Tunneling Time and Weak Measurement in Strong Field Ionization

A unified theory of tunneling times promoted by Ramsey clocks

What time does a clock tell after quantum tunneling? Predictions and indirect measurements range from superluminal or instantaneous tunneling to finite durations, depending on the specific experiment and the precise definition of the elapsed time. Proposals and implementations use the atomic motion to define this delay, although the inherent quantum nature of atoms implies a delocalization and is in sharp contrast to classical trajectories. Here, we rely on an operational approach: We prepare atoms in a coherent superposition of internal states and study the time read-off via a Ramsey sequence after the tunneling process without the notion of classical trajectories or velocities. Our operational framework (i) unifies definitions of tunneling delay within one approach, (ii) connects the time to a frequency standard given by a conventional atomic clock that can be boosted by differential light shifts, and (iii) highlights that there exists no superluminal or instantaneous tunneling.

Non-local temporal interference

Although position and time have different mathematical roles in quantum mechanics, with one being an operator and the other being a parameter, there is a space-time duality in quantum phenomena-a lot of quantum phenomena that were first observed in the spatial domain were later observed in the temporal domain as well. In this context, we propose a modified version of the double-double-slit experiment using entangled atom pairs to observe a non-local interference in the arrival time distribution, which is analogous to the non-local interference observed in the arrival position distribution. However, computing the arrival time distribution in quantum mechanics is a challenging open problem, and so to overcome this problem we employ a Bohmian treatment. Based on this approach, we numerically demonstrate that there is a complementary relationship between the one-particle and two-particle interference visibilities in the arrival time distribution, which is analogous to the complementary relationship observed in the position distribution. These results can be used to test the Bohmian arrival time distribution in a strict manner, i.e., where the semiclassical approximation breaks down. Moreover, our approach to investigating this experiment can be applied to a wide range of phenomena, and it seems that the predicted non-local temporal interference and associated complementary relationship are universal behaviors of entangled quantum systems that may manifest in various phenomena.

Quantum Measurements and Delays in Scattering by Zero-Range Potentials

Eisenbud-Wigner-Smith delay and the Larmor time give different estimates for the duration of a quantum scattering event. The difference is most pronounced in the case where the de Broglie wavelength is large compared to the size of the scatterer. We use the methods of quantum measurement theory to analyse both approaches and to decide which one of them, if any, describes the duration a particle spends in the region that contains the scattering potential. The cases of transmission, reflection, and three-dimensional elastic scattering are discussed in some detail.

Reconciling Conflicting Approaches for the Tunneling Time Delay in Strong Field Ionization

Several recent attoclock experiments have investigated the fundamental question of a quantum mechanically induced time delay in tunneling ionization via extremely precise photoelectron momentum spectroscopy. The interpretations of those attoclock experimental results were controversially discussed, because the entanglement of the laser and Coulomb field did not allow for theoretical treatments without undisputed approximations. The method of semiclassical propagation matched with the tunneled wave function, the quasistatic Wigner theory, the analytical R-matrix theory, the backpropagation method, and the under-the-barrier recollision theory are the leading conceptual approaches put forward to treat this problem, however, with seemingly conflicting conclusions on the existence of a tunneling time delay. To resolve the contradicting conclusions of the different approaches, we consider a very simple tunneling scenario which is not plagued with complications stemming from the Coulomb potential of the atomic core, avoids consequent controversial approximations and, therefore, allows us to unequivocally identify the origin of the tunneling time delay.

Full experimental determination of tunneling time with attosecond-scale streaking method

Tunneling is one of the most fundamental and ubiquitous processes in the quantum world. The question of how long a particle takes to tunnel through a potential barrier has sparked a long-standing debate since the early days of quantum mechanics. Here, we propose and demonstrate a novel scheme to accurately determine the tunneling time of an electron. In this scheme, a weak laser field is used to streak the tunneling current produced by a strong elliptically polarized laser field in an attoclock configuration, allowing us to retrieve the tunneling ionization time relative to the field maximum with a precision of a few attoseconds. This overcomes the difficulties in previous attoclock measurements wherein the Coulomb effect on the photoelectron momentum distribution has to be removed with theoretical models and it requires accurate information of the driving laser fields. We demonstrate that the tunneling time of an electron from an atom is close to zero within our experimental accuracy. Our study represents a straightforward approach toward attosecond time-resolved imaging of electron motion in atoms and molecules.

