Atomtronics: Ultracold-atom analogs of electronic devices

Atomtronics focuses on atom analogs of electronic materials, devices, and circuits. A strongly interacting ultracold Bose gas in a lattice potential is analogous to electrons in solid-state crystalline media. As a consequence of the gapped many-body energy spectrum, cold atoms in a lattice exhibit insulatorlike or conductorlike properties. $P$-type and $N$-type material analogs are created by introducing impurity sites into the lattice. Current through an atomtronic wire is generated by connecting the wire to an atomtronic battery which maintains the two contacts at different chemical potentials. The design of an atomtronic diode with a strongly asymmetric current-voltage curve exploits the existence of superfluid and insulating regimes in the phase diagram. The atom analog of a bipolar junction transistor exhibits large negative gain. Our results provide the building blocks for more advanced atomtronic devices and circuits such as amplifiers, oscillators, and fundamental logic gates.

Figure 1

(Color online) Schematic of the atomtronic many-body energy band structure of strongly interacting bosons in a lattice. The first band is made up of all states with a filling of less than one atom per site while the second band contains all states with filling between one and two atoms per site. The two bands are separated by the onsite repulsive interaction $U$.

Figure 2

(Color online) Zero temperature phase diagram of a Bose gas obtained from Eq. (2) for an infinite one-dimensional lattice using the two-state approximation (fermionized regime, $U⪢J$). At large values of $J∕U$ the gas is superfluid (SF). Below a critical value of $J∕U$ the system enters a Mott-insulator phase (MI) for integer filling, while it remains superfluid for noninteger filling. The MI1 Mott-insulator region has one atom per site and the MI2 region has two atoms per site. The width of a Mott lobe at a given $J∕U$ gives the size of the band gap in the many-body band structure while the width of a band is given by the width of the superfluid region. The Mott lobe boundaries are linear due to the use of the two-state approximation [35].

Figure 3

(Color online) Schematic of atoms in a lattice connected to an atomtronic battery. A voltage is applied by connecting the system to two reservoirs, one of higher (left) and one of lower (right) chemical potential. An excess (deficit) of atoms is generated in the left (right) part of the system, giving rise to a current from left to right.

Figure 4

(Color online) (a) Schematics of an $N$-doped lattice. The donor sites feature a level right below the first empty many-body band. An atom which occupies this level can easily be excited and move throughout the lattice. (b) Schematics of a $P$-doped lattice. Acceptor sites have a level right above the highest full band. Atoms can easily be excited into this level and allow for a hole to move throughout the lattice.

Figure 5

(Color online) Zero temperature phase diagram of a Bose gas in an undoped (blue), an $N$-doped (red) and a $P$-doped (green) lattice. The boundary of the lowest Mott insulator lobe with a filling of one atom per site is displayed as calculated using the two-state approximation. The results for the doped lattices were obtained by adding an energy of $\pm 5J$ to every third site in an infinite lattice using a modified Jordan-Wigner transformation. $N$-type doping turns an insulating state into a superfluid state with more than one atom per site. Similarly, $P$-type doping turns an insulating state into a superfluid state with less than one atom per site.

Figure 6

(Color online) Current as a function of chemical potential difference (voltage) for different materials (different fillings $n$) in the fermionization regime where bosons are impenetrable. A wire with a certain number of atoms carries the same current as a wire with that number of holes (particle-hole symmetry). The chemical potential difference is given in units of $\Delta {\mu }_{\max }$. This quantity denotes the chemical potential differences that yield the maximum currents and corresponds to the chemical potential difference at which the system enters an insulating regime at one of the two battery contacts for a given $n$. Inset: Current as a function of filling at $\Delta {\mu }_{\max }$. Maximum currents are attained at half-filling. Note the symmetry in the current due to the particle-hole symmetry.

Figure 7

(Color online) Schematic of the conduction band of an electronic (left-hand panels) and atomtronic (right-hand panels) $PN$-junction diode in (a) equilibrium, (b) reverse bias, and (c) forward bias with $N$-type materials on the left and $P$-type on the right of each configuration. Left-hand panels: The electronic system features a voltage dependent energy barrier at the junction leading to an increasing (decreasing) flux from the $N$-type to the $P$-type material as the junction is forward (reverse) biased while the current from $P$ to $N$ is independent of voltage. Right-hand panels: The operation of an atomtronic diode is based on the existence of insulating phases where $\partial n∕\partial \mu =0$, i.e., a change in voltage does not lead to a change in particle transfer between battery and system. As a consequence, only a small current can flow from $N$-component to $P$-component through an atomtronic diode in reverse bias while in forward bias the particle transfer between battery and system can be varied over a large range.

Figure 8

(Color online) Phase diagram of the $PN$-junction configuration of an atomtronic diode. The small (large) population of the second many-body band in the $P$-type ($N$-type) material yields the analog of the small (large) thermal electron population of the conduction band in a semiconductor. The states of the $P$-type and $N$-type materials at zero voltage are represented by the squares. Arrows indicate the chemical potential difference (voltage) imposed to obtain a reverse (left) and a forward biased junction (right).

Figure 9

(Color online) The characteristic current-voltage curve for an atomtronic diode. The larger the forward bias (voltage$>0$), the higher the current of atoms flowing from the $P$-component and to the $N$-component. In reverse bias (voltage$<0$) the current saturates since the particle transfer between battery and system cannot be increased beyond a certain small value when the battery pole chemical potentials enter insulating zones where $\partial n∕\partial \mu =0$.

Figure 10

Circuit schematic of an electronic or atomtronic bipolar junction transistor of the $NPN$-type. A thin $P$-type component (base) is sandwiched between two $N$-type components (emitter and collector). The key feature is that the voltage $V_{EB}$ can be used to obtain gain in the collector current $I_C$ relative to the base current $I_B$.

Figure 11

(Color online) (a) Squares: equilibrium configuration of an atomtronic bipolar junction transistor in the phase diagram. Arrows: direction in which the chemical potentials of the battery contacts are varied. (b) Differential gain $d{I}_C∕d{I}_B$ as a function of emitter-base voltage $V_{EB}$. Inset: Base current and collector current as a function of emitter base voltage. The voltage is changed by varying the emitter contact potential from $\mu (n_E=1)$ to $\mu (n_E=1.5)$. The base contact is kept at $\mu (n_B=1)$ right below its equilibrium value $\mu (n_B=1.005)$ while the collector contact is kept at the equilibrium value $\mu (n_C=1.5)$.