Squeezing of light from Planck-scale physics

The concept of a minimal length is one of the most widely examined potential manifestations of quantum characteristics of spacetime. The premise here is that there exists a finite limit to the precision at which space can be probed, anticipated to be in the vicinity of the Planck length, $l_{\text{Pl}}\approx 1.62\cdot 10^{-35}$ m. This idea originates from quantum gravity considerations, which suggest that at such extreme scales, traditional concepts of spacetime break down and give way to quantum effects.

The idea of a minimal length has been successfully integrated into a self-consistent special-relativistic framework, commonly known as doubly special relativity Magueijo:2001cr ; Amelino-Camelia:2002cqb . This paradigm introduces a new invariant scale in addition to the speed of light, typically the Planck length, to reconcile quantum mechanics with relativity. In the broader landscape of quantum gravity theories, some approaches, such as loop quantum gravity (LQG), propose a possible generalization of this concept to a general-relativistic context, thus extending its scope and potential implications. Nevertheless, the actual behavior and impact of this minimal length scale in the dynamical sector remains a subject of ongoing study.

The concept of maximal spatial resolution is naturally implementable in the realm of quantum mechanics, particularly in relation to the Heisenberg uncertainty principle. This principle dictates inherent limits to the precision with which pairs of canonical variables, such as generalized position and momentum, can be simultaneously measured. By employing a generalized uncertainty principle (GUP) Bojowald:2011jd ; Bosso:2023aht , it is feasible to integrate a minimal length scale into conventional quantum mechanics, thereby suggesting a fundamental limit to our ability to precisely measure position. Nonetheless, this adaptation inevitably implies that the commutation relation between the canonical variables, such as the generalized position and momentum, will undergo alterations.

Modifications to the commutation relation are understood to correlate with the nonlinear geometry of the corresponding phase space. Notably, the existence of a minimum length, a potential hallmark of quantum gravity theories, is intimately associated with a curvature in the momentum component of the phase space. This relationship provides a geometric interpretation for the minimal quantum length Wagner:2021bqz .

This concept is not confined to the domain of point particles – it extends into the broader field theoretical context. An instance of this expansion can be found in the LQG-inspired polymer quantization. In this scenario, studies such as those presented in Refs. Hossain:2010wy ; Hossain:2010eb analyzed cylindrical (i.e., $\mathbb{R}\times\mathbb{S}$) deformations of the scalar field phase space. Other works in the context of string theory and quantum gravity also generalize the notion of phase space to deformed (possibly curved) ones Freidel:2013zga ; Freidel:2017xsi ; Riello:2017iti ; Dittrich:2017nmq ; Haggard:2015ima ; Bonzom:2014wva ; Dupuis:2019yds . The nonlinear field space theory (NFST) research program, as outlined in Ref. Mielczarek:2016rax , is committed to methodically explore the implications of fields having a nontrivial phase space. In the context of NFST, the case of $\mathbb{S}^{2}$ phase-space generalization of the scalar field theory has been primarily considered Artigas:2020hfj ; Artigas:2020fab .

The effects of a GUP can lead to phenomenological consequences, currently under intense investigation in the context of multimessenger astronomy Addazi:2021xuf . The physical consequences that are primarily analyzed are deformed dispersion relations and vacuum birefringence.

Here, we focus our attention on the properties of quantum states of light. Specifically, we focus on single and multimode light with GUP in a field theoretical context. Potential manifestations of the GUP have been previously explored in the particle context in Ref. Pikovski:2011zk . Our aim is to build upon these existing insights and delve deeper into the understanding of light under the influence of GUP.

This article is structured as follows: In Sec. II, we introduce the model of the Generalized Uncertainty Principle (GUP) and the ensuing single-mode Hamiltonian. Following this, we carry out a perturbative analysis of the time evolution of a single-mode light in Sec. III. Based on these findings, we then examine the quantum squeezing of various states due to the GUP in Sec. IV. As a supplement to the squeezing effect, Sec. V delves into how the GUP modifies the dispersion relation for a multimode light state. In Sec. VI, we consider the potential experimental implications of our derived predictions. Last, in Sec. VII, we summarize our findings and discuss potential future directions.

The most widely studied generalized uncertainty principle leading to a minimal length is of the form Das:2008kaa :

$$\Delta Q\Delta P\geq\frac{\hslash}{2}(1+\beta\Delta P^{2}),$$

(1)

where $\Delta Q$ and $\Delta P$ are uncertainties on the generalized position and momenta, respectively. Here, $\beta$ is a small dimensional parameter, serving as a measure of the strength of quantum gravity effects. This parameter is assumed to be positive definite here. However, negative values of $\beta$ have also been studied in the literature Bojowald:2011jd ; Ong:2018zqn .

