PHYSICAL REVIEW A 81, 053609 (2010) Noise-induced dephasing in neutron interferometry G. Sulyok, 1 Y. Hasegawa, 1 J. Klepp, 2 H. Lemmel, 1,3 and H. Rauch 1 1 Atominstitut, Technische Universit¨ at Wien, 1020 Vienna, Austria 2 Faculty of Physics, University of Vienna, 1090 Vienna, Austria 3 Institut Laue-Langevin, F-38042 Grenoble, France (Received 26 February 2010; published 11 May 2010) Decoherence phenomenona in a neutron interferometer are analyzed by simulation of the effects of an environment with magnetic noise fields. Basic calculations and experiments show the validity and limitations of this model system. In particular, loss and recovery of the interference pattern with controllable noise sources in both interferometer arms are discussed in detail. In addition, the decoherence behavior at high interference order, where Schr¨ odinger-cat-like states exist in the interferometer, is investigated. While at low interference order a smearing of the interference pattern is observed, at high interference order a smearing of the modulated momentum distribution occurs. DOI: 10.1103/PhysRevA.81.053609 PACS number(s): 03.75.Dg, 03.65.Yz, 42.50.−p I. INTRODUCTION Interferometers of different types have become standard tools for the demonstration of wave properties of massive particles, underlining the validity of quantum mechanics for particles like electrons [1,2] or neutrons [3], and even for atoms and molecules with high mass numbers [4,5]. Apart from imperfections of the whole interferometer setup, observation of interference properties is complicated because of decoherence effects caused by the environment. The theory of open quantum systems [6–9] provides explanations for the associated loss of coherence. The interaction between the observed system and the environment causes their entangle- ment and destroys the unitary evolution of the system and its quantum behavior. Experimental observations of decoherence processes have been reported, for example, with electrons coupling to an electron gas inside a semiconducting plate [10], with molecules colliding with background gases [11], and with molecules decohering by thermal emission of radiation [12]. A profound understanding of these decoherence phenomena also leads to a deeper understanding of the transition between the quantum regime and the classical world. In this context, Stern et al. [13] have proved that the loss of coherence can also be described by statistically distributed phase accumulations of the interfering waves. In the case of neutrons, these phases can be caused by the magnetic dipole interaction described by the Zeeman-Hamiltonian ˆ H =−µ  σ ·  B . During their flight through the field region, the neutrons accumulate a phase given by φ = (µ/¯ h)  σ ·   Bdt . In our experiments, the phase derives from a classical noise field that is time dependent and causes energy exchange in the form of photon absorption or emission, as calculated and measured by Summhammer and co-workers [14,15]. Since the states of the magnetic field do not change in the photon exchange process, there is no entanglement between the neutron and the field, but the effects of the statistically distributed phase shifts on the observed interference pattern are equivalent to the effects of a quantum-mechanical environment [13]. For a more detailed description of the connection between noise fields and decoherence effects in the framework of Lindblad master equations, see [16]. We investigate the dephasing effect as a function of the strength of the Gaussian noise field, which shows the response to the dynamical quantum phase. The geometrical phase remains unchanged since the field acts along the same direction and no area is enclosed owing to the excursion in parameter space. A study of the stability of the geometrical phase and its contribution to the dephasing process has been published recently [17]. Another prediction of decoherence theory concerns macro- scopically distinguishable states (so-called Schr¨ odinger-cat- like states) whose sensitivity to external fluctuations increases when their spatial separation increases [8,18]. In the neutron interferometer, these states can be produced by thick phase shifters, when the phase shifts become larger than the coher- ence lengths [19,20]. The work presented is organized as follows. In Sec. II, basic formulas are developed for calculating the interfer- ometer contrast when magnetic noise fields are applied in the interferometer. Section III A focuses on the experimental results for the standard interferometric setup (phase contrast measurements). Contrast behavior for a noise field with different frequency bandwidths is investigated. Further, for noise fields applied in each interferometer arm, both the cases of correlated and uncorrelated signals are discussed. In Sec. III B, the preparation of Schr¨ odinger-cat-like states in the neutron interferometer and their properties are explained. These states are then exposed to magnetic noise and the effect on the arising momentum modulation is examined. II. THEORY Before addressing the actual experiments, we briefly review the connection between the measured interferometer contrast and the neutron state. Following the density-matrix approach [21], we write the incoming state as ρ in =|0〉〈0|⊗ ρ spin =  1 0 0 0  ⊗ ρ spin . (1) For the path degree of freedom, there are only two possible states, namely, |0〉 (denoting the direction of the incoming O beam) and |1〉 (denoting the direction of the reflected 1050-2947/2010/81(5)/053609(8) 053609-1 ©2010 The American Physical Society