Principles of an atomtronic transistor

Our semiclassical treatment of an atomtronic transistor begins by borrowing the nomenclature of the semiconductor field-effect transistor. The three regions of the triple-well potential, shown in figure 1(a), are labeled the 'source,' 'gate,' and 'drain' wells. Emulating an experimentally viable system we take the trapping potential to be cigar shaped with corresponding harmonic trap frequencies ${\omega }_{x}\ll {\omega }_{\perp }$. This potential is sectioned in the longitudinal direction by two repulsive Gaussian barriers having peak heights ${V}_{\mathrm{GS}}$ and ${V}_{\mathrm{GD}}$, respectively, to form the source, gate, and drain wells. The separation and width of the barriers determines the longitudinal gate well trap frequency. Furthermore, the longitudinal trap axis of the source well is assumed to be half harmonic for simplicity. The degree of overlap of the two barriers contributes to a potential bias in the gate, ${V}_{{\rm{G}},0}$, with respect to the source and drain. Finally, the drain well is modeled as a reflectionless output port that would feed into a subsequent circuit element.

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Previous works have modeled the flow of atoms across a potential barrier in order to study both the selective removal of atoms involved in evaporative cooling as well as the reverse process in trap loading [22, 23, 33]. Here, the expressions in [33] are modified to include the chemical potential due to the presence of a Bose–Einstein condensate (BEC) in each well, the analogue of an applied bias voltage. Thus, the atom currents in the triple-well system are defined by the set of equations:

$$\begin{eqnarray}{I}_{\mathrm{sg}} & = & {\gamma }_{{\rm{s}}}{N}_{\mathrm{th},{\rm{s}}}\mathrm{exp}[-({V}_{\mathrm{GS}}-{\mu }_{{\rm{s}}})/{{kT}}_{{\rm{s}}}],\\ {I}_{\mathrm{gs}} & = & {\gamma }_{{\rm{g}}}{N}_{\mathrm{th},{\rm{g}}}\mathrm{exp}[-({V}_{\mathrm{GS}}-{V}_{{\rm{G}},0}-{\mu }_{{\rm{g}}})/{{kT}}_{{\rm{g}}}],\\ {I}_{\mathrm{gd}} & = & {\gamma }_{{\rm{g}}}{N}_{\mathrm{th},{\rm{g}}}\mathrm{exp}[-({V}_{\mathrm{GD}}-{V}_{{\rm{G}},0}-{\mu }_{{\rm{g}}})/{{kT}}_{{\rm{g}}}],\end{eqnarray} \tag{ 1 }$$

where k is the Boltzmann constant, the product ${\gamma }_{i}{N}_{\mathrm{th},i}$ is the effective collision rate of thermally excited atoms in the ith well, and the exponential factor reflects the thermodynamic probability that an atom possesses sufficient energy to traverse the barrier. Note that the subscript order indicates the direction of current flow, e.g. ${I}_{\mathrm{sg}}$ describes the atom flux from the source to the gate well, and capitalized subscripts indicate model parameters.

We consider the range of temperatures below the critical temperature, T_c, but above the temperature associated with the interaction energy per particle, ${T}_{0}$. In this regime, ${T}_{{\rm{c}}}\gtrsim T\gt {T}_{0}$, the elastic collision rate responsible for the currents is dominated by collisions between thermally excited atoms [34]. The equilibrium collision rate ($\gamma =\sqrt{2/\pi }{n}_{\mathrm{th}}{\sigma }_{0}{\rm{\Delta }}{\rm{v}}$) then depends only on the peak density of the thermal component, ${n}_{\mathrm{th}}=\zeta (3/2)/{\lambda }_{\mathrm{th}}^{3}$, where ${\sigma }_{0}$ is the collision cross section, ${\rm{\Delta }}{\rm{v}}$ is the mean thermal velocity, $\zeta (z)$ is the Riemann zeta function, and ${\lambda }_{\mathrm{th}}$ is the thermal DeBroglie wavelength [34]. Furthermore, scattering at the energies considered is purely s-wave. Thus, the collision rate for the ith well is given by

$$\begin{eqnarray}&&{\gamma }_{i}=32{\pi }^{2}\zeta (3/2)m{({a}_{{\rm{s}}}{{kT}}_{i})}^{2}{/h}^{3},\end{eqnarray} \tag{ 2 }$$

where m is the atomic mass, a_s is the s-wave scattering length, and h is the Planck constant.

