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arXiv:0902.2908 · Physical Review A

Supercurrent induced by tunneling Bogoliubov excitations in a Bose-Einstein condensate

Shunji Tsuchiya, Yoji Ohashi

Published Jun 17, 2009• 12 citations

Abstract

We study the tunneling of Bogoliubov excitations through a barrier in a Bose-Einstein condensate. We extend our previous work [S. Tsuchiya and Y. Ohashi, Phys. Rev. A 78, 013628 (2008)] to the case when condensate densities are different between the left and right of the barrier potential. In the framework of the Bogoliubov mean-field theory, we calculate the transmission probability and phase shift, as well as the energy flux and quasiparticle current carried by Bogoliubov excitations. We find that Bogoliubov phonons twist the condensate phase due to a back-reaction effect, which induces the Josephson supercurrent. While the total current given by the sum of quasiparticle current and induced supercurrent is conserved, the quasiparticle current flowing through the barrier potential is shown to be remarkably enhanced in the low-energy region. When the condensate densities are different between the left and right of the barrier, the excess quasiparticle current, as well as the induced supercurrent, remains finite far away from the barrier. We also consider the tunneling of excitations and atoms through the boundary between the normal and superfluid regions. We show that supercurrent can be generated inside the condensate by injecting free atoms from outside. On the other hand, atoms are emitted when the Bogoliubov phonons propagate toward the phase boundary from the superfluid region.

DOI: 10.1103/physreva.79.063619

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We study the tunneling of Bogoliubov excitations through a barrier in a Bose-Einstein condensate. We extend our previous work [S. Tsuchiya and Y. Ohashi, Phys. Rev. A 78, 013628 (2008)] to the case when condensate densities are different between the left and right of the barrier potential. In the framework of the Bogoliubov mean-field theory, we calculate the transmission probability and phase shift, as well as the energy flux and quasiparticle current carried by Bogoliubov excitations. We find that Bogoliubov phonons twist the condensate phase due to a back-reaction effect, which induces the Josephson supercurrent. While the total current given by the sum of quasiparticle current and induced supercurrent is conserved, the quasiparticle current flowing through the barrier potential is shown to be remarkably enhanced in the low-energy region. When the condensate densities are different between the left and right of the barrier, the excess quasiparticle current, as well as the induced supercurrent, remains finite far away from the barrier. We also consider the tunneling of excitations and atoms through the boundary between the normal and superfluid regions. We show that supercurrent can be generated inside the condensate by injecting free atoms from outside. On the other hand, atoms are emitted when the Bogoliubov phonons propagate toward the phase boundary from the superfluid region.

We investigate tunneling properties of Bogoliubov phonons in a Bose-Einstein condensate. We find the anomalous enhancement of the quasiparticle current

J_{\rm q}

carried by Bogoliubov phonons near a potential barrier, due to the supply of the excess current from the condensate. This effect leads to the increase of quasiparticle transmission probability in the low energy region found by Kovrizhin {\it et al.}. We also show that the quasiparticle current twists the phase of the condensate wavefunction across the barrier, leading to a finite Josephson supercurrent

J_{\rm s}

through the barrier. This induced supercurrent flows in the opposite direction to the quasiparticle current so as to cancel out the enhancement of

J_{\rm q}

and conserve the total current

J=J_{\rm q}+J_{\rm s}

.

We investigate tunneling properties of Bogoliubov phonons in a Bose-Einstein condensate. We find the anomalous enhancement of the quasiparticle current

{J}_{\mathrm{q}}

carried by Bogoliubov phonons near a potential barrier, due to the supply of the excess current from the condensate. This effect leads to the increase of quasiparticle transmission probability in the low energy region found by Kovrizhin et al. We also show that the quasiparticle current twists the phase of the condensate wave function across the barrier, leading to a finite Josephson supercurrent

{J}_{\mathrm{s}}

through the barrier. This induced supercurrent flows in the opposite direction to the quasiparticle current so as to cancel out the enhancement of

{J}_{\mathrm{q}}

and conserve the total current

J={J}_{\mathrm{q}}+{J}_{\mathrm{s}}

.

We discuss tunneling properties of Bogoliubov excitations through a potential barrier. We find that the quasi-particle current carried by Bogoliubov excitations is remarkably enhanced near the potential barrier at low incident energies. This explains the increase of the transmission probability of Bogoliubov phonons in the low energy region. In addition, taking into account of the backreaction of Bogoliubov excitations on Bose condensates, we show that the quasi-particle current twists the phase of the condensate wavefunction, leading to a Josephson supercurrent through the barrier.

We investigate tunneling properties of Bogoliubov mode in a Bose-Einstein condensate. Using an exactly solvable model with a $\ensuremath{\delta}$-functional barrier, we show that each component in the two-component wave function

(u,v)

of low-energy Bogoliubov phonon has the same form as the condensate wave function in the supercurrent state. As a result, the currents

{J}_{u}

and

{J}_{v}

associated with

u

and

v

, respectively, have the same tunneling properties as those of supercurrent carried by condensate. Thus, the tunneling of low-energy Bogoliubov phonon described by the tunneling of these two currents shows perfect transmission. We also show that the supercurrent behaviors of Bogoliubov phonon still exist in the presence of supercurrent carried by condensate, except in the critical supercurrent state. In the critical current state, the perfect transmission is absent, because

{J}_{u}

or

{J}_{v}

exceeds their upper limit given by the critical value of the supercurrent associated with the condensate. Our results consistently explain the recently proposed two tunneling phenomena associated with Bogoliubov phonon, namely, the anomalous tunneling effect (perfect transmission in the low-energy limit) and the breakdown of the perfect transmission in the critical supercurrent state.

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 $\ifmmode\pm\else\textpm\fi{}q$. At $k=\ifmmode\pm\else\textpm\fi{}q$, 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.

We investigate the quasiresonant tunneling of phonon excitations across a potential barrier spatially separating two condensates with different macroscopic phases. We confirm that the quasiresonant tunneling occurs through the quasibound state. The quasibound state is induced by a potential well created by the condensate density. We analyze the relation between the phase difference φ of two condensates and the transmission coefficient T by solving the Bogoliubov-de Gennes equations. As a result, the transmission coefficient T is found to depend strongly on (p because of the φ dependence of the quasibound state.