Principles of the atomtronic diode and transistor

Introduction

The field coming to be known as 'Atomtronics' focuses on establishing ultracold atom analogs of electronic circuits and devices. Foundational work has theoretically established the possibility of transistor-like device behavior from ultracold atomic systems. In this presentation, we introduce a complete neutral atom circuits analog of the semiconductor diode as well as the first atomtronic transistor circuit to exhibit both switching behavior and linear amplification over arbitrary time scales. We also outline a method for calculating the dynamics of this specific class of open quantum systems. The circuits presented here provide the rudiments of atom signal shaping and processing that may find use in atom-based sensors and possibly enable the basic elements of cold atom-based quantum logic.

The general systems considered

Our atom-optical analogy to electronic circuits begins with the definition of the `atomtronic battery', which is composed of two reservoirs of ultracold atoms having different chemical potentials (corresponding to different electric potentials at the terminals of a conventional battery). The `wires' and atomtronic components are composed of optical lattices, and current refers to the number of atoms that pass a specific point in a given amount of time. The analog of a simple circuit composed of a diode connected to a battery is illustrated in Fig 1.

Fig 1: Atomtronic analogy to a simple diode circuit. The atomtronic analogy of a diode formed from the joining of p-type and n-type semiconductor materials. Electrons are replaced by ultracold atoms, the battery is replaced by high and low chemical potential reservoirs, and the metallic crystal lattices (the microscopic medium that the electrons traverse) are replaced by an optical lattice. The atomtronic diode is achieved by energetically shifting one half of the optical lattice with respect to the other.

The atomtronic diode

The atomtronic diode is a device that allows an atomic flux to flow across it in essentially only one direction. It is made by adding a potential step, which emulates a semiconductor junction (the boundary between p-type and n-type solid-state materials), to an energetically-flat optical lattice (Fig 1). We find that, like the semiconductor diode, the principle behavior is a direct consequence of the junction itself. In fact, the desired current response is apparent for an atomtronic junction composed of only two lattice sites offset by a potential. For this reason, we focus here on a careful study of a two lattice site system. Despite the apparent simplification, we emphasize that our full calculations for extended systems confirm that the diode behavior is maintained irrespective of the number of lattice sites around the junction. We find that the atomtronic diode conduction asymmetry becomes optimal as the height of the potential step approaches the on-site interaction energy (the interaction energy associated with putting two particles on the same lattice site). We refer to this as the `resonance condition'.

Fig 2: Current response of the atomtronic diode. Current response of the atomtronic diode. (a) Exact numerical simulation of the reverse bias characteristic of the atomtronic diode as a function of the chemical potential difference. Here the left reservoir's chemical potential is held at zero (so that the reservoir takes all atoms from the left site), and the right reservoirs chemical potential is continuously raised to the point that allows an occupancy of two atoms on the right site. The current response (here in units of the square of the coupling of the reservoir, times the average density of states of the reservoir divided by hbar) is negligible. (b) Exact numerical simulation of the forward bias characteristic of the atomtronic diode as a function of the chemical potential difference. Here the right reservoir's chemical potential is held at zero, and the left reservoir's chemical potential is continuously raised to the point that allows an occupancy of two atoms on the right site. Here we see there is a significant jump in the current response when the chemical potential matches the resonance condition. (c) The Fock energy schematic of the reverse bias scenario. The chemical potentials for the left and right reservoirs (labeled as μ_L and μ_R) are set to maintain an occupancy of zero and two atoms on the left and right lattice sites (respectively). System transitions brought on by connection of the right reservoir are labeled by the red arrows. The combined action of the reservoirs evolves the system into the decoupled state, |0 2〉. (d) The Fock energy schematic of the forward bias scenario. The chemical potentials for the left and right reservoirs aim to maintain an occupancy of two and zero atoms on the left and right lattice sites (respectively). System transitions brought on by the right reservoir are labeled by red arrows, transitions brought on by the left reservoir are labeled by blue arrows, and intra-system transitions are labeled by the green arrow. The energetically-equal states |2 0〉 and |1 1〉 allow atoms to flow across the system.

Figures 2(a) and 2(b) present our calculation of the current response of a two-site optical lattice, subject to the resonance condition, as a function of the chemical potential difference. Here the external energy of the right lattice site is assumed to be equal to the external energy of the left plus the on-site interaction energy. In Fig 2(a), the left chemical potential is set so that the left reservoir accepts all atoms on the left site, and the right chemical potential is smoothly increased to allow first zero, then one, and then two atoms on the right site. The result is a near-null current response for all values of the chemical potential difference. Conversely, in Fig 2(b), the right chemical potential remains fixed so that the right reservoir accepts all atoms on the right site, and the left chemical potential is smoothly increased to allow first zero, then one and then two atoms on the left site. Here, there is a dramatic current increase, triggered when the left chemical potential matches the resonance condition. The knee in the curve can be likened to the exponential rise in current for a semiconductor diode in forward bias, yet unlike the electronic case, the semiconductor diode the atom current saturates.

