The final link in the chain so far takes us one step further in terms of generalization. Up until this point, the pumped-up $SU(1, 1)$ was designed only for spatial interferometry and thus restricted to local channels. If, however, we are to consider frequency metrology on a BEC, there is no spatial separation between the modes. This allows for non-local channels in the imprint stage, which has been incorporated into the pumped-up $SU(1, 1)$ scheme in recent work by Howl, Fuentes et al. <cit.>. In particular, they have managed to consider not only phase imprints, but also squeezing and mode mixing and therefore a complete set of two-mode Gaussian unitary operators. In <cit.>, they have found an expression in terms of BEC parameters for what would be the sensitivity for the GW amplitude in the quasi- $(1+1)$-dimensional case of the previous section (<ref>)

\begin{equation} \label{eq:pumped-up_SU11_sensitivity}
\Delta h_{+, 0} \geq
\frac{m \pi}{4 \sqrt{2} \hbar}
\frac{1}{\sqrt{\tau N_{phon} N_d}}
\frac{\sqrt{l n} \left(l-n \right)^2}{l^2+n^2}
\frac{\alpha^3}{\theta N_0^2}
\sqrt{\frac{L^7}{a_s^3 t}} \; ,
\end{equation}

where $m$ is the particle mass, $N_d$ the number of detectors, $(l, n)$ the phonon mode numbers. $L$ is the length of the trap $\alpha = R/L$ the ratio between its length and width, $\tau$ the integration time and $t$ the duration of a single experiment. Equation (<ref>) will be used at the end of chapter <ref> to estimate the usefulness of the squeeze factor and the parameters derived there.

CHAPTER: SQUEEZING MECHANISM

Once the question of why two-mode phonon squeezing is desirable, one might naturally ask how to achieve it. This is the subject of the following section. One possible answer has its origins again in quantum metrology, albeit this time in a non-relativistic context. More specifically, the ansatz comes from a recent paper by Rätzel et al. <cit.>, who investigated whether it would be possible to detect the influence of a small oscillating mass on a BEC placed next to it. While this was motivated by the potential use of the BEC as a sensing device for other experiments, for instance the superposition of macroscopic quantum systems <cit.>, it was found that one of the effects of the oscillating Newtonian potential sourced by the mass on the phonon states populating the BEC is two-mode squeezing. In the case considered in <cit.>, the degree of squeezing is, of course, limited by the magnitude of the gravitational interaction. If the experimenter's interest, however, lies not in the detection of said potential but in its exploitation, it does not need to be gravitational in nature. Electromagnetic fields, which are already crucial to the trapping mechanism, can be used to create perturbations of the same character as that of an oscillating nearby mass, but orders of magnitude larger, such that the created two-mode squeezing becomes not only detectable, but could possibly even be used to maximize the sensitivity of the GW detector described in <cit.>.

The following chapter will follow the arguments made in <cit.> and start by extending the Hamiltonian for a weakly interacting Bose gas (<ref>) to time-dependent fields while keeping the (also time-dependent) external potential general. In the now familiar Bogoliubov approximation, the Hamiltonian can then be expanded in its fluctuations around the mean field to derive mode equations for the phonons. These are then applied to the situation considered in <cit.>: a Bose gas in an elongated box trap with a time-dependent Newtonian potential. Finally, it is shown that the resulting time evolution for the phononic state contains a squeezing operator.

§ QUANTUM FIELD THEORY ANSATZ

Let us begin by generalizing the Hamiltonian for a weakly interacting Bose gas (<ref>) to account for an external potential that includes a time-dependent perturbation $\mathcal{V} ( \mathbf{r}, t) = \mathcal{V}_0 (\mathbf{r}) + \delta \mathcal{V} ( \mathbf{r}, t)$. Taking only the zero-momentum component of the interaction potential (<ref>), $\hat{H} ( t )$ becomes

Ĥ (t)    =
∫𝐫   ψ̂^†( 𝐫, t)
- ħ^2/2m ∇^2 + 𝒱_0 (𝐫)
ψ̂ ( 𝐫, t)

∫𝐫   ψ̂^†( 𝐫, t)
δ𝒱 ( 𝐫, t)
ψ̂ ( 𝐫, t)

+ U_0/2
ψ̂^†( 𝐫, t) ψ̂^†( 𝐫, t)
ψ̂ ( 𝐫, t) ψ̂ ( 𝐫, t)   .

The aim here is again to take the Bogoliubov approximation and diagonalize the Hamiltonian up to second order in the fluctuations, but only after separating the time evolution due to $\delta \mathcal{V}$ out of the solution for the ground state. In order to do so, consider a solution $\psi_o (\mathbf{r})$ of the stationary Gross-Pitaevskii equation (<ref>) with only the constant part $\mathcal{V}$ of the external potential and with the interaction strength $g$ replaced by $U_0$.

\begin{equation} \label{eq:squeezing_mechanism_stationary_GPE}
\left( - \frac{\hbar^2}{2m} \nabla^2 + \mathcal{V}_0 (\mathbf{r}) + U_0 | \psi_0 (\mathbf{r})|^2 \right) \psi_0 (\mathbf{r}) = \mu \psi_0 (\mathbf{r})
\end{equation}

Neglecting the perturbation to the external potential, the ground state would evolve as $\psi (\mathbf{r}, t) = \psi_0 (\mathbf{r}) \mathrm{e}^{-i \mu t /\hbar}$. However, $\delta \mathcal{V}$ adds a time-dependent change to the chemical potential:

\begin{equation} \label{eq:squeezing_mechanism_chemical_potential_perturbation}
\delta \mu (t) = \frac{
\int \ud \mathbf{r} \; \psi_0^* (\mathbf{r}) \delta \mathcal{V} (\mathbf{r}, t ) \psi_0 (\mathbf{r})
\int \ud \mathbf{r} \; | \psi_0 (\mathbf{r}) |^2
\end{equation}

This definition comes natural: the numerator is the additional energy of the condensate due to $\delta \mathcal{V}$, the denominator normalizes it to the energy per particle. With (<ref>), the actual ground state of the condensate in the presence of a time-dependent perturbing potential evolves as

\begin{equation}
i \hbar \partial_t \psi (\mathbf{r}, t ) =
\left(
- \frac{\hbar^2}{2m} \nabla^2 + \mathcal{V}_0 (\mathbf{r}) + \delta \mu (t) + U_0 | \psi_0 (\mathbf{r})|^2
\right)
\psi (\mathbf{r}, t )
\end{equation}

and therefore

\begin{equation}
\psi (\mathbf{r}, t ) = \psi_0 (\mathbf{r})
\mathrm{e}^{
-\frac{i}{\hbar} \left( \mu t + \int_0^{t'} \ud t' \; \delta \mu (t') \right)
} \; .
\end{equation}

The time evolution of $\psi$ can now be separated out of the full bosonic field to define

\begin{equation}
\hat{\psi}' (\mathbf{r}, t) \equiv \hat{\psi} (\mathbf{r}, t )
\mathrm{e}^{
\frac{i}{\hbar} \left( \mu t + \int_0^{t'} \ud t' \; \delta \mu (t') \right)
} \; .
\end{equation}

In the Heisenberg picture, $\hat{\psi}$ evolves as $-i \hbar \partial_t \hat{\psi} = [\hat{H}, \hat{\psi}]$. $\hat{\psi}$ and $\hat{\psi}'$ both obey the bosonic equal time commutation relations.[
$[\hat{\psi} (\mathbf{r}, t), \hat{\psi}^\dagger (\mathbf{r}', t)] = \delta^{(3)} (\mathbf{r} - \mathbf{r}')$, $[\hat{\psi} (\mathbf{r}, t), \hat{\psi} (\mathbf{r}', t)] = [\hat{\psi}^\dagger (\mathbf{r}, t), \hat{\psi}^\dagger (\mathbf{r}', t)] = 0$
Hence, the evolution of $\hat{\psi}'$ is governed by <cit.>

-i ħ∂_t ψ̂' (𝐫, t)
( -i ħ∂_t + μ+ δμ(t) )
ψ̂ (𝐫, t )
μt + ∫_0^t' t'   δμ(t')

Ĥ (t), ψ̂' (𝐫, t)
+ ( μ+ δμ(t) ) ψ̂' (𝐫, t)

Ĥ (t) - ( μ+ δμ(t) ) N̂(t)
≡Ĥ' (t)

, ψ̂' (𝐫, t)
]   ,

where the number operator $\hat{N}$ was introduced in the second line and its equal time commutation relation with $\hat{\psi}'$ was used in the third line.[
$\hat{N} (t) \equiv \int \ud \mathbf{r} \, \hat{\psi}'^\dagger (\mathbf{r}, t) \hat{\psi}' (\mathbf{r}, t), \qquad [\hat{N} (t), \hat{\psi}' (\mathbf{r}, t)] = -\hat{\psi}' (\mathbf{r}, t)$

$H' = \hat{H} - (\mu + \delta \mu) \hat{N}$ is the grand canonical Hamiltonian. Taking the Bogoliubov approximation $\hat{\psi}' = \psi_0 + \delta \hat{\psi}$, $\hat{H}'$ expands to[
For the sake of readability, we drop the variables ($\mathbf{r}, t)$ for now.

Ĥ' [ψ̂']
-ħ^2/2m ∇^2 + 𝒱_0 + U_0/2 |ψ_0|^2
) ψ_0

-ħ^2/2m ∇^2 + 𝒱_0 + U_0 | ψ_0|^2
+ h.c.

-ħ^2/2m ∇^2 + 𝒱_0 + 2 U_0 | ψ_0|^2
+ U_0/2
ψ_0^2 δψ̂^†2 + ψ_0^*2 δψ̂^2

U_0 (
ψ_0 δψ̂^†2 δψ̂ + ψ_0^* δψ̂^†δψ̂^2
+ U_0/2 δψ̂^†2 δψ̂^2

|ψ_0|^2 δ𝒱
+ (
ψ_0^* δ𝒱 δψ̂ + ψ_0 δ𝒱 δψ̂^†)
+ δψ̂^†δ𝒱 δψ̂

+ δμ|ψ_0|^2
+ μ(
ψ_0^* δψ̂ + ψ_0 δψ̂^†)

ψ_0^* δψ̂ + ψ_0 δψ̂^†)
+ μδψ̂^†δψ̂
+ δμδψ̂^†δψ̂   ,

which looks rather complicated. Luckily, parts of the first and fifth and the entire second and sixth line can be shown to cancel out with help of the stationary GPE and the definition (<ref>) of $\delta \mu$ <cit.>. The remaining terms can be grouped by their order in $\delta \hat{\psi}$, beginning with the zeroth order term: the ground state energy

\begin{equation}
H^{(0)} = - \frac{U_0}{2} \int \ud \mathbf{r} \, | \psi_0|^4 \; ,
\end{equation}

where the hat accent has been omitted due to its classicality. Once the trapping potential is specified, $H^{(0)}$ can be used to calculate the chemical potential $\mu = \partial H^{(0)}/ \partial N_0$ where $N_0$ is the ground state particle number.

The part of the Hamiltonian mediating the phonon-condensate interaction via the perturbing potential can be defined as

\begin{equation} \label{eq:squeezing_mechanism_interaction_hamiltonian}
H^{(1)}_{int} = \int \ud \mathbf{r} \,
\left(
\delta \mathcal{V}- \delta \mu
\right)
\left(
\psi_0^* \delta \hat{\psi} + \psi_0 \delta \hat{\psi}^\dagger
\right)
\delta \hat{\psi}^\dagger
\left(
\delta \mathcal{V} - \delta \mu
\right)
\delta \hat{\psi}
\end{equation}

and will lead to the nontrivial part of the time evolution.

For the terms $\mathcal{O} (\delta \hat{\psi}^2)$, one can apply normal ordering to omit the vacuum energy <cit.> and form the Bogoliubov Hamiltonian

\begin{equation}
\hat{H}^{(2)} = \, \vcentcolon \int \ud \mathbf{r} \,
\delta \hat{\psi}^\dagger
\left(
-\frac{\hbar^2}{2m} \nabla^2 + \mathcal{V}_0 - \mu + 2 U_0 | \psi_0|^2
\right)
\delta \hat{\psi}
+ \frac{U_0}{2}
\left(
\psi_0^2 \delta \hat{\psi}^{\dagger 2} + \psi_0^{*2} \delta \hat{\psi}^2
\right) \vcentcolon
\end{equation}

Third and higher order terms

\begin{equation}
\hat{H}^{(3+)}_{int} = U_0 \int \ud \mathbf{r} \,
\psi_0 \delta \hat{\psi}^{\dagger 2} \delta \hat{\psi} + \psi_0^* \delta \hat{\psi}^\dagger \delta \hat{\psi}^2 + \frac{1}{2} \delta \hat{\psi}^{\dagger 2} \delta \hat{\psi}^2
\end{equation}

are neglected for the remainder of this chapter and accounted for in chapter <ref> by considering phonon damping and three-particle collisions.

As was stated at the beginning of the calculations above, we want to diagonalize $\hat{H}^{(2)}$ and change to the quasi-particle picture by doing so. Hence, with a Bogoliubov transformation in analogy to (<ref>), $\delta \hat{\psi}$ is expanded as <cit.>

\begin{equation}
\delta \hat{\psi} ( \mathbf{r}, t) = \sum_n u_n ( \mathbf{r}) \hat{b}_n + v_n^* ( \mathbf{r}) \hat{b}_n^\dagger \; ,
\end{equation}

with the orthonormal mode functions satisfying

\begin{equation}
\int \ud \mathbf{r} \,
u_n^* ( \mathbf{r}) u_l ( \mathbf{r}) - v_n^* ( \mathbf{r}) v_l ( \mathbf{r}) = \delta_{ln} \; .
\end{equation}

Demanding for the Bogoliubov Hamiltonian to become diagonal

\begin{equation} \label{eq:squeezing_mechanism_diagonalized_bogo_hamiltonian}
\hat{H}^{(2)} = \sum_n \hbar \omega_n \hat{b}^\dagger_n \hat{b}_n
\end{equation}

leads to the stationary Bogoliubov-de Gennes (BdG) equations <cit.>[
There are some subtleties in the derivation of Bogoliubov equations and analogously in the BdG equations, which are well covered in section 8.2 of Pethick and Smith's book <cit.> and <cit.>.

