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Journal of Modern Optics

Quantum logic in coarse grained control of wavepackets

E. A. Shapiro *, Michael Spanner, Misha Ivanov

Published Apr 15, 2005• 16 citations
Publisher

Abstract

We describe ‘coarse-grained’ control of wavepackets with finite resolution in phase space and show how this control can be approached as quantum information processing. To this end, we introduce the wavepacket analogues of qubits, designed to represent the information recorded within a wavepacket, and show how they can be addressed within a single wavepacket. The control is implemented both for non-dispersive wavepacket propagation using linear optical elements, illustrated for the classical waves, and for the dispersive evolution using fractional wavepacket revivals and AC Stark shifts as control tools, illustrated for the revivals of molecular alignment.

DOI
10.1080/09500340512331313944

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We describe ‘coarse-grained’ control of wavepackets with finite resolution in phase space and show how this control can be approached as quantum information processing. To this end, we introduce the wavepacket analogues of qubits, designed to represent the information recorded within a wavepacket, and show how they can be addressed within a single wavepacket. The control is implemented both for non-dispersive wavepacket propagation using linear optical elements, illustrated for the classical waves, and for the dispersive evolution using fractional wavepacket revivals and AC Stark shifts as control tools, illustrated for the revivals of molecular alignment.
We study control of wave packets with a finite accuracy, approaching it as quantum information processing. For a given control resolution, we define the analogs of several quantum bits within the shape of a single wave packet. These bits are based on wave packet symmetries. Analogs of one- and two-bit gates can be implemented using only free wave packet evolution and coordinate-dependent ac Stark shifts applied at the moments of fractional revivals. As in quantum computation, the gates form a logarithmically small set of basis operations which can be used to approximate any unitary transformation desired for quantum control of the wave packet dynamics. Numerical examples show the application of this approach to control vibrational wave packet revivals.
We study a finite-resolution route to decoherence-like behavior in quantum systems whose classical limit is chaotic. The mechanism does not require tracing over an external environment, adding phenomenological noise, or changing unitary quantum mechanics. Instead, chaotic phase-space transport drives Wigner-function interference fringes to progressively shorter wavelengths, while any observation, reconstruction, or deliberately coarse description has finite phase-space resolution. Modeling that resolution by Gaussian coarse-graining, we show that interference components transported below the resolution scale are exponentially attenuated in the coarse-grained phase-space description. The exact quantum state remains pure under unitary evolution; the suppression is, therefore, operational rather than objective. In the semiclassical pre-Ehrenfest window, the instability scale is set by finite-time Lyapunov exponents, while the dimensionless suppression factor is determined by the initial fringe geometry and the coarse-graining covariance. The same geometry gives a robustness criterion: interference components with strong projection onto covector-expanding Oseledets directions are suppressed first, while components avoiding those directions remain visible longer. A hyperbolic cat-map example illustrates the mechanism explicitly. We state the regime of validity, distinguish the mechanism from environment-induced decoherence, and give a practical extraction procedure for numerical or experimental tests.
Abstract Quantum state transfer is a key operation for quantum information processing. The original pitch-and-catch protocols rely on flying qubits or single photons with engineered wavepacket shapes to achieve a deterministic, fast and high-fidelity transfer. Yet, these protocols overlook two important factors, namely, the distortion of the wavepacket during the propagation and non-Markovian effects during the emission and reabsorption processes due to time-dependent controls. Here we address both difficulties in a general quantum-optical model and propose a correction strategy to improve quantum state transfer protocols. Including non-Markovian effects in our theoretical description, we show how to derive control pulses that imprint phases on the wavepacket that compensate the distortion caused by propagation. Our theoretical results are supported by detailed numerical simulations showing that a suitable correction strategy can improve state transfer fidelities up to three orders of magnitude.
An exquisite control over the dynamics of a quantum system is essential for realizing any quantum technology. This thesis presents some practical and efficient strategies for coherent quantum control: a paradigm for controlling quantum dynamics using electromagnetic fields. Coherent control is a common framework for describing the physical implementation of quantum logic gates in several quantum information processing platforms including superconducting qubits and trapped ions. For the purpose of demonstration, however, liquid-state Nuclear Magnetic Resonance (NMR) is used in this thesis as it is an ideal test-bed for developing, testing and benchmarking coherent control tools and techniques. Coherent control was revitalised in 2005 by the development of Gradient Ascent Pulse Engineering (GRAPE), a powerful optimal control technique that is widely used for designing shaped pulses for a variety of tasks. GRAPE, like any other closed-loop optimal control algorithm, requires multiple evaluations of the computationally expensive matrix exponential and the full time-evolution propagator. To this end, I present strategies to sidestep the explicit evaluation of matrix exponentials, thereby offering substantial speed-ups over conventional implementations of the GRAPE algorithm. The results are demonstrated experimentally in NMR, but also in simulations of ultracold molecules and superconducting qubits. Beyond GRAPE, this thesis also considers methods for designing spin-echo sequences for implementing any arbitrary network of controlled-Z gates in a system of qubits with zz-couplings. First, a method based on linear-programming provides near time-optimal sequences in fully-coupled systems having up to a few hundred qubits. Further, an analytic method based on graph colouring finds near time-optimal spin echo sequences in engineered systems with any number of qubits having only nearest and next-nearest neighbour-couplings.
Neutral atoms offer a promising platform for single- and many-body quantum control, as required for quantum information processing. This includes excellent isolation from the decohering influence of the environment, and the existence of well developed techniques for atom trapping and coherent manipulation. We present a review of our work to implement quantum control and measurement for ultra-cold atoms in far-off-resonance optical lattice traps. In recent experiments we have demonstrated coherent behavior of mesoscopic atomic spinor wavepackets in optical double-well potentials, and carried out quantum state tomography to reconstruct the full density matrix for the atomic spin degrees of freedom. This model system shares a number of important features with proposals to implement quantum logic and quantum computing in optical lattices. We present a theoretical analysis of a protocol for universal quantum logic via single qubit operations and an entangling gate based on electric dipole-dipole interactions. Detailed calculations including the full atomic hyperfine structure suggests that high-fidelity quantum gates are possible under realistic experimental conditions.
Atoms in high-lying Rydberg states with large values of principal quantum number n, n ≥300, form a valuable laboratory in which to explore the control and manipulation of quantum states of mesoscopic size using carefully tailored sequences of short electric field pulses whose characteristic times (duration and/or rise/fall times) are less than the classical electron orbital period. Atoms react to such pulse sequences very differently than to short laser or microwave pulses providing the foundation for a number of new approaches to engineering atomic wavefunctions. The remarkable level of control that can be achieved is illustrated with reference to the generation of localized wavepackets in Bohr-like near-circular orbits, and the production of non-dispersive wavepackets under periodic driving and their transport to targeted regions of phase space. New protocols continue to be developed that will allow even tighter control with the promise of new insights into quantum-classical correspondence, information storage in mesoscopic systems, physics in the ultra-fast ultra-intense regime, and non-linear dynamics in driven systems.