Radim Filip

Nonlinear phase gates are essential to achieve the universality of continuous-variable quantum processing and its applications. We present a deterministic protocol for generating nonlinear phase gates in trapped ion systems using simultaneous two-tone sideband drives beyond the Lamb-Dicke regime. Our approach harnesses higher-order interaction terms typically neglected or suppressed to construct nonlinear phase gates. This methodology enables high-fidelity gate engineering with a near three-fold reduction in control pulses compared to state-of-the-art theoretical proposals.

Distributed quantum sensing leverages quantum correlations among multiple sensors to enhance the precision of parameter estimation beyond classical limits. Most existing approaches target phase estimation and rely on a shared phase reference between the signal and the probe, yet many relevant scenarios deal with regimes where such a reference is absent, making the estimation of force or field amplitudes the main task. We study this phase-insensitive regime for bosonic sensors that undergo identical displacements with common phases randomly varying between experimental runs. We derive analytical bounds on the achievable precision and show that it is determined by first-order normal correlations between modes in the probe state, constrained by their average excitations. These correlations yield a collective sensitivity enhancement over the standard quantum limit, with a gain that grows linearly in the total excitation number, revealing a distributed quantum advantage even without a global phase reference. We identify families of multimode states with definite joint parity that saturate this limit and can be probed efficiently via local parity measurements already demonstrated or emerging in several quantum platforms. We further demonstrate that experimentally relevant decoherence channels favor two distinct sensing strategies: splitting of a single-mode nonclassical state among the modes, which is robust to loss and heating, and separable probes, which are instead resilient to dephasing and phase jitter. Our results are relevant to multimode continuous platforms, including trapped-ion, solid-state mechanical, optomechanical, superconducting, and photonic systems.

Quantum interference between interacting systems is fundamental to basic science and quantum technology, but it typically requires precise control of the interaction phases of lasers or microwave generators. Can interference be observed if those interaction phases are stable but unknown, usually prohibitive for complex state without active control? Here, we answer this question by experimentally preparing a Schrödinger-cat-like state of an internal qubit and a motional oscillator of a trapped $^{40}$Ca$^{+}$ ion, and its robustness to such uncontrolled phase. By applying alternating red and blue sideband pulses, we enforce a strict qubit-parity correlation and interference inherently insensitive to stable but unknown phases of the driving laser. For this qubit-parity interference, we use a minimal two-pulse interferometric sequence to demonstrate characteristic visibilities of $20\%$ and $40\%$, which approach the theoretical visibility limit, providing a scalable coherence witness without full state tomography for high-dimensional states.

We analyze dynamics of the Rabi model describing interactions of a qubit with a single soft-mode oscillator. We show that the model with a slightly violated parity generates symmetric Schr{ö}dinger cat states of the oscillator, which suddenly perish in a spontaneous unitary process similar to the measurement-induced wave-function collapse. The effect is sufficiently robust to be tested experimentally, e.g., with trapped ions, macroscopic mechanical oscillators or superconducting circuits.

Nonlinear processes with individual quanta beyond bilinear interactions are essential for quantum technology with bosonic systems. Diverse coherent splitting and merging of quanta in them already manifest in the estimation of their nonlinear coupling from observed statistics. We derive non-trivial, but optimal strategies for sensing the basic and experimentally available trilinear interactions using non-classical particle-like Fock states as a probe and feasible measurement strategies. Remarkably, the optimal probing of nonlinear coupling reaches estimation errors scaled down with $N^{-1/3}$ for overall $N$ of quanta in specific but available high-quality Fock states in all interacting modes. It can reveal unexplored aspects of nonlinear dynamics relevant to using such nonlinear processes in bosonic experiments with trapped ions and superconducting circuits and opens further developments of quantum technology with them.

