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

Fast macroscopic-superposition-state generation by coherent driving

Emi Yukawa, G. J. Milburn, and Kae Nemoto
Phys. Rev. A 97, 013820 – Published 16 January 2018
PDFHTMLExport Citation
Abstract
Authors
Article Text
  • INTRODUCTION
  • MODEL AND METHOD
  • NONCLASSICALITY WITNESS AND PRECISION…
  • DISCUSSION AND CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We propose a scheme to generate macroscopic superposition states (MSSs) in spin ensembles, where a coherent driving field is applied to accelerate the generation of macroscopic superposition states. The numerical calculation demonstrates that this approach allows us to generate a superposition of two classically distinct states of the spin ensemble with a high fidelity above 0.97 for 300 spins. For a larger spin ensemble, though the fidelity slightly declines, it maintains above 0.84 for an ensemble of 500 spins. The time to generate an MSS is also estimated, which shows that the significantly shortened generation time allows us to achieve such MSSs within a typical coherence time of the system.

    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    • Figure
    8 More
    • Received 20 March 2017

    DOI:https://doi.org/10.1103/PhysRevA.97.013820

    ©2018 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Atom opticsBose gasesEntanglement productionQuantum controlQuantum state engineering
    Atomic, Molecular & OpticalQuantum Information, Science & Technology

    Authors & Affiliations

    Emi Yukawa1,2,*, G. J. Milburn3, and Kae Nemoto1

    • 1National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
    • 2RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako-shi, Saitama 351-0198, Japan
    • 3Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia

    • *emi.yukawa@riken.jp

    Article Text (Subscription Required)

    Click to Expand

    References (Subscription Required)

    Click to Expand
    Issue

    Vol. 97, Iss. 1 — January 2018

    Reuse & Permissions
    Access Options
    CHORUS

    Article Available via CHORUS

    Download Accepted Manuscript
    Author publication services for translation and copyediting assistance advertisement

    Authorization Required

    Log In
    Other Options
    ×

    Images

    • Figure 1
      Figure 1

      (i) Plots of time evolution of the fidelity F(J;ω̃,ϕ;τ), the relative phase γ0′, and the displacement angle δ0 and (ii) the Q functions Q(α,β)≡2J+14π|〈ΦCSS(J;α,β)|Ψ(J;ω̃,ϕ;τ)〉|2 corresponding to the initial state and the first local maximum of the fidelity for J=50, ω̃=0.0204π, and ϕ=0.024π. (i) Time dependences of F(J;ω̃,ϕ;τ), γ0′, and δ0 are indicated by the black solid curve, the red dashed curve, and the green dots, respectively. The yellow and red shaded regions represent the intervals F(J;ω̃,ϕ;τ)≥0.99 and δ0≥0.95π, respectively. (ii) The color at the point indicated by the polar and azimuthal angles of (α,β) represents 4π2J+1Q(α,β) according to the right gauge. The time evolution of the fidelity and the Q functions for J=74.5 and J=200 are shown in Figs. 9 and 10, respectively.

      Reuse & Permissions
    • Figure 2
      Figure 2

      Total spin dependences of (i) the set of the rescaled driving frequency and the driving phase, (ii) the fidelity, (iii) the displacement angle, and (iv) the rescaled evolution time. In the plots (i)–(iv), there is discontinuity between J=150 and 174.5.

      Reuse & Permissions
    • Figure 3
      Figure 3

      Interference fringes of the quantum fluctuations in σ̂x⊗N. The red solid curves and the blue dashed curves indicate the interference fringes produced by the perfect MSS given in Eq. (8) and pure MSSs |Ψopt(J)〉, respectively. The black dots with error bars represent the mean values and the standard variances of the quantum fluctuations in σ̂x⊗N and τopt of |Ψ(J;τopt)〉 represents the optimized evolution time for J¯. (i),(ii) The fringes produced by the MSSs with 5% of the Gaussian fluctuations in the number of spins for N=149 and 200. (iii),(iv) The fringes produced by the MSSs with 5% of the uniform distribution in the magnitude of the driving field Ω.

      Reuse & Permissions
    • Figure 4
      Figure 4

      Discrete Fourier transformation of σ̂x⊗N−1. The red solid curves and the blue dashed curves indicate the spectra produced by the perfect MSS given in Eq. (8) and pure MSSs |Ψopt(J)〉. The black dots represent the mean values of the spectra and τopt of |Ψ(J;τopt)〉 represents the optimized evolution time for J¯. The black solid lines indicate the frequencies of the interference fringes for the perfect MSS, i.e., ω=±4Jcosβ in Eq. (8). (i),(ii) The spectra produced by the MSSs with 5% of the Gaussian fluctuations in the number of spins for N=149 and 200 at each step of rotation.

