Comment on Splitting Times of Doubly Quantized Vortices in Dilute Bose-Einstein Condensates

a r X i v :0803.2187v 1 [c o n d -m a t .o t h e r ] 14 M a r 2008

Comment on ”Splitting Times of Doubly Quantized Vortices in Dilute Bose-Einstein

Condensates”

K.Gawryluk,1T.Karpiuk,1M.Brewczyk,1and K.Rz?a ˙z ewski 2,3

1

Wydzia l Fizyki,Uniwersytet w Bia l ymstoku,ulica Lipowa 41,15-424Bia l ystok,Poland 2

Centrum Fizyki Teoretycznej PAN,Aleja Lotnik′o w 32/46,02-668Warsaw,Poland

3

Faculty of Mathematics and Sciences UKSW,Warsaw,Poland

In their recent Letter [1],Huhtam¨a ki et al.theoret-ically investigated the splitting of a topologically im-printed doubly charged vortex into two singly charged vortices as occurring in a dilute atomic Bose-Einstein condensate.They compare the results of simulation with recent experiment [2]and show that the combination of gravitational sag and the time dependence of the trap-ping potential alone are enough to explain the observed splitting times.In particular,the numerically obtained dependence of the decay time on the peak condensate density ?ts well the experimental data [2].Based on this agreement the authors of Ref.[1]claim that,contrary to previous theoretical results [3],the thermal excitations play no role in the experiment.We are going to show in this Comment that there is no con?ict between expla-nations in Ref.[1]and [3].In fact,a number of ther-mal (uncondensed)atoms appears in the system while disturbing the gas.They continue to appear afterwords during the decay of the vortex (i.e.,when the separation between two cores of singly quantized vortices is slowly increasing).The overall number of uncondensed atoms remains approximately on the level of 20%,which is al-ready at the edge of experimental detection capabilities.However,the uncondensed atoms do not form the broad cloud allowing the identi?cation by ?tting to a bimodal distribution -they are rather located in the core of the vortex and therefore are harder to detect.

To investigate the thermal excitations in a Bose gas one needs a tool allowing description of bosons at nonzero temperatures.We have developed the classical ?elds ap-proximation [4]-an approach that treats both condensed and thermal atoms at the same footing until the detec-tion time.In this method all atoms are described by the single complex ?eld which evolves according to the Gross-Pitaevskii equation.On detection,the atomic cloud is split into the condensate and the thermal (uncondensed)clouds.Technically,such a decomposition requires calcu-lation of time and space average of a one-particle density matrix built of the classical ?eld.

Therefore,we have repeated the calculations of Ref.[1].As in [1],we closely follow the experiment reported in [2]and solve the Gross-Pitaevskii equation with time-dependent trapping potential (originating from the time-dependent magnetic ?eld according to the idea of topo-logical phase imprinting)combined with the gravitational

potential.However,we interpret the solution of the Gross-Pitaevskii equation as a classical ?eld and at each time we do a ’coarse graining’and split the atomic cloud into 5.7 3.8 1.90

1.9 3.8 5.70

2468TIME [ms]

X [ m]μ

D E N S I T Y [a r b . u n i t s n 0

(e)

(d)

0102030405060

1.00.9

0.80.7

FIG.1:(Color online).The z -integrated (over [?15μm,15

μm]interval)condensate (a),thermal (b),and total (c)den-sities at 47ms for the interaction strength an z =8.5;the horizontal cuts of the condensate (dashed line),thermal (dot-ted line),and total (solid line)densities (d);the condensate fraction as a function of time (e).

Fig.1clearly shows that the uncondensed atoms ap-pear in the system during the evolution.It happens when the system looses its rotational symmetry and the split-ting of two singly quantized vortices is in progress.Final fraction of thermal atoms is about 20%.Fig.1also proves that the thermal atoms are mainly located in the vortex core.On the other hand,the thermal noise on the level of about 20%leads to the decay of a vortex in approximately 45ms (see Fig.3in Ref.[3])in agreement with the calculations in [1]and this Comment.We claim this means that,indeed,behind the origin of the split-ting of the doubly quantized vortex reported in [1](the impetus given by the time dependence of the external po-tential)lies the presence of uncondensed atoms appearing in the system as a result of condensate disturbance.The authors acknowledge support by the Polish KBN Grant No.1P03B 05130and by Polish Government research funds for 2006-2009.

[1]J.A.M.Huhtam¨a ki et al.,Phys.Rev.Lett.97,110406

(2006).

[2]Y.Shin et al.,Phys.Rev.Lett.93,160406(2004).

2 [3]K.Gawryluk,M.Brewczyk,and K.Rz?a˙z ewski,J.Phys.

40,R1(2007).

B39,L225(2006).

[4]M.Brewczyk,M.Gajda,and K.Rz?a˙z ewski,J.Phys.B