QUANTUM TRANSPORT EQUATION FOR SYSTEMS WITH ROUGH SURFACES AND ITS APPLICATION TO ULTRACOLD NEUTRONS IN A QUANTIZING GRAVITY FIELD
M. Escobar, A. E. Meyerovich*
Department of Physics, University of Rhode Island, Kingston R.I 02SS1-0S17, USA
Received May 7, 2014
We discuss transport of particles along random rough surfaces in quantum size effect conditions. As an intriguing application, we analyze gravitationally quantized ultracold neutrons in rough waveguides in conjunction with GRANIT experiments (ILL, Grenoble). We present a theoretical description of these experiments in the biased diffusion approximation for neutron mirrors with both one- and two-dimensional (ID and 2D) roughness. All system parameters collapse into a single constant which determines the depletion times for the gravitational quantum states and the exit neutron count. This constant is determined by a complicated integral of the correlation function (CF) of surface roughness. The reliable identification of this CF is always hindered by the presence of long fluctuation-driven correlation tails in finite-size samples. We report numerical experiments relevant for the identification of roughness of a new GRANIT waveguide and make predictions for ongoing experiments. We also propose a radically new design for the rough waveguide.
Cwitribvtiwi for the JETP special issue in honor of A. F. Andrew's 75th birthday
DOI: 10.7868/S0044451014120141
1. INTRODUCTION
The role of surface scattering increases dramatically with advances in micro- and nanofabrication, multilayer systems, pure materials, vacuum technology, etc. Below, we address some universal features of transport of particles or waves along random rough walls in quantum size effect conditions. As an application, we look at the gravitationally quantized ultracold neutrons in rough waveguides in conjunction with ongoing GRANIT experiments (ILL, Grenoble). This is one of the cleanest model-free testing grounds for our theory.
Intuitively, scattering by surface inhomogeneities should not be very different from scattering by other static defects such as bulk impurities. However, while the basic effects of impurity scattering are described in elementary textbooks, a similar simple general account for surface roughness has been missing. This is not entirely surprising. The underlying issue is an unusual structure of the perturbation theory. Randomly varying space inside corrugated systems makes it difficult
* E-mail: Alexander-Meyerovich'ffluri.edu
to introduce a proper set of basis wave functions which are necessary for perturbative expansions. It is not always clear when this issue is important and what to do when it is.
Recently, we developed a consistent perturbative approach within which this issue disappears, clearing the way to a rigorous impurity-like description of quantum transport of particles in systems with rough boundaries. What is more, the structure of the corrugation-driven scattering probabilities is largely universal, irrespective of particle spectra, types of surfaces, and bulk fields between them.
The next section contains a simplified outline of our general transport results for systems with slight roughness, which are relevant for further discussion. In Sec. 3, we apply these results to beams of the gravitationally quantized neutrons in rough waveguides. In Sec. 4, we discuss correlation properties of random rough surfaces. We show that the identification of the roughness correlation function (CF) is not trivial and should not be based solely on a statistical quality of the fit to some fitting function. Section 5 contains our conclusions, experimental predictions, and recommendations.
2. QUANTUM SIZE EFFECT AND TRANSPORT OF PARTICLES ALONG RANDOMLY CORRUGATED WALLS
Theoretical approaches to particle transport in systems with random rough boundaries (see, e.g., books fl 4]; a brief review can be found in Ref. [5]) can be split into two main groups. The first one deals with boundary scattering by means of an effective boundary condition. We prefer alternative approaches that incorporate the boundary scattering directly into the bulk equations and allow using powerful bulk methods to describe the surface effects in transport and interference phenomena, localization, etc.
If we ignore potential complications, the simplest bulk-like approach fC] IS E St-Fctl ghtforward perturbation expansion in small corrugation £ (y, z) of the wall. We suppose that the wall is located at x = xa + £Q (y, z) and corresponds to an abrupt change of the potential
by [U].
U=[U]0(x-x*+S*(y,z)).
The small corrugation £Q looks like a good perturbation parameter,
u = [U] 0 (x - xn) + [U] £a¿ (X - xn) + ...
The calculation of the matrix element is trivial:
l][a) = J exp (is- (q - q')) ía (s) [£/] X
X S (x - xa) = £ (q - q') [U} 'I',, (.i:Q) (,i;Q) , (1)
where <!', (x) are the wave functions in the absence of corrugation. This simple expression can be extended [5] to systems with rough external walls for which [{/] —¥ oc:
v¡ka) = -^(tl-q!)%(xa)%(xa). (2)
If we need a more rigorous perturbative approach or want to study interference effects, a better option is to map the problem with the corrugated boundaries onto a mathematically equivalent problem with flat boundaries and distorted bulk [7 10]. Such mapping for a system with two rough walls,
which makes the boundaries straight, A' = ±L/2, without even specifying the single-valued random functions £1,2. The rest is straightforward: we have to perform a conjugate transformation of momenta p > P and rewrite the original Hamiltonian H0(p,r) in terms of R and P:
Ho (p,r) = Ho (P,R) + V (P, R, {Ci,2 (R)}) • (5)
The result is the exactly equivalent problem in which the (random) bulk perturbation operator V replaces the surface inhomogeneities. In simple situations, the matrix elements of V are similar to (1) and (2). The drawback of mapping transformation (4) is that its Ja-cobian J ^ 1. When this is important, the transformation can be modified [10].
