7 Δ$_{t}$. In this figure I$_{0}$ = I c C = eρ$_{s}$/ ℏ ∼ 8 n A is the unit of current and E$_{0}$ = e $^{2}$/ϵ l is the unit of energy. Typi- cal quantum Hall superfluid experiments are performed at a magnetic field of roughly 2.1 Tesla for which the magnetic length is 17.65 nm which gives E$_{0}$ = 6 . 4 meV. In all the calculations reported on here we used a pixel area A$_{pixel}$ = 10 × 10 l 2 and the the mean-field theory estimate 7 ( ρ$_{s}$ ≃ 0 . 005 E$_{0}$ ) for the pseudospin stiffness. The calculations in Fig.( 3) are for 40 pixels in the width W direction and 50 pixels in the current L direction. For this system size , the Josephson length is approximately equal to L when 2 πl $^{2}$Δ$_{t}$ n ∼ 4 × 10 − $^{7}$E$_{0}$ . The numerical results illustrated in Fig. (3) show the crossovers from Δ$_{t}$-dependence, to Δ 1 / 2 t -dependence, to saturation as Δ$_{t}$ increases that was anticipated in our qualitative discus- sion. At small Δ$_{t}$ n the critical current is reduced by a small fraction compared to the single pixel result in ac- cord with the Fig.(1). We now examine the ingredients which enter these numerical results in greater detail. According to the schematic Fig. (1), the pseudospin transfer torque acts only near the hot spots at which current enters and ex- its the sample. Fig. (4) shows a typical numerical re- sults for the spatial distribution of the pseudospin trans- fer torque. In the present model, the area of the region in which transport current is converted into supercurrent depends on the pixel size and the pseudospin stiffness and anisotropy coefficients. In the tunneling geometry, coun- terflow supercurrent is generated near both source and drain hot spots and flows diagonally toward the center of the sample. In Fig. (5) we show a typical supercur- rent distribution for the tunneling geometry case. The corresponding distribution for an equivalent drag exper- iment (both contacts connected to the same layer) is illustrated in Fig. (6). Since the elliptic sine-Gordon equation applies locally when the pseudospin transfer torque is negligible, it follows from Green’s theorem and Eq.( 7) that in a steady state the total counterflow cur- rent injected into the interior of the sample (which for the tunnel geometry equals the total charge current flowing through the system) must match the area integration of (1 / 2)(Δ$_{t}$ n/ ℏ ) sin φ . For a drag geometry experiment the same integral should vanish. We now examine steady state condensate configura- tions at currents near the critical current for both small FIG. 5: Supercurrent distribution in system with area 500 × 400 l $^{2}$, 2 πl $^{2}$n Δ t = 10 $^{$_{−}$6}$E 0 , and I/I c = 0 . 29. This plot is for a tunneling geometry in which the source is a top layer contact and the drain is a bottom layer contact. Supercurrents are generated near both hot spots and flow diagonally toward the sample center. E 0 and l are defined as in previous figures. FIG. 4: Spatial distribution of the ˆ z component of the pseu- dospin transfer torque in a system with 500 × 400 l 2 area, 2 πl $^{2}$n Δ t = 10 $^{$_{−}$6}$E 0 , and I/I c = 0 . 29. The pseudospin trans- fer torque in the model studied here acts mainly in the hot spot pixels. E 0 and l are defined as in previous figures. FIG. 6: Supercurrent distribution in system with area 500 × 400 l $^{2}$, 2 πl $^{2}$n Δ t = 10 $^{$_{−}$6}$E 0 . This plot is for the drag geometry case in which the source and drain are both top layer contacts, but other parameters of the calculation are identical to those used in the preceding tunnel-geometry figure. E 0 and l are defined as in previous figures. y / 40  l  x / 40  l  y / 40  l  x / 40  l 