Importantly, in this article, the framework of field theory is considered, for which $[Q]=E^{-1/2}$, $[P]=E^{1/2}$ and as a consequence $[\beta]=E^{-1}$. Since the modification is considered to be due to Planck-scale physics, it is therefore expected that $\beta\sim 1/E_{\text{Pl}}$, where the Planck energy $E_{\text{Pl}}\approx 1.22\cdot 10^{19}$ GeV. Notably, this field theory scenario differs substantially from the usual case of a point particle, for which $[Q]=E^{-1}$, $[P]=E$, and consequently $[\beta]=E^{-2}$, leading to the $\beta$ parameter being $\beta\sim 1/E_{\text{Pl}}^{2}$. The differing dimensions of the canonical variables between both scenarios, therefore, suggest that the effects of new physics at the Planck scale transcribed by the GUP could have a more pronounced impact in the field theory context (where they are expected to be suppressed by $E/E_{\text{Pl}}$ to some power) compared to the point particle scenario (where they are expected to be suppressed by $E^{2}/E_{\text{Pl}}^{2}$ to some power).

Additionally, while the GUP in the context of a point particle implies the existence of a minimal physical length, the situation differs in the field theoretical framework considered here. In this setting, the GUP does not give rise to a minimal length, but rather suggests a minimal field value.

The GUP given by Eq. (1) can be derived from the deformed commutation relation:

$$[\hat{Q},\hat{P}]=i\hslash(\hat{\mathbb{I}}+\beta\hat{P}^{2}).$$

(2)

This commutation relation is predicted by different Planck-scale physics models, such as relative locality Amelino-Camelia:2011lvm or loop quantum gravity Bojowald:2011jd ; Mielczarek:2013rva .

An observation made in Pedram:2011aa is that this commutation relation (2) transforms into the standard one $[\hat{q},\hat{p}]=i\hslash\mathds{1}$ under the following change of variables:

	$\displaystyle\hat{Q}$	$\displaystyle=\hat{q},$		(3)
	$\displaystyle\hat{P}$	$\displaystyle=\frac{\tan(\sqrt{\beta}{\hat{p}})}{\sqrt{\beta}}.$		(4)

This can be proven using the fact that for any function $f(\hat{p})$, $[\hat{q},f(\hat{p})]=i\hslash\frac{df(\hat{p})}{d\hat{p}}$ if $[\hat{q},\hat{p}]=i\hslash\hat{\mathbb{I}}$. Since the commutation relation changes, the above change of variables is not a canonical transformation.

This article aims to consider the quantum properties of light, taking into account the deformed commutation relation (2). Some studies in this direction have already been made in Refs. Bosso:2017ndq ; Conti:2018hhj .

The simplest case we are going to begin with is a single-mode light for which the standard Hamiltonian takes the form

$$\hat{H}=\frac{1}{2}\left(\hat{P}^{2}+\omega^{2}\hat{Q}^{2}\right),$$

(5)

where $\omega$ denotes the frequency of the mode, so that $[\omega]=E$. Consequently, $[\beta\omega\hbar]=1$. Importantly, the polarization states of light are not considered here but the amplitude of the field solely.

Employing the change of variables (3) and (4) and expanding the obtained expression up to the linear order in $\beta$ leads to

$$\hat{H}=\frac{1}{2}\left(\hat{p}^{2}+\omega^{2}\hat{q}^{2}\right)+\frac{\beta}{3}\hat{p}^{4}+\mathcal{O}(\beta^{2}).$$

(6)

Worth mentioning is that only even powers of $\hat{p}$ are contributing to the series. From now on all $\mathcal{O}(\beta^{2})$ contributions will be neglected and we will focus only on the leading effect .

