Atomic Spin-States

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)

= (|α_x|²+|α_y|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 ˆS_i where i is the measurement basis. The observable ˆS_x 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.

used with individual atoms and capital letters are associated with collective variables.

3.3.1 Cesium

In our experiments, our alkali atom of choice is cesium. Throughout this thesis, we exclu- sively refer to the only stable isotope of cesium: ¹³³Cs with 55 protons and 78 neutrons.

The choice of cesium over other alkali atoms that can be trapped and cooled is mostly historical: other Caltech groups have used it and we have available equipment and diode lasers at the appropriate wavelength. Here I summarize a few of the properties of cesium, while leaving the details to other works, most notably [84]. The cesium atom consists of a single valence electron with spin s= 1/2 and a nucleus of spin i= 7/2. Throughout most of this thesis we consider the D2 optical transition between the ground state with orbital momentum l = 0 and the excited state with l = 1 and j = l+s = 3/2 (as opposed to the D1 line with j = l−s = 1/2). The hyperfine levels corresponding to this transition are displayed in figure 3.1. This fine structure splitting corresponds to a wavelength of 852 nm, which is the coarse setting for all of our trapping and probing lasers. Now consider the total angular momentum

ˆf = ˆs⊗1ˆl⊗i+ ˆ1s⊗ˆl⊗1i+ ˆ1s⊗lˆi (3.53)

where ˆs, ˆl, and ˆi are respectively the electron spin, orbital angular momentum, and the nuclear spin. For the ground state we have two possible spins f = 7/2±1/2 = 3,4 (each with 2f + 1 magnetic sublevels), these ground states have a hyperfine splitting of exactly 9.192631770 GHz, which defines the unit of time. For the D2 line there are four excited levels f⁰ = 2,3,4,5 (each with 2f⁰+ 1 sublevels). Thus to simulate the entire ground and excited state manifold, one needs to keep track of (2·3 + 1) + (2·4 + 1) = 16 ground state sublevels and (2·2 + 1) + (2·3 + 1) + (2·4 + 1) + (2·5 + 1) = 32 excited state sublevels.

We represent the internal state of the atom in terms of the (Zeeman degenerate) atomic hyperfine ground states,|f, mi, and excited states,|f⁰, m⁰i. Heref andf⁰ are the total spin quantum numbers for the ground and excited hyperfine levels respectively whilem andm⁰ are their projections on the z-axis. That is to say|f, mi are eigenstates of the total atomic angular momentum defined above. The quantum numbers, f, and m, are defined in the

usual manner,

ˆf²|f, mi = ~²f(f+ 1)|f, mi (3.54)

fˆz|f, mi = ~m|f, mi. (3.55)

The projector onto the ground state f is given by Pˆ_f = X

mf

|f, m_fihf, m_f| (3.56)

and the projector onto the excited state f⁰ is given by Pˆ_f⁰ = X

m⁰_f

|f⁰, m⁰_fihf⁰, m⁰_f|. (3.57)

Summing over all ground state hyperfine levels gives the ground state projector Pˆg = X

f

Pˆf (3.58)

and summing over all excited state hyperfine levels gives the excited state projector Pˆ_e = X

f⁰

Pˆ_f⁰ (3.59)

and adding these two together gives the identity ˆI = ˆPe+ ˆPg.

We leave a discussion of the fine and hyperfine energy level splittings to chapter 5. The spectrum and many other properties of cesium can be found in [84].