Observation of the Decrease of Larmor Tunneling Times with Lower Incident Energy

How much time does a tunneling particle spend in a barrier? A Larmor clock, one proposal to answer this question, measures the interaction between the particle and the barrier region using an auxiliary degree of freedom of the particle to clock the dwell time inside the barrier. We report on precise Larmor time measurements of ultracold ^{87}Rb atoms tunneling through an optical barrier, which confirm longstanding predictions of tunneling times. We observe that atoms generally spend less time tunneling through higher barriers and that this time decreases for lower energy particles. For the lowest measured incident energy, at least 90% of transmitted atoms tunneled through the barrier, spending an average of 0.59±0.02  ms inside. This is 0.11±0.03  ms faster than atoms traversing the same barrier with energy close to the barrier's peak and 0.21±0.03  ms faster than when the atoms traverse a barrier with 23% less energy.

Measurement of the time spent by a tunnelling atom within the barrier region

Tunnelling is one of the most characteristic phenomena of quantum physics, underlying processes such as photosynthesis and nuclear fusion, as well as devices ranging from superconducting quantum interference device (SQUID) magnetometers to superconducting qubits for quantum computers. The question of how long a particle takes to tunnel through a barrier, however, has remained contentious since the first attempts to calculate it¹. It is now well understood that the group delay²-the arrival time of the peak of the transmitted wavepacket at the far side of the barrier-can be smaller than the barrier thickness divided by the speed of light, without violating causality. This has been confirmed by many experiments^3-6, and a recent work even claims that tunnelling may take no time at all⁷. There have also been efforts to identify a different timescale that would better describe how long a given particle spends in the barrier region^8-10. Here we directly measure such a time by studying Bose-condensed ⁸⁷Rb atoms tunnelling through a 1.3-micrometre-thick optical barrier. By localizing a pseudo-magnetic field inside the barrier, we use the spin precession of the atoms as a clock to measure the time that they require to cross the classically forbidden region. We study the dependence of the traversal time on the incident energy, finding a value of 0.61(7) milliseconds at the lowest energy for which tunnelling is observable. This experiment lays the groundwork for addressing fundamental questions about history in quantum mechanics: for instance, what we can learn about where a particle was at earlier times by observing where it is now^11-13.

Attosecond Molecular Angular Streaking with All-Ionic Fragments Detection

Attosecond angular streaking (or "attoclock") is an insightful technique for probing the ultrafast electron dynamics in strong laser fields. Up until recently, this technique relied solely on an accurate measurement of the photoelectron momentum distribution and has remained restricted to atomic targets. Here, we propose a novel attosecond angular streaking scheme applicable to molecules, for which the ionic fragments of dissociative ionization are detected in the polarization plane of a close-to-circular polarized laser light. Our ionic attoclock measurements are consistent with theoretical results from a numerical solution of the time-dependent Schrödinger equation and an upper bound of 10 as on the tunneling time from the attoclock readings in the H_{2} molecule has been given, which is significantly smaller than any definitions of tunneling time available in the literatures.

Unifying Tunneling Pictures of Strong-Field Ionization with an Improved Attoclock

We demonstrate a novel attoclock, in which we add a perturbative linearly polarized light field at 400 nm to calibrate the attoclock constructed by an intense circularly polarized field at 800 nm. This approach can be directly implemented to analyze the recent hot and controversial topics involving strong-field tunneling ionization. The generally accepted picture is that tunneling ionization is instantaneous and that the tunneling probability synchronizes with the laser electric field. Alternatively, recently it was described in the Wigner picture that tunneling ionization would occur with a certain of time delay. We unify the two seemingly opposite viewpoints within one theoretical framework, i.e., the strong-field approximation (SFA). We illustrate that both the instantaneous tunneling picture and the Wigner time delay picture that are derived from the SFA can interpret the measurement well. Our results show that the finite tunneling delay will accompany nonzero exit longitudinal momenta. This is not the case for the instantaneous tunneling picture, where the most probable exit longitudinal momentum would be zero.

