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arXiv:2403.05471

Squeezing, trisqueezing, and quadsqueezing in a spin-oscillator system

O. Băzăvan, Sebastian Saner, Webb, D. J., E. M. Ainley, P. Drmota, D. P. Nadlinger, G. Araneda, Lucas, D. M., Christopher J. Ballance, R. Srinivas

Published Mar 8, 2024• 1 citation
arXivHTMLPDF

Abstract

Quantum harmonic oscillators model a wide variety of phenomena ranging from electromagnetic fields to vibrations of atoms in molecules. Their excitations can be represented by bosons such as photons, single particles of light, or phonons, the quanta of vibrational energy. Linear interactions that only create and annihilate single bosons can generate coherent states of light or motion. Introducing nth-order nonlinear interactions, that instead involve n bosons, leads to increasingly complex quantum behaviour. For example, second-order interactions enable squeezing, used to enhance the precision of measurements beyond classical limits, while higher-order interactions create non-Gaussian states essential for continuous-variable quantum computation. However, generating nonlinear interactions is challenging, typically requiring higher-order derivatives of the driving field or specialized hardware. Hybrid systems, where linear interactions couple an oscillator to an additional spin, offer a solution and are readily available across many platforms. Here, using the spin of a single trapped ion coupled to its motion, we employ two linear interactions to demonstrate up to fourth-order bosonic interactions; we focus on generalised squeezing interactions and demonstrate squeezing, trisqueezing, and quadsqueezing. We characterise these interactions, including their spin dependence, and reconstruct the Wigner function of the resulting states. We also discuss the scaling of the interaction strength, where we drive the quadsqueezing interaction more than 100 times faster than using conventional techniques. Our method presents no fundamental limit in the interaction order n and applies to any platform supporting spin-dependent linear interactions. Strong higher-order nonlinear interactions unlock the study of fundamental quantum optics, quantum simulation, and computation in a hitherto unexplored regime.

