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matter - wave bright solitons  stable solitary waves whose density is greater than the background one  have recently been realized in a quasi one - dimensional ( 1d ) bose - einstein condensate ( bec )  @xcite . in 1d , the quantum kinetic pressure counterbalances an attractive interatomic interaction , allowing a stable bright soliton to be formed . however , in 2d or higher dimensions , bright solitons are always unstable against collapse or expansion as long as the attractive interaction is constant in time  @xcite . it has been demonstrated by abdullaev _ et al . _ @xcite and by us  @xcite that by rapid oscillations of interaction between attractive and repulsive through , e.g. , the feshbach resonance  @xcite , a bec can be stabilized in a quasi 2d space without a radial trapping potential . the mechanism of the stabilization is similar to that of an inverted pendulum , in which a bob is stabilized above the vertically oscillating pivot  @xcite . however , the stabilization of a bec in 3d free space has been elusive due to complex dynamical instabilities inherent in nonlinear nonequilibrium systems .    in the present paper , we show that a droplet  @xcite of a bec can be stabilized in 3d free space by using oscillating interactions . in order to simulate experimental situations at finite temperature , we take into account the effect of dissipation caused by the interaction between a bec and a thermal cloud . this can be incorporated in mean field theory by a non - unitary term added to the gross - pitaevskii ( gp ) equation . this model has successfully been used to describe damping of collective modes  @xcite and vortex nucleation  @xcite . we find that the dissipation plays a crucial role in the stabilization of a bec droplet by suppressing the dynamical instabilities , and show that the droplet can indeed be stabilized for a realistic amount of dissipation . this paper is organized as follows . section  [ s : gp ] introduces the gp equation with oscillating interactions , and shows that the dependence of the dynamics on the oscillation frequency can be eliminated by the scaling property when a trapping potential vanishes . section  [ s : num ] presents the main results of this paper , in which a bec droplet is shown to be stabilized in 3d free space by use of oscillating interactions . the stability diagrams with respect to the interaction and dissipative parameters are also obtained . section  [ s : var ] studies the variational analysis , and sec . [ s : conc ] concludes this paper . we consider the gp equation in the presence of dissipation given by  @xcite @xmath0 where @xmath1 is a phenomenological dissipation constant which is to be determined experimentally , and the trapping potential @xmath2 and the s - wave scattering length @xmath3 are controlled to vary in time . the last term on the right - hand side of eq . ( [ gp ] ) guarantees the normalization @xmath4 with @xmath5 being the number of atoms , where the chemical potential is given by @xmath6    for simplicity , we assume an isotropic trapping potential @xmath7 where @xmath8 is unity at @xmath9 , then decreases towards zero , and vanishes at some time . the s - wave scattering length is made to oscillate as @xmath10 where @xmath11 ramps up from 0 to 1 . the ramp functions @xmath8 and @xmath11 are introduced to avoid initial nonadiabatic disturbances that cause dynamic instabilities . these functions must be chosen so that neither collapse nor expansion occurs in the course of the ramping . normalizing the length , time , energy , and wave function by @xmath12 , @xmath13 , @xmath14 , and @xmath15 , respectively , we obtain the normalized form of the gp equation : @xmath16 where @xmath17 , and the wave function is normalized as @xmath18 . it should be noted that once the trapping potential is switched off [ @xmath19 , eq .  ( [ gpn ] ) no longer depends on @xmath20 explicitly . this implies that the @xmath20 dependence can be eliminated by appropriate scaling of the time , length , and interaction . the dynamics at driving frequency @xmath21 can be reduced to those at @xmath20 by rescaling @xmath22 , @xmath23 , and @xmath24 . for example , if large @xmath20 is unavailable experimentally , we have only to increase the strength of interaction and enlarge the size of the bec droplet . we numerically solve the gp equation ( [ gpn ] ) using the crank - nicholson scheme  @xcite , where the following simple forms of the ramp functions are used : @xmath25 we use a quadratic function for @xmath11 because it makes initial nonadiabatic disturbances smaller than linear ramps . figure  [ f : stable ] shows the time evolution of the peak density @xmath26 and the monopole moment @xmath27 for @xmath28 and @xmath29 . ( left axis ) and monopole moment @xmath30 ( right axis ) for the oscillating interaction @xmath31 with @xmath32 . the initial state is the noninteracting ground state in a trapping potential @xmath33 . then the interaction is gradually turned on , and the trapping potential is simultaneously turned off as shown in the inset . ( b ) a magnification of ( a ) . the inset shows the density profile @xmath34 from @xmath35 to @xmath36 . the dotted line is a sine function for comparison of the phase . , width=317 ]    the dissipation constant @xmath1 is taken to be 0.03  @xcite , and the ramp parameters are @xmath37 and @xmath38 . the initial state is chosen to be the noninteracting ground state for a trapping potential with @xmath39 . the density and width oscillate at the driving frequency . [ the black bands in fig . [ f : stable ] ( a ) represent rapid oscillations of @xmath40 and @xmath41 which are beyond the resolution limit . ] we note that the amplitudes and mean values of both @xmath42 and @xmath41 converge to constant values , indicating that a bec droplet is dynamically stabilized . the initial disturbances last for a relatively long time shown as slow oscillations or ripples of @xmath42 and @xmath41 in fig . [ f : stable ] ( a ) , which gradually decay due to dissipation . from the time evolution of the density profile @xmath34 [ inset in fig . [ f : stable ] ( b ) ] , we find that the droplet exhibits breathing - mode oscillations with a large amplitude . this indicates that large oscillations of mass flow persist even after stabilization . we find from fig . [ f : stable ] ( b ) that the phase of the breathing - mode oscillations of the droplet lags slightly behind that of the oscillating interaction shown as the dotted curve . this retardation is considered to be due to dissipation because there is no phase difference in the dissipation - free 2d case  @xcite , where @xmath41 and @xmath42 are in - phase and out - of - phase with the oscillating interaction . we gradually changed @xmath43 and @xmath44 from those of the stable droplet state in fig . [ f : stable ] , and found distinct types of instabilities as shown in fig . [ f : unstable ] . ( left axis ) and monopole moment @xmath30 ( right axis ) . the initial state is prepared as in fig . [ f : stable ] ( a ) , and @xmath45 are linearly changed from @xmath46 to @xmath47 in ( a ) , @xmath48 in ( b ) , and @xmath49 in ( c ) during @xmath50 . the insets in ( b ) and ( c ) are magnifications of @xmath51 in the dashed squares and the density profiles @xmath34 at the times indicated by a and b. , width=317 ]    when @xmath52 is decreased , a droplet expands because of a decrease in an attractive interaction as shown in fig . [ f : unstable ] ( a ) . the expansion occurs also by an increase in @xmath44 , since the effective repulsive interaction due to the oscillation is proportional to @xmath53 [ see eq . ( [ eom2 ] ) ] . when @xmath52 is increased and @xmath44 is decreased , the system becomes dynamically unstable against the slow oscillation and eventually expands away as shown in fig . [ f : unstable ] ( b ) . the expansion occurs , despite the fact that an increase in @xmath52 enhances the attraction and a decrease in @xmath44 suppresses the repulsion . the slow oscillation is seen in the profile of the maxima of @xmath41 ( the ripple in the upper edge ) , while the minima is almost constant . figure  [ f : unstable ] ( c ) shows an instability that arises for large @xmath52 and @xmath44 . since the modulation at half frequency grows , this instability is considered to arise from parametric resonance . the density profiles in the inset show that higher radial modes are excited , which leads to destabilization of the droplet . we prepare a stable droplet in the same manner as in fig . [ f : stable ] , and change the parameters to various values to obtain the stability diagram shown in fig . [ f : contour ] .    , and the interaction is linearly changed during @xmath50 as in fig . [ f : unstable ] . ( a ) the black region corresponds to the one in which a droplet expands during the change of the interaction , and the white region corresponds to the one in which a droplet survives until at least @xmath54 . the behaviors in the regions `` instability i , ii , and iii '' correspond to those in fig . [ f : unstable ] ( a ) , ( b ) , and ( c ) , respectively . ( b ) the parameter @xmath1 is also linearly changed as well as the interaction during @xmath50 from @xmath55 to @xmath56 . , width=317 ]    the parameters @xmath43 and @xmath44 are changed linearly during @xmath57 . we define the lifetime of the droplet as the duration between @xmath58 and the time at which the droplet begins to expand indefinitely . the regions referred to as `` instability i , ii , and iii '' in fig . [ f : contour ] correspond to the instabilities demonstrated in fig . [ f : unstable ] ( a ) , ( b ) , and ( c ) . we find from fig . [ f : contour ] ( a ) that the stable region is surrounded by the three distinct types of instabilities . the boundary between the `` stable '' and `` instability i '' regions are sharp while the `` instability ii '' gradually sets in as @xmath52 increases and @xmath44 decreases . the collapse occurs for large @xmath52 and small @xmath44 , but the corresponding parameter regime lies outside fig . [ f : contour ] .    in fig [ f : contour ] ( b ) , @xmath1 is linearly decreased from 0.03 to 0.01 as @xmath52 and @xmath44 are ( also linearly ) changed . the instability regions enlarge as compared with those of fig  [ f : contour ] ( a ) , and the stability region disappears . from this result , we conclude that the stability region exists only for @xmath59 . however , we can not exclude the possibility that other stable states exist , since we do not investigate the entire functional space .    in numerical calculations of the time evolution , we must pay special attention to the boundary effect . since the trapping potential is absent , the atoms that escape from the droplet spread out , reflect at the boundary , and return to the droplet region , producing spurious boundary effects . for example , when the dirichlet boundary condition is imposed at @xmath60 , the lifetime of the droplet ( whose definition is the same as in fig . [ f : contour ] ) for @xmath28 and @xmath61 becomes @xmath62 , while the correct value is @xmath63 . therefore , the spatial cutoff in the numerical analysis must be much larger than the size of the droplet . the numerical simulations that we have presented so far have been carried out under the assumption of the spherical symmetry of the system . however , this geometry can not generate multipole dynamical instabilities which destroy a bec droplet . in order to check the absence of such multipole instabilities , we have performed full 3d calculations by discretizing the space of @xmath64 size in our dimensionless unit into a @xmath65 mesh . a limited spatial size due to our computational power causes spurious boundary effects and a rough mesh produces numerical errors . nevertheless , as shown in fig . [ f:3d ] , snapshots of time evolution of a bec droplet appear isotropic and do not show any multipole instability .     in units of @xmath66 . , width=317 ]    we have thus confirmed that a bec droplet can be stabilized without multipole instabilities . the gaussian variational wave function well describes the dynamical stabilization of becs in 2d free space  @xcite . we examine a variational method in 3d using the gaussian trial function as  @xcite @xmath67 where @xmath68 is the variational parameter that characterizes the size of the condensate . substituting eq . ( [ gauss ] ) into the action @xmath69 we obtain the equation of motion for @xmath70 as @xmath71 where @xmath72 . we separate @xmath70 into the slowly varying part @xmath73 and the rapidly oscillating part @xmath74 as @xmath75 . according to ref . @xcite , an effective potential for @xmath73 is given by @xmath76 , where @xmath77 is the amplitude of the oscillating `` force '' for @xmath73 and @xmath20 is its frequency . from eq . ( [ eom ] ) , @xmath77 corresponds to @xmath78 . since we are using a system of units in which @xmath79 , the effective potential becomes @xmath80 , and then the equation of motion for @xmath73 reads @xmath81 this equation agrees with that in ref . @xcite when @xmath20 is much larger than the characteristic frequencies of the system . thus , the oscillating interaction blocks the collapse by the effective potential proportional to @xmath82 . the effective potential @xmath83 has a local minimum for appropriate values of @xmath84 and @xmath85 . for instance , when @xmath86 and @xmath87 , @xmath83 has a local minimum at @xmath88 . in fact , by numerically