Single Atom Transistor in a 1D Optical Lattice

We propose a scheme utilizing a quantum interference phenomenon to switch the transport of atoms in a 1D optical lattice through a site containing an impurity atom. The impurity represents a qubit which in one spin state is transparent to the probe atoms, but in the other acts as a single atom mirror. This allows a single-shot quantum nondemolition measurement of the qubit spin.

Figure 1

(a) A spin $1/2$ impurity used as a switch: in one spin state it is transparent to the probe atoms, but in the other it acts as a single atom mirror. (b) Implementation of the SAT as a separately trapped impurity $q$ with probe atoms $b$ in an optical lattice.

Figure 2

(a) The optical Feshbach setup couples the atomic state ${\widehat{b}}_0^{†}{\widehat{q}}_{\sigma }^{†}|\mathrm{v}\mathrm{a}\mathrm{c}\rangle$ (in a particular motional state quantized by the trap) to a molecular bound state of the Born-Oppenheimer potential, ${\widehat{m}}_{\sigma }^{†}|\mathrm{v}\mathrm{a}\mathrm{c}\rangle$, with effective Rabi frequency ${\Omega }_{\sigma }$ and detuning ${\Delta }_{\sigma }$. (b) A single atom passes the impurity ($\mathrm{I}\to \mathrm{I}\mathrm{I}\mathrm{I}$) via the two dressed states (II), $|+\rangle ={\widehat{b}}_0^{†}{\widehat{q}}_{\sigma }^{†}|\mathrm{v}\mathrm{a}\mathrm{c}\rangle +{\widehat{m}}_{\sigma }^{†}|\mathrm{v}\mathrm{a}\mathrm{c}\rangle$ and $|-\rangle ={\widehat{b}}_0^{†}{\widehat{q}}_{\sigma }^{†}|\mathrm{v}\mathrm{a}\mathrm{c}\rangle -{\widehat{m}}_{\sigma }^{†}|\mathrm{v}\mathrm{a}\mathrm{c}\rangle$ and quantum interference gives rise to an effective tunneling rate $J_{\mathrm{e}\mathrm{f}\mathrm{f},\sigma }$.

Figure 3

(a) SAT transmission coefficients $T\equiv |f^{(+)}{|}^2$ for a particle $b$ as a function of its energy $\epsilon (k)$ for $\Omega /J=4,\Delta =0,U_{qb}/J=0$ (solid line), $\Omega /J=8,\Delta /J=4,U_{qb}/J=2$ (dashed line), $\Omega /J=1,\Delta =0,U_{qb}/J=2$ (dotted line), and $\Omega /J=1,\Delta =0,U_{qb}/J=0$ (dash-dotted line). (b) The number of particles to the right of the impurity, $N_R(t)$, from exact numerical calculations for bosons in the limit $U_{bb}/J\to \infty$ (dashed lines) and Fermions (solid lines) in a 1D Mott insulator state with $n=1$, for $\Delta =0$, $\Omega /J=0,1,2$.

Figure 4

(a) The steady state current of $b$ atoms through the impurity as a function of the initial filling factor $n$ for $\Delta =0$ and $\Omega /J=0,1,2$. The solid lines show the analytic result $I_0$ for fermions, whereas the dashed lines show the exact numerical result for hard-core bosons with $U_{bb}/J\to \infty$. For $\Omega =0$ these results are indistinguishable. (b),(c) The steady state current as a function of the Rabi frequency $\Omega /J$ on resonance $\Delta =0$ for (b) unit filling and (c) half filling. The solid lines show the analytic result for fermions, whereas the dashed (dotted) lines give numerical results for bosons with $U_{bb}/J=20$ ($U_{bb}/J=4$).

Figure 5

Exact numerical results showing the propagation of $N=30$ hard-core bosons in an initial MI state ($n=1$) through a SAT with (a)–(c) $\Omega =0$ and (d)–(f) $\Omega =0.5J$, with $\Delta =U_{qb}=U_{bm}=0$. The plot shows (a),(d)  the momentum distribution, (b),(e) the five largest eigenvalues ${\lambda }_m$ of the single particle density matrix $\langle {\widehat{b}}_i^{†}{\widehat{b}}_j\rangle$, and (c),(f) the spatial density of the largest eigenmode (the quasicondensate) as a function of time. Darker colors represent higher values.