For further convenience, the full Hamiltonian (6) is decomposed into a free $\hat{H}_{0}=\frac{1}{2}\left(\hat{p}^{2}+\omega^{2}\hat{q}^{2}\right)$, and an interaction $\hat{H}_{1}=\frac{\beta}{3}\hat{p}^{4}$ parts, such that

$$\hat{H}=\hat{H}_{0}+\hat{H}_{1}.$$

(7)

At this point, it is useful to introduce the standard creation and annihilation operators $\hat{a}^{\dagger}$ and $\hat{a}$ defined in the usual way:

	$\displaystyle\hat{q}$	$\displaystyle:=\sqrt{\frac{\hslash}{2\omega}}\left(\hat{a}^{\dagger}+\hat{a}\right),$		(8)
	$\displaystyle\hat{p}$	$\displaystyle:=i\sqrt{\frac{\hslash\omega}{2}}\left(\hat{a}^{\dagger}-\hat{a}\right),$		(9)

so that $[\hat{a},\hat{a}^{\dagger}]=\hat{\mathbb{I}}$. The free Hamiltonian then reads $\hat{H}_{0}=\frac{\hslash\omega}{2}\left(\hat{a}^{\dagger}\hat{a}+\hat{a}\hat{a}^{\dagger}\right)$ and verifies $[\hat{H}_{0},\hat{a}^{\dagger}]=\hslash\omega\hat{a}^{\dagger}$ and $[\hat{H}_{0},\hat{a}]=-\hslash\omega\hat{a}$. Furthermore, the interaction Hamiltonian reads

$$\hat{H}_{1}=\frac{\beta}{3}\left(\frac{\hslash\omega}{2}\right)^{2}\left(\hat{a}^{\dagger}-\hat{a}\right)^{4}.$$

(10)

By applying the standard perturbation theory, one can find that at the first order of the perturbative expansion, the Hamiltonian (7) eigenvalues are

	$\displaystyle E_{n}^{(1)}$	$\displaystyle=\bra{n^{(0)}}\hat{H}_{0}+\hat{H}_{1}\ket{n^{(0)}}$
		$\displaystyle=\bra{n}\hslash\omega\left(\hat{N}+\frac{1}{2}\hat{\mathbb{I}}\right)\ket{n}+\bra{n}\frac{\beta}{3}\hat{p}^{4}\ket{n}$
		$\displaystyle=\hslash\omega\left(n+\frac{1}{2}\right)+\beta\frac{\hslash^{2}\omega^{2}}{4}\left(2n^{2}+2n+1\right),$		(11)

in which $\hat{N}:=\hat{a}^{\dagger}\hat{a}$. One easily recovers from this expression that the difference of energy levels at zeroth order in $\beta$ is $E_{n}^{(0)}-E_{m}^{(0)}=\hslash\omega(n-m)$.

The associated eigenstates are, at first order in $\beta$,

	$\displaystyle\ket{n^{(1)}}$	$\displaystyle:=\ket{n}+\sum_{m\neq n}\frac{\bra{m}\hat{H}_{1}\ket{n}}{E_{n}^{(0)}-E_{m}^{(0)}}\ket{m}$
		$\displaystyle=\ket{n}+\frac{\beta\hslash\omega}{12}\left[-\frac{1}{4}\sqrt{\frac{(n+4)!}{n!}}\ket{n+4}\right.$
		$\displaystyle+\sqrt{n+1}\sqrt{n+2}\left(2n+3\right)\ket{n+2}$
		$\displaystyle-\sqrt{n}\sqrt{n-1}\left(2n-1\right)\ket{n-2}$
		$\displaystyle+\left.\frac{1}{4}\sqrt{\frac{n!}{(n-4)!}}\ket{n-4}\right].$		(12)

In particular, the first-order vacuum energy is

$\displaystyle E_{0}^{(1)}=\frac{\hslash\omega}{2}\left(1+\beta\frac{\hslash\omega}{2}\right),$

(13)

and the first-order vacuum state

$$\ket{0^{(1)}}=\ket{0}+\frac{\beta\hslash\omega}{12}\left(3\sqrt{2}\ket{2}-\sqrt{\frac{3}{2}}\ket{4}\right).$$

(14)

Therefore, the GUP correction slightly leverages the ground state energy.

The time evolution of a single-mode light can be studied by introducing the following operator:

$$\hat{F}(t):=\hat{U}^{-1}_{0}(t)\hat{U}(t),$$

(15)

where the unitary operator $\hat{U}_{0}(t):=\exp\left(-\frac{i}{\hslash}\hat{H}_{0}t\right)$ utilizes the free part of the Hamiltonian (7), whereas $\hat{U}(t)=\exp\left(-\frac{i}{\hslash}\hat{H}t\right)$. The operator $\hat{F}$ satisfies the equation

$$\frac{d\hat{F}(t)}{dt}=-\frac{i}{\hslash}\hat{H}_{1}^{I}(t)\hat{F}(t),$$

(16)

which has a solution in the form of a Dyson series:

