# Suppressing Recoil Heating in Levitated Optomechanics using Squeezed Light

C. Gonzalez-Ballestero  Institute for Quantum Optics and Quantum Information
of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria.  Institute for
Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria.
<EMAIL_ADDRESS>J. A. Zielińska  Photonics Laboratory,
ETH Zürich, CH-8093 Zürich, Switzerland  M. Rossi  Photonics Laboratory, ETH
Zürich, CH-8093 Zürich, Switzerland  A. Militaru  Photonics Laboratory, ETH
Zürich, CH-8093 Zürich, Switzerland  M. Frimmer  Photonics Laboratory, ETH
Zürich, CH-8093 Zürich, Switzerland  L. Novotny  Photonics Laboratory, ETH
Zürich, CH-8093 Zürich, Switzerland  P. Maurer  Institute for Quantum Optics
and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck,
Austria.  Institute for Theoretical Physics, University of Innsbruck, A-6020
Innsbruck, Austria.  O. Romero-Isart  Institute for Quantum Optics and Quantum
Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria.
Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck,
Austria.

###### Abstract

We theoretically show that laser recoil heating in free-space levitated
optomechanics can be strongly reduced by shining squeezed light onto an
optically trapped nanoparticle. The presence of squeezing modifies the quantum
electrodynamical light-matter interaction in a way that enables us to control
the amount of information that the scattered light carries about a given
mechanical degree of freedom. We analyze the trade-off between measurement
imprecision and back-action noise and show that optical detection beyond the
standard quantum limit can be achieved. We predict that, with state-of-the-art
squeezed light sources, laser recoil heating reduction of at least $60\%$ can
be attained by squeezing a single Gaussian mode with an appropriate incidence
direction. Our results, which are valid both for motional and librational
degrees of freedom, will lead to improved feedback cooling schemes as well as
boost the coherence time of optically levitated nanoparticles in the quantum
regime.

Recent experiments have achieved active feedback ground-state cooling of a
center-of-mass degree of freedom of an optically levitated dielectric
nanosphere in free space [1, 2]. This achievement requires (i) to efficiently
measure the light scattered by the nanoparticle in order to extract all the
information it carries about the center-of-mass position, and (ii) that the
motional noise is dominated by the measurement back-action, that is, laser
recoil heating [3]. After ground-state cooling, the coherence time of the
quantum mechanical degree of freedom of an optically levitated nanoparticle is
still limited by (ii), even if (i) is not needed anymore. To achieve coherence
times beyond what laser recoil heating allows, one needs to either (a) switch
off the laser light, a strategy followed in the context of macroscopic quantum
physics [4, 5, 6, 7] and hybrid particle control with electric and magnetic
forces [8, 9, 10, 11, 12, 13, 14], or (b) suppress the amount of information
about the nanoparticle position that is carried by the scattered light.

In this Letter, we propose a method to achieve (b) using squeezed light. In
particular, we theoretically show that by squeezing a particular
electromagnetic mode that interacts with the optically trapped nanoparticle,
the quantum electrodynamical light-matter interaction can be modified to
reduce the information carried by the scattered light about a given mechanical
degree of freedom. Consequently, laser recoil heating is reduced by an amount
proportional to the degree of squeezing and to the overlap between the
squeezed mode angular profile and the angular distribution of trapping beam
photons that are inelastically scattered by the nanoparticle. In addition, we
discuss the tradeoff between measurement imprecision and back-action noise as
a function of the squeezing parameters and show that the sensitivity of
optical measurements can lie beyond the standard quantum limit (SQL). The
latter opens the door to use squeezed light to reach lower temperatures via
feedback cooling of an optically levitated nanoparticle. We remark that
similar ideas have been discussed in the context of cavity optomechanics [15]
and experimentally proven in an electromechanical system [16].

Figure 1: a) We consider a sub-wavelength particle (green) is trapped by a
focused laser beam (red arrow) and interacting with an electromagnetic mode in
a squeezed vacuum state (blue arrow). The functions
$A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$ and
$A_{z}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$ describe,
respectively, the angular and polarization distribution of the squeezed beam
and of the inelastically scattered photons in the absence of squeezing. b)
Wigner function of the squeezed light mode at frequency $\omega$.

We consider a sub-wavelength dielectric particle of mass $m$ trapped at the
focus of a laser beam of frequency $\omega_{0}$ propagating along the positive
$z-$axis and polarized along the $x-$axis (Fig. 1a). We choose the origin of
coordinates at the equilibrium position of the particle, and for simplicity we
only consider its motional dynamics along the $z$ axis (see [17] for the 3D
description). In a frame rotating at frequency $\omega_{0}$, the Hamiltonian
reads

$\hat{H}^{(0)}=\hat{H}_{p}+\hat{H}_{\rm EM}\\\ +\hbar\hat{q}\int
d\mathbf{k}\sum_{\bm{\epsilon}_{\mathbf{k}}}\left[G(\mathbf{k},\bm{\epsilon}_{\mathbf{k}})\hat{a}^{\dagger}(\mathbf{k},\bm{\epsilon}_{\mathbf{k}})+\text{H.c.}\right].$
(1)

As shown in [17], the Hamiltonian has been derived by (i) expanding the light-
matter interaction Hamiltonian to second order in the mechanical displacement
operator (Lamb-Dicke expansion), (ii) coherently displacing the
electromagnetic mode describing the trapping laser field, (iii) linearizing
the optomechanical Hamiltonian, and (iv) undertaking a rotating wave
approximation. The terms
$\hat{H}_{p}\equiv\hbar\Omega_{z}\hat{b}^{\dagger}\hat{b}$ and $\hat{H}_{\rm
EM}\equiv\hbar\int
d\mathbf{k}\sum_{\bm{\epsilon}_{\mathbf{k}}}\Delta(|\mathbf{k}|)\hat{a}^{\dagger}(\mathbf{k},\bm{\epsilon}_{\mathbf{k}})\hat{a}(\mathbf{k},\bm{\epsilon}_{\mathbf{k}})$
represent the free Hamiltonian of the mechanical mode and the electromagnetic
field respectively, with $\hbar$ the reduced Planck’s constant, $\Omega_{z}$
the mechanical frequency coming from the optical restoring force,
$\Delta(|\mathbf{k}|)\equiv\omega-\omega_{0}$, and
$\mathbf{k},\bm{\varepsilon}_{\mathbf{k}}$ the wave- and polarization vectors
of each plane-wave mode with $\omega\equiv c|\mathbf{k}|$, where $c$ is the
speed of light in vacuum. Both the mechanical $(\hat{b},\hat{b}^{\dagger})$
and the electromagnetic $(\hat{a},\hat{a}^{\dagger})$ ladder operators obey
bosonic commutation relations. The third term in Eq. (1) describes the light-
matter interaction in terms of the dimensionless displacement operator
$\hat{q}\equiv\hat{z}/z_{0}=\hat{b}+\hat{b}^{\dagger}$ (with
$z_{0}\equiv[\hbar/(2m\Omega_{z})]^{1/2}$ the zero-point displacement) and the
linearized coupling rate $G(\mathbf{k},\bm{\epsilon}_{\mathbf{k}})$ in the
small-particle limit. Under the approximations performed in this work, all
relevant quantities depend on the coupling rate at the laser frequency, given
by [17, 2, 18, 19, 20]

$G(\mathbf{k},\bm{\epsilon}_{\mathbf{k}})\big{|}_{\omega=\omega_{0}}=\sqrt{c^{3}\Gamma^{(0)}/(2\pi\omega_{0}^{2})}A_{z}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}}).$
(2)

Here $\Gamma^{(0)}$ is the recoil heating rate in the absence of squeezing.
The square-normalized function
$A_{z}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$ describes the
coupling strength between the motion and the electromagnetic mode with
polarization $\bm{\varepsilon}_{\mathbf{k}}$ and propagation direction given
by the two spherical angles $\Omega_{\mathbf{k}}$. This function is core to
ground-state feedback cooling experiments [1, 2] as it is directly related to
the inelastic differential scattering cross section
$d\sigma_{z}/d\Omega_{\mathbf{k}}$ and thus determines the angular
distribution of the photons scattered by the motional degree of freedom, also
known as information radiation patterns [21, 20, 17].

Figure 2: a) Recoil heating rate as a function of the degree of squeezing for
$\phi_{s}^{\prime}\equiv\phi_{s}-2\arg(\xi)=0$ (lower three lines) and
$\phi_{s}^{\prime}=\pi$ (upper three lines). $\Gamma^{(0)}$ is the recoil
heating rate in absence of squeezing. b) Recoil heating rate as a function of
squeezing phase for $r_{s}=1.73$ ($15$ dB). In both a) and b) we show the
perfect overlap case ($|\xi|=1$, dashed) and the case of squeezed light with a
Gaussian profile and numerical apertures $0.8$ (green) and $0.5$ (purple).

We now consider the additional presence of squeezed light by assuming that a
collective electromagnetic mode is in a squeezed vacuum state. We define this
mode as a combination of plane wave modes with associated bosonic annihilation
operator

$\hat{a}_{s}(\omega)\equiv\frac{\omega}{\sqrt{c^{3}}}\int
d\Omega_{\mathbf{k}}\sum_{\bm{\varepsilon}_{\mathbf{k}}}A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})\hat{a}(\mathbf{k},\bm{\epsilon}_{\mathbf{k}}).$
(3)

The square-normalized function
$A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$ describes the
angular- and polarization distribution of the squeezed mode, and depends on
the optical elements through which the squeezed light is sent before
interacting with the particle. We assume the squeezed light is produced by a
standard cavity-enhanced spontaneous parametric down-conversion process. This
process produces a two-mode squeezed state in frequency domain between two
sideband modes equally detuned from a laser carrier [22, 23], whose frequency
we choose equal to the trapping frequency $\omega_{0}$. By applying the
corresponding squeezing transformation [17] we obtain the Hamiltonian in the
presence of squeezing, given by

$\hat{H}=\hat{H}^{(0)}+\hbar\hat{q}\int
d\omega\left[K(\omega)\hat{a}^{\dagger}_{s}(\omega)+\text{H.c.}\right],$ (4)

i.e., as the Hamiltonian in the absence of squeezing, Eq. (1), plus an
additional squeezing-induced contribution. This additional term shows that
squeezing fundamentally modifies the interaction between the particle and the
electromagnetic field. The coupling constants $K(\omega)$ are given in [17].
At the laser frequency they read

$K(\omega_{0})\\!=\\!\sqrt{\frac{\Gamma^{(0)}}{2\pi}}\left\\{[\cosh(r_{s})-1]\xi-e^{i\phi_{s}}\sinh(r_{s})\xi^{*}\right\\}.$
(5)

Here, $r_{s}\equiv r(\omega_{0})$ and $\phi_{s}\equiv\phi(\omega_{0})$, where
$r(\omega)$ and $\phi(\omega)/2$ are the degree and the phase of squeezing at
frequency $\omega$ (see Fig. 1b). We have also defined the angular and
polarization overlap between the coupling rate and squeezed light
distributions,

$\xi\equiv\int
d\Omega_{\mathbf{k}}\sum_{\bm{\varepsilon}_{\mathbf{k}}}A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})A_{z}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}}).$
(6)

This overlap is crucial for the effectiveness of the scheme. In the extreme
case where the two distributions are orthogonal, the second term in Eq. (4)
vanishes and the motion remains unaffected by the squeezing. Note that all the
modifications induced by the squeezed light are fully determined by the
parameters $r_{s},\phi_{s},$ and $\xi$.

