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arXiv:1909.02395 · Physical Review A

Numerical study of Wigner negativity in one-dimensional steady-state resonance fluorescence

Ingrid Strandberg, Yong Lu, Fernando Quijandría, Göran Johansson

Published Dec 4, 2019• 21 citations
arXivHTMLPDF

Abstract

In a numerical study, we investigate the steady-state generation of nonclassical states of light from a coherently driven two-level atom in a one-dimensional waveguide. Specifically, we look for states with a negative Wigner function, since such nonclassical states are a resource for quantum information processing applications, including quantum computing. We find that a waveguide terminated by a mirror at the position of the atom can provide Wigner-negative states, while an infinite waveguide yields strictly positive Wigner functions. Moreover, our paper reveals a connection between the purity of a quantum state and its Wigner negativity. We also analyze the effects of decoherence on the negativity of a state.

DOI
10.1103/physreva.100.063808

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In a numerical study, we investigate the steady-state generation of nonclassical states of light from a coherently driven two-level atom in a one-dimensional waveguide. Specifically, we look for states with a negative Wigner function, since such nonclassical states are a resource for quantum information processing applications, including quantum computing. We find that a waveguide terminated by a mirror at the position of the atom can provide Wigner-negative states, while an infinite waveguide yields strictly positive Wigner functions. Moreover, our paper reveals a connection between the purity of a quantum state and its Wigner negativity. We also analyze the effects of decoherence on the negativity of a state.
In this work we demonstrate numerically that the nonlinearity provided by a continuously driven two-level system allows for the generation of Wigner-negative states of the electromagnetic field confined in one spatial dimension. Wigner-negative states, also known as Wigner nonclassical states, are desirable for quantum information protocols beyond the scope of classical computers. Focusing on the steady-state emission from the two-level system, we find the largest negativity at the drive strength where the coherent reflection vanishes.
Propagating modes of light with negative-valued Wigner distributions are of fundamental interest in quantum optics and represent a key resource in the pursuit of optics-based quantum information technologies. Most schemes proposed or implemented for the generation of such modes are probabilistic in nature and rely on heralding by detection of a photon or on conditional methods where photons are separated from the original field mode by a beam splitter. In this Letter we demonstrate theoretically, using a cascaded-quantum-systems model, the possibility of deterministic generation of Wigner-negativity in temporal modes of the steady-state emission of a two-level system driven by finite-bandwidth quadrature-squeezed light. Optimal negativity is obtained for a squeezing bandwidth similar to the linewidth of the transition of the two-level system. While the Wigner distribution associated with the incident squeezed light is Gaussian and everywhere positive, the Wigner functions of the outgoing temporal modes show distinct similarities and overlap with a superposition of displaced squeezed states.
We experimentally demonstrate the steady-state generation of propagating Wigner-negative states from a continuously driven superconducting qubit. We reconstruct the Wigner function of the radiation emitted into propagating modes defined by their temporal envelopes, using digital filtering. For an optimized temporal filter, we observe a large Wigner logarithmic negativity, in excess of 0.08, in agreement with theory. The fidelity between the theoretical predictions and the states generated experimentally is up to 99%, reaching state-of-the-art realizations in the microwave frequency domain. Our results provide a new way to generate and control nonclassical states, and may enable promising applications such as quantum networks and quantum computation based on waveguide quantum electrodynamics.
The output field from a continuously driven linear parametric oscillator may exhibit considerably more squeezing than the intracavity field.Inspired by this fact, we explore the nonclassical features of the steady-state output field of a driven nonlinear Kerr parametric oscillator using a temporal wave packet mode description.Utilizing a new numerical method, we have access to the density matrix of arbitrary wave packet modes.Remarkably, we find that even though the steady-state cavity field is always characterized by a positive Wigner function, the output may exhibit Wigner negativity, depending on the properties of the selected mode.
Generating the negative Wigner functions where the corresponding Wigner states are nonclassical has been recognized as a powerful tool for successfully performing quantum information and computing protocols beyond the scope of classical computers. Here, we present the possibility to generate and engineer the negative Wigner function at a steady state using parametric (two-photon) driving and homodyne-based feedback in a quantum van der Pol (vdP) oscillator. Specifically, we employ a quantum master equation approach for calculating the Wigner function of the vdP oscillator field in phase space and, furthermore, quantifying its negativity content. We clearly show that the negative-value magnitudes, regions, and shapes of the Wigner function can be effectively tuned by the parametric driving phase and the parametric driving amplitude, as well as the feedback coefficient within a large range. We identify different contributions of these involved parameters to the Wigner-function negativity. In the present scheme, more complex quantum coherence and interference phenomena are introduced via the parametric driving and feedback, which stabilizes the phase of the vdP oscillator field and renders the capability to generate the negative Wigner function. Therefore, the enhanced Wigner-function negativity can be achieved under these optimized system parameters. Our in-depth study provides insight into the formation and in situ control of the desirable Wigner nonclassical states. The obtained results are not limited to the vdP oscillator systems and should be generally applicable to other coherent coupled systems within the reach of modern experimental facilities.
Negativity of the Wigner function is arguably one of the most striking non-classical features of quantum states. Beyond its fundamental relevance, it is also a necessary resource for quantum speedup with continuous variables. As quantum technologies emerge, the need to identify and characterize the resources which provide an advantage over existing classical technologies becomes more pressing. Here we derive witnesses for Wigner negativity of single mode and multimode quantum states, based on fidelities with Fock states, which can be reliably measured using standard detection setups. They possess a threshold expectation value indicating whether the measured state has a negative Wigner function. Moreover, the amount of violation provides an operational quantification of Wigner negativity. We phrase the problem of finding the threshold values for our witnesses as an infinite-dimensional linear optimisation. By relaxing and restricting the corresponding linear programs, we derive two hierarchies of semidefinite programs, which provide numerical sequences of increasingly tighter upper and lower bounds for the threshold values. We further show that both sequences converge to the threshold value. Moreover, our witnesses form a complete family – each Wigner negative state is detected by at least one witness – thus providing a reliable method for experimentally witnessing Wigner negativity of quantum states from few measurements. From a foundational perspective, our findings provide insights on the set of positive Wigner functions which still lacks a proper characterisation.