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Virtual potentials for feedback traps

The recently developed feedback trap can be used to create arbitrary virtual potentials, to explore the dynamics of small particles or large molecules in complex situations. Experimentally, feedback traps introduce several finite time scales: there is a delay between the measurement of a particle's position and the feedback response; the feedback response is applied for a finite update time; and a finite camera exposure integrates motion. We show how to incorporate such timing effects into the description of particle motion. For the test case of a virtual quadratic potential, we give the first accurate description of particle dynamics, calculating the power spectrum and variance of fluctuations as a function of feedback gain, testing against simulations. We show that for small feedback gains, the motion approximates that of a particle in an ordinary harmonic potential. Moreover, if the potential is varied in time, for example by varying its stiffness, the work that is calculated approximates that done in an ordinary changing potential. The quality of the approximation is set by the ratio of the update time of the feedback loop to the relaxation time of motion in the virtual potential.