Transitions and excitations in a superfluid stream passing small impurities

We analyze asymptotically and numerically the motion around a single impurity and a network of impurities inserted in a two-dimensional superfluid. The criticality for the breakdown of superfluidity is shown to occur when it becomes energetically favorable to create a doublet—the limiting case between a vortex pair and a rarefaction pulse on the surface of the impurity. Depending on the characteristics of the potential representing the impurity, different excitation scenarios are shown to exist for a single impurity as well as for a lattice of impurities. Depending on the lattice characteristics it is shown that several regimes are possible: dissipationless flow, excitations emitted by the lattice boundary, excitations created in the bulk, and the formation of large-scale structures.

Figure 1

Critical velocity of vortex nucleation as a function of obstacle radius $b/\xi .$ Solid line is the numerically determined critical velocity. The critical velocity $v_{cr}=0.36970c$ for an infinite obstacle is given by the thin red dashed line. Thick green dashed line is the analytically determined critical velocity, as explained in the text.

Figure 2

Critical velocity at which excitations are generated as a function of obstacle height $V_0$ for fixed $b=\xi .$ Dots represent the generation of vortex pairs, while triangles represent rarefaction pulses. Squares represent the depletion of the condensate (uniform density minus the minimum of the density at the center of the potential) due to the repulsive interactions with the obstacle [60].

Figure 3

Vortex pair generation due to stronger BEC-impurity interaction. Pseudocolor density plots of superfluid flow around an obstacle specified by $V_0=0.7$, $b=1$ (measured in healing lengths) at (a) $t=60$, (b) $t=84$, and (c) $t=90$ with velocity $v=0.753c.$ The more luminous the picture the higher the density. Each picture shown corresponds to an area of $15\times 15$ [61]. The dimensionless units are used as specified in the main text.

Figure 4

Rarefaction pulse generation due to weaker potential. Pseudocolor density plots of superfluid flow around an obstacle specified by $V_0=0.3$, $b=1$ at (a) $t=150$, (b) $t=175.5$, and (c) $t=186$ with flow velocity $v=0.88c.$ The more luminous the picture, the higher the density. Each picture shown corresponds to an area of $15\times 15$ [61]. The dimensionless units are used as specified in the main text.

Figure 5

Rarefaction pulse generation due to a $\delta$ function impurity. Pseudocolor density plots of superfluid flow around a delta impurity at (a) $t=21.5$, (b) $t=26.5$, and (c) $t=41.5$, with velocity $v=0.92c.$ Each picture shown corresponds to an area of $12\times 12$ [62]. The more luminous the picture the higher the density. The dimensionless units are used as specified in the main text.

Figure 6

Vortex pair generation due to a potential of diameter less than a healing length. Pseudocolor density plots of superfluid flow around a very small obstacle specified by $V_0=40$ and $\alpha =7$ at (a) $t=0$ and (b) $t=22.8$ with velocity at infinity $v=0.92c.$ Each picture shown corresponds to an area of $10\times 10$ [63]. The more luminous the picture, the higher the density. The dimensionless units are used as specified in the main text.

Figure 7

Vortex pair generation due to a potential of diameter much bigger than healing length and causing only a slight depletion of the condensate. Pseudocolor density plots of superfluid flow around an obstacle specified by (a) $V_0=0.25$ and $b=4$ at $t=57$, (b) $t=75$, and (c) $t=105$ with velocity at infinity $v=0.8c.$ Each picture shown corresponds to an area of $40\times 40$ [62]. The more luminous the picture, the higher the density is. The dimensionless units are used as specified in the main text.

Figure 8

Rarefaction pulse generation due to a small and weakly interacting potential at supersonic speed. Pseudocolor density plots of superfluid flow around a obstacle specified by $V_0=0.2$ and $b=1$ at $t=45$, (b) $t=82.5$ and (c) $t=180$ with Mach number $v=1.01c.$ The more luminous the picture, the higher the density is. Each picture shown corresponds to an area of $35\times 35$ [64]. The dimensionless units are used as specified in the main text.

