Researchers for Quantum Electronics at ETH Zürich have, for the first time, experimentally generated Schrödinger’s cat manifolds containing two, three, four and five components within a single trapped-ion oscillator. The team achieved this control by harnessing strongly non-linear atom-light interactions, a technique that typically hinders precise control protocols. This work demonstrates a novel form of non-linear reservoir engineering, opening up a new toolbox applicable to bosonic quantum error correction, computation, and sensing. The results represent the first experimental use of such high-order non-linear processes for controlling non-classical states of a quantum harmonic oscillator.
Non-Linear Reservoir Engineering Stabilizes Schrödinger’s Cat Manifolds
Generating Schrödinger’s cat manifolds with 2, 3, 4, and 5 components is now experimentally demonstrated in a single trapped-ion oscillator, a feat previously unseen in mechanical systems. This work extends beyond the standard two-component “cat” state, pushing the boundaries of quantum state manipulation and opening avenues for more sophisticated quantum technologies. The team specifically designed Hamiltonians to enable measurements of these generated state manifolds, a critical step in verifying and characterizing these fragile quantum systems.
This approach introduces a resource of non-linearity distinct from spin, allowing for the realization of high-order boson processes selected through resonance conditions. Initial demonstrations focused on cooling the motional mode into manifolds of non-Gaussian states via this non-linear reservoir engineering, simultaneously driving both excitation raising and lowering resonant sidebands while optically pumping the spin.
The resulting master equation describes the evolution of the motion, incorporating a Lindblad jump operator that combines excitation and lowering processes of varying orders. Fidelity measurements between reconstructed and simulated density matrices confirm the accuracy of the technique, yielding 89.7(4)%, 86(1)%, 87(1)%, and 82(2)% for configurations (r,l) of (0,2), (1,2), (1,3), and (2,3) respectively, with uncertainties determined through bootstrapping.
The negativity of the reconstructed Wigner functions further validates the generation of non-Gaussian states. For (r,l) = (1,2), (1,3), and (2,3), the steady state distribution occupies a subset of the total dark states, 2, 3, and 3 respectively, consistent with the symmetry of the operator used in the experiment.
Strongly Non-Linear Atom-Light Coupling Enables State Control
This advancement relies on a specific approach to harnessing strongly non-linear atom-light interactions, a regime where control protocols are typically more difficult to design and implement. The team’s work departs from traditional approaches by operating the trapped-ion setup outside the linearized Lamb-Dicke regime, fully using the non-linear nature of the atom-light coupling. This allowed for the production and control of stabilized manifolds of quantum states, a significant step toward more complex quantum systems.
The experimental setup also enabled the selection of specific Hamiltonians, designed to facilitate measurements of these generated state manifolds, and to map the symmetry of the underlying state onto a spin readout. This precise control extended to demonstrating a n mod 3 measurement, which purified mixtures of three-component manifolds.
Analysis of the coupling strength revealed that high-order non-linear processes can be strongly driven outside the linearized regime, as evidenced by data showing boson processes from order -4 to +4. Steady-state Fock state distributions were experimentally verified, with error bars representing the standard error of the mean, and corroborated by simulations of the Wigner functions. The researchers also demonstrated control over the manifold of states, obtaining steady-state Fock distributions in red and blue, respectively, through non-linear reservoir engineering with parameters (r,l) = (1,2).
Stabilization of 3-, 4-, and 5-Component Cat States
Beyond the established two-component Schrödinger’s cat state, researchers have now stabilized quantum harmonic oscillator manifolds containing three, four and five components, a first for mechanical systems. This achievement relies on a specific implementation of non-linear reservoir engineering, allowing for the preparation of these complex states with rotational symmetries dictated by the drive Hamiltonian. The ability to create and control these higher-order cat states expands the potential for applications in quantum error correction, computation, and sensing.
The process uses non-linearities extending to higher orders, unlocking a range of effects for quantum state control and stabilization. A key element is the use of an effective two-level spin derived from a linear coupling to an ancillary system with anharmonic levels selected through resonance conditions. Analysis of the system’s evolution reveals that, after stabilization, the probability of revival is dependent on the time and the range of occupied Fock states.
Extended data shows the spin population as a function of stabilization time, with error bars indicating statistical uncertainty. The researchers observed that over longer timescales, extending to a few milliseconds, population leaks from state |ψ₂| into states |ψ₀| and |ψ₁|. The uncertainty range is calculated as the standard deviation of bootstrap sample fidelities, expressed as σF = √(sum_i(F^i-F)²/B).
Lamb-Dicke Regime Limits Higher-Order Non-Linearity
Operating beyond the conventional Lamb-Dicke regime, where light wavelength significantly exceeds oscillator wavepacket size, allows for stronger coupling in boson processes, a detail often considered a limitation for quantum oscillator control. This approach, detailed in recent work, instead uses these high-order processes to drive non-linear interactions, opening pathways to manipulate quantum states in ways previously inaccessible. The ability to operate outside the Lamb-Dicke regime is particularly relevant for realizing multi-component Schrödinger’s cat codes, which are promising candidates for correcting oscillator dephasing errors.
These codes have remained unrealized until now, and the demonstrated method offers a new route to their implementation. By using non-Lamb-Dicke terms as coherent Hamiltonians, researchers can generalize Kerr-type Hamiltonians, expanding the possibilities for quantum control. While operation in this non-linear regime presents challenges in protocol design, likely requiring heuristic or numerically optimized methods, the benefits in terms of state manipulation are substantial. The work demonstrates a departure from analytical simplicity, but the resulting control over the quantum harmonic oscillator is significant.
