Tunnelling is one of the phenomena at the heart of quantum mechanics. Despite its central role in physics, the duration of the tunnelling process has been debated since the development of quantum theory without a definite answer. In this thesis, the question of how long a particle takes to tunnel is investigated experimentally using ultracold atoms. These systems provide an ideal platform to address this question due to their isolation from the external environment and the ability to create almost arbitrary potentials for the atoms. As the tunnelling time experiment requires low temperatures, the realization and characterization of wave packets below the nanokelvin regime are studied first. A 16x reduction of the effective temperature of the atoms is obtained through delta-kick cooling. However, the achieved temperatures are so low that a new method, dubbed the “knife-edge technique”, is developed to characterize the final atomic state accurately. Through this new method, a final temperature of 900 ± 200 pK is found. Finally, the tunnelling time is investigated using a Bose-Einstein condensate in a one dimensional waveguide intersected by a 1.3 μm barrier. As the atoms tunnel through the thin potential, their spins time the tunnelling event through the Larmor precession induced by a pseudo-magnetic field localized inside the barrier. The precession angle is found by performing tomography on the spin state of the tunnelling atoms. With the knowledge of the Larmor frequency and the rotation angles, we find a tunnelling time of 0.65(7) ms. This experiment achieves a long-sought tunnelling time measurement with a clear interpretation, and its results show good quantitative agreement with theory based on the weak measurement formalism.
Tunnelling is one of the phenomena at the heart of quantum mechanics. Despite its central role in physics, the duration of the tunnelling process has been debated since the development of quantum theory without a definite answer. In this thesis, the question of how long a particle takes to tunnel is investigated experimentally using ultracold atoms. These systems provide an ideal platform to address this question due to their isolation from the external environment and the ability to create almost arbitrary potentials for the atoms. As the tunnelling time experiment requires low temperatures, the realization and characterization of wave packets below the nanokelvin regime are studied first. A 16x reduction of the effective temperature of the atoms is obtained through delta-kick cooling. However, the achieved temperatures are so low that a new method, dubbed the “knife-edge technique”, is developed to characterize the final atomic state accurately. Through this new method, a final temperature of 900 ± 200 pK is found. Finally, the tunnelling time is investigated using a Bose-Einstein condensate in a one dimensional waveguide intersected by a 1.3 μm barrier. As the atoms tunnel through the thin potential, their spins time the tunnelling event through the Larmor precession induced by a pseudo-magnetic field localized inside the barrier. The precession angle is found by performing tomography on the spin state of the tunnelling atoms. With the knowledge of the Larmor frequency and the rotation angles, we find a tunnelling time of 0.65(7) ms. This experiment achieves a long-sought tunnelling time measurement with a clear interpretation, and its results show good quantitative agreement with theory based on the weak measurement formalism.
The simplicity and versatility of ultra-cold atoms make it an ideal system for studying Quantum Mechanical phenomenon like tunneling. This thesis describes progress towards experimentally measuring the time spent by a tunneling particle in the classically forbidden region. A detailed analysis of the experimental requirements to measure the tunneling time is presented along with the experimental tools developed to meet these requirements. These include a thin \SI{1.3}{\micro m} optical tunnel barrier, a smooth atomic waveguide, and a Larmor clock to measure the tunneling time. Delta-kick cooling is used to achieve the extremely low temperatures required for the tunneling time experiments. A 2.4x reduction in the atomic velocity spread is demonstrated, reducing the rms velocity spread to \SI{0.46(5)}{mm/s}. This corresponds to cooling to an effective temperature of \SI{2.0(4)}{nK}. The thin optical barrier opens up the possibility to study the decay of quasi-bound condensates via tunneling. We develop a novel trapping configuration for this purpose, in which the barrier acts as one of the walls of the trap. Inter-atomic interactions strongly dictate the escape dynamics of the condensate out of this trap, giving rise to three distinct regimes--- classical over the barrier spilling, quantum tunneling driven by interactions, and decay dominated by background losses. We show that in the tunneling regime, the decay rate depends exponentially with the chemical potential of the condensate. Experimental results show good agreement with numerical solutions of the 3D Gross-Pitaevskii equation.
Tunnelling is one of the most paradigmatic and evocative phenomena of quantum physics, underlying processes such as photosynthesis and nuclear fusion, as well as devices ranging from SQUID magnetometers to superconducting qubits for quantum computers. The question of how long a particle takes to tunnel, however, has remained controversial since the first attempts to calculate it, which relied on the group delay. It is now well understood that this delay (the arrival time of the transmitted wave packet peak at the far side of the barrier) can be smaller than the barrier thickness divided by the speed of light, without violating causality. There have been a number of experiments confirming this, and even a recent one claiming that tunnelling may take no time at all. There have also been efforts to identify another timescale, which would better describe how long a given particle spends in the barrier region. Here we present a direct measurement of such a time, studying Bose-condensed 87Rb atoms tunnelling through a 1.3-μm 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 it takes them to cross the classically forbidden region. We find a traversal time of 0.62(7) ms and study its dependence on incident energy. In addition to finally shedding light on the fundamental question of the tunnelling time, this experiment lays the groundwork for addressing deep foundational questions about history in quantum mechanics: for instance, what can we learn about where a particle was at earlier times by observing where it is now?