Let us first evaluate the motional recoil heating rate in the presence of
squeezed light. We compute it by tracing out the electromagnetic field within
the Born-Markov approximation, obtaining a master equation for the mechanical
degrees of freedom [18]. The validity of this approximation is discussed in
[17]. We assume the squeezing to be wideband, i.e.,
$\\{r(\omega_{0}\pm\Omega_{z}),\phi(\omega_{0}\pm\Omega_{z})\\}\approx\\{r(\omega_{0}),\phi(\omega_{0})\\}$.
This approximation is fulfilled for typical cavity-enhanced spontaneous down-
conversion squeezed light sources which, at infrared laser wavelengths, can
reduce electric field fluctuations by more than $15$dB ($r_{s}\approx 1.73$)
[24] and have linewidths on the order of tens of MHz [24, 25]. The recoil
heating rate reads [17]

$\\!\frac{\Gamma}{\Gamma^{(0)}}\\!=\\!1-|\xi|^{2}\Big{[}1-e^{2r_{s}}\sin^{2}\\!\left(\phi_{s}^{\prime}/2\right)-e^{-2r_{s}}\cos^{2}\\!\left(\phi_{s}^{\prime}/2\right)\\!\Big{]},$
(7)

where we define an effective squeezing phase
$\phi_{s}^{\prime}\equiv\phi_{s}-2\arg(\xi)$. Maximal reduction or increase of
$\Gamma$ requires a squeezing phase $\phi_{s}^{\prime}=0$ or
$\phi_{s}^{\prime}=\pi$ respectively, a large degree of squeezing,
$r_{s}\gtrsim 1$, and near-perfect overlap between the squeezed mode and the
information radiation pattern.

Figure 3:  a) Modification of the recoil heating of the motion along the
$z$-axis is maximized for a squeezed beam counter-propagating with respect to
the trapping light. b) Differential scattering cross section
$d\sigma_{z}/d\Omega_{\mathbf{k}}$ in the absence of squeezing (information
radiation patterns). The value of the function is encoded in both the radial
distance to the origin and the color scale encodes the relative amplitude for
each panel. c) Differential scattering cross section for squeezed light with a
Gaussian spatial profile of different numerical apertures. Upper and lower
rows correspond to $13$dB and $\phi_{s}^{\prime}=0$ and to $3$dB and
$\phi_{s}^{\prime}=\pi$, respectively.

In Fig. 2a we show the recoil heating rate as a function of the degree of
squeezing for squeezing phases $\phi_{s}^{\prime}=0$ (lower lines) and
$\phi_{s}^{\prime}=\pi$ (upper lines). The dashed curve corresponds to perfect
overlap $|\xi|=1$, whereas the colored curves correspond to a squeezed mode
with angular distribution given by a $x-$polarized Gaussian beam propagating
along the negative $z$ axis. This configuration, illustrated in Fig. 3a, is
optimal to maximize the overlap $\xi$ since the information radiation pattern
$A_{z}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$ is peaked along
this direction and polarization (see Fig. 3b). The corresponding Gaussian
functions $A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$, given in
[17], model the focusing of the squeezed light onto the particle by a lens of
numerical aperture NA. At $15$ dB squeezing, we have $\exp(-2r_{s})\ll 1$ and
the reduction of the recoil heating rate is limited by the mode overlap,
$\Gamma\approx\Gamma^{(0)}(1-|\xi|^{2})$. The reduction in this limit is of
$\sim 60\%$ for a Gaussian beam with NA$=0.9$. At the same degree of squeezing
and $\phi_{s}^{\prime}=\pi$ the heating rate is increased by a factor $\sim
20$. This pronounced increase should allow a motional reheating experiment [3]
to clearly observe the impact of light squeezing in the measurement back-
action noise.

Let us now show how the squeezing modifies the properties of the light
scattered by the nanoparticle. To do so, we cast the Heisenberg equations for
the mechanical operators, according to the Hamiltonian in Eq. (4), as

$\frac{d\hat{b}}{dt}=-i\Omega_{z}\hat{b}-i\sqrt{\Gamma^{(0)}}\left[\tilde{a}^{\rm
in}(t)+\text{H.c.}\right].$ (8)

The bosonic ladder operator $\tilde{a}^{\rm in}(t)$ corresponds to the
interacting mode [2, 19], namely the single collective electromagnetic mode
that couples to the mechanical degree of freedom. This operator reads

$\tilde{a}^{\rm
in}(t)\equiv\tilde{a}^{\text{in}(0)}(t)+\\!\sqrt{\Gamma^{(0)}}\\!\\!\\!\int\\!\\!d\omega
e^{-i(\omega-\omega_{0})(t-t_{0})}\tilde{G}^{*}(\omega)\\\
\times\\!\left[(c_{\omega}-1)\hat{a}_{s}(\omega,t_{0})-e^{i\phi(\omega)}s_{\omega}\hat{a}^{\dagger}_{s}(2\omega_{0}-\omega,t_{0})\right]\\!,$
(9)

with
$\\{s_{\omega},c_{\omega}\\}\equiv\\{\sinh[r(\omega)],\cosh[r(\omega)]\\}$,
$t_{0}\to-\infty$, the interacting mode operator in the absence of squeezing

$\tilde{a}^{\text{in}(0)}(t)\equiv\frac{1}{\sqrt{\Gamma^{(0)}}}\int
d\mathbf{k}\sum_{\bm{\epsilon}_{\mathbf{k}}}G^{*}(\mathbf{k},\bm{\epsilon}_{\mathbf{k}})\allowbreak\\\
\times\hat{a}(\mathbf{k},\bm{\epsilon}_{\mathbf{k}},t_{0})e^{-i\Delta(|\mathbf{k}|)(t-t_{0})},$
(10)

and $\tilde{G}(\omega)$ a constant given in [17] which, at the laser
frequency, takes the simple form
$\tilde{G}(\omega_{0})=\xi\big{(}\Gamma^{(0)}/2\pi\big{)}^{1/2}$. Crucially,
the interacting mode is modified by the presence of squeezing. In particular,
the total fraction of the squeezed mode $\hat{a}_{s}(\omega)$ in the
interacting mode is suppressed by a factor $\sim e^{-r_{s}}$ at
$\phi_{s}^{\prime}=0$, or enhanced by a factor $\sim e^{r_{s}}$ at
$\phi_{s}^{\prime}=\pi$, following the respective decrease or increase of the
effective light-matter coupling rate, see Eq. (5). This in turn reduces or
enhances the scattering of photons from the trapping laser into the directions
and polarizations contained in the squeezed mode $\hat{a}_{s}(\omega)$, which
is reflected by the differential scattering cross section, calculated in [17]
and shown in Fig. 3c for different numerical apertures. The increase and
decrease of the information about the motion leaked to propagating
electromagnetic modes, shown in Fig. 3c, is responsible for the respective
increase and decrease of the motional decoherence rate shown in Fig. 2(a-b).

We now analyze the impact of squeezing on the trade-off between measurement
imprecision and back-action noise. We first define the input and output
quadratures of the interacting mode as $\tilde{X}^{\rm
in/out}(t)\equiv\tilde{a}^{\dagger\rm in/out}+\tilde{a}^{\rm in/out}$ and
$\tilde{Y}^{\rm in/out}(t)\equiv i(\tilde{a}^{\dagger\rm
in/out}-\tilde{a}^{\rm in/out})$, where $\tilde{a}^{\rm out}(t)$ is given by
Eq. (9) under the substitution of $t_{0}$ by a time $t_{1}\to\infty$ [17]. The
formal solution of the Heisenberg equations provide the input-output relations
$\tilde{X}^{\rm out}(t)=\tilde{X}^{\rm in}(t)$ and $\tilde{Y}^{\rm
out}(t)=\tilde{Y}^{\rm in}(t)-\big{(}4\Gamma^{(0)}\big{)}^{1/2}\hat{q}(t)$. By
homodyne measurement of the output light one can measure the power spectral
density of the quadrature $\tilde{Y}^{\rm out}(t)$ containing the information
about the mechanical degree of freedom,

$S_{YY}^{\rm out}(\omega)=S_{YY}^{\rm
in}+\frac{4\Gamma^{(0)}}{z_{0}^{2}}\left[S^{f}(\omega)+S^{\rm
ba}(\omega)\right]-\frac{\sqrt{4\Gamma^{(0)}}}{z_{0}}S^{c}(\omega),$ (11)

with the power spectral densities defined as
$S_{AB}(\omega)\\!\equiv(2\pi)^{-1}\allowdisplaybreaks\int_{\mathbb{R}}d\tau\exp(i\omega\tau)\langle\hat{A}(\tau)\hat{B}(0)\rangle$
for $A,B\in\\{\tilde{X}^{\rm in/out},\tilde{Y}^{\rm in/out}\\}$ and with the
expected value taken over the vacuum state. The first term in Eq. (11) is the
imprecision due to photon shot noise [21]. The second term contains the
unperturbed power spectral density of the mechanical motion,
$S^{f}(\omega)\equiv(2\pi)^{-1}z_{0}^{2}\int_{\mathbb{R}}\text{d}\tau\exp(i\omega\tau)\langle\hat{q}^{f}(t+\tau)\hat{q}^{f}(t)\rangle$
with
$\hat{q}^{f}(t)\equiv\hat{b}(t_{0})\exp(-i\Omega_{z}(t-t_{0}))+\text{H.c.}$,
and is thus the relevant signal to be extracted from the output light. The
third and fourth terms in Eq. (11) describe the back-action contribution [21]
and the amplitude-phase correlations of light, respectively. The imprecision,
back-action, and correlation terms are explicitly given in [17] and are all
modified by the squeezing. The minimum detectable mechanical signal, which we
label $S_{\rm min}$, is defined as the value of $S^{f}(\omega)$ that makes the
second term in Eq. (11) (the signal) equal to the sum of the remaining terms
(the noise). For simplicity, here we focus on the low-frequency limit
$\omega\ll\Omega_{z}$, but a derivation for all $\omega$ is given in [17]. All
the results below remain valid for $|\omega-\Omega_{z}|\gtrsim\gamma_{z}$,
with $\gamma_{z}$ being the mechanical damping due to, e.g., the surrounding
gas. In this regime, the minimum detectable signal reads

$S_{\rm min}=\pi S_{\rm SQL}\bigg{[}uS_{XX}^{\rm in}+\frac{1}{u}S_{YY}^{\rm
in}-\left(S_{XY}^{\rm in}+S_{YX}^{\rm in}\right)\bigg{]}.$ (12)

Here, $u\equiv 4\Gamma^{(0)}/\Omega_{z}$ and $S_{\rm SQL}\equiv
z_{0}^{2}/(\pi\Omega_{z})$ is the SQL, i.e., the minimum detectable signal in
the absence of squeezing. Analytical expressions for the input power spectral
densities are derived in [17]. In the absence of squeezing, Eq. (12) recovers
the usual imprecision-back-action tradeoff (black dashed curve in Fig. 4a),
with a minimum detectable signal $S_{\rm min}=S_{\rm SQL}$ at $u=1$.

