# Chalcogenide perovskites: an emerging class of semiconductors for
optoelectronics

Pooja Basera<EMAIL_ADDRESS>Saswata Bhattacharya
<EMAIL_ADDRESS>[

###### Abstract

Chalcogenide perovskites have received considerable interest in photovoltaic
research community owing to their stability (thermal and aqueous), non-
toxicity and lead free composition. However, to date a theoretical study
mainly focusing on the excitonic and polaronic properties are not explored
rigorously, due to its huge computational demand. Herein, we capture the
excitonic and polaronic effects in a series of chalcogenide perovskites ABS3
where A=Ba, Ca, Sr, and B=Hf, Sn by employing state-of-the art hybrid density
functional theory and many body perturbative approaches viz. GW and BSE. We
find that these perovskites possess a large exciton binding energy than 3D
inorganic-organic hybrid halide perovskites. We examine the interplay of
electronic and ionic contribution to the dielectric screening, and conclude
that electronic contribution is dominant over the ionic contribution. Further
using Feynman polaron model, polaron parameters are computed, and we observe
that charge separated polaronic states are less stable than bound excitons.
Finally, the theoretically calculated spectroscopic limited maximum efficiency
(SLME) suggests that among all chalcogenide perovskites, CaSnS3 could serve as
a best choice for photovoltaic applications.

###### keywords:

chalcogenide perovskite, hybrid DFT, GW-BSE, excitons, polarons, exciton
binding energy, SLME

+91-11-2659 1359 Indian Institute of Technology Delhi] Department of Physics,
Indian Institute of Technology Delhi, New Delhi, India Perovskites have gained
unprecedented attention in the past few years owing to their versatile
chemical and physical properties, and lead to a breakthrough progress from
3.8% to 25.5% 1, 2. This indicates that device performance is intimately
linked with basic material properties. The suitable material characteristics
allow to significantly shorten the time period for the evolution of high-
efficiency cells. However, the toxicity, degradability and the instability are
still serious challenges for large scale industrial applications 3, 4. This
motivates researchers towards a new species of perovskites, namely,
chalcogenide perovskites that possess Pb-free composition, are earth-abundant,
and highly stable in ambient atmosphere 5, 6. Several studies have been
performed experimentally as well as theoretically that show the successful
formation of chalcogenide perovskites along with their promising properties 7,
8, 9, 10, 11, 12, 13, 14, 9, 15, 16, 17, 18. Perera $et$ $al.$ have
synthesised AZrS3 (A = Ba, Ca, Sr) using high temperature sulfurization of the
oxides with CS2 19. S. Niu $et$ $al.$ have also synthesized them as well as
probed the optoelectronic properties experimentally 20. Meng $et$ $al.$
predicted mixed perovskites chalcogenides BaZr1-xTixS3 by repeated annealing
of binary mixtures 21. Sun $et$ $al.$ proposed S and Se based perovskites and
predicted their suitable band gaps for single junction solar cell 22. However,
most of the earlier experimental and theoretical studies are mainly focused on
Zr-based chalcogenide perovskites 23. Recently, a few reports in the
literature presented a first-principles studies for Sn based chalcogenides as
a high performance thermoelectric materials 24, and Hf based chalcogenides as
green light emitting semiconductors 25. In addition, a recent experimental
studies successfully unveil the synthesis of CaSnS3 perovskite 26. In view of
this, we aim to provide a detailed theoretical study for Hf and Sn- based
chalcogenides concerning the properties suitable for optoelectronic
applications that may act as valuable guidance for the ongoing experimental
work.
Intriguingly, the charge separation in these materials are considerably
affected by the exciton formation. The operation behind the solar cell
mechanism depends on the thermally dissociate excitons into free electrons and
holes, which in turn induce free-charge transport. Therefore, the accurate
determination of the exciton binding energy is crucial for their active usage
in optoelectronic materials. Additionally, the charge carrier and exciton
dynamics are also influenced by the polaron formation. The multiple
photophysical phenomena in these materials can be explained by electron-phonon
interactions and the transport of polarons 27, 28. Regardless of several
studies on chalcogenide perovskites, a thorough understanding on excitonic and
polaronic effects on Hf and Sn-based chalcogenides are hithertho unknown, and
unveil for the first time, to the best of our knowledge.
In this Letter, we have performed a robust study of electronic, optical,
excitonic and polaronic properties of chalcogenide perovskites ABS3 (where A=
Ba, Ca, Sr, and B=Hf, Sn) within the framework of density functional theory
(DFT) 29, 30, hybrid DFT (HSE06) and beyond DFT approaches (GW and BSE).
First, we optimize the crystal structures using semi-local PBE 31 exchange-
correlation ($\epsilon_{\textrm{xc}}$) functional, and validate the optimized
lattice parameters in light of the available experimental results. Then, to
study electronic structure, atom-projected electronic partial density of
states (pDOS) are computed using hybrid $\epsilon_{\textrm{xc}}$ functional
HSE06 32. Subsequently, the optical properties are determined using many-body
perturbation theory (MBPT). Then, we solve the Bethe-Salpeter equation (BSE)
33, 34 on top of single-shot G0W0@PBE 35, 36, to obtain the exciton binding
energy as well as electronic contribution to the dielectric function. On the
other hand, the ionic contribution to the dielectric function is determined
using density functional perturbation theory (DFPT). Finally, we compute the
spectroscopic limited maximum efficiency (SLME) 37 using the quasiparticle
(QP) band gap and absorption coefficient.

Herein, we consider the distorted orthorhombic phase of chalcogenide
perovskites ABS3 (A=Ba, Ca, Sr, and B=Hf, Sn) having the space group $Pnma$ 7
(see Figure 2a). For BaSnS3 and SrSnS3, the needle-like phase is examined in
our study. Table 1 presents the PBE $\epsilon_{\textrm{xc}}$ functional
optimized lattice parameters, which are closely in agreement with experimental
results reported in literature.

Figure 1: Crystal structure (a) BaHfS3 in orthorhombic distorted phase and
BaSnS3 in needle-like phase. (b) CaHfS3 and CaSnS3 in orthorhombic distorted
phase. (c) SrHfS3 in orthorhombic distorted phase and SrSnS3 in needle-like
phase. Sulphur atom projected density of states (d) BaHfS3 and (e) BaSnS3
Table 1: Calculated lattice parameters of ABS3 (A=Ba, Ca, Sr and B = Hf, Sn)
chalcogenide perovskites. The experimental values are provided in brackets,
and are taken from Ref. 7, 38, 26. Calculated distortion parameters for BS6
octahedra in ABS3 chalcogenide perovskites.