This resonance condition leads to the desired diode-like response for a straight-forward reason, which can be understood by looking at the relative energies of the quantum states in the Fock basis. The condition sets up an exact energetic degeneracy between two of the three two-particle Fock states (either two atoms on the left site (labeled as |2 0〉) or an atom on each site, |1 1〉), and energetically-separates them from the third, |0 2〉).

Figures 2(c) and 2(d) depict the dynamics in the Fock picture as we attempt to drive current across a two site lattice under the resonance condition, from right to left, and left to right (respectively). As seen in Fig 2(c), if the right chemical potential (μ_R) is set to allow two atoms on the right site and the left (μ_L) is set to take all atoms from the left, the decoupling of the |0 2〉 state, the system evolves to a steady state in which it is most likely to be found in |0 2〉 with very little current flow. This is the reverse-bias configuration. Interchanging the chemical potentials, so that they attempt to maintain two atoms on the left and zero atoms on the right promotes the system into the |2 0〉 state, we see that the resonant coupling between |2 0〉 and |1 1〉 allows one atom to hop over to the right site, putting the system in the |1 1〉 state. At this point, the combined action of the reservoirs causes the system to undergo two independent cycles, where the net effect of each contributes to an overall current across the system (Fig 2(d)). This is the forward-bias configuration.

The atomtronic transistor

The desired function of an atomtronic transistor (Fig 3) is to enable a weak atomtronic current to be amplified or to switch,either on or off, a much larger one. Transistor action requires at least three lattice sites connected to three independent reservoirs (Fig 3(a)). The resonance condition for this device is found to be an extension of the diode case to account for the third well: the left external energy is shifted above the middle site by the on-site interaction energy and is of equal energy to that of the right site.

To investigate the device characteristics, we put a fixed bias across the transistor (arranging the left and right chemical potentials to maintain an occupancy of one atom on the left site and no atoms on the right) and monitor the system response to an increase of the middle chemical potential (Fig 3(b)).

When the middle chemical potential is set to maintain zero atoms on the middle site, we find that the system primarily remains in the |1 0 0〉 state since it can only enter the right reservoir through the |0 0 1〉 state, which requires a second-order off-resonant process.

If the middle chemical potential is set to maintain an occupancy of one atom on the middle site, a resonance is accessed between the |1 1 0〉, |0 2 0〉 and |0 1 1〉 states. This triggers a cycle that amounts to a net current across the system. Simulations, however, reveal a competing process: atoms also leave the system through the middle reservoir since its chemical potential is set to maintain an average of one atom on the middle site, not two. Identical reservoir coupling strengths lead to an inefficient transistor.

As in the electronic transistor case, we can choose the base regime to be extremely `thin' so that a very small middle current controls a much larger current from left to right. This can be done by coupling the middle reservoir to the system much more weakly than the other two reservoirs. Then, for example, for every ten atoms which transition from left to right, the weak coupling can allow only one atom to leave the system from the middle site. The end result is an appreciable differential gain that can be controlled.

Fig 3(c) shows the results of the numerical simulation for the case in which the middle reservoir coupling is one-tenth of the coupling of the reservoirs on either end. One can see the small increase in current via the middle leg (red curve) generating a substantial current across the device (blue curve). Furthermore the gain is approximately linear, an important practical requirement for a transistor to act as an operational amplifier.

Fig3: Dynamics of the atomtronic transistor.(a) A cartoon of the atomtronic transistor as a three-well system, where each well is connected to its own independent reservoir. (b) An energy schematic of the relevant states of the system under the assumed resonance condition. In both illustrated cases, there is a fixed chemical potential difference across the system. In case 1, the middle chemical potential maintains an occupancy of zero particles on the middle site and most of the population remains on the left site. In case 2, the base potential is raised to put one particle on the middle site. This triggers two competing cycles that, given weak coupling of the middle reservoir, causes an avalanche of current to flow across the system. (c) An exact calculation of the current responses of the atomtronic transistor. The middle reservoir here has one-tenth the coupling strength of the left and right reservoirs. For a fixed chemical potential difference across the device, we vary the middle potential and record the response of currents leaving the system from both the right site (blue) as well as out of the middle site (red). The differential current gain for this specific system is both large and essentially linear.

Conclusions

We have presented the characteristics of simple diode and transistor atomtronic circuits. There is clear potential for expanding the analogy to more complex circuits, such as amplifiers, constant current sources, flip flops and logic. Apart from their potential utilitarian value, it is intriguing to consider the connection of atomtronics with quantum information physics, since the atom devices operate in the coherent regime.

However, atomtronic devices do not better their electronic counterparts in every regard. For example, the overall current flow is determined principally by the tunneling rate from one site to the next, making the atomtronic current much slower than electronic current. Compensating for this is the fact that this is a completely different physical system in which to pursue device physics: the atoms have complex internal structure and internal states, the atoms can be bosons or fermions, they are massive and affected by gravitational fields, the lattice can be dynamically varied, and many undesirable effects present in solid-state systems such as crystalline impurities, dislocations, and phonon scattering are absent.