Temporal Talbot interferometer of a strongly interacting molecular Bose-Einstein condensate

The Talbot interferometer, as a periodic reproduction of momentum distribution in the time domain, finds significant applications in multiple research. The interparticle interactions during the diffraction and interference process introduce numerous many-body physics problems, leading to unconventional interference characteristics. This work investigates both experimentally and theoretically the influence of interaction in a Talbot interferometer with a ${}^6\mathrm{Li}_2$ molecular Bose-Einstein condensate in a one-dimensional optical lattice, with interaction strength directly tunable via magnetic Feshbach resonance. A clear dependence of the period and amplitude of signal revivals on the interaction strength can be observed. While interactions increase the decay rate of the signal and advance the revivals, we find that over a wide range of interactions, the Talbot interferometer remains highly effective over a certain evolutionary timescale, including the case of fractional Talbot interference. This work provides insight into the interplay between interaction and the coherence properties of a temporal Talbot interference in optical lattices, paving the way for research into quantum interference in strongly interacting systems.

Figure 1

Schematic of the experimental system. (a) Schematic of the experimental system. Feshbach mBECs are trapped in a pair of crossover dipole traps. The blue circles in the $x\text{−}y$ plane represent the Feshbach magnetic coils and the red Gaussian beams along the $x$ axis mark the crossed optical dipole trap. The two orange arrows in the $x\text{−}y$ plane stand for the lattice beams with a crossing angle of ${120}^{\circ }$. The blue arrow shows the imaging direction, which is perpendicular to the $x\text{−}y$ plane. (b) A typical experimental time sequence. The black line represents the optical lattice, and the two pulses with duration of ${\tau }_0=0.7\mu \mathrm{s}$ before and after ${\tau }_{\text{evo}}$ are Talbot pulses. The interference time $\tau ={\tau }_{\mathrm{evo}}+{\tau }_0$. The blue line and green line represent the strength of the Feshbach field and optical dipole trap, respectively. The Feshbach field is switched from $B_0$ to $B_t$ in $t_{\mathrm{switch}}$ and kept for a period of $t_{\mathrm{hold}}=100$ ms. The optical dipole trap remains constant and is turned off together with the second lattice pulse. The red line represents the absorption imaging pulse, which is applied after the time-of-flight process with time $t_{\mathrm{TOF}}=3\mathrm{ms}$. Different stages are denoted at the top with varying shades of gray. (c) $T_0$: mBEC. (d) $T_1$: the scattering pattern with pulse $U_0=50E_r$ and $t=0.7\mu \mathrm{s}$. (e) $T_2$: the interference pattern with ${\tau }_{\mathrm{evo}}=T_{\mathrm{T}}/2$. (top) Raw image after 3-ms TOF (ten images averaged) for $a_{dd}=865a_0$ and (bottom) the corresponding bimode fitting result for distinguishing between condensed and noncondensed particles. The scales on the horizontal axis represents positions at $\pm 2n\hslash k$ modes, with each division marking 48 pixels ($168\mu \mathrm{m}$).

Figure 2

Temporal Talbot interferometer of strongly interacting mBEC. The plot shows the proportion of the $0\hslash k$ particles over the total number of particles in the absorption image for different free evolution times between two pulses of a 1D optical lattice for $a_{dd}=865a_0$. The red dots are experimental results, and the error bar shows the standard error of five measurements. The blue dashed and solid lines are the theoretical results without damping and with exponential decay correction, respectively. Raw plots of three different nodes during oscillation are presented in the insets. The red box indicates the molecular statistical range of each momentum mode. The positions of $T_{\mathrm{T}}/2$ multiples are indicated by gray dotted lines.

Figure 3

The damping of the Talbot signals under different interactions. (a) The first peak of $P_0$ under different interactions. The red square dots with error bars are the results of Method 1, and the blue circular dots are the results of Method 2, fitted with the black line. (b) The decay rate of the Talbot signals under different interactions. (c) The variation of the full width at half maximum (FWHM) of the molecular cluster with momentum $0\hslash k$, under different interactions, as a function of the interference period. The length corresponding to $2\hslash k$ in the momentum space is $168\mu \mathrm{m}$.

Figure 4

The shift of Talbot time caused by interactions. The blue cross dots and the red round dots show the experimental results of the oscillation of Talbot signals near $\tau =3T_{\mathrm{T}}/2$ under $655a_0$ and $2004a_0$, fitted with the blue solid line and red solid line, respectively. Details near the peak are shown in the inset, with the simulation results marked by the blue dashed (right) line ($655a_0$) and red dashed (left) line ($2004a_0$).

Figure 5

Fractional Talbot effect of strong interacting mBEC. (a) $U_0=100E_r$. (b) $U_0=150E_r$. (a1,b1) The initial state after the first lattice pulse ($T_1$). (a2,b2) Fractional Talbot interferometer with $a_{dd}=865a_0$. The theoretical curves are marked by solid blue lines and the experimental data are marked by solid red dots with error bars. ${\tau }_{\mathrm{evo}}$ corresponding to integer multiples of $T_{\mathrm{T}}/2$ are depicted by gray dashed lines.

Figure 6

Experimental apparatus. Schematics of the experimental setup.

Figure 7

Theoretical calculation of strongly interacting Talbot interferometer. The optical lattice pulse pattern is the same with lattice depth $U_0=50E_{r0}$ and pulse duration ${\tau }_0=0.7\mu \mathrm{s}$. The particle number is set as $30000$. (a) Talbot interferometer at different scattering lengths. The blue solid line, red dashed line, yellow dotted line, and purple dash-dotted line represent the simulation results with $a_{dd}=100a_0, 1000a_0, 5000a_0$, and $10000a_0$, respectively. $N_{\mathrm{Talbot}}=2\tau /T_{\mathrm{theoretic}}$. (b) Talbot interferometer at $a_{dd}=1000a_0$ with different characteristic lattice energy $E_r$. The blue solid line, red dashed line, and green dotted line represent the simulation results with $E_r=E_{r0}, E_r=0.2E_{r0}$, and $E_r=0.1E_{r0}$, respectively. (c) The results of simulations with different interactions during the pulse stage and evolution stage. The blue solid line represents the results with constant interaction $a_{dd}=2004a_0$, while the red dashed line indicates scenarios where no interaction occurs throughout. The yellow squares and green circles denote the results obtained when interaction is absent during the pulse phase and evolution phase, respectively.

Figure 8

Talbot signal decay in Method 1 and Method 2. (a) The decay curve obtained with Method 1 under different interactions. The black squares, red circles, orange triangles, and blue pentagons represent the experimental results of $a_{dd}=655a_0, 865a_0, 1330a_0$, and $2004a_0$, respectively. The fitting results under each interaction strength are indicated by lines of the corresponding colors. (b) The exponential fitting parameters for the decay curves under different interactions. ${\tau }_{\mathrm{decay}}$ is marked by the blue square dots and $y_0$ the red round dots. (c) The decay curve is obtained with Method 2 under different interactions. The black squares, red circles, orange triangles, and blue pentagons represent the experimental results of $a_{dd}=655a_0, 865a_0, 1330a_0$, and $2004a_0$, respectively. The linear fitting results under each interaction strength are indicated by lines of the corresponding colors.