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Physical Review A

Quantum driving protocols for a two-level system: From generalized Landau-Zener sweeps to transitionless control

Nicola Malossi, Mark G. Bason, Matthieu Viteau, E. Arimondo, R. Mannella, O. Morsch, D. Ciampini

Published Jan 16, 2013• 44 citations
Publisher

Abstract

We present experimental results on the preparation of a desired quantum state in a two-level system with the maximum possible fidelity using driving protocols ranging from generalizations of the linear Landau-Zener protocol to transitionless driving protocols that ensure perfect following of the instantaneous adiabatic ground state. We also study the minimum time needed to achieve a target fidelity and explore and compare the robustness of some of the protocols against parameter variations simulating a possible experimental uncertainty. In our experiments, we realize a two-level model system using Bose-Einstein condensates inside optical lattices, but the results of our investigation should hold for any quantum system that can be approximated by a two-level system.

DOI
10.1103/physreva.87.012116

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We present experimental results on the preparation of a desired quantum state in a two-level system with the maximum possible fidelity using driving protocols ranging from generalizations of the linear Landau-Zener protocol to transitionless driving protocols that ensure perfect following of the instantaneous adiabatic ground state. We also study the minimum time needed to achieve a target fidelity and explore and compare the robustness of some of the protocols against parameter variations simulating a possible experimental uncertainty. In our experiments, we realize a two-level model system using Bose-Einstein condensates inside optical lattices, but the results of our investigation should hold for any quantum system that can be approximated by a two-level system.
We present experimental results on the preparation of a desired quantum state in a two-level system with the maximum possible fidelity using driving protocols ranging from generalizations of the linear Landau-Zener protocol to transitionless driving protocols that ensure perfect following of the instantaneous adiabatic ground state. We also study the minimum time needed to achieve a target fidelity and explore and compare the robustness of some of the protocols against parameter variations simulating a possible experimental uncertainty. In our experiments, we realize a two-level model system using Bose-Einstein condensates inside optical lattices, but the results of our investigation should hold for any quantum system that can be approximated by a two-level system.
A fundamental requirement in quantum information processing and in many other areas of science is the capability of precisely controlling a quantum system by preparing a quantum state with the highest fidelity and/or in the fastest possible way. Here we present an experimental investigation of a two level system, characterized by a time-dependent Landau-Zener Hamiltonian, aiming to test general and optimal high-fidelity control protocols. The experiment is based on a Bose-Einstein condensate (BEC) loaded into an optical lattice, then accelerated, which provides a high degree of control over the experimental parameters. We implement generalized Landau-Zener sweeps, comparing them with the well-known linear Landau-Zener sweep. We drive the system from an initial state to a final state with fidelity close to unity in the shortest possible time (quantum brachistochrone), thus reaching the ultimate speed limit imposed by quantum mechanics. On the opposite extreme of the quantum control spectrum, the aim is not to minimize the total transition time but to maximize the adiabaticity during the time-evolution, the system being constrained to the adiabatic ground state at any time. We implement such transitionless superadiabatic protocols by an appropriate transformation of the Hamiltonian parameters. This transformation is general and independent of the physical system.
Nonlinear two-level Landau-Zener type equations for systems with relevance for Bose-Einstein condensates and nonlinear optics are considered and the minimal time Tmin{T}_{\mathrm{min}}Tmin​ to drive an initial state to a given target state is investigated. Surprisingly, the nonlinearity may be canceled by a time-optimal unconstrained driving and Tmin{T}_{\mathrm{min}}Tmin​ becomes independent of the nonlinearity. For constrained and unconstrained driving explicit expressions are derived for Tmin{T}_{\mathrm{min}}Tmin​, the optimal driving, and the protocol.
It is known that high intensity fields are usually required to implement shortcuts to adiabaticity via transitionless quantum driving (TQD). Here, we show that this requirement can be relaxed by exploiting the gauge freedom of generalized TQD, which is expressed in terms of an arbitrary phase when mimicking the adiabatic evolution. We experimentally investigate the performance of generalized TQD in comparison to both traditional TQD and adiabatic dynamics. By using a Yb+171 trapped ion hyperfine qubit, we implement a Landau–Zener adiabatic Hamiltonian and its (traditional and generalized) TQD counterparts. We show that the generalized theory provides energy-optimal Hamiltonians for TQD, with no additional fields required. In addition, the optimal TQD Hamiltonian for the Landau–Zener model is investigated under dephasing. Even using less intense fields, optimal TQD exhibits fidelities that are more robust against a decohering environment, with performance superior to that provided by the adiabatic dynamics.
We investigate high-fidelity quantum driving in a nonlinear two-level system and find that nonlinear atomic interaction can break the speed limit of the linear model. We show that repulsive atomic interaction can decrease the minimal time requested for reaching target state even to zero, while attractive atomic interaction tends to increase the minimal time. There exists a critical attractive interaction beyond which the target state cannot be reached with high fidelity. Possible experimental observation of the nonlinear effects using a Bose-Einstein condensate in an accelerating optical lattice is discussed.
The Landau--Zener (LZ) model describes a two-level quantum system that undergoes an avoided crossing. In the adiabatic limit, the transition probability vanishes. An auxiliary control field HCDH_\text{CD}HCD​ can be reverse-engineered so that the full Hamiltonian H0+HCDH_0 + H_\text{CD}H0​+HCD​ reproduces adiabaticity for all parameter values. Our aim is to construct a single control field H1H_1H1​ that drives an ensemble of LZ-type Hamiltonians with a distribution of energy gaps. H1H_1H1​ works best statistically, minimizing the average transition probability. We restrict our attention to a special class of H1H_1H1​ controls, motivated by HCDH_\text{CD}HCD​. We found a systematic trade-off between instantaneous adiabaticity and the final transition probability. Certain limiting cases with a linear sweep can be treated analytically; one of them being the LZ system with Dirac δ(t)δ(t)δ(t) function. Comprehensive and systematic numerical simulations support and extend the analytic results.