Entire Wave Function Research Articles

Effect of a Moving Mirror on the Free Fall of a Quantum Particle in a Homogeneous Gravitational Field

We investigate the effect of time-dependent boundary conditions on the dynamics of a quantum bouncer—a particle falling in a homogeneous gravitational field on a moving mirror. We examine more particularly the way a moving mirror modifies the properties of the entire wavefunction of a falling particle. We find that some effects, such as the fact that a quantum particle hitting a moving mirror may bounce significantly higher than when the mirror is fixed, are in line with classical intuition. Other effects, such as the change in relative phases or in the current density in spatial regions arbitrarily far from the mirror are specifically quantum. We further discuss how the effects produced by a moving mirror could be observed in link with current experiments, in particular with cold neutrons.

Measuring the distance between quantum many-body wave functions

We study the distance of two wave functions under chaotic time evolution. The two initial states differ only by a local perturbation. To be entitled ‘chaos’ the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance d2(t) by investigating the difference of two reduced density matrices of the subsystem A that is spatially separated from the local perturbation. This distance d2(t) grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates d2(t) to the square of commutator (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem A. Among various one-dimensional spin- models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in d2(t) when the local perturbation and subsystem A are well separated. This so-called Lyapunov regime is absent in the spin- chain with local interaction only. After sufficiently long time evolution, d2(t) relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that d2(t) is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions.

Spatial transport of electron quantum states with strong attosecond pulses

This work follows up the work of Dimitrovsky, Briggs and co-workers on translated electron atomic states by a strong field of an atto-second laser pulse, also described as creation of atoms without a nucleus. Here, we propose a new approach by analyzing the electron states in the Kramers–Henneberger moving frame in the dipole approximation. The wave function follows the displacement vector . This allows arbitrarily shaped pulses, including the model delta-function potentials in the Dimitrovsky and Briggs approach. In the case of final-length single-cycle pulses, we apply both the Kramers–Henneberger moving frame analysis and a full numerical treatment of our 1D model. When the laser pulse frequency exceeds the frequency associated by the energy difference between initial and final states, the entire wavefunction is translated in space nearly without loss of coherence, to a well defined distance from the original position where the ionized core is left behind. This statement is demonstrated on the excited Rydberg states (n = 10, n = 15), where almost no distortion in the transported wave functions has been observed. However, the ground state (n = 1) is visibly distorted during the removal by pulses of reasonable frequencies, as also predicted by Dimitrovsky and Briggs analysis. Our approach allows us to analyze general pulses as well as the model delta-function potentials on the same footing in the Kramers–Henneberger frame.

On the 2-mode and k-photon quantum Rabi models

By mapping the Hamiltonians of the 2-mode and 2-photon Rabi models to differential operators in suitable Hilbert spaces of entire functions, we prove that the two models possess entire and normalizable wave functions in the Bargmann–Hilbert spaces only if the frequency [Formula: see text] and coupling strength [Formula: see text] satisfy certain constraints. This is in sharp contrast to the quantum Rabi model for which entire wave functions always exist. For model parameters fulfilling the aforesaid constraints, we determine transcendental equations whose roots give the regular energy eigenvalues of the models. Furthermore, we show that for [Formula: see text], the [Formula: see text]-photon Rabi model does not possess wave functions which are elements of the Bargmann–Hilbert space for all non-trivial model parameters. This implies that the [Formula: see text] case is not diagonalizable, unlike its RWA cousin, the [Formula: see text]-photon Jaynes–Cummings model which can be completely diagonalized for all [Formula: see text].