The diagrammatic derivation of the transport equation for systems with random surface inhomogeneities has been done in Ref. [10]. The restricted motion perpendicular to the walls is quantized, (pr)j ~ jh/L, E(p) —¥ Ej (q), where q = (py,p~) is the two-dimensional momentum. This quantization is important for ultrathin systems, multilayer media, interconnects, particles absorbed on or bound to the surfaces, quantum wells, etc. The transport equation is quantum in the direction perpendicular to the walls and is qua-siclassical along the walls. In ultrathin systems with a large separation between the minibands Ej, as well as in thick quasiclassical films, the transport equation has a usual Boltzmann-like form.
i)idii ¡ (q) + — • dTÓHj (q) + F • dqónj (q) =
/-,><}• (6)
In-between, there is an anomalous regime in which the transport equation acquires a highly unusual and complicated form [10]; we do not deal with this situation here. Since the mapping transformation approach is mathematically rigorous, it can be extended to more complex situations, including the surface-driven localization [11], interference between surface and bulk scattering processes [12], topological phase transitions [13], etc.
The perturbative collision integrals Lj are determined by the transition probabilities Wjj' (q, q') =
(3)
= 11'
between the states (j, q) —¥ (j',q'):
can be achieved [9] by the coordinate transformation r R,
-Y =
■r + Ci/2 - fr/2 l-ti/L-b/L'
Y = y, Z =
(4)
L¡ = 2tt£ / M'y (q,q'
Tlji q' TljqJ X
X 6 (Cj
,q ' ,"<1') ('2-hf
■ (7)
1283
10*
Generally, the transition probabilities Wjj> (q, q') factor into the products of the CF of surface roughness C(q-q') and the boundary values of the wave functions 'I', in the absence of corrugation. These combinations depend on the structure of the system, the number of interfaces, and the correlation between in-homogeneities from different walls.
The CF of surface inhomogeneities (|s|) and its power spectrum (|q|) are defined as
CaH (|s|) = (Ca(Sl)C^(si + s)) =
/ Ca(Sl)C;i(Si + S)(fei,
Caz* (|q|) =
(t28 exp ( ) (¡a¡:i (|s|
= (8)
(-XJ
= 2?r j Ca/i («) Jo (qs) « ds,
where ,4 is the area, and the indices a, 8 indicate the surfaces that are the sources of inhomogeneities
and £■fi.
If the system has only one rough surface at x = L + £(y, z) with the potential jump [Z7] on it, then, according to Eq. (1),
Wjf =aq^ql)[U}2\yj(L)\2\yjl(L)\
0)
If, on the other hand, there are several interfaces at .iQ = Lq + Ca (;</- ) with different discontinuities [t/]Q, then
wai! = jf
= Ro [w (q-q') [U]a PI, , (10)
where <I'a = <!' (La). The full scattering probability W is the sum of all these II"'1 \ For a system with two external walls with [i/^ 2 —¥ oc, Eq. (2), the probabilities H" are
ir'1:/" =
33 ¡,,,>
1
Re [u,^«^^^]
ajj =1,2
(ID
and the interference between inhomogeneities on the external wall (a) and the internal interface (/?) yields
(12)
Equation (10) for internal interfaces is the same irrespective of the particle spectrum, while Eqs. (11) and (12) are given for e = p2/2m (equations for arbitrary e (p) are more cumbersome [5]).
The terms with a = 3 and a ^ 3 describe the intra wall and inter wall correlations of inhomogeneities. The interwall contribution a ^ /?, when it exists [14], is nontrivial. While Waa is always positive, the sign of the interwall term II"'11 with a ^ 3 is not fixed, and the interwall interference can be constructive or destructive depending on a particular realization of the system (overall, W is positive because Caa (q) +w (q) > 2 ICa/ü (q)| for any corrugation). For illustration, here is the full roughness-driven transition probability for particles with the quadratic spectrum in a homogeneous quantum well with infinite potential walls:
W'W =
1
in2 L2
Cll + C-22
2 M)i+i' Cr.
7Tf
L
(13)
The interwall correlation term with C12 has an oscillating structure and can sometimes lead to a large increase
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