Taking the operator \(A\) in the weak measurements as a projection operator \(\hat{P}_a=|a\rangle\langle a|\) we can determine the probability of finding the quantum system in the state \(|a\rangle\): \[ P_a=|\langle a|\psi\rangle|^2 =\langle\hat{P}_a\rangle=\frac{1}{\gamma\tau}(\langle q'\rangle-\langle q\rangle)\,. \] In particular, for \(\hat{P}_x=|x\rangle\langle x|\) weak measurements can determine the probability density \(\rho(x)=|\psi(x)|^{2}\).
The Schrödinger equation for the wave function leads to the equation of continuity \[ \frac{\partial}{\partial t}\rho(x,t)+\frac{\partial}{\partial x}j(x,t)=0\,, \] where \(\rho(x,t)=|\psi(x,t)|^2\) is the probability density and \(j(x,t)\) is the probability current. From the continuity equation it follows \[ j(X,t)=\frac{\partial}{\partial t}\int_{X}^{\infty}\rho(x,t)\,\mathrm{d}x\,. \] The quantity \[ \int_X^{\infty}\rho(x,t)\,\mathrm{d}x=\langle\psi(t)|\hat{P}_X|\psi(t)\rangle \] is the probability to find a quantum system in the spatial region \(x>X\). Here \[ \hat{P}_X = \Theta(\hat{x}-X) \] is the projection operator on the subspace \(x>X\) and \(\Theta\) is the Heaviside unit-step function. Approximating the time derivative by a finite difference we get \[ j(X,t)=\lim_{\Delta t\rightarrow 0}\left(\langle\hat{P}_X(t+\Delta t)\rangle-\langle\hat{P}_X(t)\rangle\right) \approx\frac{1}{\Delta t}\left(\langle\hat{P}_X(t+\Delta t)\rangle-\langle\hat{P}_X(t)\rangle\right)\,. \] Since the weak measurements do not disturb the time evolution of the quantum system, we can measure both averages \(\langle\hat{P}_X (t)\rangle\) and \(\langle\hat{P}_X (t+\Delta t)\rangle\) using weak measurements where the quantum system interacts with two detectors at times \(t\) and \(t+\Delta t\). Alternatively, one can use a single detector with time dependent interaction strength \(\gamma\): the interaction is briefly switched on at times \(t\) and \(t+\Delta t\) with the change of the sign of interaction strength, \(\gamma(t+\Delta t)=-\gamma(t)\).
There is an interesting alternative approach to measure probability current using weak measurements on post-selected ensemble. We start from the classical definition of velocity as a position change during a time interval divided by the duration of the time interval. This classical definition of velocity cannot be transferred to quantum mechanics using usual measurements because the position measurement collapses the wave function and completely changes subsequent time evolution of the quantum system. However, using weak measurements we can employ a weak measurement of the position first, followed by an usual measurement after a short time. In this way, following the classical definition of the velocity we can define weak velocity as [1] \[ v(x)=\lim_{\Delta t\rightarrow0}\frac{1}{\Delta t}(x-\langle\hat{x}\rangle_{x,\Delta t})\,, \] where \(\langle\hat{x}\rangle_{x,\Delta t}\) is a weak value of the position in an ensemble post-selected to have position \(x\) after the time \(\Delta t\). We will show that such a definition of weak velocity is related to the probability current \(j(x)\) as \(j(x)=\rho(x)\mathrm{Re}\,v(x)\), where \(\rho(x)\) is the probability density.
