Suppose that a single particle wave function \(\psi_0\) is prepared in \(x<0\) with a detector waiting to measure the time of arrival at \(x=0\). Assuming that their interaction is negligible outside of \(x \geq 0\), the dynamics of the incoming particle is governed by a Schrödinger equation that is decoupled from the detector in \(x<0\). In contrast, a dominant interaction must take place in \(x \geq 0\) that rapidly transitions the apparatus from its “ready" state to a”post-detection" state residing in one of many channels \(\mathcal{H}_t\) that are labeled by the recorded arrival time.

The precision of the recorded arrival time depends in part on how quickly the arrival at \(x=0\) induces a transition in the state of the apparatus. However, several theoretical treatments of the measurement procedure have predicted that the probability of any registered arrival goes to zero in the limit as exact precision is achieved [1]–[7]. The dominant interaction in \(x\geq0\) has the unintended effect of reflecting a fraction of the incoming wave function, and this effect intensifies so that all wave functions are completely reflected as the arrival time measurement procedure is made arbitrarily precise. This phenomenon can be regarded as a particular instance of the quantum Zeno effect [8]–[10], or the related “watchdog" effect [11].

In this Letter, it is shown that quantum mechanics can accommodate an arbitrarily precise arrival time measurement procedure without the complete reflection of all incoming wave functions. The approach taken here differs from others who proposed normalizing the vanishing probability distribution [12], [13] or using quantum stroboscopy [14] to obtain a non-zero conditional probability distribution for the arrival time. A family of arrival time measurement procedures is constructed that admits a non-zero probability distribution, even in the limit as exact precision is achieved.

The detector model contains a two-level system with energy difference \(\hbar \omega\) that transitions to its lower energy state upon detection. The incoming particle is absorbed upon detection and, at that same instant, a clock particle of mass \(M\) is emitted. A crucial aspect of the set-up is the placement of a wall, modeled through a Neumann boundary condition near \(x=0\), to ensure that the incoming particle does not penetrate too far into the detector region (see [15] for the physical meaning of this boundary condition). It is shown that a family of exact arrival time measurement procedures emerge in the limit as \(\omega,M\to \infty\) with \(2\hbar\omega/M\to v_c^2>0\). In this limit, the emitted clock particle is classically transported with constant velocity \(v_c\) so that its position forms a permanent record of the arrival time.

It is also shown that, in this limit, the localized interaction between the incoming particle and the apparatus converges to an absorbing boundary condition (ABC) at \(x=0\). ABCs have been studied as phenomenological models of localized detectors by several authors [16], [17], thesis?, but Tumulka [18] is most notable for proposing that ABCs may describe detectors capable of registering a particle at the instant of arrival. This analysis shall justify his claim by demonstrating that ABCs emerge as the registration process of an arbitrarily precise arrival time measurement procedure.

A direct measurement of the arrival time at \(x=0\) consists of three distinct steps: the registration of the particle exiting \(x<0\); the formation of a macroscopic record of the arrival time; and the preservation of this record for future observation. Achieving exact precision requires that for any incoming wave function:

1. the registration process in \(x \geq 0\) must be sufficiently rapid, preferably instantaneous, so that parts of the wave function exiting \(x< 0\) are quickly transferred to the post-detection state space;

2. parts of the wave function exiting \(x<0\) must be brought to the correct time-labeled channel \(\mathcal{H}_t\);

3. records must remain stable. States residing in \(\mathcal{H}_t\) may not transition to any state residing in a different time-labeled channel \(\mathcal{H}_{t'}\), or back to a ready state, otherwise the registration process may be repeated. Detections must be effectively irreversible.

A family of measurement procedures that satisfy the above conditions can be constructed as follows. Motivated by Halliwell [19], the detector consists of a two-level system with states \(|\text{R}\rangle\) and \(|\text{PD}\rangle\) representing the ready state and the post-detection state, respectively. The Hamiltonian of the detector is \[\begin{align} \hat{H}_D=0 |\text{R}\rangle \langle \text{R}| -\hbar \omega |\text{PD}\rangle \langle\text{PD}|, \end{align}\] so that \(|\text{R}\rangle\) and \(|\text{PD} \rangle\) are energy eigenstates of \(\hat{H}_D\), with eigenvalues \(0\) and \(-\hbar\omega\), respectively.