	$\displaystyle\hat{F}(t)$	$\displaystyle=\hat{T}\exp\left(-\frac{i}{\hslash}\int_{0}^{t}\hat{H}_{1}^{I}(t^{\prime})dt^{\prime}\right)$
		$\displaystyle=\hat{\mathbb{I}}-\frac{i}{\hslash}\int_{0}^{t}\hat{H}_{1}^{I}(t^{\prime})dt^{\prime}+\mathcal{O}(1/\hslash^{2}),$		(17)

where

$$\hat{H}_{1}^{I}(t):=\hat{U}^{-1}_{0}(t)\hat{H}_{1}\hat{U}_{0}(t)$$

(18)

is the interaction Hamiltonian in the interaction picture. Furthermore, $\hat{T}$ is the time ordering operator, and the operator $\hat{F}(t)$ satisfies the initial condition $\hat{F}(0)=\hat{\mathbb{I}}$.

The time evolution of an initial state $|\Psi(0)\rangle$ is given by

	$\displaystyle|\Psi(0)\rangle=$	$\displaystyle\ \hat{U}(t)|\Psi(0)\rangle$
	$\displaystyle=$	$\displaystyle\ \hat{U}_{0}(t)\hat{F}(t)|\Psi(0)\rangle$
	$\displaystyle=$	$\displaystyle\ \hat{U}_{0}(t)|\Psi(0)\rangle$
		$\displaystyle-\frac{i}{\hslash}\hat{U}_{0}(t)\int_{0}^{t}\hat{H}_{1}^{I}(t^{\prime})dt^{\prime}|\Psi(0)\rangle+\mathcal{O}(1/\hslash^{2}).$		(19)

From now on, all $\mathcal{O}(1/\hslash^{2})$ contributions will be neglected and we will focus on the leading effect only.

Employing the Baker-Campbell-Hausdorff formula, one can find that

	$\displaystyle\hat{U}^{-1}_{0}(t)\hat{a}^{\dagger}\hat{U}_{0}(t)$	$\displaystyle=\hat{a}^{\dagger}e^{i\omega t},$		(20)
	$\displaystyle\hat{U}^{-1}_{0}(t)\hat{a}\hat{U}_{0}(t)$	$\displaystyle=\hat{a}e^{-i\omega t}.$		(21)

The use of the above leads to

$$\hat{H}_{1}^{I}(t)=\frac{\beta}{3}\left(\frac{\hslash\omega}{2}\right)^{2}\left(\hat{a}^{\dagger}e^{i\omega t}-\hat{a}e^{-i\omega t}\right)^{4},$$

(22)

for $\hat{H}_{1}$ given by the interaction term in Eq. (6).

When developing this expression and using the commutation relation between the creation and annihilation operators, one gets the following form of the interaction Hamiltonian in the interaction picture¹¹1It may be useful to observe that $\displaystyle\hat{a}\left(\hat{a}^{\dagger}\right)^{n}$ $\displaystyle=j\left(\hat{a}^{\dagger}\right)^{n-1}+\left(\hat{a}^{\dagger}\right)^{j}\hat{a}\left(\hat{a}^{\dagger}\right)^{n-j}\,,$ (23) $\displaystyle\left(\hat{a}\right)^{n}\hat{a}^{\dagger}$ $\displaystyle=j\left(\hat{a}\right)^{n-1}+\left(\hat{a}\right)^{n-j}\hat{a}^{\dagger}\left(\hat{a}\right)^{j}\,,$ (24) for all integers $j\in[0,n]$.:

	$\displaystyle\hat{H}_{1}^{I}(t)=$	$\displaystyle\frac{\beta\hslash^{2}\omega^{2}}{12}\big{[}\left(\hat{a}^{\dagger}\right)^{4}e^{4i\omega t}$
		$\displaystyle-\left(4\left(\hat{a}^{\dagger}\right)^{3}\hat{a}+6\left(\hat{a}^{\dagger}\right)^{2}\right)e^{2i\omega t}$
		$\displaystyle+\left(6\left(\hat{a}^{\dagger}\right)^{2}\left(\hat{a}\right)^{2}+12\hat{a}^{\dagger}\hat{a}+3\hat{\mathbb{I}}\right)$
		$\displaystyle-\left(4\hat{a}^{\dagger}\left(\hat{a}\right)^{3}+6\left(\hat{a}\right)^{2}\right)e^{-2i\omega t}$
		$\displaystyle+\left(\hat{a}\right)^{4}e^{-4i\omega t}\big{]}\,.$		(25)

