Barrier transmission for the Nonlinear Schrödinger Equation : Engineering nonlinear transport

In this communication we report on a peculiar property of barrier transmission that systems governed by the nonlinear Schrödinger equation share with the linear one: For unit transmission the potential can be divided at an arbitrary point into two sub-potentials, a left and a right one, which have exactly the same transmission. This is a rare case of an exact property of a nonlinear wave function which will be of interest, e.g., for studies of coherent transport of Bose-Einstein condensates through mesoscopic waveguides. PACS numbers: 03.65-w, 03.t5.Lm, 03.75.Kk Submitted to: J. Phys. A: Math. Gen. ar X iv :0 80 5. 20 78 v1 [ qu an tph ] 1 4 M ay 2 00 8 Barrier transmission for the Nonlinear Schrödinger Equation 2 Quantum mechanics is full of surprises, in some cases even in quite elementary situations, that are seemingly well known from textbooks. A recent example is the following remarkable observation for barrier penetration. Let us consider the transmission of a one-dimensional wavefunction through a potential V (x) with V (x) → 0 for x → ±∞. A well known example is a rectangular potential well V (x) = { −V0 , |x| ≤ a 0 , |x| > a (1) (V0 > 0) with a transmission probability [1] |T |(E) = { 1 + V 2 0 4E(E + V0) sin (2~a √ 2m(E + V0) ) }−1 (2) (m is the mass of the particle and E the energy). At certain “resonance” energies Eres the potential is 100% transparent, for example at En = −V0 + (πn/~a) > 0, n = nmin, nmin + 1, . . ., for the rectangular well (1). Such resonances have been studied for more general potentials, as for example symmetrical or unsymmetrical doublebarrier structures in connection with resonant-tunneling, where they are denoted as unit resonances (see, e.g., [2] and references therein). Chabanov and Zakhariev [3] discovered that, at such a resonant energy, the potential scattering shows a surprising symmetry: Dividing V (x) into two distinct parts at some arbitrary point x′ defining a ’left’ and a ‘right’ potential VL(x) = { V (x) x ≤ x′ 0 x > x′ , VR(x) = { 0 x < x′ V (x) x ≥ x′ , (3) the transmission probabilities |TL| for VL and |TR| for VR are equal at the resonance energy, |TL|(Eres) = |TR|(Eres) , (4) despite the fact that the left and right potentials can be very different. This is shown easily using the transfer matrix M connecting the amplitudes of the wavefunction on the left hand side, ψ(x) = A exp(ikx) +B exp(−ikx), with those on the right hand side, ψ(x) = C exp(ikx) +D exp(−ikx) (the limit |x| → ∞ is understood for a potential not vanishing outside a finite range): ( C D ) = M ( A B ) = ( α β β∗ α∗ )( A B )