Attosecond angular streaking and tunnelling time in atomic hydrogen

The tunnelling of a particle through a potential barrier is a key feature of quantum mechanics that goes to the core of wave-particle duality. The phenomenon has no counterpart in classical physics, and there are no well constructed dynamical observables that could be used to determine 'tunnelling times'. The resulting debate^1-5 about whether a tunnelling quantum particle spends a finite and measurable time under a potential barrier was reignited in recent years by the advent of ultrafast lasers and attosecond metrology⁶. Particularly important is the attosecond angular streaking ('attoclock') technique⁷, which can time the release of electrons in strong-field ionization with a precision of a few attoseconds. Initial measurements^7-10 confirmed the prevailing view that tunnelling is instantaneous, but later studies^11,12 involving multi-electron atoms-which cannot be accurately modelled, complicating interpretation of the ionization dynamics-claimed evidence for finite tunnelling times. By contrast, the simplicity of the hydrogen atom enables precise experimental measurements and calculations^13-15 and makes it a convenient benchmark. Here we report attoclock and momentum-space imaging¹⁶ experiments on atomic hydrogen and compare these results with accurate simulations based on the three-dimensional time-dependent Schrödinger equation and our experimental laser pulse parameters. We find excellent agreement between measured and simulated data, confirming the conclusions of an earlier theoretical study¹⁷ of the attoclock technique in atomic hydrogen that presented a compelling argument for instantaneous tunnelling. In addition, we identify the Coulomb potential as the sole cause of the measured angle between the directions of electron emission and peak electric field: this angle had been attributed^11,12 to finite tunnelling times. We put an upper limit of 1.8 attoseconds on any tunnelling delay, in agreement with recent theoretical findings¹⁸ and ruling out the interpretation of all commonly used 'tunnelling times'¹⁹ as 'time spent by an electron under the potential barrier'²⁰.

Direct probing of tunneling time in strong-field ionization processes by time-dependent wave packets

We propose a straightforward approach to directly probe the tunneling time by observing the transition of photoelectron wave packets in strong-field ionization processes, where Coulomb potentials do not affect the results. A circularly polarized laser pulse is used to avoid the impact of scattering electrons on the direct ionization electrons, and a pure transmission photoelectron wave packet can be obtained. Then, a positive tunneling time is extracted. The results demonstrate that the tunneling time is dominated mainly by the laser frequency in some laser intensity range. At the same time, we also investigate the tunneling time by analyzing the instantaneous ionization rate, and the consentaneous results are obtained.

Arrival Time Distributions of Spin-1/2 Particles

The arrival time statistics of spin-1/2 particles governed by Pauli’s equation, and defined by their Bohmian trajectories, show unexpected and very well articulated features. Comparison with other proposed statistics of arrival times that arise from either the usual (convective) quantum flux or from semiclassical considerations suggest testing the notable deviations in an arrival time experiment, thereby probing the predictive power of Bohmian trajectories. The suggested experiment, including the preparation of the wave functions, could be done with present-day experimental technology.

Dwell time, Hartman effect and transport properties in a ferromagnetic phosphorene monolayer

In this paper, spin-dependent dwell time, spin Hartman effect and spin-dependent conductance were theoretically investigated through a rectangular barrier in the presence of an exchange field by depositing a ferromagnetic insulator on the phosphorene layer in the barrier region. The existence of the spin Hartman effect was shown for all energies (energies lower than barrier height) and all incident angles in phosphorene. We also compared our results of the dwell time in the phosphorene structure with similar research performed on graphene. We reported a significant difference between the tunneling time values of incident quasiparticles with spin-up and spin-down. We found that the barrier was almost transparent for incident quasiparticles with a wide range of incident angles and energies higher than the barrier height in phosphorene. We also found that the maximum spin-dependent transmission probability for energies higher than barrier height does not necessarily occur in the zero incident angle. In addition, we showed that the spin conductance for energies higher (lower) than barrier height fluctuates (decays) in terms of barrier thickness. We discovered that, in contrast to graphene, the Klein paradox does not occur in the normal incident in the phosphorene structure. Furthermore, the results demonstrated the achievement of good total conductance at certain thicknesses of the barrier for energies higher than the barrier height. This study could serve as a basis for investigations of the basic physics of tunneling mechanisms and also for using phosphorene as a spin polarizer in designing nanoelectronic devices.

Under-the-Tunneling-Barrier Recollisions in Strong-Field Ionization

A new pathway of strong-laser-field-induced ionization of an atom is identified which is based on recollisions under the tunneling barrier. With an amended strong-field approximation, the interference of the direct and the under-the-barrier recolliding quantum orbits are shown to induce a measurable shift of the peak of the photoelectron momentum distribution. The scaling of the momentum shift is derived relating the momentum shift to the tunneling delay time according to the Wigner concept. This allows us to extend the Wigner concept for the quasistatic tunneling time delay into the nonadiabatic domain. The obtained corrections to photoelectron momentum distributions are also relevant for state-of-the-art accuracy of strong-field photoelectron spectrograms in general.

