Stueckelberg interferometry using periodically driven spin-orbit-coupled Bose-Einstein condensates

We study the single-particle dispersion of a spin-orbit-coupled (SOC) Bose-Einstein condensate (BEC) under the periodical modulation of the Raman coupling. This modulation introduces a further coupling of the SOC dressed eigenlevels, thus creating a second generation of modulation-dressed eigenlevels. Theoretical calculations show that these modulation-dressed eigenlevels feature a pair of avoided crossings and a richer spin-momentum locking, which we observe using BEC transport measurements. Furthermore, we use the pair of avoided crossings to engineer a tunable Stueckelberg interferometer that gives interference fringes in the spin polarization of BECs.

Figure 1

Experimental schematic. (a) Laser geometry showing both Raman beams. ${\vec{B}}_{\text{Bias}}$ denotes the bias magnetic field (of $\approx 5G$) which lifts the degeneracy of the $m_F$ spin states. The acceleration due to gravity is along the $-\widehat{y}$ direction. (b) Energy-level diagram showing the two bare spin states, $\left|F,m_F\right.〉=\left|1,-1\right.〉$ and $\left|1,0\right.〉$ (also denoted up and down, respectively), and their Raman-induced coupling. Drawing is not to scale. (c) Representative timing diagram for the dipole trapping laser (dashed-dot line) and the Raman coupling (solid line).

Figure 2

Experimental demonstration of the modified energy-momentum dispersion relation in the presence of modulated Raman coupling. (a) The unmodulated 1D SOC eigenlevels $[E_U\left(q\right)$ and $E_L\left(q\right)]$ calculated from Eq. (1) with ${\mathrm{\Omega }}_0=1.3E_r$ and $\delta =0E_r$. The dashed line shows the location of $E_U\left(q\right)$ if it had been shifted down by $f_{\mathrm{mod}}=10.56$ kHz. (b) The $E_{+}\left(q\right)$ and $E_{-}\left(q\right)$ modulation-dressed eigenlevels calculated from Eq. (3) with identical ${\mathrm{\Omega }}_0$ and $\delta$ as (a) and with ${\mathrm{\Omega }}_C=0.58E_r$ and $f_{\mathrm{mod}}=10.56$ kHz. The shifted but unmodulated dressed eigenlevels are shown by dashed lines. In (a) and (b) the blue and red colors superimposed on the eigenlevels and images represent spin down (bare spin $m_F=0$) and up (bare spin $m_F=-1$), respectively. Note the richer spin-momentum locking of the modulation-dressed SOC eigenlevels and the two avoided crossings labeled A and B. The green bars indicate the avoided crossings with gap size ${\mathrm{\Omega }}_C$ used as beam splitters in the interference experiments discussed later. (c),(d) Experimental comparison between the spin polarization of BECs transported through an unmodulated band $E_L\left(q\right)$ and a modulated band $E_{-}\left(q\right)$. Both bands used ${\mathrm{\Omega }}_0=1.3E_r$ and $\delta =0E_r$. The modulated band had ${\mathrm{\Omega }}_M=1.3E_r$ and $f_{\mathrm{mod}}=10.56$ kHz. The BECs started at $q_i\approx +1k_r$, fell under gravity with acceleration ${\alpha }_F\approx 1680k_r/\text{s}$ along the $-\widehat{y}$ direction for 1.5 ms, and reached $q_f\approx -1.5k_r$. (c) Four representative time-of-flight images taken at $q/k_r$ of 1.0, 0.8, 0.5, and $-0.3$ for both $E_L\left(q\right)$ and $E_{-}\left(q\right)$ showing their different spin-momentum lockings. In this panel, $k=q+k_r$ for $m_F=-1$ and $k=q-k_r$ for $m_F=0$. (d) Comparison between the observed and calculated spin polarizations for BECs in both $E_L\left(q\right)$ and $E_{-}\left(q\right)$. In the unmodulated case, the BEC nearly adiabatically follows the lowest-energy eigenlevel and the expected monotonic spin rotation is observed (black crosses). However, in the modulated case, an additional oscillation of the spin polarization is observed (purple circles). Solid black (purple dashed) lines are the calculated spin polarization of $E_L\left(q\right) \left[E_{-}\left(q\right)\right]$ given the experimental parameters using Eq. (1)[(3)], and ${\mathrm{\Omega }}_C=0.58E_r$. For this and following figures, a representative error bar indicates an average of 10% uncertainty in atom population in each spin due to technical noise.