Photoelectron spectra with Qprop and t-SURFF

Calculating strong-field, momentum-resolved photoelectron spectra (PES) from numerical solutions of the time-dependent Schrödinger equation (TDSE) is a very demanding task due to the large spatial excursions and drifts of electrons in intense laser fields. The time-dependent surface flux (t-SURFF) method for the calculation of PES [Tao and Scrinzi (2012)] allows to keep the numerical grid much smaller than the space over which the wavefunction would be spread at the end of the laser pulse. We present an implementation of the t-SURFF method in the well established TDSE-solver Qprop [Bauer and Koval (2006)]. Qprop efficiently propagates wavefunctions for single-active electron systems with spherically symmetric binding potentials in classical, linearly (along z) or elliptically (in the xy-plane) polarized laser fields in dipole approximation. Its combination with t-SURFF makes the simulation of PES feasible in cases where it is just too expensive to keep the entire wavefunction on the numerical grid, e.g., in the long-wavelength or long-pulse regime. Program summaryProgram title: QpropCatalogue identifier: ADXB_v2_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADXB_v2_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 12458No. of bytes in distributed program, including test data, etc.: 86258Distribution format: at.gzProgramming language: C++.Computer: x86_64.Operating system: Linux.RAM: The memory requirements for calculating PES are determined by the maximum ℓ in the spherical harmonics expansion of the wave function and the number of momentum (or energy) values for which the PES are to be calculated. The example with the largest memory demand (large-clubs) uses approximately 6GB of RAM. The size of the numerical representation of a wavefunction during propagation is modest for the examples included (53 MB for the large-club example).Number of processors used: The evaluation of the PES can be distributed over up to Nk MPI processes (Nk is the number of momentum values).Catalogue identifier of previous version: ADXB_v1_0Journal reference of previous version: Comput. Phys. Comm. 174(2006)396Classification: 2.5.External routines: GNU Scientific Library, Open MPI (optional), BOOST (optional)Does the new version supersede the previous version?: For TDDFT calculations the previous version should be used.Nature of problem:When atoms are ionized by intense laser fields electrons may escape with large momenta (especially when rescattering is involved). This translates to a rapidly spreading wavefunction in numerical simulations of these systems thus rendering the calculation of PES very costly for increasing wave lengths and peak intensities.Solution method:The TDSE is solved by propagating the wavefunction using a Crank–Nicolson propagator. The wavefunction is represented by an expansion in spherical harmonics. In order to reduce the requirements with respect to the grid size the t-SURFF method is used to calculate PES.Reasons for new version:Using the window operator method to calculate PES is increasingly costly with increasing ponderomotive energies and pulse durations. The new version of Qprop provides an implementation of the t-SURFF method which allows the use of much smaller numerical grids.Summary of revisions:An implementation of the t-SURFF method and examples for calculating PES are provided in the new release.Restrictions:The dipole approximation for the laser interaction has to be applicable. t-SURFF is only implemented for velocity gauge. Furthermore a finite cutoff for long range binding potentials has to be used in the implemented t-SURFF method.Additional comments:For additional information see www.qprop.deRunning time:Depends strongly on the laser interaction studied. The examples given in this paper have run times from a few minutes to 12.5 hours.

Stationary bubbles and their tunneling channels toward trivial geometry

In the path integral approach, one has to sum over all histories that start from the same initial condition in order to obtain the final condition as a superposition of histories. Applying this into black hole dynamics, we consider stable and unstable stationary bubbles as a reasonable and regular initial condition. We find examples where the bubble can either form a black hole or tunnel toward a trivial geometry, i.e., with no singularity nor event horizon. We investigate the dynamics and tunneling channels of true vacuum bubbles for various tensions. In particular, in line with the idea of superposition of geometries, we build a classically stable stationary thin-shell solution in a Minkowski background where its fate is probabilistically given by non-perturbative effects. Since there exists a tunneling channel toward a trivial geometry in the entire path integral, the entire information is encoded in the wave function. This demonstrates that the unitarity is preserved and there is no loss of information when viewed from the entire wave function of the universe, whereas a semi-classical observer, who can see only a definitive geometry, would find an effective loss of information. This may provide a resolution to the information loss dilemma.

Phantom of the Hartle–Hawking instanton: connecting inflation with dark energy

If the Hartle-Hawking wave function is the correct boundary condition of our universe, the history of our universe will be well approximated by an instanton. Although this instanton should be classicalized at infinity, as long as we are observing a process of each history, we may detect a non-classicalized part of field combinations. When we apply it to a dark energy model, this non-classicalized part of fields can be well embedded to a quintessence and a phantom model, i.e., a quintom model. Because of the property of complexified instantons, the phantomness will be naturally free from a big rip singularity. This phantomness does not cause perturbative instabilities, as it is an effect emergent from the entire wave function. Our work may thus provide a theoretical basis for the quintom models, whose equation of state (EoS) can cross the cosmological constant boundary (CCB) phenomenologically.

Dynamics of magnetic shells and information loss problem

We investigate dynamics of magnetic thin-shells in three dimensional anti de Sitter background. Because of the magnetic field, an oscillatory solution is possible. This oscillating shell can tunnel to a collapsing shell or a bouncing shell, where both of tunnelings induce an event horizon and a singularity. In the entire path integral, via the oscillating solution, there is a non-zero probability to maintain a trivial causal structure without a singularity. Therefore, due to the path integral, the entire wave function can conserve information. Since an oscillating shell can tunnel after a number of oscillations, in the end, it will allow an infinite number of different branchings to classical histories. This system can be a good model of the effective loss of information, where information is conserved by a solution that is originated from gauge fields.