Configurations | a (Å) | b (Å) | c (Å) |
---|---|---|---|---
BaHfS3 | 6.95 (6.99) | 6.99 (7.00) | 9.99 (9.92) |
BaSnS3 | 3.97 (3.93) | 8.48 (8.53) | 15.35 (14.51) |
CaHfS3 | 6.39 (6.52) | 6.89 (6.98) | 9.35 (9.54) |
CaSnS3 | 6.72 (6.68) | 7.06(7.08) | 9.67 (11.28) |
SrHfS3 | 6.77 (6.72) | 7.09 (7.05) | 9.77 (9.72) |
SrSnS3 | 3.78 (3.79) | 8.41 (8.74) | 14.62(14.08) |
Octahedra (BS6) | Average bond length (Å) | Bond angle variance (deg.(2)) | Polyhedral volume (Å3) | Distortion index
BaHfS3 (HfS6) | 2.51 | 0.29 | 21.04 | 0.0036
BaSnS3 (SnS6) | 2.61 | 19.61 | 23.38 | 0.024
CaHfS3 (HfS6) | 2.47 | 1.14 | 20.06 | 0.00513
CaSnS3(SnS6) | 2.59 | 3.22 | 23.23 | 0.00513
SrHfS3 (HfS6) | 2.54 | 0.15 | 21.86 | 0.00377
SrSnS3 (SnS6) | 2.59 | 15.72 | 22.92 | 0.01958

In addition, the octahedra distortions (BS6) for ABS3 chalcogenides are also
listed in Table1. The average bond length, bond angle variance, polyhedral
volume and distortion index are calculated corresponding to BS6 octahedra. The
data shows that the BS6 octahedra in chalcogenides having needle like phase
(BaSnS3 and SrSnS3) is more distorted compared to octahedra in orthorhombic
phase (see Fig 1 and Table 1). In BaSnS3 and SrSnS3, there are four types of
Sn-S bonds of different strengths, namely S1, S2, S3, S4 (see Fig 1[a-c]). We
notice that Sn-S1 bonding is relatively weaker, however Sn-S4 bonding is the
strongest one. The atom-projected density of states are calculated to
disentangle the contribution of each S atoms at valence band edges (Note that
S atom has main contribution at valence band maximum for these perovskites
(vide infra)). For BaHfS3, the S1, S2 and S3 contribution are same at valence
band edges, because of almost same binding strength with Hf atom, however for
BaSnS3, Sn-S4 has stronger binding that leads to its least contribution at
valence band edges (see SI, Fig S3, for remaining configurations). Therefore,
the differentiation of S atoms at valence band edges subsequently lead to
lighter valence bands (low m${}_{\textrm{h}}^{*}$), which eventually helps to
boost the hole mobility (Table 2).
To gain deep insights on the electronic structure, we compute the electronic
pDOS of the ABS6 chalcogenides using HSE06 $\epsilon_{\textrm{xc}}$ functional
as shown in Figure 2[a-f]. For AHfS3 perovskites, the valence band maximum
(VBM) is majorly contributed by S 3p-orbitals, whereas Hf 4d-orbitals
primarily contributed at the conduction band minimum (CBm). The remaining
orbitals have minor contribution at VBM and CBm. For ASnS3 perovskites, the
VBM is mainly contributed by S 3p -orbitals, the CBm is dominated by the
hybridization of S (3p) and Sn (5s) orbitals. The presence of hybridized
states in case of ASnS3 perovskite is responsible for its lower band gap than
AHfS3 perovskites, the latter does not contain hybridized states. The orbitals
and their hybridzation can be visualized clearly from wave function analysis.
For BaHfS3, the S p orbitals and the Hf d orbitals contribution at VBM and
CBm, respectively are shown in Figure 2[g]. For BaSnS3, the contribution of S
p orbitals at VBM and the hybridization of Sn s and S p orbitals are shown in
Figure 2[h]. From the electronic structure, we observe p-d transition in Hf-
based perovskites, where as p-p and p-s transition in Sn-based perovskites.
Notably, perovskites are well known for their transport properties. Thus, here
we compute the band structures for all the cases (see SI, Fig S1) and their
corresponding reduced mass ($\mu$). We observe $\mu$ $<$ 1 for all the cases,
which in turn indicates high carrier mobility, and therefore infers the better
charge carrier transport. However, for BaSnS3 and SrSnS3,
m${}_{\textrm{h}}^{*}$ $<$ m${}_{\textrm{e}}^{*}$ due to the difference in the
Sn-S bonding strength at valence band edge (vide supra).

Table 2: Calculated effective mass of electron (m${}_{\textrm{e}}^{*}$), hole (m${}_{\textrm{h}}^{*}$) and reduced mass ($\mu$) along a $\Gamma-$Z high symmetry path. All values are in terms of free-electron mass (m${}_{\textrm{e}}$). Configurations | m${}_{\textrm{e}}^{*}$ | m${}_{\textrm{h}}^{*}$ | $\mu$
---|---|---|---
BaHfS3 | 0.367 | 0.642 | 0.233
BaSnS3 | 0.639 | 0.233 | 0.170
CaHfS3 | 0.444 | 0.500 | 0.235
CaSnS3 | 0.385 | 0.450 | 0.207
SrHfS3 | 0.445 | 0.567 | 0.249
SrSnS3 | 0.527 | 0.213 | 0.152
Table 3: Band gap (in eV) of chalcogenide perovskites. $i$, $d$, $e$ and $t$ represent indirect, direct, experimental and theoretical band gap. Configurations | PBE | HSE06 | G0W0@PBE | Previous work
---|---|---|---|---
BaHfS3 | 1.12 | 1.97 | 2.17 | 2.17e10
BaSnS3 | 0.87i(0.95d) | 1.83i(1.96d) | 1.91i(2.03d) | 1.62t11
CaHfS3 | 1.38 | 2.23 | 2.46 | 2.21t11
CaSnS3 | 0.76 | 1.40 | 1.43 | 1.58t11
SrHfS3 | 1.47 | 2.32 | 2.59 | 2.41e11
SrSnS3 | 0.96 | 1.94 | 2.04 | 1.56t11
Figure 2: Electronic partial density of states (pDOS) of (a) BaHfS3, (b)
BaSnS3, (c) CaHfS3, (d) CaSnS3, (e) SrHfS3, (f) SrSnS3 using HSE06
$\epsilon_{\textrm{xc}}$ functional. Wave function analysis in real space of
(g) BaHfS3, and (h) BaSnS3 at VBM and CBm.

Note that, all the ABS3 perovskites have band gap less than 2.6 eV, thereby
suitable for optoelectronic applications. However, we observe a slight
indirect band gap for BaSnS3 (differ by 0.08 eV), and weakly indirect band gap
for SrSnS3 (differ by few meV’s). In other words, the direct gap is at energy
level very close to indirect gap. The reason for indirect band gap could be a
large octahedral tilting (BS6) for needle like phase, indicated from
distortion index value given in Table 1. The band gaps computed from semi-
local PBE $\epsilon_{\textrm{xc}}$ functional are underestimated due to well
known self-interaction error (see Table 3). However, hybrid
$\epsilon_{\textrm{xc}}$ functional HSE06 works very well and correct the band
gap i.e. in agreement with the previous reports (see Table 3). This shows the
reliability of hybrid functional HSE06 to predict the electronic structure
accurately. For optical properties, we go beyond the HSE06, and performed MBPT
viz. GW and BSE calculations. GW calculations determine the fundamental band
gap analogous to photoelectron spectroscopy (PES) and inverse photoelectron
spectroscopy (IPES) 35, 36, and assumed to be more accurate where as BSE
calculation predict the optical band gap akin to experimental optical
absorption spectroscopy 33, 34.

Figure 3: Imaginary [Im $(\upvarepsilon)$] part of the dielectric function
average along x, y and z directions for (a) BaHfS3, (b) BaSnS3, (c) CaHfS3,
(d) CaSnS3, (e) SrHfS3, (f) SrSnS3 obtained using G0W0@PBE and BSE@G0W0@PBE.
The oscillator strength is represented by orange color.

