Falsification of Leggett's model using neutron matter waves

According to Bell's theorem, no theory based on the joint assumption of realism and locality can reproduce certain predictions of quantum mechanics. Another class of realistic models, proposed by Leggett, that demands realism but abandons reliance on locality, is predicted to be in conflict with quantum mechanics. In this paper, we report on an experimental test of a contextual realistic model analogous to the model of Leggett performed with matter waves, more precisely with neutrons. Correlation measurements of the spin-energy entangled single-particle system show violation of a Leggett-type inequality by more than 7.6 standard deviations. Our experimental data falsify the contextual realistic model and are fully in favor of quantum mechanics.


Introduction
Although predictions of quantum theory have been verified with high accuracy, there persists a long-standing debate on whether the concepts of quantum mechanics can be extended to yield a deterministic description of nature, e.g. by introducing so-called hidden variables [1]. Experiments based on Bell's theorem [2] discard certain types of alternative theories to quantum mechanics [3][4][5][6][7][8], whereas the Kochen-Specker theorem [9][10][11][12][13][14] accentuates the incompatibility between such alternatives and quantum mechanics. In 2003, Leggett proposed a non-local realistic model [15], and a paper concerning an extension of Leggett's model appeared later [16]. The results of photon experiments are in conflict with Leggett's model [17][18][19][20]. A natural question arises as to whether it might be that nature uses different models depending on the physical systems. Thus, it is important to test a modelà la Leggett on neutrons, in addition to the ones already performed on photon pairs. In this work, we present a neutron matter-wave experiment, where correlations between spin and energy observables of single particles are measured. A clear violation of the contextual realistic model is demonstrated. This confirms that also for massive particles Leggett's assumptions cannot fully account for quantum predictions.
An EPR-Bell argument [1,2] is the best-known approach to examine whether local realism complies with quantum theory: one finds that quantum mechanical predictions cannot be reproduced by the class of local realistic theories. Although hidden-variable theories are introduced to maintain a realistic and deterministic description of nature, thereby providing alternative theories of quantum mechanics, experiments [3][4][5][6][7] show a contradiction between these theories and quantum mechanics. Another class of hidden-variable theories is noncontextual hidden-variable theories, in which the measured value v[A] of an observable A is assumed predetermined and not affected by a joint (or simultaneous) measurement of an observable B compatible with A ([A, B] = 0)-in other words, the measured value v[A] of observable A does not depend on the act and specific settings (the context) of the measurement of observable B [21]. Studies of non-contextual theories were started by Kochen and Specker [10]. Conflicts between non-contextual theories and quantum mechanics have been pointed out theoretically [22] and subsequently confirmed by experiment [8,[11][12][13][14].

Leggett's model
In 2003, Leggett proposed a class of non-local hidden-variable theories and proved that his model was incompatible with quantum predictions [15,23]. The first experimental test was carried out by extending conventional correlation measurements of linear polarizations to elliptical polarizations [17]. In a subsequent experiment, the rotational symmetry assumption of the bipartite correlation was removed [18] and a different approach to applying a finite number of measurement settings was realized [19]: both experiments clearly exhibit the incompatibility of non-local theories with quantum mechanics. Further correlation measurements [20] revealed violations of inequalities derived also from partially predetermined values.
Until now, non-local Leggett models have been examined only experimentally with photons [24,25]. In this paper we report experiments with massive particles, replacing non-local correlations by correlations between commuting and compatible observables. Starting with an observation of a violation of a Bell-like inequality [8], neutron interferometer experiments [26] have exploited entanglement between degrees of freedom for single neutrons and have accomplished studies of quantum contextuality: Kochen-Specker phenomena [13] as well as GHZ-like entanglement [27] have been demonstrated. Neutron polarimetric experiments [28] are suitable for investigating contextual modelsà la Leggett due to high intensities, efficient manipulations and insensitivity to ambient disturbances.
For a polarimetric test, the criteria of the first experimental study by Gröblacher et al [17] are used. We test the model assuming the system to possess two definite physical properties that correspond to two commuting observables (of a two-dimensional system) in quantum mechanics. The model to be tested here is based on the following assumptions. Following the development used in works dealing with non-local hidden-variable models [17,19] and assuming full rotational symmetry, a similar inequality can be employed to test Leggett's model in our experiment. Denoting the measurement settings for observables A and B by a 1 , a 2 and b 1 , b 2 , respectively (on the Bloch sphere, see figure 1(a)), the Leggett-like inequality is given by where E j ( a j ; φ), with j = 1, 2 (as in [19]), represent expectation values of joint (correlation) measurements at settings a j and b j with relative angle φ. We assume settings a 1 , a 2 and b 1 to lie in a single (equatorial) plane and b 2 to lie in a plane perpendicular to it: expectation values E 1 and E 2 are given by correlations in planes perpendicular to each other. The expectation value E j ( a j ; 0) is derived from joint measurements between a j and b j (see figure 1(a)). For a pure singlet state, quantum mechanics predicts the expectation values E j ( a j ; φ) = − a j · b j = − cos φ and for the S-function S QM (φ) = 2|1 + cos φ|. Maximum violation is expected at φ max ∼ 0.1π, resulting in a bound of the Leggett-like inequality S Legg = 3.797 and a quantum value of S QM = 3.899.