The entire system is initially prepared as a pure-product \(\Psi_0=\psi_0\otimes |\text{R}\rangle\), where \(\psi_0\) is supported in \(x<0\). \(\Psi\) evolves unitarily in a Hilbert space \(\mathcal{H}_\text{ND}\oplus\mathcal{H}_\text{PD}\) that contains two sectors. The dynamics in the ready state space \(\mathcal{H}_\text{ND}=L^2((-\infty,\varepsilon))\otimes |\text{R}\rangle\) models an incoming particle of mass \(m\) while the detector remains ready. The detection region is allowed to occupy a small interval \((0,\varepsilon)\) that will soon be localized to \(x=0\). A wall, modeled through a Neumann boundary condition, is placed at \(x=\varepsilon\) to ensure that the particle does not penetrate past that point. The dynamics in the post-detection state space \(\mathcal{H}_\text{PD}=L^2((-\infty,\varepsilon))\otimes |\text{PD}\rangle\) models an emitted clock particle of mass \(M\), so the particle Hamiltonian is \[\hat{H}_p=-\frac{\hbar^2}{2m}\partial_x^2~|\text{R}\rangle\langle \text{R}| - \frac{\hbar^2}{2M}\partial_x^2~|\text{PD}\rangle \langle\text{PD}|.\] The absorption of the incoming particle and subsequent emission of the clock particle is implemented through an interaction Hamiltonian that couples the two sectors \[\hat{H}_I= W(x)|\text{R}\rangle \langle \text{PD}|+W^*(x) |\text{PD} \rangle\langle\text{R}|.\] Here, \(W(x)\) is a complex-valued function supported in the detection region \((0,\varepsilon)\). Expanding the total wave function as \(\Psi_t=\psi_t\otimes |\text{R}\rangle + \phi_t\otimes |\text{PD}\rangle\), one finds that \(i\hbar \partial_t\Psi=(\hat{H}_D+\hat{H}_p+\hat{H}_I)\Psi\) can be written as a coupled system of one-body Schrödinger equations \[i\hbar \partial_t\begin{bmatrix} \psi\\ \phi \end{bmatrix}=\begin{pmatrix} -\frac{\hbar^2}{2m}\partial_x^2 &W(x) \\ W^*(x)&-\frac{\hbar^2}{2M}\partial_x^2-\hbar\omega \end{pmatrix}\begin{bmatrix} \psi \\ \phi \end{bmatrix}.\] These evolution equations are supplemented with the Neumann boundary conditions \(\partial_x\psi(\varepsilon)=0=\partial_x\phi(\varepsilon)\).

It shall now be described how an appropriate limit of these dynamics returns a family of exact arrival time measurement procedures. To localize the detection process at \(x=0\), set \(W(x)=W\theta(x)/\varepsilon\) and take \(\varepsilon \to 0\). After repeating similar steps to those taken in [20], [21], one finds that the limiting dynamics is governed by a system of free Schrödinger equations in \(x<0\) coupled through their boundary conditions at \(x=0\) \[\begin{equation} i\hbar \partial_t\begin{bmatrix} \psi\\ \phi \end{bmatrix}=\begin{pmatrix} -\frac{\hbar^2}{2m}\partial_x^2 &0 \\ 0&-\frac{\hbar^2}{2M}\partial_x^2-\hbar\omega \end{pmatrix}\begin{bmatrix} \psi \\ \phi \end{bmatrix}, \end{equation} \begin{eqnarray} \partial_x\psi(0)=-\frac{2m}{\hbar^2}W\phi(0), ~~ \partial_x\phi(0)=-\frac{2M}{\hbar^2}W^*\psi(0). \end{eqnarray}\] As is, this procedure admits several defects. For a particle to act as a reliable clock, it must travel with a reliable velocity. However, the velocity, loosely speaking, of the emitted particle depends on the momentum of the incoming particle according to \(v_\text{em}=p_\text{em}/M=\sqrt{(p^2_\text{in}/m + 2\hbar\omega)/M}\). Moreover, the position of the emitted particle does not act as a stable arrival time record because the post-detection dynamics allows parts of \(\phi\) to disperse and even propagate back to the non-detection state space.