solving the equation of motion ( [ eom ] ) for @xmath70 , we find that a stable solution exists . however , fig . [ f : contour ] ( b ) implies that the system is unstable against the dynamical instability for @xmath89 . this indicates that the simple gaussian function ( [ gauss ] ) is insufficient to describe the instability of the system . figure  [ f : diff ] illustrates the difference between the density profiles of the stable droplet ( @xmath90 in fig . [ f : stable ] ) and the gaussian functions fitted to them . ( solid curves ) of the stable droplet for @xmath28 and @xmath29 ( @xmath91 in fig . [ f : stable ] ) when the peak density becomes ( a ) maximal and ( b ) minimal . the dashed curves are the gaussian functions fitted to the density profiles . , width=317 ]    we adopt the least squares fitting of @xmath92 to @xmath93 with fitting parameter @xmath94 , which is an appropriate fitting in 3d . figure  [ f : diff ] shows that the numerically exact wave functions significantly deviate from the gaussian functions even in the stable droplet . we note that the gaussian function in fig  [ f : diff ] ( a ) is almost the same as that in fig  [ f : diff ] ( b ) . this indicates that the outer region , which has large weight in the fitting due to the factor @xmath95 , is almost stationary , and the mass flow occurs mainly around the center of the droplet . thus , more appropriate functions are needed to correctly describe the phenomena . we have studied a bose - einstein condensate with an oscillating interaction with dissipation , and shown that a bec droplet is stabilized in 3d free space with realistic dissipation , @xmath55 . the frequency of the oscillating interaction must be much faster than the characteristic frequency of the system , but it can be a moderate frequency according to the scaling property discussed in sec .  [ for instance , the situation in fig . [ f : stable ] can be realized in a condensate of @xmath96 @xmath97 atoms by the s - wave scattering length @xmath98 [ nm ] with @xmath99 hz , where a trapping potential with frequency 3.3 hz is removed in 25 ms , and the final size of the droplet becomes a few micrometers . such an oscillation of interaction can be easily realized in experiments using the feshbach resonance  @xcite .    under gravity , a condensate falls after the trapping potential is turned off . in the above example , the gravitational sag in the initial trapping potential is @xmath100 cm , and the condensate falls about 1 mm until the trapping potential vanishes . the effect of gravity can be canceled out using the magnetic levitation  @xcite , which enables us to observe the longtime behavior of the droplet . a `` gaseous bec droplet '' found in this paper is different from the usual condensate in that it coheres by itself without the help of the trapping potential . this self - trapped matter wave might exhibit interesting dynamics , such as in collective mode , collapsing dynamics , and vortex nucleation , which will be discussed elsewhere .    _ note added _  while preparing the present paper , a similar paper  @xcite appeared which also reports stabilization of a bec in 3d free space by an oscillating interaction . the strength of interaction studied in ref . @xcite is much larger than that of ours . in our parameter regime , a droplet can be stabilized adiabatically by simple ramp functions with realistic dissipation . on the other hand , in ref  @xcite a droplet is stabilized without dissipation by following a complicated ramp scheme to reach a final state . this work was supported by the special coordination funds for promoting science and technology , a 21st century coe program at tokyo tech `` nanometer - scale quantum physics '' , and a grant - in - aid for scientific research ( grant no . 15340129 ) from the ministry of education , science , sports , and culture of japan . an illuminating account of this stabilizing mechanism is found in : r. p. feynman , r. b. leighton , and m.  l.  sands , _ the feynman lectures on physics _ , ( addison - wesley , massachusetts , 1964 ) , vol . ii , chap . 29 .
we show that a droplet of a bose - einstein condensate can be dynamically stabilized in free space by rapid oscillations of interatomic interactions between attractive and repulsive through , e.g. , the feshbach resonance . 
 energy dissipation , which is present in realistic situations , is found to play a crucial role to suppress dynamical instabilities inherent in nonlinear nonequilibrium systems .