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The second law of thermodynamics states that the entropy of an isolated system can only increase over time. This appears to conflict with the reversible evolution of isolated quantum systems under the Schr\"odinger equation, which preserves the von Neumann entropy. Nonetheless, one finds that with respect to many observables, expectation values approach a fixed value -- their equilibrium value. This ultimately raises the question: in what sense does the entropy of an isolated quantum system increase over time? For classical systems, one introduces the assumption of a low entropy initial state along with the concept of ignorance about the microscopic details of the physical system, leading to a statistical interpretation of the second law. By considering the observables through which we examine quantum systems, both these assumptions can be incorporated, building upon recent studies of the equilibration on average of observables. While the statistical behavior of observable expectation values is well-established, a quantitative connection to entropy increase has been lacking so far. In deriving novel bounds for the equilibration of observables, and considering the entropy of the system relative to observables, we recover a variant of the second law: the entropy with respect to a given observable tends towards its equilibrium value in the course of the system's unitary evolution. These results also support recent findings which question the necessity of non-integrability for equilibration in quantum systems. We further illustrate our bounds using numerical results from the paradigmatic example of a quantum Ising model on a chain of spins. There, we observe entropy increasing up to equilibrium values, as well as fluctuations which expose the underlying reversible evolution in accordance with the derived bounds., Comment: 7+10 pages; 3 figures; update: clarifications OE, Lemma 3; comments welcome

The challenge of understanding quantum measurement persists as a fundamental issue in modern physics. Particularly, the abrupt and energy-non-conserving collapse of the wave function appears to contradict classical thermodynamic laws. The contradiction can be resolved by considering measurement itself to be an entropy-increasing process, driven by the second law of thermodynamics. This proposal, dubbed the Measurement-Equilibration Hypothesis, builds on the Quantum Darwinism framework derived to explain the emergence of the classical world. Measurement outcomes thus emerge objectively from unitary dynamics via closed-system equilibration. Working within this framework, we construct the set of \textit{`objectifying observables'} that best encode the measurement statistics of a system in an objective manner, and establish a measurement error bound to quantify the probability an observer will obtain an incorrect measurement outcome. Using this error bound, we show that the objectifying observables readily equilibrate on average under the set of Hamiltonians which preserve the outcome statistics on the measured system. Using a random matrix model for this set, we numerically determine the measurement error bound, finding that the error only approaches zero with increasing environment size when the environment is coarse-grained into so-called observer systems. This indicates the necessity of coarse-graining an environment for the emergence of objective measurement outcomes., Comment: 12 + 8 pages, 5 figures

A curious feature of quantum tunneling known as the MacColl-Hartman effect results in the numerical observation that particles can traverse a barrier with effective superluminal speed. However, because tunneling is never certain, any attempt to use this effect to send a signal faster than light would require sending many particles. In this work, we consider sending -- in parallel, without interactions between particles -- sufficiently many particles to ensure at the least one of them tunnels. In this case, in spite of the time advance of the mean time for a single tunneling particle, the mean time to send one bit of information is larger for tunneling particles than for the same number of free photons. This removes any possibility of superluminal signaling. We show that the mean time to send one bit using N particles is determined by the early-time tail of the distribution of tunneling times for one particle and that, when this early-time tail is highly accurately modeled using steepest descent, the MacColl-Hartman effect is seen to fade away., Comment: 10 pages, 3 figures, 1 appendix, 65 citations, 43 equations

The quantum eraser variant of the double-slit experiment, and its 'delayed choice' sub-variant, are considered from the perspective of weak value and weak measurement theory (which is briefly reintroduced here). The interference fringes that appear when measuring certain spin states, which can then be 'erased' when measuring other spin states, are shown to be anomalous weak values that depend on particular post-selection choices. By framing the choice of spin measurement as a weak value of a certain weak measurement, it is then made clear what physical claims can and cannot be made about what occurs in the quantum eraser experiment. Specifically, claims about the choice of spin-state `retrocausally' influencing the choice of slit(s) for the particles to travel through are discredited, and a simple framework is presented for understanding how the fringes arise and why they can be 'erased'., Comment: 9 pages, 1 figure, 26 references, submitted to European Journal of Physics

