Figure 1

One-body density matrix ${\rho }_{1A}(x)$ in a balanced system $N_A=N_B=10$ for different interaction strength ${\lambda }_A={\lambda }_B$. Thin uppermost curve, ${\lambda }_A={\lambda }_B=0$ (mixture of two ideal Bose gases), thermodynamic limit as given by Eq. (13). Thick lines, Monte Carlo results for $N_A=N_B=10$ particles and increasing interaction strength (from upper to lower curve), as labeled in legend.Reuse & Permissions

Figure 2

Inset: projected two-body density matrix ${\rho }_{2\mathit{AA}}^P(R)$, Eq. (12), in a mixture of two ideal Bose gases for $N_A=N_B=$ 80, 60, 40, 20, 10, 5 (from upper to lower curves). Main figure: the same function multiplied by $2N$.Reuse & Permissions

Figure 3

Two-body density matrix ${\rho }_{2\mathit{AA}}(r,L/4)$ in a balanced system of $N_A=N_B=10$ particles for different interaction strengths ${\lambda }_A={\lambda }_B=0$–1 (decreasing from the value at zero), as labeled in the legend. In the presence of BEC of molecules, ${\rho }_{2\mathit{AA}}(r,L/4)$ is proportional to the square of the molecular orbital. For $\lambda =0,$ the thermodynamic result $[{\varphi }_{\mathrm{mol}}^{\mathit{AA}}(r)]{}^2$ (17) is shown for comparison.Reuse & Permissions

Figure 4

Schematic phase diagram for ${\lambda }_A={\lambda }_B=0$ showing a crossover as the density imbalance $n_B/n_A$ is changed from zero to infinity. Component $A$ passes through three phases: (1) true BEC of $A$, (2) quasicondensate of $A$, BEC of large $AA$ dimers, and (3) BEC of highly localized dimers (molecules). Component $B$ passes through the same phases but in an inverted order. The system is normal in all regimes as no sound exits.Reuse & Permissions

Figure 5

Momentum distribution for a balanced system $n_A=n_B$ for characteristic values of the interaction parameter $\lambda =0$, 1, 10. The heavy solid line, $\lambda =0$, shows the thermodynamic limit of a mixture of two ideal Bose gases (14). Dashed line, $\lambda =1$, and dash-dotted line, $\lambda =10,$ are results of Monte Carlo calculation with $N=40$ particles. Thin solid line, Gaussian approximation to the momentum distribution in the “classical” regime as given by Eq. (18). Note that the height of the peak for $\lambda =1$ depends explicitly on the number of particles in the system.Reuse & Permissions

Figure 6

3D plot of the momentum distribution for a balanced system $n_A=n_B$ as a function of the momentum and the interaction parameter $\lambda$.Reuse & Permissions

Figure 7

Inset: Momentum distribution of a balanced system with ${\lambda }_A={\lambda }_B=1$ for different system sizes $N_A=N_B=10,20,40,80,160,320,1000$ as a function of momentum in units of Fermi momentum $k/k_{\mathrm{F}}$. Main figure: the same data points shown on a log-log scale as a function of the separation from the divergency point $|k-k_{\mathrm{F}}|/k_{\mathrm{F}}$. High-momentum $1/k^2$ and low-momentum $1/\left|k\right|{}^{1/4}$ asymptotics are shown for comparison.Reuse & Permissions

Figure 8

Momentum distribution of a system with $N_A=N_B=10$ particles with the $AA$ interaction strength fixed to ${\lambda }_A=1$ for different interaction strengths of the $BB$ component ${\lambda }_B$ (see legend) as a function of momentum in units of Fermi momentum $k/k_{\mathrm{F}}$. Compare the data to the inset of Fig. 7.Reuse & Permissions

Figure 9

Projected two-body density matrix ${\rho }_{2\mathit{AA}}^P(R)$, Eq. (12) for $N_A=N_B=10$. ${\lambda }_A={\lambda }_B=0$–1 (from upper to lower curves, as labeled). Main figure: logarithmic scale. Inset: closeup of the tail.Reuse & Permissions

Figure 10

Altitude of the highest peak in the momentum distribution $\underset{k}{max}n(k)$ for $N_A=10$ and $N_B=2$–$100$ particles shown as a contour plot in variables of the density imbalance $n_B/n_A$ and interaction parameter $\lambda$. The dashed lines separate different phases and correspond to parameters for which the heights of the peaks at $k=0$ and $k=k_{\mathrm{F}}$ are equal. Some important features of the phase diagram are observed in this plot. True Bose-Einstein condensation [constant ODLRO in ${\rho }_1(x)$] of $A$ ($B$) particles happens for $\lambda =0$ and $n_B=0$ ($n_A=0$). Small, but finite, values of $\lambda$ and $n_B$ ($n_A$) correspond to quasicondensate of $A$ ($B$) particles with the highest peak positioned at $k\approx 0$. By increasing $\lambda$ the position of the highest peak shifts to $k=k_{\mathrm{F}}$ which corresponds to the phase of Fermi-momentum condensation. The strongest divergence at the Fermi momentum is observed for $\lambda =1$. By increasing the value of $\lambda$ further, the position of the highest peak shifts once more to $k\approx 0,$ which marks the “classical” regime. We stop the dashed line for small $n_B$ when $k_{\mathrm{F}}^B$ is comparable to the momentum quantization $2\pi /L$. In the ideal-gas limit $n_B\to 0,$ the Fermi momentum vanishes as $k_{\mathrm{F}}^B\to 0$.Reuse & Permissions