The electron-hole interaction is explicitly considered in the BSE calculations
39. Firstly, we perform single shot GW (G0W0) on the top of PBE to determine
the optical response. Note that the quasiparticle (QP) band gap is slighly
overestimated as compare to experimental band gap, due to non-inclusion of
electron-hole interaction. Table 3 presents the QP gaps of ABS3 calculated
using GW@PBE. The experimental band gaps are nicely captured from GW
calculations, however overestimated from the reported theoretical band gap
calculations, as many of them are performed using semi-local or hybrid
$\epsilon_{\textrm{xc}}$ functionals. This indicates the advantage of
performing MBPT calculations and could be very useful for ongoing experimental
work. The imaginary part of the dielectric function [Im ($\upvarepsilon$)]
calculated using GW@PBE and BSE@GW@PBE is shown in Figure 3. The oscillator
strength corresponding to BSE@GW@PBE is also shown in Figure 3. We compute the
exciton binding energy (E${}_{\textrm{B}}$) by taking the difference of the QP
band gap (GW@PBE peak position) and the optical band gap (BSE@GW@PBE peak
position). The E${}_{\textrm{B}}$ of the first bright exciton for BaHfS3,
BaSnS3, CaHfS3, CaSnS3, SrHfS3 and SrSnS3 is 0.193 eV, 0.134 eV, 0.250 eV,
0.222 eV, 0.256 eV, and 0.148 eV, respectively. From the analysis of BSE
eigenvalues, we observe a dark exciton (optically inactive) below the first
bright exciton for the case of BaHfS3, BaSnS3, and SrSnS3 perovskites. The
E${}_{\textrm{B}}$ corresponding to the dark excitons are 0.236 eV, 0.316 eV,
and 0.248 eV for BaHfS3, BaSnS3, and SrSnS3, respectively. The oscillator
strength matches well with the excitonic peak position. The optical
transitions below the quasiparticle band gap may lead to various interesting
excitonic properties40, 41. Using aforementioned quantities such as exciton
binding energy, band gap, dielectric function, and reduced mass, we compute
several excitonic parameters such as excitonic temperature (Texc), radius
(rexc), and probability of wavefunction ($|\phi_{\textrm{n}}(0)|^{2}$) for
electron-hole pair at zero separation (see details in SI). The inverse of
$|\phi_{\textrm{n}}(0)|^{2}$) gives a qualitative description of the excitonic
lifetime ($\tau$). Therefore, the exciton lifetime ($\tau$) for the
perovskites follows an order: SrSnS3 $>$ BaSnS3 $>$ CaSnS3 $>$ BaHfS3 $>$
CaHfS3 $>$ SrHfS3. Hence, we obtain that Sn based chalcogenides have larger
exciton lifetime than Hf based chalcogenides. Among Sn-based chalcogenides,
needle-like phase show large exciton lifetime than orthorhombic phase.

Table 4: Calculated excitonic parameters for chalcogenide perovskites Excitonic parameters | BaHfS3 | BaSnS3 | CaHfS3 | CaSnS3 | SrHfS3 | SrSnS3
---|---|---|---|---|---|---
E${}_{\textrm{B}}$ (eV) | 0.193 | 0.134 | 0.250 | 0.222 | 0.256 | 0.148
T${}_{\textrm{exc}}$ (K) | 2239 | 1555 | 2901 | 2576 | 2970 | 1717
r${}_{\textrm{exc}}$ (nm) | 0.932 | 1.43 | 0.841 | 1.046 | 0.788 | 1.667
$|\phi_{\textrm{n}}(0)|^{2}(10^{27}$m${}^{-3})$ | 0.393 | 0.108 | 0.535 | 0.277 | 0.649 | 0.069
Figure 4: DFPT calculations to determine the ionic contribution to dielectric
function for (a) BaHfS3, (b) BaSnS3, (c) CaHfS3, (d) CaSnS3, (e) SrHfS3, (f)
SrSnS3.

We have obtained high E${}_{\textrm{B}}$ values compared to conventional 3D
halide perovskite42. This can be explained by the interplay of electronic and
ionic contribution to the dielectric screening. Note that, an electronic
contribution is dominant over the ionic one, if E${}_{\textrm{B}}$ is much
larger than the longitudinal optical phonon energy
($\hbar\omega_{\textrm{LO}}$) 43. In that scenario, ionic contribution can be
neglected, and as a result, E${}_{\textrm{B}}$ does not change. Following
this, in chalcogenide perovskite cases, all the dominant longitudinal optical
active phonon modes are below 50 meV (see Figure 4). Table 4 presents
E${}_{\textrm{B}}$ calculated from BSE calculations, where
E${}_{\textrm{B}}\gg\hbar\omega_{\textrm{LO}}$, hence ionic screening to
dielectric function can be ignored. Further, we have also employed Wannier-
Mott approach to validate the E${}_{\textrm{B}}$. As per this model, the
E${}_{\textrm{B}}$ is dependent on the reduced mass of the charge carriers
($\mu$) for screened Coulomb interacting e-h pairs and the effective
dielectric constant ($\varepsilon_{eff}$). The expression is given by:

$\textrm{E}_{\textrm{B}}=\frac{\mu}{\varepsilon_{eff}^{2}}\textrm{R}_{\infty}$
(1)

Table 5: The upper and lower limit of exciton binding energy (E${}_{\textrm{B}}$) for chalcogenide perovskites Configurations | Upper bound (eV) | Lower bound (meV)
---|---|---
BaHfS3 | 0.188 | 1.24
BaSnS3 | 0.149 | 19.92
CaHfS3 | 0.227 | 5.47
CaSnS3 | 0.189 | 3.61
SrHfS3 | 0.230 | 2.37
SrSnS3 | 0.138 | 17.27

where, $\textrm{R}_{\infty}$ and $\varepsilon_{eff}$ are the Rydberg constant
and the value lies in between the static value of electronic and ionic
dielectric constants, respectively. The upper and lower bound to the exciton
binding energy can be calculated from the contribution of electronic and ionic
static dielectric constants. For BaHfS3, BaSnS3, CaHfS3, CaSnS3, SrHfS3 and
SrSnS3, the corresponding static electronic dielectric constants are 4.10,
4.60, 3.73, 4.09, 3.71, and 4.79, respectively. Note that, these values are
obtained using BSE. The DFPT calculations are performed to determine static
ionic dielectric constants i.e. 50.45, 10.77, 24.16, 27.91, 37.76, and 10.94
for respective perovskites (see Figure 4). We notice that the needle-like
phase has relatively a low value of static ionic dielectric constants. Using
Equation 1, we determine the upper and lower bounds of E${}_{\textrm{B}}$ by
substituting reduced mass from Table 2, and the above mentioned static
electronic and ionic dielectric constants. We have found that the upper bounds
E${}_{\textrm{B}}$ (Table 5) agree well with the difference of GW and BSE peak
positions (listed in Table 4). This confirms that for chalcogenide
perovskites, the electronic contribution is dominant over the ionic
contribution in dielectric screening.
Further, we have used Fröhlich mesoscopic model 44, 45 to investigate the
interaction of longitudinal optical phonon modes with the carriers that
strongly influence the carrier mobility. This interaction is defined by the
dimensionless Fröhlich parameter $\alpha$, given by

