Coherent transport of single atoms in optical lattices

We describe a technique for transferring a two-level atom between two adjacent potential wells of an optical lattice, using pairs of pump and Stokes pulses, each resonantly coupling the same pair of internal atomic states to form a Raman transition. Starting from a vibrational eigenstate of one well the atom slowly moves under the action of the pulse pair, to the vibrational eigenstate with the same quantum number in the neighboring well. The transfer takes place in two stages: A conventional stimulated Raman adiabatic passage (STIRAP) technique, in which Stokes precedes pump pulse, is followed by a pulse sequence where pump precedes Stokes and with an inverted sign of the Stokes envelope. In the first step the atom is accelerated toward the adjacent well and in the second step decelerated to the initial vibrational energy. The STIRAP technique avoids the introduction of stochastic motion caused by spontaneous emission from the excited internal atomic state.

Figure 1

Coherent transport of an atom in an optical lattice due to a Raman pulse. (a) Sketch of the generic setup. The optical lattice of period $a$, Eq. (1), is created by two laser beams 1 and 2 propagating and crossing at angle $2\alpha$. Application of the pump and Stokes pulses produces a force which moves the atom from the left lattice site to the right one. (b) Quantization of the atomic motion in the optical lattice. (1) The state selective optical potential, Eq. (2). The potential for the atom in the excited state is shifted by $a∕2$ with respect to the potential for the atom in the ground state. The driving lasers with transverse profile $g\left(x\right)$ illuminate predominantly one period $\left(a\right)$ of the optical lattice. In bold are shown three potential wells involved in the transfer, two in the atomic ground state $\mid g⟩$ and one in the excited state $\mid e⟩$. For better selectivity the transverse beam shape $g\left(x\right)$ should be chosen with flat top and steep wings like a super-Gaussian shape. (2) Quantization of the atomic center of mass motion within each lattice site. Dashed vertical lines indicate that each well is treated separately. (3) Approximation of each ${\sin }^2$-shape well by a truncated parabola. Corresponding harmonic oscillator energy levels are shown by horizontal equidistant lines. The Raman pump-Stokes pair of pulses realizes the transfer. (c) The Franck-Condon factors $d_{\overline{m}n}^2={⟨G,\overline{m}\mid A,n⟩}^2$ calculated as the transition matrix elements from the chosen vibrational state $\overline{m}=43$ in the left manifold to a number of vibrational states in the excited manifold; see Eq. (38). The factors are evaluated in the harmonic approximation where the harmonic potential matches the ${\sin }^2$ potential in the vicinity of the central eigenstate $\mid G,\overline{m}⟩$.

Figure 2

Atom transport as viewed in the language of a degenerate $\Lambda$ system. (a) Level structure of Fig. 1b showing a few relevant neighboring states around central states $\mid G,\overline{m}⟩$, $\mid A,\overline{n}⟩$, and $\mid G^{\prime },\overline{m}⟩$ denoted by the corresponding probability amplitudes $g_0$, $a_0$, and $g_0^{\prime }$. (b) Dressed state picture of the $\Lambda$ system composed of central states $g_0$, $a_0$, and $g_0^{\prime }$. In bare state basis fields ${\Omega }_P$ and ${\Omega }_S$ have the same frequency and are therefore indistinguishable. In the dressed state picture the indistinguishability appears as the single bright field ${\Omega }_B$ coupling $a_0$ and $B$, while the dark state $D$ stays fully decoupled. (c) Evolution of populations ${\mid g_0\mid }^2$, ${\mid a_0\mid }^2$, and ${\mid g_0^{\prime }\mid }^2$ of the three central states under the action of a Gaussian pulse $\Omega \left(t\right)={\Omega }_0\exp [-t^2∕2\Delta t^2]$ using the relation $\Delta t={\left(R\nu \right)}^{-1}$ between the temporal width $\Delta t$ and the nonmonochromaticity parameter $R$. Parameters are nonmonochromaticity parameter $R=0.1$ and peak Rabi frequency ${\Omega }_0=0.28$. From here on time and Rabi frequency are given in units of the vibrational period $T\equiv 2\pi ∕\nu$ and $1∕T$, respectively. Note that the complete population transfer to the target state $g_0^{\prime }$ is accompanied by a substantial transit of population through the excited state $a_0$. (d) Final (after the passage of the Raman pulse) population ${\mid g_0^{\prime }\mid }^2$ of the target state as a function of peak Rabi frequency for two values of $R$. Well pronounced Rabi oscillations are obtained for $R=0.02$ (dashed line) over a wide range of Rabi frequencies, whereas for larger pulse bandwidth ($R=0.1$, solid line) the oscillations degrade fast.

Figure 3

STIRAP in a nondegenerate $\Lambda$ system. (a) Bare states are coupled by the resonant pump and Stokes pulses of different carrier frequencies. (b) Dressed state picture of the $\Lambda$ system. (c) Example of the STIRAP—the population transfer by the pump and Stokes pulses arranged in the counterintuitive order (Stokes precedes pump) as shown in the inset. Note the greatly suppressed population transit through the upper state as compared to the degenerate case in Fig. 2c.

Figure 4

The nine-level system for the population transfer between the three manifolds. (a) Bare state picture of the nine-level system with couplings produced by the pump (solid line) and Stokes (dashed line) pulse. Thick (thin) lines indicate strong (weak) transitions. Also shown are surrounding vibrational levels which ideally (for $R\to 0$) do not participate in the dynamics. The black circle represents the population initially concentrated in the $g_0$ state. (b) The nine-level system in the rotated basis, see Eq. (55), for odd offset $\Delta n$. The initial condition of ${\mid g_0\mid }^2=1$ corresponds to the population equally distributed between states $X_0$ and $Y_0$. (c) Same as in (b) for even $\Delta n$. (d) The pulse sequence for the two-stage STIRAP: The conventional STIRAP sequence (Stokes precedes pump) is followed by the inverted STIRAP (pump precedes Stokes while the Stokes features an inverted sign). (e) Evolution of the state vector $\mid \Psi ⟩$ in the Hilbert space spanned by basis vectors $\mid X_0⟩$, $\mid a_1⟩$, $\mid a_0⟩$, $\mid a_2⟩$, and $\mid Y_2⟩$. Under ideal STIRAP conditions the evolution of $\mid \Psi ⟩$ takes place mainly in the plane spanned by vectors $\mid X_0⟩$ and $\mid Y_2⟩$ with vanishing admixture of excited states. Note that the Hilbert space spanned by the five vectors is only a subspace of the original nine-dimensional Hilbert space. Since the projection on $\mid Y_0⟩$ state (not shown) is constant in time and equal to $1∕\sqrt{2}$, the length of the evolving reduced state vector in the five-dimensional space is $1∕\sqrt{2}$. In the first stage the state vector is rotated from $X_0(-\infty )=1∕\sqrt{2}$ toward $\mid Y_2⟩$. In the second stage the rotation proceeds farther toward $X_0(+\infty )=-1∕\sqrt{2}$.

Figure 5

Dynamics of populations of relevant vibrational states in the optical lattice. Parameters are ${\Omega }_{P0}={\Omega }_{S0}=0.07$, $T_P=16$, $T_S=6.4$, and $\Delta t=11$. Franck-Condon factors are chosen as explained in the text. (a) Nine-level system of Fig. 4a, $R\to 0$. (b) Same as (a) taking into account 30 surrounding vibrational states in each well of the optical lattice, $R=0.01$. Amplitude $a_2$ is not shown; no population is transiting via $a_2$ due to absence of coupling; see Fig. 4b.