Steady-state circuit operation is analyzed by enforcing particle number and energy conservation, expressed using analogues of Kirchhoff's current and voltage laws:

$$\begin{eqnarray}&&{I}_{\mathrm{sg}}={I}_{\mathrm{gs}}+{I}_{\mathrm{gd}},\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{I}_{\mathrm{sg}}({V}_{\mathrm{GS}}+{\kappa }_{\mathrm{GS}}{{kT}}_{{\rm{s}}})={I}_{\mathrm{gs}}({V}_{\mathrm{GS}}+{\kappa }_{\mathrm{GS}}{{kT}}_{{\rm{g}}})+{I}_{\mathrm{gd}}({V}_{\mathrm{GD}}+{\kappa }_{\mathrm{GD}}{{kT}}_{{\rm{g}}}),\end{eqnarray} \tag{ 4 }$$

where the κ's, which indicate the average excess energy of atoms traversing the barriers, are of order unity [33]. It is supposed that some external reservoir supplies atoms to the source well, maintaining a fixed chemical potential, ${\mu }_{{\rm{s}}}$, and temperature, T_s. For barrier heights that are large compared to the chemical potential and thermal energies, the three wells are weakly coupled by thermal atoms that have sufficient energy to traverse the barriers. Thus, we assume that the gate acquires a well-defined chemical potential, ${\mu }_{{\rm{g}}}$, and temperature, T_g, once a steady-state is reached. Additionally, we impose that current into the drain is removed from the system such that no current flows from the drain back towards the gate. An illustration of the model system is shown in figure 2.

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With the source well ensemble held constant, we seek steady-state values of the gate chemical potential and temperature in terms of ${\mu }_{{\rm{s}}}$ and T_s. It proves useful to define a temperature drop τ and barrier height difference υ, both normalized to the source temperature:

$$\begin{eqnarray}&&\tau \equiv ({T}_{{\rm{s}}}-{T}_{{\rm{g}}})/{T}_{{\rm{s}}}\equiv {\rm{\Delta }}T/{T}_{{\rm{s}}},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\upsilon \equiv ({V}_{\mathrm{GD}}-{V}_{\mathrm{GS}})/{{kT}}_{{\rm{s}}}\equiv {\rm{\Delta }}V/{{kT}}_{{\rm{s}}}.\end{eqnarray} \tag{ 6 }$$

The value υ is referred to as the feedback parameter. Using the currents defined in equation (1), along with equations (3) and (4), it is possible to derive relations for both the temperature drop

$$\begin{eqnarray}&&\tau ={{\rm{e}}}^{-\upsilon /(1-\tau )}\displaystyle \frac{\upsilon +({\kappa }_{\mathrm{GD}}-{\kappa }_{\mathrm{GS}})}{{\kappa }_{\mathrm{GS}}+{\kappa }_{\mathrm{GD}}{{\rm{e}}}^{-\upsilon /(1-\tau )}}\end{eqnarray} \tag{ 7 }$$

and the normalized chemical potential drop

$$\begin{eqnarray}{\hat{\mu }}_{\mathrm{sg}} & = & (1-\tau )\mathrm{ln}\left[{(1-\tau )}^{4}\left(1+{{\rm{e}}}^{-\frac{\upsilon }{1-\tau }}\right)\right]-\tau ({\hat{V}}_{\mathrm{GS}}-{\hat{\mu }}_{{\rm{s}}})\\ & & +(1-\tau )\mathrm{ln}\left[\displaystyle \frac{1-\tau +\displaystyle \frac{\zeta (2)}{\zeta (3)}({\hat{\mu }}_{{\rm{s}}}-{\hat{V}}_{{\rm{G}},0}-{\hat{\mu }}_{\mathrm{sg}})}{1+\displaystyle \frac{\zeta (2)}{\zeta (3)}{\hat{\mu }}_{{\rm{s}}}}\right],\end{eqnarray} \tag{ 8 }$$

