Researchers Characterise Quantum States Via Intensity Measurements in Matter-Wave Interferometry

Ismaele V. Masiello and colleagues at TU Wien, in collaboration with Vienna Centre for Quantum Science and Technology (VCQ), have developed a technique to fully characterise path weak-values in a generalised Mach-Zehnder interferometer using only intensity measurements and controlled phase shifts. The technique determines weak values in quantum systems without the need for complex measurements. This advancement bypasses conventional methods requiring weak interactions and metre states, simplifying experimental setups and reducing measurement times. Their demonstration in matter-wave interferometry reveals anomalous weak values and negative quasiprobability distributions, confirming the nonclassical behaviour of the quantum system and offering a flexible approach for two-level quantum systems.

Interferometric determination of weak values using post-selection and phase shifting

Intensity measurements at the output ports of the interferometer, combined with controlled relative phase shifts between the paths, are uniquely relied upon. A measurement outcome known as the weak value arises when studying the physical properties of a quantum system between a pre-selected and a post-selected ensemble, specifically between its initial state |ψin⟩ and final state |ψfi⟩, while examining an observable A with weak measurements. The weak value is defined as ⟨ψfi| A|ψin⟩ ⟨ψfi|ψin⟩, and unlike the expectation value, this quantity can be complex and may lie outside the eigenvalue range of the observable.

Since the first implementation of a weak measurement, the physical relevance of weak values has been confirmed across a range of impactful experiments. For example, weak-value amplification enabled the observation of tiny pointer’s shifts, leading to the first measurement of the spin Hall effect of light. They are also employed for the direct measurement of the wave function and for probing nonlocal observables in Hardy’s paradox. Experiments extending beyond photonic systems, including observations of the quantum Cheshire cat and the quantum pigeonhole effect performed using neutron interferometry, demonstrate the technique’s application to diverse systems.

While the interpretation of weak values remains a topic of discussion, they are found to play a role in several fundamental theoretical studies of quantum mechanics, such as quantum paradoxes, uncertainty relations, quasiprobability distributions, and nonclassicality. These far-reaching implications confirm that weak values constitute an extremely valuable extension of the standard measurement framework when one seeks to characterise quantum systems between preparation and post-selection, both conceptually and experimentally. Effective and accurate methods of extraction are essential to exploit the advantages of weak values experimentally.

Conventionally, weak values are obtained from weak measurements in which a metre system interacts weakly with the target system, followed by a readout of the metre. More recently, alternative schemes departed from this standard approach, including employing a classical parameter instead of a metre state while still maintaining a weak interaction, as well as using strong interaction while still retaining a metre state. The employment of metre states or the requirement of weak interactions can increase the experimental complexity.

Increasing the manipulation of additional degrees of freedom demands more experimental resources, such as additional instrumentation and calibration procedures, while weak interactions typically necessitate longer measurement times due to the limited information gained per detection event. In this paper, a method to extract the complete path weak-values in a generalised Mach-Zehnder interferometer is proposed, requiring neither metre states nor weak interactions. This approach relies only on intensity measurements at the output ports of the interferometer combined with a controlled relative phase shift between the paths.

Experimental verification performs using a neutron Mach-Zehnder interferometer, enabling the study of the most fundamental element of quantum mechanics, the two-level quantum system, under conditions that challenge classical models: a single massive spin-1/2 particle in a superposition of spatially separated paths. Compared with typical photonic implementations, neutron interferometry is less susceptible to classical interpretations of the results. The setup requires only a few commonly used optical elements: two phase shifters, one for state preparation and the other for state manipulation, an absorber to control the relative path intensities, and a beam blocker.

It is further shown that the same result can be obtained using a single phase shifter, reducing the number of required elements. Compared with previous neutron interferometer experiments extracting weak values using metre states or weak interactions, this method achieves equal or higher accuracy while substantially reducing the number of optical elements and sharply shortening the measurement time. The results confirm the presence of anomalous weak values, i.e., weak values whose real part exceeds the eigenvalue range, which provide insight into the nonclassical behaviour of the quantum system.