DOI
10.48550/arxiv.2403.05471

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Quantum harmonic oscillators model a wide variety of phenomena ranging from electromagnetic fields to vibrations of atoms in molecules. Their excitations can be represented by bosons such as photons, single particles of light, or phonons, the quanta of vibrational energy. Linear interactions that only create and annihilate single bosons can generate coherent states of light or motion. Introducing nth-order nonlinear interactions, that instead involve n bosons, leads to increasingly complex quantum behaviour. For example, second-order interactions enable squeezing, used to enhance the precision of measurements beyond classical limits, while higher-order interactions create non-Gaussian states essential for continuous-variable quantum computation. However, generating nonlinear interactions is challenging, typically requiring higher-order derivatives of the driving field or specialized hardware. Hybrid systems, where linear interactions couple an oscillator to an additional spin, offer a solution and are readily available across many platforms. Here, using the spin of a single trapped ion coupled to its motion, we employ two linear interactions to demonstrate up to fourth-order bosonic interactions; we focus on generalised squeezing interactions and demonstrate squeezing, trisqueezing, and quadsqueezing. We characterise these interactions, including their spin dependence, and reconstruct the Wigner function of the resulting states. We also discuss the scaling of the interaction strength, where we drive the quadsqueezing interaction more than 100 times faster than using conventional techniques. Our method presents no fundamental limit in the interaction order n and applies to any platform supporting spin-dependent linear interactions. Strong higher-order nonlinear interactions unlock the study of fundamental quantum optics, quantum simulation, and computation in a hitherto unexplored regime.
Quantum harmonic oscillators model phenomena from electromagnetic fields to molecular vibrations, with excitations represented by bosons such as photons or phonons. Linear interactions that create or annihilate single bosons generate coherent states of light or motion. Introducing higher-order nonlinear interactions produces richer quantum behaviour: second-order interactions enable squeezing, whereas higher-order interactions generate non-Gaussian states useful for continuous-variable quantum computation. However, such interactions are usually weak or require specialized hardware. Hybrid systems, where a linear interaction couples an oscillator to a spin, offer an alternative. Here we combine two spin-dependent linear bosonic interactions to implement up to fourth-order nonlinear bosonic interactions in a single trapped ion, focusing on generalized squeezing. We demonstrate and characterize squeezing, trisqueezing and quadsqueezing; reconstruct the Wigner functions of the resulting states; and achieve quadsqueezing over 100 times faster than conventional methods. The approach has no fundamental limit on the interaction order and applies to any platform supporting spin-dependent linear interactions.
Full coherent control and generation of nonclassical superpositions of the quantum harmonic oscillator are not only of fundamental interest but are crucial for applications in quantum simulations, quantum-enhanced metrology, and continuous-variable quantum computation. Here, we create arbitrary superpositions of nonclassical and non-Gaussian states of a quantum harmonic oscillator using the motion of a trapped ion coupled to its internal spin states. We interleave spin-dependent nonlinear interactions with midcircuit spin measurements that herald the probabilistic preparation of superpositions of squeezed, trisqueezed, and quadsqueezed states, which have not previously been experimentally demonstrated. We achieve independent control over the complex-valued squeezing parameter and the probability amplitude of each constituent, as well as their spatial separation. We directly observe the nonclassical nature of these states in the form of Wigner negativity following a full-state reconstruction. Our methods apply to any system where a quantum harmonic oscillator is coupled to a spin.
Theoretical models of spins coupled to bosons provide a simple setting for studying a broad range of important phenomena in many-body physics, from virtually mediated interactions to decoherence and thermalization. In many atomic, molecular, and optical systems, such models also underlie the most successful attempts to engineer strong, long-ranged interactions for the purpose of entanglement generation. Especially when the coupling between the spins and bosons is strong, such that it cannot be treated perturbatively, the properties of such models are extremely challenging to calculate theoretically. Here, exact analytical expressions for nonequilibrium spin-spin correlation functions are derived for a specific model of spins coupled to bosons. The spatial structure of the coupling between spins and bosons is completely arbitrary, and thus the solution can be applied to systems in any number of dimensions. The explicit and nonperturbative inclusion of the bosons enables the study of entanglement generation (in the form of spin squeezing) even when the bosons are driven strongly and near resonantly, and thus provides a quantitative view of the breakdown of adiabatic elimination that inevitably occurs as one pushes towards the fastest entanglement generation possible. The solution also helps elucidate the effect of finite temperature on spin squeezing. The model considered is relevant to a variety of atomic, molecular, and optical systems, such as atoms in cavities or trapped ions. As an explicit example, the results are used to quantify phonon effects in trapped ion quantum simulators, which are expected to become increasingly important as these experiments push towards larger numbers of ions.
Mechanical degrees of freedom are natural candidates for continuous-variable quantum information processing and bosonic quantum simulations. These applications, however, require the engineering of squeezing and nonlinearities in the quantum regime. Here we demonstrate ground state squeezing of a gigahertz-frequency mechanical resonator coupled to a superconducting qubit. This is achieved by parametrically driving the qubit, which results in an effective two-phonon drive. In addition, we show that the resonator mode inherits a nonlinearity from the off-resonant coupling with the qubit, which can be tuned by controlling the detuning. We thus realize a mechanical squeezed Kerr oscillator, where we demonstrate the preparation of non-Gaussian quantum states of motion with Wigner function negativities and high quantum Fisher information. This shows that our results also have applications in quantum metrology and sensing.
Using discrete and continuous variable subsystems, hybrid approaches to quantum information could enable more quantum computational power for the same physical resources. Here, we propose a hybrid scheme that can be used to generate the necessary Gaussian and non-Gaussian operations for universal continuous variable quantum computing in trapped ions. This scheme utilizes two linear spin-motion interactions to generate a broad set of nonlinear effective spin-motion interactions including one- and two-mode squeezing, beam splitter, and trisqueezing operations in trapped ion systems. We discuss possible experimental implementations using laser-based and laser-free approaches.
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.