Weak values obtained in matter-wave interferometry

Abstract

Weak values of the spin operatorŜz of massive particles, more precisely neutrons, have been experimentally determined by applying a novel measurement scheme. This is achieved by coupling the neutron's spin weakly to its spatial degree of freedom in a single-neutron interferometer setup. The real and imaginary parts as well as the modulus of the weak value are obtained by a systematical variation of pre-and post-selected ensembles, which enables to study the complex properties of spin weak values.The meaning of weak values has been an issue of heated debates ever since the concept was introduced by Aharonov, Albert and Vaidman (AAV) in 1988 [1]. Unlike in the von Neumann measurement approach, where the outcome of the measurement is an eigenvalue of the observable and the premeasurement state collapses into the corresponding eigenstate [2], the obtained values, so called weak values, may lie far outside the range of the observable's eigenvalues. A weak measurement of an ob-servable invokes three steps: (i) preparation of an initial quantum state |ψ i (pre-selection), (ii) a unitary coupling of with a dynamical variable of a probe system, via a coupling Hamiltonian, with the coupling being weak so that the system is minimally disturbed. (iii) postselection of the final quantum state |ψ f , by performing a standard projective measurement of another observablê B of the quantum system. Finally, the probe system is read out, yielding the weak value given byExperimental realizations of the procedure proposed by AAV have been performed using various optical setups [3-9]. More recently experiments addressing average trajectories of single photons in a two-slit interferometer [10], direct measurement of the quantum wavefunction [11], or violation of Heisenberg's measurementdisturbance relationship [12-14] have been performed. Measurements of weak values in at least partially nonphotonic schemes are extremely recent, involving transmons in superconducting circuits [15] and spontaneous emission of photons from atoms [16]. Moreover, most experiments have either measured purely real or purely imaginary weak values, not complex quantities. Quantum paradoxes such as the three-box problem [17], Hardy's paradox [18,19] or the quantum cheshire cat [20,21] have been demonstrated by using weak values.In this paper we report the full determination of the weak value of the Pauli spin operatorσ z , i.e., its real and imaginary part, as well as its absolute value (for simplicityσ z is the relevant observable rather thanŜ z = /2σ z ). The real and imaginary contributions of the weak value of σ z are measured using matter wave interferometry with neutrons. Our experiment illuminates a peculiarity of the weak value, namely that it is a complex number in general. Purely imaginary weak values have been utilized to observe amplification effects in weak-measurement-based quantum metrology [8,22,23]. However, the physical account of the imaginary part of the weak value is still under active discussion [24][25][26]. Therefore a systematic invest...

Confined contextuality in neutron interferometry: Observing the quantum pigeonhole effect

“…The infidelity I = sin 2 α between the partial path states corresponding to z = ±1 quantifies the measurement strength. Our weak measurement has infidelity I = 0.067 [19], compared to a strong measurement with α = 90 • and I = 1.…”

“…The neutron's path is used as a pointer to measure both the real and imaginary parts of Z w . This approach has already been successfully used to completely determine weak values of massive systems [19]. The experiment was conducted at the instrument S18 at the high flux research reactor of the Institute Laue-Langevin (ILL) in Grenoble, France.…”

“…A perfect silicon crystal selects neutrons with a wavelength of λ 0 = 1.91Å (λ/λ 0 ∼ 0.02) by Bragg reflection from a white neutron beam [19]. Between the monochromator and the interferometer crystal, two magnetically birefringent prisms (P) split the unpolarized beam in two beams, one with the neutron spin aligned parallel to the positive z-direction and one aligned antiparallel.…”

“…However, neither the BKS theorem, nor these experiments, specify which contexts are contradictory. In this article, using recently developed weak measurement techniques in neutron interferometry [14][15][16][17][18][19][20], we experimentally demonstrate which specific measurement context within a BKS-set (Fig. 1a) must contain contradictory value assignments, essentially confining the contextuality [21].…”

“…We measured the spin using neutron interferometry by performing path-dependent spin rotations, making the path a weakly-coupled meter for the spin (Fig. 2); conditioning the path measurements on spin postselections then reveals the desired weak values [19]. We collected seventeen inde- Figure 1.…”