The question of how long a tunneling particle spends in the forbidden region of a barrier has been a perplexing puzzle in foundational quantum mechanics for many decades. The Larmor time, one definition of tunneling time, uses an auxiliary degree of freedom of the tunneling particles to clock the time spent inside the barrier. Recently, we made our first measurement of the Larmor time for Bose-condensed 87Rb atoms tunneling through an optical barrier.1 Here, we report on follow up measurements with improved precision in the measured times and also study the time for the reflected atomic cloud. We observe significant discrepancies between our results and a simple theory based on weak measurement. We discuss our findings, hypothesize explanations for our results, and suggest future studies.
The precise definition and physical interpretation of a tunneling time is a fundamental problem in quantum mechanics. The lack of a well-defined time operator in quantum theory precludes the calculation of time in terms of an expectation value. Consequently, model calculations and simulation studies have been proposed and used to determine the time for a particle to traverse a quantum-mechanical barrier. The results offer both qualitatively and quantitatively, disparate predictions. One of the more commonly used approaches is the phase method, which we have applied to several one-dimensional model potentials to show that the tunneling time is characterized by the ratio of the typical decay length in the barrier to the incident velocity, τ∼(1/κ)/v0. We also show that the phase and the spin precession methods are equivalent when the magnetic field is applied throughout the space containing the particle wave function and the tunneling barrier. In principle, the spin precession method can be regarded as an operational definition of the tunneling time. It also has been shown that a tunneling time can be defined operationally for a time-dependent barrier, for which the model calculations predict a tunneling time given by the mean barrier width divided by the magnitude of the imaginary velocity of the particle in the barrier, τ∼d/‖v‖. This pseudokinematic result has been inferred by magnetic field dependent tunneling through heterostructures and computer simulation of wave packet tunneling through one-dimensional square barriers. Other tunneling time theories predict one or the other of the above mentioned characteristic times; several theories predict both. We interpret these different results for τ to mean that a tunneling time is not uniquely defined as a function of particle energy and the barrier shape unless the process in which the tunneling is involved is also specified. That is, the tunneling time is a meaningful and unique concept only for particular tunneling processes or definite operational procedures.
Introduction In classical physics it is meaningful to ask the question, ‘How much time does a particle take to pass through a given region?’ The interesting question in quantum mechanics is: does a particle take a definite time to tunnel through a classically forbidden region? The question has been debated ever since the idea that there was such a time in quantum theory was first put forward by MacColl way back in 1932 [253]. A plethora of times has since then been proposed, and the answer seems to depend on the interpretation of quantum mechanics one uses. A reliable answer is clearly of great importance for the design of high-frequency quantum devices, tunnelling phenomena (as, for example, in scanning tunneling microscopy), nuclear and chemical reactions and, of course, for purely conceptual reasons. Most of the controversies centre around simple and intuitive notions in idealized one-dimensional models in a scattering configuration in which a particle (usually represented by a wave packet) is incident on a potential barrier localized in the interval [ a, b ]. Three kinds of time have been defined in this context. One, called the transmission time τ T ( a, b ), is the average time spent within the barrier region by the particles that are eventually transmitted. Similarly, the reflection time τ R ( a, b ) is the average time spent within the barrier region by the particles that are eventually reflected.
The three experiments described in this thesis investigate fundamental properties of ultracold atoms. Using laser cooling and evaporative cooling, a dilute gas of sodium atoms is cooled to ~100 nK. Under these circumstances a Bose-Einstein condensate (BEC) forms, where millions of atoms collapse into the lowest energy state of the system and share a macroscopic wavefunction. The experiments are done in an ultrahigh vacuum and the atoms are manipulated remotely using laser beams, magnetic fields, and RF-fields. Using ultracold atoms allows unprecedented control over the internal and external degrees of freedom of the system, making it very suitable for precision measurements. Ultracold atoms also serve as a model system for more complicated or remote fields such as solid-state physics and cosmology. In the first experiment the refractive index of an ultracold bosonic gas is studied. The atomic clouds are analyzed for temperature and number of particles using nearly non-destructive phase-contrast imaging. After each pulse of probe light a small fraction of the atoms is lost, while the cloud is simultaneously slightly heated. This makes it possible to study the scattering rate as a function of temperature using only a single sample. It is observed that the scattering rate increases below the critical temperature for BEC by more than a factor of 3 compared to the classical value. This enhancement is in good agreement with the predictions by theoretical work that takes two-body correlation and the Lorentz-Lorenz correction into account. The method is also used to perform calorimetry on the Bose gas and directly measure the heat capacity of the BEC. The second chapter deals with magnetism in ultracold gases. When cooling an ultracold gas containing different spin components below the critical temperature for BEC, stable magnetic domains form. The appearance of these domains and the dynamics of the domain walls are studied. The spin mixture is prepared by performing an RF-sweep on ground state atoms in the presence of a magnetic field. Spin dynamics are induced by applying magnetic gradients. Spin drag, which is the transport coefficient responsible for damping spin currents, was measured in a two-component system. Furthermore a new method for spin resolved phase contrast imaging is presented, which is used to study time evolution in situ non-destructively. The last experiment focuses on spontaneous symmetry breaking. When passing a continuous phase transition, spontaneous symmetry breaking causes the development of domains with an independently chosen order parameter. The domains are separated by boundary defects. In this experiment the spontaneous creation of solitons was observed, which are boundary defects in the phase of the condensate, as well as magnetic domain walls, which are the boundary defects in the magnetization. Through the Kibble-Zurek mechanism it is possible to determine the critical exponents that characterize the phase transition. The critical exponents are predicted to be universal, i.e. do not depend on the microphysics of the system, so the results from our ultracold atoms lab can help answer universal questions for example from cosmology and from the early universe.