where ${\mu }_{\mathrm{sg}}\equiv {\mu }_{{\rm{s}}}-{\mu }_{{\rm{g}}}$ and the hat ($\hat{\;}$) indicates quantities normalized to the source temperature. Within these equations, ${{\rm{e}}}^{-\upsilon /(1-\tau )}$ represents the ratio of forward current into the drain relative to current fed back into the source, ${I}_{\mathrm{gd}}/{I}_{\mathrm{gs}}$. In deriving equation (8) we have used the fact that the thermal and chemical potential drops are zero at $\upsilon =\infty$ along with [34] to determine the ratio of ${\gamma }_{{\rm{g}}}{N}_{\mathrm{th},{\rm{g}}}{/\gamma }_{{\rm{s}}}{N}_{\mathrm{th},{\rm{s}}}$ that satisfies equation (3). In the absence of an external gate input, equation (8) shows that the source–gate junction is self-biased to some Q-point characterized by ${\mu }_{\mathrm{sg}}$. The magnitude of this bias is primarily dependent on the feedback parameter, both directly and through the temperature drop. This bias is illustrated in figure 1(a) within the atomic system and figure 1(b) for an equivalent, simplified electronic circuit. Equations (7) and (8) form the basis of our model and can be solved self-consistently in order to characterize the thermodynamic variables of the triple-well system in steady-state.

To calculate the values of τ and ${\hat{\mu }}_{\mathrm{sg}}$, the κ parameters of the trapping potential must be determined. Trap geometry has a strong effect on the average energy removed by an atom leaving via a controlled trajectory. For atoms escaping isotropically from a potential well, as in typical evaporative cooling schemes, $\kappa \simeq 1$ [22, 23]. However, limiting the escape trajectory to purely 1D, for instance along the loose axis of a cigar-shaped trap, $\kappa \simeq 2.9$ for truncation parameters $\eta \equiv V/kT$ = 4–7 [33]. The difference in κ factors is a direct result of allowed escape trajectories. Molecular dynamics simulations akin to those used in [33, 35] were performed to confirm $\kappa \simeq 2.9$ for the geometry and feedback parameters modeled here.

The temperature and chemical potential drops are shown as functions of the feedback parameter in figure 3, with ${\kappa }_{\mathrm{GS}}={\kappa }_{\mathrm{GD}}=2.9$. The behavior shown by the temperature drop is somewhat non-intuitive in that for values of positive feedback (${V}_{\mathrm{GD}}\gt {V}_{\mathrm{GS}}$ ), the temperature of the gate is actually lower than the source, despite current from the source into the gate having an average energy ${V}_{\mathrm{GS}}+{\kappa }_{\mathrm{GS}}{{kT}}_{{\rm{s}}}$. Furthermore, figure 3(b) shows that there will generally exist a range of feedback parameters for which the chemical potential drop is negative, meaning the source–gate junction is reverse-biased. The inset of figure 3(b) shows the gate–drain current normalized to the source–gate current versus the feedback parameter. The ratio of currents is independent of η, but the absolute magnitude of the I_gd depends strongly on the source ensemble and η given the exponential nature of the currents in equations (1).

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Also shown in figure 3(b) are two threshold feedback parameters. The first, ${\upsilon }_{\mathrm{BEC}}$, indicates the formation of a BEC in the gate well. For $\upsilon \geqslant {\upsilon }_{\mathrm{BEC}}$ the balance of particle and energy currents leads to a steady-state gate well ensemble with sufficiently high phase-space density to condense. The second threshold, ${\upsilon }_{\mathrm{TC}}$, indicates the feedback parameter above which negative transconductance occurs for the steady-state parameters. Section 3.2 provides an in-depth discussion of this threshold.