$\alpha=\left(\frac{1}{\upvarepsilon_{\infty}}-\frac{1}{\upvarepsilon_{\textrm{static}}}\right)\sqrt{\frac{\textrm{R}_{\infty}}{ch\omega_{\textrm{LO}}}}\sqrt{\frac{\textrm{m}^{*}}{\textrm{m}_{\textrm{e}}}}$
(2)

where $h$ and $c$ are Planck’s constant and speed of light, respectively. The
Fröhlich parameter $\alpha$ is dependent on the $\upvarepsilon_{\infty}$
(electronic dielectric constant), $\upvarepsilon_{\textrm{static}}$ (ionic
static dielectric constant), carrier effective mass (m∗), and phonon frequency
$\omega_{\textrm{LO}}$. The characteristic frequency $\omega_{\textrm{LO}}$
has been calculated using athermal ‘B’ scheme of Hellwarth et al 46, where
spectral average of infrared active optical phonon modes are considered. The
computed $\alpha$ values are listed in Table 6, the $\alpha$ parameter for
hole is given in SI. By knowing $\alpha$, we compute the polaron energy (Ep)
using Equation 3. Note that the polaron formation may lead to reduction of
quasiparticle energy of electron and hole, and is given by the following
equation 43, 47:

$\textrm{E}_{\textrm{p}}=(-\alpha-0.0123\alpha^{2})\hbar\omega_{\textrm{LO}}$
(3)

For BaHfS3, BaSnS3, CaHfS3, CaSnS3, SrHfS3, SrSnS3, the QP gap is lowered by
0.17, 0.09, 0.21, 0.15, 0.19, 0.08 eV, respectively. By comparing these values
with E${}_{\textrm{B}}$, we obtain that the charge separated polaronic state
is less stable than the bound exciton. Other polaron parameters such as
effective polaron masses ($\textrm{m}_{\textrm{P}}$), radii
(l${}_{\textrm{p}}$) and polaron mobilities ($\mu_{\textrm{P}}$) are
determined using Hellwarth polaron model 46. These parameters determine an
upper limit for charge carrier mobilities by assuming only the involvement of
optical phonons (see details in SI). However, one may expect a lower value of
polaron carrier mobility, if local distortions due to polarons and defect
scattering due to acoustic phonons are taken into account. From Table 6, we
have found that large polarons (l${}_{\textrm{p}}$) are formed, and Sn-based
chalcogenides possess a large value of polaron mobility than Hf based ones.

Table 6: Polaron parameters for electrons in chalcogenide perovskites Configurations | 1/$\epsilon^{*}$ |  $\omega_{\textrm{LO}}$ (cm-1) |  $\alpha$ e | $\textrm{m}_{\textrm{P}}$ | l${}_{\textrm{P}}$ (Å) |  $\mu_{\textrm{P}}$ (cm2V-1s-1)
---|---|---|---|---|---|---
BaHfS3 | 0.223 | 148.53 | 3.68 | 0.76 | 35.75 | 15.82
BaSnS3 | 0.124 | 195.32 | 2.36 | 1.05 | 55.26 | 16.41
CaHfS3 | 0.226 | 253.92 | 3.13 | 0.976 | 57.96 | 13.84
CaSnS3 | 0.208 | 167.54 | 3.31 | 0.766 | 41.36 | 17.01
SrHfS3 | 0.242 | 158.47 | 4.26 | 1.10 | 34.71 | 9.58
SrSnS3 | 0.117 | 199.23 | 1.99 | 0.80 | 61.06 | 25.08
Figure 5: Electronic band structure and transition probability from VBM to CBm
(square of dipole transition matrix elements) for CaSnS3. Spectroscopic
limited maximum efficiency of ABS3 (A=Ca, Sr, and B=Hf, Sn).

Notably, ABS3 perovskites can be considered as good solar absorbers as they
exhibit large absorption coefficient and most of them show a direct band gap
in visible region range. Therefore, to quantify this, we choose a parameter
introduced by Yu et al. known as spectroscopic limited maximum efficiency
(SLME) 37, 48. SLME determines the maximum limit of the solar power conversion
efficiency of an absorber material. It incorporates the shape of absorption
spectra, band gap and its nature (direct or indirect), thickness of the thin
film absorber layer and the material dependent non-radiative recombination
losses. This is considered as an improved version of Shockley and Queisser
(SQ) efficiency 49. From above discussion we know that SLME depends on the
nature of the band gap, however for various perovskites 50, it has been seen
that even after having a direct electronic band gap, the optical allowed
dipole transition from VBM to CBm was forbidden. This happens because of the
presence of inversion symmetry that leads to VBM and CBm at the same parity.
We, therefore, check the possibility of the optical transition from VBM to CBm
for our systems. For this, we calculate the dipole transition matrix, defined
as the electric dipole moment related with a transition between the initial
state and the final state. Its square gives the probability of transition
between the two states. We have computed the square of the dipole transition
matrix elements (p2) for all the systems. We have found that BaHfS3 has
forbidden dipole transition at $\Gamma$, regardless of direct electronic band
gap (see SI, Fig S1). Figure 5, shows an optically allowed dipole transition
for CaSnS3 perovskites (for other configurations, see SI, Fig S1). Thus,
keeping this in mind, we proceed next for SLME calculations, only for systems,
that posseses direct band gap along with optically allowed dipole transitions.
Note that, SrSnS3 has also been considered, as the difference of indirect and
direct band gap of only few meV’s, that does not impact the absorption
coefficient. For SLME, the input parameters are band gap, thickness,
absorption coefficient and the standard solar spectrum. Figure 5 presents the
calculated SLME of ABS3 (A=Ca, Sr, and B=Hf, Sn). SLME values at 10 $\mu$m
absorber layer thickness are 13.26%, 32.45%, 10.56%, and 21.80% for CaHfS3,
CaSnS3, SrHfS3, SrSnS3 respectively (see Figure 5). We have achieved the
highest value of SLME for CaSnS3, that could serve as a best material for
photovoltaic applications.

In summary, we have determined the electronic, optical (excitonic and
polaronic) properties of chalcogenide perovskites ABS3 (A= Ba, Ca, Sr, and
B=Hf, Sn) under the framework of DFT and MBPT. We have obtained reduce mass,
$\mu$ $<$ 1 for all the cases, that confirms high carrier mobility. For BaSnS3
and SrSnS3, the difference in Sn-S bonding strength of SnS6 octahedra leads to
lighter valence bands, which in turn boost up the hole mobility. The large
value of exciton binding energy is noted for chalcogenide perovskites i.e.,
0.193, 0.134,0.250, 0.222, 0.256, and 0.148 eV for BaHfS3, BaSnS3, CaHfS3,
CaSnS3, SrHfS3 and SrSnS3, respectively. The exciton binding energy computed
from BSE calculations are well in agreement with Wannier-Mott approach,
thereby indicate that electronic contribution is dominant over ionic
contribution in dielectric screening. Further, from Fröhlich’s mesosocopic
model, we have found that the charge-separated polaronic states are less
stable than the bound excitons. Lastly, the spectroscopic limited maximum
efficiency (SLME) suggest that among all studied chalcogenide perovskites,
CaSnS3 could be a best choice for efficient photovoltaic applications.

