Fractality and macroscopic entanglement in two-component Bose-Einstein condensates

Spin coherent states are the matter equivalent of optical coherent states, where a large number of two component particles form a macroscopic state displaying quantum coherence. Here we give a detailed study of entanglement generated between two spin-$1/2$ Bose-Einstein condensates (BECs) due to an $S_1^z{S}_2^z$ interaction. The states that are generated show a remarkably rich structure showing fractal characteristics. In the limit of large particle number $N$, the entanglement shows a strong dependence upon whether the entangling gate times are a rational or irrational multiple of $\pi /4$, with a fractal dimension of $d\approx 1.7$. We discuss the robustness of various states under decoherence and show that despite the large number of particles in a typical BEC, entanglement on a macroscopic scale should be observable as long as the gate times are less than $\hslash /J\sqrt{N}$, where $J$ is the effective BEC-BEC coupling energy. Such states are anticipated to be useful for various quantum information applications such as quantum teleportation and quantum algorithms.

Figure 1

Entangling operation between two spinor Bose-Einstein condensates. (a) A schematic of the operation considered in this paper. The $Q$ functions for the initial spin coherent state $|\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\rangle \rangle$ on each BEC are shown. The $S^z{S}^z$ and $S^x{S}^z$ interactions induce a fanning of coherent states in the direction shown by the arrows. (b) The von Neumann entropy normalized to the maximum entanglement [$E_{\max }={\log }_2(N+1)$] between two BEC qubits for $N=1$ and $N=100$ after the operation of $S_1^z{S}_2^z$ for a time $\tau$. (c) Entanglement at times $\tau =\frac{\pi }{4N},\frac{1}{\sqrt{2N}},\frac{\pi }{4}$ for various boson numbers $N$. Inset: The same data for $\tau =\frac{1}{\sqrt{2N}}$, but normalized to $E_{\max }$.

Figure 2

Visualization of the entangled state (2) for gate times (a) $\tau =\frac{\pi }{4N}$, (b) $\tau =\frac{1}{\sqrt{2N}}$, (c) $\tau =\frac{\pi }{4}$, and (d) $\tau =\frac{\pi }{5}$. Eigenstates of the $S^z$ operator, $|k\rangle$ are entangled with spin coherent states of the same color as given in the key. The radii of the circles correspond to approximate distances for diminishing overlap of the spin coherent states. Colors give the entangled states given in the key, and the transparency of the circles reflects the size of the coefficients in (2). All plots correspond to $N=50$.

Figure 3

The entanglement as given by the approximate formula Eq. (5) (valid for times $\tau$ that are a rational multiple of $\pi /4$) for (a) $N=100$ and (b) $N=10000$. (c) The self-similar behavior of the entanglement can be observed in a zoomed-in view of (b). (d) Logarithmic plot of the number of boxes $N_{\varepsilon }$ versus the size of the box $\varepsilon$ in the box-counting method to determine the Haussdorff (fractal) dimension of the exact entanglement curve for $N=200$.

Figure 4

Entanglement between two BECs in the presence of decoherence as quantified by the logarithmic negativity $E_{\mathrm{neg}}$. (a) The entanglement including $z$ dephasing for $N=20$ and the parameter values as shown. (b) Dependence of the entanglement on the number of particles $N$ in each BEC for entangling times as shown for $z$ (solid lines) and $x$ dephasing (dotted lines) for ${\Gamma }_{z,x}=0.1$. (c) The ratio of the entanglement for dephased and ideal cases $R=E_{\mathrm{neg}}({\Gamma }_j=0.1)/E_{\mathrm{neg}}({\Gamma }_j=0)$ versus the inverse particle number for various $\tau$ as shown. Dotted lines show calculations for $x$ dephasing while solid lines are for $z$ dephasing. (d) Log-log plot of $R$ and $N$ showing power law behavior for $\tau =\frac{1}{\sqrt{2N}}$ for $z$ and $x$ dephasing for ${\Gamma }_{z,x}=0.1$.