To better understand the steady-state results, one can consider the power dissipated within the gate well of the transistor, ${P}_{\mathrm{sg}}={I}_{\mathrm{sg},\mathrm{net}}^{2}{R}_{\mathrm{sg}}$. Here ${I}_{\mathrm{sg},\mathrm{net}}={I}_{\mathrm{sg}}-{I}_{\mathrm{gs}}\gt 0$ is the total atom current flowing into the gate. Note that the power dissipated within the gate well does not include the gate to drain current, as power output to the drain is available to do work on a connected load. The static source–gate resistance in steady-state is

$$\begin{eqnarray}&&{R}_{\mathrm{sg}}\equiv {\mu }_{\mathrm{sg}}/{I}_{\mathrm{sg},\mathrm{net}}.\end{eqnarray} \tag{ 9 }$$

Given the sign of the chemical potential drop, ${R}_{\mathrm{sg}}$ can be negative indicating an ohmic cooling synonymous with the evaporative cooling process, which leads to positive $\tau$.

Within the steady-state model presented, current gain is the most readily studied quantity since voltage and power gain require more explicit knowledge of the connected 'load.' To study gain, we use a method similar to determining the two-port admittance parameters of an electronic circuit, which is commonly used to describe transistor action in electronic transistors [36]. The coupled system of equations for the triple-well atomic system is given by

$$\begin{eqnarray}&&\left(\begin{array}{l}{\rm{d}}{I}_{{\rm{N}},{\rm{g}}}\\ {\rm{d}}{I}_{{\rm{Q}},{\rm{g}}}\end{array}\right)=\left(\begin{array}{rl}{a}_{11} & {a}_{12}\\ {a}_{21} & {a}_{22}\end{array}\right)\;\left(\begin{array}{l}{\rm{d}}{\mu }_{{\rm{g}}}\\ {\rm{d}}{T}_{{\rm{g}}}\end{array}\right),\end{eqnarray} \tag{ 10 }$$

where ${I}_{{\rm{N}},{\rm{g}}}$ and ${I}_{{\rm{Q}},{\rm{g}}}$ are the particle and heat currents into the gate well. The 'a' parameters relate the currents to either a change in chemical potential or temperature assuming the other is constant, e.g. ${a}_{11}={(\partial {I}_{{\rm{N}},{\rm{g}}}/\partial {\mu }_{{\rm{g}}})}_{{\rm{d}}{T}_{{\rm{g}}}=0}$. Equation (10) is akin to the Onsager relations [16, 37] for particle and heat diffusion that have been utilized to describe non-equilibrium transport dynamics in atomic systems [26, 27]. Here, they are used to describe the particle and heat currents applied to the gate well in order to elicit the changes ${\mu }_{{\rm{g}}}\to {\mu }_{{\rm{g}}}+{\rm{d}}{\mu }_{{\rm{g}}}$ and ${T}_{{\rm{g}}}\to {T}_{{\rm{g}}}+{\rm{d}}{T}_{{\rm{g}}}$. The particle and heat currents on the left-hand side of equation (10) are given by the differential of the sum of terms in equations (3) and (4), respectively, with current flow into (out of) the gate taken to be positive (negative). In steady-state, ${\rm{d}}{I}_{{\rm{Q}},{\rm{g}}}=0;$ thus, the temperature response to a change in chemical potential is given by $\partial {T}_{{\rm{g}}}/\partial {\mu }_{{\rm{g}}}={-a}_{21}/{a}_{22}$. It is not surprising that this quantity is negative, as the chemical potential varies inversely with temperature at constant atom number. For small deviations from the steady-state, this relation is used in conjunction with equation (10) to determine the input current required to change the gate chemical potential by some ${\rm{d}}{\mu }_{{\rm{g}}}$:

$$\begin{eqnarray}&&{\rm{d}}{I}_{{\rm{g}}}={\rm{d}}{\mu }_{{\rm{g}}}\left({a}_{11}-{a}_{12}\frac{{a}_{21}}{{a}_{22}}\right),\end{eqnarray} \tag{ 11 }$$

