Strengthening weak-value amplification with recycled photons

We consider the use of cyclic weak measurements to improve the sensitivity of weak-value amplification precision measurement schemes. Previous weak-value experiments have used only a small fraction of events, while discarding the rest through the process of “postselection.” We extend this idea by considering recycling of events which are typically unused in a weak measurement. Here we treat a sequence of polarized laser pulses effectively trapped inside an interferometer using a Pockels cell and polarization optics. In principle, all photons can be postselected, which will improve the measurement sensitivity. We first provide a qualitative argument for the expected improvements from recycling photons, followed by the exact result for the recycling of collimated beam pulses, and numerical calculations for diverging beams. We show that beam degradation effects can be mitigated via profile flipping or Zeno reshaping. The main advantage of such a recycling scheme is an effective power increase, while maintaining an amplified deflection.

Figure 1

Simple cyclic weak-measurement scheme. A laser emits a pulse of horizontal (H) polarization through a polarizing beam splitter (PBS), which travels through an active Pockels cell (PC) that rotates the polarization to vertical (V), after which it enters a Sagnac interferometer through a 50:50 beam splitter. Inside the interferometer the combination of a half-wave plate (HWP) and Soleil-Babinet compensator (SBC) rotates the pulse polarization back to H with a relative tunable phase shift of $\varphi$ between the clockwise- ($⥁$) and counterclockwise- ($⥀$) traveling paths. The piezo-driven mirror imparts a transverse momentum kick $k$ that differs by a sign for the $⥁$ and $⥀$ paths. A split detector is placed at the dark port to measure a resulting pulse deflection. The H-polarized part of the pulse that exits the bright port is rotated again by the active PC back to V before being confined by the PBS and mirror to return the pulse to the interferometer through the PC, now inactive.

Figure 2

Strongly misaligned regime with $\varphi <1<k\sigma$. Left: the transverse pulse profile $n_r\left(x\right)$ that impacts the dark-port detector on the first (blue, solid) and second (red, dashed) traversals. Right: the accumulated transverse pulse profile ${\overline{n}}_r\left(x\right)$ on the dark-port detector after a single (blue, solid) and ten (red, dashed) traversals. For this regime the interference pattern covers the entire profile, subsequent pulses are strongly attentuated, and the interference of the accumulated profile is slowly filled in.

Figure 3

Weak-value regime with $k\sigma <\varphi <1$, with parameters chosen to exaggerate the walk-off effect. Left: the transverse single-lobed pulse profile $n_r\left(x\right)$ that impacts the dark-port detector on the first four traversals in order of (blue, solid), (red, dashed), (purple, dot-dashed), and (brown, dotted). Right: the accumulated transverse pulse profile ${\overline{n}}_r\left(x\right)$ on the dark-port detector after the first four traversals, with the same color coding. For this regime, the dark-port profile resembles a single shifted Gaussian that gradually walks back toward the center on multiple traversals with some attenuation, eventually recovering the original profile. For more realistic parameters, such as those used in [5], the walk-off effect is smaller per traversal.

Figure 4

Inverse weak-value regime with $\varphi <k\sigma <1$, with parameters chosen to exaggerate the walk-off effect. Left: the transverse double-lobed pulse profile $n_r\left(x\right)$ that impacts the dark-port detector on the first four traversals in order of (blue, solid), (red, dashed), (purple, dot-dashed), and (brown, dotted). Right: the accumulated transverse pulse profile ${\overline{n}}_r\left(x\right)$ on the dark-port detector after the first four traversals, with the same color coding. For this regime, there are two lobes that very gradually walk back toward the center on multiple traversals with some attenuation to eventually recover the original beam profile. For more realistic parameters, such as those used in [11], the walk-off effect is smaller per traversal.

Figure 5

Normalized split-detection response for a collimated pulse. Left: response for the weak-value regime as a function of $k\sigma$ with fixed $\varphi$, where $k\sigma <\varphi <1$, and with parameters consistent with Ref. [5]. The accumulated signal is shown for traversal numbers $r=1$ (blue, solid), $r=5$ (red, dashed), $r=10$ (purple, dot-dashed), and $r=20$ (brown, dotted). Right: response for the inverse weak-value regime as a function of $\varphi /\pi$ with fixed $k$, where $\varphi <k\sigma <1$, and with parameters consistent with Ref. [11]. The accumulated signal is shown for the same traversal numbers and color coding. Although the walk-off effects at large traversal numbers change the slope in both regimes, the linear response is preserved. Hence, one can calibrate the slope through repeated experiments with a fixed number of traversals per laser period. The slope is negative here since the signal is negative in Eqs. (20).

Figure 6

SNR gain versus traversal number $r$ for a collimated pulse. Left: the SNR gain for the weak-value regime $k\sigma <\varphi <1$ with $k\sigma ={10}^{-4}$ and $\varphi =\pi /16$. The uncorrected beam with walk-off (blue, solid) shows clear degradation with traversal number, while the beam corrected with parity flipping (red, dashed) as in Sec. 3b4 matches the simple power-scaling law exactly (black, dashed) from Eq. (2). Note that Zeno stabilization will identically produce this power-scaled SNR by construction. Right: the SNR gain for the inverse weak-value regime $\varphi <k\sigma <1$ with $k\sigma =0.1$ and $\varphi =\pi /256$. The uncorrected double-lobed beam with walk-off (blue, solid) shows somewhat less degradation than the weak-value regime due to the forced zero in the profile; however, the double-lobed beam corrected with parity flipping (red, dashed) manages to exceed the simple power-scaling law (black, dashed), as well as the scheme with Zeno stabilization.

Figure 7

Normalized split-detection response for a collimated pulse stabilized by parity flipping, using the same color coding as in Fig. 5. Left: response for the weak-value regime as a function of $k\sigma$ for fixed $\varphi$, where $k\sigma <\varphi <1$. All traversal numbers $r$ have the same linear response due to the parity-flip stabilization; Zeno stabilization produces the same result. Right: response for the inverse weak-value regime as a function of $\varphi /\pi$ for fixed $k$, where $\varphi <k\sigma <1$. The linear response acquires a steeper slope for larger traversal number using parity-flip stabilization; Zeno stabilization, however, would result in identical slopes for any $r$.

Figure 8

The effect of propagation and different diverging lens choices on the SNR of the weak-value regime with $k\sigma ={10}^{-6}$ and $\varphi =\pi /16$. In order of (blue, solid), (red, dashed), (purple, dot-dashed), and (brown, dotted) we show weak initial diverging lenses with extreme focal lengths $s_i=-1\text{m,}-5\text{m,}-10\text{m,}$ and no lens. The (black, dashed) curve shows the scaling given by power considerations in Eq. (2). Here ${\mathcal{R}}_1$ refers to the SNR of a single unrecycled pulse with an optimally chosen focal length of $s_i=-0.5\text{m}$, while ${\mathcal{R}}_r$ is the accumulated SNR over $r$ traversals for the indicated lens choices.