      Reuse & Permissions
    • Figure 5
      Figure 5

      Dependence of the gap Δ̃(J;τ) between |ɛ1(J;τ)〉 and |ɛ2(J;τ)〉 on the phase of the driving field ω̃optτ+ϕopt. (i) The gap Δ̃(J;τ) with respect to ω̃optτ+ϕopt for J=50–250. The dots, the triangles, and the thin diamonds mark τ=0, τ=0.2τopt, τ=0.4τopt, τ=0.6τopt, τ=0.8τopt, τ=τopt on the curves representing Δ̃(J;τ). (ii) The gap functions Δ̃(J;τ) for J=50–250 coincide with each other by shifts in the phase of the driving field and enlargements (or shrinks) of the magnitude of the gap energy.

      Reuse & Permissions
    • Figure 6
      Figure 6

      Time evolution of the Q functions of |ɛ1(J;τ)〉 and |ɛ2(J;τ)〉 for J=74.5 and J=200. The color hue represents 4π2J+1Q(α,β) whose gauge is the same as Figs. 1. The dotted white curves indicate the contours of h̃opt(J;τ) in the mean-field limit, which is given by Ẽopt(J;α,β;τ)=J[12cos2β+rcosαsinβcos(ω̃optτ+ϕopt)], and the values on the contours represent 2JẼopt(J;α,β;τ). The solid white curves represent the energy contours Ẽopt(J;α,β;τ)=rJcos(ω̃optτ+ϕopt).

      Reuse & Permissions
    • Figure 7
      Figure 7

      J dependence of the probability distribution of the initial state |ΦCSS(J;0,π2)〉 on the highest-energy eigenstate |ɛ1(J;0)〉 (black solid curve with dots) and the second-highest-energy eigenstate |ɛ2(J;0)〉 (red dashed curve with triangles) of the Hamiltonian h̃(J;0) and the other eigenstates (blue dotted curve with thin diamonds). The probability on |ɛ1(J;0)〉 monotonically and slowly decreases with respect to J and converges toward ∼0.5, while the probability on |ɛ2(J;0)〉 stays at zero. The probability distributing on the other eigenstates monotonically and slowly increases and converges toward ∼0.5.

      Reuse & Permissions
    • Figure 8
      Figure 8

      Plots of the relative phases γM′'s of |ɛ1(J;τ)〉 and |ɛ2(J;τ)〉 as functions of τ and MJ for (i) J=74.5 and (ii) J=200. The red dots and the blue triangles represent γM′'s for |ɛ1(J;τ)〉 and |ɛ2(J;τ)〉, respectively. The right-hand sides of blue shaded planes parallel to the MJ−γM′ planes are the time regions where Δ̃(J;τ)<O(10−6), i.e., the region where the gaps can be regarded to be closed. We note that we plot γM′'s for every five points for J=74.5 and for every 20 points for J=200 with respect to M/J for the sake of visibility of the points. For both J=74.5 and 200, γM′=0 for |ɛ1(J;τ)〉 and γM′=π for |ɛ2(J;τ)〉 when the gaps are open. When J=200 and the gap closes, γM′ can be considered to be indefinite, since the Q functions of |ɛ1(J=200;τ=τopt)〉 and |ɛ2(J=200;τopt)〉 in Fig. 6 imply that they are close to coherent spin states, which are separable, and the probabilities either on |J;±M〉 become ∼0.

      Reuse & Permissions
    • Figure 9
      Figure 9

      Rescaled-time dependences of fidelity F(J;ω̃,ϕ;τ), relative phase γ0′, and displacement angle δ0(α0,β0) for (i) J=74.5 and (ii) J=200. The vertical scales on the left-hand sides and the right-hand sides are for F(J;ω̃,ϕ;τ) and the two angles, γ0′ and δ0(α0,β0), respectively. The black solid curves, the green dots, and the red dashed curves represent F(J;ω̃,ϕ;τ), γ0′, and δ0(α0,β0), respectively. The yellow shaded regions and the red shaded region with a left-right arrow express the intervals satisfying F(J;ω̃,ϕ;τ)≥0.99 and the interval where an almost perfect cat state 12(|J,J〉+|J,−J〉) is generated, i.e., the region with F(J;ω̃,ϕ;τ)≥0.99 and δ0(α0,β0)≥0.95. The Q functions at which the black and thin dotted lines, (a)–(e), are illustrated in Figs. 10 and τ=τmax, at which the first local maximum of the fidelity Fmax(J;ω̃,ϕ) is achieved, is indicated by the black and thin dashed line. The driving-field parameters are set to be ω̃=0.0204π and ϕ=0.024π for J=50, ω̃=0.0174π, and ϕ=0.012π for J=74.5, and ω̃=0.0151π and ϕ=−0.0128π for J=200.

