Experimental study of matter-wave four-wave mixing in $^{39}$K Bose-Einstein condensates with tunable interaction

TL;DR

This work addresses how tunable interatomic interactions influence matter-wave four-wave mixing (FWM) in potassium Bose-Einstein condensates. It compares two configurations: a square geometry for a single spin and a collinear geometry implemented with a spin-dependent lattice for two spins. The main findings show that single-spin FWM yield increases with the scattering length and can reach about 5% before losses suppress growth, while two-spin FWM yields peak near the gas–droplet boundary where quantum fluctuations enhance nonclassical effects. The results suggest practical pathways for matter-wave amplification and entangled-atom generation with potential impact on quantum information processing and precision measurement.

Abstract

We experimentally investigate four-wave mixing (FWM) of matter waves in two geometric configurations in $^{39}$K Bose-Einstein condensates with the atomic interaction tuned via Feshbach resonances. For one configuration with the single-spin component, the FWM yield increases with a larger scattering length. For the two-spin component configuration, we specifically investigate FWM in both the droplet and gas parameter regimes. We find that the FWM yield reaches its maximum near the critical parameter region between the gas and droplet phases. Our research can help to optimize the FWM yield for matter-wave amplification and entangled atom pair generation, making it conducive to applications in quantum information processing and precision measurement.

Figures (6)

• Figure 1: (a) Experimental optical configuration for FWM of a single-spin component. Two sets of Bragg pulses $L_1+L_2$ and $L_1+L_3$ are applied sequentially to generate three distinct momentum states $\mathbf{p}_1$, $\mathbf{p}_2$, and $\mathbf{p}_3$, as shown in (b). The three light beams have nearly identical wavelength of $790.00\ \mathrm{nm}$. (b) The left panel shows a typical absorption image of a single-spin component FWM in a square geometric configuration, where the collision of three wave packets leads to the generation of a fourth wave packet. The right panel shows the corresponding schematic diagram of the FWM processes in which the solid arrows and dashed arrows represent two separated processes of atomic momentum transition. (c) Experimental optical configuration for FWM of two-spin components. The laser for the lattice beams operates at the tune-out wavelength of $769.35\ \mathrm{nm}$, and the bias magnetic field is oriented parallel to the laser propagation direction, which ensures that atoms in state $\lvert\uparrow\rangle$ experience the lattice potential $V_0$, while those in state $\lvert\downarrow\rangle$ are unaffected. (d) The left panel shows a typical absorption image of a two-spin component FWM in a collinear geometric configuration. When the initial atomic spin state is a mixture of $\lvert\uparrow\rangle$ and $\lvert\downarrow\rangle$, the $\lvert\downarrow\rangle$ state exhibits two scattered momentum states at $\lvert\pm 2 \hbar k\rangle$. The right panel shows the corresponding schematic diagram of the FWM processes in which the solid arrows and dashed arrows represent two symmetric FWM processes, respectively. In (b, d), the green dashed circles denote the wave packets generated by FWM.
• Figure 2: Intraspin and interspin scattering lengths of $^{39}$K atoms as a function of Feshbach magnetic field, with the spin $\lvert\uparrow\rangle$ and $\lvert\downarrow\rangle$ labeling the $\lvert1,-1\rangle$ and $\lvert1,0\rangle$ states, respectively. The light blue shaded region denotes the magnetic field range employed for FWM in single-spin BECs. The top-left inset, a magnification of the dashed rectangle, denotes the magnetic field range employed for FWM in two-spin BECs, where the shaded region represents the gas regime with $\delta a > 0$ and the remaining region designates the droplet regime with $\delta a < 0$.
• Figure 3: (a) Schematic illustration of interrupting the FWM process. Pulses I and II prepare three momentum states for FWM, while pulse III (30 $\mu$s) removes atoms from momentum state $\mathbf{p}_1$ to $-\mathbf{p}_3$, thereby interrupting the process. The optical dipole trap (ODT) is turned off after pulse I and II, to allow the wave packets to evolve in free space for a duration of $T$. (b) The growth curves of the FWM processes with different scattering lengths. The solid lines represent sigmoidal fits to the experimental data. The fitting formula is $f(T) = A / \{1 + \exp[-K(T - T_c)]\}$, where $A$ represents the maximum value of the curve, $T_c$ denotes the time at half-maximum, and $K$ indicates the growth rate. The fitted $T_c$ values are $0.11\ \mathrm{ms}$, $0.12\ \mathrm{ms}$, and $0.11\ \mathrm{ms}$, respectively. Each data point represents an average over five experimental runs, and the error bars indicate standard deviations of repeated experimental measurements.
• Figure 4: The FWM yield of the single-spin components under various scattering lengths. The FWM yield vanishes as the scattering length approaches zero, reaches its maximum in the scattering length around $118\ a_0$, and then decreases with further increased scattering length due to enhanced three-body losses. Each data point represents an average over five experimental runs, and the error bars indicate standard deviations of repeated experimental measurements. The blue solid line presents the fit of experimental data based on an empirical formula $f(x) = ax\exp(-x^b / c)$. Corresponding to the units of the horizontal and vertical axes in the figure, we obtain the dimensionless fitting parameters as $a = 7.0435 \times 10^{-4}$, $b = 1.6550$, and $c = 5401.0$.
• Figure 5: The FWM yield of two-spin components under various Feshbach magnetic field strengths. The inset shows the curves of the FWM yield and $\delta a$ versus the scattering length $a_{\downarrow\downarrow}$. The FWM yield reaches a maximum around $\delta a = -6\ a_0$. Each data point represents an average over five experimental runs, and the error bars indicate standard deviations of repeated experimental measurements.