Neutron interferometry offers several advantages, such as macroscopic beam separation, individual control of the sub-beams, and long interaction and coherence times at room temperature and ambient pressure. The monolithic crystal structure of the interferometer provides stable experimental conditions, enabling interference contrast that can exceed 90%. Because of the fermionic nature of the neutron, the observed interference is intrinsically a single-particle phenomenon. The possibility to control additional degrees of freedom beyond the path states, such as spin and energy, makes neutron interferometry suitable for the investigation of nonclassical phenomena such as Bell-like inequalities and entanglement.

For these reasons, neutron interferometry has played a central role in experimental tests of fundamental aspects of quantum mechanics. The proposed scheme is not restricted to the path observables and post-selections of a generalised neutron Mach-Zehnder interferometer. The derivation is independent of the specific choice of two-level quantum system, observables, and post-selections, and therefore applicable to any quantum systems in which similar measurements and controlled phase shifts can be implemented.

This makes the presented method suitable for applications in a variety of experiments across different fields, for instance in atom-optics experiments using Rabi oscillations or atom interferometers, as well as polarimeter experiments. Consider the generalised Mach-Zehnder interferometer scheme. No assumptions are made about the nature of the interfering quantum particle (e.g., photon, neutron, atom), and the first beam splitter is allowed to have arbitrary transmission and reflection coefficients.

After passing through this beam splitter, the particle is prepared in a superposition of the path states |1⟩ and |2⟩, given by |ψin⟩ = cos θ 2 |1⟩ + eiφ sin θ 2 |2⟩, where cos2 θ 2 and sin2 θ 2 denote the relative intensities in path 1 and 2, respectively, and φ is the initial relative phase between the path states. The interferometer operates in the balanced configuration when θ = π 2, corresponding to the conventional 50/50 beam splitter. For any other value of θ, the configuration is referred to as unbalanced.

Next, an additional phase shift δ is introduced in path 1 transforming the initial state according to |ψin⟩ → e−iδ Π1 |ψin⟩, where Πj = |j⟩⟨j| is the path projector with j = {1, 2}. Finally, at the second beam-splitter of the interferometer, the system is projected onto the final states |ψ+⟩ = (|1⟩ + |2⟩) / √2 and |ψ−⟩ = (|1⟩ − |2⟩) / √2, corresponding to the two exit ports of the interferometer. This last beam splitter is assumed to be 50/50. The intensities I±(δ) measured at the output ports are proportional to the detection probability at the corresponding port I±(δ) = A ⟨ψ±|e−iδ Π1|ψin⟩ 2 = A 1/2 ±1/2 sin θ cos (φ + δ). This standard treatment of a generalised Mach-Zehnder interferometer predicts the complementary sinusoidal oscillations of the intensities measured at the two outgoing beams. For fixed relative path intensities, defined by θ, and initial phase shift φ, these oscillations depend on the phase δ and the obtained interference pattern is generally referred to as an interferogram.

Note that the amplitude of the interferogram is not proportional to the product of the relative path intensities cos2 θ 2 sin2 θ 2 = 1/4 sin2 θ, but on its square root. This distinction is key in differentiating between particles being in a quantum superposition of both paths and particles that are deterministically travelling through either one path or the other with a given probability. The path weak-values w±,j, characterised by the initial state |ψin⟩, the observable Πj and the final states |ψ±⟩, is a complex quantity defined as w±,j = ⟨ψ±|Πj|ψin⟩ ⟨ψ±|ψin⟩ = wR ±,j + i wI ±,j, where the terms wR ±,j and wI ±,j are, respectively, their real and imaginary part.