##  1 Computational Methods

The DFT 29, 30 calculations are performed as implemented in the Vienna ab
initio simulation package (VASP) 51, 52. Projector-augmented wave (PAW)
pseudopotentials 53, 52 are used to describe the ion-electron interactions.
The valence states considered for Ca, Sr, Ba, Hf, Sn and S are 3s23p64s2,
4s24p65s2, 5s25p66s2, 5p66s26d4, 4d105s25p2, and 3s23p4, respectively. PBE 31
exchange-correlation ($\epsilon_{\textrm{xc}}$) functional has been used for
the structural optimization. The structural relaxation are performed until the
forces are smaller than 0.001 eV/Å. The plane wave basis set expansion or
kinetic energy cutoff is set to 500 eV. For single point energy calculations,
the tolerance criteria for energy convergence is set to 0.001 meV. For
Brillouin zone integration, a $k$-grid of $7\times 7\times 5$ is used,
generated from Monkhorst-Pack 54 scheme. The advanced hybrid
$\epsilon_{\textrm{xc}}$ functional HSE06 32 and GW calculations are performed
for the better estimation of the electronic band gap. To determine excitonic
properties, Bethe-Salpeter equation (BSE) 33, 34 calculations are performed on
top of single shot GW 35, 36 (G0W0). PBE $\epsilon_{\textrm{xc}}$ functional
is considered as a starting point for single shot GW calculations. A grid of
50 frequency points is used for the polarizability calculations. We have
considered sufficient number of unoccupied bands i.e. nine times the number of
occupied bands. The convergence for number of unoccupied bands is shown in SI.
For BSE calculations, $\Gamma$-centered $3\times 3\times 2$ $k$-grid has been
used. The electron-hole kernel for BSE calculations are constructed by 24
occupied and 24 unoccupied bands (for convergence see SI). Density functional
perturbation theory (DFPT) calculations are performed to get the ionic
contribution to dielectric screening using $7\times 7\times 5$ $k$-grid. Note
that spin-orbit coupling (SOC) has not been taken into account, as it does not
alter the electronic structure (see SI). PB acknowledges UGC, India, for the
senior research fellowship [grant no. 20/12/2015 (ii) EUV]. SB acknowledges
the financial support from SERB under core research grant (grant no.
CRG/2019/000647). We acknowledge the High Performance Computing (HPC) facility
at IIT Delhi for computational resources. (I) Electronic band structure and
transition probability. (II) Effect of spin-orbit coupling on band gap. (III)
Convergence of number of bands used in single shot GW calculations. (IV)
Convergence of number of occupied (NO) and unoccpied bands (NV) used in
electron-hole interaction kernel for BSE calculations. (V) Excitonic
parameters: exciton binding energy (E${}_{\textrm{B}}$), dielectric constant
($\upvarepsilon_{\textrm{eff}}$), band gap (Eg) and reduced mass ($\mu$). (VI)
Polaron parameters of chalcogenide perovskites. (VII) pDOS of S atoms of
chalcogenide perovskites.

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# Coherence properties of a spin in a squeezed resonator

Inbar Shani  Emanuele G. Dalla Torre  Michael Stern  Department of Physics and
Center for Quantum Entanglement Science and Technology, Bar-Ilan University,
52900, Ramat Gan Israel

###### Abstract

A promising venue for hybrid quantum computation involves the strong coupling
between impurity spins and superconducting resonators. One strategy to control
and enhance this coupling is to prepare the resonator in a non-classical
state, such as a squeezed state. In this work, we theoretically study the
effects of these states on the coherence properties of the spin. We develop an
analytic approach based on the Schrieffer-Wolf transformation that allows us
to quantitatively predict the coupling and the dephasing rate of the spin, and
we numerically confirm its validity. We find that squeezing can enhance the
coupling between the resonator and the spin. However, at the same time, it
amplifies the photon noise and enhances the spin decoherence. Our work
demonstrates a major impediment in using squeezing to reach the strong-
coupling limit.

##  I Introduction

Impurity spins in semiconductors are quantum entities with a long coherence
time, which enables them to store safely quantum information [1].
Unfortunately, the weakness of their interaction with the environment hinders
our ability to control them directly. An appealing road towards a spin-based
quantum processor consists of combining the impurity spins with
superconducting circuits [2, 3]. To realize this kind of hybrid system, one
needs to reach the strong coupling regime, where the coupling between the
spins and the superconducting circuit is much larger than the decoherence
rates. In recent years, such a regime was reached for large ensemble of spins
[4, 5], and for spin-like macroscopic structures [6, 7]. Yet, the coupling
constant of a single microscopic spin with a superconducting resonator is
extremely small (of the order of a few kHz [8]), and reaching the strong
coupling, in these conditions, remains a great challenge [9]. Recently, Ref.
[10] suggested to increase artificially the coupling by squeezing the
resonator. In a squeezed state, the fluctuations of the electromagnetic field
in a given quadrature can be controlled and made arbitrarily large [11, 12,
13]. Thus, increasing the coupling by squeezing seems interesting and even
promising. On the other hand, the large number of photons that characterizes a
squeezed state leads to large fluctuations, and may compromise the coherence
of the spin. In this work, we study the interplay between these two effects
and show that the beneficial effects of squeezing are strongly suppressed by
the noise in the photon number.

##  II Physical model

We consider the quantum circuit illustrated in Fig. 1, which contains a lumped
element LC resonator of resonance frequency $\omega_{r}$, coupled inductively
by a coupling constant $g$ to a spin of transition frequency $\omega_{s}$. As
mentioned earlier, the coupling between the two systems is intrinsically small
and it is necessary to increase it by at least one order of magnitude in order
to reach the strong coupling regime. Following Ref. [10], we explore the
possibility of squeezing the resonator to enhance its coupling with the spin.
To achieve this goal, we connect the resonator to a non-linear element, namely
a superconducting quantum interference device (SQUID). The Hamiltonian of the
system can be written as

Figure 1: Circuit diagram showing a lumped element LC resonator (red) of
resonance frequency $\omega_{r}$ and coupled inductively by coupling constant
$g$ to a spin (gray). In order to increase the coupling $g$, one drives a
SQUID (blue) with a parametric drive (yellow) at frequency
$\omega_{p}=2\omega_{r}$.

$\displaystyle\begin{split}H/\hbar&=\underbrace{\omega_{r}a^{\dagger}a}_{\text{LC
resonator}}+\;\;\;\underbrace{\tfrac{1}{2}\omega_{s}\sigma_{z}}_{\text{spin}}\;\;+\;\;\underbrace{g(a+a^{\dagger})\sigma_{x}}_{\text{coupling}}\\\
&\;\;\;\;\;\;\;\;\;\;\;\underbrace{-2\frac{E_{J}}{\hbar}\cos\left(\frac{\Phi_{\rm
ext}}{2\varphi_{0}}\right)\cos\left(\varphi\right)}_{\text{SQUID}}\end{split}$
(1)

where $a^{\dagger}$ ($a$) is the creation (annihilation) operator of a photon
in the resonator, $\\{\sigma_{z},\sigma_{x}\\}$ are the Pauli matrices,
$E_{J}$ is the Josephson energy of the junctions, $\Phi_{\rm ext}$ is the flux
threading the loop of the SQUID, $\varphi_{0}=\hbar/2e$ is the reduced flux
quantum and $\varphi$ is the superconducting phase difference between the
terminals of the inductor L [14]. The flux $\Phi_{\rm ext}$ is varied over
time according to

$\displaystyle\Phi_{\rm ext}=\Phi_{\rm DC}+\Phi_{\rm
AC}\cos\left(\omega_{p}t\right).$ (2)

If $\Phi_{\rm AC}\ll\varphi_{0}$, it is possible to expand the cosine term in
Eq. 1 to the first order around $\Phi_{\rm DC}$, such that