where the quantity in the parentheses is equivalent to the gate well input admittance. The differential current gain about the Q-point is then defined as the ratio of the resulting change in gate–drain current and the gate well input current:

$$\begin{eqnarray}&&{A}_{{\rm{I}}}\equiv \displaystyle \frac{{\rm{d}}{I}_{\mathrm{gd}}}{{\rm{d}}{I}_{{\rm{g}}}},\end{eqnarray} \tag{ 12 }$$

where ${\rm{d}}{I}_{\mathrm{gd}}={I}_{\mathrm{gd}}^{\prime }-{I}_{\mathrm{gd}}$ and the primed current is evaluated at ${\mu }_{{\rm{g}}}^{\prime }={\mu }_{{\rm{g}}}+{\rm{d}}{\mu }_{{\rm{g}}}$ and ${T}_{{\rm{g}}}^{\prime }={T}_{{\rm{g}}}+(\partial {T}_{{\rm{g}}}/\partial {\mu }_{{\rm{g}}}){\rm{d}}{\mu }_{{\rm{g}}}$.

Figure 4 shows the current gain for the range of feedback parameters shown in figure 3, illustrating the current gain at various Q-points. Two complementary operating regimes arise in which the transistor provides either positive or negative differential current gain. For feedback parameters near zero, A_I is positive, indicating that a positive (negative) ${\rm{d}}{I}_{{\rm{g}}}$ leads to an increase (decrease) in I_gd. For increasingly negative feedback parameters, i.e., $\upsilon \lt {\upsilon }_{\mathrm{BEC}}$, the current gain becomes nonphysical as ${\rm{d}}{\mu }_{{\rm{g}}}$ becomes ill-defined in the absence of a condensate in the gate well. At $\upsilon \sim 0.5$, the sign of A_I flips where the changes in I_gd due to ${\rm{d}}{\mu }_{{\rm{g}}}$ and ${\rm{d}}{T}_{{\rm{g}}}$ become equal. Finally, at $\upsilon \sim 1.6$, the gain reaches a maximum, negative amplitude. Positive and negative differential current gain regimes arise due to the interplay between ${\mu }_{{\rm{g}}}$ and T_g. To better understand each regime, consider the scaling of the chemical potential with respect to the number of condensed atoms in the Thomas–Fermi limit, $\mu \propto {N}_{{\rm{c}}}^{2/5}$. In the negative differential current gain regime, steady-state ${\mu }_{{\rm{g}}}$ is larger than in the positive gain regime. Therefore, a larger number of atoms at $T\lt {T}_{{\rm{g}}}$ must be injected into the gate well to elicit a positive ${\rm{d}}{\mu }_{{\rm{g}}}$. The resulting decrease in T_g causes a reduction of I_gd that is more substantial than the increase of I_gd due to the additional chemical potential. Thus, ${\rm{d}}{\mu }_{{\rm{g}}}$ is positive, but the net change in the gate–drain current is negative. The opposite effect is responsible for positive differential current gain.

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With the sign of the current gain understood, the magnitude of the output current is determined. Multiple traces are shown in figure 4 that indicate the gain for different gate well geometric trap frequencies, ${\bar{\omega }}_{{\rm{g}}}$, relative to the trap frequencies of the external well that injects the control current, ${\bar{\omega }}_{{\rm{c}}}$, where $\bar{\omega }={({\omega }_{x}{\omega }_{\perp }^{2})}^{1/3}$. The enhancement in the magnitude of the current gain arises due to the scaling of the specific heat, $C\propto 1/{\bar{\omega }}^{3}$ and chemical potential, $\mu \propto \bar{\omega }$, of a tight well [34]. More specifically, as $\bar{\omega }$ increases, the scaling in $\mu$ and C reduce the number of atoms needed to alter the chemical potential and temperature of the steady-state gate well population. In the case that the trap frequencies are equal, one recovers the expected result that ${A}_{{\rm{I}}}=1/2$ at $\upsilon =0$, as there are two equally probable output channels from the gate. Thus, the ratio of trap frequencies is a key design parameter for achieving greater than unity gain.