Figure 9

Schematics of the distinct regions in superfluid flow around regular lattices on a rectangular area. The boundaries are approximate and are sensitive to the parameters of the impurity network. Here, $a$ denotes the distance between nearest neighbors, which ranges from 0 to 12 healing lengths, and $v$ is the absolute value of the superfluid velocity at infinity ranging up to the speed of sound $c.$ Hatched areas $\mathrm{II}$ and $\mathrm{III}$ show regions where excitations are outside the lattice and $\mathrm{V}$, $\mathrm{IV}$ indicate the regimes where excitations are within the lattice as well.

Figure 10

Pseudocolor density plots ${\left|\psi \right|}^2(x,y)$ of qualitatively distinguishable phases in superfluid flow passing a triagonal lattice of impurities. Shown is the condensate density at (a) velocity $v=0.5c$ and (b) $v=0.555c$. Other parameters of the system are array size $A=240\times 72$, frame shown $480\times 120$, $a=5$, $b=1.5$,$V_0=1$, and all snapshots are at $t=125$ [67]. The more luminous the picture, the higher the density is. The dimensionless units are used as specified in the main text.

Figure 11

Critical velocities as functions of the distance $a$ between impurity to its nearest neighbors. For this comparison the remaining free parameters is chosen to be $V_0=1$, $b=1.5$ while the area at which the triangular lattice of atoms is present is $200\times 120$ (measured in healing lengths). Triangles represent data points of the first critical velocity and squares represent data points of the second critical velocity. The interpolation between data points is linear [67].

Figure 12

Pseudocolor density plots of superfluid flow around a obstacles on arranged on a triangular lattice specified by $V_0=10$ and $b=2$ at (a) $t=310$, (b $t=312.5$, (c) $t=315$, (d) $t=317.5$, (e) $t=320$, and (f) $t=325$ with velocity at infinity $v=0.67c.$ Each picture shown corresponds to an area of $30\times 30$ [68]. The more luminous the picture, the higher the density is. The dimensionless units are used as specified in the main text.

Figure 13

Pseudocolor density plots ${\left|\psi \right|}^2(x,y)$ of superfluid flow passing dense triagonally distributed atoms with velocity $v=0.5c.$ The size of the atom array is $290\times 180$, the frame size shown in picture (a) is $220\times 220$ at $t=522.5.$ Picture (b) is a detail of frame size $70\times 70$ at $t=522.5$ and picture (c) is at $t=525$ (with frame size $70\times 70$). The atom parameters were chosen to be $V_0=1$, $a=3$, and $b=1.5$ [68]. The more luminous the picture, the higher the density is. The dimensionless units are used as specified in the main text.

Figure 14

Pseudocolor density plots ${\left|\psi \right|}^2(x,y)$ of superfluid flow passing a very dense triangular lattice of impurities with velocity $v=0.2c.$ The frame size shown is $150\times 150$ at $t=539.25.$ The impurity parameters were chosen to be $V_0=1$, $a=1.6$, and $b=1.5$ [68]. The more luminous the picture the higher the density is. Arrows point towards the direction of motion of the encircled vortices, and the big arrow indicates the direction of the superfluid mainstream. The dimensionless units are used as specified in the main text.

Figure 15

Pseudocolor density plots ${\left|\psi \right|}^2(x,y)$ of phases in superfluid flow passing uniformly distributed atoms with different velocities. The atom number is $N_{\mathcal{A}}=80$, size of possible atom positions $120\times 130$ (measured in healing length), the frame size shown is $210\times 210$ in the same units, and the atom parameters were chosen to be $V_0=2$ and $b=2.$ Picture (a) shows the flow at $v=0.2c$, (b) at $v=0.32c$, (c) at $v=0.39c.$ All pictures show the same (random) distribution of atoms and represent a snapshot at $t=165$ [72]. The more luminous the picture, the higher the density is. The dimensionless units are used as specified in the main text.