ħω_n u_n    =
-ħ^2/2m ∇^2 + 𝒱_0 - μ+ 2 U_0 | ψ_0|^2
) u_n
U_0 ψ_0^2 v_n

- ħω_n v_n    =
-ħ^2/2m ∇^2 + 𝒱_0 - μ+ 2 U_0 | ψ_0|^2
) v_n
U_0 ψ_0^* 2 u_n

§ SIMPLIFICATIONS AND TIME EVOLUTION IN A UNIFORM TRAP

For the situation described in <cit.>, the above expressions and the BdG equations simplify quite substantially (for more detailed derivations, the reader is referred to appendix <ref>). Consider an elongated box type trap with length $L$ and cross section $A$ (and thus volume $V=L A$) and a sinusoidally time-dependent perturbing potential[
In the original paper, a spatially uniform $\Phi_{0, \Omega}$ term is included in $\delta \mathcal{V}$ as well as a proportionality to the particle mass, since it is considered to be a gravitational potential. However, the spatially uniform term does not result in any squeezing and we are not specifying a particular type of force here. Thus, $\Phi_{0, \Omega}$ and the mass factor are omitted in the following description.

\begin{equation} \label{eq:squeezing_mechanism_perturbing_potential}
\delta \mathcal{V} (t) = \left( a_\Omega x + \mathfrak{G}_\Omega \frac{x^2}{2} \right) \sin \Omega t \equiv \delta \bar{\mathcal{V}}  \sin \Omega t \; .
\end{equation}

The perturbation to the chemical potential (<ref>) must then share the same time dependence

\begin{equation}
\delta \mu (t) = \frac{\mathfrak{G}_\Omega L^2}{24} \delta \bar{\mu} \equiv \delta \bar{\mu} \sin \Omega t \; .
\end{equation}

From section <ref>, we know that for an interacting Bose gas in a box trap, the ground state becomes uniform $\psi_0 = \sqrt{n_0}$ with the condensate particle density $n_0$ and the chemical potential (<ref>) thus turns out as $\mu = U_0 n_0$. The stationary BdG equations for modes with nonvanishing momenta only in the direction of the potential reduce to

ω_n u_n ( x)    =    ħ/2m [ ( -  ∇^2 + ξ) u_n ( x) + ξn_0 v_n ( x) ]

- ω_n v_n ( x)    =    ħ/2m [ ( -  ∇^2 + ξ) v_n ( x) + ξn_0 u_n ( x) ]

with von Neumann boundary conditions at $x= \pm L/2$. The resulting mode solutions are

u_n    =    α_n φ_n

v_n    =    β_n φ_n

α_n    =    √(1/V) √(1/√(2)k_n ξ + 1)
β_n    =    - √(1/V) √(1/√(2)k_n ξ - 1)

φ_n    =    cos( k_n (x+L/2) )

k_n    =    n π/L

and the phonon dispersion relation $\omega_n = c_s k_n$ (<ref>) is recovered in the limit $k_n \xi \ll 1$ <cit.>.

We are trying to show that the time evolution due to $\delta \mathcal{V}$ contains a squeezing operator. Therefore, we need the corresponding Hamiltonian $H^{(1)}_{int}$ (<ref>) in the interaction picture (see appendix <ref>, <cit.>), which can be composed in terms of matrix elements.

H^(1)_int, I
ℳ_0n (u_n + v_n)
e^-i ( ω_n - Ω) t b̂_n - e^i ( ω_n - Ω) t b̂_n^†)

+ ∑_n, l
b̂_n b̂_l e^-i ( ω_n + ω_l - Ω) t
- b̂_n^†b̂_l^†e^i ( ω_n + ω_l - Ω) t

+ ∑_n>l
𝒜_nl + ℬ_nl
e^-i ( ω_n - ω_l - Ω) t b̂_l^†b̂_n
- e^i ( ω_n - ω_l - Ω) t b̂_n^†b̂_l

ℳ_0n     ≡
- i/2 √(n_0) A ∫_-L/2^L/2 x
δ𝒱̅ - δμ̅
( u_n + v_n )

ℳ_nl     ≡
- i/2 A ∫_-L/2^L/2 x
δ𝒱̅ - δμ̅
u_n v_l

𝒜_nl     ≡
- i/2 A ∫_-L/2^L/2 x
δ𝒱̅ - δμ̅
u_n u_l

ℬ_nl     ≡
- i/2 A ∫_-L/2^L/2 x
δ𝒱̅ - δμ̅
v_n v_l   .

Using (<ref>) and to first order in $k_n \xi$, the matrix elements become (<cit.>, appendix <ref>)

i √(
L^3 N_0 ξ/( √(2) n π)^3
1- (-1)^n
) a_Ω/L
1+ (-1)^n
) 𝔊_Ω/2

-i  ( l^2 + n^2 ) L^3/
2 √(2 l n) ( l^2 - n^2 )^2 π^3 ξ
( 1 - (-1)^l+n ) a_Ω/L
( 1 + (-1)^l+n ) 𝔊_Ω/2
for  n ≠l

i 𝔊_ΩL^3/16 √(2) n^3 π^3 ξ
for  n=l

ℬ_nl ≈- ℳ_nl   .

For processes on resonance such that $n = n_\Omega$ in $\mathcal{M}_{0n}$ (i.e. $l+n = n_\Omega$) in the other matrix elements with $n_\Omega \equiv L \Omega / (\pi c_s)$, the time evolution is then

Û_I, res    =
exp     [
(-1)^n_Ω α_n_Ω (t) ( b̂_n_Ω^†- b̂_n_Ω )

1 + (-1)^n_Ω/2 r_n_Ω/2/2 (
b̂_n_Ω/2^†2 - b̂_n_Ω/2^2

- ∑_n < n_Ω/2
(-1)^n_Ω r_n, n_Ω-n (t)
b̂_n^†b̂_n_Ω-n^†- b̂_n b̂_n_Ω-n

+ ∑_n > n_Ω
(-1)^n_Ω Θ_n, n-n_Ω (t)
b̂_n^†b̂_n_Ω-n - b̂_n_Ω-n^†b̂_n

with the coefficients

α_n (t)     ≡    1/ħ |ℳ_0n| t

r_n (t)     ≡    2/ħ | ℳ_nn | t

r_n, l (t)     ≡    2/ħ | ℳ_nl | t

Θ_n, l (t)     ≡    2/ħ | 𝒜_nl | t   .

As can be seen from (<ref>), these coefficients parametrize (in the order they are written in above)

* Displacement (see (<ref>))

* Single-mode squeezing (see (<ref>))

* Two-mode squeezing

* Mode mixing

Thus, it is shown that the perturbation with a potential of the type (<ref>) on a BEC in a box-type cavity produces the desired two-mode squeezing. To get a better impression of the behaviour of its parameter $r_{n, l} (t)$ which we would like to maximize, note first from (<ref>) that for an odd and for an even sum of mode numbers, only the $a_\Omega$ and the $\mathfrak{G}_\Omega$ term contributes to $r_{n, l}$, respectively. In other words: for $l+n = $odd, the linear periodic perturbation produces squeezing and for $l+n=$even, the harmonic periodic perturbation does. At first glance, this seems to indicate that for odd $l+n$, $r_{n, l}$ scales with the third power of the trap length, whereas it is only proportional to $L^2$ for an even sum of mode numbers. As we shall see in the next chapter, the proportionality of the upper bounds for $a_\Omega$ and $\mathfrak{G}_\Omega$ partly resolves this peculiarity.

By choosing $l+n=$odd, as is necessary for the proposed GW detector described in the previous chapter, one can also avoid single-mode squeezing by setting $\mathfrak{G}_\Omega=0$.

For a fixed difference between the mode numbers $l-n$, all the parameters in (<ref>) are monotonously decreasing for higher modes. $r_{n, l}$ is therefore maximal in $l=1$, $n=2$ or, for even $n_\Omega$, $l=1$, $n=3$.

CHAPTER: IMPLEMENTATION

When trying to estimate which experimental setups would be best suited to maximize the two-mode squeezing induced by the mechanism depicted in the previous chapter, the work on this thesis took a turn that was, to the author, surprising, although it was to be expected in hindsight.

Naively, one can derive upper bounds for the magnitudes of the perturbing potential $a_\Omega$ and $\mathfrak{G}_\Omega$ in (<ref>) by demanding that $\delta \mathcal{V}$ only evokes a negligible deformation of the condensate and use these upper bounds to derive a maximum value for the two-mode squeeze factor $r_{n, l}$ given a particular set of modes $(n, l)$ and experimental parameters.

The application of the relations found in this manner to the experiment described in <cit.>, however, led to a squeeze factor $r_{1, 2} \sim 2000$, corresponding to the production of $\sim 10^{868}$ phonons. This is of course unreasonable and would result in the immediate destruction of the condensate. However, some considerations made on the path to this result are still relevant for the work that followed. It will therefore be outlined in the next section.

The overly large squeeze factor from this naive approach made it clear that the limitations to the magnitude of the perturbation must lie not only in the deformation of the ground state, but also in the number of phonons produced: once it assumes the same order as the number $N_0$ of particles in the ground state, the Bose gas ceases to be a condensate. In this sense, $N_0$ can be seen as a budget of quasiparticles that are allowed to be excited by the squeezing mechanism. One can then apply the unitary transformation (<ref>) to an initial state, which is taken to be thermal, and derive the phonon number in the resulting state to find the maximum values for the coefficients (<ref>) and with them $a_\Omega$ and $\mathfrak{G}_\Omega$. The quality of squeezing in the final state is estimated by comparison to a squeezed vacuum state (<ref>). A description of this procedure can be found in section. <ref>.

In sections <ref> and <ref>, the results are then applied to existing experiments (or approximations thereof) with the outcome that they were obviously not designed with our squeezing scheme in mind and hence an experiment that would be better suited is proposed.

§ NAIVE APPROACH

Suppose the magnitudes $a_\Omega$ and $\mathfrak{G}_\Omega$ of the linear and the harmonic perturbation are limited solely by the requirement that the perturbed condensate wave function $\psi_p$ deviates from the unperturbed one to an extent that is negligible.

As a first approximation (and to get simple, analytical expressions for the bounds we want to derive), we can formulate this requirement by demanding that the mean deviation of the condensate density should be much smaller than the condensate particle number. For a box trap of volume $V=LA$, this means

\begin{equation} \label{eq:naive_approach_bound_condition}
\int_V \ud \mathbf{r} \left| |\psi_0 |^2 - | \psi_{p} ( \mathbf{r} ) |^2 \right| \ll N_0 \; .
\end{equation}

We thus need to find the condensate wave functions for a box-type cavity with an additional linear and harmonic potential. To do so, we make use of the GPE in the Thomas-Fermi approximation (<ref>) with the chemical potential $\mu = g n_0$ of the uniform ground state and the condition that the condensate particle number and thus the integral of $| \psi_{p} ( \mathbf{r} ) |^2$ over the trap volume remains $N_0$.

Consider first the linear perturbation, where the condensate density then fulfills

\begin{equation}
| \psi_{p, lin} (x)|^2 = \frac{1}{g} \left( \mu - a_\Omega x \right)
\end{equation}

while the unperturbed ground state wave function is still uniform. Plugging this into the condition (<ref>) yields

A/g ∫_-L/2^L/2 | N_0/V - N_0/V + a_Ωx |
A a_Ω/g L^2/4

⇒a_Ω    ≪    N_0/V 4g/L    = 4 μ/L   .

For the harmonic perturbation parametrized by $\mathfrak{G}_\Omega$, the same conditions lead to

\begin{equation}
| \psi_{p, harm} (x)|^2 = \frac{1}{g} \left( \mu + \mathfrak{G}_\Omega (-x^2 + \frac{L^2}{12}) \right)
\end{equation}

A/g ∫_-L/2^L/2 | 𝔊_Ω( -x^2 + L^2/12 ) |
2A 𝔊_ΩL^3/9 √(3) g

⇒𝔊_Ω    ≪    N_0/V 9 √(3) g/2 L^2    = 9 √(3) μ/2 L^2   .

Due to the relative abundance of Bose-Einstein condensate experiments making use of harmonic traps in comparison to box type cavities, the author has decided to assume the existence of experiments of the latter nature with their remaining parameters (such as temperature, interaction strength, and particle number) being the same as in existing harmonic trap setups. This assumption, however, comes with some intricacies: if we imagine a BEC with repulsive interactions in an $x^2$ potential (or any non-hard-wall potential, for that matter), it is not hard to see that as its scattering length is increased, the gas will expand in its trap. The spatial dimensions of a gas in a quasi-infinite square potential well, on the other hand, are fixed. This means that the size of a box cavity approximating a harmonic trap depends not only on the trap frequencies, but also on the interaction strength, as well as the particle number and mass.

As an estimate for the size of the condensate, the Thomas-Fermi radius (<ref>) is used, assuming a cylindrical hard-wall potential. For the overall life time of both the condensate and the phonons to use in (<ref>), we can take the minimum of the inverse of the sum of the Landau and the Beliaev damping rates (<ref>)[We now denote phonon modes not by their momentum $\mathbf{p}$, but by their mode number.] and the half-life time due to three-body losses (<ref>)

\begin{equation} \label{eq:phonon_life_time_overall}
t_{life} = \mathrm{min} \left( \frac{1}{\gamma^{La}_l + \gamma^{Be}_l}, \, t_{1/2} \right) \; .
\end{equation}

Because we are squeezing in two modes and both $\gamma^{La}_l$ and $\gamma^{Be}_l$ are monotonously increasing with the mode number, the index $l$ denotes the higher one of the two modes considered.

§.§ Example: the Hulet Group $^7$Li Experiment

The $^7$Li experiment described in <cit.>, performed by the Hulet group ad Rice University with the intention of finding signatures of Efimov physics provided a suitable candidate for a first test of the considerations described above due to the completeness of the set of experimental parameters given in the publication as well as the vast range of tunability of the interaction strength.

Parameters and approximate deduced characteristics for a BEC in a box trap with a test interaction strength $a_s=100 \, a_0$[$a_0 \approx 5.29 \cdot 10^{-11} \,$m is the Bohr radius.] are given in table <ref>.