Figure 4: a) Minimum detectable signal in units of its SQL as a function of
$4\Gamma^{(0)}/\Omega_{z}$, in the absence of squeezing (dashed) and for $15$
dB squeezing and perfect overlap, $|\xi|=1$, along different quadratures
(colored). Inset: Wigner function of the input light for these curves (see
[17] for details). The black circle indicates the Wigner function size in the
absence of squeezing. b) Minimum signal $(S_{\rm min}/S_{\rm
SQL})|_{\phi_{s}^{\prime}=3\pi/2}$ as a function of degree of noise reduction
$e^{2r_{s}}$ and mode overlap. The upper ticks mark the value of the overlap
$\xi$ corresponding to a squeezed beam with Gaussian angular profile and
NA$=0.3$, $0.5$, $0.7$, and $0.9$. Constant-value isolines are marked by
dashed curves.

In Fig. 4(a) we show $S_{\rm min}$ in the presence of squeezing. For
$\phi_{s}^{\prime}=0$ the back-action, and thus the recoil heating rate, is
reduced, and the minimum signal shifts to a higher value of $u$. For
$\phi_{s}^{\prime}=\pi$ the opposite behavior occurs. At
$\phi_{s}^{\prime}=3\pi/2$ the minimum falls below the SQL. At this squeezing
phase, the amplitude-phase correlations $S^{c}(\omega)$ in the input light
minimize the detection noise [26, 19, 27], and the sensitivity reads

$\min_{u}(S_{\rm min}/S_{\rm SQL})=1-|\xi|^{2}\left(1-e^{-2r_{s}}\right).$
(13)

This expression, shown in Fig. 4b, is identical to the minimum achievable
value of $\Gamma/\Gamma^{(0)}$. The results of Fig. 4 extend the concept of
enhancement of displacement sensitivity via injection of squeezed vacuum light
[28, 26, 29] to free-space levitated optomechanics. Note that this enhancement
can also be used to overcome technical limitations in the optical measurement
of the mechanical displacement, which unavoidably reduce the detection
efficiency below unity. Our results show that, by properly tuning the
squeezing parameters, this efficiency can be effectively increased, which can
lead to performance improvements for measurement-based cooling schemes [30].

To conclude, we have theoretically shown that squeezed light can be used to
control the information that the light scattered by an optically levitated
nanoparticle carries about its mechanical degrees of freedom. While we have
focused on center-of-mass motion of a nanosphere, in [17] we show that the
formalism detailed in this paper can be directly extended to librational
degrees of freedom in optically trapped nanorotors [31, 32, 33, 34, 35, 36,
37, 38] by just substituting the angular functions
$A_{z}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$ by their
librational analogues. Our results predict that one can either reduce
measurement back-action noise to boost the coherence time of the quantum
mechanical degree of freedom or reduce imprecision noise to perform more
efficient displacement measurements in levitated free-space optomechanics. We
show that this method can be promptly used to reduce displacement measurement
sensitivities below the standard quantum limit. Our results are timely as they
provide a new tool to improve the quantum control of current experiments
cooling optically levitated nanoparticles [39, 2, 1, 40, 41]. This in turn
paves the way to preparing more complex quantum states (e.g., mechanical
squeezed states, non-Gaussian quantum states) which is key to test quantum
mechanics and develop quantum-enhanced sensing schemes [8, 42].

We acknowledge discussions with F. Tebbenjohanns. This research was supported
by the European Union’s Horizon 2020 research and innovation programme under
grant agreement No. [863132] (IQLev) and by the European Research Council
(ERC) under the grant Agreement No. [951234] (Q-Xtreme ERC-2020-SyG). This
work has been partly supported by ETH Grant No. ETH-47 20-2.

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Supplemental Material

##  S1 Detailed derivations

This section of the supplementary material contains detailed derivations of
central quantities of the main article. In Sec. S1.1 we derive the Hamiltonian
in the presence of squeezed light (Eq. (4)), which we then use to summarize
the derivation of the recoil heating rate (Eq. (7)) in Sec. S1.2. Then, in
Sec. S1.3, we derive the input-output relations and an expression for the
power spectral densities of the interacting modes (Eq. (9)) in terms of the
squeezing parameters. Finally, in Sec. S1.4, we derive an expression for the
differential scattering cross section that are shown in Fig. 3.

###  S1.1 Derivation of the Hamiltonian in the presence of squeezed light

We consider a sub-wavelength spherical particle of relative permittivity
$\epsilon$, volume $V$, and mass density $\rho\equiv m/V$ situated at the
origin of coordinates and interacting with the electromagnetic field. The
total Hamiltonian $\hat{H}=\hat{\mathbf{p}}^{2}/(2m)+\hat{H}_{\rm
EM}+\hat{H}_{\rm int}$ contains the kinetic energy of the particle with
center-of-mass momentum $\hat{\mathbf{p}}$, the Hamiltonian of the
electromagnetic field $\hat{H}_{\rm EM}$, and the interaction between the
field and the center-of-mass motion $\hat{H}_{\rm int}$. In the small-particle
limit the Hamiltonian of the field can be well approximated by $\hat{H}_{\rm
EM}\approx\hbar\sum_{\kappa}\omega_{\kappa}\hat{a}^{\dagger}_{\kappa}\hat{a}_{\kappa}$,
i.e. by a diagonal Hamiltonian in terms of plane-wave modes labelled by a
multi-index $\kappa=(\mathbf{k},\bm{\varepsilon}_{\mathbf{k}})$ denoting the
wave-vector $\mathbf{k}\in\mathds{R}^{3}$ and the two orthogonal polarization
vectors $\bm{\varepsilon}_{\mathbf{k}}\perp\mathbf{k}$, and the frequency
$\omega_{\kappa}=c|\mathbf{k}|$ [43]. In the small-particle limit the light-
matter coupling is sufficiently well described by the interaction Hamiltonian
of a polarizable point-dipole in an electric field $\hat{H}_{\rm
int}=-(\alpha/2)\hat{\mathbf{E}}^{2}(\hat{\mathbf{r}})$ [44, 45, 46, 18], with
$\hat{\mathbf{r}}$ the center-of-mass position operator, $\alpha\equiv
3\epsilon_{0}V(\epsilon-1)/(\epsilon+2)$ the particle polarizability, and
$\hat{\mathbf{E}}(\mathbf{r})$ the electric field operator that, expressed in
plane-wave modes, reads

$\hat{\mathbf{E}}(\mathbf{r})=i\sum_{\kappa}\sqrt{\frac{\hbar\omega_{\kappa}}{2\epsilon_{0}(2\pi)^{3}}}\left(\bm{\varepsilon}_{\mathbf{k}}e^{i\mathbf{k}\mathbf{r}}\hat{a}_{\kappa}-\text{H.c.}\right).$
(S1)

We remark that sums over a multi-index $\kappa$ must be understood as
integrals over continuous indices and sums over discrete ones.

Let us now expand the interaction Hamiltonian $\hat{H}_{\rm int}$ to second-
order in center-of-mass-position operators $\hat{r}_{\mu}$ with
$\mu\in\\{x,y,z\\}$ (Lamb-Dicke expansion). The zeroth-order term is
negligible in the small-particle limit [43], whereas the second-order term
provides the trapping potential. The total Hamiltonian is then written as

$\hat{H}\approx\hat{H}_{p}+\hat{H}_{\rm
EM}-\frac{\alpha}{2}(\hat{\mathbf{r}}\cdot\nabla)\hat{\mathbf{E}}^{2}(\mathbf{0}),$
(S2)

with
$\hat{H}_{p}=\hbar\sum_{\mu}\Omega_{\mu}\hat{b}^{\dagger}_{\mu}\hat{b}_{\mu}$,
$\Omega_{\mu}$ the mechanical frequency, and $\hat{b}_{\mu}^{\dagger}$ and
$\hat{b}_{\mu}$ the motional creation and annihilation operators defined
through the center-of-mass position operators $\hat{r}_{\mu}\equiv
r_{0\mu}(\hat{b}_{\mu}+\hat{b}_{\mu}^{\dagger})$ with
$r_{0\mu}\equiv[\hbar/(2m\Omega_{\mu})]^{1/2}$ the zero-point displacement
along axis $\mu$.

We aim at studying the behavior of the system in the presence of (i) an
optical trapping laser, here modelled as a single electromagnetic plane-wave
mode $\kappa_{0}$ whose occupation is very large, and (ii) the squeezed state
of a collective plane-wave mode. We first account for (i) by applying the
following unitary transformation to the Hamiltonian,

$\hat{T}_{1}(t)\equiv\exp\left(\alpha_{0}^{*}\hat{a}_{\kappa_{0}}-\text{H.c.}\right)\exp\left(i\omega_{0}t\sum_{\kappa}\hat{a}^{\dagger}_{\kappa}\hat{a}_{\kappa}\right),$
(S3)

i.e. a transformation into a frame rotating at the laser frequency
$\omega_{0}$ and a coherent displacement of mode $\kappa_{0}$ by an amplitude
$\alpha_{0}\in\mathds{C}$. The bosonic electromagnetic operators are
transformed as
$\hat{T}_{1}(t)\hat{a}_{\kappa}T_{1}^{\dagger}(t)=e^{-i\omega_{0}t}(\alpha_{0}\delta_{\kappa\kappa_{0}}+\hat{a}_{\kappa})$,
where $\delta_{\kappa\kappa_{0}}$ denotes a Dirac (Kronecker) delta for each
continuous (discrete) index. The transformed Hamiltonian can be split into
three terms of different orders in $|\alpha_{0}|$. For $|\alpha_{0}|\gg 1$ we
can neglect the $\mathcal{O}(1)$ term thus linearizing the Hamiltonian.
Furthermore, assuming the trapping laser frequency $\omega_{0}$ is optical, we
can neglect rapidly rotating terms at frequencies $\pm 2\omega_{0}$ under a
rotating wave approximation. This results in a linearized Hamiltonian

$\hat{H}=\hbar\sum_{\kappa}\Delta_{\kappa}\hat{a}^{\dagger}_{\kappa}\hat{a}_{\kappa}+\hbar\sum_{\mu}\Omega_{\mu}\hat{b}^{\dagger}_{\mu}\hat{b}_{\mu}+\hbar\sum_{\mu\kappa}\hat{q}_{\mu}\left(G_{\kappa\mu}\hat{a}^{\dagger}_{\kappa}+\text{H.c.}\right),$
(S4)

with $\hat{q}_{\mu}\equiv\hat{b}^{\dagger}_{\mu}+\hat{b}_{\mu}$,
$\Delta_{\kappa}\equiv\omega_{\kappa}-\omega_{0}$, and $G_{\kappa\mu}$ the
linearized coupling rates. Hereafter we focus on a laser mode $\kappa_{0}$
propagating along the positive $z-$axis and polarized along the $x-$axis, for
which the coupling rates read

$G_{\kappa\mu}\equiv
i\alpha_{0}\alpha\frac{\sqrt{\omega_{\kappa}\omega_{0}}}{2\epsilon_{0}(2\pi)^{3}}(\bm{\varepsilon}_{\mathbf{k}}^{*}\cdot\mathbf{e}_{x})r_{0\mu}[(\mathbf{k}-k_{0}\mathbf{e}_{z})\cdot\mathbf{e}_{\mu}].$
(S5)

The above expression assumes that the gradient of the trapping laser phase at
the trap center is equal to $k_{0}$, an approximation valid in the Lamb-Dicke
regime [18]. At the trapping lasers frequency, the coupling rates can be
expressed as

