Atom transistor from the point of view of quantum nonequilibrium dynamics

We analyze the atom field-effect transistor scheme [J. A. Stickney, D. Z. Anderson and A. A. Zozulya, Phys. Rev. A 75, 013608 (2007)] using the standard tools of nonequlilibrium dynamics. In particular, we study the deviations from the Eigenstate Thermalization Hypothesis, quantum fluctuations, and the density of states, both ab initio and using their mean-field analogues. Having fully established the quantum vs. mean-field correspondence for this system, we attempt, using a mean-field model, to interpret the off-on threshold in our transistor as the onset of ergodicity---a point where the system becomes able to visit the thermal values of the former integrals of motion in principle, albeit not being fully thermalized yet.


I. INTRODUCTION
Similarly to the conventional electronic transistors, in atomtronics, a transistor is a nonlinear device where a small atom current or atom number controls a large current. The existing proposals for an atom transistor can be divided into open and closed architectures. The former [1][2][3] is conceptually close to the conventional electronic devices (the work [2] also contains a design for a diode). The latter, in turn, can be separated into the schemes based on an adiabatic population transfer protocol [4][5][6] (along with the similar atom diode proposals [7,8]), and the schemes where the (large) base current is induced by the difference in chemical potentials between the "electrodes". The base current can be controlled by either the internal state of a localized impurity [9][10][11] or by the number of (strongly interacting) atoms in the middle site [12,13]. This latter scheme is nothing else but an atomic version of the field-effect transistor (FET). Here, the large atomic current from the "source" to the "drain" is controlled by a a small number of atoms in the "gate." The interatomic repulsion in the gate is so strong that the chemical potential is large even when the number of atoms in the gate is small.
Here we show that the off-on threshold in our transistor corresponds to the onset of ergodicity-the point where the system becomes able to visit the thermal values of the former integrals of motion in principle, albeit not being fully thermalized yet.

II. SYSTEM OF INTEREST
The paradigmatic atom transistor is realized in a tight binding model for three coupled bosonic wells on a line, the "source", the "gate", and the "drain" [12]. The Hamiltonian of the system readŝ where the indices "s", "g", and "d" stand for source, gate and drain, respectively, ε η and U η are the site-dependent one-body energy and the strength of the on-site two-body interactions for a site η = s, g, d, and J ηη is the hopping constant between sites η and η . In general, the purpose of the atom FET transistor is to control the current of atoms from the source to the drain using small variations in the chemical potential of the gate, µ g = ε g − U g n g , controlled in turn by its population, n g . According to the original proposal [12], the transition of atoms between the source and the gate plays the role of a "bottleneck," activated only when the source and gate chemical potentials become close; in contrast, the gate-drain link is made insensitive to the gate-drain chemical potential difference, transmitting every atom that happens to appear at the gate. This can be achieved by providing a comparatively large on-site interaction strength for the gate, subsequently detuned from the energy of the source. One then chooses the hopping constants and the on-site energies in such a way that the source-gate transition exhibits a narrow resonance, and the gate-drain transition a broad one. We will be using the following representative set of parameters: J sg = −0.1, U g N = 100, ε g = −1.3, J gd = −1.0, ε d = 0.5 and N = 30, with the rest of the parameters set to zero: U s N = ε s = U d N = 0. Figure 1a shows the energy spectrum E α of the transistor for the chosen set of parameters. Here and below, α is the index of the eigenstate |α of the Hamiltonian in Eq. (1):Ĥ|α = E α |α . The spectrum, bounded from both below and above, contains 496 eigenstates. Note that the density of states decreases with energy. On the upper end of the spectrum, the dominant contribution to the energy is provided by the interactions between the atoms in the gate, n g ; this energy increases quadratically with n g , leading to the decreasing (with energy) energy spacings.
Most of the spectrum turns out to be within the limits of applicability of the semiclassical approximation. To verify this assertion, we check that Weyl's law holds in our system. According to Weyl's law, if one computes how the number of quantum states below an energy E arXiv:1506.02467v1 [cond-mat.quant-gas] 8 Jun 2015 The ab initio energy spectrum (496 eigenstates) for an atom transistor with N = 30 atoms. (b) The meanfield emulation of the quantum energy spectrum. 496 initial conditions were uniformly distributed through the whole available phase space. The vertical axis gives the temporal average of total energy, for each realization. The realizations were permuted in such a way that the energies increase along the list. The vertical axis reflects the position of the realizations in this list. depends on the energy E, and then takes the smooth envelope of this dependence, the result is proportional to the classical phase space volume occupied by these states: is the corresponding mean-field Hamiltonian, expressed through the action-angle variables of the zero-hopping analogue of the Hamiltonian in Eq. (1), , and Θ(x) is the Heaviside stepfunction. Figure 1b shows the inverse of the dependence (2), computed using a Monte-Carlo method: 496 phase-space points were distributed uniformly within an intersection of the whole available volume of the phase space and a narrow shell corresponding a constant window of the values of the norm (where norm ≡ ), around norm = N . Notice that for this calculation, we have chosen as many Monte-Carlo realization as there are quantum eigenstates of the system. A priori, one number is not related to the other, and more realizations would produce a smoother semi-clalssical curve. However, it can be argued that Fig. 1b can serve as a semicalssical emulation of the quantum spectrum Fig. 1a, where the fluctuations of the eigenenergies around a smooth envelope have at least the same order of magnitude as their semiclassical counterparts. Indeed, on the one hand, the Monte-Carlo method produces energy values through a Poisson process; on the other hand, it is known that in quantum integrable systems, eigenenergies are distributed as if they are the outcome of a Poisson process [14,15]. In contrast, quantum-ergodic systems with time-reversal invariance exhibit the type-1 Wigner-Dyson statistics [14,15]. In that case, the standard deviation for the spacing between two neighboring levels is 1/ (4/π) − 1 ≈ 2 times lower than in the case of Poisson statistics.  [16][17][18][19]-the expectation values of the relevant observables; in our case, this is the relative occupation of the drain: α|n d |α . Notice that this expectation value, when plotted as a function of α, does not collapse to a single line; this means our system has not reached eigenstate thermalization. Let Figure 2b shows the classical infinite time averages of the drain occupation, where the pseudo-eigenstates of Fig. 1b are used as the initial conditions. The overall behavior of the quantum and classical expectation values stand in good correspondence (see comment 24 in Ref. [20]).
Quantum fluctuations of n d in the eigenstates |α , are shown in Fig. 3a. In accordance with the conjecture The system parameters are the same as Fig. 1(b) . expressed in Ref. [21], the classical temporal variance, shown in Fig. 3b, constitutes a faithful classical analogue of the quantum fluctuations. Figure 4 shows the detuning of the source-gate transition as a function of the relative gate occupation. The resonance occurs at n g = 0.013. Recall that in our case, µ s = ε s and µ g = ε g − U g n g .
In all numerical experiments below, the initial value of the drain occupation n d is zero or very close to zero.
At zero n g initially, only a minimal conductance is expected, and this is what we observe (Fig. 5a). To the contrary, at the resonant point, n g (t = 0) = 0.013, the transistor is supposed to provide the maximal source-drain conductance, and this is indeed what we find (Fig. 5b).  Fig. 2(a). (b) Temporal variance for the initial states generated, in the mean-field approximation. The solid horizontal line reflects the average over all realizations. The parameters are the same as in Fig. 2(b) The conventional figure of merit is the current to the drain, characterized by the initial slope in the dependence of n g vs. time (in our case, the slope of the red line between t = 0 and t ≈ 20, in Fig. 5b). Below, we will use an integral figure of merit-the infinite time average of  Fig. 2(b). the drain occupation, n d t -to facilitate the analysis of the system from the point of view of ergodicity or its absence.
In Fig. 6, we trace the n d t vs. n g (t = 0) dependence, along with the temporal variation of n d . The slope of this dependence, β ≈ 16, we identify as gain. Unexpectedly, we find that even at the resonant point, the average drain occupation does not reach its ensemble average (also represented by the short horizontal line in Fig. 2). Instead, as the system approaches the resonant point n g (t = 0) = 0.013, the ensemble average starts falling within the range of the temporal fluctuations; not enough to be predominantly within the range, but enough for this effect to be detectable.

