Research ArticleOPTICS

Shaping quantum photonic states using free electrons

See allHide authors and affiliations

Science Advances  10 Mar 2021:
Vol. 7, no. 11, eabe4270
DOI: 10.1126/sciadv.abe4270
  • Fig. 1 Shaping photonic states of novel quantum statistics using QPINEM interactions.

    (A) Schematic for a physical realization of a QPINEM interaction. Input electron and photonic states, with specific energy distributions, are generated by the electron gun and some external light source (e.g., a laser pulse), respectively. The interaction takes place between the cavity mode’s field and the electron pulse, where the output states are now, usually, entangled. Lastly, the electron is measured by an electron energy loss spectrometer. The inset lists several optional photonic structures suitable for high-g QPINEM interactions, as in (39, 40, 58, 74). To characterize the resulting light state, one could use conventional detection schemes from quantum optics, such as coincidence counting (75) and homodyne detection (7678). (B) Interaction scheme of a single QPINEM interaction. Photonic states can be described by a half-infinite energy ladder (a quantum harmonic oscillator), with respect to a single frequency 𝜔. Electron states can be described by an infinite energy ladder with the same energy steps. Example input and output states are drawn with a photon-electron state probability map at the output, where k and n are the electron and photon number states, respectively.

  • Fig. 2 Electron interaction with a coherent state: Controlling the photonic state by postselection of the free electron.

    (A) Interaction scheme. Similar to the conventional PINEM case, our input photonic state is a coherent state ∣α⟩p and our input electron state is a baseline energy electron, denoted ∣δ⟩e ≜ ∣0⟩e. At the output of the system, we postselect a specific electron energy. (B) Output photon-electron probability map. Showing the entanglement between the two subsystems. Slices (red and purple) visualize the act of postselection, as shown in the interaction scheme. (C) Resulting photonic states after postselection. We get Poissonian-looking probabilities, around varying mean photon numbers. Bar colors match the previous plot. Red is for postselected k = − 2 and purple is for k = 2. The initial photonic state used was ∣α2 = 50⟩p, and the interaction strength gQu = 0.25i. (D and E) The same as (B) and (C) but for an interaction strength of gQu = 1i and postselected energies k = ± 6 (similar to before, red denotes electron energy loss, and purple denotes gain). We get much richer entanglement for the stronger interaction, as well as more complex photonic states, after postselection.

  • Fig. 3 Creation of a photonic Fock state.

    Iterative projection of QPINEM interactions until the desired Fock state is achieved. (A) Interaction scheme. The input photonic state is that of an empty cavity, the vacuum state. The input electron state is a delta. After each interaction, we measure the electron energy gain/loss, which equals the photonic energy lost or gained, respectively. Thus, a desired Fock state may be achieved, after enough iterations. (B) Two examples of the shaping process. The black dashed lines represent the goal Fock state. The red bars show the current photon number, per interaction, and the solid black lines show the “mean process,” if the photon could go up exactly ∣gQu2 energy steps every interaction. The left process reaches the goal state faster than the average expected growth, and the right reaches it more slowly. Both plots simulate gQu = 1i and a goal Fock state ∣Ngoal = 100⟩. For these values, we expect an average of 100 steps to achieve the goal state.

  • Fig. 4 Thermalization of photonic coherent states.

    (A) Interaction scheme. Input coherent photonic state and a delta electron. After each interaction, we trace out the electron state. (B) Photonic state evolution. Top, linear scale; bottom, log scale. Both plots depict an initial photonic coherent state ∣α2 = 10⟩ and a coupling strength gQu = 0.1i. After many interactions, the statistics converge to thermal (easily seen by the linearity under log scale). (C) Photonic state properties. Top: The effective θ = ℏω/kBT of the photonic state (computed as minus the mean slope in log scale) for different initial coherent states. For the early interactions, this effective slope has no special meaning, as the state has not converged to a thermal state yet. However, it is plotted, nevertheless, for completeness. Bottom: Mandel Q parameter (48) of the evolving states for the same three different initial coherent states. Both of these properties visibly converge to that of a true thermal state with the same 〈n〉. Both plots simulate an interaction strength of gQu = 1i.

  • Fig. 5 Displacement of photonic coherent states.

    (A) Interaction scheme. Input photonic state is a coherent state. Input electron state is a comb. After each interaction, we trace out the electron and introduce a new electron comb. (B) Photonic state evolution. Evolves from red to purple, using an electron comb of 30 states. The state maintains its Poissonian distribution quite well while gradually gaining energy from each interaction. (C) Top: The effective α evolution (computed as n) of the photonic state, which evidently grows linearly (with a slope of 0.0153), very close to the theoretical slope of βgQu = 0.0158. Bottom: The Mandel Q parameter, for different comb lengths (number of electron states), which remains very low throughout the whole process, for long enough combs. All of the simulations above are for an initial α=1000, a comb with eigenvalue β = − i, and an interaction strength gQu = 0.0158i.

  • Fig. 6 Creation of displaced Fock states.

    (A) Interaction scheme. Input photonic state is a Fock state. Input electron state is a comb. After each interaction, we trace out the electron and introduce a new electron comb. (B) Photonic state evolution. Plotted for various initial photonic Fock states, where for each initial Fock state Ni, we end up with Ni + 1 roughly Poissonian-looking peaks. In all of the above plots, the interaction strength used is gQu = 0.5i.

Supplementary Materials

  • Supplementary Materials

    Shaping quantum photonic states using free electrons

    A. Ben Hayun, O. Reinhardt, J. Nemirovsky, A. Karnieli, N. Rivera, I. Kaminer

    Download Supplement

    This PDF file includes:

    • Sections S1 to S5

    Files in this Data Supplement:

Stay Connected to Science Advances

Navigate This Article