We begin by describing the electric field representation of the probe beam with two trans- verse polarization modes, and then we derive several polarization operators and some of their properties. The coherent state of the probe beam is specified and related to the polarimeter detection described in subsequent chapters.
3.2.1 Electric Field Representations
The classical electric field describing a single spatial mode of light with two orthogonal polarization components σ= 1,2 is denoted
E=E X
σ=1,2
σeσexp[iκz−iωt] + c.c. (3.1)
We always consider the beams to be propagating along the z-direction. The power in this beam is then given by
P/A= 0cE2
2 =0cE2. (3.2)
The quantized version of this field [68] is defined as Eˆ =p
~g X
σ=1,2
ˆ
aσeσexp[iκz−iωt] + h.c. (3.3)
where g = ω/(20V) and V is the volume of the mode. (Note that, in this thesis, I use a notation where vectors are given boldface and operators are given hats.) The ˆaσ are the annihilation operators satisfying the usual commutation relation [ˆai,ˆa†j] = δij. Unless otherwise noted, we will ignore the spatial dependence and only consider ˆE at the spatial locationz= 0. However, this simplification would need to be removed in the consideration of spatially extended scattering media imparting significant polarization rotation (on the order of radians).
In quantum optics, it is often convenient to move into the frame rotating at the optical frequency to remove the time dependence of a particular operator. Explicitly, we transform
the field operator as
EˆR = exp[iωt(ˆa†1ˆa1+ ˆa†2ˆa2)] ˆEexp[−iωt(ˆa†1ˆa1+ ˆa†2ˆa2)] (3.4)
= Eˆ(−)+ ˆE(+). (3.5)
where
Eˆ(−) = p
~gh ˆ
a†1e∗1+ ˆa†2e∗2i
(3.6) Eˆ(+) = p
~g[ˆa1e1+ ˆa2e2]. (3.7) From now on we will assume the rotating frame is being used and remove the R subscript from ˆE.
3.2.2 Polarization Basis
Now we discuss ways of representing the polarization of the optical beam propagating along the z-direction. We start in the basis given by ex and ey. In general one can represent the polarization of the light in an elliptical superposition of these elements [68], but we choose to limit ourselves to three distinct basis pairs. We denote byex0 andey0 the basis pair that is also real, but at 45 degrees relative to ex and ey. The circular basis is then defined as the complex superposition of the real basis and denoted by e+ and e−. More specifically, these vectors are related to each other through the following transformations:
ex = √1
2(ex0 −ey0) = 1
√2(−e++e−) (3.8)
ey = √1
2(ex0 +ey0) = i
√2(e++e−) (3.9)
ex0 = √1
2(ex+ey) = 1
2((−1 +i)e++ (1 +i)e−) (3.10) ey0 = √1
2(−ex+ey) = 1
2((1 +i)e++ (−1 +i)e−) (3.11) e+ = √1
2(−ex−iey) = 1
2((−1−i))ex0 + (1−i)ey0) (3.12) e− = √12(ex−iey) = 1
2((1−i)ex0 + (−1−i)ey0). (3.13)
3.2.3 Spherical Basis and Tensors
The basisex0 and ey0 is a trivial real rotation of the original basisex and ey, however the circular basis is somewhat less intuitive, so we review some of its properties here. When the circular basis is combined with the propagation vector along z, we get the spherical basis
e+ = −(ex+iey)/√
2 (3.14)
e− = (ex−iey)/√
2 (3.15)
ez = ez, (3.16)
which is often used in atomic physics due to its symmetry properties [84]. When making the associations + (q = +1),−(q =−1), and z(q = 0) the elements of the spherical basis have the properties
e∗q =e−q(−1)q (3.17)
eq·e∗q0 =δq,q0. (3.18)
For an arbitrary vector A we have Aq = eq·A so that A = P
qAqe∗q =P
q(−1)qAqe−q. Due to the elementez this basis describes any field for any propagation direction and not just a beam propagating along z.