$[2 \pi \mathrm{s}^{-1}]$

$[2 \pi \mathrm{s}^{-1}]$

$[ \, ]$

$[ \, ]$

density[Calculated by the author in the Thomas-Fermi approximation]

temperature[The temperature is given in terms of the critical temperature $T_c$, which in turn is taken at $a_s=200 \, a_0$.]

$4 \cdot 10^5$

$3.6 \cdot 10^5$

$\approx 1.8 \cdot 10^{13}$

$[\mu \mathrm{m}]$

$[\mu \mathrm{m}]$

$[ \, ]$

$[ \, ]$

$[n \mathrm{K}]$

$\approx 280$

$\approx 9$

$4 \cdot 10^5$

$3.6 \cdot 10^5$

$\approx 5.2 \cdot 10^{12}$

$\approx 22$

speed of

$[\mathrm{kg \ m}/\mathrm{s}^2]$

$[ \, ]$

$\approx 5$

$\approx 58$

$\approx 10$

$\approx 3 \cdot 10^{-25}$

$\approx 4 \cdot 10^{-29}$

$\approx 2000$

Approximation of the BEC in <cit.> to a box trap of comparable parameters. The peak density in the harmonic trap was calculated by the author in the Thomas-Fermi approximation. The temperature was given in terms of the critical temperature $T_c$ at a scattering length $a_s = 200 a_0$ and the absolute temperature was deduced by the author following <cit.>. For the calculations involving phonons, the first two modes were considered. The phonon life time was calculated considering Landau and Beliaev damping (<ref>) as well as the limit to the overall condensate life time by three-body losses. Note the overly large squeeze factor which led to the considerations that follow in section <ref>.

§ PHONON BUDGET AND STATE TRANSFORMATION

In order to still be able to meaningfully speak of a Bose-Einstein condensate, we set the maximum number of phonons in the final state to a small –albeit, admittedly, somewhat arbitrary– fraction of the condensate particle number $N_{phon} \leq N_0 / 50$. The task is then to find out how many phonons the time evolution (<ref>) creates given an initial thermal state. In order to do so, the covariance matrix formalism presents itself as a valuable tool once more.[
In contrast to chapter <ref>, the complex picture of the CM formalism is applied here (see appendix <ref>).
] This means that we need the CM of an infinite-mode thermal state (<ref>), its particle number and a transformation in phase space reflecting the unitary time evolution (<ref>). The thermal occupation number of a mode $l$ with energy $\hbar \omega_l$ is $(\mathrm{e}^{\beta \hbar \omega_l}-1)^{-1}$ (see appendix <ref>). In a box trap, we have $\omega_l = l \omega_1$ and therefore

\begin{equation} \label{eq:thermal_state_infinite_mode_particle_number}
N_{phon, i} = \sum_{l=1}^\infty \frac{1}{\mathrm{e}^{\beta \hbar l \omega_1}-1} = \frac{1}{2} - \frac{1}{\beta \hbar \omega_1} \left( \ln (\mathrm{e}^{\beta \hbar n \omega_1} - 1) + \psi_{\mathrm{Exp}(\beta \hbar \omega_1)} (1) \right) \; ,
\end{equation}

where $\psi_q (z)$ is the $q$-digamma function <cit.> and a transformation in phase space reflecting the unitary time evolution (<ref>).

In general, the transformation of the first and second moments corresponding to a Gaussian unitary transformation in the Hilbert space is given by <cit.>

Û    =    e^
i/2 𝐀̂^†W 𝐀̂ + 𝐀̂^†K γ

S_U    =    e^i K W

𝐛 = ( ∫_0^1 t e^i K W t ) γ   ,

where $\hat{\mathbf{A}}$ is the vector of creation and annihilation operators, $K$ is the symplectic form and $W$ and $\boldsymbol{\gamma}$ encode the change in first and second moments, respectively. The CM $\sigma$ and the vector $\mathbf{d}$ of first moments then transform as

\begin{equation}
\sigma \rightarrow S_U \sigma S_U^\dagger \; , \quad \mathbf{d} \rightarrow S_U \mathbf{d} + \mathbf{b} \; .
\end{equation}

The particle number of the final state can then be calculated with <cit.>

\begin{equation} \label{eq:CM_particle_number_expectation_value}
\langle \hat{N} \rangle = \frac{1}{4} \left(
\mathrm{Tr} \left( \sigma \right) + \mathbf{d}^\dagger \mathbf{d} - 2N
\right) \; .
\end{equation}

The derivation of the symplectic transformation corresponding to (<ref>) for all sets of modes at once would result in an infinite-dimensional matrix exponential and is not the way to go. Nonetheless, for a given $n_\Omega$, the part of our transformation that creates phonons can be split up into three types living on subspaces that are mutually noninteracting:

Coherent displacement in mode $n_\Omega$, which gets mixed into mode $2n_\Omega$

Single-mode squeezing (for even $n_\Omega$) in mode $n_\Omega/2$, mixing into $3n_\Omega/2$

Two-mode squeezing in modes $k$, $l$ with $k+l=n_\Omega$, mixing with modes $k+n_\Omega$ and $l+n_\Omega$

The maximum number $N_{phon, f}$ of phonons in the final state can thus be split up into the initial phonon number $N_{phon, i}$ and the phonons created be the three mechanisms above and we have

\begin{equation} \label{eq:implementation_phonon_budget}
N_{phon, f} = N_{phon, i} + \Delta N_{phon, d} + \Delta N_{phon, 1-sq} + \Delta N_{phon, 2-sq} \leq \frac{N_0}{50}
\end{equation}

For a given set of experimental parameters,[That is, $L$ $N_0$ and $\xi$.] and a resonance mode $n_\Omega$, all coefficients in (<ref>) are uniquely related and can be expressed in terms of one two-mode squeeze factor. For later use in the subsequent sections, let us denote the modulus of the matrix elements in (<ref>) without the factors corresponding to the perturbing potential as

m_0n    =    √(
L^3 N_0 ξ/( √(2) n π)^3

m_nl ≈a_nl ≈b_nl    =

( l^2 + n^2 ) L^3/
2 √(2 l n) ( l^2 - n^2 )^2 π^3 ξ
for  n ≠l

𝔊_ΩL^3/16 √(2) n^3 π^3 ξ
for  n=l   .

With this and the transformations described below, the maximum squeeze factor $r_{l, n_\Omega-l}$ satisfying (<ref>) has been computed for a set of box traps approximating existing experiments with a number of specific $n_\Omega$. At first glance, it may seem as squeezing at lower resonances such as $n_\Omega=3$ or $n_\Omega=4$ might be advantageous. For higher $n_\Omega$, there is more than one pair of modes whose sum is at resonance and therefore there will be multiple subspaces where two-mode squeezing occurs. In all of these subspaces, quasi-particles are created, accelerating the exhaustion of our phonon budget. Closer inspection shows this thought to be wrong: while there are multiple instances of two-mode squeezing for $n_\Omega \geq 5$, the amount of particles created strongly depends on the initial (thermal) occupation $(\mathrm{e}^{\beta \hbar \omega_l}-1)^{-1}$ of the mode $l$ in question, which decreases significantly with higher $l$. In addition, lower thermal occupation numbers also mean that the fidelity of the final state in comparison to the desired two-mode squeezed vacuum is drastically improved, as is shown section <ref>.

For arbitrary $n_\Omega$ the mode pair $(l, n_\Omega-l)$ subject to the highest squeeze factor is the one with the largest $l$ satisfying $l<n_\Omega/2$ and thus the smallest distance $(n_\Omega-l)-l$. This can be seen by comparing their respective matrix elements and finding that

\begin{equation}
m_{l, n_\Omega-l}>m_{l', n_\Omega-l'} \quad \forall \; l' \, \in \, \mathbb{N} \, <l \; .
\end{equation}

We will refer to this pair of modes $(l_{max}, n_\Omega-l_{max})$ as the balanced pair and choose the highest two-mode squeeze factor $r_{l_{max}, n_\Omega-l_{max}}$ as the one parameter[The calculations to express all the other parameters in terms of $r_{l_{max}, n_\Omega-l_{max}}$ can be found in the Mathematica file that was used to determine the symplectic transformation in appendix <ref>.] determining the strength of the transformation to express all other $r_{l', n_\Omega-l'}$ as

\begin{equation}
r_{l', n_\Omega-l'} = \frac{m_{l', n_\Omega-l'}}{m_{l_{max}, n_\Omega-l_{max}}} r_{l_{max}, n_\Omega-l_{max}} \; .
\end{equation}

§.§ Coherent Displacement and Mode Mixing

The state transformation in the modes $(n_\Omega, 2n_\Omega)$ consists of displacement and mode mixing

\begin{equation}
U = \mathrm{e}^{
(-1)^{n_\Omega} \left(
\alpha_{n_\Omega} (\hat{b}_{n_\Omega}^\dagger - \hat{b}_{n_\Omega} )
\Theta_{2n_\Omega, n_\Omega} (\hat{b}_{2n_\Omega}^\dagger \hat{b}_{n_\Omega} - \hat{b}_{n_\Omega}^\dagger \hat{b}_{2n_\Omega} )
\right)
} \; ,
\end{equation}

W    =
- (-1)^n_Ω  Θ_2n_Ω, n_Ω
( σ_z ⊗σ_y )

S_U    =    1 ⊗
cosφ    sinφ

-sinφ    cosφ

γ    =    (-1)^n_Ω α_n_Ω

𝐛	    =    α_n_Ω/ Θ_2n_Ω, n_Ω

- sinφ

1 - cosφ   ,

where $\sigma_z$ and $\sigma_y$ are Pauli matrices. The number of phonons created by the transformation above can then be found by applying it to the corresponding modes of a thermal state and recalling (<ref>). For the overall phonon number in a transformed thermal state, one finds

\begin{equation}
\langle \hat{N} \rangle_{th, U} = \langle \hat{N} \rangle_{vac, U} + \langle \hat{N} \rangle_{th} \; .
\end{equation}

$\langle \hat{N} \rangle_{vac, U}$ and $\langle \hat{N} \rangle_{th}$ are the numbers of quasiparticles that would have been produced by the same transformation on a vacuum state and the initial thermal occupation numbers, respectively. The phonons created by displacement and mode mixing are then

\begin{equation}
\Delta N_{phon, disp} = \langle \hat{N} \rangle_{vac, U} = 2
\left( \frac{\alpha_{n_\Omega}}{ \Theta_{2n_\Omega ,n_\Omega}} \right)^2
\sin^2 \left( \frac{ \Theta_{2n_\Omega, n_\Omega}}{2} \right) \; .
\end{equation}

As mentioned before, the ratio between the ratio $\alpha_{n_\Omega}/ \Theta_{2n_\Omega ,n_\Omega}$ is fixed for a given set of experimental parameters and $n_\Omega$:

Given such a set, equations (<ref>) and  (<ref>) can be used to set a fixed ratio

\begin{equation}
x_{disp} \equiv \frac{ \Theta_{2n_\Omega ,n_\Omega}}{\alpha_{n_\Omega}}= 2 \left| \frac{a_{2n_\Omega, n_\Omega}}{m_{0, n_\Omega}} \right|
\frac{5}{18} \frac{1}{\sqrt{N_0}}
\left(
\frac{\sqrt{2} L}{\pi n_\Omega \xi}
\right)^{3/2}
\end{equation}

between the strengths of the displacement and the mode mixing and simplify

\begin{equation}
\Delta N_{phon, disp} = 2
x_{disp}^2 \,
\sin^2 \left( \frac{x_{disp} \alpha_{n_\Omega}}{2} \right) \; .
\end{equation}

§.§ Single-mode Squeezing and Mode Mixing

We start again by picking out the single-mode squeezing part of (<ref>) and the term of the exponent responsible for mixing with the mode that gets squeezed

\begin{equation} \label{eq:implementation_single-squeezing-mixing_unitary}
U = \mathrm{e}^{
- \frac{r_{n_\Omega/2}}{2} \left(
\hat{b}_{n_\Omega/2}^{\dagger 2} - \hat{b}_{n_\Omega/2}^2
\right)
\Theta_{3n_\Omega/2, n_\Omega/2}
\left(
\hat{b}_{3n_\Omega/2}^\dagger \hat{b}_{n_\Omega/2} - \hat{b}_{n_\Omega/2}^\dagger \hat{b}_{3n_\Omega/2}
\right)
} \; .
\end{equation}

While the transformation of the second moments

\begin{equation}
W= - \frac{r_{n_\Omega/2}}{i}
\begin{pmatrix}
0 & 0 & 1 & 0
\\
0 & 0 & 0 & 0
\\
-1 & 0 & 0 & 0
\\
0 & 0 & 0 & 0
\end{pmatrix}
- \Theta_{3n_\Omega/2, n_\Omega/2} \: \sigma_z \otimes \sigma_y
\end{equation}

is nonvanishing, both squeezing and mode mixing leave the first moments of a state with $\mathbf{d}=0$ invariant since $\boldsymbol{\gamma}$ is zero. The symplectic transformation $S_U$ is a rather unwieldy expression and will not be displayed here.[The interested reader is again referred to appendix <ref>.] However, as in the previous section, we can take the ratio between the $r_{n_\Omega/2}$ and $\Theta_{3n_\Omega/2, n_\Omega/2}$, which is constant in this instance,

\begin{equation}
x_{1sq} \equiv \frac{\Theta_{3n_\Omega/2, n_\Omega/2}}{r_{n_\Omega/2}}
\left|
\frac{a_{3n_\Omega/2, n_\Omega/2}}{m_{n_\Omega/2, n_\Omega/2}}
\right| = \frac{5}{4 \sqrt{3}}
\end{equation}

and use it to simplify the final quasiparticle number in the modes $n_\Omega/2$, $3n_\Omega/2$ after the transformation (<ref>) as far as possible to

⟨N̂ ⟩_th, U    =    1/4 ( Tr (S_U σ_th S_U^†) - 4 )

-1 + 2
βħω/2/1 + 2 coshβħω ×

√(3/13) sin(
√(13/3) r/2
) sinhr
⟨N̂ ⟩_vac, U + 1
]  ,

where we have dropped the index $n_\Omega/2$ for $r$ and $\omega$ and the number of phonons created by (<ref>) from a vacuum state is

\begin{equation}
\langle \hat{N} \rangle_{vac, U} = \frac{1}{13}
\left[
25 - 12 \cos
\left(
\frac{1}{2} \sqrt{\frac{13}{3}} r
\right)
\right]
\cosh r - 1
\end{equation}