Measurements on the reality of the wavefunction

Quantum mechanics is an outstandingly successful description of nature, underpinning fields from biology through chemistry to physics. At its heart is the quantum wavefunction, the central tool for describing quantum systems. Yet it is still unclear what the wavefunction actually is: does it merely represent our limited knowledge of a system, or is it an element of reality? Recent no-go theorems argued that if there was any underlying reality to start with, the wavefunction must be real. However, that conclusion relied on debatable assumptions, without which a partial knowledge interpretation can be maintained to some extent. A different approach is to impose bounds on the degree to which knowledge interpretations can explain quantum phenomena, such as why we cannot perfectly distinguish non-orthogonal quantum states. Here we experimentally test this approach with single photons. We find that no knowledge interpretation can fully explain the indistinguishability of non-orthogonal quantum states in three and four dimensions. Assuming that some underlying reality exists, our results strengthen the view that the entire wavefunction should be real. The only alternative is to adopt more unorthodox concepts such as backwards-in-time causation, or to completely abandon any notion of objective reality.

Reflection and transmission at the apparent horizon during gravitational collapse

We examine the wave-functionals describing the collapse of a self-gravitating dust ball in an exact quantization of the gravity-dust system. We show that ingoing (collapsing) dust shell modes outside the apparent horizon must necessarily be accompanied by outgoing modes inside the apparent horizon, whose amplitude is suppressed by the square root of the Boltzmann factor at the Hawking temperature. Likewise, ingoing modes in the interior must be accompanied by outgoing modes in the exterior, again with an amplitude suppressed by the same factor. A suitable superposition of the two solutions is necessary to conserve the dust probability flux across the apparent horizon, thus each region contains both ingoing and outgoing dust modes. If one restricts oneself to considering only the modes outside the apparent horizon then one should think of the apparent horizon as a partial reflector, the probability for a shell to reflect being given by the Boltzmann factor at the Hawking temperature determined by the mass contained within it. However, if one considers the entire wave function, the outgoing wave in the exterior is seen to be the transmission through the horizon of the interior outgoing wave that accompanies the collapsing shells. This transmission could allow information from the interior to be transferred to the exterior.

Transmission microscopy without lenses for objects of unlimited size

We demonstrate experimentally, for the first time, a new form of lensless microscopy. The image we obtain contains the entire wavefunction emanating from the sample. Large scale, quantitative phase information can be measured, unlike in conventional (Zernike) methods. For light optical experiments, we can dispense with expensive high-quality lenses and the very large working distances available would allow remote monitoring of e.g., environmental cells without compromising resolution. In short wavelength microscopy (X-rays and electrons), where lens components are of very limited numerical aperture, the technique has revolutionary implications: objects of any lateral size or shape can be used and, for transmission electron imaging, resolution down to the scale of the wavelength is likely to be limited only by the presence of atomic vibrations.

The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities

We show that the group delay in tunnelling is not a traversal time but a lifetime of stored energy or stored probability escaping through both ends of the barrier. Because it is a lifetime associated with both forward (transmitted) and backward (reflected) fluxes, it cannot be used to define a group velocity for forward transit in cases where a wavepacket is mostly reflected. For photonic tunnelling barriers the group delay is identical to the dwell time which is also a property of an entire wavefunction with reflected and transmitted components. Theoretical predictions and experimental reports of superluminal group velocities in barrier tunnelling are re-interpreted.

Guiding paths and time-dependent basis sets for wavefunction propagation

Two methods are explored for evolving wavefunctions using short-time propagators of Herman–Kluk form [M.F. Herman, E.K. Kluk, Chem. Phys. 91 (1984) 271]. At each time step the wavefunction is expanded in a set of coherent states distributed in phase space about a guiding trajectory. A harmonic approximation for the potential allows the stability analysis to be done analytically. A Monte Carlo `path integral' form for the long-time propagator is derived and tested for a simple harmonic system. Another approach, where the entire wavefunction is evolved one step at time, is applied to wavepacket motion in a Morse potential.

Quasiclassical wave function in multidimensional quantum decay problems

The quantum decay rate of metastable state is calculated in quasiclassical accuracy. Specifically, the calculation is based entirely on the standard quasiclassical ansatz, where the wave function is put equal to exp-(action)/ħ, and where the action is expanded in powers of ħ. Such a calculation is well known in the case of one-dimensional systems, and not so well known for multidimensional systems. Therefore, this paper intends to give a comprehensive presentation of the quasiclassical method. A straightforward procedure leads to a decay rate which agrees with the one obtained by the instanton technique. However, a calculation of the entire wave function shows that the procedure is consistent only if the metastable state is dominated by an internal degree of freedom which is slow in comparison with the remaining degrees of freedom.

On the derivation of ℏ ⋅ d k/d t=F, the k-space form of Newton’s law for Bloch waves

A new derivation of ℏ ⋅ dk/dt=F is given that is both rigorous and comparatively simple. The central idea is to consider the time evolution of the expectation value of the translation operator of a state that is initially a pure Bloch wave, rather than attempting to solve the more complicated but unnecessary problem of the actual evolution of the entire wave function. Because interband transitions take place between states that have the same expectation value of the translation operator, a consideration of the time evolution of the latter makes it unnecessary to consider interband transitions at all.