^{3}.

We apply our general theory of transport in systems with random rough boundaries to gravitationally quantized ultracold neutrons in rough waveguides as in GRANIT experiments (ILL, Grenoble). We consider waveguides with roughness in both two and one dimensions (2D and 1D). In the biased diffusion approximation the depletion times for the gravitational quantum states can be easily expressed via each other irrespective of the system parameters. The calculation of the exit neutron count reduces to evaluation of a single constant which contains a complicated integral of the correlation function of surface roughness. In the case of 1D roughness (random grating) this constant is calculated analytically for common types of the correlation functions. The results obey simple scaling relations which are slightly different in 1D and 2D. We predict the exit neutron count for the new GRANIT cell.

One of the most interesting recent achievements in neutron physics is a series of GRANIT [

Another potential application area is the use of such experiment as a test for a quantum transport theory in systems with random rough boundaries (see, for example, [

Recently we developed a consistent perturbative approach to quantum transport along rough surfaces [

In the case of neutron beams propagating between one rough mirror and one flat mirror, the transition probabilities have the form

We use the dimensionless variables, which are common to the field (for details see [

Diffusion of neutrons between discrete states

This strong upward bias has two consequences. First, almost all the time

The ratios

In [

The fact that the depletion times

The value of the dimensionless constant

The dependence of the exit neutron count

In Figure

The only remaining task is to calculate

We start from waveguides with 1D roughness for which most of the calculations can be carried out analytically. Towards the end of the paper we will mention why 1D roughness is important though the existing rough mirrors exhibit 2D roughness [

It is more convenient to start not from (

In our dimensionless variables the scattering probabilities (

Since the transition rate

In these notations our main parameter

If, as it is often assumed, the correlation function is Gaussian,

The Fourier image of a power law correlation function,

In the opposite case, when the power spectrum of roughness

The purely exponential correlation functions in configuration or momentum spaces emerge from (

These equations allow one to find the depletion times

The inverse depletion times

The depletion times

In contrast to systems with 1D roughness, most of the calculations in 2D cases can be done only numerically. We start from (

The calculation of the zeroth angular harmonic of the 2D correlation function in momentum space

In contrast to the earlier experiments, the roughness correlation function for the ongoing GRANIT experiments in a new cell has been measured [

As in the case of 1D roughness, both

Figure

The same as Figure

The critical values of

Our prediction for the neutron count

The prediction for the neutron count for the new waveguide with 2D exponential roughness with parameters

The presence of well-developed steps on the curve, which correspond to consecutive depletion of the gravitational quantum states, is explained mostly by a relatively large amplitude of roughness

In summary, we developed a quantitative theory of propagation of ultracold neutrons through a rough waveguide. The immediate applications are the ongoing GRANIT experiments at ILL (Grenoble) aimed at analysis of quantization of neutrons by the gravity field. There are also other experimental groups exploring similar setups. If successful, these experiments will produce neutrons in well defined ultralow energy states in the peV range which can be used for precise measurements of fundamental forces.

We analyzed waveguides with 1D and 2D roughness. The ratios of the depletion times (line broadenings) in the biased diffusion approximation were the universal functions of the waveguide width and did not depend on the waveguide parameters. All relevant waveguide and roughness parameters collapsed into a single constant (essentially, a linewidth of the lowest quantum state), which was responsible for the exit neutron count. This constant strongly depended on the functional form of the roughness correlation function. We calculated this constant for various waveguides. In waveguides with 1D roughness the calculations could be carried out analytically for the most common types of the correlation functions; the 2D calculations were mostly numerical.

Our results were in good agreement with earlier experimental data despite the lack of experimental information about many important parameters. The predicted neutron count for the new experimental setup, for which the roughness profile was accurately measured, exhibited well-developed quantum step corresponding to consecutive depletion of the lower and lower gravitational states. Large amplitude of roughness in this setup could degrade the usability of the results. One of the possible ways to circumvent these difficulties and produce a much more controllable environment would be the use of a radically new design for a rough mirror which we called an Ising mirror [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The results of this paper were reported at GRANIT-2014 Workshop (Les Houches, France, March 2–7, 2014). One of the authors (A. E. Meyerovich) is grateful to the organizers and other members of GRANIT collaboration for support and hospitality during the workshop.