Numerical analysis of coherent many-body currents in a single atom transistor

We study the dynamics of many atoms in the recently proposed single-atom-transistor setup [A. Micheli, A. J. Daley, D. Jaksch, and P. Zoller, Phys. Rev. Lett. 93, 140408 (2004)] using recently developed numerical methods. In this setup, a localized spin-$1∕2$ impurity is used to switch the transport of atoms in a one-dimensional optical lattice: in one state the impurity is transparent to probe atoms, but in the other acts as a single-atom mirror. We calculate time-dependent currents for bosons passing the impurity atom, and find interesting many-body effects. These include substantially different transport properties for bosons in the strongly interacting (Tonks) regime when compared with fermions, and an unexpected decrease in the current when weakly interacting probe atoms are initially accelerated to a nonzero mean momentum. We also provide more insight into the application of our numerical methods to this system, and discuss open questions about the currents approached by the system on long time scales.

Figure 1

(Color online) A single-atom-transistor in a 1D optical lattice. A single spin-$1∕2$ impurity atom $q$ separately trapped at a particular lattice site is transparent to probe atoms $b$ in one state (“on”), but in the other acts as a single atom mirror (“off”). The probe atoms can either diffuse past the impurity site with mean initial momentum ${⟨\widehat{k}⟩}_{t=0}=0$ or can be accelerated to a finite initial momentum ${⟨\widehat{k}⟩}_{t=0}\ne 0$ by a kick of strength $p_k$.

Figure 2

(Color online) An optical Feshbach resonance for a single spin channel ($\Omega ={\Omega }_{\sigma }$, $\Delta ={\Delta }_{\sigma }$). One probe atom and the impurity atom, in an atomic state ${\widehat{b}}_0^{†}{\widehat{q}}_{\sigma }^{†}\mid \mathrm{vac}⟩$ which is quantized by the trapping potential of the lattice site, are coupled by an optical Feshbach setup to a bound molecular state $m_{\sigma }^{†}\mid \mathrm{vac}⟩$ of the Born-Oppenheimer potential $V\left(r\right)$ (note that here the Born-Oppenheimer potential is modified by the trapping potential of the lattice site [17]). The coupling has the effective two-photon Rabi frequency $\Omega$ and detuning $\Delta$.

Figure 3

(Color online) The sequence as (left) a probe atom approaches the impurity site and is located on site $i=-1$, (center) the probe atom is on the impurity site $i=0$, and (right) the probe atom has tunneled past the impurity and is located on site $i=1$. Quantum interference occurs in this process because the two dressed states on the impurity site, $\mid \pm ⟩=({\widehat{b}}_0^{†}{\widehat{q}}_{\sigma }^{†}\mid \mathrm{vac}⟩\pm m_{\sigma }^{†}\mid \mathrm{vac}⟩)∕\sqrt{2}$, gives rise to two separate paths with equal and opposite amplitudes.

Figure 4

(Color online) The number of atoms to the right of the impurity site $N_R$ as a function of dimensionless time $tJ$ for a Bose gas in the Tonks limit $(U_{bb}∕J\to \infty )$ with $\Omega ∕J=1$, $n=N∕M=1$, and varying number of states retained in the method, $\chi =10,20,30,40,50,60,70$ (lines from bottom to top). These results are from the original simulation method.

Figure 5

(Color online) The number of atoms to the right of the impurity site $N_R$ as a function of dimensionless time $tJ$ for a Bose gas in the Tonks limit $(U_{bb}∕J\to \infty )$ with $\Omega ∕J=1$, $n=N∕M=1$. This plot shows a comparison of results from the original method (dashed lines, $\chi =50,70$; cf. Fig. 4), and from the number conserving method (solid lines, $\chi =50,100,200,300$).

Figure 6

(Color online) Comparison as a function of dimensionless time $tJ$ of the number of atoms to the right of the impurity site $N_R$ (solid line), and the sum of squares of the discarded Schmidt eigenvalues ${\epsilon }_{\lambda }={\sum }_{\beta >\chi }{\lambda }_{\beta }^2$ (dashed line). These results are taken from the number conserving simulation method with $\chi =100$, for a Bose gas in the Tonks limit $(U_{bb}∕J\to \infty )$ with $\Omega ∕J=1$, $n=N∕M=1$.