§.§ Two-mode Squeezing and Mode Mixing

As it was the case for single-mode squeezing, if the first moments of the initial state vanish, they remain zero after the transformation. For an individual squeeze factor $r_{l, n_\Omega-n}$ the entire subspace that has to be considered due to mixing consists of the four modes $(l, n_\Omega-l, 2n_\Omega-l, n_\Omega+l)$ and in this subspace, we have the unitary

U   = exp [
- (-1)^n_Ω r_l, n_Ω-l
b̂_l^†b̂_n_Ω-l^†- b̂_l b̂_n_Ω-l

(-1)^n_Ω Θ_n_Ω+l, l
b̂_n_Ω+l^†b̂_l - b̂_l^†b̂_n_Ω+l

(-1)^n_Ω Θ_2n_Ω-l, n_Ω-l
b̂_2n_Ω-l^†b̂_n_Ω-l - b̂_n_Ω-l^†b̂_2n_Ω-l

and with it the $W$ matrix[Using the shortened notation $r_{l, n_\Omega-l}=r$, $\Theta_{n_\Omega +l, l}=\Theta_1$ and $\Theta_{2n_\Omega -l, n_\Omega-l}=\Theta_2$.]

\begin{equation}
W= \frac{(-1)^{n_\Omega}}{2i}
\begin{pmatrix}
0 & 0 &  \Theta_1 & 0 & 0 & r & 0 & 0
\\
0 & 0 & 0 &  \Theta_2 & r & 0 & 0 & 0
\\
- \Theta_1 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & - \Theta_2 & 0 & 0 & 0 & 0 & 0 & 0
\\
0 & -r & 0 & 0 & 0 & 0 & - \Theta_1 & 0
\\
-r & 0 & 0 & 0 & 0 & 0 & 0 & - \Theta_2
\\
0 & 0 & 0 & 0 &  \Theta_1 & 0 & 0 & 0
\\
0 & 0 & 0 & 0 & 0 &  \Theta_2 & 0 & 0
\end{pmatrix} \; .
\end{equation}

Though the author was not able to find a reasonably short analytical expression for the symplectic transformation $S_U=\mathrm{e}^{iKW}$ or the resulting phonon number, both results can be evaluated for a given set of squeeze factors and thermal occupation numbers[Albeit with exceedingly long computation times for resonance conditions $n_\Omega \gtrsim 41$, see again appendix <ref>.] and simplified using the ratios between the squeeze factor and the mode mixing parameters

\begin{equation}
x_{2sq, 1} \equiv \frac{\Theta_{n_\Omega +l, l}}{r_{l, n_\Omega-l}}
\left|
\frac{a_{n_\Omega +l, l}}{m_{l, n_\Omega-l}}
\right|
\sqrt{\frac{n-l}{n+l}} \frac{
(n-2l)^2 (n^2 + 2ln + 2l^2)
(n+2l)^2 (n^2 - 2ln + 2l^2)
\end{equation}

\begin{equation}
x_{2sq, 1} \equiv \frac{\Theta_{2n_\Omega -l, n_\Omega-l}}{r_{l, n_\Omega-l}}
\left|
\frac{a_{2n_\Omega -l, n_\Omega-l}}{m_{l, n_\Omega-l}}
\right|
\sqrt{\frac{l}{2n-l}} \frac{
(n-2l)^2 (5n^2 - 6ln + 2l^2)
(3n-2l)^2 (n^2 - 2ln + 2l^2)
} \, .
\end{equation}

For the full transformation, $r_{n_\Omega/2}$ and $\alpha_{n_\Omega}$ were expressed in terms of the two-mode squeeze factor as

r_n_Ω/2    =
m_n_Ω/2, n_Ω/2/m_l, n_Ω-l r_l, n_Ω-l
√(l (n_Ω-l)) (n_Ω-2l)^2/n_Ω(n_Ω^2 - 2 l n_Ω+ 2l^2) r_l, n_Ω-l

α_n_Ω    =    m_0, n_Ω/m_l, n_Ω-l r_l, n_Ω-l
√(N_0 ξ^3 π^3/√(2) L^3) √(l n_Ω(n_Ω-l)) (n_Ω-2l)^2/(n_Ω^2 - 2 l n_Ω- 2l^2) r_l, n_Ω-l

§ QUALITY OF THE FINAL STATE

The initial state considered in the GW detector proposal <cit.> from chapter <ref> is a two-mode squeezed vacuum state. The result of the mechanism we are considering, however, is a thermal state that was subjected to squeezing and mode mixing. In order to quantify its value for quantum metrology applications that assume a squeezed vacuum, it is therefore necessary to not only provide the maximum squeeze factor that is allowed for by the phonon budget and the highest possible perturbation, but one also has to consider the overlap between the final state and the ideal one of the squeezed vacuum. As a measure for this overlap, we can make use of the Uhlmann fidelity once again.

It then quickly becomes apparent that a quantity decisive for the quality of the final state is the thermal occupation number before the transformation, which we will express by $\beta \hbar \omega_1/2$.[
$\omega_1$ is the first phonon frequency and the factor $1/2$ is for convenience because of the CM elements of the thermal state, see (<ref>). We keep considering a box trap where $\omega_n = n \omega_1$.
] For a first impression, one can calculate the fidelity between a squeezed vacuum and a squeezed thermal state in the first two phonon modes with equal squeeze factors to find

\begin{equation}
\mathcal{F} (\sigma_{sq2}, \sigma_{therm, sq}) = 4 \mathrm{e}^{-3 \beta \hbar \omega_1/2} \ \sinh \left( \frac{\beta \hbar \omega_1}{2} \right) \sinh \left( \beta \hbar \omega_1 \right) \; .
\end{equation}

Owing to the rather involved nature of the state transformation, an expression for the fidelity between the actual final state and the squeezed vacuum is not quite as easy to find. Nonetheless, we can maximize $\mathcal{F} \left( \sigma_{out} (r_{l, n_\Omega-l}, \beta \hbar \omega_1/2), \, \sigma_{sq2} (r_{vac}) \right)$ numerically over $r_{vac}$ in order to make some qualitative observations. For this purpose, see figure <ref>. One such observation is that while the parameter $r_{l, n_\Omega-l}$ of the state transformation might increase, the corresponding squeeze factor of the nearest squeezed vacuum seems to asymptotically reach a maximum even for large $r_{l, n_\Omega-l}$ at low resonance conditions. Furthermore, at low-lying resonances, the reachable fidelity not only strongly depends on $\beta \hbar \omega_1/2$, but also on the squeeze factor. This is most probably a consequence of the imbalanced initial population of the modes in the thermal state. Two-mode squeezing, particularly strong squeezing with $r_{l, n_\Omega-l} \gtrsim 2$ thus becomes efficient only for higher mode pairs.

On the quality of the final state in the balanced mode pair after the full symplectic transformation of the squeezing scheme. The three columns correspond to the three resonance conditions $n_\Omega=3$, $n_\Omega=11$ and $n_\Omega=31$ and the rows are

– the Uhlmann fidelity between the final state and the nearest squeezed vacuum state as a function of the two-mode squeeze factor $r_{l, n_\Omega-l}$ of the transformation for various frequency-temperature ratios $\beta \hbar \omega_1/2$

– the Uhlmann fidelity as a function of the frequency-temperature ratio over a range of squeeze factors $r_{l, n_\Omega-l}$

– the squeeze factor of the nearest vacuum state as a function of the frequency-temperature ratio over a range of squeeze factors. $r_{l, n_\Omega-l}$

Note how at low resonances, $r_{vac}$ reaches an asymptotic limit that is seemingly independent of both the squeeze factor of the transformation and $\beta \hbar \omega_1/2$. From the second row and particularly (e), we can see that in the temperature-frequency range $\beta \hbar \omega_1/2 \approx 0.22$ of the experiment proposed in section <ref>, squeezing with factors $r \gtrsim 4$ becomes efficient only for higher-lying resonances $n_\Omega \gtrsim 11$.

§ SOME GUIDELINES FOR OPTIMIZATION

The complicated nature of the state transformation and with it the long computation times of final states and their properties in connection with the vast parameter space makes it difficult to optimize over experimental parameters and modes to squeeze in. Nonetheless, we can make some qualitative observations on which degrees of freedom could be utilized to maximize the allowed squeeze factor and the quality of the final state.

At the same time, we must keep in mind the limitations to the phonon and condensate life times, posed by three-body losses[In order to keep expressions simple, we assume for now to be far from possible Efimov resonances.] (<ref>), Landau damping (<ref>) and Beliaev damping (<ref>). Requiring reasonable phononic life times could then be used to bound the parameter space in which the final states can be optimized.

Finally, we have to remind ourselves of the assumptions that we have made along the way and that have to be fulfilled in order for the derivations so far to be applicable: in section <ref>, we assumed to be dealing with a dilute and weakly interacting Bose gas, where $|a_s| n_0^{1/3} \ll 1$. Further on, we have considered the ultracold regime $k_B T \ll \mu$ and lastly, the quasi-particles were treated as phonons, implying $k_n \xi \ll 1$.

To gain some more intuition on how an experiment could be improved for squeezing, let us consider the cases where one of the possible causes of phonon decoherence becomes dominant and derive the resulting maximum squeeze factors (<ref>) from the naive approach <ref> in each case as well as the conditions listed above. For a better overview, these quantities are expressed in terms of typical characteristics that are given when describing an experiment in table <ref>.

Low damping rates and low thermal occupation numbers call for high particle densities, which are, on the other hand, bounded from above by the diluteness condition and three-body losses.

High mode numbers $l$ also have the advantage of low quasi-particle numbers in the initial state and thus the quality of the state after the transformation. They are, however, limited by their shorter life time as well as the condition that they can be approximated as phonons.

Assuming that the scattering length can be tuned using a Feshbach resonance and in the absence of an Efimov resonance, increasing $a_s$ will lead to drastically enhanced three-body losses, which scale as $a_s^4$. A compromise must therefore be found when maximizing $a_s$ to reduce the thermal population .

An increase in the trap length $L$ will lead to higher squeeze factors permitted by the naive approach, but at the same time it reduces the phonon frequencies and thereby lowers the energy levels of the quasi-particle state. This means, again, that occupation numbers in the initial state are increased, lowering the amount of phonons we are allowed to create and the fidelity of the final state.

The temperature, finally, is simple: improved cooling is beneficial in every single aspect considered in table <ref>.

$(\gamma^{La}_{k_B T \ll \mu})^{-1}$ (<ref>)

$(\gamma^{La}_{k_B T \gg \mu})^{-1}$ (<ref>)

$(\gamma^{Be, 0})^{-1}$ (<ref>)

$t_{1/2}$ (<ref>)

$t \approx$

$\frac{640}{3 \pi} \frac{\hbar^7 \beta^4 L n_0^3 a_s^2}{l m^3}$

$\frac{8}{\pi} \frac{\beta L}{a_s l}$

$\frac{640}{3 \pi^6} \frac{m L^5 n_0}{\hbar l^5}$

$\frac{1}{50} \frac{m}{\hbar n_0^2 a_s^4}$

$r \propto$

$\frac{\hbar^8 \beta^4 L^2 n_0^{9/2} a_s^{7/2}}{l^2 m^4}$

$\frac{\hbar^2 L^2 \beta n_0^{3/2} a_s^{1/2}}{l^2 m}$

$\frac{L^6 n_0^{5/2} a_s^{3/2}}{l^6}$

$\frac{L}{l n_0^{1/2} a_s^{5/2}}$

$\beta \omega_l$

$2 \sqrt{\pi} \frac{\hbar l \sqrt{n_0 a_s}}{L m}$

$|a_s| n_0^{1/3} \ll 1$

$\sqrt{\frac{\pi l^2}{L^2 n_0 a_s}} \ll 1$

$\frac{1}{4 \pi} \frac{m}{\hbar^2 \beta \, n_0 \, a_s} \ll 1$

Approximate values for the limitations to the phonon and condensate life times and resulting proportionalities of the maximum two-mode squeeze factor from SECTION <ref> in the case where the respective damping or particle losses become dominant.

The two bottom rows show a measure $\beta \omega_l$ for the thermal occupation of the initial state, which should be minimal, and restrictions from the diluteness condition and the assumptions of the phononic and the ultracold regime.

To see where one could optimize and which boundaries will be encountered, all quantities above are given in terms of the parameters characterizing an individual experiment.

§ EXISTING EXPERIMENTS

In face of the sheer number of BEC experiments around the globe, the task of picking a representative selection to examine with respect to the possible application of the squeezing scheme presented was not quite easy. The choice was made with the intention of giving an overview of various species, where each particular one of the experiments listed below has a distinguishing property, be it the measurement of Efimov parameters <cit.>, extremely low absolute temperatures <cit.> or high particle numbers <cit.> or because they were, in fact, performed in box traps <cit.>. For experiments using harmonic potentials, a condensate of comparable size in a cylindrical hard-wall potential was assumed by deriving the length and diameter from the Thomas-Fermi radius (<ref>).

After the investigation of the selected experiments for their potential for phonon squeezing, it became clear that they were not designed with our method in mind, albeit for varying reasons. The following paragraphs provide a brief description of each condensate, including the (approximate) numbers most decisive for their squeezability:

* The temperature $T$

* The initial particle number  $N_0$

* The initial number of thermal phonons $N_{phon, i}$

* A resonance $n_\Omega$ whose balanced pair (see section <ref>) has reasonable life times $t_{life}$ while still being in the phonon regime

* The highest possible corresponding two-mode squeeze factor $r_{l_{max}, n_\Omega -l_{max}}$, which is the lower one of the squeeze factors permitted for by the naive ansatz and the phonon budget.

* The Uhlmann fidelity $\mathcal{F}$ of the closest squeezed vacuum state and its squeeze factor $r_{vac}$

$N_{phon, i}$ was calculated using (<ref>), $t_{life}$ by means of (<ref>), where no low or high temperature limits were taken and the integrals for the Landau and Beliaev damping rates were computed numerically. Computation of $\mathcal{F}$ was done by transforming the initial thermal state, dropping all subspaces except the one with the highest two-mode squeeze factor and numerically maximizing $\mathcal{F}$ between the final and a squeezed vacuum state over its squeeze factor $r_{vac}$.

