ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

162 9. Two-mode Gaussian states in the lab
OPO / OPA
one-mode
thermal state
one-mode
squeezed beam
BS
50:50
two-mode
GLEMS
Figure 9.2. Possible scheme for the generation of Gaussian least entangled
mixed states (GLEMS), introduced in Sec. 4.3.3.1. A single-mode squeezed
state (obtained, for example, by an optical parametric oscillator or amplifier)
interferes with a thermal state through a 50:50 beam-splitter. The resulting
two-mode state is a minimally entangled mixed Gaussian state at given global
and marginal purities.
The state at the output of the beam-splitter will be a correlated two-mode Gaussian
state with CM σout that reads
⎛
⎞
σout = 1
2
⎜
⎝
n + k 0 n − k 0
0 n + k −1 0 n − k −1
n − k 0 n + k 0
0 n − k −1 0 n + k −1
⎟
⎠ ,
with k = e2r and n = µ −1 . By immediate inspection, the symplectic spectrum of
this CM is ν− = 1 and ν+ = 1/µ. Therefore the output state is always a state with
extremal generalized entropy at a given purity (see Sec. 2.3). Moreover, the state
is entangled if
cosh(2r) > µ2 + 1
2µ = exp (2ω/kBT ) + 1
. (9.3)
exp (2ω/kBT ) − 1
Tuning the experimental parameters to meet the above condition, indeed makes the
output state of the beam-splitter a symmetric GLEMS. It is interesting to observe
that nonsymmetric GLEMS can be produced as well by choosing a beam-splitter
with transmittivity different from 1/2.
9.2. Experimental production and manipulation of two-mode entanglement
Experimentally, CV entanglement can be obtained directly by type-II parametric
interaction deamplifying either the vacuum fluctuations as was demonstrated in the
seminal experiment by Ou et al. [171] (or in recent experiments [139, 254]) or the
fluctuations of a weak injected beam [281]. It can also be obtained indirectly by
mixing on a beam-splitter two independent squeezed beams, as shown in Fig. 9.1.
The required squeezing can be produced by Kerr effects — using optical fibers [217]
or cold atoms in an optical cavity [129] — or by type-I parametric interaction in
a cavity [275, 33]. Single-pass type-I interaction in a non-collinear configuration
can also generate directly entangled beams as demonstrated recently by Wenger et