Experiment
The measurement is based on joint measurements of two commuting observables, A spin for the neutron's spin and B energy for the neutron's total energy (the sum of kinetic and potential energies), representing two degrees of freedom. Coherent manipulations of the energy degree of freedom and a realization of a triply entangled GHZ-like state (additionally using the path degree of freedom) have already been reported [27]. In the present experiment, a maximally entangled Bell-like state with spin basis states |⇑ and |⇓ , as well as energy basis states |E 0 and |E 0 −hω , is generated and subjected to successive energy and spin measurements. The experimental setup is displayed in figure 2.
The experiment was carried out at the research reactor facility TRIGA Mark II of the Vienna University of Technology. The incident beam from the reactor is monochromatized to λ = 1.96 Å by a precisely oriented pyrolytic graphite monochromator and propagates in the +y-direction. Passing through a bent Co-Ti super mirror array, the beam is highly polarized. The same technique is employed to analyze the polarization. A high-efficiency (nearly 100% [29]) BF 3 detector records the expectation values given in equation (1).
Two identical radio-frequency (RF) spin rotators are employed, each producing a sinusoidally oscillating magnetic field (∼1 G for a π/2 rotation) with ω = 40 kHz. They are about 20 cm long and made of an enameled copper wire wound on PVC pipes (diameter ∼4 cm). Both RF spin rotators are put in a homogeneous and static magnetic guide field (∼13 G) supplied by two rectangular coils. Under these conditions, the rotating-wave approximation is well justified. After tuning the guide field strength, scans of the magnetic field amplitude exhibit sinusoidal intensity modulations with more than 99% contrast. While the position of the first spin rotator (RF1) is fixed, the second RF spin rotator (RF2) is mounted on a translation table: this enables precise adjustment of neutron flight time between the two spin rotators.
By tuning the rotation angle of RF1 to π/2, a maximally entangled Bell-like state | Bell N is generated. The amplitude and phase of the oscillating magnetic field generated by RF2 are directly associated with the parameters of the measurement: the former enables tuning of the polar angle α and the latter the azimuthal angle β of the spin measurement. In addition,

