Landau-Zener model: Effects of finite coupling duration

Vitanov, Nikolay V.; Garraway, B. M.

doi:10.1103/physreva.53.4288

Cited by 238 publications

(324 citation statements)

References 25 publications

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“…First of all, it is desired to re-analyze more strictly the LZ dynamics, to get a systematic control of the intuitive results obtained above. This work can be done utilizing the results from [13]. Second, it seems to be very interesting to investigate how the adiabatic/impulse simplification of system dynamics works in other quantum mechanical systems, for instance those which exhibit faster increase of the gap with the distance from anti-crossing.…”

mentioning

confidence: 99%

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

2005

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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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confidence: 99%

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

2005

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“…In the vicinity of these points from (3.6) 19) i.e. the crossing points are square root bifurcation points for the function Ω 12 (t).…”

Section: Adiabatic Perturbation Theorymentioning

confidence: 99%

“…where s 0 and s stem for the initial and final points of the contour, and the solution (2.18) determines uniquely the transformation matrixT for the curl-less fieldÂT 19) and the diagonal matrixD can be found from (2.17) and can be expressed in terms of the geometric phase factor…”

Section: Born -Oppenheimer Approximationmentioning

confidence: 99%

Instanton versus traditional WKB approach to the Landau-Zener problem

Benderskiĭ¹,

Vetoshkin²,

Kats³

2003

J. Exp. Theor. Phys.

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Different theoretical approaches to the famous two state Landau -Zener problem are briefly discussed. Apart from traditional methods of the adiabatic perturbation theory, Born -Oppenheimer approximation with geometric phase effects, two-level approach, and momentum space representation, the problem is treated semiclassically also in the coordinate space. Within the framework of the instanton approach we present a full and unified description of 1D Landau-Zener problem of level crossing. The method enables us to treat accurately all four transition points (appearing at two levels crossing), while the standard WKB approach takes into account only two of them. The latter approximation is adequate for calculating of the transition probability or for studying of scattering processes, however it does not work for finding corresponding chemical reactions rates, where very often for typical range of parameters all four transition points can be relevant. Applications of the method and of the results may concern the various systems in physics, chemistry and biology.

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“…For large-order value of the Weber functions with a phase of argument | arg z k | < π/2 the leading terms are [46,47] …”

Section: Some Important Properties Of the Weber Functionsmentioning

confidence: 99%

Non-Hermitian quantum annealing in the ferromagnetic Ising model

Nesterov¹,

Zepeda²,

Berman³

2013

Phys. Rev. A

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We developed a non-Hermitian quantum optimization algorithm to find the ground state of the ferromagnetic Ising model with up to 1024 spins (qubits). Our approach leads to significant reduction of the annealing time. Analytical and numerical results demonstrate that the total annealing time is proportional to ln N , where N is the number of spins. This encouraging result is important in using classical computers in combination with quantum algorithms for the fast solutions of NPcomplete problems. Additional research is proposed for extending our dissipative algorithm to more complicated problems.

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Landau-Zener model: Effects of finite coupling duration

Cited by 238 publications

References 25 publications

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

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

Instanton versus traditional WKB approach to the Landau-Zener problem

Non-Hermitian quantum annealing in the ferromagnetic Ising model

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