      Reuse & Permissions
    • Figure 10
      Figure 10

      Time evolution of the Q functions Q(α,β)≡2J+14π|〈ΦCSS(J;α,β)|Ψ(J;ω̃,ϕ;τ)〉|2 for J=50, J=74.5, and J=200. The driving-field parameters are set to be the same as Figs. 9. The color at the point indicated by the polar and azimuthal angles of (α,β) represents 4π2J+1Q(α,β) according to the gauge shown in Fig. 1. For J=50, the driving field parameters are set to be ω̃=0.0204π and ϕ=0.024π and the snapshots are taken at the rescaled elapsed times of (a) τ=0, (b) 4, (c) 8, (d) 12, and (d) 16. For J=74.5 and J=200, the driving field parameters are given by ω̃=0.0174π and ϕ=−0.012π and ω̃=0.0151π and ϕ=−0.0128π, respectively, and the snapshots are taken at (a) τ=0, (b) 5, (c) 10, (d) 15, and (d) 20.

      Reuse & Permissions
    • Figure 11
      Figure 11

      Fidelity Fmax(J;ω̃,ϕ) of the first local maximum and its corresponding displacement angle δmax(J;ω̃,ϕ) as functions of the driving-field frequency and phase, ω̃ and ϕ, for (i) J=50, (ii) J=74.5, (iii) J=100, (iv) J=124.5, (v) J=150, (vi) J=174.5, (vii) J=200, (viii) J=224.5, and (ix) J=250. The z axes represent Fmax(J;ω̃,ϕ) and the color on the Fmax(J;ω̃,ϕ) surface and the plane at the bottom of the plot indicate the magnitude of the displacement angle whose gauge is shown on the right-hand side of (ix). There are two parameter regions with high fidelity with δmax(J;ω̃,ϕ)≳0.4π for J=100−150, while δmax(J;ω̃,ϕ) decreases to be δmax(J;ω̃,ϕ)∼0 in the region with smaller ω̃ for J=174.5−250.

      Reuse & Permissions
    • Figure 12
      Figure 12

      Plots of J dependence of the angles αopt, βopt, and γopt′, which are represented by the red dots, the blue triangles, and the green thin diamonds, respectively. The relative phase γopt′=0 for all J.

      Reuse & Permissions
    • Figure 13
      Figure 13

      Fringes of 〈(Δσ̂x⊗N)2〉 generated by |Ψopt(J)〉 with the spin-number fluctuations of 2% and 10% for J¯=74.5 and J¯=200. The red solid curves and the blue dashed curves express the fringes produced by the perfect MSS |ΦMSS(J;αopt,βopt,γopt)〉 and |Ψopt(J)〉 without spin-number fluctuation. The black dots with the black-solid error bars respectively represent the mean values and the standard deviations of 250 trials of fringe experiments, where the spin-number fluctuation is given by Eq. (C9) and |Ψ(J;τopt)〉 is prepared after the optimized evolution time τopt for J¯.

      Reuse & Permissions
    • Figure 14
      Figure 14

      Fringes of 〈(Δσ̂x⊗N)2〉 generated by |Ψopt(J)〉 with the fluctuation in the driving-field magnitude of 2% and 10% for J¯=74.5 and J¯=200. The red solid curves and the blue dashed curves express the fringes produced by the perfect MSS |ΦMSS(J;αopt,βopt,γopt)〉 and |Ψopt(J)〉 without the fluctuation in Ω, i.e., Ω=Ω¯. The black dots with the black-solid error bars respectively represent the mean values and the standard deviations of 250 trials of fringe experiments with the Ω distributed randomly between (1±σ)Ω¯.

      Reuse & Permissions
    • Figure 15
      Figure 15

      Fringes of 〈(Δσ̂x⊗N)2〉 generated by |Ψopt(J)〉 with the fluctuation in the nonlinear interaction energy λ of 2%, 5%, and 10% for J¯=74.5 and J¯=200. The red solid curves and the blue dashed curves express the fringes produced by the perfect MSS |ΦMSS(J;αopt,βopt,γopt)〉 and |Ψopt(J)〉 without the fluctuation in λ, i.e., λ=λ¯. The black dots with the black-solid error bars respectively represent the mean values and the standard deviations of 250 trials of fringe experiments with the λ distributed randomly between (1±σ)λ¯.

      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
    ×