$G_{\kappa\mu}\big{|}_{k=k_{0}}=\sqrt{\frac{c^{3}\Gamma_{\mu}^{(0)}}{2\pi\omega_{0}^{2}}}A_{\mu}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}}),$
(S6)

with $\Gamma_{\mu}^{(0)}$ the recoil heating in the absence of squeezing,
$\Omega_{\mathbf{k}}$ the two spherical angles describing the direction of the
unit vector $\mathbf{e}_{\mathbf{k}}\equiv\mathbf{k}/k$, and
$A_{\mu}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$ an orthonormal
angular distribution given by

$A_{\mu}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})\equiv
ie^{i\arg(\alpha_{0})}\sqrt{\frac{3}{8\pi
l_{\mu}}}(\bm{\varepsilon}_{\mathbf{k}}\cdot\mathbf{e}_{x})\left[(\mathbf{e}_{\mathbf{k}}-\mathbf{e}_{z})\cdot\mathbf{e}_{\mu}\right],$
(S7)

with $l_{\mu}\equiv(1,2,7)\cdot\mathbf{e}_{\mu}/5$. As shown below, this
function determines the probability amplitude for the process where a trapping
laser photon scatters off the motional degree of freedom $\mu$ into a free-
space plane-wave mode propagating along the direction $\Omega_{\mathbf{k}}$
and with polarization $\bm{\varepsilon}_{\mathbf{k}}$. The two remaining
undefined quantities, namely the displaced mode amplitude $|\alpha_{0}|$ and
the mechanical frequency $\Omega_{\mu}$, can be expressed in the paraxial
approximation in terms of the trapping laser power $P_{t}$ and its waist at
the focus $W_{t}$ as [18]

$\displaystyle|\alpha_{0}|^{2}$ $\displaystyle=\frac{16\pi^{2}P_{t}}{\hbar
c^{2}k_{0}W_{t}^{2}},$ (S8)
$\displaystyle\Omega_{x}=\Omega_{y}=\Omega_{z}\frac{\sqrt{2}\pi
W_{t}}{\lambda_{0}}$
$\displaystyle=\sqrt{\frac{\epsilon-1}{\epsilon+2}\frac{12P_{t}}{\pi c\rho
W_{t}^{4}}}.$ (S9)

Let us now turn our attention on describing the single-mode squeezing. We
assume light is squeezed via a parametric down-conversion process, where the
electromagnetic quantum correlations are modified in frequency domain [22,
23]. It is thus convenient to define the following frequency-domain operators,

$\displaystyle\hat{a}_{s}(\omega)$ $\displaystyle\equiv\frac{k}{\sqrt{c}}\int
d\Omega_{\mathbf{k}}\sum_{\bm{\varepsilon}_{\mathbf{k}}}A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})\hat{a}_{\kappa},$
(S10) $\displaystyle\hat{a}_{j}(\omega)$
$\displaystyle\equiv\frac{k}{\sqrt{c}}\int
d\Omega_{\mathbf{k}}\sum_{\bm{\varepsilon}_{\mathbf{k}}}A_{j}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})\hat{a}_{\kappa},$
(S11)

where $s$ ($j$) is a single (continuous) index and we have rewritten the
plane-wave mode indices as
$\kappa\to\\{\omega,\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}}\\}$,
with $\omega=ck=c|\mathbf{k}|$ the frequency and $\Omega_{\mathbf{k}}$ the
propagation direction. Each of the above modes is by construction a linear
combination of plane waves with the same frequency and with a distribution of
polarizations and propagation directions given by the functions
$A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$ and
$A_{j}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$, respectively. The
mode $\hat{a}_{s}(\omega)$ is identified with the mode whose fluctuations will
be modified by the squeezing, i.e. the function
$A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$ describes the
angular and polarization distribution of the squeezed light for each
particular implementation. This mode is complemented by the set of modes
$\hat{a}_{j}(\omega)$, chosen such that the set
$\\{\hat{a}_{j}(\omega),\hat{a}_{s}(\omega)\\}$ forms a complete and
orthonormal basis in the subspace of electromagnetic modes with frequency
$\omega$, that is
$[\hat{a}_{\beta}(\omega),\hat{a}_{\beta^{\prime}}^{\dagger}(\omega^{\prime})]=\delta_{\beta\beta^{\prime}}\delta(\omega-\omega^{\prime})$
for $\beta\in\\{s,j\\}$. Note that the definition of these modes amounts to a
change of basis with

$\hat{a}_{\kappa}=\frac{\sqrt{c}}{k}\sum_{\beta}A_{\beta}^{*}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})\hat{a}_{\beta}(\omega_{\kappa}),$
(S12)

and the state of the electromagnetic field remains yet undefined and
orthonormality of this basis is guaranteed by choosing the weight functions to
fulfill

$\int
d\Omega_{\mathbf{k}}\sum_{\bm{\varepsilon}_{\mathbf{k}}}A_{\beta}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})A_{\beta^{\prime}}^{*}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})=\delta_{\beta\beta^{\prime}}.$
(S13)

In order to describe the squeezing of mode $\hat{a}_{s}(\omega)$, we apply the
unitary transformation

$\hat{T}_{s}(\eta,\omega)\equiv\exp\left[\int
d\omega\frac{\eta(\omega)}{2}\hat{a}_{s}^{\dagger}(\omega)\hat{a}_{s}^{\dagger}(2\omega_{0}-\omega)-\text{H.c.}\right],$
(S14)

which describes light generated by parametric down-conversion under a pump
beam of frequency $\omega_{0}$ [23]. The modulus and phase of the parameter
$\eta(\omega)=r(\omega)e^{i\phi(\omega)}$ quantify the degree of squeezing and
the squeezing phase of electromagnetic modes with frequency $\omega$,
respectively. Without loss of generality we assume
$\eta(2\omega_{0}-\omega)=\eta(\omega)$. To simplify integral expressions
below, we also assume the squeezing parameter to be peaked around
$\omega_{0}$, i.e. $r(0),r(2\omega_{0})\approx 0$. This approximation is
typically fulfilled as squeezed light is generated by an optical cavity with a
linewidth $\kappa_{c}\ll\omega_{0}$ [24, 25]. The squeezing transformation
transforms the operator $\hat{a}_{s}(\omega)$ as

$\hat{T}_{s}(\eta,\omega)\hat{a}_{s}(\omega)\hat{T}_{s}^{\dagger}(\eta,\omega)=\cosh[r(\omega)]\hat{a}_{s}(\omega)-e^{i\phi(\omega)}\sinh[r(\omega)]\hat{a}_{s}^{\dagger}(2\omega_{0}-\omega),$
(S15)

and leaves the modes $\hat{a}_{j}(\omega)$ unaffected, as they remain in their
vacuum state. Combing the above identities we express the Hamiltonian in the
presence of squeezed light as

$\displaystyle\hat{H}$ $\displaystyle=\hat{H}_{p}+\hbar\int
d\omega\Delta(\omega)\sum_{\beta}\hat{a}^{\dagger}_{\beta}(\omega)\hat{a}_{\beta}(\omega)$
$\displaystyle+\hbar\sum_{\mu}\hat{q}_{\mu}\int d\omega\left\\{\tilde{G}_{\mu
s}(\omega)\left[c_{\omega}\hat{a}^{\dagger}_{s}(\omega)-e^{-i\phi(\omega)}s_{\omega}\hat{a}_{s}(2\omega_{0}-\omega)\right]+\text{H.c.}\right\\}$
$\displaystyle+\hbar\sum_{\mu}\hat{q}_{\mu}\int
d\omega\sum_{j}\left[\tilde{G}_{\mu
j}(\omega)\hat{a}^{\dagger}_{j}(\omega)+\text{H.c.}\right],$ (S16)

with $\Delta(\omega)\equiv\omega-\omega_{0}$, and we introduce the short-hand
notation
$\\{s_{\omega},c_{\omega}\\}\equiv\\{\sinh[r(\omega)],\cosh[r(\omega)]\\}$.
The modified coupling rates read

$\tilde{G}_{\mu\beta}(\omega)\equiv\frac{\omega}{\sqrt{c^{3}}}\int
d\Omega_{\mathbf{k}}\sum_{\bm{\varepsilon}_{\mathbf{k}}}A_{\beta}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})G_{\kappa\mu}\big{|}_{k=\omega/c}.$
(S17)

Rearranging the terms in the Hamiltonian Eq. (S16) and using the above
assumptions about $r(\omega)$ and $\phi(\omega)$ one readily obtains Eq. (4)
in the main text, with coupling rates $K_{\mu}(\omega)\equiv\tilde{G}_{\mu
s}(\omega)(c_{\omega}-1)-\tilde{G}^{*}_{\mu
s}(2\omega_{0}-\omega)e^{i\phi_{\omega}}s_{\omega}$ and $\mu=z$.

###  S1.2 Recoil Heating Rates

Using the Hamiltonian Eq. (S16) we can derive the effective dynamics of the
mechanical degrees of freedom under the influence of the squeezed free-space
electromagnetic modes, considered as an external bath. We do so by tracing out
these modes under the Born-Markov approximation with the following assumptions
[47]

  1. 1.

In the displaced and squeezed frame, the electromagnetic modes are at zero
temperature, since the main contributions to the dynamics will stem from
optical frequency modes.

  2. 2.

The system (mechanical degrees of freedom) and the bath (electromagnetic
modes) are weakly coupled, i.e., $|G_{\kappa\mu}|e^{\pm r(\omega_{0})}$ is
sufficiently small. This condition is safely fulfilled since: (i) for the
parameters considered in this work $e^{\pm r(\omega_{0})}\lesssim 6$, and (ii)
the bare coupling rates $|G_{\kappa\mu}|$ are small enough, that is, the
recoil heating rates calculated in the absence of squeezing and within the
weak-coupling approximation are consistent with experimentally measured values
[18, 3]. Note that the validity of this approximation can also be certified a
posteriori as, according to Fig. 1c, the maximum recoil heating we obtain, for
perfect overlap, is $\Gamma_{\mu}\approx 30\Gamma_{\mu}^{(0)}$ which fulfills
the condition $\Gamma_{\mu}\lesssim\Omega_{\mu}$ [47]. Note also that, within
the wideband approximation (see next point), the weak-coupling approximation
is automatically fulfilled if the squeezed beam is tuned to reduce the recoil
heating, regardless of the amount of squeezing, since at this configuration
the total coupling rate to the squeezed mode is reduced, i.e. $\tilde{G}_{\mu
s}(\omega_{0})+K_{\mu}(\omega_{0})=\tilde{G}_{\mu
s}(\omega_{0})e^{-r(\omega_{0})}$.

  3. 3.

The coupling rates $\tilde{G}_{\mu\beta}(\omega)$ are smooth functions of
$\omega$ around the trapping laser frequency $\omega_{0}$. More specifically,
we assume

$\tilde{G}_{\mu\beta}(\omega_{0}\pm\Omega_{\mu})e^{\pm
r(\omega_{0}\pm\Omega_{\mu})}\approx\tilde{G}_{\mu\beta}(\omega_{0}),$ (S18)

and a similar condition including the phase factor $\exp[\pm i\phi(\omega)]$
on both sides. Since this condition is known to be fulfilled for the bare
coupling rates $\tilde{G}_{\mu\beta}(\omega)$ [20], the above expression is
equivalent to the “wideband squeezing” assumption
$\\{r(\omega_{0}\pm\Omega_{\mu}),\phi(\omega_{0}\pm\Omega_{\mu})\\}\approx\\{r(\omega_{0}),\phi(\omega_{0})\\}.$
The validity of this assumption is discussed in the main text.

