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arXiv:2505.20844 · Physical Review Letters

Engineering Continuous-Variable Entanglement in Mechanical Oscillators with Optimal Control

Maverick J. Millican, Vassili G. Matsos, Christophe H. Valahu, Tomas Navickas, Liam J. Bond, Ting Rei Tan

Published Nov 12, 2025• 3 citations
arXivHTMLPDF

Abstract

We demonstrate an optimal quantum control strategy for the deterministic preparation of entangled harmonic oscillator states in trapped ions. The protocol employs dynamical phase modulation of laser-driven Jaynes-Cummings and anti-Jaynes-Cummings interactions. We prepare two-mode squeezed vacuum states in the motions of a trapped ion and characterize the states with phase-space tomography. We verify continuous-variable entanglement by measuring an Einstein-Podolsky-Rosen entanglement parameter of 0.0132(7), which surpasses the threshold of 0.25 for Reid's Einstein-Podolsky-Rosen criterion. We also perform a continuous-variable Bell test and find a violation of the Clauser-Horne-Shimony-Holt inequality, measuring 2.26(3), exceeding the entanglement threshold of 2. We also demonstrate the flexibility of our method by preparing a non-Gaussian entangled oscillator state-a superposition of two-mode squeezed vacuum states.

DOI
10.1103/2ntn-qh3t

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We demonstrate an optimal quantum control strategy for the deterministic preparation of entangled harmonic oscillator states in trapped ions. The protocol employs dynamical phase modulation of laser-driven Jaynes-Cummings and anti-Jaynes-Cummings interactions. We prepare two-mode squeezed vacuum states in the motions of a trapped ion and characterize the states with phase-space tomography. We verify continuous-variable entanglement by measuring an Einstein-Podolsky-Rosen entanglement parameter of 0.0132(7), which surpasses the threshold of 0.25 for Reid's Einstein-Podolsky-Rosen criterion. We also perform a continuous-variable Bell test and find a violation of the Clauser-Horne-Shimony-Holt inequality, measuring 2.26(3), exceeding the entanglement threshold of 2. We also demonstrate the flexibility of our method by preparing a non-Gaussian entangled oscillator state-a superposition of two-mode squeezed vacuum states.
Quantum harmonic oscillators are a central resource in quantum technologies. Coupling this resource to a two-level system gives rise to spin-oscillator dynamics that can be used for system calibrations, enhanced sensitivity to signals, and the preparation of oscillator entangled states. Chapter 4 introduces a motional-frequency calibration protocol based on a time-reversal, state-dependent force sequence that maps the effects of motional-frequency miscalibration onto a narrow spin-response feature. The chapter also discusses how thermally occupied motional states enhance sensitivity to particular Hamiltonian terms, making them effective probe states. A two-point feedback servo with processing performed on a field-programmable gate array, or FPGA, is used to track the interaction resonance in real time, enabling sub-second updates and continuous correction of frequency drift during data acquisition. Chapter 5 realizes phase-insensitive displacement sensing by embedding spin-dependent squeezing in a time-reversal interferometry sequence. The method prepares a spin-correlated superposition of orthogonally squeezed motional states, yielding a spin-observable response used to estimate displacement amplitudes while remaining agnostic to the signal phase. Experiments with vacuum and number-state probes demonstrate enhanced sensitivity with increasing motional energy. Chapter 6 extends the spin-oscillator toolkit to multi-oscillator control. Optimized phase modulations applied to Jaynes-Cummings and anti-Jaynes-Cummings couplings between a single spin and two motional modes synthesize an effective two-mode squeezing interaction, enabling preparation of two-mode squeezed vacuum states and a non-Gaussian superposition of those states. Joint phase-space tomography reveals multi-mode correlations, and continuous-variable entanglement is certified by the Einstein-Podolsky-Rosen criterion and a Clauser-Horne-Shimony-Holt-type continuous-variable Bell test.