$\displaystyle\begin{split}\cos\left(\frac{\Phi_{\rm
ext}}{2\varphi_{0}}\right)\approx\cos\left(\frac{\Phi_{\rm
DC}}{2\varphi_{0}}\right)-\frac{\Phi_{\rm
AC}}{2\varphi_{0}}\sin\left(\frac{\Phi_{\rm
DC}}{2\varphi_{0}}\right)\cos(\omega_{p}t).\end{split}$ (3)

This leads to

$\displaystyle\begin{split}H/\hbar&=\omega^{\prime}_{r}a^{\dagger}a+\tfrac{1}{2}\left(\omega^{\prime}_{r}-\omega_{r}\right)\left(a^{2}+{a^{\dagger}}^{2}\right)+\tfrac{1}{2}\omega_{s}\sigma_{z}\\\
&+g(a+a^{\dagger})(\sigma_{+}+\sigma_{-})-\lambda\cos(\omega_{p}t)\left(a+{a^{\dagger}}\right)^{2},\end{split}$
(4)

where $\omega^{\prime}_{r}=\omega_{r}+4\varphi_{\rm
ZPF}^{2}\left(E_{J}/\hbar\right)\cos\left[\Phi_{\rm DC}/(2\varphi_{0})\right]$
and

$\displaystyle\lambda=\frac{E_{J}}{\hbar}\Phi_{\rm AC}\;\frac{\varphi_{\rm
ZPF}^{2}}{\varphi_{0}}\sin\left(\frac{\Phi_{\rm DC}}{2\varphi_{0}}\right).$
(5)

We now move to a frame that rotates at the pump frequency, by applying the
unitary transformation
$U(t)=\exp\left[i\omega_{p}t/2\left(a^{\dagger}a+\sigma_{z}/2\right)\right]$,
leading to the transformed Hamiltonian
$\tilde{H}=UHU^{\dagger}-i\hbar\dot{U}U^{\dagger}$. When $\omega_{p}\approx
2\omega^{\prime}_{r}$, one can neglect the quickly rotating terms and obtain

$\displaystyle\begin{split}\tilde{H}/\hbar=&\tilde{\omega}_{r}a^{\dagger}a+\tfrac{1}{2}\tilde{\omega}_{s}\sigma_{z}-\tfrac{1}{2}\lambda\left(a^{2}+a^{\dagger
2}\right)\\\ &+g\left(a\sigma_{+}+a^{\dagger}\sigma_{-}\right),\end{split}$
(6)

where $\tilde{\omega}_{r}={\omega^{\prime}}_{r}-\omega_{p}/2$ and
$\tilde{\omega}_{s}=\omega_{s}-\omega_{p}/2$. Eq. (6) describes the coupling
between a spin and a squeezed resonator and is the focus of the present study.
In order to characterize the effective coupling between the spin and the
squeezed resonator, we diagonalize the latter using a Bogoliubov
transformation. To perform this task, we introduce the canonical operators
$\gamma$ and $\gamma^{\dagger}$, defined as

$\displaystyle\begin{cases}\gamma=a\cosh(r)-a^{\dagger}\sinh(r)\\\
\gamma^{\dagger}=a^{\dagger}\cosh(r)-a\sinh(r)\end{cases},$ (7)

such that

$\displaystyle\gamma^{\dagger}\gamma=\cosh(2r)a^{\dagger}a-\tfrac{1}{2}\sinh(2r)\left(a^{2}+{a^{\dagger}}^{2}\right).$
(8)

The Hamiltonian (6) becomes

$\displaystyle\begin{split}\tilde{H}/\hbar=&\Omega_{r}\gamma^{\dagger}\gamma+\tfrac{1}{2}\tilde{\omega}_{s}\sigma_{z}+\tfrac{1}{2}ge^{r}(\gamma^{\dagger}+\gamma)(\sigma_{+}+\sigma_{-})\\\
&-\tfrac{1}{2}ge^{-r}(\gamma^{\dagger}-\gamma)(\sigma_{+}-\sigma_{-}).\end{split}$
(9)

where $\Omega_{r}=\tilde{\omega}_{r}/{\cosh(2r)}$ and
$r={\tanh^{-1}}(\lambda/\tilde{\omega}_{r})/2$. The Hamiltonian Eq. 9
describes two main effects of squeezing: First, the frequency of the resonator
is reduced from $\tilde{\omega}_{r}$ to $\Omega_{r}$ and, second, the coupling
between the spin and the resonator is enhanced by a factor of $e^{r}/2$. At
first sight, this factor can be arbitrarily large and thus brings the system
to the strong coupling regime [10]. However, as we will see in the following,
this effect is impaired by the enhanced decoherence of the spin.

To study the interplay between squeezing and decoherence, we consider the
combined action of the Hamiltonian $\tilde{H}$ and the decay of photons from
the resonator, described by the quantum master equation

$\displaystyle\frac{d}{dt}\rho=-\frac{i}{\hbar}\left[\tilde{H},\rho\right]+L\rho
L^{\dagger}-\tfrac{1}{2}\left(L^{\dagger}L\rho+\rho L^{\dagger}L\right),$ (10)

where $\rho(t)$ is the density matrix and $L=\sqrt{\kappa}a$ is a Lindblad
superoperator. The operator $L$ originates from the coupling between the
superconducting resonator and the external environment, and drives the
resonator to its vacuum (zero photons) state.

##  III Numeric simulation of a squeezed resonator

Figure 2: (a) Frequency of the squeezed resonator, $\Omega_{r}$, in the frame
rotating at angular frequency $\omega_{p}/2$ relative to the laboratory frame,
as a function of the squeezing parameter $r$. The resonator frequency is set
to $\tilde{\omega}_{r}=1$ MHz and spin’s frequency changes between
$\tilde{\omega}_{s}=0.9,0.6,0.3$ MHz. Accordingly, the crossing point between
the resonator and the spin changes between $r_{c}\approx 0.16,0.61,0.89$. The
inset shows a schematic picture of the energy levels, illustrating the
resonance condition in the presence of finite squeezing. (b) Spectrum of the
resonator $S_{r}(\omega)$, in the vicinity of the crossing conditions
($\Omega_{r}=\tilde{\omega}_{s}$) for $g=5$ kHz and $\kappa=40$ kHz.

To quantify the coupling between the spin and the resonator, we numerically
compute the power-spectrum of the resonator $S_{r}\left[\omega\right]$,
defined as

$\displaystyle S_{r}[\omega]=\int_{-\infty}^{\infty}\langle
a^{\dagger}(t)a(0)\rangle_{\infty}e^{-i\omega t}dt$ (11)

where the sub-index $\infty$ implies that we compute the expression in the
steady state 111The numerical calculations presented in this work were
obtained using the QuTiP python package [26]. All the codes used to generate
the figures in this article can be found online at
https://github.com/InbarShani2610/Coherence_Properties_of_a_Spin_in_a_Squeezed_Resonator.
Since we work with ladder operators of an harmonic oscillator, and these
cannot be described by a finite matrix, the precision of the numerical
calculation depends on the truncation of the matrix operator which represents
them. We truncate the matrix to a maximal number of photons, denoted by $N$,
leading to density matrices of size $(2N)^{2}$. In the presence of squeezing,
large values of $N$ are required to obtain results that coincide with the
exact solution (see Fig. 7 in Appendix B). To overcome this difficulty, we
perform the numerical calculations in the squeezed frame, see Eq. 9, where
smaller values of $N$ are sufficient to obtain good numerical results. Note
that the Lindblad superoperator $L$ must transform accordingly:

