Figure.1. (Left) Selection of deuterons by $z$ (left) at different momentum ($\rm{p}$). TPC is used when ${\rm p}<3$\,GeV/$c$, TPC+TOF are used when ${\rm p}>3$\,GeV/$c$. (Right) The mass distribution of $p-\pi^-$ pair for the reconstruct $\Lambda$ at different transverse momentum ($p_{T}$). Red curves show the selected particles.

Figure.2. $d-\Lambda$ correlation function ($C(k^*)$) shown as a function of relative momentum ($k^*$) at different centralities. Statistical and systematical uncertainties are shown with bars and boxes, respectively. Dashed black lines are fitting curves with LL approach (from 0 to 200 MeV/c). Blue and orange dashed lines show the contribution from the D-state and Q-state, respectively. The dashed pattern shows the SMASH+LL model. See text for details.

Figure.3. The extracted final state interaction parameters, scattering length ($f_0$) and effective range ($d_0$), for $d-\Lambda$  interaction shown as contours of the probability distribution for both doublet (D) (blue) and quartet (Q) (orange) spin states. The unconstrained fit results are shown in panel (a) and panel (b) for Q state and D state, respectively. The constrained fit results for both states are shown in panel (c). The purple triangles show the different model calculations. 

Figure.4. $\Lambda$-separation energy of $^{3}_{\Lambda}{\mathrm{H}}$. The upper panel shows the probability distribution of the $\Lambda$-separation energy of $^{3}_{\Lambda}{\mathrm{H}}$ from \dla{} correlation with the Bethe formula. The corresponding radius is shown at the top. The lower panel compares the results obtained from this study with worldwide data. The errors combined both statistical and systematical uncertainties. The shaded band shows the world average.

Conclusion:the first observation of \dla{} correlation in Au+Au collisions at $\sqrt{s_{\rm NN}}$ = 3\,GeV is reported in this letter. With the Lednick\'y-Lyuboshitz approach, the spin-dependent strong interaction parameters are separated from the emission source. It is observed that from central to peripheral collisions, the Gaussian source size parameter ($R_G$) decreases.  Calculations from the hadronic transport model SMASH coupled with a coalescence procedure for deuteron formation underestimated the source size parameters for the \dla{} source. While the unconstrained fit results show attractive potential for both spin states ($f_0>0$), a clear hint of $f_0<0$ for the D-state is seen. These results reconfirmed with the $f_0 ({\rm D})$ from the constrained fit. The final state interaction parameters are determined to be $f_0({\rm D})=-25.2 \pm 5.7$\,fm, $d_0({\rm D})=2.9\pm1.7$\,fm for D-state, and $f_0({\rm Q})=18.9\pm2.6$\,fm, and $d_0({\rm Q})=1.5\pm 0.9$\,fm for Q-state under constrained fit. %The results included both statistical and systematical uncertainties. 

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From the scattering parameters of the D-state, the $\Lambda$-separation energy of $^{3}_\Lambda \mathrm{H}$ is obtained to be $0.04_{-0.02}^{+0.03}$\,MeV.