ENTANGLEMENT OF GAUSSIAN STATES Gerardo Adesso

160 9. Two-mode Gaussian states in the lab
generally mixed, Gaussian resources [GA9], which will be established in Chapter
12.
Before turning to the operational interpretation of entanglement, we judge of
interest to discuss, at this point of the Dissertation, the issue of providing efficient
recipes to engineer, in the lab, the various classes of Gaussian states that we have
singled out in the previous Parts for their remarkable entanglement properties.
These optimal production schemes (among which we mention the one for all pure
Gaussian states exhibiting generic entanglement [GA14], presented in Chapter 11)
are of inherent usefulness to experimentalists who need to prepare entangled Gaussian
states with minimal resources. Unless explicitly stated, we will always consider
as preferred realistic setting for Gaussian state engineering that of quantum optics
[65].
In this Chapter, we thus begin by first completing the analysis of Chapter 4
on the two special classes of two-mode “extremally” entangled Gaussian states that
have arisen both in the negativity versus purity analysis, and in the comparison
between Gaussian entanglement measures and negativities, namely GMEMS and
GLEMS [GA3]. We discuss how both families of Gaussian states can be obtained
experimentally, adding concreteness to the plethora of results previously presented
on their entanglement characterization. After that, we move more deeply into the
description of the experiment concerning production and optimization of entanglement
in two-mode Gaussian states by optical means, reported in [GA8].
State engineering of Gaussian states of more than two modes will be addressed
in the next two Chapters.
9.1. Schemes to realize extremally entangled states in experimental settings
We discuss here how to obtain the two-mode states introduced in Sec. 4.3.3.1 in a
practical setting.
9.1.1. GMEMS state engineering
As we have seen, GMEMS are two-mode squeezed thermal states, whose general
CM is described by Eqs. (2.54) and (4.29). A realistic instance giving rise to
such states is provided by the dissipative evolution of an initially pure two-mode
squeezed vacuum with CM Eq. (2.22). The latter may be created e.g. by mixing
two independently squeezed beams (one in momentum and one in position, with
equal squeezing parameter r) at a 50:50 beam-splitter B1,2(1/2), Eq. (2.26), as
shown in Fig. 9.1. 25
Let us denote by σr the CM of a two mode squeezed vacuum with squeezing
parameter r, Eq. (2.22), derived from Eqs. (4.29) with ν∓ = 1. The interaction
of this initial state with a thermal noise results in the following dynamical map
describing the time evolution of the CM σ(t) [210]
σ(t) = e −Γt σr + (1 − e −Γt )σn1,n2 , (9.1)
where Γ is the coupling to the noisy reservoir (equal to the inverse of the damping
time) and σn1,n2 = ⊕ 2 i=1 ni2 is the CM of the thermal noise (see also Sec. 7.4.1.1).
25 See also Sec. 2.2.2. A more detailed discussion concerning the production of two-mode
squeezed states is deferred to Sec. 9.2.