$\displaystyle
L=\sqrt{\kappa}\cosh(r)\gamma+\sqrt{\kappa}\sinh(r)\gamma^{\dagger}$ (12)

Equation (12) represents a major difference between the present work and Ref.
[10], where the Lindblad operator was assumed to be proportional to the
annihilation operator in the squeezed frame, namely
$L^{\prime}=\sqrt{\kappa}\gamma$. The physical realization of the Lindblad
operator $L^{\prime}$ requires one to squeeze the vacuum outside the resonator
by exactly the same amount as the squeezing inside the resonator. While
theoretically possible, this situation is unrealistic in an actual experiment.
The substitution of $L^{\prime}$ with $L$ has dramatic implications. In
particular, this substitution leads to the disappearance of the level
splitting shown in Fig. 1 of Ref. [10] (see Appendix A).
To study the effect of squeezing on the effective coupling between the
resonator and the spin in a controlled manner, we consider a system where the
bare frequency of the resonator is larger than the frequency of the spin
$\tilde{\omega}_{r}>\tilde{\omega}_{s}$. For concreteness, throughout the
article we consider a system with $\tilde{\omega}_{r}=1$ MHz,
$\tilde{\omega}_{s}=0.6$ MHz, $g=5$ kHz, and $\kappa=40$ kHz (unless
explicitly mentioned otherwise). By introducing a squeezing term, we
effectively reduce the resonator frequency to $\Omega_{r}$ (see Eq. 9), until
the resonance condition is matched ($\Omega_{r}=\tilde{\omega}_{s}$), as
illustrated in Fig. 2(a). Changing the bare frequency of the spin
$\tilde{\omega}_{s}$ modifies the value of the squeezing parameter $r=r_{c}$
at resonance condition. The power spectrum near $r_{c}$, Fig. 2(b), has the
typical structure of an avoided level-crossing. At a fixed value of $r$, the
spectrum shows two peaks, whose frequencies correspond to the energy levels of
the mixed resonator-spin states (see inset in Fig. 2(a)). We use the frequency
difference between the maximum and the local minimum of the spectrum at
resonance, i.e. for $r=r_{c}$, to estimate the coupling strength between the
spin and the resonator. In analogy to the Jaynes-Cummings model, we denote
this distance as $\chi$ and plot its value as a function of $r_{c}$ in Fig.
3(a). For small bare detunings
($\left|\tilde{\omega}_{r}-\tilde{\omega}_{s}\right|\ll\tilde{\omega}_{r}$),
the size of the anticrossing $\chi$ increases as a function of $r_{c}$ until
it reaches a maximal value. This result is in stark contrast to the case of
the squeezed vacuum operators of Ref. [10], where arbitrarily large couplings
can be obtained.

The upper limit of $\chi$ is due to the back-action of the squeezed resonator
on the spin, leading to its fast dephasing: this process can be identified by
observing the broadening of the spectrum at resonance (see Fig. 2(b)). We
quantify this effect by measuring the ratio between the maximal intensity of
the spectrum. We denote as $S_{\rm min}$ (resp. $S_{\rm max}$) the values of
the spectrum at its local minimum (resp. maximum) and show in Fig. 3(b) that
the contrast of the anticrossing, defined as $1-S_{\rm min}/S_{\rm max}$, is a
monotonously decreasing function of $r_{c}$. When the resonator is squeezed
significantly ($r_{c}\sim 1$) the contrast vanishes, indicating that the spin
is completely dephased. To analyze the physical origin of this effect, we
repeat the same calculations for different values of the loss rate $\kappa$,
which controls the number of photons occupying the resonator (see the inset of
Fig. 3(b)). The maximum size of the anticrossing $\chi$ is shifted towards
higher values when the decay rate of the resonator $\kappa$ is increased.
Eventually, for large values of the decay rate $\kappa$, $\chi$ begins to
deteriorate. The maximal value of $\chi$ is only 20$\%$ larger than its
initial value and is insufficient to reach the strong coupling regime.

Figure 3:  (a) Dispersive shift $\chi$ determined from the spectrum of the
resonator $S_{r}[\omega]$, as a function of $r_{c}$. (b) Contrast of the
spectrum $1-S_{min}/S_{max}$. The suppression of the contrast at large
squeezing indicates that enhanced decoherence of the spin. Inset: photon
number occupying the resonator.

##  IV Schrieffer-Wolff Transformation for a Squeezed resonator

Motivated by the numerical results of the previous section, we now introduce a
systematic approach to describe the effects of the squeezed resonator on the
spin. Our approach generalizes the Schrieffer-Wolf (SW) transformation [16] to
a squeezed resonator and allow us to compute the energy shift and the
decoherence of the spin analytically. As a first step, we write the
Hamiltonian (6) as $\tilde{H}=\tilde{H}_{0}+V$, with

$\displaystyle\tilde{H}_{0}/\hbar$
$\displaystyle=\tilde{\omega}_{r}a^{\dagger}a+\tfrac{1}{2}\tilde{\omega}_{s}\sigma_{z}-\tfrac{1}{2}\lambda\left(a^{2}+{a^{\dagger}}^{2}\right)$
(13) $\displaystyle V/\hbar$
$\displaystyle=g(a^{\dagger}\sigma_{-}+a\sigma_{+}).$ (14)

Next, we apply the unitary transformation $U=e^{S}$, where $S$ is the anti-
Hermitian operator

$\displaystyle S$
$\displaystyle=C_{1}(a\sigma_{+}-a^{\dagger}\sigma_{-})+C_{2}(a\sigma_{-}-a^{\dagger}\sigma_{+}),\leavevmode\nobreak\
{\rm with}\leavevmode\nobreak\ $ (15) $\displaystyle C_{1}$
$\displaystyle=\frac{g}{\Delta}\frac{\tilde{\omega}_{s}+\tilde{\omega}_{r}}{\tilde{\omega}_{s}+\Omega_{r}},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
C_{2}=\frac{g}{\Delta}\frac{\lambda}{\tilde{\omega}_{s}+\Omega_{r}}$ (16)

and $\Delta=\tilde{\omega}_{s}-\Omega_{r}$. This transformation satisfies the
condition $[\tilde{H}_{0},S]=V$ and leads to a transformed Hamiltonian
$\tilde{H}_{\rm SW}=e^{S}\tilde{H}e^{-S}$ that does not include linear terms
in $V$. By expanding this expression to second order in $V$, we obtain
$\tilde{H}_{\rm
SW}/\hbar=\tilde{H}_{0}/\hbar+\delta\tilde{\omega}_{s}\sigma_{z}/2$, with a
dispersive shift

$\displaystyle\delta\tilde{\omega}_{s}$
$\displaystyle=\chi_{r}(2\tilde{n}+1),$ (17)

where we defined $\chi_{r}={g}^{2}/\Delta$, and

$\displaystyle\tilde{n}+\tfrac{1}{2}=\frac{\left(\tilde{\omega}_{s}+\tilde{\omega}_{r}\right)\left(2a^{\dagger}a+1\right)-\lambda\left(a^{2}+{a^{\dagger}}^{2}\right)}{2(\tilde{\omega}_{s}+\Omega_{r})}$
(18)

For $\lambda=0$, Eqs. 17 and 18 recover the well known Lamb shift of a spin
coupled to a resonator

