Quantizing Lévy Flights: Extending the Caldeira-Leggett model for velocity-dependent coupling forces in dissipative systems
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Brownian motion is the random-walk behavior exhibited by a particle when subjected to a bath composed of smaller particles that constantly undergo collisions with the Brownian particle. Its trajectory is described by the Langevin equation, which has several technical problems for constructing a quantum version of the model. Most notably, there is a velocity dependence in the equation of motion, which makes direct canonical quantization problematic; energy is not conserved due to diffusion. To solve this problem, Anthony James Leggett and Amir Ordaçgi Caldeira, modeled the bath as a collection of harmonic oscillators, which allowed them to close the system, derive the Langevin equation in the appropriate (classical) limit for the ohmic regime, and investigate quantum aspects of Brownian motion. Building on this Caldeira-Leggett model, this research suggests changing the interaction between the Brownian particle and the bath to depend on the velocity of the particle in a general way. We derive a generalized velocity-dependent Langevin equation with memory effects and multiplicative noise. First, when we reduce to a coupling linear in velocity the memory disappears and we find a Langevin-type equation that resembles the equation for the self-interaction of an electron with its own radiation field, i.e. the Abraham-Lorentz equation, in the superohmic regime. Second, when we make the approximation that the second derivative of the coupling is negligible, a particular nonpolynomial coupling reproduces a force term that gives rise to Lévy flights as encountered in ultracold-atoms experiments, e.g. Sisyphus laser cooling. We also use the path-integral quantization method to construct a quantum version for our generalized velocity-dependent Caldeira-Leggett model. After tracing out the bath, it turns out to be possible to find an effective action without a special choice for the form of the coupling; hence, it remains completely general. Reducing to the linear-velocity case, we obtain an effective action that can be interpreted as a bath-induced resistance to a change in velocity of the Brownian system---on top of classical inertia terms. For the effective action corresponding with the Lévy flight model, further numerical research is desirable. Last, we will also discuss future possibilities of engineering environments in such a way that they are friendly to coherence, which has many applications---for example, suppressing decoherence for quantum computers.