Absorbing Schroedinger

Absorbing Schroedinger (or Absorbing Schrödinger) is phenomenological modification of the Schrödinger equation, that corresponds to absorption of particles.

The Absorbing Schroedinger can be written as follows:
 * \( \displaystyle \!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \mathrm i \dot \Psi = \hat H ~ \Psi -\mathrm i ~ \hat V ~ \Psi \)

where \(\Psi\) is wave function, the dot differentiats it with respect to time, \(\hat H\) is the conventional, Hermitian part Hamiltonian, and \(\hat V\) is the Hermitian operators that represents the entanglement of the state with some other degrees of freedom, that are taken into account only phenomenologocally.

The Absorbing Schroedinger describes the evolution of the quantum state of a particle slightly coupled to other particles, environment, giving evolution of the component of the wave function that "did not yet interact" with the environment.

WARNING! This article is UNDER CONSTRUCTION! The formulas should be checked before to use.

Absorption in some part of space
In the simplest case of movement of a single particle in vicinity of some device, that tries to observe it, the Nonlinear Schroedinger can be written as follows:
 * \( \displaystyle \!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \mathrm i ~ \hbar ~ \dot \Psi =  \frac{-\hbar^2 \nabla^2}{2m} \Psi + U(\vec x) ~ \Psi  -\mathrm i ~ V(\vec x) ~ \Psi \)

where \(\Psi=\Psi(\vec x, t)\) is wave function of a particle in the coordinate representation, \(m\) is its mass, \(\hbar\) is the Planck constant, \(U\) is conventional potential, and \(V(x)\) describes the probability of absorption ("registration") of the particle at the point with coordinate \(x\)

The distortion of the wave function of the particle due to the observation (or attempt of its observation) is called Zeno effect . The Absorbing Schroedinger describes the quantum reflection of particles from the region of their absorption.

Often, there is no ab initio description for the relaxation, absorption of the particle; in this case, it is convenient to use the "normalized" absorpbig schroedinger, setting the Plansk constant \(\hbar\) to unity. Also, in the simplest case, the uniform absorption takes place in the part of space. Let \(\theta(\vec x)=1\) in the regions where the particle can be observed, and let \(\theta(\vec x)=0\) in the regions where the particle cannot be observed. For the dimention-less case, the Absorbing Schroedinger can be written as follows:
 * \( \displaystyle \!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ \mathrm i \dot \Psi +\Psi '' = U(\vec x) \Psi -\mathrm i \theta(\vec x) \gamma \Psi \)

where the double primes represent the second ferivatives with respect to the coordinates., and positive parameter \(\gamma\) describes the efficiency of absorption. (Any observation is also interpreted as absorption.)

Paraxial propagation
In the simple paraxial case, the particle propagates mainly along some coordinate, let it be \(z\), and the potential depends only on the transversal coordinates. Neglecting the conventional potential \(U\), and assuming the exponential decay of the wave function along the propagation, the Absorbing Schroedinger can be written for
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~ \Psi=\mathrm e ^{- \mathrm i \omega t} \psi\)

in the following form:
 * \( \displaystyle \!\!\!\!\!\!\!\!\!\! (5) ~ ~ ~ \omega \psi + \psi '' = -\mathrm i \gamma \theta(\vec x)  \psi \)

Uniform absorption
The simplest and important case is the uniform absorption (detection) of the particle. This case reveals the physical sense of constant \(\gamma\) in the equaiton (5) above. Let
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\! (6) ~ ~ ~ \psi=\mathrm e ^{\mathrm i (c+\mathrm i s) z }\)

where \(c\) and \(s\) are real numbers, and let \(\theta(\vec x)=1\) in the whole space. The substitution into (5) gives
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\! (7) ~ ~ ~ \omega - (c+\mathrm i s)^2 = -\mathrm i \gamma\)
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\! (8) ~ ~ ~ (c+\mathrm i s)^2  = \omega + i \gamma\)
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\! (9) ~ ~ ~  c^2- s^2+ 2\, \mathrm i \,c\, s  = \omega + i \gamma\)

which means that
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (10) ~ ~ ~ c^2-s^2= \omega\)
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (11) ~ ~ ~ 2~c~s= \gamma\)

In these expressions, \(c\) has physical sense of wavenumber of the wave, and \(s\) means the absorption rate. Both parameters allow the direct measurements in the experiments with photons, atoms or any other kinds of waves. In the physical applications, the case \(~ s/c \ll 1 ~\) is of the most interest; in this case, there is no need to make difference between \(c\) and wavenumber of particle before the entry in the absorbing (detecting) medium.

However, one may consider as given between the "initial" wavenumber \(\sqrt{\omega}\) and absorption \(s\); then
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (12) ~ ~ ~ c=\sqrt{\omega+s^2} ~ ~\), \(~ ~ ~ \gamma= 2\, s\, \sqrt{\omega+s^2}\)

In this notations, \(\sqrt{\omega}\) has sense of the initial wavenumber, before the particle enters the area with absorption, and \(s\) is jus the absorption.