After this careful enumeration of all relevant assumptions and approximations,
let us now sketch the derivation of the recoil heating rates. By tracing out
the electromagnetic modes under the above approximations we obtain the
following master equation for the mechanical density matrix, $\hat{\rho}$

$\frac{\text{d}\hat{\rho}}{\text{d}t}=\frac{1}{\text{i}\hbar}\left[\hat{H}_{p},\hat{\rho}\right]-\sum_{\mu\mu^{\prime}}\frac{\Gamma_{\mu\mu^{\prime}}}{2}\left[\hat{q}_{\mu},\left[\hat{q}_{\mu^{\prime}},\hat{\rho}\right]\right].$
(S19)

We have neglected an additional coherent contribution to the mechanical
Hamiltonian. This contribution is independent of the squeezing and thus is the
same as in free-space levitodynamics in the absence of squeezing, where it is
known to be negligible. The dissipative rates are given by

$\Gamma_{\mu\mu^{\prime}}=\Gamma_{\mu}^{(0)}\delta_{\mu\mu^{\prime}}+2\sqrt{\Gamma_{\mu}^{(0)}\Gamma_{\mu^{\prime}}^{(0)}}|\xi_{\mu}\xi_{\mu^{\prime}}|\left[s_{0}^{2}-s_{0}c_{0}\cos(\phi_{s}-\psi_{\mu}-\psi_{\mu^{\prime}})\right].$
(S20)

Here, we have defined, $\psi_{\mu}=\text{arg}(\xi_{\mu})$ , the squeezing
functions at the trapping lasers frequency, $s_{0}\equiv\sinh(r_{s})$,
$c_{0}\equiv\cosh(r_{s})$ , $r_{s}\equiv r(\omega_{0})$,
$\phi_{s}\equiv\phi(\omega_{0})$, and the overlap integral

$\xi_{\mu}\equiv\int
d\Omega_{\mathbf{k}}\sum_{\bm{\varepsilon}_{\mathbf{k}}}A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})A_{\mu}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}}).$
(S21)

The recoil heating along axis $\mu$ in the absence of squeezing is

$\Gamma_{\mu}^{(0)}=\frac{2\pi}{c}|\alpha_{0}|^{2}\left(\frac{\alpha}{2\epsilon_{0}(2\pi)^{3}}\right)^{2}\omega_{0}^{2}Z_{\mu}^{2}\frac{8\pi
k_{0}^{4}}{3}l_{\mu},$ (S22)

with $l_{\mu}\equiv(1/5,2/5,7/5)\cdot\mathbf{e}_{\mu}$, coinciding with the
known expression [18, 20]. Finally, note that for the examples in the main
text we choose the angular distribution of the squeezed light,
$A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$, to overlap as much
as possible with one of the scattering amplitude functions
$A_{\mu}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$. Since these
functions are orthogonal, this choice of maximizing $|\xi_{\mu}|$ makes the
remaining overlap functions small, allowing to neglect the cross-couplings in
the master equation and approximate

$\frac{\text{d}\hat{\rho}}{\text{d}t}\approx\frac{1}{\text{i}\hbar}\left[\hat{H}_{p},\hat{\rho}\right]-\sum_{\mu}\frac{\Gamma_{\mu}}{2}\left[\hat{q}_{\mu},\left[\hat{q}_{\mu},\hat{\rho}\right]\right],$
(S23)

with recoil heating rate

$\Gamma_{\mu}=\Gamma_{\mu}^{(0)}\left\\{1+2|\xi_{\mu}|^{2}\left[s_{0}^{2}-s_{0}c_{0}\cos(\phi_{s}-2\psi_{\mu})\right]\right\\},$
(S24)

for the chosen motional degree of freedom (in the main text, the motion along
$z$), and $\Gamma_{\mu}\approx\Gamma_{\mu}^{(0)}$ for the remaining two. By
regrouping the terms in Eq. (S24) we obtain Eq. (7) in the main text.

In the main text we consider for
$A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})$ a Gaussian beam
polarized along $x-$axis and propagating along the negative $z-$axis. We model
the angular distribution of such Gaussian beam as

$A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})=Ce^{-(\sin\theta_{\mathbf{k}}/\text{NA})^{2}}\left[-\delta_{\bm{\varepsilon}_{\mathbf{k}},\mathbf{e}_{\theta_{\mathbf{k}}}}\cos\theta_{\mathbf{k}}\cos\phi_{\mathbf{k}}+\delta_{\bm{\varepsilon}_{\mathbf{k}},\mathbf{e}_{\phi_{\mathbf{k}}}}\sin\phi_{\mathbf{k}}\right],$
(S25)

with NA the numerical aperture through which the squeezed beam is focused,
$\mathbf{e}_{\theta_{\mathbf{k}}}$ and $\mathbf{e}_{\phi_{\mathbf{k}}}$ the
polar and azimuthal spherical vectors associated with the direction
$\mathbf{k}$, and a normalization constant $C$ that is computed numerically.

###  S1.3 Input-Output Relations and Power Spectral Densities

Let us now derive the input-output relations and power spectral densities of
the interacting modes in Eq. (9) in the main text. The Heisenberg equations of
motion derived from the Hamiltonian in the presence of squeezed light in Eq.
(S16), read

$\displaystyle\frac{d\hat{b}_{\mu}}{dt}$
$\displaystyle=-i\Omega_{\mu}\hat{b}_{\mu}-i\int
d\omega\bigg{\\{}\sum_{j}\tilde{G}_{\mu
j}(\omega)\hat{a}^{\dagger}_{j}(\omega)+\tilde{G}_{\mu
s}(\omega)\left[c_{\omega}\hat{a}^{\dagger}_{s}(\omega)-e^{-i\phi(\omega)}s_{\omega}\hat{a}_{s}(2\omega_{0}-\omega)\right]+\text{H.c.}\bigg{\\}},$
(S26) $\displaystyle\frac{d\hat{a}_{j}(\omega)}{dt}$
$\displaystyle=-i\Delta(\omega)\hat{a}_{j}(\omega)-i\sum_{\mu}\hat{q}_{\mu}\tilde{G}_{\mu
j}(\omega),$ (S27) $\displaystyle\frac{d\hat{a}_{s}(\omega)}{dt}$
$\displaystyle=-i\Delta(\omega)\hat{a}_{s}(\omega)-i\sum_{\mu}\hat{q}_{\mu}P_{\mu}(\omega),$
(S28)

with $P_{\mu}(\omega)\equiv\tilde{G}_{\mu
s}(\omega)c_{\omega}-\tilde{G}^{*}_{\mu
s}(2\omega_{0}-\omega)s_{\omega}e^{i\phi(\omega)}$. We can formally solve Eq.
(S27) and Eq. (S28) in terms of an initial condition, namely a time
$t_{0}\to-\infty$ in the distant past, or a final condition, namely a time
$t_{1}\to+\infty$ in the distant future. We obtain

$\displaystyle\hat{a}_{j}(\omega,t)$
$\displaystyle=\hat{a}_{j}(\omega,t_{l})e^{-i\Delta(\omega)(t-t_{l})}-i\int_{t_{l}}^{t}ds\sum_{\mu}\hat{q}_{\mu}(s)e^{-i\Delta(\omega)(t-s)}\tilde{G}_{\mu
j}(\omega),$ (S29) $\displaystyle\hat{a}_{s}(\omega,t)$
$\displaystyle=\hat{a}_{s}(\omega,t_{l})e^{-i\Delta(\omega)(t-t_{l})}-i\int_{t_{l}}^{t}ds\sum_{\mu}\hat{q}_{\mu}(s)e^{-i\Delta(\omega)(t-s)}P_{\mu}(\omega),$
(S30)

for $l=0,1$. By inserting Eqs. (S29)-(S30) with $l=0$ in Eq. (S26) and using
the expression of the rates $\tilde{G}_{\mu\beta}(\omega)$ it is possible to
cast Eq. (S26) in the form

$\frac{d\hat{b}_{\mu}}{dt}=-i\Omega_{\mu}\hat{b}_{\mu}+\hat{A}_{\mu
1}+\hat{A}_{\mu 2}.$ (S31)

Here, we have defined $\hat{A}_{\mu 1}$ that contains the contribution from
the electromagnetic operators at the initial time $t_{0}$, namely

$\hat{A}_{\mu 1}=-i\int
d\omega\left\\{e^{i\Delta(\omega)(t-t_{0})}\tilde{G}_{\mu
s}(\omega)\left[c_{\omega}\hat{a}^{\dagger}_{s}(\omega,t_{0})-e^{-i\phi(\omega)}s_{\omega}\hat{a}_{s}(2\omega_{0}-\omega,t_{0})\right]\right.\\\
+e^{i\Delta(\omega)(t-t_{0})}\sum_{j}\tilde{G}_{\mu
j}(\omega)\hat{a}^{\dagger}_{j}(\omega,t_{0})+\text{H.c.}\Big{\\}},$ (S32)

and $\hat{A}_{\mu 2}$ that depends on the center-of-mass dynamics and reads

$\hat{A}_{\mu 2}=2i\text{Im}\int
d\omega\int_{t_{0}}^{t}ds\sum_{\mu^{\prime}}\hat{q}_{\mu^{\prime}}(s)e^{i\Delta(\omega)(t-s)}\bigg{\\{}\tilde{G}_{\mu
s}(\omega)\left[c_{\omega}P^{*}_{\mu^{\prime}}(\omega)+s_{\omega}e^{-i\phi(\omega)}P_{\mu^{\prime}}(2\omega_{0}-\omega)\right]\\\
+\sum_{j}\tilde{G}_{\mu
j}(\omega)\tilde{G}_{\mu^{\prime}j}^{*}(\omega)\bigg{\\}}.$ (S33)

To simplify these and other expressions throughout this section, we use that
for a function $f(\omega)$ which evolves slowly around $\omega_{0}$, we can
approximate

$\int d\omega e^{i\Delta(\omega)T}f(\omega)\approx\delta(T)2\pi\int
d\omega\delta(\omega-\omega_{0})f(\omega).$ (S34)

In the light of this approximation, and the wideband approximation Eq. (S18)
one can show that Eq. (S31) can be written as

$\frac{d\hat{b}_{\mu}}{dt}=-i\Omega_{\mu}\hat{b}_{\mu}-i\sqrt{\Gamma_{\mu}^{(0)}}\left[\tilde{a}_{\mu}(t)+\text{H.c.}\right],$
(S35)

where we have defined the interacting mode operator [2, 19]

$\tilde{a}_{\mu}(t)\equiv\frac{1}{\sqrt{\Gamma_{\mu}^{(0)}}}\int d\omega
e^{-i\Delta(\omega)(t-t_{0})}\left\\{\tilde{G}_{\mu
s}^{*}(\omega)\left[c_{\omega}\hat{a}_{s}(\omega,t_{0})-e^{i\phi(\omega)}s_{\omega}\hat{a}^{\dagger}_{s}(2\omega_{0}-\omega,t_{0})\right]+\sum_{j}\tilde{G}^{*}_{\mu
j}(\omega)\hat{a}_{j}(\omega,t_{0})\right\\}.$ (S36)

