## Phase slips in space-time crystals

##### Summary

In this work, we attempt to find a suitable approach to study phaseslip switching between time crystal modes. These time crystal are density oscillations in Bose Einstein condensates. Our goal is to determine the parameter space in which dynamics between crystal modes arise, characterise this behaviour, and provide a basis for experimental work. Numerical evaluation of the Fokker-Planck equation requires a long evaluation time. Combined with the large parameter space of our system, faster running approximations are desirable. We focus on two models. The first method is the line model, an one-dimensional approximation to the Fokker Planck equation. The second is a quantum trajectory method, based on the the equations of motion of our system. We compare these to the (two dimensional) Fokker-Planck equation, evaluated using the MacCormack method.
The line model reproduces the correct equilibrium distribution at high number of quanta in the crystal, but fails close to the origin in the phase space of the crystal. We observe that flux current terms in the Fokker-Planck equation become dominant in this region, preventing probability flow between modes. This is further verified
by the dynamics observed using the other methods, the origin is always avoided. We conclude this model to be unsuitable. The quantum trajectory method becomes nonphysical in the limit of no damping, otherwise it reproduces the same qualitative behaviour as the 2d Fokker-Planck method. At higher noise levels we find both over and under-damped oscillations between crystal modes. Based on the 2d Fokker-Planck method, the type of oscillation depends largely on the detuning of the crystal with respect to the breathing mode. Both the frequency and damping parameters of the oscillations scale with the noise parameter. Additional simulations are required to verify if the quantum trajectory method produces the same scaling of the oscillation parameters. Even if this is not the case, the quantum trajectory method does provide a useful way to rapidly search
the parameter space. Experimental observation of a mode switch constitutes a π shift in the phase of
the time crystal, after the system is allowed to go unmeasured for some time. Apart from sufficient isolation, observation of mode switching requires control to some extend over the detuning and noise term. The first could be realized by changing the breathing mode frequency temporarily. The noise term needs to be two orders of magnitude stronger than experimental realisations so far. To this end, the influence of the thermal cloud on the noise term needs to be investigated. The parameter space of the driving parameters in which mode switching occurs, match that of current experimental realisations.