (cid:19) The total Hamiltonian of the system can be written Hˆ =Hˆ +Hˆ +Hˆ +Hˆ . (3.8) QQ PP QP PQ The energyeigenstates, E ,satisfythe eigenvalueequationHˆ E =E E . The | i | i | i energy eigenstates can be decomposed into their contributions from the two regions of configuration space, so that E = γ Qˆ φ +Pˆ E , (3.9) j j | i | i | i j X where γ = φ Qˆ E . The eigenvalue equation then takes the form j j h | | i HˆQQ 0 ... HˆQP γ1Qˆ φ1 γ1Qˆ φ1 0 Hˆ ... Hˆ γ Qˆ|φ i γ Qˆ|φ i QQ QP 2 2 2 2  ... ... ... ...  ...| i=E ...| i (3.10) HˆPQ HˆPQ ... HˆPP  Pˆ|Ei   Pˆ|Ei  This yields a series of equations Hˆ φ γ +Hˆ E =EQˆ φ γ , (3.11) QQ j j QP j j | i | i | i for j =1,2,... and Hˆ E + H φ γ =EPˆ E . (3.12) PP PQ j j | i | i | i j X Before we can proceed further, we must find conditions for Hermiticity of the Hamiltonian, Hˆ. 9 4 Hermiticity Condition Consider the arbitrary states, Ψ and Ξ . The condition for Hermiticity of | i | i these states is that ΞHˆ Ψ ΨHˆ Ξ ∗ =0. (4.1) h | | i−h | | i We can decompose the states Ψ and Ξ into their contributions to the two | i | i disjoint configuration space regions and write them in the form, Ψ =Qˆ Ψ +Pˆ Ψ = β Qˆ φ +Pˆ Ψ , (4.2) j j | i | i | i | i | i j X where β = φ Qˆ Ψ and j j h | | i Ξ =Qˆ Ξ +Pˆ Ξ = α Qˆ φ +Pˆ Ξ , (4.3) j j | i | i | i | i | i j X where α = φ Qˆ Ξ . In the interior region, we have expanded Ψ and Ξ j j h | | i | i | i in terms of the complete set of energy eigenstates, Qˆ φ , of the Hamiltonian, j | i Hˆ . To simplify the notation, let ψ β Qˆ φ , ξ α Qˆ φ , Pˆ Ψ = Ψ , QQ j j j j j j P | i≡ | i | i≡ | i | i | i and Pˆ Ξ = Ξ . The states can be written P | i | i ξ ψ 1 1 | i | i ξ ψ 2 2 Ξ = | .i  and Ψ = | . i . (4.4) | i .. | i ..  Ξ   Ψ   P   P  | i | i If we note that HˆQQ 0 ... HˆQP ψ1 0 Hˆ ... Hˆ |ψ i hΞ|Hˆ|Ψi=(hξ1|,hξ2|,...hΞP|) ... Q...Q ... Q...P  | ...2i  HˆPQ HˆPQ ... HˆPP |ΨPi = ξ Hˆ ψ + ξ Hˆ Ψ + Ξ Hˆ ψ + Ξ Hˆ Ψ , j QQ j j QP P P PQ j P PP P h | | i h | | i h | | i h | | i j j j X X X (4.5) then the condition for Hermiticity is ΞHˆ Ψ ΨHˆ Ξ ∗ = [ ξ Hˆ ψ ψ Hˆ ξ ∗] j QQ j j QQ j h | | i−h | | i h | | i−h | | i j X 10