Rearranging the terms above, and identifying
$\tilde{a}_{\mu}^{\text{in}}(t)=\tilde{a}_{\mu}(t)$ we recover Eq. (9) in the
main text with $\tilde{G}(\omega)\equiv\tilde{G}_{\mu s}(\omega)$, and
$\mu=z$. In the absence of squeezing we recover

$\lim_{r(\omega)\to
0}\tilde{a}^{\text{in}}_{\mu}(t)=\tilde{a}_{\mu}^{\text{in}(0)}(t)\equiv\frac{1}{\sqrt{\Gamma_{\mu}^{(0)}}}\sum_{\kappa}G_{\kappa\mu}^{*}e^{-i\Delta_{\kappa}(t-t_{0})}\hat{a}_{\kappa}(t_{0}).$
(S37)

The formulation in terms of interacting modes, that can be shown to fulfill
$\left[\tilde{a}_{\mu}(t),\tilde{a}_{\mu^{\prime}}(t^{\prime})\right]=\delta(t-t^{\prime})\delta_{\mu\mu^{\prime}}$,
allows to analyze the modification imparted by the mechanical motion on the
light and the subsequent optical detection of such mechanical motion.
Specifically, we can formally solve the equation of motion for the mechanical
mode in Eq. (S35) to obtain

$\hat{b}_{\mu}(t)=\hat{b}_{\mu}(t_{0})e^{-i\Omega_{\mu}(t-t_{0})}-i\sqrt{\Gamma_{\mu}^{(0)}}\int_{t_{0}}^{t}ds\left[\tilde{a}^{\dagger}_{\mu}(s)+\tilde{a}_{\mu}(s)\right]e^{-i\Omega_{\mu}(t-s)}\equiv\hat{b}_{\mu}^{f}(t)+\hat{b}_{\mu}^{\rm
ba}(t).$ (S38)

Here, we have split the time-dependence of this operator into its unperturbed,
free evolution
$\hat{b}_{\mu}^{f}(t)\equiv\hat{b}_{\mu}(t_{0})e^{-i\Omega_{\mu}(t-t_{0})}$,
which is the signal we aim at detecting, and its modification due to the
interaction with the light, which constitutes the back-action term
$\hat{b}_{\mu}^{\rm ba}(t)$. To analyze the optical detection of the
mechanical motion, we relate the input interacting mode operator
$\tilde{a}_{\mu}^{\rm in}(t)$ to the output interacting mode operator
$\tilde{a}_{\mu}^{\rm out}(t)$ given by Eq. (S36) under the substitution
$t_{0}\to t_{1}$. Finally, we define the amplitude quadrature
$\tilde{X}_{\mu}(t)\equiv\tilde{a}^{\dagger}_{\mu}+\text{H.c.}$ and phase
quadrature $\tilde{Y}_{\mu}(t)\equiv i\tilde{a}^{\dagger}_{\mu}+\text{H.c.}$,
and connect them using Eqs. (S29)-(S30) into the following input-output
relations

$\displaystyle\tilde{X}_{\mu}^{\rm out}(t)$
$\displaystyle=\tilde{X}_{\mu}^{\rm in}(t),$ (S39)
$\displaystyle\tilde{Y}_{\mu}^{\rm out}(t)$
$\displaystyle=\tilde{Y}_{\mu}^{\rm
in}(t)-\sqrt{4\Gamma_{\mu}^{(0)}}\hat{q}_{\mu}^{f}(t)-\sqrt{4\Gamma_{\mu}^{(0)}}\hat{q}_{\mu}^{\rm
ba}(t).$ (S40)

To model photodetection we compute the power spectral densities (PSDs) of the
input and output quadratures, defined as

$S_{\mu,AB}(\omega)\equiv\int\frac{d\tau}{2\pi}e^{i\omega\tau}\langle\hat{A}_{\mu}(t+\tau)\hat{B}_{\mu}(t)\rangle.$
(S41)

for two arbitrary quadratures $A_{\mu},B_{\mu}\in\\{\tilde{X}^{\rm
in/out}_{\mu},\tilde{Y}^{\rm in/out}_{\mu}\\}$, with the expected value taken
over the vacuum state. The PSDs of the input quadratures reflect the modified
fluctuations present in the squeezed state. In the wideband squeezing
approximation they read

$\displaystyle 2\pi S_{\mu,XX}^{\rm in}$
$\displaystyle=1+2|\xi_{\mu}|^{2}s_{0}(s_{0}-c_{0}\cos(\phi_{s}-2\psi_{\mu}))=\frac{\Gamma_{\mu}}{\Gamma_{\mu}^{(0)}},$
(S42) $\displaystyle 2\pi S_{\mu,YY}^{\rm in}$
$\displaystyle=1+2|\xi_{\mu}|^{2}s_{0}(s_{0}+c_{0}\cos(\phi_{s}-2\psi_{\mu})),$
(S43) $\displaystyle 2\pi\left(S_{\mu,XY}^{\rm in}-S_{\mu,YX}^{\rm in}\right)$
$\displaystyle=2i,$ (S44) $\displaystyle 2\pi\frac{S_{\mu,XY}^{\rm
in}+S_{\mu,YX}^{\rm in}}{2}$
$\displaystyle=-2|\xi_{\mu}|^{2}s_{0}c_{0}\sin(\phi_{s}-2\psi_{\mu}).$ (S45)

Let us now compute the PSD of the output phase-quadrature $\tilde{Y}^{\rm
out}_{\mu}$ that contains all the information about the mechanical dynamics.
Assuming no initial correlations between the electromagnetic field and the
mechanical motion
($\langle\hat{q}_{\mu}^{f}(t_{0}+\tau)\hat{a}_{\kappa}(t_{0})\rangle=0$), this
PSD takes the form

$S_{\mu,YY}^{\rm out}(\omega)=S_{\mu,YY}^{\rm
in}+\frac{4\Gamma_{\mu}^{(0)}}{r_{0\mu}^{2}}S_{\mu}^{f}(\omega)+\frac{4\Gamma_{\mu}^{(0)}}{r_{0\mu}^{2}}S_{\mu}^{\rm
ba}(\omega)-\frac{\sqrt{4\Gamma_{\mu}^{(0)}}}{r_{0\mu}}S_{\mu}^{\rm
c}(\omega).$ (S46)

The first term in Eq. (S46) describes the original fluctuations of the light
field, namely the original imprecision, and is determined by the fluctuations
of the imprecision quadrature. The second term contains the unperturbed PSD of
the mechanical mode,

$S_{\mu}^{\text{f}}(\omega)\equiv\frac{r_{0\mu}^{2}}{2\pi}\int_{\mathbb{R}}\text{d}\tau\langle\hat{q}^{f}_{\mu}(t+\tau)\hat{q}^{f}_{\mu}(t)\rangle
e^{i\omega\tau},$ (S47)

and corresponds to the signal to be detected in the output light. The third
term in Eq. (S46) describes the disturbance in the measurement caused by the
back-action of the light on the mechanical mode,

$S_{\mu}^{\text{ba}}(\omega)\equiv\frac{r_{0\mu}^{2}}{2\pi}\int_{\mathbb{R}}\text{d}\tau\langle\hat{q}^{\text{ba}}_{\mu}(t+\tau)\hat{q}^{\text{ba}}_{\mu}(t)\rangle
e^{i\omega\tau}=\frac{r_{0\mu}^{2}\Gamma_{\mu}}{2\pi}|2m\Omega_{\mu}\chi_{0\mu}(\omega)|^{2},$
(S48)

where we define the zero-loss susceptibility

$\chi_{0\mu}(\omega)\equiv\lim_{\gamma_{\mu}\to
0^{+}}\chi_{\mu}(\omega)\equiv\lim_{\gamma_{\mu}\to
0^{+}}\frac{1}{m}\frac{1}{\Omega_{\mu}^{2}-\omega^{2}-i\gamma_{\mu}\omega},$
(S49)

with $\chi_{\mu}(\omega)$ the mechanical susceptibility. The last term in Eq.
(S46) describes the amplitude-phase correlations of the light,

$S^{c}_{\mu}(\omega)\equiv\frac{r_{0\mu}}{2\pi}\int d\tau
e^{i\omega\tau}\bigg{[}\langle\tilde{Y}_{\mu}^{\rm
in}(t+\tau)\hat{q}_{\mu}^{\rm ba}(t)\rangle+\langle\hat{q}_{\mu}^{\rm
ba}(t+\tau)\tilde{Y}_{\mu}^{\rm in}(t)\rangle\bigg{]}\\\
=r_{0\mu}\sqrt{\Gamma_{\mu}^{(0)}}\text{Re}[2m\Omega_{\mu}\chi_{0\mu}(\omega)]\left(S_{\mu,XY}^{\rm
in}+S_{\mu,YX}^{\rm in}\right).$ (S50)

The expressions of Eq. (S48) and (S50) have been derived within the wideband
squeezing approximation. In the rest of this section we generalize the PSDs
containing zero-loss susceptibility to the case of a non-negligible mechanical
damping, due to e.g. collisions of the particle with the surrounding gas, by
substituting

$\chi_{0\mu}(\omega)\to\chi_{\mu}(\omega)\equiv\frac{1}{m\Omega_{\mu}^{2}}\tilde{\chi}_{\mu}(\omega),$
(S51)

were the adimensional susceptibility $\tilde{\chi}_{\mu}(\omega)$ is defined
for convenience.

To detect the mechanical motion in the PSD of the output light, the second
term in Eq. (S46) (the signal) must be larger than the sum of the remaining
terms (imprecision, back-action, and correlations). We define the minimum
detectable signal as

$S_{\mu,\rm
min}(\omega)\equiv\frac{r_{0\mu}^{2}}{4\Gamma_{\mu}^{(0)}}S_{\mu,YY}^{\rm
in}+S_{\mu}^{\rm
ba}(\omega)-\frac{r_{0\mu}}{\sqrt{4\Gamma_{\mu}^{(0)}}}S^{c}_{\mu}(\omega).$
(S52)

Using all the above expressions we can cast it as

$S_{\mu,\rm min}(\omega)=\frac{S_{\mu,\rm
SQL}(\omega)}{2}\left\\{|u\tilde{\chi}_{\mu}(\omega)|\left(2\pi
S_{\mu,XX}^{\rm in}\right)+\frac{1}{u|\tilde{\chi}_{\mu}(\omega)|}\left(2\pi
S_{\mu,YY}^{\rm
in}\right)-\frac{\text{Re}(\tilde{\chi}_{\mu}(\omega))}{|\tilde{\chi}_{\mu}(\omega)|}2\pi(S_{\mu,XY}^{\rm
in}+S_{\mu,YX}^{\rm in})\right\\}.$ (S53)