Westbrook Group $^4$He$^*$ Experiment (table <ref>)

The 2012 experiment by Jaskula et al. <cit.> with metastable $^4$He$^*$ has some similarities with the mechanism presented in this thesis: the stiffness of the harmonic trapping potential was modulated to produce pairs of squeezed phonons (although they are, in the spirit of chapter <ref>, referred to as twin beam states). In the axial direction, the potential is very shallow (axial and radial trapping frequencies are $7$ Hz and $1500$ Hz), which, in combination with the relatively high temperature and low particle number, takes it far from the ultracold regime and makes it an improper candidate. As a consequence, the number of thermal phonons before attempting to squeeze would already exceed our bound of $N_0/50$ for  a similarly dimensioned box cavity.

Hulet Group $^7$Li Experiment (table <ref>)

The setup used in <cit.> was chosen as an example primarily because of the completeness of the data given in the publication and its measurement of Efimov parameters by (for positive scattering lengths) observing local minima of three-body losses. For the computation of the values in  table <ref>, $a_s$ was assumed to be tuned to the first minimum of the three-body recombination rate at $\approx 117 a_0$, with $a_0$ being the Boltzmann radius. The elongation of the potential in question conforms well with the assumptions made in section <ref>. Because the temperature in <cit.> was given only in terms of a fraction of the critical temperature, the formula derived by Giorgini et al. <cit.> for the critical temperature in interacting Bose gases was applied to derive the absolute temperature.

Ketterle Group $^{23}$Na Experiment (table <ref>)

On the search for extremely cold Bose gases, the 2003 experiment by Wolfgang Ketterle's group <cit.> was brought to the author's attention, but it does come with some caveats: while the absolute temperature reached was a record setting $450$pK, the trap was also very shallow and pancake-shaped. This means that the state before the transformation was highly populated and transverse phonon modes would have to be accounted for, which was not done in this thesis. In addition, even at this low temperature, $k_B T/\mu \approx 30$.

van der Straten Group $^{23}$Na Experiment (table <ref>)

In the early phase of discussing this thesis, $^{23}$Na presented itself as a promising candidate for squeezing due to the large particle numbers that have been achieved, see for example <cit.>. The data that the results in table <ref> are based on were kindly provided in personal correspondence by Peter van der Straten. The extensive size of the condensate and the low, non-tunable scattering length keep collective excitations well in the phonon regime, but their life time is strongly limited by high temperature Landau damping and thermal population numbers do not allow for much squeezing.

Hadzibabic Group $^{39}$K Experiment (table <ref>)

The $^{39}$K and $^{87}$Rb setups (of which only the former is still in use) located at the University of Cambridge are two of the rare examples of BECs in a uniform potential and thus of particular interest for this thesis. Given the parameters provided in <cit.>, they are, however, not well suited for our purposes: the comparably low particle density $n_0 \approx 10^{12} \, \mathrm{cm}^{-3}$ increases the healing length and thus pushes quasi-particles out of the phononic regime at a test scattering length $a_s = 100 a_0$. Increasing the interaction strength, on the other hand, would further reduce phonon life times.

Hadzibabic Group $^{87}$Rb Experiment (table <ref>)

Since the early days of BECs, $^{87}$Rb has been the workhorse among the atomic species. This box trap setup, though, is now discontinued. In the light of our intentions, it is far from the ultracold limit at $k_B T/\mu \approx 13$ and suffers from short phonon life times, which are again, a consequence of Landau damping and prevent us from going beyond $n_\Omega=3$.

Grimm Group $^{133}$Cs Experiment (table <ref>)

The Innsbruck $^{133}$Cs BEC <cit.>, like the Houston $^7$Li one <cit.>, was used successfully to measure Efimov parameters. In <cit.>, only an upper bound for $\eta_+<2$ could be found for the decay parameter at positive scattering lengths. In the calculations made here, we will follow the argument made in the paper cited and take the theoretical assumption from <cit.> that $\eta_+ = \eta_-$ as a rough estimate. Dips in $C_3$ (<ref>) are then not pronounced enough to result in local minima of the three-body loss rate. Nonetheless, the scattering length is assumed to be tuned to an Efimov resonance at $a_s = 210 a_0$.

$T \, [ \mathrm{nK} ]$

$N_{phon, i}$

$r_{l_{max}, n_\Omega -l_{max}}$

$t_{life} \, [\mathrm{s}]$

$ 1$

Westbrook Group $^4$He$^*$ Experiment: even very high quasi-particle modes can still be considered phonons here, but high temperature Landau damping with its $l^{-1}$ dependence already limits the life time of the lowest mode to $\approx 2$s. The large value of $N_{phon, i}$ does not allow for the creation of additional phonons within the bounds we have set.

$T \, [ \mathrm{nK} ]$

$N_{phon, i}$

$r_{l_{max}, n_\Omega -l_{max}}$

$ 22$

$3.6 \cdot 10^5$

$ 220$

$ 2.9$

$t_{life} \, [\mathrm{s}]$

$ 3$

$ 2.9$

Hulet Group $^7$Li Experiment:
the gas is not in the ultracold regime with $k_B T/\mu \approx 0.96$. The low fidelity is due to the initial thermal population. Squeezing in higher modes is not an option due to the shorter phonon life times, which are limited by Landau damping. Both of these issues could potentially be resolved by tightening the trap in the axial direction.

$T \, [ \mathrm{nK} ]$

$N_{phon, i}$

$r_{l_{max}, n_\Omega -l_{max}}$

$t_{life} \, [\mathrm{s}]$

Ketterle Group $^{23}$Na Experiment:
the quasi-particles have reasonably long life times, but their large wave numbers and healing length of the condensate place even the lowest modes outside the phonon regime. The small number of atoms in the condensate does not allow for the additional creation of quasi-particles.

$T \, [ \mathrm{nK} ]$

$N_{phon, i}$

$r_{l_{max}, n_\Omega -l_{max}}$

$t_{life} \, [\mathrm{s}]$

van der Straten Group $^{23}$Na Experiment:
due to the high temperature, there is not much room for squeezing and the final state is far from the ideal of a squeezed vacuum. Targeting higher modes is not an option due to Landau damping.

$T \, [ \mathrm{nK} ]$

$N_{phon, i}$

$r_{l_{max}, n_\Omega -l_{max}}$

$4 \cdot 10^5$

$t_{life} \, [\mathrm{s}]$

$2 \cdot 10^{-3}$

Hadzibabic Group $^{39}$K Experiment:
the values above were computed assuming collective excitations can still be treated as phonons. Low life times keep us from squeezing in higher modes. Improved cooling and higher particle densities could help cure this problem. The room for improvement is limited by three-body losses.

$T \, [ \mathrm{nK} ]$

$N_{phon, i}$

$r_{l_{max}, n_\Omega -l_{max}}$

$4 \cdot 10^5$

$t_{life} \, [\mathrm{s}]$

$4 \cdot 10^{-5}$

Hadzibabic Group $^{87}$Rb Experiment:
potential for optimization for the purposes of our squeezing mechanism lies in increasing the particle density and possibly the trap length, mostly though in lowering the temperature.

$T \, [ \mathrm{nK} ]$

$N_{phon, i}$

$r_{l_{max}, n_\Omega -l_{max}}$

$t_{life} \, [\mathrm{s}]$

Grimm Group $^{133}$Cs Experiment: in contrast to the experiments above, $r_{l_{max}, n_\Omega -l_{max}}$ is bounded by the naive ansatz rather than the phonon budget in this instance. Phonons soon fall victim to Landau damping. Choosing $n_\Omega=3$ instead would bring life times up to $\approx 1$s, but at the same time decrease the fidelity by more than an order of magnitude.

§ PROPOSAL: A UNIFORM $^7$LI BEC

Clearly, experiments similar to the ones listed above leave little chance to produce states that come close to our goals of reasonable life times, high squeeze factors and good overlap with a squeezed vacuum state. This poses the question what a setup that can achieve these goals could look like and brings us back to the considerations on optimization and constraints from table <ref>.

As far as the atomic species is concerned, there are two features of $^7$Li that make it a promising candidate: its low mass, which allows for higher phonon frequencies, and the fact that the Efimov dip in $L_3$ at $a_s \approx 117 a_0$[
Do not let the fact that $a_+$ was measured to be $\approx 119 a_0$ confuse you. $a_+$ is the scattering length at which $C_3$ is minimized, but due to the additional factor $a_s^4$ in the rate coefficient $L_3$, the actual minimum of the three-body loss rate is at slightly lower scattering lengths.]
with a decay parameter $\eta_+ \approx 0.008$ measured in <cit.> can be exploited to achieve high densities without suffering from excessive three-body decay.

Because the balanced mode pair is more balanced the closer the modes are and also in order to avoid producing unwanted phonons by single-mode squeezing, we will restrict ourselves to targeting only odd $n_\Omega$.

The crucial quantity, as before, will be the temperature of the condensate. We will assume the low, but reasonable value of $T=10$nK. If we then imagine the same number $N_0 \approx 3.6 \cdot 10^5$ of atoms that was quoted in <cit.> in an elongated cylindrical box trap with length $L= 150 \, \mu$m and radius $r = 5 \, \mu$m, all the conditions in table <ref> are met for balanced phonon pairs up to $n_\Omega=41$.

To squeeze to the highest factor $r_{15, 16}$ allowed for by the particle budget with the resonance condition $n_\Omega=31$, we can remind ourselves of equations (<ref>) and (<ref>) to find the appropriate linear perturbation $a_\Omega$ after choosing a squeezing time $t$

\begin{equation}
a_\Omega = \frac{\hbar \sqrt{2ln} \left(l^2-n^2 \right)^2 \pi^2 \xi \, r_{ln}}{2 \left(l^2 + n^2 \right) L^2 t} \; .
\end{equation}

For the setup in question, the healing length is $\xi \approx 450$nm. With the choice to take $t=1$s to squeeze, we get for the magnitude of the linear perturbation $a_\Omega \approx 6.9 \cdot 10^{-30} \mathrm{J} \mathrm{m}^{-1}$, which is two orders of magnitude below the maximum derived in section <ref> and hence produces only negligible perturbations in the ground state.

In the light of the gravitational wave detector proposal described in chapter <ref>, we can now get an estimate for the sensitivity using eq. (<ref>) with a single detector $N_d=1$, integration time $\tau$ of one year and a squeeze angle $\theta = \sqrt{0.092}$ <cit.>. As an ad-hoc ansatz to incorporate the overlap of the produced state with the ideal input state of a squeezed vacuum, we will take the Uhlmann fidelity of the final state after squeezing and a squeezed vacuum as an additional factor on the quantum Fisher information of the GW detector scheme. For $n_\Omega=31$ and thus the balanced pair of modes $(15, 16)$, we find

\begin{equation}
\Delta h_{+, 0} \sim 10^{-16} \; .
\end{equation}

Mind that this is before serious attempts of optimizing the squeezing scheme and before any attempt to improve the experimental parameters with respect to the GW detector. Thus, even though the strain sensitivity above does not allow for measurement of the sources considered in <cit.>, there is a vast parameter space that has to be explored.

$T = 10 \mathrm{nK}$

$N_{phon, i}$

$\omega_{n_\Omega} [\mathrm{Hz}]$

$r_{l_{max}, n_\Omega -l_{max}}$

$t_{life} [\mathrm{s}]$

$n_\Omega = 3$

$n_\Omega = 11$

$n_\Omega = 31$

$T = 5 \mathrm{nK}$

$N_{phon, i}$

$\omega_{n_\Omega} [\mathrm{Hz}]$

$r_{l_{max}, n_\Omega -l_{max}}$

$t_{life} [\mathrm{s}]$

$n_\Omega = 3$

$n_\Omega = 11$

$n_\Omega = 31$

Results on two-mode squeezing in the proposed $^7$Li BEC for the initial thermal phonon population number $N_{phon, i}$, the resonance frequency $\omega_{n_\Omega} [\mathrm{Hz}]$, the two-mode squeeze factor for the balanced mode pair $r_{l_{max}, n_\Omega -l_{max}}$, the life time $t_{life}$ of said mode pair and the Uhlmann fidelity $\mathcal{F}$ between the transformed state and a two-mode squeezed vacuum state.

For $T=10 \mathrm{nK}$, the state after the transformation begins to become useful with regard to quantum metrology applications at $n_\Omega \gtrsim 11$, which puts the resonance frequencies in the kHz range. While in lower modes, three-body losses are the dominant limit to the phonon life times, at higher modes Landau damping starts to become relevant.

If cooling can be improved to reach $T=5 \mathrm{nK}$, Landau damping loses its relevance up to $n_\Omega = 31$ and the $t_{life}$ becomes independent of the mode number. More importantly, though, thermal phonons are reduced further and thus the fidelity of the output state becomes useful for even lower lying resonances.

With both temperatures, the most striking difference between the three resonances depicted lies in the overlap of the final state with a squeezed vacuum state.

CHAPTER: CONCLUSION AND OUTLOOK

As does a majority of the work in present-day physics, this thesis stands on the shoulders of giants. This is pointed out here because the thesis builds upon the findings of an extensive range of fields, ranging from ultracold atoms and general relativity over quantum optics and quantum metrology to Efimov physics, some of them prominent and well-established, others recent and more obscure. It was therefore the intention of the author to provide the reader with the concepts that proved most essential for finding the presented results and refer to comprehensive sources wherever this would have been beyond the scope of the thesis.

The process leading up to it involved multiple unexpected turns, most notably the relevance of Efimov physics. Although at first, it presented itself merely as a technique to find the necessary relation between three-body losses and the interaction strength, its compelling features seeming out of reach, it turned out to be a promising tool retain reasonable life times of a condensate while simultaneously minimizing the thermal population of phonon modes and thus allowing for stronger squeezing and higher fidelities of the states produced.