3.3.2 Ground State Spin Operators

In most of our experiments, the atomic population is primarily among the sublevels of one of the ground states, usually f = 4. The excited states are populated transiently via the probing process as discussed in subsequent chapters, and this will eventually lead to state decay to both ground state levels. However, at any one time all of the spins are in one level, therefore we restrict the following analysis to a single ground state spin of sizef. The

Figure 3.1: CesiumD₂ transition levels as described in the text. The probe and pump laser configurations are also shown.

individual spin vector operator for an atom labelediis denoted

ˆf⁽ⁱ⁾= [ ˆf_x⁽ⁱ⁾,fˆ_y⁽ⁱ⁾,fˆ_z⁽ⁱ⁾] (3.60)

and these spin operators obey the usual commutation relations hfˆ_x,fˆ_yi

= i~fˆ_z (3.61)

hfˆ_z,fˆ_xi

= i~fˆ_y (3.62)

hfˆ_y,fˆ_zi

= i~fˆ_x. (3.63)

In terms of these individual operators, we now define the collective spin vector operator as the sum of all of the individual operators

Fˆ =

N

X

i

ˆf⁽ⁱ⁾= [ ˆF_x,Fˆ_y,Fˆ_z] (3.64)

and these operators satisfy the same commutation relations as above.

In fact, as is clear from above, the “individual spin” is individual from the atomic

perspective, but also collective because it is composed of the electron spin, orbital and nuclear spin degrees of freedom. Thus, for larger than spin-1/2 states, like the f = 4 used here, one can speak of spin-squeezing an individual atom, which corresponds to entangling the atom’s inner components.

3.3.3 Spin Uncertainty Relations

Now we translate the commutation relations to uncertainty relations via the usual analysis.

From [87], for two operators ˆA and ˆB we have the general uncertainty relation

h∆ ˆA²ih∆ ˆB²i ≥ |h∆ ˆA∆ ˆBi|², (3.65) which can be rewritten as

h∆ ˆA²ih∆ ˆB²i ≥ 1

4|h[ ˆA,B]i|ˆ ²+1

4|h{∆ ˆA,∆ ˆB}i|². (3.66) Usually the cross-correlation terms (diagonals of the covariance matrix) can be ignored, but we include them here for the sake of completeness. In the case of the collective spin operators, this leads to the uncertainty relations

h∆ ˆF_x²ih∆ ˆF_y²i ≥ ~²

4 |hFˆzi|²+1

4|h{∆ ˆFx,∆ ˆFy}i|² (3.67) h∆ ˆF_y²ih∆ ˆF_z²i ≥ ~²

4 |hFˆxi|²+1

4|h{∆ ˆFy,∆ ˆFz}i|² (3.68) h∆ ˆF_z²ih∆ ˆF_x²i ≥ ~²

4 |hFˆyi|²+1

4|h{∆ ˆFz,∆ ˆFx}i|². (3.69) Below we use these relations to get a sense of the moment distribution on the Bloch sphere of several representative states, including spin-squeezed states.

3.3.4 Symmetric States and Moments

3.3.4.1 Dicke States

The Dicke states [88, 68], |F, mi are simply the eigenstates of ˆFz whether or not it is a collective operator

Fˆz|F, mi=~m|F, mi. (3.70)

In this section we will calculate some moments of these states for future reference. The collective spin raising and lowering operators are useful for this purpose and are defined as

Fˆ±= ˆF_x±iFˆ_y (3.71)

with the inverse being

Fˆx = ( ˆF++ ˆF−)/2 (3.72)

Fˆy = −i( ˆF+−Fˆ−)/2. (3.73)

The property that makes ˆF± so useful is that it acts as a raising/lowering operator Fˆ±|F, mi=~

pF(F+ 1)−m(m+ 1)|F, m±1i, (3.74)

which can be used in expectation value calculations. Now the first-order moments of the Dicke states are

hF, m|Fˆz|F, mi = ~m (3.75)

hF, m|Fˆx|F, mi = hF, m|Fˆy|F, mi= 0 (3.76) and the second-order moments are

hF, m|Fˆ_z²|F, mi = ~²m² (3.77)

hF, m|Fˆ_x²|F, mi = hF, m|Fˆ_y²|F, mi= ~²

2 [F(F+ 1)−m²]. (3.78) HerehFˆ_x²iis calculated by expanding ˆF_xinto the raising/lowering operators and using their properties to simplify.

