Weak measurement: Effect of the detector dynamics

The contribution of the detector dynamics to the weak measurement is analyzed. According to the usual theory [Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988)] the outcome of a weak measurement with preselection and postselection can be expressed as the real part of a complex number: the weak value. By accounting for the Hamiltonian evolution of the detector, here we find that there is a contribution proportional to the imaginary part of the weak value to the outcome of the weak measurement. This is due to the coherence of the probe being essential for the concept of complex weak value to be meaningful. As a particular example, we consider the measurement of a spin component and find that the contribution of the imaginary part of the weak value is sizable.

Figure 1

A schematic view of the measurement with preselection and postselection, the horizontal direction representing increasing time.

Figure 2

$⟨A⟩$ as a function of $\gamma$ for $\theta =\pi /2$, and $\varphi =\pi /2$ (top figure), $\varphi ={\varphi }^{\ast }=\pi /4$ (bottom figure). (The inset shows how the angles were defined.) The full line depicts the exact value, the dotted one depicts the approximate value including the contribution of $\text{Im}\left\{A_w\right\}$, and the dashed line depicts the contribution of $\text{Re}\left\{A_w\right\}$. The thin vertical and horizontal lines correspond to ${\gamma }^{\ast }=\pi -\lambda /\nu$, ${⟨A⟩}_m=\epsilon \left(\varphi \right)\nu /\lambda$ [see Eqs. (17, 18)]. In the range $\gamma =\left[0,3\right]$, the dotted and the full line practically coincide. We assumed $\lambda /\delta p=0.01,\lambda /p_{\varphi }=0,\delta p=\Delta P=p_H$.

Figure 3

(a) The variance of $A$ as a function of $\gamma$ for fixed $\theta =\pi /2$, $\varphi =\pi /4$. The dashed vertical and horizontal lines correspond to the positions and the values of the extrema [Eqs. (20, 21)]. (b) The probability distribution for $\theta =\pi /2$, $\varphi =\pi /4$ for some significant values of $\gamma$. The parameters are the same as those of Fig. 2.