Motivated by the initial proposal <cit.>, the ensuing debate <cit.> the ongoing work <cit.> on a BEC-based gravitational wave detector and with it the potential expansion of GW astronomy into previously inaccessible realms with apparatus that are more compact than the currently active interferometric GW detectors, the aim was to contribute to this debate and development by pointing out the details of the production of the necessary squeezed states. This was accomplished by adapting an idea originally intended to detect a potential by the transformation it evokes <cit.> to controlling said potential to produce squeezed phonons. Though hope might have originally been to simply maximize the squeeze factor contained in said transformation, it soon became apparent that the situation required a more intricate look. This led to what is, in the eyes of the author, the major achievement of this thesis: the consideration of the full transformation of the phononic states populating a uniform BEC under a periodic harmonic or linear perturbation.

With these theoretical findings, a mechanism for two-mode phonon squeezing was presented, which, albeit not well suited for the existing experiments that were taken as examples, can be highly effective in setups that seem within reach of present-day technology. It can thus be incorporated into the efforts going towards gravitational wave detectors, but conceivable applications also span over the entire field of frequency metrology.

A central question that remains to be addressed is the one of optimization. While qualitative arguments on where experiments could be improved and where the boundaries of the parameter space lie were given, the involved nature of the state transformation and with it exceedingly long evaluation times did not allow the author to tackle this task. Potential answers are twofold: one direction might be to find simplifications that are applicable in certain limiting cases, the other is for a more experienced programmer to streamline the computation process, allowing for quicker evaluation of the transformation.

For the incorporation of the results presented into GW detectors, the necessity of squeezed vacuum states as initial phonon states remains to be cleared. Work on this issue has been done and can be found in <cit.>. Furthermore, optimization of BEC parameters for the proposed detector scheme is currently being pursued and can now be done while at the same time taking phonon squeezing into account.

CHAPTER: BOGOLIUBOV TRANSFORMATIONS

The general definition of a Bogoliubov transformation is more broad than the one introduced in <ref>. While only the most important points for this work are given here, the interested reader is referred to Birrell and Davies' book <cit.> for a far more in-depth picture and, put into the context of relativistic quantum metrology, <cit.>.

Imagine a massless, real scalar quantum field (a boson field) $\hat{\Phi}$ satisfying the Klein-Gordon equation

\begin{equation} \label{eq:app_bogo_klein-gordon_massless}
\Box \hat{\Phi} = 0
\end{equation}

with periodic boundary conditions. One can then find a complete, discrete set of modes solving (<ref>) such that $\hat{\Phi}$ can be expanded as

\begin{equation} \label{eq:app_bogo_field_expansion}
\hat{\Phi} = \sum_i \hat{a}_i \varphi_i + \hat{a}^\dagger_i \varphi^*_i \; ,
\end{equation}

with the creation and annihilation operator satisfying the canonical commutation relations

[ â_i,   â^†_j ]    =    δ_ij

[ â_i,   â_j ]    =    0 = [ â^†_i,   â^†_j ]

The modes $\varphi_i$ are orthonormal

\begin{equation}
\left( \varphi_i , \varphi_j \right) = \delta_{ij}
\end{equation}

with respect to the Klein-Gordon inner product[The integral here goes over a spatial hypersurface for a fixed time t] <cit.>

( φ_i , φ_j )
- i ∫𝐱    φ_i ∂_t φ_j^* - ( ∂_t φ_i ) φ_j^*

- i ∫𝐱   φ_i ∂_t φ_j^*   .

The expansion (<ref>) is, in general, not unique. $\hat{\Phi}$ can be defined in terms of another set of ladder operators satisfying the same commutation relations and orthonormal modes

Φ̂    =    ∑_j b̂_j φ̅_j + b̂^†_j φ̅^*_i

=     ∑_i â_i φ_i + â^†_i φ^*_i   .

The modes and ladder operators are then connected by the Bogoliubov transformation <cit.>

φ̅_j    =    ∑_i
α_ji φ_i
β_ji φ^*_i

φ_i    =    ∑_j
α^*_ji φ̅_j
β_ji φ̅^*_i

α_ji ≡( φ̅_j, φ_i )
β_ji ≡- ( φ̅_j, φ_i^* )

⇒  â_i    =    ∑_j
α_ji b̂_j
β_ji^* b̂_j^†

b̂_j    =    ∑_i
α_ji^* â_i
β_ji^* â_i^†  .

$\alpha_{ji}$ and $\beta_{ji}$ are called the Bogoliubov coefficients. A Bogoliobov transformation can —but does not have to— correspond to a coordinate change between different observers, an illustrative example for which is the Unruh effect <cit.>, where an inertial observer's vacuum state appears as a thermal state to a uniformly accelerated observer. In general nonvanishing coefficients $\beta_{ji} \neq 0$ are responsible for particle creation, as is particularly well illustrated in (<ref>): while the vacuum state $\hat{b}_i | \bar{0} \rangle = 0$ has a zero eigenvalue for the annihilation operator $\hat{b}_j$, for another observer's annihilation operator related to $\hat{b}_j$ by (<ref>), we have $\hat{a}_i | \bar{0} \rangle \neq 0$ if $\beta_{ji} \neq 0$.

CHAPTER: THE COVARIANCE MATRIX FORMALISM

When dealing with continuous variable systems and especially with Gaussian states, the covariance matrix (CM) formalism <cit.> is a remarkably helpful tool. For once, this is because it lives on the phase space which admits to a direct sum structure as opposed to the tensor product structure of the Hilbert space. As a result, when considering only subsystems, there is no need for infinite-dimensional traces.

Gaussian states are uniquely determined by their first and second moments. In the CM formalism, they are therefore described by a vector and a covariance matrix encoding the former and the latter, respectively. Gaussian transformations[
Gaussian transformations preserve the “Gaussianity” of a state. They are given by unitaries whose exponents are at most quadratic in the creation and annihilation operators.
are then given by the set of symplectic transformations, which can be derived from their corresponding Hilbert space representation.

The representation of the CM formalism is, however, not unique. First and second moments can be defined in terms of expectation values of the creation and annihilation operators (this is referred to as the complex picture) or the quadratures[Even within the real picture, there are ambiguities in the definition of the quadratures and their ordering in the CM.] (leading to the real picture). The following sections will give the identities most important for the purposes of this thesis and conclude with the reconciliation of the pictures presented in <cit.>.

§ THE SYMPLECTIC FORM

Consider the bosonic creation and annihilation operators $\hat{a}_n$, $\hat{a}_n^\dagger$ for the mode $n$. The operators for all modes in an $N$-dimensional Hilbert space can then be collected in the vector $\hat{\mathbf{A}} \equiv \left( \hat{a}_1, \ldots \hat{a}_N, \hat{a}_1^\dagger , \ldots , \hat{a}_N^\dagger \right)^T$ and their commutation relations are encoded in the symplectic form $K$ <cit.>

\begin{equation}
K_{ij} \equiv \left[ \hat{\mathbf{A}}_i, \hat{\mathbf{A}}_j^\dagger \right] \; \Rightarrow \; K =
\begin{pmatrix}
1 & 0
\\
0 & -1
\end{pmatrix}
\otimes \mathbb{1}_N \; .
\end{equation}

$K$ is unitary and hermitian and thus $KK=\mathbb{1}$.

§ GAUSSIAN STATES: FIRST AND SECOND MOMENTS

In the complex picture of the CM formalism, the first and second moments of a state are expressed in terms of expectation values of the creation and annihilation operators as

𝐝     ≡    ⟨𝐀̂ ⟩

σ_ij     ≡
⟨𝐀̂_i 𝐀̂_j^†+
2 ⟨𝐀̂_i ⟩⟨𝐀̂_j^†⟩

The first moments, encoded in the vector $\mathbf{d}$ can be set to zero using local unitary operations and are thus mostly ignored in quantum metrological considerations. The relevant properties for these applications, however, lie in the covariance matrix $\sigma$.

As for the Hilbert space, where the density matrix $\rho$ of a physical state must fulfill

\begin{equation}
\mathrm{Tr} \, \rho = 1 \, , \; \rho \geq 0 \; ,
\end{equation}

there is the bona fide criterion

\begin{equation} \label{eq:CM_bona_fide}
\sigma + K \geq 0
\end{equation}

for covariance matrices, which also contains information about the commutation relations of the creation and annihilation operators and thereby includes the uncertainty relation.

Once the question of how to represent states in the phase space is settled, the natural next question is how they transform. The unitary transformations $\rho \rightarrow U \rho U^\dagger$ on the Hilbert space that preserve the Gaussian nature of a state are mirrored in the phase space by

\begin{equation} \label{eq:CM_symplectic_trafo_definition_new}
\sigma \, \rightarrow \, S \sigma S^\dagger \; ,
\end{equation}

where the linear transformation $S$ leaves the symplectic form $K$ invariant

\begin{equation} \label{eq:CM_symplectic_form_invariance_new}
S K S^\dagger = K
\end{equation}

and thus keeps the canonical commutation relations intact. Note that (<ref>) also implies $|S|=1$. The symplectic transformations in the equations above form are a complex representation of the real symplectic group $Sp (2N, \mathbb{R})$ <cit.>.

As previously mentioned, Gaussian transformations are unitaries whose exponents are, at most, quadratic in the creation and annihilation operators.

\begin{equation}
\hat{U} = \mathrm{e}^{
\frac{i}{2} \hat{\mathbf{A}}^\dagger W \hat{\mathbf{A}} + \hat{\mathbf{A}}^\dagger K \gamma
} \quad \mathrm{with} \quad
W = W^\dagger =
\begin{pmatrix}
X & Y
\\
Y^* & X^*
\end{pmatrix}
\; , \;
\gamma =
\begin{pmatrix}
\tilde{\gamma}
\\
\tilde{\gamma}^*
\end{pmatrix}
\end{equation}

Under the transformation represented by $\hat{U}$, $\sigma$ and $\mathbf{d}$ become <cit.>

\begin{equation} \label{eq:CM_transformation_behaviour}
\sigma \rightarrow S_U \sigma S_U^\dagger \; , \quad \mathbf{d} \rightarrow S_U \mathbf{d} + \mathbf{b}
\end{equation}

\begin{equation} \label{eq:CM_symplectic-unitary-correspondence}
S_U = \mathrm{e}^{i K W} \quad \mathrm{and} \quad \mathbf{b} = \left( \int_0^1 \ud t \mathrm{e}^{i K W t} \right) \gamma \; .
\end{equation}

The first part of (<ref>) is used chapter <ref> to derive the symplectic transformation corresponding to the time evolution (<ref>).

One symplectic transformation of particular importance is the diagonalization of $\sigma$ by virtue of Williamson's theorem <cit.>:

\begin{equation}
\sigma = S_D D S_D^\dagger \quad \mathrm{such \, that} \quad
D = \mathbb{1}_2 \otimes \mathrm{diag} \left( \lambda_1, \ldots \, \lambda_N \right) \, ,
\end{equation}

where $\{ \lambda_i \}$ are the symplectic eigenvalues. $D$ corresponds to a diagonal product state in the Hilbert space.

The bona fide condition (<ref>) also implies an expression of the uncertainty principle in terms of the symplectic spectrum of a CM <cit.>.

\begin{equation}
\lambda_i \geq 1 \; \forall \, i = 1,  \ldots, n
\end{equation}

Pure states saturate the uncertainty principle and hence all the symplectic eigenvalues of a pure state are equal to one.

§ QUANTUM METROLOGY IN THE CM FORMALISM

When trying to estimate a parameter $\epsilon$ that gets encoded into a state by a transformation, the central quantity is the quantum Fisher information (QFI) $H_\epsilon$. It puts an upper bound on the precision of the estimation of $\epsilon$ in the form of the quantum Cramér-Rao bound <cit.>

\begin{equation}
\langle \left( \Delta \hat{\epsilon} \right)^2 \rangle \geq \frac{1}{M H_\epsilon} \; .
\end{equation}

In the equation above, $\hat{\epsilon}$ is an estimator for the parameter in question, $M$ is the number of independent measurements and the bound is meant after optimization over all possible measurements.

The QFI tells us how far two states that are separated by an infinitesimal change $\ud \epsilon$ lie from each other and it is thus not surprising that it is related to a fidelity between $\sigma_\epsilon$ and $\sigma_{\epsilon + \ud \epsilon}$, more precisely the Uhlmann fidelity $\mathcal{F} \left( \sigma_\epsilon, \sigma_{\epsilon + \ud \epsilon} \right)$ <cit.>

\begin{equation} \label{eq:RQM_QFI_from_uhlmann_fidelity_new}
H_\epsilon = \frac{
8 \left( 1 - \sqrt{
\mathcal{F} \left( \sigma_\epsilon, \sigma_{\epsilon + \ud \epsilon} \right)
} \right)
\ud \epsilon^2
} \; .
\end{equation}

In the Hilbert space, the Uhlmann fidelity between two states $\rho_1$ and $\rho_2$ is defined as

\begin{equation}
\mathcal{F} (\rho_1, \rho_2) \equiv \left(
\mathrm{Tr} \sqrt{
\sqrt{\rho_1} \rho_2 \sqrt{\rho_1}
\right)^2
\end{equation}

For two-mode Gaussian states, an explicit expression for the equivalent $\mathcal{F} ( \sigma_1, \sigma_2)$ has been given in <cit.>. In the complex picture of the CM formalism, we will use the expression given in <cit.>:

ℱ ( σ_1, σ_2 )

4 e^
δ𝐝^†( σ_1 + σ_2 )^-1 δ𝐝

√(Γ) + √(Λ) - √(( √(Γ) + √(Λ) )^2 - Δ)

with   the   determinants

Δ    ≡    | σ_1 + σ_2 |

Γ    ≡    |
1 + K σ_1 K σ_2

Λ    ≡    | σ_1 + K |   | σ_2 + K |

δ𝐝     ≡    𝐝_1 - 𝐝_2   .

§ EXAMPLES OF COVARIANCE MATRICES AND SYMPLECTIC TRANSFORMATIONS

The vacuum state, whose first moments vanish, is a pure, completely uncorrelated state. Its CM must therefore be diagonal with all symplectic eigenvalues $\lambda_i=1$, which is simply an identity matrix. In $N$ modes, we have

\begin{equation}
\sigma_{vac} = \mathbb{1}_{2N} \; .
\end{equation}

This also means that any state that is derived by a Gaussian transformation of the vacuum (such as the squeezed vacuum state, see chapter <ref>) is found by taking $\sigma=SS^\dagger$.

