Figure 1

Noise correlations in different phases, derived from Luttinger liquid theory. (The analogous results from using TEBD are given in Fig. 3.) In the MI state [column (a)], $\delta$-function-like correlations along $k=k^{\prime }$ in ${\mathcal{G}}_{11}(k,k^{\prime })$ are visible, whereas ${\mathcal{G}}_{12}$ nearly vanishes. In the SF state [column (b)], with Luttinger parameters $K_A=1.03$ and $K_S=0.96$, we find various contributions in ${\mathcal{G}}_{11}$. There is a singularity along $k=k^{\prime }$, which diverges as $k$ and $k^{\prime }$ approach zero. In Fig. 2, we show the contour plots for ${\mathcal{G}}_{11}$ and ${\mathcal{G}}_{12}$ for the same state, where we can see the negative correlations at $k=0$ and $k^{\prime }=0$, as well as pairing correlation along $k=-k^{\prime }$, which is similar to the single-species result in Ref. [5]. ${\mathcal{G}}_{12}$ shows similar features, but the bunching contribution is an algebraic peak, rather than a $\delta$ function. (c) An example for the PSF phase, with $K_A=0.01$ and $K_S\simeq 1.3$. (d) An example for the CFSF phase, with $K_S=0.01$ and $K_A\simeq 1.2$. In the PSF state, the interspecies correlation ${\mathcal{G}}_{12}(k,k^{\prime })$ has strong correlations along $k=-k^{\prime }$, a reflection of pairing. In the CFSF state, the peak is formed along the $k=k^{\prime }$ direction, an indication of the antipairing (particle-hole) formation in the CFSF state.Reuse & Permissions

Figure 2

Noise correlations, ${\mathcal{G}}_{11}(k,k^{\prime })$ (a) and ${\mathcal{G}}_{12}(k,k^{\prime })$ (b), in the SF state. The data used here are the same as those used in Fig. 1b. We create the nonlinear gray scales by plotting functions $\tanh (500{\mathcal{G}}_{11})$ (a) and $\tanh ({\mathcal{G}}_{12}\times {10}^5)$ (b), in order to magnify the details around $k=k^{\prime }=0$. The labels of the color-bar reflect the values of ${\mathcal{G}}_{11}(k,k^{\prime })$ and ${\mathcal{G}}_{12}(k,k^{\prime })$. In these plots, we can clearly see the negative correlations between the quasi-condensate ($k=0$ and $k^{\prime }=0$) and the higher momentum states at the quantum depletion, as well as the antipairing correlation along $k=k^{\prime }$ and the pairing correlation along $k=-k^{\prime }$.Reuse & Permissions

Figure 3

Noise correlations for a homogeneous system of 40 lattice sites, calculated with the TEBD method. These plots should be compared to the LL results given in Fig. 1. Frames (a)–(d) correspond to the examples (a)–(d) in Table . (a) The noise correlations of a MI state. In the plot of ${\mathcal{G}}_{11}(k,k^{\prime })$, there is a strong correlation along the direction $k=k^{\prime }$, whereas the noise correlation function ${\mathcal{G}}_{12}(k,k^{\prime })$ essentially vanishes. (b) The noise correlations of a SF state. Here, we can see the peak around $k=k^{\prime }$ corresponding to the $\delta$-function bunching peak predicted by LL theory [see also Fig. 1b]. For ${\mathcal{G}}_{12}$, we find a negative value at $k=k^{\prime }=0$, which is different from the LL result [Fig. 1b]. Other structures predicted by LL theory can be seen in Fig. 4, where ${\mathcal{G}}_{12}$ and ${\mathcal{G}}_{11}$ are plotted in a nonlinear color scale to magnify the structures around $k=k^{\prime }=0$. In (c) and (d), we show the noise correlations of the PSF and CFSF states, respectively. In the PSF state (c), the interspecies correlation ${\mathcal{G}}_{12}(k,k^{\prime })$ has strong correlations along $k=-k^{\prime }$, a consequence of pairing [see also Fig. 1c]. In the CFSF state (d), the peak is formed along the direction $k=k^{\prime }$, an indication of antipairing in the CFSF state (see also Fig. 1).Reuse & Permissions