Before the measurement the quantum system is in the state \(|\psi\rangle\). After the interaction with the detector the state of the quantum system and the detector becomes \(|\Psi'\rangle\) and after time \(\Delta t\) this state evolves to \[ |\Psi'(\Delta t)\rangle=U_{\mathrm{S}}(\Delta t)|\Psi'\rangle\,. \] Here \(U_{\mathrm{S}}(t)\) is the unitary evolution operator of the quantum system obeying the equation \[ \mathrm{i}\hbar\frac{\partial}{\partial t}U_{\mathrm{S}}(t)=H_{\mathrm{S}}U_{\mathrm{S}}(t)\,, \] where \[ H_{\mathrm{S}}=\frac{p^{2}}{2M}+V(x) \] is the Hamiltonian of the quantum system. We see that projection of the state \(|\Psi'(\Delta t)\rangle\) on the state \(|x\rangle\) is the same as projection of the state \(|\Psi'\rangle\) on the state evolved backwards in time \(U_{\mathrm{S}}^{\dagger}(\Delta t)|x\rangle\). Thus the weak value of the postion is \[ \langle\hat{x}\rangle_{x,\Delta t}= \frac{\langle x|U_{\mathrm{S}}(\Delta t)\hat{x}|\psi\rangle}{\langle x|U_{\mathrm{S}}(\Delta t)|\psi\rangle}\,. \] Since \(\Delta t\) is small, keeping only the terms up to first-order in \(\Delta t\) we have \[ U_{\mathrm{S}}(\Delta t)\approx 1-\frac{\mathrm{i}}{\hbar}\Delta t H_{\mathrm{S}} \] and \[ \langle\hat{x}\rangle_{x,\Delta t}\approx x +\frac{\mathrm{i}}{\hbar}\Delta t\frac{1}{\langle x|\psi\rangle}\langle x|[\hat{x},H_{\mathrm{S}}]|\psi\rangle \] Using \[ [\hat{x},H_{\mathrm{S}}]=\frac{\mathrm{i}\hbar}{M}p \] we obtain \[ \langle\hat{x}\rangle_{x,\Delta t}\approx x-\frac{\Delta t}{M}\frac{\langle x|p|\psi\rangle}{\langle x|\psi\rangle} =x+\mathrm{i}\frac{\hbar\Delta t}{M}\frac{1}{\psi(x)}\frac{\partial}{\partial x}\psi(x)\,, \] therefore the weak velocity is \[ v(x)=\frac{\hbar}{Mi}\frac{1}{\psi(x)}\frac{\partial}{\partial x}\psi(x)\,. \] The real part takes the form \[ \mathrm{Re}\,v(x)= \frac{1}{|\psi(x)|^{2}}\frac{\hbar}{2Mi}\left(\psi^{*}(x)\frac{\partial}{\partial x}\psi(x) -\psi(x)\frac{\partial}{\partial x}\psi^{*}(x)\right) =\frac{j(x)}{\rho(x)}\,. \]
The third way to use weak measurements for determination of the probability current is weak measurement of the momentum \(p\) in an ensemble post-selected on position \(x\), yielding the weak value \[ \langle p\rangle_x = \frac{\langle x|p|\psi\rangle}{\langle x|\psi\rangle}\,. \] The probability current then can be expressed as \[ j(x)=\frac{1}{M}\rho(x)\mathrm{Re}\,\langle p\rangle_x\,. \] This method has been used in an experiment described in Ref. [2].
Let us consider a state \(|\psi(t)\rangle\) of a quantum system that evolves in time. In the ordinary quantum mechanics there is no general method for the determination of durations. We can define a dwell time [3] in the state \(|a\rangle\) as \[ \tau_{\mathrm{Dw}}(T)=\int_0^T |\langle a|\psi(t)\rangle|^2\,\mathrm{d}t\,. \] The dwell time can be interpreted as a duration of time that the system spends in the state $|a\rangle$ before time \(T\). As a particular case, for spatial region between coordinates \(x_1\) and \(x_2\) the dwell time is \[ \tau_{\mathrm{Dw}}(T)=\int_0^T\mathrm{d}t\int_{x_1}^{x_2}\mathrm{d}x\,|\psi(x,t)|^2\,. \] We will show that the dwell time can be measured using weak measurements.
If the interaction strenght is small, to obtain meaningful information about the quantum system the process of interaction has to be performed on an ensemble of identical quantum systems. Each system with its own detector is prepared in the same initial state, interacts with the detector, and after the interaction time \(T\) the readings of the detectors are collected and averaged. Let us consider the time evolution of the quantum system together with the microscopic part of the detector. This time evolution is described by the Hamiltonian \[ H=H_{\mathrm{S}}+H_{\mathrm{I}}\,. \] Here \(H_{\mathrm{S}}\) is the Hamiltonian of the quantum system and \[ H_{\mathrm{I}}=\gamma p_q\hat{P}_a \] describes the interaction with the detector as in a simple von Neumann measurement. The operator \[ \hat{P}_a=|a\rangle\langle a| \] is the projection operator projecting into the sate \(|a\rangle\), \(p_q\) is the momentum of the detector conjugate to the coordinate of the detector \(q\), and the parameter \(\gamma\) describes the strength of the interaction. In order to not disturb the evolution of the quantum system, the parameter \(\gamma\) is very small. The interaction Hamiltonian \(H_{\mathrm{I}}\) results in a small shift of the detector's position \(q\). If the initial state of the detector has a width much larger than this shift, the evolution of the quantum system is only slightly disturbed.