The thermal state is also completely uncorrelated and thus diagonal. It is, however, a mixed state. To find the symplectic eigenvalues, consider the substate of a single mode $i$

\begin{equation}
\sigma_{th, i} = \begin{pmatrix}
\lambda_i & 0
\\
0 & \lambda_i
\end{pmatrix} \; ,
\end{equation}

take the expectation value of the particle number operator $\hat{N} \equiv \sum_{i=1}^N \hat{a}_i^\dagger \hat{a}_i$ <cit.>

\begin{equation} \label{eq:CM_particle_number_expectation_value_appendix}
\langle \hat{N} \rangle = \frac{1}{4} \left(
\mathrm{Tr} \left( \sigma \right) + \mathbf{d}^\dagger \mathbf{d} - 2N
\right)
\end{equation}

And remember the average occupation number for a single-mode thermal state in mode $i$, $\langle \hat{N} \rangle_{th} = 1/(\mathrm{e}^{\beta \hbar \omega_i}-1)$ with the inverse temperature $\beta$ and mode frequency $\omega_i$. Equating this and (<ref>), we find

⟨N̂ ⟩_th    =    1/4 ( 2 λ_i - 2 )

λ_i    =    2 ⟨N̂ ⟩_th + 1

⇒λ_i    =    2/e^βħω_i-1 + 1

and we have derived the CM of an $N$-mode thermal state

\begin{equation} \label{eq:CM_formalism_thermal_state_CM}
\sigma_{th} = \mathbb{1}_2 \otimes \mathrm{diag}
\left(
\coth \frac{\beta \hbar \omega_1}{2}, \ldots, \coth \frac{\beta \hbar \omega_N}{2}
\right) \; .
\end{equation}

Single-mode squeezing: starting with the real single-mode squeezing operator (see eq.(<ref>)) in the Hilbert space $\hat{U} = \mathrm{Exp} \left( -\frac{r}{2} ( \hat{a}^{\dagger 2} - \hat{a}^2 ) \right)$, we want to apply (<ref>) and find

\begin{equation}
W_{sq1}= i \, r \begin{pmatrix}
0 & 1
\\
-1 & 0
\end{pmatrix}
\end{equation}

and the symplectic transformation

\begin{equation}
S_{sq1} = \mathrm{Exp} \left[
-r \begin{pmatrix}
0 & 1
\\
-1 & 0
\end{pmatrix}
\right]
\begin{pmatrix}
\cosh r & -\sinh r
\\
-\sinh r & \cosh r
\end{pmatrix} \; .
\end{equation}

As mentioned above, the covariance matrix of the squeezed vacuum state is obtained by simply taking $S_{sq1} S_{sq1}^\dagger$:

\begin{equation}
\sigma_{sq1} =
\begin{pmatrix}
\cosh 2r & -\sinh 2r
\\
-\sinh 2r & \cosh 2r
\end{pmatrix} \; .
\end{equation}

Two-mode squeezing is produced by the two-mode squeezing operator (<ref>) which we take to be real for simplicity's sake to get

\begin{equation}
W_{sq2}= i \, r \begin{pmatrix}
0 & 0 & 0 & 1
\\
0 & 0 & 1 & 0
\\
0 & -1 & 0 & 0
\\
-1 & 0 & 0 & 0
\end{pmatrix}
\end{equation}

\begin{equation}
S_{sq2}= \begin{pmatrix}
\cosh r & 0 & 0 & -\sinh r
\\
0 & \cosh r & -\sinh r & 0
\\
0 & -\sinh r & \cosh r & 0
\\
-\sinh r & 0 & 0 & \cosh r
\end{pmatrix} \; .
\end{equation}

The squeezed vacuum state, finally, has the CM

\begin{equation}
\sigma_{sq2}= \begin{pmatrix}
\cosh 2r & 0 & 0 & -\sinh 2r
\\
0 & \cosh 2r & -\sinh 2r & 0
\\
0 & -\sinh 2r & \cosh 2r & 0
\\
-\sinh 2r & 0 & 0 & \cosh 2r
\end{pmatrix} \; .
\end{equation}

Bogoliubov transformations can be easily derived by considering (<ref>) and remembering the transformation behaviour of $\mathbf{d}$ (<ref>). One then finds

\begin{equation} \label{eq:CM_symplectic_trafo_from_bogo_new}
S_{Bog} = \begin{pmatrix}
\alpha & \beta
\\
\beta^* & \alpha^*
\end{pmatrix}
\; ,
\end{equation}

where $\alpha$ and $\beta$ are the matrices made up of the Bogoliubov coefficients $\alpha_{ij}$, $\beta_{ij}$.

§ THE REAL AND THE COMPLEX PICTURE

When reading about the CM formalism and its applications in different sources, one encounters different pictures of it, which are nonetheless equivalent. Primarily, one can divide them into the real and the complex picture as described in <cit.>. However, there is still ambivalence in both the definition of the quadratures in the real picture as well as the ordering of the subspaces, as can be seen by comparing the pictures listed above with the one found in Adesso's dissertation <cit.>.

The table on the next page shows how these pictures are reconciled and connected. The indices $r$ and $ad$ denote the real pictures in $\cite{safranek_quantum_2015}$ and <cit.>, respectively. Note that, due to the differing norm between their quadratures, the transformation $T_{ad \rightarrow r}$ from Adesso's formalism to the $r$ picture is not unitary.

Complex formalism

$U_{r \rightarrow c} = \frac{1}{\sqrt{2}} \begin{pmatrix}
\mathbb{1} & i \mathbb{1}
\\
\- \mathbb{1} & -i \mathbb{1}
\end{pmatrix}$
Real formalism

$T_{ad \rightarrow r, 2} = \frac{1}{\sqrt{2}}
\begin{pmatrix}
1 & 0 & 0 & 0
\\
0 & 0 & 1 & 0
\\
0 & 1 & 0 & 0
\\
0 & 0 & 0 & 1
\end{pmatrix}
Adesso's formalism <cit.>

$\hat{x}_{r, i} = \frac{1}{\sqrt{2}} (\hat{a}_i + \hat{a}_i^\dagger )$

$\hat{p}_{r, i} = -\frac{i}{\sqrt{2}} (\hat{a}_i - \hat{a}_i^\dagger )$

$\hat{x}_{ad, i} = \hat{a}_i + \hat{a}_i^\dagger$

$\hat{p}_{ad, i} = -i ( \hat{a}_i + \hat{a}_i^\dagger )$

$1^{st}$ order vector

$\hat{\mathbf{A}} = \left( \hat{a}_1, \ldots \hat{a}_N, \hat{a}_1^\dagger , \ldots , \hat{a}_N^\dagger \right)^T$

$\hat{\mathbf{A}} = U_{r \rightarrow c} \hat{\mathbf{Q}}$

$\hat{\mathbf{Q}} = \left( \hat{x}_{r, 1}, \ldots, \hat{x}_{r, N}, \hat{p}_{r, 1}, \ldots, \hat{p}_{r, N} \right)^T$

$\hat{\mathbf{Q}} = T_{ad \rightarrow r} \hat{\mathbf{R}}$

$\hat{\mathbf{R}} = \left(\hat{x}_{ad, 1}, \hat{p}_{ad, 1}, \ldots \hat{x}_{ad, N}, \hat{p}_{ad, N} \right)^T$

$1^{st}$ moments

$\mathbf{d} = \langle \hat{\mathbf{A}} \rangle$

$\mathbf{d}_r = \langle \hat{\mathbf{Q}} \rangle$

$\mathbf{R} = \langle \hat{\mathbf{R}} \rangle$

$K_{ij} \equiv \left[ \hat{\mathbf{A}}_i, \hat{\mathbf{A}}_j^\dagger \right]$

$K= \begin{pmatrix}
\mathbb{1}_N & 0
\\
0 & \mathbb{1}_N
\end{pmatrix}$

$i \Omega_{r, ij} \equiv \left[ \hat{\mathbf{Q}}_i, \hat{\mathbf{Q}}_j \right]$

$\Omega_{r, ij}= \begin{pmatrix}
0 & \mathbb{1}_N
\\
-\mathbb{1}_N & 0
\end{pmatrix}$

$2 i \Omega_{ad, ij} \equiv \left[ \hat{\mathbf{R}}_i, \hat{\mathbf{R}}_j \right]$

$\Omega_{ad} = \bigoplus_{i=1}^N \omega$

$\omega = \begin{pmatrix}
0 & 1
\\
-1 & 0
\end{pmatrix}$

$\sigma_{ij}= \left\langle
\hat{\mathbf{A}}_i \hat{\mathbf{A}}_j^\dagger + \hat{\mathbf{A}}_j^\dagger \hat{\mathbf{A}}_i
\right\rangle$

$- 2 \left\langle \hat{\mathbf{A}}_i \right\rangle \left\langle \hat{\mathbf{A}}_j^\dagger \right\rangle$

$\sigma = U_{r \rightarrow c} \sigma_r U_{r \rightarrow c}^\dagger$

$\sigma_{r, ij}= \left\langle
\hat{\mathbf{Q}}_i \hat{\mathbf{Q}}_j + \hat{\mathbf{Q}}_j \hat{\mathbf{Q}}_i
\right\rangle$

$- 2 \left\langle \hat{\mathbf{Q}}_i \right\rangle \left\langle \hat{\mathbf{Q}}_j \right\rangle$

$\sigma_r = T_{ad \rightarrow r} \sigma_{ad} T_{ad \rightarrow r}^{-1}$

$\sigma_{ad, ij}= \frac{1}{2} \left\langle
\hat{\mathbf{R}}_i \hat{\mathbf{R}}_j + \hat{\mathbf{R}}_j \hat{\mathbf{R}}_i
\right\rangle$

$-  \left\langle \hat{\mathbf{R}}_i \right\rangle \left\langle \hat{\mathbf{R}}_j \right\rangle$

of 1$^{st}$

moments, $\sigma$

$\mathbf{d} = \begin{pmatrix}
\tilde{\mathbf{d}}
\\
\tilde{\mathbf{d}}^*
\end{pmatrix},
\;
\sigma = \begin{pmatrix}
X & Y
\\
Y^* & X^*
\end{pmatrix}$

$\sigma^\dagger = \sigma$

\Leftrightarrow \,
X^\dagger = X, \, Y^T=Y$

\mathbf{d}_r= \begin{pmatrix}
\mathbf{x}
\\
\mathbf{p}
\end{pmatrix}, \;
\sigma_r = \begin{pmatrix}
X_r & Y_r
\\
Y_r^T & Z_r
\end{pmatrix}

\sigma_r^T = \sigma_r

\Leftrightarrow \; X_r^T = X_r, \, Z_r^T=Z_r

\mathbf{d}_{ad} = \bigoplus_{i=1}^N
\begin{pmatrix}
\hat{x}_{ad, i}
\\
\hat{p}_{ad, i}
\end{pmatrix}, \;
\sigma_{ad} = \begin{pmatrix}
X_{ad} & Y_{ad}
\\
Y_{ad}^T & Z_{ad}
\end{pmatrix}

\sigma_{ad}^T = \sigma_{ad}

\Leftrightarrow \; X_{ad}^T = X_{ad}, \, Z_{ad}^T=Z_{ad}

$\sigma = SDS^\dagger, \;
S= \begin{pmatrix}
\alpha & \beta
\\
\beta^* & \alpha^*
\end{pmatrix}$

D= \mathbb{1} \otimes L

L= \mathrm{diag} \left( \lambda_1 , \ldots \lambda_N \right)

S \tilde{\in} \mathrm{Sp} (2N, \mathbb{R})

$SKS^\dagger = K$

$S=U_{r \rightarrow c} S_r U_{r \rightarrow c}^\dagger$

$\sigma_r = S_r D_r S_r^T$

$D_r = D$

S_r =
\begin{pmatrix}
\alpha_r & \beta_r
\\
\gamma_r & \delta_r
\end{pmatrix}

\begin{pmatrix}
\mathrm{Re} ( \alpha + \beta ) & -\mathrm{Im} (\alpha - \beta )
\\
\mathrm{Im} ( \alpha + \beta ) & \mathrm{Re} (\alpha - \beta)
\end{pmatrix}

$S_r \in \mathrm{Sp} (2N, \mathbb{R})$

$S_r \Omega_r S_r^T = \Omega_r$

$S_r = T_{ad \rightarrow r, 2} S_{ad} T_{ad \rightarrow r, 2}^{-1}$

\sigma_{ad} = S_{ad}^T \nu S_{ad}

\nu = \bigoplus_{i=1}^N
\begin{pmatrix}
\nu_i & 0
\\
0 & \nu_i
\end{pmatrix}

S_{ad} = \begin{pmatrix}
\alpha_{ad, 11} & \alpha_{ad, 12}
\\
\alpha_{ad, 21} & \alpha_{ad, 22}
\end{pmatrix}

$\alpha_{ad, ij}=
\begin{pmatrix}
\mathrm{Re} (\alpha_{ij} + \beta_{ij}) & - \mathrm{Im} (\alpha_{ij} - \beta_{ij})
\\
-\mathrm{Im} (\alpha_{ij} + \beta_{ij}) & \mathrm{Re} (\alpha_{ij} - \beta_{ij})
\end{pmatrix}

$S_{ad} \in \mathrm{Sp} (2N, \mathbb{R})$

$S_{ad} \Omega_{ad} S_{ad}^T = \Omega_{ad}$

The complex representation of the covariance matrix formalism, the real representation and a variant of the latter as it is used in eg. <cit.>. Columns between the descriptions of the representations describe how to transform between them. The $2$ in the index of the transformation $T_{ad \rightarrow r, 2}$ denotes the number of modes that the transformation is displayed for. The tilde in $S \tilde{\in} \mathrm{Sp} (2N, \mathbb{R})$ indicates that in the complex formalism, $S$ is an element of a complex representation of the real symplectic group.

