Shortcut to Adiabaticity in the Lipkin-Meshkov-Glick Model

Abstract

We study transitionless quantum driving in an infinite-range many-body system described by the LipkinMeshkov-Glick model. Despite the correlation length being always infinite the closing of the gap at the critical point makes the driving Hamiltonian of increasing complexity also in this case. To this aim we develop a hybrid strategy combining a shortcut to adiabaticity and optimal control that allows us to achieve remarkably good performance in suppressing the defect production across the phase transition. DOI: 10.1103/PhysRevLett.114.177206 PACS numbers: 75.10.Jm, 05.30.Rt, 64.60.Ht, 73.43.Nq The dynamical evolution of a quantum system often has to be tailored so that a given initial state is transformed into a suitably chosen target one. In cases such as this, the use of techniques for quantum optimal control can be key in engineering an efficient protocol. Over the years, formal control methods have been devised, both in the classical and quantum scenario [1]. To date, optimal control has been proven beneficial in a multitude of fields, ranging from molecular physics to quantum information processing or high precision measurements [2]. Only very recently, however, this framework has been extended so as to cope with the rich phenomenology and complexity of quantum many-body systems [3,4]. In this context, quantum optimal control has been shown to be crucial for the design of schemes for the preparation of many-body quantum states [3,5,6], the exploration of the experimentally achievable limits in quantum interferometry [7], and the cooling of quantum systems [8].Needless to say, quantum optimal control is not the only way to design the dynamical evolution of a quantum system, and one could consider simpler (suboptimal) ways to drive the desired dynamics. For instance, using the adiabatic theorem we are able to constrain a quantum system to remain in an eigenstate during any evolution. However, in order for such a technique to be accurate, it should operate on a rather long time scale. Unwanted transitions between the state we would like to confine the system into and other ones in its spectrum, which are induced by the unavoidably finite-speed nature of an evolution and ultimately limit the precision of the adiabatic dynamics, can be suppressed by adding suitable corrections to the Hamiltonian guiding the evolution [9,10]. This form of quantum control, named the shortcut to adiabaticity (STA), has been considered in a variety of different situations, and recently reviewed in Ref. [11]. An experimental implementation using cold atomic gases has been reported in Ref. [12].Recently, the idea of a STA has been extended to quantum many-body systems, a context where it can be potentially very beneficial. The STA was first employed in the suppression of defects produced when crossing a quantum phase transition in the paradigm model embodied by the one-dimensional Ising model [13]. Despite such potential, a crucial feature that emerges from the use of a STA in many-body scenarios is the inherent complexity of the ...

Local shortcut to adiabaticity for quantum many-body systems

“…This results in breakdown of the adiabatic theorem of quantum mechanics near the the critical point thus invariably resulting in non-adiabatic excitations, which vanish only in the ideal, and experimentally unachievable, limit of τ → ∞ [25,26]. It is possible to cross a critical point without defect generation either by optimal control [8,9] or by means of STA [22,23]. In both cases, in different ways, the existence of the critical point manifest in the need of an increasing complex control protocol.…”

“…Shortcut to adiabaticity was also extended to control the dynamics of many-body system governed by the transverse Ising [22] and Lipkin-Meshkov-Glick [23] Hamiltonians. A distinct feature emerging from these works is the increasing complexity of the Hamiltonian H c as a result of gaps closing in the spectrum.…”

Cost of counterdiabatic driving and work output

“…On the other hand, it has been shown for both classical and quantum systems that external drivings can allow a system to evolve adiabatically even when driven in finite time [32][33][34][35][36][37][38][39][40][41][42][43][44]. This has applications in quantum control, and can be performed in three ways: (i) Driving of a system such that, instantaneously, a state evolves adiabatically (e.g.…”

Shortcut to Adiabaticity for an Anisotropic Gas Containing Quantum Defects

“…They can be formulated as counterdiabatic driving [14,15]. They are being considered for the preparation of many-body states [16][17][18], they have motivated an exploration of the quantum speed limit [19] and reflection on the third law of thermodynamics [20].…”

“…They can be formulated as counterdiabatic driving [14,15]. They are being considered for the preparation of many-body states [16][17][18], they have motivated an exploration of the quantum speed limit [19] and reflection on the third law of thermodynamics [20].The construction of an STA relies on the existence of a scaling solution to the equation describing the manybody dynamics of the system in a time-dependent trap. Such a solution exists for the ideal gas in a harmonic trap and has been used to construct an STA solution [10] implemented experimentally on a Bose cloud above T c [21].…”