From these moments, we see that we can visualize each Dicke state m as having a per- fectly determined z-component ~m and a completely undetermined transverse component while still respecting the extent of the total sphere. Them= 0 state lives along the equator of the sphere, while the m = ±F states represent the polar caps, which still have finite lateral extent. The single atom Dicke states corresponding to the sublevels of the f = 4 ground state of cesium are depicted visually in figure 3.2.

Figure 3.2: Graphical depiction of Dicke States,|f, mi_z, for the cesiumf = 4 ground state.

The highlighted uppermost state is a single-atom coherent spin-state.

We define these extended Dicke states with m =F as the coherent spin-states (CSS), which obey

hF, F|Fˆz|F, Fi = ~F (3.79)

hF, F|Fˆ_x²|F, Fi = hF, F|Fˆ_y²|F, Fi= ~²

2F (3.80)

h{∆ ˆFx,∆ ˆFy}i = 0. (3.81)

The CSS is a minimum uncertainty state because it satisfies the equality of the uncertainty inequality

h∆ ˆF_x²ih∆ ˆF_y²i = ~²

4 |hFˆzi|²+1

4|h{∆ ˆFx,∆ ˆFy}i|². (3.82) As opposed to all other Dicke states, the CSS is an unentangled (separable) state, with all spins pointing in the same direction, i.e., |CSSi =|F, Fi=| ↑ · · · ↑i. The entanglement of non-CSS Dicke States is simply seen by the two spin example in chapter 1 and discussed more thoroughly in chapter 4.

As discussed in the introduction, the CSS can be turned into a spin-squeezed state if the x variance can be reduced (squeezed) while the y variance is increased (antisqueezed)

to maintain the validity of the uncertainty relation. Below we discuss the definition of spin-squeezing in more detail.

The two single atom states that are easiest to prepare via optical pumping are the individual CSS |f, fi, and the individual m= 0 state |f, mi. From the above relations we can see that the difference in variance one expects to see, for f = 4, is

hf, f|fˆ_x²|f, fi = ~²

2 f =~²2 (3.83)

hf,0|fˆ_x²|f,0i = ~²

2 f(f+ 1) =~²10. (3.84)

Another important reference state is the completely mixed state defined by incoherently populating all of the sublevels equally

ˆ ρ=X

m

1

2f+ 1|f, mihf, m|. (3.85)

With this state, one can then show that the mean square of any moment is

hfˆ_i²i = Tr[ ˆf_i²ρ] =ˆ 2~² 2f+ 1

f

X

m=1

m² (3.86)

and for f = 4 this gives

hfˆ_i²i = Tr[ ˆf_i²ρ] =ˆ ~²20/3 =~²6.67. (3.87) From above, we see that this variance is intermediate between that of a coherent state and them= 0 state as expected.

3.3.4.2 Gaussian States

The Hilbert space describing an ensemble of N spins of size f is ridiculously large, and without any constraints the state vector would consist of (2f + 1)^N complex elements.

When we deal with an ensemble of more than a billion spins, there is no hope of using the full state space in our description. One option is to restrict the state description to the symmetric subspace, as discussed in the next chapter, which only has size (2N f+1), however this is still a potentially large number. Fortunately, we can use a Gaussian description of

the state under certain circumstances where we only have to keep track of a few moments of the distribution. In general, this kind of simplification of the Hilbert space description is a kind of “model reduction,” used to make simulations and estimation more efficient by removing unnecessary parts of the full model.

For example, consider the collective CSS just described with a large number of spins.

If this state is aligned along z and we represent it in a perpendicular basis, x or y, then the populations among the collective sublevels will be distributed in a Gaussian manner, centered at zero with a variance given by hF, F|Fˆ_x²|F, Fi = ^~₂²F. Thus the width of this distribution grows as the square root of the number as the collective Bloch sphere grows linearly with the number. The more atoms we have, the more localized the state is on the sphere, and the “flatter” the space on which it lives. Formally, these ideas can be made precise via the Holstein-Primakoff transformation [89], which is commonly used in the con- densed matter physics literature and makes it possible to derive the Gaussian approximation as an expansion in 1/N.