Hope, the recovery of dimensions for the atom optics causes no problems.

Reflection of particle from the absorption wall
The zeno-reflection of particle crom absorbing wall had been presented in 2005 showing very good agreement with experiments on quantum reflection of particles invoming on the ridged mirror at the grazing angle, . The reflection coefficient is estimated as
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (13) ~ ~ ~ r_{\mathrm{ zeno}}=\frac

{\sqrt{\sqrt{1/\chi^4+1}+1}-\sqrt{2}} {\sqrt{\sqrt{1/\chi^4+1}+1}+\sqrt{2}} \approx \exp\big(-\sqrt{8}\, \chi \Big) \) where
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (14) ~ ~ ~ \chi=\sqrt{KL} \theta\)

\(K=\sqrt{\omega}\) is wavenumber, \(L=1/s\) is absorption length and \(\theta\) is the grazing angle. In the experiments, \(L\) corresponds to the distance between absorbing ridges. Such an absorption is not continuous, so, initially only a qualitative agreement was expected. Instead, the quantitative agreement has been observed for various sets of parameters, varying for orders of magnitude, so, the approximation (13) happens to be pretty universal.

Channeling of particle between absorbing walls
The important case of Absorbing Schroedinger describes the channeling of particles between absorbing walls.

Euristic estimate
For the good channeling conditions, the effective absorption can be approximated with
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (15) ~ ~ ~ a=\frac{1-r_{\rm zeno}}{2 \,d /\theta}\)

where \(d\) is half-width of the channel and \(\theta=p/K\) is ratio of the transversal wavenumber
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (16) ~ ~ ~ p=\frac{\pi}{2\,d}\)

to the wavenumber \(K\). At the reflection of wave from a ridged mirror, \(\theta=p/K\) has sense of the grazing angle.

Substitution of (13) and (16) into (15) gives the following expression for the absorption
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (17) ~ ~ ~

a \approx \sqrt{8 K L} \frac{\theta^2}{2d}\)

The grazing angle can be approximated with
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (18) ~ ~ ~

\theta \approx \frac{\pi}{2\, d\, K}\) giving the estimate for the efficient absorption
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (19) ~ ~ ~

a \approx \sqrt{8 K L} \frac{\pi^2}{4 d^3 K^2} \approx \sqrt{ \frac{L}{2 K^3} } \frac{\pi^2}{d^3}\) For the higher transversal modes, the absorption should scale proportionally to the square of the transversal wavenumber, so,
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (20) ~ ~ ~

a_n\approx a~ (1+2n)^2\) for the even modes modes and
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (21) ~ ~ ~

a_n\approx a~ (2n)^2\) for the odd mode; it is assumed that numeration of even modes begins with zero and that for the odd modes begins with unity.

Modal analysis
At the naive construction of the absorbed mode, the resulting expression for the effective absorption deviates from the estimate (19) above.

The example of the mode \(f\) guided by the absorbing walls is shown in figure at right. The mode is constructed in the following way.

Instead of the plane decaying wave (6), consider equation (5) with
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (22) ~ ~ ~

\psi=\exp( \mathrm i \kappa z) f(x)\) where \(\kappa \in \mathrm C\) is constant, and \(x\) is transversal coordinate (for simplicity, just one); let
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (23) ~ ~ ~

\theta(x)= \mathrm{UnitStep} \big( |x|-d \Big)\) where \(d>0\) has sense of half–width of the channel. The substitution hives the equation for \(f\):
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (24) ~ ~ ~

\omega f -\kappa^2 f +f= -\mathrm i ~ \gamma ~ \mathrm{UnitStep} \Big( |x|-d \Big) ~ f \) where \(f=f(x)\) and \(f=f''(x)\). The efficient channeling is expected for the case of \(\omega \approx \kappa^2\). Search the solution \(f\) of equation (24) in the following form:
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (25) ~ ~ ~

f(x)=\left\{ \begin{array}{ccc} \cos(px) &,& |x|\le d \\ r \exp\Big( q~ \big(|x|\!-\!d\big) \Big) &,& |x|\ge d \end{array}\right.\) Substitution of (25) into (24) gives the equations for the complex parameters \(p\) and \(q\):
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (26) ~ ~ ~ \omega~ - ~ \kappa^2 - p^2 =0 \)
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (27) ~ ~ ~ \omega~ -  ~\kappa^2 + q^2 = - \mathrm i \gamma\)

Conditions of continuity of function \(f\) and its first derivative give equations
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (28) ~ ~ ~ ~ ~\cos(pd)= ~ ~ r \)
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (29) ~ ~ ~ - p ~ \sin(pd)= - q ~ r\)

Complex parameters \(p\), \(q\), \(r\), \(\kappa\) are determined by equations (26)–(29). Subtraction of (26) from (27) gives the equation
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (30) ~ ~ ~ p^2+q^2= - \mathrm i \gamma \)

