The Simplest Quantum Model Supporting the Kibble-Zurek Mechanism of Topological Defect Production: Landau-Zener Transitions from a New Perspective

Abstract

It is shown that dynamics of the Landau-Zener model can be accurately described in terms of the Kibble-Zurek theory of the topological defect production in nonequilibrium phase transitions. The simplest quantum model exhibiting the Kibble-Zurek mechanism is presented. A new intuitive description of Landau-Zener dynamics is found.PACS numbers: 03.75.Lm,32.80.Bx,05.70.Fh In this Letter we present a successful combination of the Kibble-Zurek (KZ) [1, 2] theory of topological defect production and quantum theory of the Landau-Zener (LZ) model [3]. Both theories play a prominent role in contemporary physics. The KZ theory predicts production of topological defects (vortices, strings) in the course of nonequilibrium phase transitions. This prediction applies to phase transitions in liquid 4 He and 3 He, liquid crystals, superconductors, ultracold atoms in optical lattices [4,5], and even to cosmological phase transitions in the early Universe [1,2]. The Landau-Zener theory has even broader applications. It has already become a standard tool in quantum optics, atomic and molecular physics, and solid state physics. The list of important physical systems governed by the LZ model grows. For instance, recent investigations point out that the smallest quantum magnets, Fe 8 clusters cooled below 0.36K, are successfully described by the LZ model [6].This Letter constructs the simplest quantum model whose dynamics remarkably resembles dynamics of topological defect production in nonequilibrium second order phase transitions. The model is built on the basis of LZ theory and allows us to study the KZ mechanism of topological defect production in a truly quantum case. Such a quantum insight into the KZ theory was up to now inaccessible except for the recent study of KZ theory in optical lattices filled with ultracold atoms [5]. In addition, we present a simple, intuitive, and accurate description of LZ model dynamics.For the rest of the Letter it is essential to introduce briefly the KZ theory. Consider a pressure quench that drives liquid 4 He from a normal phase to a superfluid one at a finite rate. Suppose the transition point is crossed at time t = 0, while time evolution starts at t ≪ 0. As long as the liquid is far away from the transition point its time evolution is adiabatic. In other words, the relaxation time scale τ , which tells how much time the system needs to adjust to new thermodynamic conditions, is small enough. As the transition is approached the critical slowing down occurs, i. e. τ → ∞, so that at the instant −t the system leaves adiabatic regime and enters an impulse one where its state is effectively frozen-see Fig. 1a for an illustration of these concepts. The timet is called the freeze-out time and was introduced by Zurek [2]. As the quench proceeds after crossing the transition point, the relaxation time scale decreases. At the instantt, the system goes back into an adiabatic regime. The freezeout time is determined by the Zurek's equation: τ (t) =t [2]. For the case of liquid 4 He it was found experi...

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“…The KZM approach was shown by Damski to provide an excellent approximation to LZF (Damski 2005; see also Damski & Zurek (2006 for extensions). Using LZF, we compute the sizeÑ of the spin chain that will probably remain in the ground state in the course of the quench with probability pZ0.5.…”

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“…Following Refs. [4,21], we note that the system enters the impulse region where excitation production occurs around t 0k 0 = b 1/α k 0 /ω 1 . Following Ref.…”

“…(10) can also be understood from a simple adiabatic-impulse argument provided in Refs. [4,21] as follows. For small ω 1 , the system enters the impulse region, where defect production occurs, around t = t 0k 0 = b k 0 /ω 1 .…”