IV. CONCLUSION
In this work, we studied the atom FET transistor scheme suggested in Ref. [12,13] using the tools of quantum non-equilibrium dynamics [20]. We first justified the applicability of the semiclassical approximation applied to the standard measures of the non-equilibrium. We then focused, using a semiclalssical model (equivalent to a mean-field one), on the initial conditions with zero drain occupation, and used the gate occupation as the knob that controls further dynamics. Instead of the , along with its temporal standard deviation (error bars), in the meanfield approximation, as a function of the initial occupation of the gate. 1015 initial conditions were uniformly distributed inside a phase space volume corresponding to the lowest (in energy) 1/20 of the total available phase space volume. The corresponding window of energies is between E min /N ≈ 0.08 and E max /N ≈ 0.13, out of the full available energy range of E min,full /N = E min /N ≈ 0.08 and E max,full = 47. Subsequently, only the initial conditions corresponding to low initial conditions (N d (t = 0) ≤ 0.05N ) of drain occupation were selected. A linear fit (dashed line) gives an estimate of the gain, β ≡ N d t /N g (t = 0), as β ≈ 16. Solid horizontal line reflects the average over all the realizations in the window, with no selection of the initial values of the drain occupation N d . Note that only when the initial gate occupation approaches the resonant value of n g = .013 (see Fig. 4), the mobility in the phase space becomes sufficient for the drain occupation to reach its thermal value (solid line), at least at some instances of time. The system parameters are the same as in Fig.2(b) traditional source-to-drain current, as the transistor output we choose the infinite time average of the drain occupation: this change brings us closer to the conventional measures of non-equilibrium. We then studied how the drain occupation depends on the initial gate occupation. At zero gate occupation (transistor "off"), no atoms are transmitted to the source. To the contrary, at a point where the gate's chemical potential levels with that of the source (transistor "on"), the drain becomes populated. One may expect that at this point, the transistor is fully thermalized, i.e. that the infinite time averages of the observables approach their (microcanonical) ensemble averages. However, it turned out that even at the resonant point, the drain occupation is still far below its thermal value. Instead, we found that the "on" regime is characterized by the accessibly of the thermal values of the drain in "principle". Namely, around the "on" value of the initial gate occupation, the temporal fluctuations of the drain occupation become capable of "touching" the thermal values for some periods of time, without spending much time there. The overall conclusion is that the ergodicity is not a necessary condition for the operation of atom transistor (while the conventional semiconductor devices do operate close to a thermal equilibrium): this conclusion may allow one to broaden the search for an optimal configuration of an atom transistor.