Many other papers, such as [39, 86], make use of this approximation and divide by the

“classical” length of the vector to create effective “position” and “momentum” operators Fˆ_x

q hFˆ_zi

→Xˆ (3.88)

Fˆ_y q

hFˆ_zi

→P ,ˆ (3.89)

which then follows the usual commutation and uncertainty relationship. In this thesis, we make this analogy in appendix D to simulate a quantum harmonic oscillator with a field along the same direction as the CSS.

These concepts are also discussed in the context of measurement and control in both chapter 9 and chapter 10. In section 9.7, I describe how to expand the Gaussian description naturally in terms of cumulants, which stochastically evolve due to the quantum measure- ment in an interdependent way.

So far we have only discussed the Dicke states and the CSS, which is an unentangled Dicke state. As seen in figure 1.1, the spin-squeezed state is merely a perturbation of the CSS for small degrees of squeezing and is also a Gaussian state. Thus, for much of the work described in this thesis where the spin-squeezing is relatively small, the Gaussian

approximation will be used.

3.3.5 Spin-Squeezed States

The notion of spin-squeezing has been introduced via the uncertainty relations in chapter 1 and the cartoon in figure 1.1. In this section, we technically define a spin-squeezed state and analyze some of its properties.

For a collective spin-state polarized in the x-direction (hFˆyi = hFˆzi = 0) and with minimum transverse uncertainty in the z-direction, the spin-squeezing parameterξis defined as

ξ² = 2Fh∆ ˆF_z²i

hFˆ_xi² . (3.90)

If a state hasξ² <1, then that state is referred to as aspin-squeezed state(SSS). Notice that the coherent spin-state satisfiesξ² = 1. The spin-squeezing parameter was introduced in the context of improving the performance of atomic clocks in [4, 2, 5]. In the next section, we prove that spin-squeezed states (composed of spin-1/2 particles) are necessarily entangled.

3.3.5.1 Spin-Squeezing Implies Entanglement (Spin-1/2)

Following [6], we now show that any state composed of spin-1/2 particles with ξ² < 1 is nonseparable (entangled). First we define some of the spin-operators for a spin-1/2 particle.

The Pauli angular momentum operators for the individual spins are fˆ_x = 1

2(|0ih1|+|1ih0|) (3.91)

fˆ_y = i

2(|0ih1| − |1ih0|) (3.92)

fˆ_z = 1

2(|1ih1| − |0ih0|). (3.93)

The joint/collective angular momentum variables are the sums of the microscopic variables:

Fˆi = PN

n=1fˆ_i⁽ⁿ⁾. The total angular momentum vector will be denoted ˆF = [ ˆFx,Fˆy,Fˆz].

With the definition ˆFij =PN

n |ii_nhj|_n,i, j= 0,1 we see that Fˆx = 1

2( ˆF01+ ˆF10) (3.94)

Fˆ_y = i

2( ˆF₁₀−Fˆ₀₁) (3.95)

Fˆz = 1

2( ˆF00−Fˆ11) (3.96)

N = Fˆ00+ ˆF11. (3.97)

Now, to begin the proof, we use the fact that any separable state can be written in the form

ˆ ρ=X

k

P_kρˆ⁽¹⁾_k ⊗ · · · ⊗ρˆ^(N)_k . (3.98) We can then rewrite the variance for a separable state into the useful form

hFˆ_z²i = Tr ˆ ρFˆ_z²

(3.99)

= Tr



 X

k

P_kρˆ⁽¹⁾_k ⊗ · · · ⊗ρˆ^(N)_k

N

X

n=1

fˆ_z⁽ⁿ⁾

!2

 (3.100)

= X

k

P_k

* _N X

n=1

fˆ_z⁽ⁿ⁾²+

N

X

n6=m

fˆ_z⁽ⁿ⁾fˆ_z^(m) +

k

(3.101)

= X

k

P_k



 N

4 +

* _N X

n6=m

fˆ_z⁽ⁿ⁾fˆ_z^(m) +

k



 (3.102)

= N

4 +X

k

P_k hFˆzi²_k−

N

X

n=1

hfˆ_z⁽ⁿ⁾i²_k

!

