Are collapse models testable with quantum oscillating systems? The case of neutrinos, kaons, chiral molecules

SUBJECT AREAS: QUANTUM MECHANICS THEORETICAL PHYSICS QUANTUM INFORMATION Are collapse models testable with quantum oscillating systems? The case of neutrinos, kaons, chiral molecules M. Bahrami1,2,9, S. Donadi2,3, L. Ferialdi2,3, A. Bassi2,3, C. Curceanu4, A. Di Domenico5,6 & B. C. Hiesmayr7,8 STATISTICAL PHYSICS 1 Received 3 May 2013 Accepted 13 May 2013 Published 6 June 2013 Correspondence and requests for materials should be addressed to A.B. () Department of Chemistry, K. N. Toosi University of Technology, 1587-4416 Tehran, Iran (on leave), 2Department of Physics, University of Trieste, Strada Costiera 11, 34014 Trieste, Italy, 3Istituto Nazionale di Fisica Nucleare, Trieste Section, Via Valerio 2, 34127 Trieste, Italy, 4Laboratori Nazionali di Frascati dell’INFN, Via E. Fermi 40, 00044 Frascati, Italy, 5Department of Physics, Sapienza University of Rome, P. le Aldo Moro 5, 00185 Rome, Italy, 6Istituto Nazionale di Fisica Nucleare, Sezione di Roma, P. le Aldo Moro 5, 00185 Rome, Italy, 7Masaryk University, Department of Theoretical Physics and Astrophysics, Kotlářśka 2, 61137 Brno, Czech Republic, 8University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, Austria, 9The Abdus Salam ICTP, Strada Costiera 11, 34151 Trieste, Italy. Collapse models provide a theoretical framework for understanding how classical world emerges from quantum mechanics. Their dynamics preserves (practically) quantum linearity for microscopic systems, while it becomes strongly nonlinear when moving towards macroscopic scale. The conventional approach to test collapse models is to create spatial superpositions of mesoscopic systems and then examine the loss of interference, while environmental noises are engineered carefully. Here we investigate a different approach: We study systems that naturally oscillate–creating quantum superpositions–and thus represent a natural case-study for testing quantum linearity: neutrinos, neutral mesons, and chiral molecules. We will show how spontaneous collapses affect their oscillatory behavior, and will compare them with environmental decoherence effects. We will show that, contrary to what previously predicted, collapse models cannot be tested with neutrinos. The effect is stronger for neutral mesons, but still beyond experimental reach. Instead, chiral molecules can offer promising candidates for testing collapse models. A great variety of important physical phenomena can be effectively described in a two-dimensional Hilbert space, when the system’s dynamics effectively involves only two relevant states. The most common examples include oscillatory, decaying and/or relaxation effects in: elementary particles (e.g., neutrino and kaon oscillation1,2), atoms (e.g., Rabi oscillation and spontaneous emission3), molecules (e.g., tunnelling in double-well potentials, like Ammonia inversion4–6), and crystals (e.g., spin relaxation7,8). In such systems, oscillations occur because the relevant states are not eigenstates of the system’s Hamiltonian. To be definite, and without loss of generality within the two dimensional formalism, let us take the eigenstates j1æ ^ 0 ~vx s ^z operator as the relevant states, and H ^x =2 as the Hamiltonian, where vx is the charand j2æ of the s acteristic oscillating frequency (for example, for Ammonia, vx 5 24 GHz is the inversion frequency). If we start ^z , we observe the coherent oscillation between j1æ and j2æ with frequency vx. In this from any eigenstate of s idealised situation, temporal oscillations remain coherent in time, with a constant amplitude. However, in practice they lose coherence and decay more or less rapidly, because the system is exposed to external noises. Such environmental effects can be effectively described by Lindblad-type master equations9–11. Oscillations become of great conceptual importance when the two relevant states j1æ and j2æ become ‘‘macroscopically’’ distinct. This is the typical situation with chiral molecules, as we will see. The observation of oscillations between two such states is directly connected with the highly-debated problem (both theoretically and experimentally) of the quantum-to-classical transition: how linear quantum mechanics copes with macroscopic classical variables, where ‘‘classical’’ implies no superposition9–12. The fundamental question is whether such ‘‘macroscopic oscillations’’ persist when the system increases in size (and assuming that environmental sources of noise are kept under control) as predicted by quantum mechanics, or alternatively if they unavoidably decay in time because of intrinsic nonlinear effects in the dynamics. This second possibility is predicted by collapse models12–27. Collapse models have been extensively studied in the literature. There has been also a rapid progress in experimental searches of nonlinear effects predicted by collapse models23, in particular by delocalizing large SCIENTIFIC REPORTS | 3 : 1952 | DOI: 10.1038/srep01952 1 www.nature.com/scientificreports massive objects with matter-wave interferometry and optomechanical techniques28–33. In order to motivate further experimental searches of such nonlinear effects, here we follow a different approach by studying how collapse affects naturally oscillating quantum systems. In these cases, it is not necessary to create the superpositions in the laboratory, as they appear spontaneously from the dynamics. Collapse models add stochastic and nonlinear terms to the Schrödinger dynamics, which induce the collapse of the wave function. In the most well-studied collapse models (CSL16, QMUPL20), a noise-field is nonlinearly coupled to the spatial degrees of freedom of any massive system, inducing the suppression of spatial coherence. These models are discussed later in the text. Here it suffices to say that, when restricting to a 2D Hilbert space, where the states j1æ and j2æ describe two different spatial configurations, then the collapse dynamics takes the form34: h v pffiffiffi x ^ z { hs ^z iÞdWt ^x dtz lðs d jyt i~ {i s 2  ð1Þ l ^z iÞ2 dt jyti, ^ z { hs { ðs 2 with Wt a standard Wiener process, and l the collapse rate depending on the size of system and the nature of oscillation. The last two terms of Eq. (1) induce the collapse of the wave function either to j1æ or j2æ, according to the Born probability rule. In experimental situations, only averages over the noise are relevant. These can be com^t :E½jyt ihyt j, where E½: denotes puted from the density matrix r ^t obeys the the stochastic average. It is not difficult to prove that r following Lindblad-type equation34: d vx l ^t ~{i ½s ^z ,^ ^z ,½s ^x ,^ rt { ½s rt , r dt 2 2 ð2Þ Quantum linearity (manifested by the oscillatory behavior) is well preserved when vx ? l, while nonlinearity (i.e., no quantum superposition) becomes dominant when l ? vx. In this way collapse models provide a quantitative description for the transition from the microscopic quantum world to the macroscopic classical one. For any give (...truncated)