Quantum simulations provide means to probe challenging problems within controllable quantum systems. However, implementing or simulating deep-strong nonlinear couplings between bosonic oscillators on physical platforms remains a challenge. We present a deterministic simulation technique that efficiently and accurately models nonlinear bosonic dynamics. This technique alternates between tunable Rabi and SNAP gates, both of which are available on experimental platforms such as trapped ions and superconducting circuits. Our proposed simulation method facilitates high-fidelity modeling of phenomena that emerge from higher-order bosonic interactions, with an exponential reduction in resource usage compared to other techniques. We demonstrate the potential of our technique by accurately reproducing key phenomena and other distinctive characteristics of ideal nonlinear optomechanical systems. Our technique serves as a valuable tool for simulating complex quantum interactions, simultaneously paving the way for new capabilities in quantum computing through the use of hybrid qubit-oscillator systems.

We show that the methods for quantification of system-environment entanglement that were recently developed for interactions that lead to pure decoherence of the system can be straightforwardly generalized to time-dependent Hamiltonians of the same type. This includes the if-and-only-if criteria of separability, as well as the entanglement measure applicable to qubit systems, and methods of detection of entanglement by operations and measurements performed solely on the system without accessing the environment. We use these methods to study the nature of the decoherence of a qubit-oscillator system. Qubit-oscillator entanglement is essential for developing bosonic quantum technology with quantum non-Gaussian states and its applications in quantum sensing and computing. The dominating bosonic platforms, trapped ions, electromechanics, and superconducting circuits, are based on the time-dependent gates that use such entanglement to achieve new quantum sensors and quantum error correction. The step-like time-dependence of the Hamiltonian that is taken into account allows us to capture complex interplay between the build-up of classical and quantum correlations, which could not be replicated in time-independent scenarios.

Quanta splitting is an essential generator of Gaussian entanglement, exemplified by Einstein-Podolsky-Rosen states and apparently the most commonly occurring form of entanglement. In general, it results from the strong pumping of a nonlinear process with a highly coherent and low-noise external drive. In contrast, recent experiments involving efficient trilinear processes in trapped ions and superconducting circuits have opened the complementary possibility to test the splitting of a few thermal quanta. Stimulated by such small thermal energy, a strong degenerate trilinear coupling generates large amounts of nonclassicality, detectable by more than 3 dB of distillable quadrature squeezing. Substantial entanglement can be generated via frequent passive linear coupling to a third mode present in parallel with the trilinear coupling. This new form of entanglement, outside any Gaussian approximation, surprisingly grows with the mean number of split thermal quanta; a quality absent from Gaussian entanglement. Using distillable squeezing we shed light on this new entanglement mechanism for nonlinear bosonic systems.

Quantum non-Gaussian states of photons and phonons are conclusive and direct witnesses of higher-than-quadratic nonlinearities in optical and mechanical processes. Moreover, they are proven resources for quantum sensing, communication and error correction with diverse continuous-variable systems. This review introduces theoretical analyses of nonclassical and quantum non-Gaussian states of photons and phonons. It recapitulates approaches used to derive operational criteria for photons tolerant to optical losses, their application in experiments and their nowadays extension to quantum non-Gaussian photon coincidences. It extends to a recent comparison of quantum non-Gaussianity, including robustness to thermal noise, and sensing capability for high-quality phononic Fock states of single trapped cooled ions. The review can stimulate further development in the criteria of quantum non-Gaussian states and experimental effort to prepare and detect such useful features, navigating the community to advanced quantum physics and technology.