Nonadiabatic effects in electronic and nuclear dynamics

Due to their very nature, ultrafast phenomena are often accompanied by the occurrence of nonadiabatic effects. From a theoretical perspective, the treatment of nonadiabatic processes makes it necessary to go beyond the (quasi) static picture provided by the time-independent Schrödinger equation within the Born-Oppenheimer approximation and to find ways to tackle instead the full time-dependent electronic and nuclear quantum problem. In this review, we give an overview of different nonadiabatic processes that manifest themselves in electronic and nuclear dynamics ranging from the nonadiabatic phenomena taking place during tunnel ionization of atoms in strong laser fields to the radiationless relaxation through conical intersections and the nonadiabatic coupling of vibrational modes and discuss the computational approaches that have been developed to describe such phenomena. These methods range from the full solution of the combined nuclear-electronic quantum problem to a hierarchy of semiclassical approaches and even purely classical frameworks. The power of these simulation tools is illustrated by representative applications and the direct confrontation with experimental measurements performed in the National Centre of Competence for Molecular Ultrafast Science and Technology.

Exploring tunneling time by instantaneous ionization rate in strong-field ionization

A quantum approach is presented to investigate tunneling time by supervising the instantaneous ionization rate. We find that the ionization rate peak appearance lags behind the maximum of electric field intensity for a linearly polarized pulse. This time delay interval can be taken to characterize the tunneling time. In addition, if an atom with anisotropic electronic distribution is exposed to a circular polarized pulse, the tunneling time can also be measured and defined as the time difference between the instant of the largest ionization rate and the moment when the electric field points in the maximum of the bound electron density.

Emergence of a Higher Energy Structure in Strong Field Ionization with Inhomogeneous Electric Fields

Studies of strong field ionization have historically relied on the strong field approximation, which neglects all spatial dependence in the forces experienced by the electron after ionization. More recently, the small spatial inhomogeneity introduced by the long-range Coulomb potential has been linked to a number of important features in the photoelectron spectrum, such as Coulomb asymmetry, Coulomb focusing, and the low energy structure. Here, we demonstrate using midinfrared laser wavelength that a time-varying spatial dependence in the laser electric field, such as that produced in the vicinity of a nanostructure, creates a prominent higher energy peak. This higher energy structure (HES) originates from direct electrons ionized near the peak of a single half-cycle of the laser pulse. The HES is separated from all other ionization events, with its location and width highly dependent on the strength of spatial inhomogeneity. Hence, the HES can be used as a sensitive tool for near-field characterization in the "intermediate regime," where the electron's quiver amplitude is comparable to the field decay length. Moreover, the large accumulation of electrons with tuneable energy suggests a promising method for creating a localized source of electron pulses of attosecond duration using tabletop laser technology.

Experimental Evidence for Quantum Tunneling Time

The first hundred attoseconds of the electron dynamics during strong field tunneling ionization are investigated. We quantify theoretically how the electron's classical trajectories in the continuum emerge from the tunneling process and test the results with those achieved in parallel from attoclock measurements. An especially high sensitivity on the tunneling barrier is accomplished here by comparing the momentum distributions of two atomic species of slightly deviating atomic potentials (argon and krypton) being ionized under absolutely identical conditions with near-infrared laser pulses (1300 nm). The agreement between experiment and theory provides clear evidence for a nonzero tunneling time delay and a nonvanishing longitudinal momentum of the electron at the "tunnel exit."

Quantum Tunneling: The Longer the Path, the Less Time it Takes

The standard approaches to tunneling times are replaced by considering time correlation functions. A class of correlation functions that is always positive is identified and used to define quantum mechanical transition time probability distributions. The formalism is used to study the quantum dynamics of a thermal position correlation function of a parabolic barrier Hamiltonian. The transition time probability distribution between two points distributed symmetrically about the barrier top shifts to shorter times as the temperature is reduced and tunneling is increased. A study of the mean transition time as a function of the distance between the center of the initial and final densities shows that when the temperature is sufficiently low and tunneling dominates the dynamics, increasing the length of the path traversed decreases the mean transition time. The introduction of friction to the dynamics does not "destroy" this phenomenon, except when the friction coefficient is very large.