Here, $u\equiv 4\Gamma_{\mu}^{(0)}/\Omega_{\mu}$ is the adimensional ratio
between the bare recoil heating rate (i.e. in the absence of squeezing) and
the mechanical frequency, and $S_{\mu,\rm
SQL}\equiv|\tilde{\chi}_{\mu}(\omega)|r_{0\mu}^{2}/(\pi\Omega_{\mu})$ is the
absolute minimum signal that can be detected in the absence of squeezing,
namely the Standard Quantum Limit. Indeed, in the absence of squeezing, Eq.
(S53) reduces to $S_{\mu,\rm min}=(1/2)S_{\mu,\rm
SQL}[(u|\tilde{\chi}_{\mu}(\omega)|)+(u|\tilde{\chi}_{\mu}(\omega)|)^{-1}]$
which, as a function of $u$, has a minimum value $\min_{u}(S_{\mu,\rm
min})=S_{\mu,\rm SQL}(\omega)$ at $u=|\tilde{\chi}_{\mu}(\omega)|^{-1}$. In
the presence of squeezing, Eq. (S53) reaches a minimum value at
$u=|\tilde{\chi}_{\mu}(\omega)|^{-1}[S_{\mu,YY}^{\rm in}/S_{\mu,XX}^{\rm
in}]^{1/2}$ which is given by

$\min_{u}\left[\frac{S_{\mu,\rm min}(\omega)}{S_{\mu,\rm
SQL}(\omega)}\right]=\sqrt{\left(2\pi S_{\mu,XX}^{\rm in}\right)\left(2\pi
S_{\mu,YY}^{\rm
in}\right)}-\frac{\text{Re}(\tilde{\chi}_{\mu}(\omega))}{|\tilde{\chi}_{\mu}(\omega)|}2\pi\frac{S_{\mu,XY}^{\rm
in}+S_{\mu,YX}^{\rm in}}{2}.$ (S54)

At resonance, $\omega=\Omega_{\mu}$, the above expression is always larger
than $1$, and the minimum signal remains above the Stardard Quantum Limit even
in the presence of squeezing. For $\omega\neq\Omega_{\mu}$, Eq. (S54) is
minimized at a squeezing angle $\phi_{s}-2\psi_{\mu}=3\pi/2$ if
$\text{Re}(\tilde{\chi}_{\mu}(\omega))>0$ (or equivalently, for
$\omega<\Omega_{\mu}$), and at a squeezing angle $\phi_{s}-2\psi_{\mu}=\pi/2$
if $\text{Re}(\tilde{\chi}_{\mu}(\omega))<0$ (or equivalently, for
$\omega>\Omega_{\mu}$). In both cases the minimum detectable signal at the
optimum squeezing phase is given by

$\min_{u}\left[\frac{S_{\mu,\rm min}(\omega)}{S_{\mu,\rm
SQL}(\omega)}\right]_{\rm
opt}=1-|\xi_{\mu}|^{2}\left\\{1-\frac{e^{2r_{0}}}{2}\left[1-\frac{|\text{Re}(\tilde{\chi}_{\mu}(\omega))|}{|\tilde{\chi}_{\mu}(\omega)|}\right]-\frac{e^{-2r_{0}}}{2}\left[1+\frac{|\text{Re}(\tilde{\chi}_{\mu}(\omega))|}{|\tilde{\chi}_{\mu}(\omega)|}\right]\right\\}.$
(S55)

The sensitivity is thus maximized
$|\text{Re}(\tilde{\chi}_{\mu}(\omega))|/|\tilde{\chi}_{\mu}(\omega)|\to 1$,
which corresponds to frequencies $|\omega-\Omega_{\mu}|\gg\gamma_{\mu}$. This
condition is fulfilled at almost all frequencies since in typical
levitodynamics experiments $\gamma_{\mu}\ll\Omega_{\mu}$. The expressions in
the main text correspond to the particularly simple limit
$\omega\ll\Omega_{\mu}$.

###  S1.4 Information Radiation Pattern

Let us now derive an explicit expression for the information radiation
pattern. Our starting point is the linearized Hamiltonian Eq. (S4), i.e., the
Hamiltonian before applying the squeezing transformation. We aim at computing
the transition amplitude for the process of a trapping laser photon scattering
into a plane wave by exciting a phonon in the mechanical mode $\mu$. Since a
squeezed vacuum has non-zero occupation we need to account for the two
possible processes, namely (i) the creation of a phonon by scattering
(creating) a photon in the free-space mode $\kappa$, with transition amplitude

$\mathcal{T}_{\kappa\mu}^{+}\equiv\alpha_{0}^{*}\langle{S}|\hat{a}_{\kappa}\hat{b}_{\mu}\hat{U}(t_{1},t_{0})|{S}\rangle,$
(S56)

and (ii) the creation of a phonon by absorbing (annihilating) a photon from
the free-space mode $\kappa$, with transition amplitude

$\mathcal{T}_{\kappa\mu}^{-}\equiv\alpha_{0}^{*}\langle{S}|\hat{a}_{\kappa}^{\dagger}\hat{b}_{\mu}\hat{U}(t_{1},t_{0})|{S}\rangle.$
(S57)

In both expressions, $|S\rangle=\hat{T}_{s}(\eta,\omega)|0\rangle$ represents
the squeezed state and $\hat{U}(t_{1},t_{0})$ is the time evolution operator
for the linearized Hamiltonian evaluated at times $t_{1}\to\infty$ and
$t_{0}\to-\infty$. Introducing the following short-hand notation

$\displaystyle\hat{a}_{\kappa}^{+}$
$\displaystyle\equiv\hat{T}_{s}(\eta,\omega)\hat{a}_{\kappa}^{\dagger}\hat{T}_{s}^{\dagger}(\eta,\omega),$
(S58) $\displaystyle\hat{a}_{\kappa}^{-}$
$\displaystyle\equiv\hat{T}_{s}(\eta,\omega)\hat{a}_{\kappa}\hat{T}_{s}^{\dagger}(\eta,\omega),$
(S59)

we can rewrite the above transition amplitudes as

$\mathcal{T}_{\kappa\mu}^{\pm}=e^{-i(t_{1}-2t_{0})[\pm\Delta(\omega_{\kappa})+\Omega_{\mu}]}\alpha_{0}^{*}\langle{0}|a_{\kappa}^{\pm}\hat{b}_{\mu}\hat{U}^{\prime}_{I}(t_{1},t_{0})|{0}\rangle,$
(S60)

with the vacuum state $|{0}\rangle$. The interaction picture time-evolution
operator in Eq. (S60) is given by
$\hat{U}^{\prime}_{I}(t_{1},t_{0})=\exp(i\hat{H}_{0}t_{1}/\hbar)\hat{U}^{\prime}(t_{1},t_{0})\exp(-i\hat{H}_{0}t_{0}/\hbar)$
with $\hat{U}^{\prime}(t_{1},t_{0})$ the time-evolution operator associated to
the transformed Hamiltonian Eq. (S16) and the free Hamiltonian
$\hat{H}_{0}\equiv\hat{H}_{p}+\hbar\int
d\omega\Delta(\omega)\sum_{\beta}\hat{a}^{\dagger}_{\beta}(\omega)\hat{a}_{\beta}(\omega)$.
Under the weak-coupling assumption, we can compute the above transition
amplitudes to first order in time-dependent perturbation theory, i.e.
approximating

$\hat{U}^{\prime}_{I}(t_{1},t_{0})\approx
1-i\int_{t_{0}}^{t_{1}}\\!\\!ds\sum_{\mu}\left[\hat{b}_{\mu}e^{-i\Omega_{\mu}(s-t_{0})}+\text{H.c.}\right]\int
d\omega\Bigg{\\{}e^{i\Delta(\omega)(s-t_{0})}\sum_{j}\tilde{G}_{\mu
j}(\omega)\hat{a}^{\dagger}_{j}(\omega)+\tilde{G}_{\mu s}(\omega)\\\
\times\left[c_{\omega}\hat{a}^{\dagger}_{s}(\omega)-e^{-i\phi(\omega)}s_{\omega}\hat{a}_{s}(2\omega_{0}-\omega)\right]+\text{H.c.}\Bigg{\\}}.$
(S61)

Under the wideband squeezing approximation, this allows us to express the
transition amplitudes

$\mathcal{T}_{\kappa\mu}^{\pm}=\frac{if^{\pm}_{\kappa\mu}}{2\pi
k_{0}}\delta(k-k_{0}\pm\Omega_{\mu}/c)e^{-i(t_{1}-t_{0})(\pm\Delta(\omega_{\kappa})+\Omega_{\mu})}\approx\frac{if^{\pm}_{\kappa\mu}}{2\pi
k_{0}}\delta(k-k_{0}),$ (S62)

in terms of the scattering amplitudes

$\displaystyle f_{\kappa\mu}^{+}$
$\displaystyle=f_{\kappa\mu}^{(0)}-\sqrt{\frac{\Gamma_{\mu}^{(0)}(2\pi)^{3}}{c}}\alpha_{0}^{*}A_{s}^{*}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})s_{0}\left(s_{0}\xi_{\mu}-c_{0}e^{i\phi_{s}}\xi_{\mu}^{*}\right),$
(S63) $\displaystyle f_{\kappa\mu}^{-}$
$\displaystyle=-\sqrt{\frac{\Gamma_{\mu}^{(0)}(2\pi)^{3}}{c}}\alpha_{0}^{*}A_{s}^{*}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})s_{0}\left(s_{0}\xi_{\mu}^{*}-c_{0}e^{-i\phi_{s}}\xi_{\mu}\right),$
(S64)

that are defined in terms of the scattering amplitude in the absence of
squeezing,

$f_{\kappa\mu}^{(0)}\equiv\frac{4\pi^{2}k_{0}\alpha_{0}^{*}}{c}G_{\kappa\mu}\big{|}_{k=k_{0}}.$
(S65)

The total scattering amplitude of emitting a photon in mode $\kappa$ can then
be expressed as the difference between the scattering amplitude associated to
the creation and annihilation of a photon in mode $\kappa$ respectively,
namely

$|f_{\kappa\mu}|^{2}\equiv|f_{\kappa\mu}^{+}|^{2}-|f_{\kappa\mu}^{-}|^{2}=|f_{\kappa\mu}^{(0)}|^{2}-\frac{(2\pi)^{3}}{c}\Gamma_{\mu}^{(0)}|\alpha_{0}|^{2}2s_{0}\text{Re}\left[\left(s_{0}\xi_{\mu}-c_{0}e^{i\phi_{s}}\xi_{\mu}^{*}\right)A_{s}^{*}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})A_{\mu}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})\right].$
(S66)

Using the scattering amplitude we can finally evaluate the differential
scattering cross section [48]

$\frac{d\sigma_{\mu}}{d\Omega_{\mathbf{k}}}=\sum_{\bm{\varepsilon}_{\mathbf{k}}}|f_{\kappa\mu}|^{2},$
(S67)

which can easily be translated into the information radiation pattern
$\mathcal{I}_{\mu}(\Omega_{\mathbf{k}})\equiv\sigma_{\mu}^{-1}d\sigma_{\mu}/d\Omega_{\mathbf{k}}$
.

