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arXiv:1005.5547 · Physical Review Letters

Observing the Phase Space Trajectory of an Entangled Matter Wave Packet

Ulrich Poschinger, Andreas Walther, Kilian Singer, Ferdinand Schmidt‐Kaler

Published Dec 20, 2010• 40 citations
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Abstract

We observe the phase space trajectory of an entangled wave packet of a trapped ion with high precision. The application of a spin-dependent light force on a superposition of spin states allows for coherent splitting of the matter wave packet such that two distinct components in phase space emerge. We observe such motion with a precision of better than 9% of the wave packet extension in both momentum and position, corresponding to a 0.8 nm position resolution. We accurately study the effect of the initial ion temperature on the quantum entanglement dynamics. Furthermore, we map out the phonon distributions throughout the action of the displacement force. Our investigation shows corrections to simplified models of the system evolution. The precise knowledge of these dynamics may improve quantum gates for ion crystals and lead to entangled matter wave states with large displacements.

DOI
10.1103/physrevlett.105.263602

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We observe the phase space trajectory of an entangled wave packet of a trapped ion with high precision. The application of a spin-dependent light force on a superposition of spin states allows for coherent splitting of the matter wave packet such that two distinct components in phase space emerge. We observe such motion with a precision of better than 9% of the wave packet extension in both momentum and position, corresponding to a 0.8 nm position resolution. We accurately study the effect of the initial ion temperature on the quantum entanglement dynamics. Furthermore, we map out the phonon distributions throughout the action of the displacement force. Our investigation shows corrections to simplified models of the system evolution. The precise knowledge of these dynamics may improve quantum gates for ion crystals and lead to entangled matter wave states with large displacements.
We directly observe the quantum interference between two well-separated trapped-ion mechanical oscillator wave packets. The superposed state is created from a spin-motion entangled state using a heralded measurement. Wave packet interference is observed through the energy eigenstate populations. We reconstruct the Wigner function of these states by introducing probe Hamiltonians which measure Fock state populations in displaced and squeezed bases. Squeezed-basis measurements with 8 dB squeezing allow the measurement of interference for Δα=15.6, corresponding to a distance of 240 nm between the two superposed wave packets.
We explore how recent advances in the manipulation of single-ion wave packets open new avenues for detecting weak magnetic fields sourced by ultralight dark matter. A trapped ion in a “Schrödinger’s cat” state can be prepared with its spin and motional degrees of freedom entangled and be used as a matter-wave interferometer that is sensitive to the Aharonov-Bohm-like phase shift accumulated by the ion over its trajectory. The result of the spin-motion entanglement is a parametrically enhanced sensitivity to weak magnetic fields as compared with an unentangled ion in a trap. Taking into account the relevant boundary conditions, we demonstrate that a single trapped ion can probe unexplored regions of kinetically mixed dark-photon dark matter parameter space in the 10 − 15 eV ≲ m A ′ ≲ 10 − 14 eV mass window. We also show how such a table-top quantum device will also serve as a complementary probe of axionlike particle dark matter in the same mass window.
Achieving control over physical systems down to individual quanta is central for advancing our understanding of nature and enabling emerging quantum technologies. Trapped-ion systems provide a powerful platform offering exceptional control over internal and external degrees of freedom and allowing the study of both fundamental and applied aspects of quantum dynamics. This thesis introduces an experimental toolbox, centered on problem-tailored modifications of Ramsey interferometry, for observing and manipulating dynamic processes with high temporal and spatial resolution, showcased in a trapped-ion quantum simulator. Using this versatile approach, we bridge quantum metrology and quantum thermodynamics – from tracking the motional wave packet evolution of a single ion in phase space to observing time-(in)dependent energy-level renormalizations in ultra-strongly coupled open systems. A central innovation of this thesis is the development of a stroboscopic measurement technique that establishes and maintains coherence among four oscillators: a global microwave reference, an ultraviolet polarization-gradient traveling-wave light field, and the spin and motional states of a single trapped ion. This architecture, spanning a phase-stable frequency range of more than nine orders of magnitude, enables superresolved, back-action-free tracking of motional wave packets in phase-space. In this way, we achieve noise floors of 1.8(2) nm for position and 8(2) zN μs for momentum on sub-100-nanosecond timescales. This precision enables quantum-enhanced diagnostics, non-classical state preparation, and fine-grained phase-space control, with broad implications for entanglement generation, quantum interferometry, and quantum sensing. In a complementary study, we exploit the trapped-ion system as a testbed for probing fundamental quantum interactions: We experimentally simulate a strongly coupled open quantum system, where dressed electronic states represent the system and a single motional mode serves as its environment. We observe time-dependent energy shifts up to 15% of the bare transition frequency, induced solely by ultra-strong system-environment coupling. These shifts, anticipated to be a general feature of all quantum platforms across all coupling regimes, are accurately described by the minimal-dissipation ansatz and interpreted as generalized, time-resolved Lamb shifts. Together, the results of this thesis extend the frontier of trapped-ion quantum simulation: The integration of Ramsey interferometry, stroboscopic techniques, and strong-coupling simulations offers a high-resolution window into diverse aspects of quantum dynamics. This work enables advanced studies – including squeezed states, analogues of relativistic effects such as Hawking radiation and the physics of the early universe, and autonomous quantum engines.
Using numerical simulations of the time-dependent Schrödinger equation, we study the full quantum dynamics of the motion of an atomic ion in a linear Paul trap. Such a trap is based on a time-varying, periodic electric field and hence corresponds to a time-dependent potential for the ion, which we model exactly. We compare the center-of-mass motion with that obtained from classical equations of motion, as well as to results based on a time-independent effective potential. We also study the oscillations of the width of the ion’s wave packet, including close to the border between stable (bounded) and unstable (unbounded) trajectories. Our results confirm that the center-of-mass motion always follows the classical trajectory, that the width of the wave packet is bounded for trapping within the stability region, and therefore that the classical trapping criterion is fully applicable to quantum motion.
We propose a scheme to generate a superposition of motional coherent states with arbitrary coefficients on a line in phase space and implement a quantum controlled phase-gate for multiple trapped ions with a single standing-wave laser pulse whose carrier frequency is tuned to the ions transition. In the scheme each ion does not need to be exactly positioned at the node of the standing wave, which is very important from viewpoint of experiment. Furthermore, our scheme may allow the generation of a superposition of coherent states with large mean phonon number for a large number of trapped ions in a fast way by choosing suitable laser intensity. We show that it can also be used to generate maximally entangled states of multiple trapped ions.
Favored schemes for trapped-ion quantum logic gates use bichromatic laser fields to couple internal qubit states with external motion through a "spin-dependent force." We introduce a new degree of freedom in this coupling that reduces its sensitivity to phase decoherence. We demonstrate bichromatic spin-dependent forces on a single trapped 111Cd+ ion, and show that phase coherence of the resulting entangled states of spin and motion depends critically upon the spectral arrangement of the optical fields. This applies directly to the operation of entangling gates on multiple ions.