Time Derivatives of Weak Values

TL;DR

The paper addresses how the time derivative of weak values relates to gauge-invariant physical information. It develops left- and right-hand derivatives, introduces explicit gauge-invariance conditions (C1–C5) to identify when derivatives yield observable weak values, and derives a local Ehrenfest-like theorem for weak values under these conditions. Finite-difference schemes (FDLHD/FDRHD) are proposed to implement the derivatives experimentally, with numerical and analytical examples linking the position weak value to local velocity, acceleration, and electromagnetic fields. The work provides practical guidelines for quantum sensing using weak-value time derivatives and highlights the privileged role of position- and velocity-post-selected weak values in attaining time-dependent consistency, while acknowledging the broader landscape of modal theories. The findings offer a framework for translating weak-value theory into laboratory protocols and for exploring local field measurements in quantum systems under semi-classical electromagnetism.

Abstract

The time derivative of a physical property often gives rise to another meaningful property. Since weak values provide empirical insights that cannot be derived from expectation values, this paper explores what physical properties can be obtained from the time derivative of weak values. It demonstrates that, in general, the time derivative of a gauge-invariant weak value is neither a weak value nor a gauge-invariant quantity. Two conditions are presented to ensure that the left- or right-time derivative of a weak value is also a gauge-invariant weak value. Under these conditions, a local Ehrenfest-like theorem can be derived for weak values giving a natural interpretation for the time derivative of weak values. Notably, a single measured weak value of the system's position provides information about two additional unmeasured weak values: the system's local velocity and acceleration, through the first- and second-order time derivatives of the initial weak value, respectively. These findings also offer guidelines for experimentalists to translate the weak value theory into practical laboratory setups, paving the way for innovative quantum technologies. An example illustrates how the electromagnetic field can be determined at specific positions and times from the first- and second-order time derivatives of a weak value of position.

Figures (8)

• Figure 1: (a): The weak value in Eq. (\ref{['ewv']}) involves five steps: preparation of $|\psi\rangle$ (red) at the initial time $t=0$, unitary evolution $\hat{U}_{{t_R}}$, weak perturbation linked to $\hat{{O}}$ (green) at time $t=t_R$, unitary evolution $\hat{U}_{{t_L}}$ and strong measurement linked to $\hat{{F}}$ (violet) at the final time $t=t_R+t_L$. The weak value can be roughly understood as the property of the system during the intermediate green times. (b): The LHD computed when $t_L\to0$ while $t_R$ is keep constant. (c): The FDLHD is evaluated at the final time and computed from the difference of the weak value $W_1$ and $W_2$, which are computed using two different values of $t_L$ (i.e., $t_L=0$ and $t_L>0$). (d): The RHD computed when $t_R\to0$ while $t_L$ is keep constant. (e): The FDRHD is evaluated at the initial time and computed from the difference of the weak value $W_1$ and $W_2$, which are computed using two different values of $t_R$ (i.e., $t_R>0$ and $t_R=0$).
• Figure 2: Time derivative of the weak value of the position for different pre- and post-selected states defined in Table \ref{['table1']} at different angles $\theta$ that parametrize different gauge functions $g(\textbf{x},t)$. At $t_L=t_R=0$, LHD (red) and RHD (blue); at negative $t_L$, FDLHD (red); at positive $t_R$, FDRHD (blue) are plotted. (a):$|\Phi_{1}\rangle$ for the pre-selected state and $\langle \Phi_{2}|$ for the post-selected state satisfying condition C4. (b):$|\Phi_{2}\rangle$ for the pre-selected state and $\langle \Phi_{1}|$ for the post-selected state satisfying condition C5. (c):$|\Phi_{3}\rangle$ for the pre-selected state and $\langle \Phi_{4}|$ for the post-selected state without satisfying neither condition C4 nor C5. The sum LHD+RHD (green) at $t_L=t_R=0$ and FDLHD+FDRHD (green) at positive $t_R$ and negative $t_L$ are also plotted.
• Figure 3: Time evolution of the modulus squared of $\langle x|\Phi_{5}(t) \rangle$ (blue) in a time-independent potential energy profile (green). Different Bohmian trajectories $x_B^j(t)$ for $j=1,..,10$ are computed by integrating the $x$-component of the weak value of the velocity in Eq. (\ref{['weakvvelo']}) as indicated in Eq. (\ref{['apnum7']}). The different $x_B^j(t)$ will be used later to evaluate the weak values in different relevant positions and times.
• Figure 4: In blue, mean value and standard deviation of the $x$-component of the weak value of the kinetic velocity of the 10 trajectories plotted in Fig. \ref{['f1']} at 20 different times computed from Eq. (\ref{['power']}) (left axis). In orange, the mean value and standard deviation of the $x$-component of the weak value of the velocity in Eq. (\ref{['weakvvelo']}) of the 10 trajectories in Fig. \ref{['f1']} at 20 different times (right axis), which corresponds to the velocity $v_{B}^{|\Phi_{5}\rangle}(t)$.
• Figure 5: In blue, mean value and standard deviation of the time derivative of Eq. (\ref{['wvk']}) of the 10 trajectories in Fig. \ref{['f1']} at 20 different times (left axis). In orange, the mean value and standard deviation of the electrical field of the 10 trajectories in Fig. \ref{['f1']} at 20 different times computed from Eq. (\ref{['power']}) (right axis).