From the classical point of view, the change of the detector's position is proportional to the duration of time that the quantum system spends in the state \(|a\rangle\). It is natural to extend the classical method of the determination of the duration into the quantum mechanics too. Therefore we assume that the change of the mean coordinate of the detector is proportional to the time that the quantum system interacts with the detector and this interaction time is the same as the time the system spends in the state \(|a\rangle\). Thus we can define the duration of time that the quantum system spends in the state \(|a\rangle\) before time \(T\) as \[ \tau(T)=\frac{1}{\gamma}(\langle q(T)\rangle-\langle q\rangle)\,, \] where \(\langle q\rangle\) and \(\langle q(T)\rangle\) are the mean initial detector's coordinate and the mean coordinate after time \(T\). We will show that such definition gives us the dwell time.
Before the interaction the detector is in the pure state \(|\phi\rangle\) and the quantum system in the state \(|\psi(0)\rangle\), thus the initial state of system and the detector is \[ |\Psi(0)\rangle=|\psi(0)\rangle|\phi\rangle\,. \] At time \(t\) the state becomes \[ |\Psi(t)\rangle=U(t)|\psi(0)\rangle|\phi\rangle\,, \] where \(U(t)\) is the unitary evolution operator obeying the equation \[ \mathrm{i}\hbar\frac{\partial}{\partial t}U(t)=(H_{\mathrm{S}}+H_{\mathrm{I}})U(t) \] with the initial condition \(U(0)=I\). The time evolution of the unperturbed quantum system is described by the evolution operator \(U_{\mathrm{S}}(t)\) obeying the equation \[ \mathrm{i}\hbar\frac{\partial}{\partial t}U_{\mathrm{S}}(t)=H_{\mathrm{S}}U_{\mathrm{S}}(t)\,. \] We can expand the operator \(U(t)\) into the power series of the powers of parameter \(\gamma\), assuming that \(\gamma\) is small. Introducting the projection operator in the interaction representation \[ \tilde{P}_a(t)=U_{\mathrm{S}}^{\dagger}(t)\hat{P}_a U_{\mathrm{S}}(t) \] in the first-order approximation we have \[ U(t)\approx U_{\mathrm{S}}(t)\left(1-\frac{\mathrm{i}}{\hbar}\gamma p_q \int_0^t\tilde{P}_a(t')\,\mathrm{d}t'\right)\,. \] As a consequence, \[ \langle q|U(t)|\psi\rangle|\phi\rangle \approx|\psi(t)\rangle\langle q|\phi\rangle -\frac{\mathrm{i}}{\hbar}\gamma\langle q|p_q|\phi\rangle \int_0^t\mathrm{d}t'\,U_{\mathrm{S}}(t-t')\hat{P}_a|\psi(t')\rangle\,. \] The duration of the interaction with the detector is \(T\). After interaction, applying the projective measurement of the postion \(q\) by the rest of the detector with the result \(q'\) yields the probability to obtain the outcome \(q'\) \[ P(q')=\langle\phi|\langle\psi(0)|U^{\dagger}(T)|q'\rangle\langle q'|U(T)|\psi(0)\rangle|\phi\rangle\,. \] Keeping only the terms up to the first-order in interaction strength \(\gamma\) we have \[ P(q')\approx|\langle q'|\phi\rangle|^2 +\frac{\mathrm{i}}{\hbar}\gamma\langle\phi|[p_{q},|q'\rangle\langle q']|\phi\rangle \int_0^T\langle\psi(t)|\hat{P}_a|\psi(t)\rangle\,\mathrm{d}t\,. \] The average of the post-measurement detector's coordinate \(q'\) is \[ \langle q(T)\rangle=\int q'P(q')\,\mathrm{d}q' \approx\langle q\rangle +\frac{\mathrm{i}}{\hbar}\gamma\langle\phi|[p_q, q]|\phi\rangle \int_0^T\langle\psi(t)|\hat{P}_a|\psi(t)\rangle\,\mathrm{d}t =\langle q\rangle+\gamma\tau_{\mathrm{Dw}}(T)\,. \] Thus the dwell time can be determined from the shift of the average coordinate \[ \tau_{\mathrm{Dw}}(T)=\frac{1}{\gamma}(\langle q(T)\rangle-\langle q\rangle)\,. \]