Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                

Experimental investigation of a multiphoton Heisenberg-limited interferometric scheme: The effect of imperfections

Shakib Daryanoosh, Geoff J. Pryde, Howard M. Wiseman, and Sergei Slussarenko
Phys. Rev. A 110, 012614 – Published 19 July 2024
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • THEORY
  • THE EFFECT OF IMPERFECTIONS
  • EXPERIMENTAL REALIZATION OF THE PROBE…
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References
    This article was published on 19 July 2024. Please update your links.

    Abstract

    Interferometric phase estimation is an essential tool for precise measurements of quantities such as displacement, velocity, and material properties. The lower bound on measurement uncertainty achievable with classical resources is set by the shot-noise limit (SNL) that scales asymptotically as 1/N, where N is the number of resources used. The experiment of Daryanoosh et al. [Nat. Commun. 9, 4606 (2018)] showed how to achieve the ultimate precision limit, the exact Heisenberg limit (HL), in ab initio phase estimation with N=3 photon-passes, using an entangled biphoton state in combination with particular measurement techniques. The advantage of the HL over the SNL increases with the number of resources used. Here we present, and implement experimentally, a scheme for generation of the optimal N=7 triphoton state. We study experimentally and theoretically the generated state quality and its potential for phase estimation. We show that the expected usefulness of the prepared triphoton state for HL phase estimation is significantly degraded by even quite small experimental imperfections, such as optical mode mismatch and unwanted higher-order multiphoton terms in the states produced in parametric downconversion.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    9 More
    • Received 4 April 2024
    • Revised 3 July 2024
    • Accepted 3 July 2024

    DOI:https://doi.org/10.1103/PhysRevA.110.012614

    ©2024 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Quantum circuitsQuantum computationQuantum controlQuantum engineeringQuantum feedbackQuantum information processingQuantum metrologyQuantum parameter estimation
    Quantum Information, Science & Technology

    Authors & Affiliations

    Shakib Daryanoosh*, Geoff J. Pryde, Howard M. Wiseman†, and Sergei Slussarenko‡

    • Centre for Quantum Dynamics and Centre for Quantum Computation and Communication Technology, Griffith University, Yuggera Country, Brisbane, Queensland 4111, Australia

    • *Contact author: shakib.daryanoosh@uwa.edu.au
    • †Contact author: h.wiseman@griffith.edu.au
    • ‡Contact author: s.slussarenko@griffith.edu.au

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 110, Iss. 1 — July 2024

    Reuse & Permissions
    Access Options
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      (a) Schematic representation of the modified Mach-Zehnder interferometer allowing for multiple applications of the phase element ϕ. The optical mode in path “I” passes multiple (p) times (here p=4) such that a total phase shift of pϕ is acquired. Also the reference phase in path “II” is set so that it imparts θ phase shift. (b) Quantum circuit illustration for the interferometric phase estimation scheme with one qubit using the interferometer from (a).

      Reuse & Permissions
    • Figure 2
      Figure 2

      (a) Circuit diagram for the Heisenberg-limited interferometric phase estimation binary encoding for probe qubits with N=7 quantum resources. (b) State preparation for the QPEA: a Hadamard operation is applied on each qubit in the same way as depicted in Fig. 1. (c) Circuit representation for creating the three-photon optimal state, Eq. (10). The input state, |ψin〉=|ψa〉|ψbc〉, is transformed into the optimal state, after application of two consecutive cnot gates.

      Reuse & Permissions
    • Figure 3
      Figure 3

      Probability distribution function for (a) the QPEA and (b) the HPEA for two different numbers of quantum resources N=3 (dashed green) and N=7 (solid gold). It can be clearly seen that employing more resources results in a localized distribution function around ϕest=ϕ. Optimizing the input state to the QPEA, the impact of rather high tails on phase estimation can be alleviated.

      Reuse & Permissions
    • Figure 4
      Figure 4

      Conceptual circuit diagram for creating the optimal three-photon state, Eq. (10), consisting of two probabilistic cnot gates. The dashed-red panel indicates a nonuniversal cnot (NCN) gate operating between qubits labeled “a” and “b,” where each is modeled by two polarization modes H and V. The gold diamonds are beam splitters with reflectivity coefficient η1=12; the dashed lines inside the beam splitters show that a photon reflected off that side acquires π phase shift. Upon successful coincidence detection of photons, this gates produces the state |ψ1〉, Eq. (15), with probability ℘NCN=12. The dashed-blue panel indicates a universal cnot (CN) gate acting between qubits labeled “a” and “c.” Each of these qubits has a vacuum port with an appropriate annihilation operator v̂a and v̂c, respectively. The green diamonds are beam splitters with reflectivity η2=13. The gate successfully operates with probability ℘CN=19 due to postselection. The black and gray diamonds illustrate beam splitters with reflectivity ζ and ξ, respectively, for modeling total inefficiency in detecting photons and mode mismatch, respectively. Each of these BSs is treated in the same way explained in Sec. 3.

      Reuse & Permissions
    • Figure 5
      Figure 5

      Variation of the phase-dependent deviation, Eq. (18), as a function of phase for the optimal three-photon state ρopt, for the K=2 (N=7) HPEA (circular orange). For ab initio phase estimation, the Holevo deviation, Eq. (1), is used, an average that corresponds to erasing any prior information about the phase. For measurements performed on the ideal state, this is depicted by a purple horizontal line. The ab initio SNL is shown by a dashed red line. Note that since the probe state assumed here is perfect, the estimate from the HPEA would be optimal, Eq. (2), so the Holevo deviation is equal to the Holevo variance, Eq. (3).