Weak values are usually described in terms of weak conditioned von Neumann measurements, with a metre state that is first weakly coupled to the target observable and then measured to extract the components of the weak value. In this treatment, no metre state is used, and the weak values can be extracted directly from measurements of the intensities of the outgoing beams, which depend on the phase shift parameter δ. In fact, the intensities I±(δ) can be rewritten as I±(δ) = A ⟨ψ±| h 1 + e−iδ −1 Π1 i |ψin⟩ 2 = A ⟨ψ±|ψin⟩ h 1 + 2 |w±,1|2 −wR ±,1 (1 −cos δ) + 2 wI ±,1 sin δ i = A |⟨ψ±|ψin⟩|2 h 1 + 2 |w±,2|2 −wR ±,2 (1 −cos δ) −2 wI ±,2 sin δ i, where the first line is obtained from the relation e−iδ Π1 = e−iδ Π1 + Π2 = e−iδ Π1 + Π2 = 1 + e−iδ −1 Π1, and the last line is derived by using the identity Π1 = 1 − Π2. This suggests the direct extraction of the path weak-values from the intensities I±(δ). For simplicity of notation, only the extraction of the weak values w+,j is focused on, whose explicit expressions are given by w+,1 = 1 / (1 + tan θ / 2) eiφ and w+,2 = 1 / (cotan θ / 2) e−iφ + 1. The derivation of the real and imaginary components of w−,j can be obtained analogously. In the previous subsection, the outgoing intensity I+(δ) of the generalised Mach, Zehnder interferometer was expressed in terms of the path weak-values. Each term in can be determined by appropriately tuning δ and exploiting the interferometer property I+(δ + π) = I−(δ), namely I+ = A |⟨ψ+|ψin⟩|2, I− − I+ / 4I+ = |w+,1|2 − wR +,1 = |w+,2|2 − wR +,2, I+( π / 2 ) − I−( π / 2 ) / 4I+ = wI +,1 = −wI +,2. The imaginary part wI +,j is directly obtained. The real part wR +,j enters quadratically, since |w+,j|2 = wR +,j 2 + wI +,j 2. This leads to two possible solutions: wR +,j = 1/2 ±1/2 √ (I− / I+ − 2 wI +,j 2) = 1/2 ±1/2 √ (I− / I+ − (I+( π / 2 ) − I−( π / 2 )) / 2I+). The two solutions, corresponding to the different signs of the square root, must be assigned to either wR +,1 or wR +,2. At this stage, however, the assignment remains ambiguous.

Rapid weak-value characterisation via accelerated neutron interferometry

Researchers at TU Wien and the Vienna Centre for Quantum Science and Technology have achieved a major advance in neutron interferometry, substantially reducing measurement times while maintaining accuracy. This new method fully characterises path weak-values, a measure of a quantum property between preparation and measurement, in a generalised Mach-Zehnder interferometer without employing conventional metre states or weak interactions. The technique relies on measuring the intensity of the neutron beams at the detector, combined with precisely controlled shifts in the phase of the waves, simplifying the experimental process and reducing data acquisition time.

The team demonstrated their technique using a neutron Mach-Zehnder interferometer, a device where a neutron beam is split and recombined to observe quantum interference effects. This setup allowed examination of a single massive spin-1/2 particle existing in a superposition of spatially separated paths. The team achieved interference contrast exceeding 90%, a level of stability facilitated by the monolithic crystal structure of the interferometer and its operation at room temperature and ambient pressure.

Direct path weak-value determination without conventional weak measurement techniques

Scientists have long sought methods to precisely characterise quantum systems, yet conventional approaches demand intricate setups and time-consuming measurements. The team at TU Wien and the Vienna Centre for Quantum Science and Technology have now bypassed the need for delicate weak interactions and metre states, simplifying the process of determining path weak-values, a measure of a quantum property between preparation and measurement. By removing the need for delicate ‘weak measurements’, the team has created a streamlined approach applicable to diverse quantum systems.

The researchers successfully characterised path weak-values in a matter-wave interferometer without using conventional weak measurements or meter states. This simplification reduces experimental complexity and measurement times while maintaining comparable accuracy in determining these quantum properties. The technique relies on intensity measurements and controlled phase shifts, offering a more efficient method for studying quantum systems. This approach is directly applicable to experiments involving two-level quantum systems, providing a versatile tool for future investigations.