CHAPTER: EXAMPLE OF THE VARIANCE OF COMBINATIONS OF OBSERVABLES IN TWO-MODE SQUEEZED STATES

As an example, take the combination of quadratures $\hat{x}_\gamma^A \pm \hat{x}_\gamma^B$ as they are defined in chapter <ref> and where the upper indices $A, B$ denote the respective subspace. Since $\langle mm | (\hat{x}_\gamma^A \pm \hat{x}_\gamma^B) |nn \rangle = 0$, the expectation value for a squeezed state $ _2\langle \zeta | ( \hat{x}_\gamma^A \pm \hat{x}_\gamma^B ) | \zeta \rangle_2$ must vanish as well. For the variance, we thus only need to calculate $ \langle ( \hat{x}_\gamma^A \pm \hat{x}_\gamma^B )^2 \rangle_{\zeta_2}$. To do so, expand the square of the quadratures and use the bosonic commutation relations for creation and annihilation operators to find[
$\hat{a}^{(\dagger)}$ and $\hat{b}^{(\dagger)}$	denote the creation and annihilation operators in modes $A$ and $B$.

( x̂_γ^A ±x̂_γ^B )^2    =
e^-2i γ
1/2 ( â^2 + b̂^2 )
±â b̂
e^2i γ
1/2 ( â^†2 + b̂^†2 )

1 + â^†â + b̂^†b̂
â b̂^†+ â^†b̂
]   .

The only nonvanishing elements $\langle mm | (\hat{x}_\gamma^A \pm \hat{x}_\gamma^B)^2 |nn \rangle$ are then

\begin{equation}
\langle mm | (\hat{x}_\gamma^A \pm \hat{x}_\gamma^B)^2 |nn \rangle
\pm \left[
n \mathrm{e}^{-2i \gamma} \delta_{m, n-1} + (n+1) \mathrm{e}^{2i \gamma} \delta_{m, n+1}
\right]
(1+2n) \delta_{n, n}
\end{equation}

and the variance becomes

Δ^2 ( x̂_γ^A ±x̂_γ^B )
= 1/cosh^2 r [
±e^i(θ- 2 γ) ∑_n=1^∞ (-1)^2n-1 n tanh^2n-1r

±e^-i(θ- 2 γ) ∑_n=0^∞ (-1)^2n+1 (n+1) tanh^2n+1r

+ ∑_n=0^∞ (-1)^2n (2n+1) tanh^2nr

= 1/cosh^2 r [
e^i(θ- 2 γ) + e^-i(θ- 2 γ)
∑_n=0^∞ (-1)^2n+1 (n+1) tanh^2n+1r

+ ∑_n=0^∞ (-1)^2n (2n+1) tanh^2nr

cosh(2r) ∓cos(θ- 2 γ) sinh(2r)   .

For $\theta = 2 \gamma$, the variance is thus minimized to

\begin{equation}
\Delta^2 ( \hat{x}_{\theta/2}^A \pm \hat{x}_{\theta/2}^B ) = \mathrm{e}^{\mp 2r} \; .
\end{equation}

Analogously, for the joint measurement of the quadratures $( \hat{p}_\gamma^A \pm \hat{p}_\gamma^B )$, we get

\begin{equation}
\Delta^2 ( \hat{p}_\gamma^A \pm \hat{p}_\gamma^B ) = \cosh(2r) \pm \cos( \theta - 2 \gamma ) \sinh (2r)
\end{equation}

and at $\theta = 2 \gamma$

\begin{equation}
\Delta^2 ( \hat{p}_{\theta/2}^A \pm \hat{p}_{\theta/2}^B ) = \mathrm{e}^{\pm 2r} \; .
\end{equation}

CHAPTER: DERIVATIONS FOR THE SQUEEZING MECHANISM

§ BDG EQUATIONS IN A BOX TRAP

For a box type cavity of length $L$, cross section $A$ and volume $V=LA$, the condensate inside the trap is uniform $\psi_0 = \sqrt{n_0}$ and the chemical potential is $\mu = U_0 n_0$ (see section <ref>). The BdG equations (<ref>) thus become

ħω_n u_n ( 𝐫 )    =    ( - ħ^2/2m ∇^2 - U_0 n_0 + 2 U_0 n_0 ) u_n ( 𝐫 ) + U_0 n_0 v_n ( 𝐫 )

-ħω_n v_n ( 𝐫 )    =    ( - ħ^2/2m ∇^2 - U_0 n_0 + 2 U_0 n_0 ) v_n ( 𝐫 ) + U_0 n_0 u_n ( 𝐫 )

ω_n u_n ( 𝐫 )    =    ħ/2m [ ( -  ∇^2 + 2m U_0 n_0/ħ^2 ) u_n ( 𝐫 ) + 2m U_0 n_0/ħ^2 n_0 v_n ( 𝐫 ) ]

- ω_n v_n ( 𝐫 )    =    ħ/2m [ ( -  ∇^2 + 2m U_0 n_0/ħ^2 ) v_n ( 𝐫 ) + 2m U_0 n_0/ħ^2 n_0 u_n ( 𝐫 ) ]

with von Neumann boundary conditions at $x= \pm L/2$. $2m U_0 n_0/ \hbar^2$ can be recognized as the healing length $\xi$ (<ref>). Furthermore, the perturbing potential $\delta \mathcal{V}$ depends only on $x$ (and on the time, of course). Therefore, only modes in the $x$-direction are considered <cit.> and the stationary BdG equations simplify to

ω_n u_n ( x)    =    ħ/2m [ ( -  ∇^2 + ξ) u_n ( x) + ξn_0 v_n ( x) ]

- ω_n v_n ( x)    =    ħ/2m [ ( -  ∇^2 + ξ) v_n ( x) + ξn_0 u_n ( x) ]

with the mode solutions <cit.>

u_n    =    α_n φ_n

v_n    =    β_n φ_n

α_n    =    √(1/V) √(1/√(2)k_n ξ + 1)
β_n    =    - √(1/V) √(1/√(2)k_n ξ - 1)

φ_n    =    cos( k_n (x+L/2) )

k_n    =    n π/L

from <ref>, the Bogoliubov dispersion relation can be recovered in the limit $k_n \xi \ll 1$

ω_n    =    ħ/2 m α_n ξ^2 [

(k_n ξ)^2
) α_n + β_n

=    ħ/2m ξ^2
1 +

-1 + √(2)k_n ξ+ 𝒪(k_n ξ)^2

⇒ω_n    =    c_s k_n

by comparison to (<ref>).

§ DERIVATION OF THE INTERACTION PICTURE HAMILTONIAN AND THE TIME EVOLUTION OPERATOR FOR THE SQUEEZING MECHANISM

Take the interaction Hamiltonian before changing into the interaction picture

\begin{equation}
H^{(1)} = A \sin \Omega t \int \ud x \, \left( \delta \bar{\mathcal{V}} - \delta \bar{\mu} \right) \left( \sqrt{n_0} \left( \delta \hat{\psi} + \delta \hat{\psi}^\dagger \right) + \delta\hat{\psi}^\dagger \delta \hat{\psi} \right)
\end{equation}

and expand $\delta \hat{\psi}, \delta \hat{\psi}^\dagger$ in the mode solutions

H^(1)_int = A sinΩt
∫x ( δ𝒱̅ - δμ̅ ) [
√(n_0) ∑_n ( u_n + v_n ) ( b̂_n + b̂_̂n̂^†)

∑_n, m u_n v_m ( b̂_n b̂_m + b̂_n^†b̂_m^†)

∑_n, m u_n u_m b̂_n b̂_m^†+ v_n v_m b̂_n^†b̂_m ]   .

Dropping the ground state energy and thereby a global phase in the time evolution, we can use the diagonalized Bogoliubov Hamiltonian (<ref>) to get for the quasi-particle creation and annihilation operators in the interaction picture $\hat{b}_{n, I}= \mathrm{e}^{-i \omega_n t} \hat{b}_n$ and $\hat{b}_{n, I}^\dagger = \mathrm{e}^{i \omega_n t} \hat{b}_n^\dagger$ and therefore

H^(1)_int, I
= A sinΩt [
√(n_0) ∑_n ∫x   (
δ𝒱̅ - δμ) ( u_n + v_n )
e^-i ωt b̂_n + e^i ωt b̂_n^†)

∑_n, m ∫x
δ𝒱̅ - δμ)
{ u_n v_m
e^-i ( ω_n + ω_n ) t b̂_n b̂_m + e^i ( ω_n + ω_m ) t b̂_n^†b̂_m^†)

u_n u_m e^-i ( ω_n - ω_m ) t b̂_n b̂_m^†+ v_n v_m e^i ( ω_n - ω_m ) t b̂_n^†b̂_m
]   .

The index $I$ denotes the interaction picture here. In order to take the rotating wave approximation (RWA), pick out the term containing the $u_n u_m$ factor, use $\sin x = -\frac{i}{2} (e^{ix} + \mathrm{e}^{-ix})$ and split the sum up into contributions with $n>m$, $n<m$ and $n=m$. For the latter, one can immediately see that it is neglected in the RWA because the phase from the mode frequencies alone cancels out. For the former two parts of the sum, we can group the pre-factors to $\tilde{\mathcal{A}}_{nm} \equiv A \int \ud x \, \left(
\delta \bar{\mathcal{V}} - \delta \mu
\right) u_n v_m = \tilde{\mathcal{A}}_{mn}$. With that, we can write

∑_n>m 𝒜̃_nm e^-i ( ω_n - ω_m )t b̂_n b̂_m^†sinΩt
∑_n<m 𝒜̃_nm e^-i ( ω_n - ω_m )t b̂_n b̂_m^†sinΩt

∑_n>m 𝒜̃_nm e^-i ( ω_n - ω_m )t b̂_n b̂_m^†sinΩt
∑_m<n 𝒜̃_mn e^-i ( ω_m - ω_n )t b̂_m b̂_n^†sinΩt

∑_n>m 𝒜̃_nm e^-i ( ω_n - ω_m )t b̂_n b̂_m^†sinΩt
∑_n>m 𝒜̃_nm e^i ( ω_n - ω_m )t b̂_n^†b̂_m sinΩt

- i/2 𝒜̃_nm

e^-i ( ω_n - ω_m - Ω) t b̂_m^†b̂_n
- e^i ( ω_n - ω_m - Ω) t b̂_n^†b̂_m

An analogous procedure is necessary for the term proportional to $v_n v_m$. For the first two lines in (<ref>), the mode frequencies are either separate or added in the exponents and the sum does not need to be split up to take the RWA. One then gets

H^(1)_int, I
ℳ_0n (u_n + v_n)
e^-i ( ω_n - Ω) t b̂_n - e^i ( ω_n - Ω) t b̂_n^†)

+ ∑_n, m
b̂_n b̂_m e^-i ( ω_n + ω_m - Ω) t
- b̂_n^†b̂_m^†e^i ( ω_n + ω_m - Ω) t

+ ∑_n>m
𝒜_nm + ℬ_nm
e^-i ( ω_n - ω_m - Ω) t b̂_m^†b̂_n
- e^i ( ω_n - ω_m - Ω) t b̂_n^†b̂_m

with the matrix elements defined as

ℳ_0n     ≡
- i/2 √(n_0) A ∫_-L/2^L/2 x
δ𝒱̅ - δμ̅
( u_n + v_n )

ℳ_nm     ≡
- i/2 A ∫_-L/2^L/2 x
δ𝒱̅ - δμ̅
u_n v_m

𝒜_nm     ≡
- i/2 A ∫_-L/2^L/2 x
δ𝒱̅ - δμ̅
u_n u_m

ℬ_nm     ≡
- i/2 A ∫_-L/2^L/2 x
δ𝒱̅ - δμ̅
v_n v_m   .

In the phonon regime $k_n \ll \xi^{-1}$, we have $u_n \approx - v_n \approx (\sqrt{2} k_n \xi)^{-1/2} \cos (k_n (x+L/2))$ and therefore $\mathcal{M}_{nm} \approx - \mathcal{B}_{nm} \approx - \mathcal{B}_{nm}$

For a uniform ground state, the time-independent part of perturbation to the chemical potential (<ref>) reduces to

δμ̅    =

A ∫_-L/2^L/2 x
n_0 δ𝒱̅ (x)
A ∫x   n_0

∫_-L/2^L/2 x
a_Ωx + 𝔊_Ωx^2/2

=    𝔊_ΩL^2/24   .

For the matrix element $\mathcal{M}_{0n}$, this means

ℳ_0n    =
- i/2 √(n_0) A ( α_n +  β_n ) ·
·∫_-L/2^L/2 x
( a_Ωx + 𝔊_Ω( x^2/2 + L^2/24 ) ) cos( n π/L ( x+ L/2 ) )

i √(
L^3 N_0 ξ/( √(2) n π)^3
1- (-1)^n
) a_Ω/L
1+ (-1)^n
) 𝔊_Ω/2
)   ,

where the coefficient $(\alpha_n + \beta_n )$ has been expanded up to $\mathcal{O}(k_n \xi)^{1/2}$.

Apart from differing coefficients, the matrix elements $\mathcal{M}_{nm}$, $\mathcal{A}_{nm}$ and $\mathcal{B}_{nm}$ all share the same integral which we shall denote by $\mathcal{C}_{nm}$.

𝒞_nm    =
∫_-L/2^L/2 x
a_Ωx + 𝔊_Ω( x^2/2 + L^2/24 )
cos( n π/L ( x+ L/2 ) )
cos( m π/L ( x+ L/2 ) )

- ( m^2 + n^2 ) L^3/( m^2 - n^2 )^2 π^2
( 1 - (-1)^m+n ) a_Ω/L
( 1 + (-1)^m+n ) 𝔊_Ω/2
)     for  n ≠m

𝔊_ΩL^3/8 n^2 π^2     for  n=m

Expanding again for small $k_n \xi$ and $k_m \xi$, the coefficients of the remaining matrix elements become

\begin{equation}
\alpha_n \beta_m \approx - \alpha_n \alpha_m \approx - \beta_n \beta_m \approx - \frac{1}{\sqrt{2 m n} \pi \xi A} \; .
\end{equation}

Combine with (<ref>) and (<ref>) to get

-i  ( m^2 + n^2 ) L^3/
√(2 m n) ( m^2 - n^2 )^2 π^3 ξ
( 1 - (-1)^m+n ) a_Ω/L
( 1 + (-1)^m+n ) 𝔊_Ω/2
for  n ≠m

i 𝔊_ΩL^3/16 √(2) n^3 π^2 ξ
for  n=m

ℬ_nm ≈- ℳ_nm

CHAPTER: MATHEMATICA CODE FOR THE SYMPLECTIC TRANSFORMATION AND THE RESULTING PHONON NUMBERS AND UHLMANN FIDELITY

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