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Surface Science

Quantum theory of energy dissipation in non-contact atomic force microscopy in Markovian approximation

Lev N Kantorovich

Published Dec 1, 2002• 8 citations
Publisher
DOI
10.1016/s0039-6028(02)02308-7

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arXiv:2110.07714•Physical Review A•Jan 19, 2022•1 citation
We apply the non-equilibrium statistical operator method within the mixed quantum–classical representation to the non-contact atomic force microscopy system in which a macroscopic tip is made to oscillate above a sample surface. In our treatment the tip is treated classically, whereas a microscopic system comprising atoms in the surface and in the nano-tip is considered quantum mechanically. Using a very general non-Markovian approach, we obtain both the Fokker–Planck equation for the tip and, independently, its equation of motion with friction terms and memory effects completely accounted for. In the classical limit (high temperatures) our equations turn exactly into the classical counterparts derived previously.
We explore the quantum dynamics of a bright matter-wave soliton in a quasi-one-dimensional bosonic superfluid with attractive interactions. Specifically, we focus on the dissipative forces experienced by the soliton due to its interaction with Bogoliubov excitations. Using the collective coordinate approach and the Keldysh formalism, a Langevin equation of motion for the soliton is derived from first principles. The equation contains a stochastic Langevin force (associated with quantum noise) and a nonlocal in time dissipative force, which appears due to inelastic scattering of Bogoliubov quasiparticles off of the moving soliton. It is shown that Ohmic friction (i.e., a term proportional to the soliton's velocity) is absent in the integrable setup. However, the Markovian approximation gives rise to the Abraham-Lorentz force (i.e., a term proportional to the derivative of the soliton's acceleration), which is known from classical electrodynamics of a charged particle interacting with its own radiation. These Abraham-Lorentz equations famously contain a fundamental causality paradox, where the soliton (particle) interacts with excitations (radiation) originating from future events. We show, however, that the causality paradox is an artifact of the Markovian approximation, and our exact non-Markovian dissipative equations give rise to physical trajectories. We argue that the quantum friction discussed here should be observable in current quantum gas experiments.
We propose a two-frequency driving scheme in dynamic atomic force microscopy that maximizes the interaction time between tip and sample. Using a stochastic description of the cantilever dynamics, we predict large classical squeezing and a small amount of skewness of the tip's phase-space probability distribution. Strong position squeezing will require close contact between tip and surface, while momentum squeezing would also be possible in the van der Waals region of the tip-surface force. Employing a generalized Caldeira-Leggett model, we predict that surface-dependent dissipative forces may be the dominant source of quantum effects and propose a procedure to isolate quantum effects from thermal fluctuations.