The time integral contributing to the series formula (19) can be analytically evaluated and decomposed into real and imaginary parts:

$$\int_{0}^{t}\hat{H}_{1}(t^{\prime})dt^{\prime}=\beta\frac{\hslash^{2}\omega}{24}\left(\hat{\mathcal{R}}(t)+i\hat{\mathcal{I}}(t)\right),$$

(26)

with

	$\displaystyle\hat{\mathcal{R}}(t)=$	$\displaystyle 2\omega t\left[6\left(\hat{a}^{\dagger}\right)^{2}\left(\hat{a}\right)^{2}+12\hat{a}^{\dagger}\hat{a}+3\hat{\mathbb{I}}\right]$		(27)
	$\displaystyle+$	$\displaystyle\frac{1}{2}\sin\left(4\omega t\right)\left[\left(\hat{a}^{\dagger}\right)^{4}+\left(\hat{a}^{4}\right)\right]$
	$\displaystyle-$	$\displaystyle\sin\left(2\omega t\right)\left[4\left(\hat{a}^{\dagger}\right)^{3}\hat{a}+6\left(\hat{a}^{\dagger}\right)^{2}+4\hat{a}^{\dagger}\left(\hat{a}\right)^{3}+6\left(\hat{a}\right)^{2}\right],$

and

	$\displaystyle\hat{\mathcal{I}}(t)=$	$\displaystyle\frac{1}{2}\left(1-\cos\left(4\omega t\right)\right)\left[\left(\hat{a}^{\dagger}\right)^{4}-\left(\hat{a}^{4}\right)\right]$		(28)
	$\displaystyle+$	$\displaystyle\left(1-\cos\left(2\omega t\right)\right)$
	$\displaystyle\times$	$\displaystyle\left[-4\left(\hat{a}^{\dagger}\right)^{3}\hat{a}-6\left(\hat{a}^{\dagger}\right)^{2}+4\hat{a}^{\dagger}\left(\hat{a}\right)^{3}+6\left(\hat{a}\right)^{2}\right].$

In the phase-space formulation of quantum mechanics the system is described by a quantum state in a space defined by generalized position and momentum coordinates, known as phase space. Quantum squeezing corresponds in this context to the distortion of the quantum state, represented, e.g., by its Wigner quasiprobability function $W(q,p)$, in this phase space. More specifically, it refers to the reduction/spread of the uncertainty in one direction (say position) at the expense of increasing/decreasing it in the conjugate dimension (momentum), ensuring compliance with the Heisenberg or more generally the Robertson-Schrödinger uncertainty relation. The squeezed state is represented as an ellipse rather than a circle (the latter corresponding to equal uncertainty in both dimensions) in the phase space. Worth emphasizing is that quantum squeezing is of particular interest in quantum optics and quantum information science, where it can improve measurement precisions and information processing capabilities by reducing quantum noise and enhancing signal strength in a specific direction (see, e.g., Refs. Lawrie ; Frascella ).

The squeezing prompted by the GUP given by Eq. (1) is examined in the next section, focusing on two classes of states: quantum harmonic oscillator energy eigenstates $\ket{n}$ and Glauber’s coherent states $\ket{\alpha}$, which play key roles in quantum optics and quantum information sciences.

Squeezing properties are quantified by the first and second moments of the $\hat{q}$ and $\hat{p}$ operators. We introduce

	$\displaystyle\langle\hat{q}\rangle$	$\displaystyle:=\bra{\Psi(t)}\hat{q}\ket{\Psi(t)},$		(29)
	$\displaystyle\langle\hat{p}\rangle$	$\displaystyle:=\bra{\Psi(t)}\hat{p}\ket{\Psi(t)},$		(30)
	$\displaystyle\Delta\hat{q}$	$\displaystyle:=\sqrt{\bra{\Psi(t)}\hat{q}^{2}\ket{\Psi(t)}-\left(\bra{\Psi(t)}\hat{q}\ket{\Psi(t)}\right)^{2}},$		(31)
	$\displaystyle\Delta\hat{p}$	$\displaystyle:=\sqrt{\bra{\Psi(t)}\hat{p}^{2}\ket{\Psi(t)}-\left(\bra{\Psi(t)}\hat{p}\ket{\Psi(t)}\right)^{2}},$		(32)
	$\displaystyle C_{qp}$	$\displaystyle:=\bra{\Psi(t)}(\hat{q}-\langle\hat{q}\rangle)(\hat{p}-\langle\hat{p}\rangle)\ket{\Psi(t)}_{\text{Weyl}}$
		$\displaystyle=\frac{1}{2}\bra{\Psi(t)}(\hat{q}\hat{p}+\hat{p}\hat{q})\ket{\Psi(t)}-\langle\hat{q}\rangle\langle\hat{p}\rangle,$		(33)