. (3.103)

The relations P

kP_k = 1, ˆf_i² = ¹₄, and hPN

n6=mfˆz⁽ⁿ⁾fˆz^(m)i = PN

n6=mhfˆz⁽ⁿ⁾ihfˆz^(m)i = hFˆ_zi²− PN

n=1hfˆz⁽ⁿ⁾i² have all been used, and the last equation follows from separability. So the variance is now

h∆ ˆF_z²i = hFˆ_z²i − hFˆ_zi² (3.104)

= N

4 +X

k

Pk hFˆzi²_k−

N

X

n=1

hfˆ_z⁽ⁿ⁾i²_k

!

− hFˆzi². (3.105)

This is equality 1.

Another conditionhfˆx⁽ⁿ⁾i²_k+hfˆy⁽ⁿ⁾i²_k+hfˆz⁽ⁿ⁾i²_k≤ ¹₄ implies that N

4 −X

k

P_kX

n

hfˆ_z⁽ⁿ⁾i²_k ≥ X

k

P_kX

n

hfˆ_x⁽ⁿ⁾i²_k+hfˆ_y⁽ⁿ⁾i²_k

. (3.106)

This is inequality 1.

Now we use the Cauchy-Schwarz inequality to derive two more useful inequalities. First notice that

hFˆ_ii² = X

k

P_khFˆ_ii_k

!2

(3.107)

= X

k

pP_k p

P_khFˆ_ii_k

!2

(3.108)

≤ X

k

Pk

! X

k

PkhFˆii²_k

!

(3.109)

= X

k

P_khFˆ_ii²_k. (3.110)

This is inequality 2. Now we continue this inequality and use Cauchy-Schwarz again to get hFˆii² ≤ X

k

PkhFˆii²_k (3.111)

= X

k

P_k X

n

hfˆ_i⁽ⁿ⁾i_k

!2

(3.112)

= X

k

P_k X

n

1× hfˆ_i⁽ⁿ⁾i_k

!2

(3.113)

≤ X

k

P_k X

n

1

! X

n

hfˆ_i⁽ⁿ⁾i²_k

!

(3.114)

= X

k

PkNX

n

hfˆ_i⁽ⁿ⁾i²_k. (3.115)

This is inequality 3.

Finally, we combine these relations to derive an inequality for theξ²of a separable state:

ξ² = Nh∆ ˆF_z²i

hFˆxi²+hFˆyi² (3.116)

= N

N 4 +P

kP_k

hFˆ_zi²_k−PN

n=1hfˆz⁽ⁿ⁾i²_k

− hFˆ_zi²

hFˆxi²+hFˆyi² (3.117)

≥ N

N 4 +P

kPk

hFˆzi²_k−PN

n=1hfˆz⁽ⁿ⁾i²_k

− hFˆzi² P

kP_kNP

n(hfˆ_x⁽ⁿ⁾i²_k+hfˆ_y⁽ⁿ⁾i²_k) (3.118)

≥

N 4 +P

kP_k

hFˆ_zi²_k−PN

n=1hfˆ_z⁽ⁿ⁾i²_k

−P

kP_khFˆ_zi²_k P

kP_kP

n

hfˆ_x⁽ⁿ⁾i²_k+hfˆ_y⁽ⁿ⁾i²_k (3.119)

≥

N 4 −P

kPkPN

n=1hfˆz⁽ⁿ⁾i²_k P

kPkP

n

hfˆx⁽ⁿ⁾i²_k+hfˆy⁽ⁿ⁾i²_k (3.120)

≥ 1 (3.121)

where equality 1 is used in 3.117, inequality 3 is used in 3.118, inequality 2 is used in 3.119, and inequality 1 is used in 3.121. Thus separability implies the absence of spin-squeezing.

