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Journal of Physics A Mathematical and Theoretical

Transitionless quantum driving

Michael Berry

Published Aug 20, 2009• 1,029 citations
Publisher

Abstract

For a general quantum system driven by a slowly time-dependent Hamiltonian, transitions between instantaneous eigenstates are exponentially weak. But a nearby Hamiltonian exists for which the transition amplitudes between any eigenstates of the original Hamiltonian are exactly zero for all values of slowness. The general theory is illustrated by spins driven by changing magnetic fields, and implies that any spin expectation history, including those where the spin never precesses, can be generated by infinitely many driving fields, here displayed explicitly. Asymptotically, the absence of transitions is explained by continuation to complex time, where the complex degeneracies in the transitionless driving fields have a nongeneric structure for which there is no Stokes phenomenon; this is analogous to the explanation of reflectionless potentials.

DOI
10.1088/1751-8113/42/36/365303

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For a general quantum system driven by a slowly time-dependent Hamiltonian, transitions between instantaneous eigenstates are exponentially weak. But a nearby Hamiltonian exists for which the transition amplitudes between any eigenstates of the original Hamiltonian are exactly zero for all values of slowness. The general theory is illustrated by spins driven by changing magnetic fields, and implies that any spin expectation history, including those where the spin never precesses, can be generated by infinitely many driving fields, here displayed explicitly. Asymptotically, the absence of transitions is explained by continuation to complex time, where the complex degeneracies in the transitionless driving fields have a nongeneric structure for which there is no Stokes phenomenon; this is analogous to the explanation of reflectionless potentials.
We apply the method of transitionless quantum driving for time-dependent quantum systems to spin systems. For a given Hamiltonian, the driving Hamiltonian is constructed so that the adiabatic states of the original system obey the Schrödinger equation. For several typical systems such as the XY spin chain and the Lipkin-Meshkov-Glick model, the driving Hamiltonian is constructed explicitly. We discuss possible interesting situations when the driving Hamiltonian becomes time independent and when the driving Hamiltonian is equivalent to the original one. For many-body systems, a crucial problem occurs at the quantum phase transition point where the energy gap between the ground and first excited states becomes zero. We discuss how the defect can be circumvented in the present method.
Quantum adiabatic evolution is a dynamical evolution of a quantum system under slow external driving. According to the quantum adiabatic theorem, no transitions occur between nondegenerate instantaneous energy eigenstates in such a dynamical evolution. However, this is true only when the driving rate is infinitesimally small. For a small nonzero driving rate, there are generally small transition probabilities between the energy eigenstates. We develop a classical mechanics framework to address the small deviations from the quantum adiabatic theorem order by order. A hierarchy of Hamiltonians is constructed iteratively with the zeroth-order Hamiltonian being determined by the original system Hamiltonian. The k th-order deviations are governed by a k th-order Hamiltonian, which depends on the time derivatives of the adiabatic parameters up to the k th-order. Two simple examples, the Landau–Zener model and a spin-1/2 particle in a rotating magnetic field, are used to illustrate our hierarchical theory. Our analysis also exposes a deep, previously unknown connection between classical adiabatic theory and quantum adiabatic theory.
Different methods have been recently put forward and implemented experimentally to inverse engineer the time-dependent Hamiltonian of a quantum system and accelerate slow adiabatic processes via nonadiabatic shortcuts. In the ``transitionless quantum driving'' proposed by Berry, shortcut Hamiltonians are designed so that the system follows exactly, in an arbitrarily short time, the approximate adiabatic path defined by a reference Hamiltonian. A different approach is based on first designing a Lewis-Riesenfeld invariant to carry the eigenstates of a Hamiltonian from specified initial to final configurations, again in an arbitrary time, and then constructing from the invariant the transient Hamiltonian that connects these boundary configurations. We show that the two approaches, apparently quite different in form and so far in results, are, in fact, strongly related and potentially equivalent, so that the inverse-engineering operations in one of them can be reinterpreted and understood in terms of the concepts and operations of the other one. We study, as explicit examples, expansions of time-dependent harmonic traps and the state preparation of two-level systems.
We study quantum dynamics of the adiabatic search algorithm with the equivalent two-level system. Its adiabatic and non-adiabatic evolution is studied and visualized as trajectories of Bloch vectors on a Bloch sphere. We find the change in the non-adiabatic transition probability from exponential decay for the short running time to inverse-square decay in asymptotic running time. The scaling of the critical running time is expressed in terms of the Lambert W function. We derive the transitionless driving Hamiltonian for the adiabatic search algorithm, which makes a quantum state follow the adiabatic path. We demonstrate that a uniform transitionless driving Hamiltonian, approximate to the exact time-dependent driving Hamiltonian, can alter the non-adiabatic transition probability from the inverse square decay to the inverse fourth power decay with the running time. This may open up a new but simple way of speeding up adiabatic quantum dynamics.
Keeping a quantum system in a given instantaneous eigenstate is a control problem with numerous applications, e.g., in quantum information processing. The problem is even more challenging in the setting of open quantum systems, where environment-mediated transitions introduce additional decoherence channels. Adiabatic passage is a well-established solution but requires a sufficiently slow evolution time that is dictated by the adiabatic theorem. Here we develop a systematic projection theory formulation for the transitionless evolution of general open quantum systems described by time-local master equations. We derive a time-convolutionless dynamical equation for the target instantaneous eigenstate of a given time-dependent Hamiltonian. A transitionless dynamics then arises in terms of a competition between the average Hamiltonian gap and the decoherence rate, which implies optimal adiabaticity timescales. We show how eigenstate tracking can be accomplished via control pulses, without explicitly incorporating counter-diabatic driving, thus offering an alternative route to accelerate adiabaticity. We examine rectangular pulses, chaotic signals, and white noise, and find that, remarkably, the effectiveness of eigenstate tracking hardly depends on the details of the control functions. In all cases the control protocol keeps the system in the desired instantaneous eigenstate throughout the entire evolution, along an accelerated adiabatic path.
Transitionless quantum driving achieves adiabatic evolution in a hurry, using a counterdiabatic Hamiltonian to stifle nonadiabatic transitions. Here this shortcut to adiabaticity is cast in terms of a generator of adiabatic transport. This yields a classical analog of transitionless driving, and provides a strategy for constructing quantal counterdiabatic Hamiltonians. As an application of this framework, exact classical and quantal counterdiabatic terms are obtained for a particle in a box and for even-power-law potentials in one degree of freedom.