Comment on "Negative Differential Conductivity in an Interacting Quantum Gas."
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y Comment on “Negative Differential Conductivity in an Interacting Quantum Gas.”
M. K. Olsen and J. F. Corney
School of Mathematics and Physics, University of Queensland, Brisbane, QLD 4072, Australia. (Dated: September 5, 2018)Labouvie et al. (Phys. Rev. Lett. , 050601, (20015)) recently demonstrated negative differentialconductivity (NDC) in a multi-well Bose-Einstein condensate. They stated “we demonstrate thatNDC originates from a nonlinear, atom number dependent tunneling coupling in combination withfast collisional decoherence.” We show theoretically how the essential feature of NDC, a reduction inatomic current caused by an increase in chemical potential, is present in unitary dynamics throughthe well-known mechanism of macroscopic self-trapping (MST), and that the collisional decoherencemerely serves as a quantitative modification of this.
PACS numbers: 05.60.Gg, 03.65.Xp, 03.75.Lm, 37.10.Jk
NDC is an unusual phenomenon in electronics whichrequires a strongly nonlinear device. In ultracold-atom transport, nonlinearity is readily available throughatomic collisions. An atomic implementation of NDC byLabouvie et al. [1] found that, if chemical potential dif-ference, ∆ µ , between wells is considered as analogous tovoltage, the proportionality between ∆ µ and tunnelingcurrent can be negative. We show here how the quali-tative effects of this phenomenon can be ascribed to thewell known MST phenomenon (MST) [2], with the col-lisional decoherence causing only quantitative changes.This means that NDC is available in a lattice ultracoldatomic system purely through coherent effects.We analyse a three-well Bose-Hubbard model [3], de-scribed by the Hamiltonian H = ¯ hχ X i =1 ˆ a † i ˆ a i − ¯ hJ (cid:16) ˆ a † ˆ a + ˆ a † ˆ a + ˆ a † ˆ a + ˆ a † ˆ a (cid:17) , (1)using the truncated Wigner representation [4, 5]. χ isthe collisional nonlinearity, J is the tunneling parameter,and phase diffusion is included by the same Louivillianproportional to Γ as used by Labouvie et al. We define∆ N ≡ N (0) − N (0), approximately proportional to thedifference in chemical potential between the wells andanalogous to voltage since it drives the atomic current.We calculate the maximum of the tunnelling current, I = − i h ˆ a † ˆ a − ˆ a † ˆ a + ˆ a † ˆ a − ˆ a † ˆ a i , for different ∆ N and two values of χ , with initial coherent states [6]. Theresults are shown in Fig. 1, with (Γ = 1 . J ) and withoutphase noise in the central well. The stronger nonlinearityleads to a decrease of current with increasing ∆ N , whichis an example of NDC. We see that the presence of phasenoise does not qualitatively change the results.These numerical results show that phase diffusion isnot necessary for negative differential conductivity, whichcan arise solely from macroscopic self-trapping, a fullycoherent process. We note here that the direct currentreported by Labouvie et al. does depend on the presenceof phase diffusion, but does not exist for the parameters we consider here. In our view, the new effect that has ∆ N
50 60 70 80 90 100 I m a x χ =0.01 χ =0.1 FIG. 1: (Colour online) The maximum currents into the mid-dle well as a function of ∆ N , for J = 1, χ = 0 .
01 and 0 . N (0) = N (0) = 100, with N (0) = N (0) − ∆ N . Thesolid lines are the results for initial coherent states with Γ = 0while the dashed lines include phase diffusion with Γ = 1 . J .Each result is the average of the order of a million stochastictrajectories and sampling errors are within line thicknesses. been discovered is this DC manifestation of NDC, andnot NDC itself.This research was supported by the Australian Re-search Council under the Future Fellowships Program(Grant ID: FT100100515). [1] R. Labouvie, B. Santra, S. Heun, S. Wimberger, and H.Ott, Phys. Rev. Lett. , 050601 (2015).[2] G.J. Milburn, J.F. Corney, E.M. Wright and D.F. Walls,Phys. Rev. A , 4318, (1997).[3] H. Gersch and G. Knollman, Phys. Rev. , 959 (1963).[4] M.J. Steel, M.K. Olsen, L.I. Plimak, P.D. Drummond,S.M. Tan, M.J. Collett, D.F. Walls, and R. Graham, Phys.Rev. A , 4824 (1998).[5] C.V. Chianca and M.K. Olsen, Phys. Rev. A , 043636(2011).[6] M.K. Olsen and A.S. Bradley, Opt. Commun.282