Partial-coherence weak measurement of polarization qubits with a temporal pointer: theory and experiment

Abstract

Weak measurement has proven useful for metrology, for probing quantum paradoxes, and for the direct measurement of quantum states. Weak measurements of polarization qubits with temporal pointers provide a direct route to weak-value characterization, including in regimes where the pointer is only partially coherent. We derive a closed-form expression for the post-selected temporal shift that interpolates between the incoherent and coherent (ideal) weak-measurement limits through a single parameter $ν$, the pointer visibility. The expression is experimentally tested for a range of polarization states spanning three trajectories corresponding to rotations about the principal axes of the Poincaré sphere. Two of the trajectories provide parameter-free benchmarks, with the real part of the weak value following $\cos^2θ$ and $1/2$. In contrast, the weak value along the third trajectory is sensitive to incoherence and is well described by the single-parameter partial-coherence model, from which a reduced pointer visibility is obtained. Time shifts are extracted from the smoothed derivative of the pulse leading edge with internal calibration, reaching few-picosecond precision on 10 ns optical pulses using commercial components. The same apparatus also gives qualitative access to the imaginary quadrature through the frequency centroid, showing that temporal weak measurements remain accessible and quantitative in partially coherent pulsed systems.

The paper examines how partial temporal coherence modifies post-selected weak-measurement signals for polarization qubits. For an input $a|H\rangle+b|V\rangle$, a polarization-dependent delay $\tau$, and diagonal post-selection, it models the normalized temporal centroid as $\widetilde W=(|a|^2+\nu\operatorname{Re}(a\bar b))/(1+2\nu\operatorname{Re}(a\bar b))$, where $\nu$ describes pointer overlap/coherence at the imposed delay. This interpolates between the incoherent population response and the ideal coherent weak-value response. The experiment uses a commercial multimode 640-nm laser producing 10-ns pulses, a polarization interferometer with approximately 177-ps differential delay, and photodiode/oscilloscope detection. Three polarization trajectories test the model: two predict visibility-independent responses, $\cos^2\theta$ and $1/2$, while the third constrains $\nu$. Timing is extracted from the maximum of a smoothed leading-edge derivative, rather than directly from the temporal centroid, with horizontal/vertical reference calibration. Repeated measurements give few-picosecond timing reproducibility. The $\cos^2\theta$ trajectory has an RMS residual of approximately 0.04; the constant trajectory averages $0.50\pm0.01$. A single-parameter fit to the coherence-sensitive trajectory yields $\nu=0.09\pm0.05$. Fourier analysis of the recorded intensity traces also gives a qualitative angular dependence associated with the magnitude of the imaginary weak value, without a calibrated magnitude or recovered sign. Stated limitations include slow acquisition drifts, angle-calibration uncertainty excluded from per-point statistical errors, and the restriction of a single visibility parameter to a fixed delay. More coherent sources and phase-preserving spectral detection are proposed for approaching ideal weak-value readout and recovering the imaginary-quadrature sign.