Tunneling properties of Bogoliubov mode and spin-wave modes in supercurrent states of a spin-1 ferromagnetic spinor Bose-Einstein condensate

Abstract

We investigate the tunneling properties of collective excitations in the ferromagnetic phase of a spin-1 spinor Bose-Einstein condensate (BEC). In addition to the Bogoliubov mode, this superfluid phase has two spin excitations, namely, the gapless transverse spin wave and the quadrupolar mode with a finite excitation gap. In the mean-field theory at T = 0, we examine how these collective modes tunnel through a barrier potential that couples to the local density of particles. In the presence of supercurrent with a finite momentum q, while the Bogoliubov mode shows the so-called anomalous tunneling behavior (which is characterized by perfect transmission) in the low-energy limit, the transverse spin wave transmits perfectly only when the momentum k of this mode coincides with ±q. At k = ±q, the wave function of this spin wave has the same form as the condensate wave function in the current-carrying state, so that the mechanism of this perfect transmission is found to be the same as tunneling of the supercurrent. Using this fact, the perfect transmission of the spin wave is proved for a generic barrier potential. We show that such perfect transmission does not occur in the quadrupolar mode. Further, we consider the effects of potentials breaking U(1) and spin-rotation symmetries on the transmission properties of excitations. Our results would be useful for understanding excitation properties of spinor BEC's, as well as the anomalous tunneling phenomenon in Bose superfluids.

Transmission of excitations in a spin-1 Bose-Einstein condensate through a barrier

“…The tunneling properties depend on nonuniform effects, so that we expect that a nonuniform magnetic field changes the present results. We report this effect in a separate paper [26].…”

“…This property is strongly related to the fact that the transverse spin wave is a Nambu-Goldstone mode. The barrier coupled only to the density does not lift the degeneracy of the ground state and it results in perfect transmission in the symmetric case [26]. From another point of view, we comment that the wave functionφ 0 behaves like an interacting Bose gas, where the potential effect disappears far from the potential 053624-11 barrier [27].…”

“…In the current carrying state of the spinor BEC, it is also unknown what happens at the critical current for the tunneling problem. We report on these issues in a separate paper [26]. …”

Counting rule of Nambu–Goldstone modes for internal and spacetime symmetries: Bogoliubov theory approach

“…The corresponding perturbation theory for a large Jordan block is given in Appendix F. We give a general treatment of "massive" NGMs [50,51,52,53] in Bogoliubov theory in Subsec. 7.2.2 and Appendix G. The resulting perfect tunneling properties [54,55,56,57,58,59,60,61] of these NGMs are discussed in Subsec. 7.2.3. This paper is organized as follows.…”

Anomalous tunneling of collective excitations and effects of superflow in the polar phase of a spin-1 spinor Bose-Einstein condensate

“…Perfect transmission through a potential barrier, which is frequently referred to as the anomalous tunneling phenomenon, is now widely recognized as a fundamental property of the Bogoliubov excitation in a Bose superfluid [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. This phenomenon is quite different from the simple tunneling problem discussed in one-particle quantum mechanics, where the transmission probability always vanishes in the low-energy limit.…”