These defects can be cured by taking the limit as \(\omega,M\to \infty\) with \(\hbar \omega=Mv_c^2/2\) for some fixed velocity \(v_c>0\). Given that \(\psi\big{|}_{t=0}=\psi_0\) and \(\phi\big{|}_{t=0}=0\), it is shown in the appendix that the limiting dynamics converge and are governed by \[\label{stopwatch} \begin{equation} i\hbar \partial_t\begin{bmatrix} \psi\\ \phi \end{bmatrix}=\begin{pmatrix} -\frac{\hbar^2}{2m}\partial_x^2 &0 \\ 0&i\hbar v_c\partial_x-2\hbar \omega \end{pmatrix}\begin{bmatrix} \psi \\ \phi \end{bmatrix}, \end{equation} \begin{eqnarray} \partial_x\psi(0)=i\kappa\psi(0), \quad \phi(0)&=-i\frac{2W^*}{\hbar v_c}\psi(0), \end{eqnarray}\tag{1}\] where \(\kappa=4m|W|^2/(v_c\hbar^3)>0\). The detection process is now effectively irreversible, and its singular interaction with the incoming particle is encapsulated by the absorbing boundary condition \(\partial_x\psi(0)=i\kappa\psi(0)\). This boundary condition ensures that probability irreversibly flows out of \(x<0\) through \(x=0\) with rate \[\begin{align} -\frac{d}{dt}||\psi||_{L^2}^2&&=j_x(0)=\frac{\hbar}{m}\text{Im}(\psi^*(0)\partial_x\psi(0)) \nonumber \\ &&=\frac{\hbar \kappa}{m}\left |\psi(0)\right|^2\geq0. \end{align}\] Parts of \(\psi\) that exit \(x<0\) are instantly transferred to the post-detection state space, and there, they undergo a classical transport to the left with constant velocity \(v_c\) \[\phi_t=\begin{cases} \frac{-2iW^*}{\hbar v_c}e^{-2i\omega x/v_c}\psi_{t+x/v_c}(0) \quad &\text{for }x>-v_ct \\ 0 \quad &\text{for }x<-v_ct \end{cases}\] The position of the emitted clock particle precisely encodes the time of arrival, in that the distance of the emitted particle from \(x=0\) divided by \(v_c\) records the time that has passed since the incoming particle arrived at \(x=0\). This record remains stable for all time since the transport equation does not allow interference between post-detection states with disjoint support. Moreover, for any time \(t>0\), the Born rule for the probability that the clock is “on the record" for an arrival having occurred at \(\tau\in(t_1,t_2)\subset (0,t)\) agrees with the probability flux distribution \[\begin{align} \mathbb{P}(\tau&\in (t_1,t_2))=\mathbb{P}(x\in (v_c(t_1-t),v_c(t_2-t)) \nonumber \\ &=\int_{v_c(t_1-t)}^{v_c(t_2-t)}dx ~|\phi_t(x)|^2 \nonumber \\ &=\int_{t_1}^{t_2}d\tau ~\frac{\hbar \kappa}{m}|\psi_\tau(0)|^2=\int_{t_1}^{t_2}d\tau ~j_x(0). \end{align}\] The infinitely oscillating term appearing in the solution formula for \(\phi_t\) may raise eyebrows. This term does not influence the statistics of recorded arrival outcomes and is a consequence of the obscene amount of energy that the two-level system dumps onto the emitted particle. It also ensures that the probability current for the post-detection state converges with \[\frac{\hbar}{M} \text{Im}\left(\phi^* \partial_x\phi \right)\xrightarrow[M=2\hbar \omega/v_c^2]{\omega \to \infty} -v_c|\phi|^2.\]

It has been shown that quantum mechanics can accommodate an exact arrival time measurement procedure without the almost sure reflection of every incoming particle. For the arrival time context, this analysis answers the question posed by Misra and Sudarshan [8]: there is no fundamental principle in quantum mechanics that denies the possibility of precise “continuous observation".

The procedure constructed here does admit some reflection. To quantify this, consider an incoming plane wave of momentum \(p\). Then, the absorbing boundary condition \(\partial_x\psi(0)=i\kappa \psi(0)\) generates a reflected plane wave with amplitude \[\psi=e^{ipx/\hbar}+r(p)e^{-ipx/\hbar} \quad \Rightarrow \quad r(p)=\frac{p-\hbar \kappa}{p+\hbar \kappa}.\] So, this measurement procedure admits almost no reflection of incoming wavepackets closely centered around \(p=\hbar \kappa=4m|W|^2/(v_c\hbar^2)\). Note that Aharonov et al. [4] conjectured the possibility of such an exact arrival time measurement procedure. However, they proposed using an energy booster that artificially preselects a preferred momentum for detection, while in this model \(\hbar\kappa\) is an apparatus-dependent quantity that emerges naturally from the particle-detector interaction.