The precise determination of a displacement of a mechanical oscillator or a microwave field in a predetermined direction in phase space can be carried out with trapped ions or superconducting circuits, respectively, by coupling the oscillator with ancilla qubits. Through that coupling, the displacement information is transferred to the qubits which are then subsequently read out. However, unambiguous estimation of displacement in an unknown direction in the phase space has not been attempted in such oscillator-qubit systems. Here, we propose a hybrid oscillator-qubit interferometric setup for the unambiguous estimation of phase space displacements in an arbitrary direction, based on feasible Rabi interactions beyond the rotating-wave approximation. Using such a hybrid Rabi interferometer for quantum sensing, we show that the performance is superior to the ones attained by single-mode estimation schemes and a conventional interferometer based on Jaynes-Cummings interactions. Moreover, we find that the sensitivity of the Rabi interferometer is independent of the thermal occupation of the oscillator mode, and thus cooling it to the ground state before sensing is not required. We also perform a thorough investigation of the effect of qubit dephasing and oscillator thermalization. We find the interferometer to be fairly robust, outperforming different benchmark estimation schemes even for large dephasing and thermalization.

Quantum non-Gaussian mechanical states from inherently nonlinear quantum processes are already required in a range of applications spanning from quantum sensing up to quantum computing with continuous variables. The discrete building blocks of such states are the energy eigenstates - Fock states. Despite the progress in their preparation, the remaining imperfections can still invisibly cause loss of the critical quantum non-Gaussian aspects of the phonon distribution relevant in the applications. We derive the most challenging hierarchy of quantum non-Gaussian criteria for the individual mechanical Fock states and demonstrate its implementation on the characterization of single trapped-ion oscillator states up to 10~phonons. We analyze the depth of quantum non-Gaussian features under mechanical heating and predict their application in quantum sensing. These results uncover that the crucial quantum non-Gaussian features are demanded to reach quantum advantage in the applications.

Laser cooled ions trapped in a linear Paul trap are long-standing ideal candidates for realizing quantum simulation, especially of many-body systems. The properties that contribute to this also provide the opportunity to demonstrate unexpected quantum phenomena in few-body systems. A pair of ions interacting in such traps exchange vibrational quanta through the Coulomb interaction. This linear interaction can be anharmonically modulated by an elementary coupling to the internal two-level structure of one of the ions. Driven by thermal energy in the passively coupled oscillators, which are themselves coupled to the internal ground states of the ions, the nonlinear interaction autonomously and unconditionally generates entanglement between the mechanical modes of the ions. We examine this counter-intuitive thermally induced entanglement for several experimentally feasible model systems, and propose parameter regimes where state of the art trapped ion systems can produce such phenomena. In addition, we demonstrate a multiqubit enhancement of such thermally induced entanglement.

We present a protocol for transferring arbitrary continuous-variable quantum states into a few discrete-variable qubits and back. The protocol is deterministic and utilizes only two-mode Rabi-type interactions which are readily available in trapped-ion and superconducting circuit platforms. The inevitable errors caused by transferring an infinite-dimensional state into a finite-dimensional register are suppressed exponentially with the number of qubits. Furthermore, the encoded states exhibit robustness against noise, such as dephasing and amplitude damping, acting on the qubits. Our protocol thus provides a powerful and flexible tool for discrete-continuous hybrid quantum systems.

Squeezed states of harmonic oscillators are a central resource for continuous-variable quantum sensing, computation and communication. Here we propose a method for the generation of very good approximations to highly squeezed vacuum states with low excess anti-squeezing using only a few oscillator-qubit coupling gates through a Rabi-type interaction Hamiltonian. This interaction can be implemented with several different methods, which has previously been demonstrated in superconducting circuit and trapped-ion platforms. The protocol is compatible with other protocols manipulating quantum harmonic oscillators, thus facilitating scalable continuous-variable fault-tolerant quantum computation.

We devise a new class of criteria to certify the nonclassicality of photon- and phonon-number statistics. Our criteria extend and strengthen the broadly used Klyshko's criteria, which require knowledge of only a finite set of Fock-state probabilities. This makes the criteria well-suited to experimental implementation in realistic conditions. Moreover, we prove the completeness of our method in some scenarios, showing that, when only two or three Fock-state probabilities are known, it detects all finite distributions incompatible with classical states. In particular, we show that our criteria detect a broad class of noisy Fock states as nonclassical, even when Klyshko's do not. The method is directly applicable to trapped-ion, superconducting circuits, and optical and optomechanical experiments with photon-number resolving detectors. This work represents a significant milestone towards a complete characterisation of the nonclassicality accessible from limited knowledge of the Fock-state probabilities.

