kBT ≪ (kBTC, µ), we are confident that phase spread due to temperature
has only a minor effect in our experiments.
2.2.8
Relation between squeezing and entanglement
The density matrix of any non-entangled, separable state of N atoms can be written in the formρ=∑ k pkρ (k) 1 ⊗ρ (k) 2 ...⊗ρ (k) N (2.43)
where ρ(ik) is the density matrix of the ith particle in the kth therm of the weighted sum. A state is m-particle entangled if ρ cannot be decomposed into a sum where each density matrix involves less thanmparticles. In other words, the sum contains at least one m-particle density matrixρ(i...ik)+m.
It is shown in [16] that a spin state satisfying ξ2 < 1 is at least bi-
partite entangled. In [46], a more general method for the identification of
m-particle entanglement through measurement of the collective variables ∆S2
θ,min and ⟨Sx⟩2 is deduced: using the Heisenberg uncertainty relation
∆Sy∆Sz ≥ ⟨Sx⟩/2 and ⟨Sx2⟩+⟨Sy2⟩+⟨Sz2⟩ ≤S(S+ 1), one can derive
∆Sθ,2min ≥ 1 2 ( S(S+ 1)− ⟨Sx⟩2− √ [S(S+ 1)− ⟨Sx⟩2]2− ⟨Sx⟩2 ) . (2.44) For large S and ⟨Sx⟩ this inequality gives an approximate bound for the
maximum squeezing achievable in a spin S system. The main result of [46] is that in a system of N spin-12 particles, for a measured set
(
⟨Sx⟩,∆Sθ,2min
)
, there is a minimum spinS satisfying equation (2.44) and the system is thus at least (2S+ 1)-particle entangled. For low entanglement (Ssmall) the ana- lytical formula (2.44) does not give a tight bound and a numerical calculation produces the corresponding limit (see figure 2.6).
2.3
State tomography
To measure the degree of spin squeezing for a given state, it is sufficient to measure its mean spin and the spin fluctuations along the angle θmin. How-
ever, much more information about the state can be gained by measuring the mean spin and its variance not only along one direction but along many an- glesθ0 ≤θ <(θ0+π). From such a state tomography one can gain a complete
Spin squeezing theory
0
0.1
0.2
0.3
0.4
0.5
0
0.2
0.4
0.6
0.8
1
∆
S
θ, min 2/S
S
x/S
10 S = 1/2Figure 2.6: Maximal squeezing for different values of S. Curves indicating the maximal possible reduction in spin variance as a function of contrast ⟨Sx⟩/S for different spins S. The black lines are numerical calcula-
tions for S = (1 2,1,
3
2,2,3,4,5,10) (top to bottom), the colored lines are the
solutions of the analytical formula (2.44) forS = 12 (green) andS = 10 (red). The analytical approximation is close to the numerical calculation only for largeS. If a measured data point
(
⟨Sx⟩,∆Sθ,2min
)
lies below a (black) S-line it implies that the ensemble is at least (2S + 1)-particle entangled. Figure adapted from [46].
Wigner function
The description of mixtures and fluctuations usually requires the use of the density operatorρwhich provides the most general description of a quantum system. There exists however a different but completely equivalent descrip- tion in phase space in the form of the quasi-probability distributions such as the Glauber-Sudarshan P distribution or the Wigner distribution (or Wigner function) [14, 98, 99]. For an harmonic oscillator the Wigner function is defined as W(q, p) = 1 π~ +∞ ∫ −∞ ⟨q−y|ρ|q+y⟩ei2py/~dy, (2.45)
2.3 State tomography
where q and p are position and momentum quadratures, respectively. For a pure state ρ=|Ψ⟩⟨Ψ| it becomes
W(q, p) = 1 π~ +∞ ∫ −∞ Ψ(x+y)Ψ∗(x−y)ei2py/~dy. (2.46) Its physical meaning is straight forward: although the Wigner function itself can have negative values, its marginals are always positive and correspond to probability distributions. To find the quadrature component distribution
p(x, θ) along an angle θ one simply integrates the Wigner function along a direction perpendicular to θ [100]:
p(x, θ) =
+∞
∫
−∞
W(xcosθ−ysinθ, xsinθ+ycosθ)dy. (2.47) On the other hand, when the probability functionsp(x, θ) for all angles within a π-interval are known, the Wigner function can be calculated as [100]
W(q, p) = 1 4π2 +∞ ∫ −∞ dx +∞ ∫ −∞ dη θ∫0+π θ0
dθ p(x, θ)|η|exp(iη(x−qcosθ−psinθ)).
(2.48) This is the inverse Radon transform [101], which is well known from classical tomography (for example from image reconstruction in X-ray tomographs). It was for example employed to reconstruct the Wigner function of squeezed light using optical homodyne detection [102].
The continuous Wigner function, as described above, is defined in a 2- dimensional, continuous variable phase space. To describe a quantized spin ‘living’ on the Bloch sphere, one generally needs to utilize a discrete Wigner formalism [103, 104]. However, for large atom numbers and coherent or mildly squeezed spin states (as it is the case in our experiment), the Bloch sphere can locally be approximated by a plane and the spin components as continuous variables (see figure 2.7) so that equation (2.48) is suitable to reconstruct the approximate Wigner functions.
After we have produced the squeezed spin state we can thus not only measure ∆S2
θ,min but the probability distribution p(Sθ) along many angles θ
in the yz-plane and use this information to reconstruct the Wigner function
W(Sy, Sz) = 1 4π2 +∞ ∫ −∞ dSθ +∞ ∫ −∞ dη θ∫0+π θ0 dθ p(Sθ)|η|exp (
iη(Sθ−Sycosθ−Szsinθ)
) .
Spin squeezing theory
z
y
Figure 2.7: Local Bloch sphere approximation. For large atom num- bers and if the spin state does not ‘wrap around’ the Bloch sphere too much, the Bloch sphere can be locally approximated by a tangent plane and the spin components as continuous variables. In this plane the Wigner function can be reconstructed using the inverse Radon transformation (2.49).
In practice we measure p(Sθ) only for a discrete set of angles, with limited
atom number resolution (thus limited resolution of Sθ), and with a limited
amount of data per angle. This imposes limits on the reconstruction accuracy which will be discussed in chapter 4.
Quantum state tomography is of interest because it gives access to mea- sures of entanglement, such as the quantum Fisher information [105, 106], which characterize a more general class of states (including states withξ2 >
Chapter 3
Experimental setup
Our experimental setup is similar to previous and existing atom chip exper- iments in the groups of Jakob Reichel and Philipp Treutlein. Many details can therefore be found in the PhD theses of previous group members. The fabrication and characterization of our microwave atom chip are covered in Philipp Treutlein’s [49] and Pascal B¨ohi’s [45] theses. The laser and vacuum systems are treated in great detail in the diploma thesis of Johannes Hof- frogge [107], who built them together with myself. Pascal B¨ohi also describes the computerized control of our experiment and the hardware used to gener- ate radiofrequency and microwave currents and radiation in our experiment. For completeness, I briefly cover all of these topics. The main part of this chapter is dedicated to the absorption imaging system used to achieve the high atom number resolution, needed to experimentally demonstrate spin squeezing. In the last part, I present a typical experimental sequence for the production of mesoscopic BECs and lifetime measurements in the trap used for the squeezing experiments.