We investigate the dynamics of entanglement between two continuous variable quantum systems. The model system consists of two atoms in a harmonic trap which are interacting by a simplified s-wave scattering. We show, that the dynamically created entanglement changes in a steplike manner. Moreover, we introduce local operators which allow us to violate a Bell-Clauser-Horne-Shimony-Holt inequality adapted to the continuous variable case. The correlations show nonclassical behavior and almost reach the maximal quantum-mechanical value. This is interesting since the states prepared by this interaction are very different from any Einstein-Podolsky-Rosen--like state.
The efficient initialization of a quantum system is a prerequisite for quantum technological applications. Here we show that several classes of quantum states of a harmonic oscillator can be efficiently prepared by means of a Jaynes-Cummings interaction with a single two-level system. This is achieved by suitably tailoring external fields which drive the dipole and/or the oscillator. The time-dependent dynamics that leads to the target state is identified by means of optimal control theory (OCT) based on Krotov's method. Infidelities below ${10}^{\ensuremath{-}4}$ can be reached for the parameters of the experiment of Raimond, Haroche, Brune and co-workers, where the oscillator is a mode of a high-Q microwave cavity and the dipole is a Rydberg transition of an atom. For this specific situation we analyze the limitations on the fidelity due to parameter fluctuations and identify robust dynamics based on pulses found using ensemble OCT. Our analysis can be extended to quantum-state preparation of continuous-variable systems in other platforms, such as trapped ions and circuit QED.
Quantum correlations are essential for achieving quantum advantage in computing, communication and sensing. Moreover, their observation challenges and constrains our fundamental understanding of nature. Mechanical oscillators in the quantum regime provide an appealing platform for preparing and investigating quantum correlations at macroscopic scales. Despite substantial progress, however, continuous-variable quantum correlations stronger than entanglement have not yet been observed in this macroscopic regime. Here, we report the experimental observation of continuous-variable Einstein-Podolsky-Rosen correlations between two spatially-separated mechanical oscillators with an effective mass of ∼16 μg\sim 16 \,μg∼16μg each. This is achieved by coupling them to a superconducting qubit which allows for engineering a two-mode squeezing interaction when parametrically driven. Crucially, we show that this interaction can be used to witness quantum correlations through the realization of a mechanical SU(1,1) interferometer. Our results expand the toolbox of operations in circuit quantum acoustodynamics and demonstrate that quantum correlations stronger than entanglement can also be observed in macroscopic systems, thereby shedding light on the boundary between quantum and classical regimes.
We demonstrate control of a trapped-ion quantum harmonic oscillator in a squeezed Fock state basis, using engineered Hamiltonians analogous to the Jaynes-Cummings and anti-Jaynes-Cummings forms. We demonstrate that for squeezed Fock states with low n the engineered Hamiltonians reproduce the sqrt[n] scaling of the matrix elements which is typical of Jaynes-Cummings physics, and also examine deviations due to the finite wavelength of our control fields. Starting from a squeezed vacuum state, we apply sequences of alternating transfer pulses which allow us to climb the squeezed Fock state ladder, creating states up to excitations of n=6 with up to 8.7 dB of squeezing, as well as demonstrating superpositions of these states. These techniques offer access to new sets of states of the harmonic oscillator which may be applicable for precision metrology or quantum information science.
We propose a dissipative method for preparation of a maximally entangled steady state of two trapped ions in the Lamb-Dicke limit. By addressing the trapped-ion system with a monochromatic standing-wave laser pulse of frequency resonant with the ionic transition and a microwave field coupled to the ground-state transitions, we obtain an effective coupling between two particles, which is independent of the phonon-number fluctuations. Meanwhile, the controlled spontaneous emission of trapped ions is implemented via pumping the metastable states upwards to the short-lived ionic states by an auxiliary laser field. Combining the unitary processes with the engineered dissipation, a deterministic Bell state can be produced irrespective of the initial states of systems. Moreover, our result shows that the Clauser-Horne-Shimony-Holt inequality can be violated for a wide range of decoherence parameters, even at finite temperature.