Dividing (29) by (28) gives
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (31) ~ ~ ~ p \tan(pd)= q \)

Combination of (30) and (31) gives
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (32) ~ ~ ~ p^2(1+\tan(pd)^2)= - \mathrm i \gamma \)

Using relation \(1+\tan(z)^2=\frac{1}{\cos(z)^2}\), equation (32) can be rewritten as follows:
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (33) ~ ~ ~ \frac{p^2}{\cos(pd)^2}= - \mathrm i \gamma \)
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (34) ~ ~ ~ \frac{\cos(pd)^2}{p^2 d^2}=\) \( \frac{\mathrm i }{ \gamma ~ d^2} \)
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (35) ~ ~ ~ \mathrm{cosc}(pd) = \sqrt{ \frac{\mathrm i ~ d^2}{ \gamma} } = \exp(\mathrm i \pi/4) \frac{1}{d ~\sqrt{\gamma}}=\exp(\mathrm i \pi/4) \alpha\)

where cosc\((x)=\cos(x)/x\) and
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (36) ~ ~ ~ \alpha = \frac{1}{d ~\sqrt{\gamma}}\)

is damping parameter that determines the damping, decay of the guided mode and, therefore, the efficiency of the guiding. Then, the transversal wavenymber \(p\) can be expressed through the ArcCosc function, \(\mathrm{acosc}=\mathrm{cosc}^{-1}\):
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (37) ~ ~ ~ pd = \mathrm{acosc}(pd) = \mathrm{acosc}\Big( \exp(\mathrm i \pi/4) ~ \alpha\Big)= \mathrm{acosq}(\alpha)\)

where acosq is modified acosc function; properties of function acosq are known. Then, \(q\) is determined with equation (30) or (31). For the case \(\alpha=1/4\), the construction of the guided mode is shown in the figure above.

The effective absorption of the mode is
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (38) ~ ~ ~ A = \Im(\kappa)\)

where \(\kappa\) is determined by, for example, (26), id est,
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (39) ~ ~ ~ \kappa=\sqrt{\omega ~–~ p^2} \approx \sqrt{\omega} \Big(1-\frac{p^2}{2\omega}\Big)

\approx\sqrt{\omega} - \frac{p^2}{2\sqrt{\omega}} \) then
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (40) ~ ~ ~ A \approx - \Re(p) \Im(p)/ \sqrt{\omega}\)

In order to provide the flux of probability from the center of mode to the absorbing walls, the imaginary part of the transversal wavenumber should be negative, \(\Im(p)<0\).

Asymptotic estimate of the guiding efficiency
The guiding efficiency can be characterizes with effective length of absorption \(S=1/A\). For the physical application, the case of weak effective absorption is especially interesting; this correspond to small values of \(A\) and \(\alpha\), and large local absorption in the walls of the channel. In this section, this case is considered asymptotically.

The expansion of function acosq (which appears in equation (37)) at zero has to following form:
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (41) ~ ~ ~  \mathrm{acosq}(z) =

\frac{\pi}{2}-\frac{\pi}{2} \mathrm e^{\mathrm i \pi/4} z +O(z^2)\) This gives the approximation for the tranzversal wavenumber \(p\) in the following form:
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (42) ~ ~ ~  pd \approx \frac{\pi}{2}  -\frac{\pi}{2} \mathrm e ^{\mathrm i \pi/4} \alpha\)

and, from (40), the estimate for the effective absorption
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (43) ~ ~ ~ A\approx \frac{\pi}{2d} \times \frac{\pi}{d\sqrt{2}} \alpha

\times \frac{1}{\sqrt{\omega}}\) Then, \(~\alpha\!=\!\frac{1}{d \sqrt{\gamma}}~\) by (36) and \(~\gamma\approx 2 s \sqrt{\omega} \approx 2 s c~\) should be used, giving \(\displaystyle ~\alpha \approx \frac{1}{d \sqrt{2 s c}}~\). Then, the effective absorption of the mode
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (44) ~ ~ ~ A\approx \frac{\pi^2}{\sqrt{8}d^2} \times \frac{1}{d\sqrt{2sc}} \times \frac{1}{c} = \frac{\pi^2}{4~ d^3 c^{3/2} s^{1/2}}

\) where \(c\) has sense of wavenumber, and \(s\) is the absorption in the wall.

This estimate should be compared to (19), where \(L\) should be replaced to \(1/(2s)\) and \(K\) should be replaced to \(c\), giving the absorption by probapility
 * \(\displaystyle \!\!\!\!\!\!\!\!\!\!\!\! (45) ~ ~ ~ a\approx \frac{\pi^2}{2~d^3 c^{3/2} s^{1/2}}

\) However, the amplitude decays half slower than probability, so, \(A=a/2\). In such a way, the asymptotic estimates (44) and (45) agree.

In the first approximation, the consideration of the multiple reflection from absorbing walls and the consideration of mode guided between the absorbing walls give the same prediction about effective absorption of this mode .