In this work, our purpose is to show how the symmetry of identical particles can influence the time evolution of free particles in the nonrelativistic and relativistic domains. For this goal, we consider a system of either two distinguishable or indistinguishable (bosons and fermions) particles. Two classes of initial conditions have been studied: different initial locations with the same momenta, and the same locations with different momenta. The flight time distribution of particles arriving at a `screen' is calculated in each case. Fermions display broader distributions as compared with either distinguishable particles or bosons, leading to earlier and later arrivals for all the cases analyzed here. The symmetry of the wave function seems to speed up or slow down propagation of particles. Due to the cross terms, certain initial conditions lead to bimodality in the fermionic case. Within the nonrelativistic domain and when the short-time survival probability is analyzed, if the cross term becomes important, one finds that the decay of the overlap of fermions is faster than for distinguishable particles which in turn is faster than for bosons. These results are of interest in the short time limit since they imply that the well-known quantum Zeno effect would be stronger for bosons than for fermions.Fermions also arrive earlier than bosons when they are scattered by a delta barrier. Furthermore, the particle symmetry does not affect the mean tunneling flight time and it is given by the phase time for the distinguishable particle., Comment: 25 pages, 1 table, 5 figures

Results for elastic atom-atom scattering are obtained as a first practical application of RmatReact, a new code for generating high-accuracy scattering observables from potential energy curves. RmatReact has been created in response to new experimental methods which have paved the way for the routine production of ultracold atoms and molecules, and hence the experimental study of chemical reactions involving only a small number of partial waves. Elastic scattering between argon atoms is studied here. There is an unresolved discrepancy between different argon-argon potential energy curves which give different numbers of vibrational bound states and different scattering lengths for the argon-argon dimer. Depending on the number of bound states, the scattering length is either large and positive or large and negative. Scattering observables, specifically the scattering length, effective range, and partial and total cross-sections, are computed at low collision energies and compared to previous results. In general, good agreement is obtained, although our full scattering treatment yields resonances which are slightly lower in energy and narrower than previous determinations using the same potential energy curve., Comment: 26 pages, 9 figures, 3 tables

Experiments are starting to probe collisions and chemical reactions between atoms and molecules at ultra-low temperatures. We have developed a new theoretical procedure for studying these collisions using the R-matrix method. Here this method is tested for the atom -- atom collisions described by a Morse potential. Analytic solutions for continuum states of the Morse potential are derived and compared with numerical results computed using an R-matrix method where the inner region wavefunctions are obtained using a standard nuclear motion algorithm. Results are given for eigenphases and scattering lengths. Excellent agreement is obtained in all cases. Progress in developing a general procedure for treating ultra-low energy reactive and non-reactive collisions is discussed., Comment: 18 pages, 6 figures, 3 tables, conference

Techniques for producing cold and ultracold molecules are enabling the study of chemical reactions and scattering at the quantum scattering limit, with only a few partial waves contributing to the incident channel, leading to the observation and even full control of state-to-state collisions in this regime. A new R-matrix formalism is presented for tackling problems involving low- and ultra-low energy collisions. This general formalism is particularly appropriate for slow collisions occurring on potential energy surfaces with deep wells. The many resonance states make such systems hard to treat theoretically but offer the best prospects for novel physics: resonances are already being widely used to control diatomic systems and should provide the route to steering ultracold reactions. Our R-matrix-based formalism builds on the progress made in variational calculations of molecular spectra by using these methods to provide wavefunctions for the whole system at short internuclear distances, (a regime known as the inner region). These wavefunctions are used to construct collision energy-dependent R-matrices which can then be propagated to give cross sections at each collision energy. The method is formulated for ultracold collision systems with differing numbers of atoms., Comment: Presented at Faraday Discussion on the Theory of Chemical Reactions Published in Faraday Discussions

Accurate and complete rotational, rotational-vibrational and rotational-vibrational-electronic line lists are calculated for sodium hydride: both the NaH and NaD isotopologues are considered. These line lists cover all ro-vibrational states of the ground ($X$~$^1\Sigma^+$) and first excited ($A$~$^1\Sigma^+$) electronic states. The calculations use available spectroscopically-determined potential energy curves and new high-quality, \textit{ab initio} dipole moment curves. Partition functions for both isotopologues are calculated and the effect of quasibound states is considered. The resulting line lists are suitable for temperatures up to about 7000~K and are designed for studies of exoplanet atmospheres, brown dwarfs and cool stars. In particular, the NaH $A-X$ band is found to show a broad absorption feature at about 385 nm which should provide a signature for the molecule. All partition functions, lines and transitions are available as Supplementary Information to this article and at \url{www.exomol.com}., Comment: 8 pages, 5 figures, 6 tables