Detection of continuous variable entanglement without coherent local oscillators

We propose three criteria for identifying continuous variable entanglement between two many-particle systems with no restrictions on the quantum state of the local oscillators used in the measurements. Mistakenly asserting a coherent state for the local oscillator can lead to incorrectly identifying the presence of entanglement. We demonstrate this in simulations with 100 particles and also find that large number fluctuations do not prevent the observation of entanglement. Our results are important for quantum information experiments with realistic Bose-Einstein condensates or in optics with arbitrary photon states.

Figure 1

Homodyne measurements of the two systems. Signal modes 1 and 2 are mixed with local oscillators; the resulting number differences are a measurement of the quadratures ${\widehat{X}}_1$ and ${\widehat{X}}_2$.

Figure 2

(Color online) The population dynamics and entanglement criteria for the case of (a)–(c) exactly 100 particles and (d)–(f) a thermal distribution with a mean of 100 particles. In (a) and (d) we see that particles are transferred from the condensate to modes 1 and 2. In (b), (c), (e), and (f) the entanglement inequalities are violated when the solid line enters the shaded areas enclosed by the dashed lines, as described in the text. The horizontal dotted lines indicate the inequality violation boundary for the canonical quadratures [9, 10, 11].