Figure 3

Measurement of Stueckelberg interference. (a) Two representative modulation induced spin-orbit eigenlevels with ${\mathrm{\Omega }}_0=1.4E_r$ (blue dashed line) and $1.7E_r$ (black solid line). Both eigenlevel calculations used ${\mathrm{\Omega }}_C=0.3E_r, f_{\mathrm{mod}}=8$ kHz, and $\delta =0E_r$. (b) Measured Stueckelberg interference fringes in the BEC spin polarization vs $f_{\mathrm{mod}}$ at ${\alpha }_F=1680k_r/\text{s}$ for ${\mathrm{\Omega }}_M=0.7E_r$ (blue) and $0.8E_r$ (black). The theoretical curves were calculated from Eq. (6) with ${\mathrm{\Omega }}_C=0.3E_r, {\sigma }_{\alpha }=0.07{\alpha }_F, f_{np}=0.4, \delta =0$, and the same ${\mathrm{\Omega }}_0$ as in the experiment ($1.7E_r$ for black and $1.4E_r$ for blue).

Figure 4

Stueckelberg interference fringes in the spin polarization of BECs at various values of the initial BEC acceleration, ${\alpha }_F$. Varying ${\alpha }_F$ changes the transport time through $E_{+}\left(q\right)$ and $E_{-}\left(q\right)$. This experiment used ${\mathrm{\Omega }}_M=0.7E_r, {\mathrm{\Omega }}_0=1.4E_r, f_{\mathrm{mod}}=8.5$ kHz, and $\delta =0E_r$. The theoretical curves were calculated from Eq. (6) and used ${\mathrm{\Omega }}_C=0.33E_r, {\sigma }_{\alpha }=0.07\times (1680k_r/\text{s})$, and $f_{np}=0.3$. (The fringe contrast is strongly reduced at smaller ${\alpha }_F$ as the dephasing effect of ${\sigma }_{\alpha }$ gets larger with longer total time.)

Figure 5

Effect of ${\mathrm{\Omega }}_M$ on Stueckelberg interference. (a) Observed Stueckelberg interference fringes for ${\mathrm{\Omega }}_M=0.3E_r$ and ${\mathrm{\Omega }}_0=1.3E_r, {\mathrm{\Omega }}_M=0.7E_r$ and ${\mathrm{\Omega }}_0=1.4E_r$, and ${\mathrm{\Omega }}_M=1.2E_r$ and ${\mathrm{\Omega }}_0=1.3E_r$ for black circles, blue squares, and red triangles; all used $\delta =0E_r$ and ${\alpha }_F=1680k_r/\text{s}$. All the theoretical curves were obtained from Eq. (6) with $\delta =0E_r, {\sigma }_{\alpha }=0.07{\alpha }_F$, and the experimental values of ${\mathrm{\Omega }}_0$. The black, blue, and red lines used ${\mathrm{\Omega }}_C=0.14,0.31,0.52E_r$ and $f_{np}=0.4,0.45,0.5$, respectively. (b) Measured fringe contrast $M$ vs modulation amplitude ${\mathrm{\Omega }}_M$. The theory curve was generated assuming a linear relation of ${\mathrm{\Omega }}_C={\mathrm{\Omega }}_M/2.25, {\mathrm{\Omega }}_0=1.4E_r, f_{np}=0.4$, and ${\sigma }_a=0.07{\alpha }_F$. Error bars indicate numerical fitting uncertainty of the fringes. (c) Theoretical calculation of ${\mathrm{\Omega }}_C$ vs ${\mathrm{\Omega }}_M$ with $\delta =0E_r$ and ${\mathrm{\Omega }}_0=1.33E_r$. A linear fit gives ${\mathrm{\Omega }}_C={\mathrm{\Omega }}_M/2.27$. This calculated ratio between ${\mathrm{\Omega }}_C$ and ${\mathrm{\Omega }}_M$ is found to change by less than $10\%$ in the range ${\mathrm{\Omega }}_0$ experimentally accessed.