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Feedback is a general idea of modifying system behaviour depending on themeasurement outcomes. It spreads from natural sciences, engineering, andartificial intelligence to contemporary classical and rock music. Recently,feedback has been suggested as a tool to induce phase transitions beyond thedissipative ones and tune their universality class. Here, we propose andtheoretically investigate a system possessing such a feedback-induced phasetransition. The system contains a Bose-Einstein condensate placed in an opticalpotential with the depth that is feedback-controlled according to the intensityof the Bragg-reflected probe light. We show that there is a critical value ofthe feedback gain where the uniform gas distribution loses its stability andthe ordered periodic density distribution emerges. Due to the externalfeedback, the presence of a cavity is not necessary for this type of atomicself-organization. We analyze the dynamics after a sudden change of thefeedback control parameter. The feedback time constant is shown to determinethe relaxation above the critical point. We show as well that the controlalgorithm with the derivative of the measured signal dramatically decreases thetransient time.

Cavityless self-organization of ultracold atoms due to the feedback-induced phase transition