Research ArticlePHYSICS

An experimental test of the geodesic rule proposition for the noncyclic geometric phase

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Science Advances  28 Feb 2020:
Vol. 6, no. 9, eaay8345
DOI: 10.1126/sciadv.aay8345
  • Fig. 1 Illustration of the geodesic rule and the experimental sequence.

    (A) An illustration of the geodesic rule (7, 10) on the Bloch sphere representing the two-dimensional space defined by our physical two-level system. The green and red arrows represent the internal states A and B of the two spatially separated wave packets, ΨA and ΨB (see Eq. 1). The rotation angle from the north pole θ and the rotation Δϕ along the latitude (continuous purple) represent the SU(2) operations applied in the experiment, where the former requires an RF pulse, while the latter requires a magnetic gradient. When θ = π/2, the arrows lie on the equator of the Bloch sphere (A0 and B0). The dashed purple curve is the geodesic joining points A (A′) and B (B′). The GP is equal to one-half of the blue area enclosed by the latitude and geodesic. The area’s orientation (indicated by the arrows) is determined by the geodesic rule. It is negative, counterclockwise (northern hemisphere) and positive, clockwise (southern hemisphere). (B) Experimental sequence (not to scale) of the longitudinal interferometer. The experiment is performed in free fall. The final interference pattern (from which the total phase is obtained) develops after time-of-flight (TOF) free evolution, in which the two wave packets expand and overlap. The pattern is then recorded by a CCD camera. (C) Evolution of the states during the sequence. After the preparation of two coherent wave packets at different locations, an RF pulse of duration TR is applied to manipulate θ, and a magnetic field gradient of duration TG is applied to manipulate Δϕ.

  • Fig. 2 Population transfer and its connection to the π phase jump.

    (A) Population transfer to state ∣1⟩ versus the duration of the RF radiation pulse TR, for which 20 μs corresponds to total population transfer (θ = π in Fig. 1). With this independent measurement, we determine θ for our SU(2) operations. (B) Averaged CCD image of interference when the Bloch vectors are all in the northern hemisphere [NH data points specified in (A)], with Δϕ ≃ π. The high visibility indicates the existence of phase rigidity, namely, that the phase is independent of θ. The phase returned by the fit is 1.13 ±0.02 rad relative to a fixed reference point, and the visibility is 0.55 ± 0.01 (see Methods for the definition). (C) Averaged picture of the second half of the data, in which the Bloch vectors are all pointing in the southern hemisphere [SH data points specified in (A)], with Δϕ ≃ π. A phase jump is clearly visible. The phase is 4.34 ± 0.03 rad relative to the fixed reference point, which is common to both pictures, and the visibility is 0.52 ± 0.01. The phase difference between (B) and (C) is thus 3.21 ± 0.05 rad, close to π. The data included in these images (in total, about 330 consecutive experimental shots without post-selection or post-correction) are presented in Fig. 3B. (D) Averaged picture of all the data for Δϕ ≃ π. The visibility is 0.03 ± 0.01. The low visibility shows that the phase jump has a value close to π. Single-shot data are presented in Fig. 3B, and single-shot images are presented in Fig. 6.

  • Fig. 3 The phase of the interference pattern: Phase jump and rigidity.

    (A to D) Total phase Φ as a function of TR (θ) for TG is equal to 6, 17, 32, and 40 μs. Each data point is an average of six experimental cycles (errors are SEM). The dashed lines are a fit to Eq. 2, which allows us to determine Δϕ for our SU(2) operations. The fit returns the values Δϕ = 2.24 (A), Δϕ = 3.14 (B), Δϕ = 5.31 ≡ 2π − 0.97 (C), and Δϕ = 6.23 ≡ 2π − 0.05 (D) radians, respectively (manifested in the graph as the peak-to-valley amplitude if we consider the periodicity of 2π when defining a phase). The fit also returns a baseline phase ϕ0. Last, the phase rigidity and the phase jump observed in Fig. 2 are clearly visible in (B). (E) Linear mapping from TG to Δϕ. As seen in the graph (TG = 0), we have a fixed background gradient equivalent to Δϕ = 1.35.