Violation of the Bell-Clauser-Horne-Shimony-Holt inequality
A measurement of the Bell-CHSH-like inequality is carried out first. The Bell-CHSH inequality (see, for instance, [3]) is written in the form where E( a j , b k ) denotes the expectation value of joint measurements with settings a j and b k (see figure 1(b)). The rotation angles of both spin rotators are set to π/2, and thus all settings of a j and b k lie in the equatorial plane of the Bloch sphere (see figure 1(b)). The maximum violation is expected for directions a 1 [π/2, 0], a 2 [π/2, π/2], b 1 [π/2, π/4] and b 2 [π/2, 3π/4]. For our measurements, the azimuthal angles γ j of the energy measurement directions a j [π/2, γ j ] are fixed at values γ 1 = 0, γ 2 = π/2 and γ j = γ j + π , whereas the azimuthal angles β k of the spin measurement directions b k [π/2, β k ] are scanned. In practice, this means that the phase of RF2 is scanned at fixed positions of RF2. Typical intensity oscillations are displayed in figures 3(a) and (b). With a maximum intensity of about 300 neutrons per second, sinusoidal oscillations with extremely high contrast, all above 98% and some reaching almost 99%, are obtained. Following the procedure described in appendix B, the four expectation values of the joint measurements are extracted from the intensity modulations at angles β 1 = π/4, β 2 = 3π/4 and β k = β k + π . The experimental value of the Bell-CHSH-like inequality is S CHSH = 2.781 (15), which is clearly above the boundary 2 by about 53 standard deviations. This experiment confirms a high signal-to-noise ratio and the reliability of our setup.
In order to see the tendency of the violations, the parameter φ is tuned to eight different values between 0 and 0.226π . Again, the azimuthal angle β k of the sets b k is scanned. The S Legg -value of the Leggett-like inequality is determined as described above. Figure 4 shows a plot of the experimentally determined S Legg , together with the limit of the contextual model as well as the quantum mechanical prediction, calculated for a contrast of 99%. The experimental values follow the quantum mechanical prediction, and this clearly confirms the violation of Leggett's model for matter waves.
Our result derived from the neutron polarimetric experiment is in excellent agreement with quantum theoretical predictions. All errors include statistical and systematic errors: the systematic errors in these experiments are much smaller than those for interferometric experiments [8,13], where systematic errors mainly result from phase instability of the interferograms. It should be mentioned that, as in other experimental tests of Leggett's and Bell's inequalities, all expectation values (as well as intensity modulations in our experiments) are measured successively by using 'equivalently prepared' samples.
It is instructive to recall the well-known property of compatible measurements of (commuting, [A, B] = 0) observables: the second (B) measurement does not destroy the previous information obtained in the first (A) measurement. From this, one can conclude that 'A and B measurements do not interfere-the term compatible is indeed deemed appropriately' [30]. Quantum indefiniteness, i.e. that individual properties cannot be defined, fully [17][18][19] or partially [20], under non-local conditions is confirmed in experiments with correlated photon pairs. In our experiment such an indefiniteness becomes explicitly visible under contextual conditions, which refer to compatible measurements: correlations between measurements are observed even though they are compatible and non-interfering.
It is an open question how these correlations come from indefinite properties through non-interfering measurements.

Conclusion
In summary, we present a polarimetric experiment with neutron matter waves to study Leggett's model. The correlations for conventional Bell-CHSH settings, where all vectors representing observables are lying in a single plane, have been measured: the mean contrast reached 98.5%, which is the highest correlation obtained between commuting observables of massive particles. Furthermore, we measured correlations between observables not lying in a single plane in order to study a Leggett-like inequality. The parameter φ for the deviation angle was varied close to the point of maximum violation: the values follow the quantum mechanical predictions and clearly violate the Leggett-like inequality. This, in turn, confirms quantum indefiniteness under the contextual condition for massive particles.

2
⇑| + e −iβ sin where α and β denote the polar and the azimuthal angle of the spin measurement direction, respectively. For the energy measurement we use the observable where |E 1 ≡ |E 0 −hω and γ gives the azimuthal angle in the equatorial plane. The experimentally measured intensity I corresponds to the expectation value of the joint measurement observable A spin ⊗ B energy with respect to the state | Bell N , given by equation (2). We find that This theoretically derived intensity function can be compared to the intensity function calculated from plane wave theory for the experimental setup shown in figure 2 (we work in units of h = 2m = 1, with m the mass and |µ| the magnetic moment of the neutron, see also [31]). The first RF flipper generates the magnetic field B (1) = 1 |µ| (ω (1) r cos(ωt + ζ ), ω (1) r sin(ωt + ζ ), ω z ) T , consisting of a guide field ω z /|µ| in the z-direction and a rotating field in the x y-plane of strength ω (1) r /|µ|, frequency ω and fixed phase ζ induces a π/2 spin-flip (2 (1) r 1 = /2, where 1 denotes the time of flight through the first coil). The incoming plane wave | 0 = e ik 0 y |E 0 |⇑ is transformed into the state with k + = k 2 0 − ω z and k − rf = k 2 + + 2ω z − ω. The second RF flipper coil produces the magnetic field B (2) = 1 |µ| (ω (2) r cos(ωt +β), ω (2) r sin(ωt +β), ω z ) T , whereβ indicates the adjustable phase of the oscillating field. The distance between the two RF flippers is denoted by L. Behind RF2 we obtain where the rotation angle α/2 = ω (2) r τ 2 is tuned by adjusting the field strength ω (2) r (τ 2 is the time of flight through the second coil) and y = y + L. Passing the analyzer, which projects onto the |⇑ -component of the state, leads to where we have used the approximations k where a j = [π/2, γ j ] = [π/2, γ j + π] and b k = [α, β k ] = [α, β k + π ]. Using four expectation values as defined in equation (B.1), S Legg and S CHSH are calculated according to equations (1) and (3), respectively.