Bessel Functions Describe Resonant Sideband Coupling
Bessel functions provide the mathematical description for coupling resonant sidebands, a critical element in controlling a quantum harmonic oscillator operating far from the standard linearized regime. Beyond the typical Lamb-Dicke regime, where light fields are considered linear, the researchers found that the coupling strength of boson processes up to fourth order demonstrably increases.
This observation is formalized by a Hamiltonian describing a single resonant sideband driving Rabi oscillations between states, with a rate dependent on the Fock occupation number k. The rate, g f(k), is a function of the Fock occupation, allowing for tailored control over the quantum system. To characterize the motional-state populations, the researchers drove a single resonant sideband for varying durations and measured the resulting spin-excitation probability.
Error bars, propagated from the fitting process, correspond to the standard error of the mean (SEM), providing a quantitative measure of uncertainty in the extracted parameters. “Error bars are propagated from the fit and correspond to the SEM,” the researchers state, highlighting the rigor of their statistical analysis.
Non-Gaussian States Expand Quantum Technology Applications
The ability to independently vary both the number and squeezing of quantum states, from 5.1 to 10.8 and 0.1 to 1.1 respectively, represents a significant expansion of control over quantum harmonic oscillators, a development with implications for quantum technologies. The work demonstrates a pathway to generating non-Gaussian states, essential resources for realizing the full potential of continuous variable quantum computers and advanced sensors.
The team’s approach allows for the creation of localized multi-component Schrödinger’s cat manifolds, states that exhibit more exotic quantum mechanical features than those described by Gaussian statistics. These states are important for quantum error correction with cat codes, where number squeezing has been shown to reduce logical error rates, a key challenge in building stable quantum computers.
The method employed relies on a parity measurement technique, deriving a revival time for parity readout using non-linear interaction between the quantum harmonic oscillator and a spin. By imposing a sign constraint on undetermined coherences, specifically, Im(ρ_(kd,(k + 1)d)) = + √(ρ_(kd, kd)) √(ρ_((k + 1)d, (k + 1)d)), researchers were able to identify the manifold with θ = 0. This selection process is critical for accurate state characterization and manipulation. The demonstrated control over both squeezing and number of photons within the oscillator’s steady state promises advancements in probing fundamental physics and information technologies.
n mod 3 Measurement Purifies Multi-Component Manifolds
A measurement technique using the remainder of a division by three, n mod 3, allows for purification of complex quantum states, specifically mixtures of three-component Schrödinger’s cat manifolds. This purification process relies on discerning subtle relationships within the density matrix of the quantum system, revealing information about coherences that would otherwise remain hidden. Researchers demonstrated this capability by applying Hamiltonians comprised of a single high-order resonant process, exhibiting a near-linear dependence on matrix elements across the occupied states and enabling a mapping of state symmetry onto a spin readout.
The team’s approach focuses on extracting information from even and odd coherences within the density matrix, using the real and imaginary parts of a non-linear characteristic function, ξ(α), as a diagnostic tool. Measurement of the real component of ξ(α) reveals information only about even off-diagonal coherences, while the imaginary component exposes odd coherences.
This selective access to coherence information is critical, as it allows for identification of a d-symmetric manifold, a state with ‘d’ components, up to a rotation of π/d through the measurement of Re[ξ(α)]. The researchers detail how knowledge of Re[ξ(α)] determines the steady state of the non-linear reservoir engineering operator for specific parameters, up to a phase-space rotation of π/3. This method extends beyond simply characterizing the states; it actively purifies mixtures.
The work demonstrates that a single eigenstate can be selected via measurement, with a resulting upward shift of four phonons due to the use of a fourth-order sideband. Post-selection involves ensuring no photon scattering during detection, requiring a spin-inversion prior to measurement when targeting a specific state. The equation Re[ξ_(i,i + d)(α)] = (-1)^dRe[ξ_(i + d,i)(α)] emphasises the symmetry inherent in the measurement process, allowing for the extraction of information from a limited set of matrix elements. The researchers state, highlighting the mathematical foundation of their purification technique.
Trapped-Ion Oscillator and Coupling Hamiltonian Details
Earlier theoretical work considered this non-linear nature, but experimental realization of high-order non-linearities with controllable quantum effects has remained a challenge across multiple platforms. The strength of the coupling between Fock states, quantified by Bessel functions, differs significantly depending on the Lamb-Dicke parameter, η, which represents the ratio of the wavevector to the oscillator’s zero-point extent. When η is small, approximating the light field as linear across the oscillator wavefunction suppresses higher-order processes; however, with η reaching 0.5 in this experiment, these processes exhibit non-negligible coupling.
The experimental setup centers on a quantized mechanical oscillator, the motion of a trapped ion along a magnetic field axis with a frequency of approximately 5.3 MHz, and the ion’s spin, encoded in two energy levels denoted |g⟩ and |e⟩. A pair of orthogonal Raman beams at 313 nm mediates the spin-oscillator coupling, establishing an η value around 0.5.
Optical pumping from the |e⟩ to |g⟩ state, facilitated by a short-lived internal state, ensures the system remains prepared for subsequent manipulation. The matrix elements governing resonant sideband coupling between Fock states are directly determined by these Bessel functions, revealing the precise strength of the non-linear interactions.
See today’s quantum computing news on Quantum Zeitgeist for the latest breakthroughs in qubits, hardware, algorithms, and industry deals.
