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 Dec 13 comment Is quantum tunneling related to imaginary time? Thanks. I guess I'm a little bit confused by the saddle point approximation. Can we show numerically that the direct integration give almost the same results as the saddle point approximation? Dec 13 comment Is quantum tunneling related to imaginary time? I'm wondering what it is plotted after in the second figure after this sentence "A typical example looks like this:". Is that the expression in the integral or $b(p,t)$ itself? Oct 30 awarded Tumbleweed Oct 23 asked Mapping the first Brillouin zone to the nth Brillouin zone Oct 18 comment Is quantum tunneling related to imaginary time? Thank you so much for posting this. It's very helpful. I also work in strong field physics. Jan 28 awarded Notable Question Sep 24 awarded Autobiographer Sep 5 awarded Teacher Jul 27 comment Optical Bloch Oscillation I don't know about optical Bloch oscillation, but Bloch oscillation in general can be understand in these two pictures: the Bloch oscillation and Wannier-Stark pictures. These two pictures corresponding two independent choices of gauges, the velocity gauge and the length gauge. In the Bloch oscillation picture, the eigenfunctions of the system are the Houston states (accelerated Bloch states). In the Wannier-Stark picture, the eigenfunctions are the Wannier-Stark states, and the Bloch oscillation can be understand as the quantum beating of a superposition of two eigenstates of the system. Jul 23 awarded Good Question Jun 9 comment Particle in a box under harmonic driving Thanks, but I'm interested in exactly the non-perturbative limit, where the Rabi oscillation can happen. Jun 9 asked Particle in a box under harmonic driving Apr 7 comment Adjoint of momentum operator The adjoin of the differential operator $\nabla^\dagger = -\nabla$ can be proved by integration by part. Mar 6 awarded Informed Mar 6 comment The gauge-invariance of the probability current Great prove ! +1 Mar 6 asked group velocity VS probability current Dec 20 awarded Yearling Nov 30 awarded Scholar Nov 30 accepted Why are rockets so big? Nov 28 awarded Popular Question