3.2.4 Stokes Representation
Given these basis pairs, we define the Stokes operators (also known as Schwinger boson operators) as
Sˆx = 1
2(ˆa†yˆay−aˆ†xaˆx) (3.19) Sˆy = 1
2(ˆa†y0ˆay0−aˆ†x0aˆx0) (3.20) Sˆz = 1
2(ˆa†+ˆa+−aˆ†−ˆa−) (3.21) Sˆ0 = 1
2(ˆa†yˆay+ ˆa†xaˆx). (3.22) Clearly, ˆSx measures in one linearly polarized basis, ˆSy measures in the other linearly polarized basis, and ˆSz measures in the circular basis. Each of these components can be
measured in a polarimeter using at most two waveplates and a polarizing beamsplitter as described below. Note that if two polarizations are perpendicular in real space (e.g., along ex andey), they will be represented by vectors that are pointing opposite directions on the Stokes sphere. Using the above transformations of basis, one can show that these operators can be defined in terms of the other basis pairs as
Sˆx = 12(ˆa†yˆay−aˆ†xˆax) = 1
2(ˆa†y0ˆax0 + ˆa†x0aˆy0) = 1
2(ˆa†+ˆa−+ ˆa†−ˆa+) (3.23) Sˆy = 12(−ˆa†yˆax−ˆa†xˆay) = 1
2(ˆa†y0ˆay0−aˆ†x0aˆx0) = i
2(ˆa†−ˆa+−aˆ†+ˆa−) (3.24) Sˆz = 2i(ˆa†yˆax−aˆ†xaˆy) = i
2(ˆa†y0ˆax0 −ˆa†x0aˆy0) = 1
2(ˆa†+ˆa+−aˆ†−ˆa−) (3.25) Sˆ0 = 12(ˆa†yˆay+ ˆa†xaˆx) = 1
2(ˆa†y0ˆay0+ ˆa†x0aˆx0) = 1
2(ˆa†+ˆa++ ˆa†−ˆa−). (3.26) Using [ˆai,ˆa†j] = δij one can show that the Stokes operators have the same commutation relations as angular momentum operators
hSˆx,Sˆy
i
= iSˆz (3.27)
hSˆy,Sˆz
i
= iSˆx (3.28)
hSˆz,Sˆx
i
= iSˆy (3.29)
hSˆi,Sˆ0
i
= 0. (3.30)
All of the usual angular momentum relations immediately follow. Other means of describing the polarization state, e.g., the Jones vector, often assume further constraints and can be found in [85].
3.2.5 Arbitrary Vector Operator Rotations
Imagine a polarized beam of light propagating through, or reflecting off, some material. If the beam is not attenuated, then the material will rotate the polarization of the beam about some, possibly arbitrary, direction on the Stokes sphere. Thus we can represent the unitary behavior of the evolution (whether it be due to a waveplate, a mirror, a piece of glass, cold atoms, etc.) simply by a rotation vector. In this section, we are interested in evaluating the operation of rotating a Stokes vector about an arbitrary direction by an arbitrary angle.
Consider the rotation of the vector operator Sˆ =
hSˆx,Sˆy,Sˆz
i
(3.31) in Cartesian coordinates about an arbitrary direction n = [γx, γy, γz]/γ by the angle γ = q
γx2+γy2+γz2. Using the rotation vector γ = γn = [γx, γy, γz], this rotation can be represented in the Heisenberg picture as
Sˆi0 = ˆUSˆiUˆ† (3.32)
where
Uˆ = exp[−iSˆ·γ] = exp[−i(γxSˆx+γzSˆz+γzSˆz)]. (3.33) The ˆSi0 can be derived explicitly using the following equation for the arbitrary rotation of any vector
Sˆ0i = (ˆS·i) cosγ+ (n·i)(n·S)(1ˆ −cosγ) +
(n×i)·Sˆ
sinγ. (3.34)
(See, for example, eq. (3.16) ofhttp://www.phys.vt.edu/~mizutani/quantum/rotations.
pdf.) Expanding and rearranging terms we get
Sˆx0 = Sˆx γx2
γ2(1−cosγ) + cosγ
+ ˆSy
γxγy
γ2 (1−cosγ) + γz
γ sinγ
+ ˆSz
γxγz
γ2 (1−cosγ)−γy
γ sinγ
(3.35) Sˆy0 = Sˆx
γyγx
γ2 (1−cosγ)−γz γ sinγ
+ ˆSy γy2
γ2(1−cosγ) + cosγ
!