##  S2 Extension to libration of nanorotors

In this section we show how the formalism developed for the center-of-mass
dynamics can also be applied to the librational degrees of freedom of
nanorotors [31, 32, 33, 34, 35, 36, 37, 38]. We consider a sub-wavelength
anisotropic particle (e.g. a nanodumbbell) interacting with an optical
trapping laser that is linearly polarized along the $z-$axis and propagating
along the positive $x-$axis. The linear electromagnetic response of the
particle is modelled by an isotropic relative permittivity $\epsilon$ and a
polarizability tensor $\overline{\bm{\alpha}}$ which, in the body frame, reads
$\overline{\alpha}_{ij}=\alpha_{i}\delta_{ij}$ with
$\alpha_{1}=\alpha_{2}\neq\alpha_{3}$. We assume that the position of the
center of mass of the particle is fixed at the origin, but that the particle
is free to rotate about its three axes. The trapping laser aligns the long
principal axis of the particle with its polarization axis. We therefore assume
that the particle is aligned along the $z-$axis and can undergo small angular
displacements $\eta_{\mu}$ along the axes $\mu\in\\{x,y\\}$. Rotations along
the particle’s main axis can be neglected for thin rotors [49]. The
Hamiltonian is then given by

$\hat{H}=\sum_{\mu}\frac{\hat{L}_{\mu}^{2}}{2I}+\hat{H}_{\rm
EM}-\frac{1}{2}\hat{\mathbf{E}}(\mathbf{0})\cdot\overline{\bm{\alpha}}_{\rm
LF}\cdot\hat{\mathbf{E}}(\mathbf{0}),$ (S68)

with $I$ the moment of inertia, assumed equal for rotations along $x$ and $y$
axes, $\hat{L}_{\mu}$ the $\mu-$component of the angular momentum operator,
$\hat{H}_{\rm EM}$ the electromagnetic field Hamiltonian given in Sec. S1.1,
and $\overline{\bm{\alpha}}_{\rm LF}$ the polarizability tensor in the
laboratory frame.

We now proceed in similar steps as for the center-of-mass motion. First, we
assume the angular displacements are small, and hence expand the
polarizability tensor to second order in the angular displacements. The
zeroth-order term is again negligible in the small-particle limit which leads
to $\overline{\bm{\alpha}}_{\rm
LF}(\hat{\eta}_{x},\hat{\eta}_{y})\approx(\Delta\alpha)\overline{M}(\hat{\eta}_{x},\hat{\eta}_{y})$,
with $\Delta\alpha\equiv\alpha_{3}-\alpha_{1}$ and

$\overline{M}(\hat{\eta}_{x},\hat{\eta}_{y})\equiv\left[\begin{array}[]{ccc}\hat{\eta}_{x}^{2}&\hat{\eta}_{x}\hat{\eta}_{y}&\hat{\eta}_{x}\\\
\hat{\eta}_{y}\hat{\eta}_{x}&\hat{\eta}_{y}^{2}&\hat{\eta}_{y}\\\
\hat{\eta}_{x}&\hat{\eta}_{y}&-(\hat{\eta}_{x}^{2}+\hat{\eta}_{y}^{2})\end{array}\right].$
(S69)

In spherical coordinates and to first order in the polar angle $\theta$, the
small displacements read $\eta_{x}\approx\theta\cos\phi$ and
$\eta_{y}\approx\theta\sin\phi$, and the angular momenta in the kinetic term
can be written as
$\hat{L}_{x}^{2}+\hat{L}_{y}^{2}\approx-\hbar^{2}[\partial_{\eta_{x}}^{2}+\partial_{\eta_{y}}^{2}]$.
Then, we apply the unitary operator in Eq. (S3) and linearize the Hamiltonian
assuming that the coherent amplitude is large $\alpha_{0}\gg 1$. Next, we
apply the rotating wave approximation and neglect terms oscillating at
frequencies $\pm 2\omega_{0}$. The remaining terms contain (i) a harmonic
trapping potential $(I/2)\Omega^{2}(\hat{\eta}_{x}^{2}+\hat{\eta}_{y}^{2})$
which allows us to define bosonic ladder operators $\hat{b}_{\mu}$ for the
rotational degrees of freedom, via
$\hat{\eta}_{\mu}=r_{0\mu}(\hat{b}_{\mu}+\hat{b}_{\mu}^{\dagger})$, with the
zero-point angular displacement $r_{0\mu}\equiv[\hbar/(2I\Omega)]^{1/2}$ and
the libration frequency

$\Omega=\sqrt{\frac{\Delta\alpha}{I}\frac{\hbar\omega_{0}|\alpha_{0}|^{2}}{\epsilon_{0}(2\pi)^{3}}},$
(S70)

and (ii) a linearized interaction between the rotational degrees of freedom
and the electromagnetic field. The total Hamiltonian can be written in
analogous form to Eq. (1), namely

$\hat{H}^{(0)}=\hat{H}_{p}+\hat{H}_{\rm
EM}+\hbar\sum_{\kappa\mu}\hat{q}_{\mu}\left(G_{\kappa\mu}\hat{a}_{\kappa}^{\dagger}+\text{H.c.}\right),$
(S71)

with
$\hat{H}_{p}=\hbar\sum_{\mu}\Omega_{\mu}\hat{b}^{\dagger}_{\mu}\hat{b}_{\mu}$,
and the linearized coupling rates

$G_{\kappa\mu}\equiv-\alpha_{0}(\Delta\alpha)\frac{\sqrt{\omega_{\kappa}\omega_{0}}}{2\epsilon_{0}(2\pi)^{3}}r_{0\mu}(\bm{\varepsilon}_{\mathbf{k}}\cdot\mathbf{e}_{\mu}).$
(S72)

In deriving this Hamiltonian we have also neglected a term that reads
$\hbar(\hat{\eta}_{x}^{2}+\hat{\eta}_{y}^{2})\sum_{\kappa}(g^{\prime}_{\kappa}\hat{a}^{\dagger}_{\kappa}+\text{H.c.})$
as the associated coupling rates are much smaller for typical parameters,
namely $|g^{\prime}_{\kappa}|/|G_{\kappa\mu}|\sim r_{0\mu}\ll 1$.

At the frequency of the trapping laser light, the coupling rate Eq. (S72) also
takes the form of Eq. (2), i.e.,

$G_{\kappa\mu}\big{|}_{k=k_{0}}=\sqrt{\frac{c^{3}\Gamma_{\mu}^{(0)}}{2\pi\omega_{0}^{2}}}A_{\mu}^{l}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}}),$
(S73)

which can be shown to cause a recoil heating rate in the absence of squeezing
given by

$\Gamma_{\mu}^{(0)}=\frac{V}{4\pi^{2}c}|\alpha_{0}|^{2}\left(\frac{\Delta\alpha}{2\epsilon_{0}(2\pi)^{3}}\right)^{2}\frac{8\pi
k_{0}^{2}}{3}r_{0\mu}^{2}\omega_{0}^{2},$ (S74)

which agrees with known results [49], and a libration angular distribution
(information radiation pattern) given by

$A_{\mu}^{l}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})\equiv-e^{i\text{arg}(\alpha_{0})}\sqrt{\frac{3}{8\pi}}(\bm{\varepsilon}_{\mathbf{k}}\cdot\mathbf{e}_{\mu}),$
(S75)

which are also orthonormal. Once expressed in this form, the whole derivation
in all the sections above remains valid as it only assumes that the form of
the Hamiltonian is given by Eq. (1) and that the coupling rates fulfill the
identity Eq. (2). All the expressions in the article are thus directly
generalizable to libration by just substituting the overlap $\xi_{\mu}$ by its
corresponding form in terms of the librational information radiation patterns:

$\xi_{\mu}\to\xi_{\mu}^{l}\equiv\int
d\Omega_{\mathbf{k}}\sum_{\bm{\varepsilon}_{\mathbf{k}}}A_{s}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}})A_{\mu}^{l}(\Omega_{\mathbf{k}},\bm{\varepsilon}_{\mathbf{k}}).$
(S76)

##  S3 Wigner function of input light

In this section we derive the Wigner function of the input and output light.
We define detection modes in terms of the interacting modes as [19]

$\hat{a}_{\mu}\equiv\int_{-\infty}^{\infty}dth(t)\tilde{a}_{\mu}(t),$ (S77)

with a square-normalized function $h(t)$ guaranteeing bosonic commutation
relations $[\hat{a}_{\mu},\hat{a}_{\mu}^{\dagger}]=1$ for both input and
output detection modes. For the following, we choose for $h(t)$ a modulated
step function,

$h(t)=\Theta(-t+T/2)\Theta(t+T/2)e^{i\tilde{\Omega}t}/\sqrt{T},$ (S78)

with $\tilde{\Omega}$ an arbitrary frequency and $1/T$ a detection bandwidth
which, in the spirit of the wideband-squeezing and the slow varying couplings
approximation, is taken much wider than the chosen frequency,
$\tilde{\Omega}T\ll 1$. We can now construct the Wigner function of this mode
recalling the definition of the amplitude and phase quadrature
$\hat{X}_{\mu}=\hat{a}^{\dagger}_{\mu}+\text{H.c.}$,
$\hat{Y}_{\mu}=i\hat{a}_{\mu}^{\dagger}+\text{H.c.}$, and using the
commutation relation of the interacting modes [50], namely

$W_{\mu}^{\rm
in}(X,Y)=\frac{1}{2\pi\sqrt{\text{det}[\overline{C}_{\mu}]}}\exp\left[-\frac{\mathbf{X}_{\mu}\cdot\overline{C}_{\mu}^{-1}\cdot\mathbf{X}_{\mu}}{2}\right],$
(S79)

with $\mathbf{X}_{\mu}\equiv[X_{\mu},Y_{\mu}]^{T}$, and $\overline{C}_{\mu}$
is the unnormalized covariance matrix,

$\overline{C}\equiv\left[\begin{array}[]{cc}2\pi S_{\mu,XX}^{\rm in}&-2\pi
S_{\rm cross}^{\rm in}\\\ -2\pi S_{\rm cross}^{\rm in}&2\pi S_{\mu,YY}^{\rm
in}\end{array}\right],$ (S80)

where $S_{\mu,\rm cross}^{\rm in}\equiv(S_{\mu,XY}^{\rm in}+S_{\mu,YX}^{\rm
in})/2.$ The Wigner function Eq. (S79) is displayed in the inset of Fig. 4(a)
in the main text. A similar expression can be derived using the bare squeezed
mode

$\hat{a}(\omega)=\frac{1}{\sqrt{\epsilon}}\int_{\omega-\epsilon/2}^{\omega+\epsilon/2}d\omega^{\prime}\hat{a}_{s}(\omega^{\prime}),$
(S81)

with $\epsilon$ small enough such that $r(\omega\pm\epsilon/2)\approx
r(\omega)$ and $\phi(\omega\pm\epsilon/2)\approx\phi(\omega)$. We define
quadratures $\hat{Q}(\omega)\equiv\hat{a}^{\dagger}(\omega)+\text{H.c.}$ and
$\hat{P}(\omega)\equiv i[\hat{a}^{\dagger}(\omega)-\text{H.c.}]$, and compute
the following expected values in the squeezed vacuum state,

$\displaystyle\langle\hat{Q}^{2}(\omega)\rangle$
$\displaystyle=e^{2r(\omega)}\cos^{2}[\phi(\omega)/2]+e^{-2r(\omega)}\sin^{2}[\phi(\omega)/2],$
(S82) $\displaystyle\langle\hat{P}^{2}(\omega)\rangle$
$\displaystyle=e^{2r(\omega)}\sin^{2}[\phi(\omega)/2]+e^{-2r(\omega)}\cos^{2}[\phi(\omega)/2],$
(S83)
$\displaystyle\langle\hat{Q}(\omega)\hat{P}(\omega)+\hat{P}(\omega)\hat{Q}(\omega)\rangle$
$\displaystyle=-2\sinh[2r(\omega)]\sin[\phi(\omega)].$ (S84)

Using these equations we can construct the Wigner function of the
electromagnetic field at frequency $\omega$ in full analogy with Eq. (S79).
This function is shown in Fig. 1b in the main text.