Atom-optics approach to studying transport phenomena

We present a simple experimental scheme, based on standard atom-optics techniques, to design highly versatile model systems for the study of single-particle quantum transport phenomena. The scheme is based on a discrete set of free-particle momentum states that are coupled via momentum-changing two-photon Bragg transitions, driven by pairs of interfering laser beams. In the effective lattice models that are accessible, this scheme allows for single-site detection, as well as site-resolved and dynamical control over all system parameters. We discuss two possible implementations, based on state-preserving Bragg transitions and on state-changing Raman transitions, which, respectively, allow for the study of nearly arbitrary single-particle Abelian U(1) and non-Abelian U(2) lattice models.

Figure 1

Experimental scheme for studying lattice-driven momentum-space dynamics. (a) Atomic matter waves are driven by a pair of counterpropagating laser fields, one of which is composed of several different frequency components, with controllable phase, frequency, and amplitude. (b) Energy-momentum dispersion. All laser fields are far-detuned by an amount $\mathrm{\Delta }$ from atomic resonance between the ground $|g\rangle$ and excited $|e\rangle$ states. Stimulated two-photon Bragg transitions are driven by the pairs of interfering laser fields, coherently coupling plane-wave momentum states separated by two-photon momenta $\left(2\hslash k\right)$. The quadratic free-particle dispersion defines a unique two-photon Bragg resonance condition $\hslash {\tilde{\omega }}_n=(2n+1)4E_R$ for each link between neighboring states. Each frequency component of the multifrequency field addresses a unique state-to-state link.

Figure 2

A two-dimensional lattice system with spectrally resolved link resonances. (a) Two pairs of interfering laser fields (set 1 shown in red, set 2 shown in blue), intersecting in a plane at an angle $\theta$, are shone onto a collection of atomic matter waves. Cross interferences between the two pairs of beams can be avoided by choice of laser polarizations or by introducing a frequency offset between the two pairs. (b) The discrete “lattice” of momentum states that can be populated by stimulated two-photon Bragg transitions, starting from zero momentum. (c) The spectral positions of the nearest-neighbor Bragg resonances ${\tilde{\omega }}_{m,n}^x$ driven by the laser pair 1, relating to the finite-sized set of states labeled $(m,n)$ with momenta ${\mathbf{p}}_{m,n}=2\hslash (m{\mathbf{k}}_1+n{\mathbf{k}}_2)$ as shown in (b). (d) Same as in (c), but for the Bragg transition resonances ${\tilde{\omega }}_{m,n}^y$ addressed by the second pair of lasers.

Figure 3

Simulated momentum-space dynamics of a small (6 site $\times$ 6 site) two-dimensional (2D) lattice system. (a)–(d) Probability distributions of the different momentum modes $|m,n\rangle$, following dynamics initiated from a state $|{\psi }_{\mathrm{in}}\rangle$, at times of 0, 1.875, 3.75, and 7.5 in units $\hslash /t$, with $t$ the tunneling energy. (a) Shown for a regular lattice with homogeneous tunneling energies $t$ and no tunneling phases, starting from $|{\psi }_{\mathrm{in}}\rangle =|0,0\rangle$. The dynamics shown relate to evolution governed by Eq. (11) from the text. (b) Same, but for an enclosed synthetic magnetic flux of $2\pi /3$ per lattice plaquette, set through a nontrivial tunneling phase along one direction, ${\phi }_{m,n}^y=2m\pi /3$. In this case, the particles avoid entering the bulk or interior of the system, and instead propagate along the system boundaries. Because state preparation is based on mode projection, with no explicit energy dependence, a combination of clockwise and counterclockwise propagating edge states are populated. (c) As in (b), but starting from the state $|{\psi }_{\mathrm{in}}\rangle =\left(\right|0,0\rangle +i|0,1\rangle )/\sqrt{2}$. The populated state propagates with essentially only one chirality. (d) Exactly as in (c), but including all tunneling contributions due to the entire sideband spectrum [i.e., with dynamics governed by the 2D equivalent of Eq. (5) and not Eq. (11), with $t/E_R=0.01$]. (e) Energy spectrum relating to the systems of (b) and (c), with $2\pi /3$ flux enclosed per lattice plaquette. The system is split into three bulk energy bands, and features additional dispersive edge states (shaded in blue as a guide to the eye). Insets show the modal distribution of different energy eigenstates. (f) Probability distribution of eigenstates populated by projection from $|{\psi }_{\mathrm{in}}\rangle =\left(\right|0,0\rangle$ (solid blue) and $|{\psi }_{\mathrm{in}}\rangle =\left(\right|0,0\rangle +i|0,1\rangle )/\sqrt{2}$ (dashed red). (g) and (h) Center-of-mass position dynamics, in terms of mode numbers $m$ and $n$ along the two directions, for enclosed flux and initial state as in (c) and (d). The black lines show the exact dynamics as in (c). From darker to lighter colors, the red (blue) lines in (g) [(h)] show dynamics for $t/E_R=0.01$, 0.04, 0.08. The smallest energy gap between spectral resonances ${\tilde{\omega }}_{m,n}^{x\left(y\right)}$ in the system is $0.97E_R$.

Figure 4

Laser driving scheme for studying U(2) lattice dynamics. (a) Counterpropagating laser fields drive a sample of atomic matter waves, where the field along one direction is composed of multiple spectral components, having frequencies ${\omega }_n^{-,\alpha }$. (b) Energy-momentum dispersion diagram. Two low-energy internal states $|g_1\rangle$ and $|g_2\rangle$ are coupled through stimulated state- and momentum-changing two-photon Raman transitions. All laser fields are far-detuned by an amount $\mathrm{\Delta }\gg {\omega }_{12}$ from atomic resonance, so that the excited state $|e\rangle$ is only virtually driven. For the left-traveling, multifrequency field, two distinct sets of frequency components (labeled $\alpha =1$ and 2), are used in conjunction with the right-traveling field to drive unique state- and momentum-changing transitions that depend on the initial internal state, as described in the text and shown in the figure.