Quantum computing potentially offers exponential speed-ups over classical computing for certain tasks. A central, outstanding challenge to making quantum computing practical is to achieve fault tolerance, meaning that computations of any length or size can be realised in the presence of noise. The Gottesman-Kitaev-Preskill code is a promising approach towards fault-tolerant quantum computing, encoding logical qubits into grid states of harmonic oscillators. However, for the code to be fault tolerant, the quality of the grid states has to be extremely high. Approximate grid states have recently been realized experimentally, but their quality is still insufficient for fault tolerance. Current implementable protocols for generating grid states rely on measurements of ancillary qubits combined with either postselection or feed forward. Implementing such measurements take up significant time during which the states decohere, thus limiting their quality. Here we propose a measurement-free preparation protocol which deterministically prepares arbitrary logical grid states with a rectangular or hexagonal lattice. The protocol can be readily implemented in trapped-ion or superconducting-circuit platforms to generate high-quality grid states using only a few interactions, even with the noise levels found in current systems.

The demonstration of optical multipath interference from a large number of quantum emitters is essential for the realization of many paradigmatic experiments in quantum optics. However, such interference remains still unexplored as it crucially depends on the sub-wavelength positioning accuracy and stability of all emitters. We present the observation of controlled interference of light scattered from strings of up to 53~trapped ions. The light scattered from ions localized in a harmonic trapping potential is collected along the ion crystal symmetry axis, which guarantees the spatial indistinguishability and allows for an efficient scaling of the contributing ion number. We achieve the preservation of the coherence of scattered light observable for all the measured string sizes and nearly-optimal enhancement of phase sensitivity. The presented results will enable realization and control of directional photon emission, direct detection of enhanced quadrature squeezing of atomic resonance fluorescence, or optical generation of genuine multi-partite entanglement of atoms.

The vast majority of physical objects we are dealing with are almost exclusively made of atoms. Due to their discrete level structure, single atoms have proved to be emitters of light which is incompatible with the classical description of electromagnetic waves. We demonstrate this incompatibility for atomic fluorescence when scaling up the size of the source ensemble, which consists of trapped atomic ions, by several orders of magnitude. The presented measurements of nonclassical statistics on light unconditionally emitted from ensembles containing up to more than a thousand ions promise further scalability to much larger emitter numbers. The methodology can be applied to a broad range of experimental platforms focusing on the bare nonclassical character of single isolated emitters.

Single atoms or atom-like emitters are the purest source of on-demand single photons, they are intrinsically incapable of multi-photon emission. To demonstrate this degree of purity we have realized a tunable, on-demand source of single photons using a single ion trapped at the common focus of high numerical aperture lenses. Our trapped-ion source produces single-photon pulses at a rate of 200 kHz with g$^2(0) = (1.9 \pm 0.2) \times 10^{-3}$, without any background subtraction. The corresponding residual background is accounted for exclusively by detector dark counts. We further characterize the performance of our source by measuring the violation of a non-Gaussian state witness and show that its output corresponds to ideal attenuated single photons. Combined with current efforts to enhance collection efficiency from single emitters, our results suggest that single trapped ions are not only ideal stationary qubits for quantum information processing, but promising sources of light for scalable optical quantum networks.

A generic model of measurement device which is able to directly measure commonly used quantum-state characteristics such as fidelity, overlap, purity and Hilbert-Schmidt distance for two general uncorrelated mixed states is proposed. In addition, for two correlated mixed states, the measurement realizes an entanglement witness for Werner's separability criterion. To determine these observables, the estimation only one parameter - the visibility of interference, is needed. The implementations in cavity QED, trapped ion and electromagnetically induced transparency experiments are discussed.