The probability of non-arrival \(|r(p)|^2\) becomes strictly greater than \(1/9\) for \(p\) outside the interval \([\hbar\kappa/2, 2\hbar \kappa]\), and it is visible from Figure 1 that this probability increases to \(1\) as \(p\) is taken to \(0\) or \(\infty\). Thus, this analysis ultimately serves to reinforce the conjecture of Allcock [1] and Aharonov et al. [4] that exceptionally precise arrival time measurement admit a partial quantum Zeno effect that distorts the arrival outcomes in an apparatus-dependent manner. There are indications that these partial reflections may be reduced with a more sophisticated set-up. The model presented here remains amenable to modification and it is possible that the strategic use of singular interaction potentials, such as those in [22]–[24], could lead to other exact measurement procedures with greater absorption.

1 Appendix↩︎

Let \(\kappa\geq0\). The Laplace transform can be applied to derive an explicit solution to the Schrödinger equation in \(x<0\) with an absorbing boundary condition at \(x=0\) \[\label{ABC32Schrod} i\hbar\partial_t\psi=-\frac{\hbar^2}{2m}\partial_x^2\psi, \quad \partial_x\psi(0)=i\kappa \psi(0).\tag{2}\] Set \(\tilde{\psi}_s:=\mathcal{L}\psi=\int_0^\infty dt ~e^{-st}\psi_t\) for \(\text{Re}(s)>0\). Taking the Laplace transform of Eq. (2 ) returns \[i\hbar(s\tilde{\psi}_s-\psi_0)=-\frac{\hbar^2}{2m}\partial_x^2\tilde{\psi}_s, \quad \partial_x \tilde{\psi}_s(0)=i\kappa \tilde{\psi}_s(0).\] This equation is subject to an additional constraint \(\lim_{x \to -\infty}\tilde{\psi}_s=0\). An explicit solution formula for \(\tilde{\psi}_s\) can be derived and expressed in terms of \(\psi_0\) as \[\tilde{\psi}_s=G_F+\frac{p-\hbar \kappa}{p+\hbar\kappa}G_R,\] where \(p:=\sqrt{2m\hbar i s}\) is defined via the principal square root, so \(\text{Re}(ip)<0\), and \[\label{Greens} \begin{eqnarray} G_F:=&\int_{-\infty}^0 dx'~ \psi_0(x')~\frac{m}{p}e^{ip|x-x'|/\hbar}, \\ G_R:=&\int_{-\infty}^0 dx'~\psi_0(x')~\frac{m}{p}e^{-ip(x+x')/\hbar}. \end{eqnarray}\tag{3}\] The first term corresponds to the free evolution, while the second term contains the reflected contributions. The solution for \(\psi_t\) follows from the inverse Laplace transform

\[\label{ABC32Solution} \psi_t=\frac{1}{2\pi \hbar}\int_{-\infty}^{\infty}dE ~e^{-iEt/\hbar}\left(G_F + \frac{p-\hbar\kappa}{p+\hbar\kappa}G_R\right).\tag{4}\] Here, the substitution \(E=i\hbar s\) has been performed so that \(p(E)=\sqrt{2mE}\).