      Reuse & Permissions
    • Figure 6
      Figure 6

      (Normalized) probability distribution, Eq. (17), of different measurement results as a function of phase for the HPEA for K=2 (N=7). It is expected that when the true phase, ϕ, is equal to one of the eight possible binary digits sequences, (ϕ0ϕ1ϕ2)∈{000,001,⋯,111}, the phase estimation algorithm's precision is at its best. The green dots are the results of numerical simulations, and the solid gold curves are obtained via Eq. (5) with the corresponding Cn for the HPEA. In all plots, nens=50×103.

      Reuse & Permissions
    • Figure 7
      Figure 7

      Conceptual diagram of Hong-Ou-Mandel interferometer for modeling spatial optical mode matching. The imperfect overlap of modes is modeled by splitting the input beams (modes â and b̂), via the gray BSs with reflectivities ξj, into two modes each such that only some portion of them interfere. v̂j represent vacuum mode annihilation operators. In this case, we show a 50:50 beamsplitter.

      Reuse & Permissions
    • Figure 8
      Figure 8

      Impact of optical mode mismatch on the overall performance of the HPEA. The result of numerical simulations of the Holevo deviation, Eq. (1), is illustrated using golden solid points (the solid golden line is a guide for the eyes). The green dashed line depicts the SNL. Heralding efficiency is set at 13% for all simulations, and each point is obtained using 104 runs.

      Reuse & Permissions
    • Figure 9
      Figure 9

      The HPEA performance in the presence of higher-order terms in the SPDC process. The Holevo deviation, Eq. (1), is plotted by (a) varying the overall efficiency of the first SPDC source, while ε2=0.05, and (b) swapping the role of ε1 and ε2. The heralding efficiency is fixed at 13% for both plots. Each data point was obtained using 50×103 simulation runs.

      Reuse & Permissions
    • Figure 10
      Figure 10

      Experimental setup arrangement. Blue region: Single and entangled photons at 820nm are generated via two type-I SPDC sources. The SPDC crystals are pumped by pulsed UV light produced through a second-harmonic-generation process. Photons are guided using single-mode fibers into the entangling gate to create the optimal state. Green region: The desired probed state is post-selectively generated by realizing two entangling gates: the probabilistic nonuniversal cnot gate acting between modes “a” and “b” composed of four HWPs and one PBS (equivalent to the red dashed box in Fig. 4), and the nondeterministic universal cnot gate operating between modes “a” and “c” made up of three PPBSs where the central one (nonflipped) coherently combines the control and target photons and two HWPs set at 22.5∘ with respect to the optical axis (corresponding to the blue dashed box in Fig. 4). Gray region: quantum state reconstruction tomography stage. Photons are directed to polarization analysis units consisting of a QWP, HWP, and PBS followed by a 2 nm spectral filter and SPCM.

      Reuse & Permissions
    • Figure 11
      Figure 11

      Experimental HOM interference of photons produced via two independent SPDC sources. (a) Photons in modes “a” and “b” nonclassically interfere in the PBS; see Fig. 10. The observed visibility is ν=0.97±0.03, and the dip width is 229±19µm. (b) Interference between photons in modes a and c in the central PPBS. The observed dip visibility is ν=0.77±0.025, and the dip width is 228±14µm. See text for details.

      Reuse & Permissions
    • Figure 12
      Figure 12

      (a) Real (left) and imaginary (right) parts of the state matrix ρopt reconstructed from polarization state tomography, and (b) the optimal state ρexp=|ψexp〉〈ψexp|, Eq. (10). The fidelity of the experimental state with respect to the optimal state is F=0.810±0.014, and the purity is equal to P=0.75±0.02, calculated from approximately 4200 fourfold coincidence photodetection.

      Reuse & Permissions
    • Figure 13
      Figure 13

      Profile of the phase-dependent Holevo deviation, Eq. (18), as a function of phase for the experimental state ρexp (rectangular green points). The Holevo deviation, Eq. (3), is depicted by a blue horizontal line. The dashed red line illustrates the SNL as in Fig. 5.

      Reuse & Permissions
    • Figure 14
      Figure 14

      (Normalized) probability distribution as in Fig. 6, but the green dots are simulation results using the reconstructed state ρexp. The solid gold curves are obtained via Eq. (5) with the corresponding Cn for the HPEA. In all plots, nens=50×103.

      Reuse & Permissions
    • Figure 15
      Figure 15

      A diagram of the Heisenberg-limited phase estimation algorithm with N=7 resources. The gray panel depicts the optimal state generation as in Fig. 10. The rest shows applications of phase shift gate (large green HWP) and classical conditional control operations (shown by orange and purple boxes for rotation by π/4 and π/2, respectively).

      Reuse & Permissions
    • Figure 16
      Figure 16

      Experimental HOM interference of two dependent photons. (a) A photon in one mode interferes with a photon in the other mode in the PPBS of the probabilistic universal cnot gate. The visibility is (79±0:5)%, and the dip width is 234±5µm. (b) Shows fluctuations in the position of the HOM-dip. We repeated the same experiment some 60 times where each took 600s. A maximum deviation of 5µm around the average was observed.

      Reuse & Permissions
    ×

    Sign up to receive regular email alerts from Physical Review A

    Log In

    Cancel
    ×

    Search

    Article Lookup

    Paste a citation or DOI
    Enter a citation
    ×