Optical Larmor clock: Measurement of the photonic tunneling time

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.

Direct Tunneling Delay Time Measurement in an Optical Lattice

We report on the measurement of the time required for a wave packet to tunnel through the potential barriers of an optical lattice. The experiment is carried out by loading adiabatically a Bose-Einstein condensate into a 1D optical lattice. A sudden displacement of the lattice by a few tens of nanometers excites the micromotion of the dipole mode. We then directly observe in momentum space the splitting of the wave packet at the turning points and measure the delay between the reflected and the tunneled packets for various initial displacements. Using this atomic beam splitter twice, we realize a chain of coherent micron-size Mach-Zehnder interferometers at the exit of which we get essentially a wave packet with a negative momentum, a result opposite to the prediction of classical physics.

Time-domain measurements of reflection delay in frustrated total internal reflection

We present experimental evidence that the contribution of the Goos-Hänchen shift to tunneling delay is suppressed in frustrated total internal reflection. We use a Hong-Ou-Mandel interferometer to perform direct time measurements of reflection delays with femtosecond resolution at optical frequencies, and take advantage of a liquid-crystal-filled double-prism structure to dynamically change the refractive index of the barrier region.

Superluminal interactions in near-field optics

The fact that photons emitted from an electric-dipole active atom cannot be spatially localized better than to the near-field zone of the atom is seen as the origin of genuine superluminality. By means of a simple model dipole current density the general theory is used to demonstrate numerically how superluminality enters the near-field dynamics, and how from a measurement one could be tempted to believe that superluminal propagation effects occur. Furthermore, it is shown how for source-detector distances larger than a pulse length one should be able to divide the pulse into two separate parts: one purely superluminal part arising solely in the non-local generation process of the field, and another part seemingly propagating with superluminal speed. We comment on different velocity analyses, and we argue that the only fundamental velocity entering the problem is the vacuum velocity of light, which in a measurement would appear as the velocity of the trailing edge of the pulse.

"Superluminal" transmission of light pulses through optically opaque barriers

Using simple considerations of causal electrodynamics we analyze the occurrence of superluminal transmission of light pulses through optically opaque barriers. We find that the phenomenon appears whenever the main frequency components of the pulse are confined to frequency regions where the presence of the barrier decreases the density of states of the electromagnetic modes of the system. We also show that these frequency regions correspond to the transmission gaps of sufficiently wide barriers. We discuss a simple theory for the density of states of the barrier system and compare the results of such a theory with exact numerical calculations.