so that the Robertson-Schrödinger uncertainty principle holds:

$$(\Delta\hat{q})^{2}(\Delta\hat{p})^{2}-C_{qp}^{2}\geq\hslash^{2}/4.$$

(34)

In the definition of the covariance $C_{qp}$ the Weyl symmetrization is applied. Furthermore, we will add superscript ⁽⁰⁾ in case of the zeroth-order formulas ($\beta\rightarrow 0$).

Alternatively, the Robertson-Schrödinger uncertainty can be written as

$$(\Delta\hat{q})^{2}(\Delta\hat{p})^{2}(1-\rho^{2})\geq\hslash^{2}/4,$$

(35)

where $\rho$ is the dimensionless correlation coefficient

$$\rho:=\frac{C_{qp}}{\Delta\hat{q}\Delta\hat{p}}.$$

(36)

When $\rho$ is different from zero, the semiaxes of the ellipsoid of covariance do not overlap with the directions $q$ and $p$.

To quantify this effect, it is convenient to introduce the covariance matrix

$${\bf\Sigma}:=\left[\begin{array}[]{cc}\omega(\Delta\hat{q})^{2}&C_{qp}\\ C_{qp}&(\Delta\hat{p})^{2}/\omega\end{array}\right],$$

(37)

where the $\omega$ factor has been introduced for dimensional reasons. The eigenvalues of the previous matrix are

$\displaystyle\lambda_{\pm}=\frac{1}{2}\left[\text{tr}{\bf\Sigma}\pm\sqrt{(\text{tr}{\bf\Sigma})^{2}-4\det{\bf\Sigma}}\right],$

(38)

where $\text{tr}{\bf\Sigma}=\omega(\Delta\hat{q})^{2}+(\Delta\hat{p})^{2}/\omega$ and $\det{\bf\Sigma}=(\Delta\hat{q})^{2}(\Delta\hat{p})^{2}(1-\rho^{2})$. Because of the square root in Eq. (38), the $\mathcal{O}(\beta^{2})$ factors could in principle bring a contribution of the $\beta$ order in the squeezing amplitude. However, the terms of the order $\beta^{2}$ in $(\Delta\hat{q})^{2}$, in $(\Delta\hat{p})^{2}$, and in $C_{qp}$ bring no contribution of the order $\beta^{2}$ in $\left[(\text{tr}{\bf\Sigma})^{2}-4\det{\bf\Sigma}\right]$ whatever the state considered (see Appendix A). In consequence, the unknown factors do not contribute to the linear in $\beta$ expressions for the eigenvalues $\lambda_{\pm}$.

Importantly, the square roots of the eigenvalues have interpretations of major and minor diameters of the ellipsoid of covariance, respectively, so that the uncertainty relation (34) takes the form

$$\sqrt{\lambda_{+}}\sqrt{\lambda_{-}}\geq\hslash/2.$$

(39)

In the eigenframe the correlation vanishes, and the relative values of $\sqrt{\lambda_{+}}$ and $\sqrt{\lambda_{-}}$ can be used to quantify the squeezing of the state. Specifically, after suitable normalization of the variables, we can write

	$\displaystyle\sqrt{\lambda_{+}}$	$\displaystyle=$	$\displaystyle\sqrt{\frac{\hslash}{2}}e^{r},$		(40)
	$\displaystyle\sqrt{\lambda_{-}}$	$\displaystyle=$	$\displaystyle\sqrt{\frac{\hslash}{2}}e^{-r},$		(41)

where $r$ is the squeezing amplitude related to the complex squeezing parameter $\xi=|r|e^{i\gamma}$. Geometrically, $\gamma/2$ is the angle between the minor axes of the ellipsoid and the $q$ axis. The $\xi$ parameter enters the squeezing operator $\hat{S}(\xi)$ as follows:

$$\hat{S}(\xi):=\exp\left(\frac{1}{2}(\xi^{*}\hat{a}^{2}-\xi\hat{a}^{\dagger 2})\right).$$

(42)