Figure 4

Noise correlations, ${\mathcal{G}}_{11}(k,k^{\prime })$ (a) and ${\mathcal{G}}_{12}(k,k^{\prime })$ (b), in the SF state of a homogeneous system. The values of ${\mathcal{G}}_{11}(k,k^{\prime })$ and ${\mathcal{G}}_{12}(k,k^{\prime })$ are exactly the same as in Fig. 3b. We create nonlinear gray scales by plotting $\tanh (10{\mathcal{G}}_{11})$ and $\tanh (200{\mathcal{G}}_{12})$ in linear scales. The labels of the color bar reflect the values of ${\mathcal{G}}_{11}(k,k^{\prime })$ and ${\mathcal{G}}_{12}(k,k^{\prime })$. The features around $k=k^{\prime }=0$ are magnified as a result of the nonlinear scale. In (a), we find the features predicted by LL calculations [Fig. 2a]. In addition, we can see a weak correlation around $k=k^{\prime }\pm 2k_F$, where $k_F=\nu \times \pi /a_L=0.5\pi /a_L$. This is where a strong correlation (cusps) will develop if CDW order is present. This feature has also been shown in LL calculations at around $k\approx 0$ and $k^{\prime }\approx 2k_F$ [Eq. (24)]. In (b), we find that the structures along $k=k^{\prime }$ are similar to the ones in LL calculations; however, the structures along $k=-k^{\prime }$ are negative, different from the LL predictions (see also Fig. 2). The difference may be understood as a result of different boundary conditions used for the finite-size calculations: The numerical calculations use a “hard-wall” boundary condition, whereas the LL calculations assume a periodic boundary condition.Reuse & Permissions

Figure 5

Left: The correlation function $C_{11}(q)$, as defined in Eq. (2). Right: The structure factor $S(k)$ for a quasi-supersolid state. The parameters are given in example (e) of Table . The Luttinger parameters are $K_A\approx 1.4$ and $K_S\approx 0.57$. The filling fraction is $\nu =0.2$, and hence the Fermi wave vector $k_F$ is $\pi \times 0.2$. At momentum $2k_F$, both quantities develop cusps, indicating the presence of CDW order.Reuse & Permissions

Figure 6

Noise correlations in the trapped system. The system size is 80 sites and $t/U=0.02$. In (a), the system is in the SF state. The particle number of each species is 30, the trap frequency is $8\times {10}^{-5}U$, and $U_{12}/U=0.01$. In (b), the particle number of each species is 40, the trap frequency is $1\times {10}^{-4}U,$ and $U_{12}/U=-0.11$. The system has both MI and PSF orders. The MI state forms a plateau at unit filling at the center of the trap and the PSF is formed at the edge. The PSF state at the edge causes the small peak along the $k=-k^{\prime }$ direction, similar to the one in (c). However, this peak is at a much smaller amplitude than the one shown in (c), where the whole system is a PSF state. In (c), the particle number of each species is 20, the trap frequency is $1\times {10}^{-5}U,$ and $U_{12}/U=-0.11$. The whole system is in the PSF state. A strong pairing correlation is formed along the $k=-k^{\prime }$ direction. In (d), the particle number of each species is 30, the trap frequency is $8\times {10}^{-5}U,$ and $U_{12}/U=0.2$. The system has both CFSF and SF order. The CFSF order forms a plateau at half-filling at the center of the trap and the SF state toward the edges of the trap. The CFSF order causes a strong antipairing (particle-hole) correlation along $k=k^{\prime }$ direction. At the same time, the SF order adds to the dips along $k=0$ and $k^{\prime }=0$.Reuse & Permissions

Figure 7

Correlations $C_{12}(q)$ and $D_{12}(q)$ for the states that are described in Fig. 6. In (a), we show the behavior of $C_{12}(q)$ [Eq. (2)] in SF, MI, PSF, and CFSF states. The strong antipairing (particle-hole) correlations in the CFSF state gives a strong signal around $q=0$ in $C_{12}(q)$. This strong signal is also unique to the CFSF state and therefore can be used to detect the CFSF order. In (b), we show the behavior of $D_{12}(q)$ [Eq. (3)] in SF, MI, PSF, and CFSF states. The strong pairing correlation in the PSF state is the reason for the high peak around $q=0$ in $C_{12}(q).$ This suggests that measuring $D_{12}(q)$ is a good way of detecting the PSF order.Reuse & Permissions

Figure 8

Noise correlation $C_{11}(q)$ and structure factor $S(k)$ in a PSF/CDW state. The system size is 80 sites and there are 20 particles of each species ($\nu =0.25$). The trap frequency is $\Omega ={10}^{-5}U$, the hopping $t=0.02U,$ and the interspecies interaction is $U_{12}=-0.11U$. The density at the center of the trap is roughly $0.45$ per site and the cusps are developed around $\pm 0.9\pi$. The inhomogeneity of a trapped system means that the Fermi wave vector $k_F$ is no longer $\pi \nu$, where $\nu$ is the average filling of the system. Instead, $k_F$ can be evaluated as $\pi n_{\mathrm{center}}$, where $n_{\mathrm{center}}$ is the density at the center of the trap.Reuse & Permissions