Conversely, spin-squeezing implies entanglement.

3.3.5.2 Heisenberg Limit of Spin-Squeezing

Optical squeezed states can in principle be infinitely squeezed, although this takes an infinite amount of energy. For a fixed number of atoms, spin-squeezed states cannot be infinitely squeezed because of the fact that the collective Bloch sphere is finite in extent, and if one squeezes too much then the distribution begins to “wrap” around the sphere.

We can describe this quantitatively as follows. Assume the sample is polarized along x: hFˆxi = F = ^N₂. By standard angular momentum commutation relations we know h∆ ˆF_y²ih∆ ˆF_z²i ≥ ^hFˆx₄ⁱ². Soh∆ ˆF_z²i ≥ ^F2

4h∆ ˆF_y²i. But because the variance cannot be bigger than the entire Bloch sphere we have h∆ ˆF_y²i ≤ Fˆ², which implies h∆ ˆF_z²i ≥ ¹₄. This gives us ξ² = ^{Nh∆ ˆFz2i}

hFˆxi²+hFˆyi² ≥ ^N/4_F2 = _N¹. This is the so called Heisenberg limit of spin-squeezing.

3.3.5.3 Spin-Squeezing Under Particle Loss

Here we analytically investigate the spin-squeezing parameter as particles are removed. We find that the spin-squeezing parameter of a state withNrspins remaining (ξ_N²_r) is dependent

on the initial squeezing parameter (ξ_N²) and polarization of the state with all spins remaining in the following way

ξ_N²_r =ξ₁²+ (ξ²_N−ξ₁²)Nr−1

N −1 (3.122)

whereξ²₁ ≡N²/(4hFˆzi²_N). So the loss of squeezing is only linear in the number of particles that are traced out. Hence the entanglement is rather robust, especially as compared to the extreme fragility of cat states, described below, to particle loss. The idea of entangled states that are robust to particle loss is also discussed in chapter 4.

3.3.5.4 Squeezing Generation

As mentioned in the introduction, there are many ways to imagine creating a spin-squeezed state. Numerically, the easiest way to spin-squeeze a coherent state is to apply either a twisting Hamiltonian ( ˆH = ˆF_x²) or a countertwisting Hamiltonian ( ˆH =−i( ˆF₊² −Fˆ₋²)) [2].

The latter is designed to ensure that the squeezed axis stays constant in time and does not

“twist” out of alignment.

On the experimental side, one can either apply controlled Hamiltonians to entangle the internal degrees of freedom as has been achieved for several ions in a trap [40]. One could also create the correlations in the atoms via absorption of a beam of squeezed light [39].

Finally, one could use QND measurement to prepare spin-squeezed states conditionally [16, 37, 36] as we mostly discuss in this thesis.

3.3.6 Cat States

As we have mentioned, the Hilbert space describing the collective spin-states is a large place. So far we have only described a very small but useful subsample of the vast number of potential states. Other states in the symmetric subspace we have yet to consider include the GHZ state

|GHZi= (| ↑↑ · · · ↑i+| ↓↓ · · · ↓i)/√

2, (3.123)

which is alternately known as a cat state, or an EPR state. One can also prepare similar coherent superpositions of CSS states but with a smaller angle between the two [90]. The entanglement properties of these states and the states mentioned previously are analyzed in the next chapter.

Chapter 4

Entanglement of Collective Spin-States

This chapter provides a highly abbreviated summary of [17], which discusses the efficient calculation of entanglement measures for symmetric spin-states. For the sake of brevity, I left out most of the details and proper references, with the aim of getting across the few core ideas. Although this work is interesting from a theoretical entanglement perspective, it is largely independent of the rest of this thesis and can safely be skipped or read separately.