Abstract
We discuss the dynamics of a quantum phase transition in a spin-1 Bose–Einstein condensate when it is driven from the magnetized broken-symmetry phase to the unmagnetized ‘symmetric’ polar phase. We determine where the condensate goes out of equilibrium as it approaches the critical point, and compute the condensate magnetization at the critical point. This is done within a quantum Kibble–Zurek scheme traditionally employed in the context of symmetry-breaking quantum phase transitions. Then we study the influence of the non-equilibrium dynamics near a critical point on the condensate magnetization. In particular, when the quench stops at the critical point, nonlinear oscillations of magnetization occur. They are characterized by a period and an amplitude that are inversely proportional. If we keep driving the condensate far away from the critical point through the unmagnetized ‘symmetric’ polar phase, the amplitude of magnetization oscillations slowly decreases reaching a nonzero asymptotic value. That process is described by an equation that can be mapped onto the classical mechanical problem of a particle moving under the influence of harmonic and `anti-friction' forces whose interplay leads to surprisingly simple fixed-amplitude oscillations. We obtain several scaling results relating the condensate magnetization to the quench rate, and verify numerically all analytical predictions.
Highlights
Any symmetry breaking phase transition can be traversed in two opposite directions
According to our numerical simulations there are three stages of such evolution depicted in Fig. 7: (i) the system is driven to the critical point through the entire broken symmetry phase and its magnetization at the critical point scales as |fT |2 ∼ τQ−2/3; (ii) the condensate is driven by a change of q, Eq (10), in the polar phase and we observe there damped magnetization oscillations; (iii) the free evolution after we stop ramping up q(t) takes place and periodic magnetization oscillations appear
Applying the basic assumptions of the adiabatic-impulse approach [3, 5], which originates from the Kibble-Zurek theory of nonequilibrium dynamics of classical phase transitions [1, 2], we have explained why the transverse magnetization of the condensate driven to the critical point scales as τQ−2/3 as well
Summary
Any symmetry breaking phase transition can be traversed in two opposite directions. The usual way is to start in the symmetric phase, and move into the phase with a broken symmetry. As in [17], we are interested in slow transitions that exhibit adiabatic-impulse behavior so that the condensate goes out of equilibrium close to the critical point In this case the condensate excitation in the polar phase reflects the scalings of the critical regime. System properties near the critical point are determined by its ground state at q = qc − q, where qis computed from the quench time τQ and the product of the critical exponents: Eq (3) This approach was already successfully used to study the Landau-Zener dynamics [5], the dynamics of the quantum Ising model [3], and the ferromagnetic condensate dynamics during the symmetry-breaking transitions from the polar to the broken-symmetry phase [17].
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