TY - JOUR

T1 - The Kostin Equation, the Deceleration of a Quantum Particle and Coherent Control

AU - Losert, Harald

AU - Ullinger, Freyja

AU - Zimmermann, Matthias

AU - Efremov, Maxim A.

AU - Rasel, Ernst M.

AU - Schleich, Wolfgang P.

N1 - Funding Information: We thank R. Folman, M. Freyberger, A. Friedrich, N. Gaaloul, E. Giese, F. Narducci, G.G. Rozenman and L. Wörner for many stimulating discussions on this topic, and M.C. Downer and P. Hommelhoff for their help with the references. W.P.S. is grateful to Texas A&M University for a Faculty Fellowship at the Hagler Institute for Advanced Study at the Texas A&M University as well as to the Texas A&M AgriLife Research.

PY - 2023/1

Y1 - 2023/1

N2 - Fifty years ago Kostin (J Chem Phys 57(9):3589–3591, 1972. https://doi.org/10.1063/1.1678812) proposed a description of damping in quantum mechanics based on a nonlinear Schrödinger equation with the potential being governed by the phase of the wave function. We show for the example of a moving Gaussian wave packet, that the deceleration predicted by this equation is the result of the same non-dissipative, homogeneous but time-dependent force, that also stops a classical particle. Moreover, we demonstrate that the Kostin equation is a special case of the linear Schrödinger equation with three potentials: (i) a linear potential corresponding to this stopping force, (ii) an appropriately time-dependent parabolic potential governed by a specific time dependence of the width of the Gaussian wave packet and (iii) a specific time-dependent off-set. The freedom of the width opens up the possibility of engineering the final state by the time dependence of the quadratic potential. In this way the Kostin equation is a precursor of the modern field of coherent control. Motivated by these insights, we analyze in position and in phase space the deceleration of a Gaussian wave packet due to potentials in the linear Schrödinger equation similar to those in the Kostin equation.

AB - Fifty years ago Kostin (J Chem Phys 57(9):3589–3591, 1972. https://doi.org/10.1063/1.1678812) proposed a description of damping in quantum mechanics based on a nonlinear Schrödinger equation with the potential being governed by the phase of the wave function. We show for the example of a moving Gaussian wave packet, that the deceleration predicted by this equation is the result of the same non-dissipative, homogeneous but time-dependent force, that also stops a classical particle. Moreover, we demonstrate that the Kostin equation is a special case of the linear Schrödinger equation with three potentials: (i) a linear potential corresponding to this stopping force, (ii) an appropriately time-dependent parabolic potential governed by a specific time dependence of the width of the Gaussian wave packet and (iii) a specific time-dependent off-set. The freedom of the width opens up the possibility of engineering the final state by the time dependence of the quadratic potential. In this way the Kostin equation is a precursor of the modern field of coherent control. Motivated by these insights, we analyze in position and in phase space the deceleration of a Gaussian wave packet due to potentials in the linear Schrödinger equation similar to those in the Kostin equation.

KW - Coherent control

KW - Cooling

KW - Gaussian wave packet

KW - Kostin equation

KW - Nonlinear Schrödinger equation

KW - Quantum dynamics

KW - Wigner phase space

UR - http://www.scopus.com/inward/record.url?scp=85138297716&partnerID=8YFLogxK

U2 - 10.1007/s10909-022-02857-y

DO - 10.1007/s10909-022-02857-y

M3 - Article

AN - SCOPUS:85138297716

VL - 210

SP - 4

EP - 50

JO - Journal of low temperature physics

JF - Journal of low temperature physics

SN - 0022-2291

IS - 1-2

ER -