Weak values obtained in matter-wave interferometry

Weak values, introduced more than 25 years ago, underwent a metamorphosis from a theoretical curiosity to a powerful resource in photonics for exploring foundations of quantum mechanics, as well as a practical laboratory tool. Due to the tiny coherence volume of particles used in matter-wave optics, a straightforward implementation of weak measurements is not feasible. We have overcome this hurdle by developing a method to weakly measure a massive particle's spin component. A neutron optical approach is realized by utilizing neutron interferometry, where the neutron's spin is coupled weakly to its spatial degree of freedom. Here, we present how one can fully characterize the weak value of the Pauli spin operator ${\widehat{\sigma }}_z$ of neutrons by extracting its real and imaginary components, as well as its modulus. The results show good agreement with theoretical predictions and demonstrate that the (spin) weak value is actually accessible for a purely quantum system of massive particles.

Figure 1

Schematic illustration of experimental apparatus for full determination of the weak value of the Pauli spin operator ${\widehat{\sigma }}_z$ of system $S$. (a) Triple Laue (LLL) neutron interferometer (IFM) setup for determination of the real component and modulus of ${\langle {\widehat{\sigma }}_z\rangle }_{\mathrm{w}}$. (b) Polarimetric setup for the imaginary part of ${\langle {\widehat{\sigma }}_z\rangle }_{\mathrm{w}}$, where no interference effects are necessary. The procedure consists of three stages: (i) preselection (green); using a polarizer (initial polarization is along $+z$ direction) and a $\pi /2$ spin rotator the initial state $|{\psi }_{\mathrm{i}}\rangle =|S_x;+\rangle$ is prepared. (ii) Weak interaction (blue); in case (a) after the beam splitting a weak spin rotation by $\pm \alpha$ is applied in arms I and II. In case (b) a spin rotation by $\pm \alpha$ is applied directly after the preparation of the initial state $|{\psi }_{\mathrm{i}}\rangle$. (iii) Postselection (yellow); a combination of a spin rotator and an analyzing supermirror is used to postselect the final spin state $|{\psi }_{\mathrm{f}}(\theta ,\varphi )\rangle =cos\left(\frac{\theta }{2}\right)|S_z;+\rangle +sin\left(\frac{\theta }{2}\right)e^{i\varphi }|S_z;-\rangle$, followed by a count rate detection. In case (a) the pointer readout, i.e., tuning of the path state $|{\phi }_{\mathrm{O}}\rangle$, being the two-level probe system $P$, is achieved by a rotating phase-shifter plate, inducing a relative phase $\chi$ between the path eigenstates $|\mathrm{I}\rangle$ and $|\mathrm{II}\rangle$.

Figure 2

Typical interference pattern recorded in the O detector without ($\alpha =0$, orange) and with ($\alpha ={15}^{\circ }$, green) weak spin rotations, which is applied in arms I and II, respectively. Right next to the plot of the interference fringes the corresponding Bloch-sphere representations of the respective pre- and postselected states, denoted as $|{\psi }_{\mathrm{i}}\rangle$ and $|{\psi }_{\mathrm{f}}\rangle$ are given. For polar angle (a) $\theta =\frac{5\pi }{6}$, (b) $\theta =\frac{\pi }{2}$, and (c) $\theta =\frac{4\pi }{3}$ with azimuthal angle $\varphi =0$ of postselected spin state $|{\psi }_{\mathrm{f}}(\theta ,\varphi )\rangle$. From the former the phase-shifter positions for $\frac{-\pi }{2},0,\frac{\pi }{2}$, and $\pi$ phase shifts are extracted and from the intensities $I_{-y},I_{+x},I_{+y}$, and $I_{-x}$, the real part and modulus of the weak value of ${\widehat{\sigma }}_z$ are determined. The relevant phase-shifter positions are indicated by dashed orange lines.

Figure 3

(a) Typical intensity oscillations for different values of $\theta$ and $\varphi =\frac{\pi }{2}$ of postselected spin state $|{\psi }_{\mathrm{f}}(\theta ,\varphi )\rangle$, recorded in the $P$ detector for paths I (cyan) and II (purple). From the values of the intensities $I_{+z}$ and $I_{-z}$ (paths I and II) the imaginary part of the weak value of ${\widehat{\sigma }}_z$ is determined. (b) Bloch-sphere representation of the pre- and postselected states, denoted as $|{\psi }_{\mathrm{i}}\rangle$ and $|{\psi }_{\mathrm{f}}(\theta ,\varphi )\rangle$.

Figure 4

Experimentally determined real (top panel) and modulus (bottom panel) and a direct measurement of the imaginary component (central panel), of the weak value of ${\widehat{\sigma }}_z$, together with the theoretical predictions (blue line). Bloch-sphere representations are given for the pre- and postselected spin states. (a) For $\varphi =0$ the real part of ${\langle {\widehat{\sigma }}_z\rangle }_{\mathrm{w}}$ exhibits values lying outside the usual range of spin eigenvalues, i.e., $\pm 1$, while the imaginary part remains zero. (b) For $\varphi =\frac{\pi }{2}$ real and imaginary components of ${\langle {\widehat{\sigma }}_z\rangle }_{\mathrm{w}}$ oscillate in quadrature yielding a constant value of $|{\langle {\widehat{\sigma }}_z\rangle }_{\mathrm{w}}|$ irrespective of the polar angle $\theta$ of the postselected state $|{\psi }_{\mathrm{f}}(\theta ,\frac{\pi }{2})\rangle$. For $\varphi =0,\theta =\pi /2$ the weak value of ${\widehat{\sigma }}_z$ equals the expectation value of ${\widehat{\sigma }}_z$ (red arrow).

Figure 5

Effect of the spin operators weak value on the wave function, depicted on the path degrees of freedom Bloch sphere: $\mathrm{Re}{\langle {\widehat{\sigma }}_z\rangle }_{\mathrm{w}}$ rotates the state vector in the $xy$ plane (left). $\mathrm{Im}{\langle {\widehat{\sigma }}_z\rangle }_{\mathrm{w}}$ tilts the state vector out of the $xy$ plane towards the poles (middle); as expected the modulus is simply a combination of both (right).