+ ˆSz
γyγz
γ2 (1−cosγ) +γx γ sinγ
(3.36) Sˆz0 = Sˆx
γzγx
γ2 (1−cosγ) +γy
γ sinγ
+ ˆSy γzγy
γ2 (1−cosγ)−γx
γ sinγ
+ ˆSz γz2
γ2(1−cosγ) + cosγ
. (3.37)
As an example, a half-waveplate with its primary axis rotated by an angle φ from the x-axis will correspond to a rotation vector of
γ=π[cos(2φ),sin(2φ),0]. (3.38) Similarly a quarter-waveplate with its primary axis rotatedφfrom the x-axis will correspond to a rotation vector of
γ= π
2[cos(2φ),sin(2φ),0]. (3.39)
3.2.6 Fock States and Coherent States
We work with the convention that Fock states are represented as |ni wheren are integers and coherent states are represented as|αiwhereα are complex numbers. The type of state should usually be clear by context so distinguishing subscripts are not used.
Fock states of a particular mode have the properties [68]
ˆa|ni = √
n|n−1i (3.40)
ˆ
a†|ni = √
n+ 1|n+ 1i (3.41)
ˆ
a†ˆa|ni = n|ni. (3.42)
For the same mode, we define the coherent state [68] in terms of the Fock states as
|αi= exp −|α|2/2
∞
X
n=0
αn
√
n!|ni (3.43)
and it quickly follows from the above that we have
a|αiˆ = α|αi (3.44)
hα|ˆa† = hα|α∗ (3.45)
hα|ˆa†a|αiˆ = |α|2. (3.46)
The power represented in a beam sliced into modes of length cdt (where c is the speed of light) is
P = hˆa†aiˆ ~ω/dt=|α|2~ω/dt (3.47) For computational reasons we will sometimes take the time slice dt small such thatα 1 and expand the coherent state to first-order
|αi ≈(1− |α|2/2)(|0i+α|1i). (3.48) For an arbitrarily polarized beam propagating along one direction we will represent the optical state as the tensor product
|Φi=|α1i1⊗ |α2i2 (3.49)
where 1 and 2 label orthogonal polarization modes. One can then use the representations of the ˆSi operators and the properties of the coherent state above to calculatehSˆiiin terms
of the αi. The power of this beam is then simply the sum of the powers in each mode P = hˆa†xˆax+ ˆa†yˆayi~ω/dt (3.50)
= 2hSˆ0i~ω/dt (3.51)
= (|αx|2+|αy|2)~ω/dt. (3.52) Throughout this thesis we focus mostly on using coherent states of probe light to create spin-squeezed states. It turns out that one can also discuss the creation of polarization squeezed light by multipassing the probe beam through the unsqueezed atomic ensemble [86], although we do not consider this effect here.
3.2.7 Polarimetry
One can measure any of the above bases or any linear combination thereof (simultaneously with ˆS0) with a fixed combination of a quarter-waveplate, a half-waveplate, a polarizing beamsplitter (PBS), and two independent detectors. The sum current (optical power) will be ˆS0 and the difference will be ˆSi where i is the measurement basis. The observable ˆSx is measured with just a PBS with axes along the directions x and y. The observable ˆSy is measured with a half-waveplate prior to the PBS with primary axis at 45 degrees relative to the PBS (x-y) axes (see figure 1.2). The observable ˆSz is measured with a quarter-waveplate prior to the PBS with primary axis also at 45 degrees relative to the PBS (x-y) axes. We discuss the theory of polarimeters more in section 8.1 and the experimental implementation in section 11.6.