Transistorlike behavior of a Bose-Einstein condensate in a triple-well potential

In the last several years considerable efforts have been devoted to developing Bose-Einstein-condensate-based devices for applications such as fundamental research, precision measurements, and integrated atom optics. Such devices, capable of complex functionality, can be designed from simpler building blocks as is done in microelectronics. One of the most important components of microelectronics is a transistor. We demonstrate that a Bose-Einstein condensate in a three-well potential structure where the tunneling of atoms between two wells is controlled by the population in the third shows behavior similar to that of an electronic field-effect transistor. Namely, it exhibits switching and both absolute and differential gain. The role of quantum fluctuations is analyzed, and estimates of the switching time and parameters for the potential are presented.

Figure 1

(Color online) The geometry of a BEC transistor. When the number of atoms in the middle well is small, tunneling from the left into the right well is negligible (a). This is due to the fact that the chemical potential of the middle well does not match that of the two other wells (c). Placing atoms in the middle well increases the chemical potential due to interatomic interactions (d) and enables tunneling then atoms tunnel from the left into the right well. This happens because atom-atom interactions increase the energy of the middle guide (b).

Figure 2

A schematic of a multiwell nonsymmetric potential structure with two adjacent wells shown. The points $x_{k-1}$, $x_k$, and $x_{k+1}$ are chosen between the wells where the eigenmodes ${\psi }_k$ are exponentially small.

Figure 3

An example of the calculation of local modes. Graphs (a)–(c) show the three lowest global eigenmodes of the potential $V$ (dotted line). These eigenmodes are nonlocal with large probability density in two or more wells. Graphs (d)–(f) show local modes, which are linear combinations of the nonlocal eigenmodes.

Figure 4

The number of atoms in the right well as a function of interaction time for different initial number of atoms in the middle well. The dotted curve corresponds to initially empty middle well, $b_m\left(0\right)=0$, and the solid curve to $b_m\left(0\right)=D$. The dashed curve corresponds to the initial condition $b_m\left(0\right)=D\exp (i\pi ∕2)$. For all curves ${\omega }_m=-1.3$, ${\omega }_r=0.5$, $Z_m{D}^2=1$, and $Z_r{D}^2=0$.

Figure 5

The number of atoms in the middle (dashed) and right (solid curve) well as a function of interaction time for ${\omega }_m=-1.3$, ${\omega }_r=0.5$, $Z_m{D}^2=1$, $Z_r{D}^2=0$, and $b_m=D$.

Figure 6

The output number of atoms in the right well $(\tau =20)$ as a function of the number of atoms initially placed in the middle well.

Figure 7

The output number of atoms in the right well $(\tau =20)$ as a function of the dimensionless frequency of the middle well ${\omega }_m$. The dotted line corresponds to the initially empty middle well, $b_m\left(0\right)=0$, the solid line to $b_m=D$, and the dashed line to $b_m=D\exp (i\pi ∕2)$.

Figure 8

The output number of atoms in the right well as a function of the relative phase of the atoms placed in the middle well.

Figure 9

The average number of atoms in the right well as a function of the interaction time for three different values of the coupling between the left and middle wells. Atoms in the middle well are initially in a coherent state with $⟨N_m⟩\left(0\right)=D^2$ and zero phase. The solid line is the result of the mean-field calculation, the dotted line is the result of the second-quantization calculation with $D^2=1$, the dashed line corresponds to $D^2=4$, and the dash-dotted line corresponds to $D^2=8$.

Figure 10

The probability $P\left(N_r\right)$ of finding $N_r$ atoms in the right well $(\tau =10)$. Atoms in the middle well are initially in a coherent state with $\alpha =D$. The dash-dotted line corresponds to $D^2=1$, the dashed line to $D^2=4$, and the solid line to $D^2=8$.

Figure 11

The average number of atoms in the right well as a function of the interaction time for three different values of the coupling between the left and middle wells. The middle well is initially empty. The solid curve is the result of the mean-field calculation, the dotted curve is the result of the second-quantization calculation with $D^2=1$, the dashed curve corresponds to $D^2=4$, and the dash-dotted curve corresponds to $D^2=8$.

Figure 12

The probability $P\left(N_r\right)$ of finding $N_r$ atoms in the right well $(\tau =10)$. The middle well is initially empty. The dash-dotted line corresponds to $D^2=1$, the dashed line to $D^2=4$, and the solid line to $D^2=8$.