Now, let \(W \in \mathbb{C}\) and \(\omega\geq0\). The Laplace transform can also be applied to study the system of Schrödinger equations in \(x<0\) coupled through their boundary conditions \(x=0\) \[\begin{equation} i\hbar \partial_t\begin{bmatrix} \psi\\ \phi \end{bmatrix}=\begin{pmatrix} -\frac{\hbar^2}{2m}\partial_x^2 &0 \\ 0&-\frac{\hbar^2}{2M}\partial_x^2-\hbar\omega \end{pmatrix}\begin{bmatrix} \psi \\ \phi \end{bmatrix}, \end{equation} \begin{eqnarray} \partial_x\psi(0)=-\frac{2m}{\hbar^2}W\phi(0), ~ \partial_x\phi(0)=-\frac{2M}{\hbar^2}W^*\psi(0), \end{eqnarray}\] with \(\psi\big{|}_{t=0}=\psi_0\) and \(\phi \big{|}_{t=0}=0\). Setting \(\tilde{\psi}_s:=\mathcal{L}\psi\) and \(\tilde{\phi}_s:=\mathcal{L}\phi\) for \(\text{Re}(s)>0\), the transformed equations read \[\begin{equation} i\hbar \begin{bmatrix} s \tilde{\psi}_s-\psi_0\\ s\tilde{\phi}_s \end{bmatrix}=\begin{pmatrix} -\frac{\hbar^2}{2m}\partial_x^2 &0 \\ 0&-\frac{\hbar^2}{2M}\partial_x^2-\hbar\omega \end{pmatrix}\begin{bmatrix} \tilde{\psi}_s \\ \tilde{\phi}_s \end{bmatrix}, \end{equation} \begin{eqnarray} \partial_x\tilde{\psi}_s(0)=-\frac{2m}{\hbar^2}W\tilde{\phi}_s(0),~~ \lim_{x \to -\infty}\tilde{\psi}_s=0, \\ \partial_x\tilde{\phi}_s(0)=-\frac{2M}{\hbar^2}W^*\tilde{\psi}_s(0),~~ \lim_{x \to -\infty}\tilde{\phi}_s=0. \end{eqnarray}\] Now, the general solution for \(\tilde{\phi}_s\) takes the form \[\tilde{\phi}_s=Te^{-iqx/\hbar},\] where \(q:=\sqrt{2M\hbar(is+\omega)}\) and \(T \in \mathbb{C}\). The boundary condition for \(\tilde{\phi}_s\) returns \[T=\frac{-2iM}{\hbar q}W^*\tilde{\psi}_s(0),\] which reduces the boundary value problem for \(\tilde{\psi}_s\) to \[\begin{equation} i\hbar (s\tilde{\psi}_s - \psi_0)=-\frac{\hbar^2}{2m}\partial_x^2\tilde{\psi}_s \end{equation} \begin{equation} \partial_x\tilde{\psi}_s(0)=i\frac{4mM}{\hbar^3q}|W|^2\tilde{\psi}_s(0), \quad \lim_{x \to -\infty}\tilde{\psi}_s=0. \end{equation}\] Introducing \(\alpha:=4mM|W|^2/(\hbar^3 q)\), the solution formula for \(\tilde{\psi}_s\) can now be expressed as \[\tilde{\psi}_s=G_F+\frac{p-\hbar\alpha}{p+\hbar\alpha}G_R,\] where \(p=\sqrt{2m\hbar i s}\) and \(G_F\) and \(G_R\) are defined in Eq. (3 ). The inverse Laplace transform returns \[\label{dynamics} \psi_t=\frac{1}{2\pi \hbar}\int_{-\infty}^{\infty}dE ~e^{-iEt/\hbar}\left(G_F+\frac{p-\hbar\alpha}{p+\hbar\alpha}G_R \right).\tag{5}\] Here, \(p(E)=\sqrt{2mE}\) and \(q(E)=\sqrt{2M(E+\hbar \omega)}\). Now, let \(v_c>0\) and set \(M=2\hbar\omega/v_c^2\). Then \[q/M=\sqrt{2(E+\hbar\omega)/M}=\sqrt{v_c^2 +(Ev_c^2/\hbar\omega}).\]By dominated convergence theorem, the limit as \(\omega \to \infty\) of the solution formula (5 ) converges to Eq. (4 ) with \(\kappa=4m|W|^2/(\hbar^3 v_c)\), as desired.

To analyze the dynamics of the post-detection state \(\phi_t\), one may employ the stationary phase method. Recall that \(\tilde{\phi}_s\) was given by \[\tilde{\phi}_s= \left(\frac{M}{q}e^{-iqx/\hbar}\right) \left( \frac{-2i}{\hbar}W^*\tilde{\psi}_s(0)\right),\] where \(-iq/\hbar=-i\sqrt{2M(is+\omega)/\hbar}=\sqrt{-2Mi(s-i\omega)/\hbar}\) since \(\text{Re}(s)>0\). Then, setting \(M=2\hbar\omega/v_c^2\), the post-detection state can be expressed as a convolution \[\phi_t=-\frac{2W^*}{\hbar v_c}\sqrt{\frac{i\omega}{\pi}}\int_{0}^t d\tau ~\frac{\psi_{t-\tau}(0)}{\sqrt{\tau}}~e^{i\omega( \tau + x^2/(v_c^2\tau))}.\] For \(\omega\) sufficiently large, this can be approximated using the stationary phase method as \[\phi_t\approx\begin{cases} \frac{-2iW^*}{\hbar v_c}e^{-2i\omega x/v_c}\psi_{t+x/v_c}(0) \quad &x>-v_ct \\ 0 \quad &x<-v_ct \end{cases}\] plus terms of order \(O(\omega^{-1})\). Hence, the limiting dynamics for \(\psi\) and \(\phi\) is governed by Eq (1 ) as desired.

References↩︎