Figure 7

(Color online) Steady-state currents through the SAT $I_{\mathrm{SS}}$ as a function of the coupling between probe atoms and the impurity $\Omega ∕J$. These plots show the comparison of a Bose gas with different interaction strengths $U∕J=4$ (dotted line), $U∕J=10$ (dashed), and $U∕J\to \infty$ (solid), and a Fermi gas (dash-dotted), with $n=\left(\mathrm{a}\right)$ $1∕2$ and (b) 1. In both cases, $\Delta =U_{qb}=U_{bm}=0$.

Figure 8

(Color online) Steady-state currents through the SAT $I_{\mathrm{SS}}$ as a function of the interaction strength $U∕J$ for bosonic probe atoms initially at half filling, $n=1∕2$ (squares), and at unit filling, $n=1$ (diamonds), with $\Omega ∕J=1$. The equivalent results for the Tonks gas $(U∕J\to \infty )$ and fermions are marked on the right-hand site of the plot. $\Delta =U_{qb}=U_{bm}=0$.

Figure 9

(Color online) Plot showing the average steady-state occupation of the molecular state (dashed lines) and the average steady-state atomic occupation of the impurity site (solid lines) for $U∕J=\left(\mathrm{a}\right)$ 4 and (b) 10, as a function of (a1), (b1) $\Omega ∕J$ with $n=1$ and (a2), (b2) $n$ with $\Omega ∕J=0.5$. In all cases, $\Delta =U_{qb}=U_{bm}=0$, and calculations were performed for $M=30$.

Figure 10

(Color online) Steady-state currents through the SAT $I_{\mathrm{SS}}$ as a function of the initial density of atoms $n=N∕M$. These plots show the comparison of a Bose gas with different interaction strengths $U∕J=4$ (dotted line), $U∕J=10$ (dashed), and $U∕J\to \infty$ (solid), and a Fermi gas (dash-dotted), with $\Omega ∕J=\left(\mathrm{a}\right)$ 0.5 and (b) 1. In both cases, $\Delta =U_{qb}=U_{bm}=0$.

Figure 11

(Color online) Steady-state currents $I_{\mathrm{SS}}$ through the SAT for fermions as a function of the kick parameter $p_k$, for (a) $\Omega =0$ and (b) $\Omega ∕J=1$. In each plot the lines from bottom to top sequentially correspond to filling of $N=3,6,9,12,15,20,25,30$ particles initially on $M=30$ lattice sites. Note that the scales are different for (a) and (b), and also that the results in (a) are exactly the same as those for a Tonks gas of bosons. In both cases, $\Delta =U_{qb}=U_{bm}=0$.

Figure 12

(Color online) Steady-state currents with coupling to the SAT, $\Omega =0$, $I_{\mathrm{SS}}$ as a function of the kick parameter $p_k$ for varying interaction strengths, $U∕J=4$ (dotted), 7 (solid), and 10 (dashed), for $N=\left(\mathrm{a}\right)$ 5, (b) 15, and (c) 30 particles initially situated on $M=30$ lattice sites. In all cases, $\Delta =U_{qb}=U_{bm}=0$.

Figure 13

(Color online) Steady-state currents through the SAT $I_{\mathrm{SS}}$ as a function of the kick parameter $p_k$, for $\Omega ∕J=1$. These plots show the comparison of a Bose gas with different interaction strengths $U∕J=4$ (dotted line), $U∕J=10$ (dashed), and $U∕J\to \infty$ (solid), and a Fermi gas (dash-dotted), with $n=\left(\mathrm{a}\right)$ $1∕2$ and (b) 1. In both cases, $\Delta =U_{qb}=U_{bm}=0$.

Figure 14

(Color online) A comparison of steady-state currents through the SAT, $I_{\mathrm{SS}}$, as a function of $\Omega ∕J$ for $p_k=0$ (solid line), $\pi ∕4$ (dashed), and $\pi ∕2$ (dotted). Here we consider a Tonks gas $(U∕J\to \infty )$ of bosonic probe atoms that is initially at half half filling, $N=15$, $M=30$. $\Delta =U_{qb}=U_{bm}=0$.