Despite a lack of explicit knowledge regarding the load circuit connected to the drain well, it is possible to determine the maximum power delivered to an impedance matched load. Stated generally, impedance matching is the process of selecting the input impedance of the load circuit to be equal to the output impedance of the device such that reflection of the output signal is minimized. The reflection of particles with a given energy impinging upon a potential landscape can be calculated using a number of methods (e.g., [38, 39]) and has been experimentally studied using cold [40] and ultracold atoms [41]. For a matched load, the electronics definition for the maximum power at the transistor output [42] is:

$$\begin{eqnarray}&&{P}_{\mathrm{max}}=\displaystyle \frac{{g}_{{\rm{m}}}{\rm{d}}{\mu }_{{\rm{g}}}^{2}}{4},\end{eqnarray} \tag{ 13 }$$

where ${\rm{d}}{\mu }_{{\rm{g}}}$ is the amplitude of the gate chemical potential modulation and g_m is the transconductance, given by

$$\begin{eqnarray}&&{g}_{{\rm{m}}}\equiv \frac{{\rm{d}}{I}_{\mathrm{gd}}}{{\rm{d}}{\mu }_{\mathrm{sg}}}=\left(\displaystyle \frac{\partial {T}_{{\rm{s}}}}{\partial {\mu }_{\mathrm{sg}}}\right)\;\left(\displaystyle \frac{\partial {I}_{\mathrm{gd}}}{\partial {T}_{{\rm{s}}}}\right),\end{eqnarray} \tag{ 14 }$$

where the partial derivatives are evaluated at constant ${\mu }_{{\rm{s}}}$. Transconductance is useful in describing operation of the devices as an active element, i.e., one that supplies power. From equation (13), it can be seen that if g_m is negative, the power dissipated at the transistor output is negative. Effectively, the transistor converts power supplied by the source well into power output from the drain controlled by ${\rm{d}}{\mu }_{{\rm{g}}}$. As alluded to in figure 3, there is a threshold feedback parameter, ${\upsilon }_{\mathrm{TC}}$, above which the system exhibits negative transconductance in steady-state. The threshold value is determined from the inflection point of g_m. The maximum power dissipated at the output of the transistor is shown in figure 5 as a function of the fraction above threshold, $(\upsilon -{\upsilon }_{\mathrm{TC}})/{\upsilon }_{\mathrm{TC}}$. The values of P_max are calculated using the same fractional value of ${\rm{d}}{\mu }_{{\rm{g}}}$ with respect to the steady-state ${\mu }_{{\rm{g}}}$ for all υ. Both the sign and magnitude of g_m are determined from the derivative of the transfer characteristic curves. Examples of these curves are shown in the inset of figure 5 for $\upsilon =10\%,\;50\%$, and 100% above ${\upsilon }_{\mathrm{TC}}$. The absolute magnitude of the power depends on the value of ${\rm{d}}{\mu }_{{\rm{g}}};$ therefore, to better illustrate the behavior of P_max it is scaled by ${P}_{\mathrm{max}}(\upsilon ={\upsilon }_{\mathrm{TC}})$ in figure 5. The magnitude of the maximum power dissipated is seen to peak at $\upsilon \approx 1.75\times {\upsilon }_{\mathrm{TC}}\approx 2.6$. As was the case for the current gain, the power delivered by the transistor is ultimately limited by the gate–drain current, which decreases exponentially with increasing υ. However, the power is delivered in the form of higher energy thermal atoms. As a result, the maximum power dissipation peaks at a higher value than the current gain.

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Negative power dissipation, a familiar concept introduced early in the literature on electronic oscillators, originates due to the negative transconductance or transresistance of a device [30]. When coupled to a circuit containing frequency dependent elements, an active device that exhibits negative transresistance cancels the resistive power loss in the load, resulting in the buildup of a resonant, oscillatory signal [29, 30]. The oscillator concept is attractive in the field of atomtronics, given the historical impact of RF and higher frequency signal generation within the field of analog electronics.