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arXiv:2205.11444 · Physical Review Letters

Determination of Multimode Motional Quantum States in a Trapped Ion System

Zhubing Jia, Ye Wang, Bichen Zhang, Jacob Whitlow, Chao Fang, Jungsang Kim, Kenneth R. Brown

Published Aug 31, 2022• 11 citations
arXivHTMLPDF

Abstract

Trapped atomic ions are a versatile platform for studying interactions between spins and bosons by coupling the internal states of the ions to their motion. Measurement of complex motional states with multiple modes is challenging, because all motional state populations can only be measured indirectly through the spin state of ions. Here we present a general method to determine the Fock state distributions and to reconstruct the density matrix of an arbitrary multimode motional state. We experimentally verify the method using different entangled states of multiple radial modes in a five-ion chain. This method can be extended to any system with Jaynes-Cummings-type interactions.

DOI
10.1103/physrevlett.129.103602

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Trapped atomic ions are a versatile platform for studying interactions between spins and bosons by coupling the internal states of the ions to their motion. Measurement of complex motional states with multiple modes is challenging, because all motional state populations can only be measured indirectly through the spin state of ions. Here we present a general method to determine the Fock state distributions and to reconstruct the density matrix of an arbitrary multimode motional state. We experimentally verify the method using different entangled states of multiple radial modes in a five-ion chain. This method can be extended to any system with Jaynes-Cummings-type interactions.
We reconstruct the density matrices and Wigner functions for various quantum states of motion of a harmonically bound 9Be+{}^{9}{\mathrm{Be}}^{+}9Be+ ion. We apply coherent displacements of different amplitudes and phases to the input state and measure the number state populations. Using novel reconstruction schemes we independently determine both the density matrix in the number state basis and the Wigner function. These reconstructions are sensitive indicators of decoherence in the system.
Summary form only given, as follows. The ability to create and completely characterize a variety of fundamental quantum states has long been sought after in the laboratory since it brings to the forefront issues involving the relationship between quantum and classical physics. The interaction of a trapped atom with classical laser light constitutes a simple and well-isolated quantum system that makes experiments along these lines possible. The motion of an ion trapped in a rf-trap can be described as a quantum harmonic oscillator. If the ion is driven by a classical light field, several interactions similar to and beyond the simple Jaynes-Cummings coupling can be realized. These couplings allow for precisely controlled engineering and measurement of quantum states. We create states having both classical and nonclassical character including thermal, number, coherent, squeezed, and Schrodinger cat states. The motional quantum state is fully reconstructed with use of two novel schemes that determine the density matrix in the number state basis and the Wigner function. Our techniques allow well-controlled experiments on decoherence and related phenomena on the quantum-classical borderline.
Trapped ions are one of the leading platforms for quantum technologies due to their excellent control over their internal electronic and external motional degrees of freedom. Many techniques and interesting effects rely on the coupling of the ions to their motion, which makes precise control of them paramount. In this work, two novel techniques for motional state detection are presented. They allow the probing of certain motional states with a single measurement, without affecting the motion of the measured ion. These techniques are employed to generate and detect a highly entangled state between two motional modes in a novel manner. Furthermore, we harness the rich and intricate dynamics arising from the coupling of the ion's motion to its electronic degrees of freedom to study the emergence of interference. Theoretical predictions, describing interference both in the quantum and the classical regimes, are verified, offering a new and more intuitive description of interference, not just for trapped ions, but for a variety of systems that can be described in a similar way. On the other hand, Rydberg excitation with trapped ions enables new interaction mechanisms and makes them extremely sensitive to their surroundings. This thesis presents advances towards Rydberg experiments with longer ion strings and concludes with a first demonstration of coherent population transfer between Rydberg states in trapped ions. The methods developed significantly increase the toolbox for trapped Rydberg ions, enabling more sophisticated experiments, especially when multiple different Rydberg states are involved, and allow more flexibility when using longer ion strings.
Summary form only given. Determining the unknown state of a quantum system from measurements of observables, a question first posed by Pauli, is a nontrivial task, which does not always permit a solution. The few examples where this inverse problem was experimentally solved include light fields, molecular vibrations, the motion of trapped ions and free atomic waves. We describe a method to reconstruct the complete density matrix of a generally mixed spin state of trapped neutral atoms. Compared to spin- 1/2 systems, the reconstruction of the spin density matrix for a large angular momentum is a much more challenging task. In our system with cesium atoms in the hyperfine ground state, a reconstruction of the 9/spl times/9 density matrix requires 80 real numbers to be experimentally determined. This can be achieved by measuring the 2F+1=9 populations along 4F+1=17 carefully chosen orientations of a Stern-Gerlach apparatus. A numerical simulation of the reconstruction shows this method to be robust against expected experimental uncertainties. We report on an experiment in progress to implement this measurement protocol in a standard laser cooling and trapping arrangement of neutral cesium atoms.
We propose and demonstrate a scalable scheme for the simultaneous determination of internal and motional states in trapped ions with single-site resolution. The scheme is applied to the study of polaritonic excitations in the Jaynes-Cummings-Hubbard model with trapped ions, in which the internal and motional states of the ions are strongly correlated. We observe quantum phase crossovers of polaritonic excitations in two ions by directly evaluating their variances per ion site. Our work establishes an essential technological method for large-scale quantum simulations of polaritonic systems.
Dispersive shifts are a widely used tool for bosonic readout and control in circuit quantum electrodynamics, yet they remain relatively unexplored in trapped-ion motional systems. Here we introduce a unified framework for multimode phonon-number measurement and nondestructive single-shot superparity measurement, i.e., phonon-number measurement modulo 2^k, using dispersive shifts in the far-detuned multimode Jaynes-Cummings interaction of a trapped ion system. We implement a Ramsey sequence that realizes a multimode spin-dependent rotation (SDR) together with a selective decoupling scheme that cancels the phase induced by the carrier AC-Stark shift while preserving the phonon-number-dependent phase induced by the dispersive shift. Within this framework, we infer single-mode and two-mode Fock-state distributions from spin-population dynamics, use SDR-based conditional parity operators with postselection to generate cat states and entangled coherent states, and realize nondestructive single-shot measurements of phonon number modulo 2, 4, and 8 in the single-mode setting. These results open a new avenue for the use of multimode parity operators in trapped-ion bosonic systems.