  • Fig. 4 Geometric SU(2) phase jump and sign flip, experiment (dots) versus theory (Eq. 3, dashed lines).

    (A) Total phase and dynamical phase for Δϕ = π as a function of TR (θ). The total phase is directly measured from the imaged interference pattern (Fig. 3), and the dynamical phase Δϕ2(1cosθ) is deduced from the independently measured values of θ and Δϕ. (B) GP ΦG determined as the difference between the two sets of points appearing in (A). The predicted sign change as the latitude crosses the equator is clearly visible. The evident phase jump is due to the geodesic rule. When Δϕ = π, the geodesic must go through the Bloch sphere pole for any θ ≠ π/2. As the latitude approaches the equator (i.e., increasing θ), the blue area in Fig. 1 (twice ΦG) continuously grows to reach a maximum of π in the limit of θ = π/2. As the latitude crosses the equator, the geodesic jumps from one pole to the other pole, resulting in an instantaneous change of sign of this large area and a phase jump of π. This plot exactly confirms the prediction in (10). (C) Total phase and dynamical phase for Δϕ = 2.24 rad (0.71π). (D) ΦG, determined as the difference between the two sets of points appearing in (C). The predicted sign change is again visible. However, in the case of Δϕ = 2.24 rad (0.71π), the geodesic line does not go through the pole, and as the latitude approaches the equator, ΦG actually reduces (after reaching its maximum for an intermediate θ), so no abrupt phase jump is expected.

  • Fig. 5 The population transfer versus TR is measured in an independent experiment by applying a strong magnetic gradient after TR.

    Because of the Stern-Gerlach effect, the mF = 1 and mF = 2 parts are shifted to different regions of space when the absorption imaging is performed to evaluate the atom number. The absorption imaging is based on the comparison between the intensity I of a light pulse going through the atoms and the intensity I0 of a reference light pulse that propagates in the absence of atoms and the Beer’s law, I(xi, zj) = I0(xi, zj)eOD(xi, zj). The optical density (OD) is proportional to the column density of the atoms at a given position ∫n(x, y, z)dy, where x and z are the object plane positions corresponding to xi and zj, respectively. The number of atoms N(xi, zj) imaged by the pixel is N(xi,zj)=Aσ0OD(xi,zj), where A is the pixel area in the object plane, σ0 = 3λ2/2π is the cross section for resonant atom-light scattering, and λ ≈ 780 nm is the optical transition wavelength. The total atom number is equal to ∫N(x, z)dxdz. We can then reliably determine the relation between population transfer and TR as presented in Fig. 2A, e.g., 10 μs corresponds to θ = π/2, 20 μs corresponds to θ = π, and 40 μs corresponds to θ = 2π.

  • Fig. 6 The interference pattern versus TR when TG = 17 μs (Δϕ ≃ π).

    The number in each subfigure indicates the duration of TR in microseconds. When the Bloch vectors are in the northern hemisphere, the interference phase is seen to be rigid (fixed). When the Bloch vectors cross the equator at TR = 10 μs, there is a π phase jump. The interference phase will jump by another π when the vectors cross the equator again at TR = 30 μs. Namely, phase rigidity appears when the Bloch vectors are located in either the northern or southern hemisphere, with a π phase jump in between, as presented in Fig. 2 (B to D) and Fig. 3B. The fluctuations in the interference pattern’s location are due to fluctuations in the initial conditions from shot to shot, while the inferred phase is stable, as explained in (34).

Supplementary Materials

  • Supplementary Materials

    This PDF file includes:

    • Fig. S1. Detailed scheme of the spatial SU(2) interferometer.
    • Fig. S2. Theoretical curves of the GP ΦG versus θ for different values of Δϕ.

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