$\displaystyle\delta\tilde{\omega}_{s}=$
$\displaystyle\chi_{0}\left(2a^{\dagger}a+1\right)\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\chi_{0}=\frac{g^{2}}{\tilde{\omega}_{s}-\tilde{\omega}_{r}}.$ (19)

####

The SW transformation creates an effective channel of dissipation for the
spin, in addition to its intrinsic dissipation. In order to consider this
effect, one needs to transform the dissipation operator $L$ using the SW
transformation. Up to second order in $V$,

$\displaystyle\begin{split}&\tilde{L}_{\rm
SW}=e^{S}Le^{-S}=L+\left[S,L\right]+\tfrac{1}{2}\left[S,\left[S,L\right]\right]+...\end{split}$
(20)

Using Eq. 15, we obtain

$\displaystyle L_{\rm
SW}=\sqrt{\kappa}a+\sqrt{\kappa}C_{1}\sigma_{-}+\sqrt{\kappa}C_{2}\sigma_{+}+{\rm
O}\left(\frac{g^{2}}{\Delta^{2}}\right)$ (21)

After the transformation, the resonator and the spin are decoupled, such that
the coherence of the spin is entirely dictated by the second and third terms
of Eq. 21. If we consider these two terms as independent channels, we obtain
an effective decoherence rate

$\displaystyle\begin{split}\Gamma_{\rm
Purcell}&=\Gamma_{1}+\Gamma_{2}=\kappa\left(C_{1}^{2}+C_{2}^{2}\right).\end{split}$
(22)

In the limit of $\lambda=0$, this expression reproduces the well-known Purcell
decay, $\Gamma_{\rm Purcell}=g^{2}\kappa/\Delta^{2}$ [17, 5].

##  V Photon noise

In addition to the Purcell effect, the squeezing of the resonator also affects
the pure dephasing rate of the spin. Stochastic fluctuations in the number of
photons in the resonator create a random dispersive shift which translates
into a dephasing of the spin. The rate of this process can be determined by
the decay of the expectation value $\langle\sigma_{x}(t)\rangle$ in a system
initialized in the $|+\rangle=\left(|0\rangle+|1\rangle\right)/\sqrt{2}$ state
[18],

$\displaystyle\langle\sigma_{x}(t)\rangle=\frac{1}{2}\left(e^{i\int_{0}^{t}\delta\tilde{\omega}_{s}(t^{\prime})dt^{\prime}}+e^{-i\int_{0}^{t}\delta\tilde{\omega}_{s}(t^{\prime})dt^{\prime}}\right).$
(23)

Assuming that $\tilde{n}(t)=\tilde{n}_{0}+\delta\tilde{n}{(t)}$, where the
fluctuation $\delta\tilde{n}(t)$ is a random variable with zero average, one
can rewrite the exponent as

$\displaystyle e^{\pm
i\int_{0}^{t}\delta\tilde{\omega}_{s}(t^{\prime})dt^{\prime}}=e^{\pm
i\chi_{r}\left(2{\tilde{n}_{0}}+1\right)t}e^{\pm
i2\chi_{r}\int_{0}^{t}\delta\tilde{n}(t^{\prime})dt^{\prime}}.$ (24)

By expanding the exponent to a Taylor series, one obtains

$\displaystyle\langle e^{\pm
i\chi_{r}\int_{0}^{t}\delta\tilde{n}(t^{\prime})dt^{\prime}}\rangle$
$\displaystyle\approx
1-2\chi_{r}^{2}\int_{0}^{t}dt^{\prime}\int_{0}^{t}dt^{\prime\prime}\leavevmode\nobreak\
\langle\delta\tilde{n}(t^{\prime})\delta\tilde{n}(t^{\prime\prime})\rangle$
$\displaystyle\approx
1-2\chi_{r}^{2}t\int_{-\infty}^{\infty}\langle\delta\tilde{n}(\tau)\delta\tilde{n}(0)\rangle_{\infty}d\tau.$
(25)

Here, the last identity is valid in the limit of $t\to\infty$, under the
assumption that the resonator is found in a steady state, where two-time
correlations depend on the time difference only. In the case of Gaussian
fluctuations, the higher order terms can be re-summed exactly leading to

$\displaystyle\langle\sigma_{x}(t)\rangle=$
$\displaystyle\cos\left[\chi_{r}\left({2\tilde{n}_{0}}+1\right)t\right]e^{-\Gamma_{\phi}^{\rm
photon}t},$ (26)

where we defined

$\displaystyle\Gamma_{\phi}^{\rm
photon}=2\chi_{r}^{2}\tilde{\eta},\leavevmode\nobreak\ {\rm
and}\leavevmode\nobreak\
\tilde{\eta}=\int_{-\infty}^{\infty}\langle\delta\tilde{n}(\tau)\delta\tilde{n}(0)\rangle
d\tau.$ (27)

In Appendix D, we compute the correlations $\langle\delta n(\tau)\delta
n(0)\rangle$ for a thermal state and for a coherent state and reproduce known
results for the photon noise dephasing of the spin.

##  VI Steady state correlations of a squeezed resonator

In the previous sections we expressed the dispersive shift and the decoherence
rates of the spin in terms of physical observables of the resonator, see Eqs.
17, 22 and 27 . In the following, we compute these quantities in the steady
state of a squeezed resonator described by the Hamiltonian (13) (with $g=0$).
According to the Lindblad master equation (10), the expectation value of a
generic operator $O(t)$ is determined by the differential equation

$\displaystyle\begin{split}\frac{d}{dt}\left<O(t)\right>&=\frac{i}{\hbar}\left<\left[H,O(t)\right]\right>+\left<L^{\dagger}O(t)L\right>\\\
&-\tfrac{1}{2}\left(\left<L^{\dagger}LO(t)\right>+\left<O(t)L^{\dagger}L\right>\right).\end{split}$
(28)

This equation can be explicitly solved if one finds a set of variables
$\vec{O}=(O_{1},O_{2},...)$ whose expectation values satisfy the closed-form
recursive relation

$\displaystyle\frac{d}{dt}\left<O_{i}(t)\right>=\sum_{j}G_{i,j}\left<O_{j}(t)\right>,$
(29)

where $G_{i,j}$ is a time independent scalar. For a squeezed resonator, this
condition is satisfied by the operator $\vec{O}=\begin{pmatrix}a^{\dagger}a,\
a^{2},\ {a^{\dagger}}^{2},\ \mathds{1}\end{pmatrix}^{T}$ with

$\displaystyle G=\left(\begin{matrix}-\kappa&-i\lambda&i\lambda&0\\\
i2\lambda&-i2\tilde{\omega}_{r}-\kappa&0&i\lambda\\\
-i2\lambda&0&i2\tilde{\omega}_{r}-\kappa&-i\lambda\\\
0&0&0&0\end{matrix}\right).$ (30)

The steady-state expectation values of $\vec{O}$ can be found by equating the
left-hand side of Eq. 29 to 0, or equivalently by demanding that
$\sum_{j}G_{i,j}\left<O_{j}(t)\right>_{\infty}=0$. Using this approach, we
find

$\displaystyle\langle a^{\dagger}a\rangle_{\infty}$
$\displaystyle=\frac{2\lambda^{2}}{4\Omega_{r}^{2}+\kappa^{2}}$ (31)
$\displaystyle\langle a^{2}\rangle_{\infty}$
$\displaystyle=\frac{\lambda(2\tilde{\omega}_{r}+i\kappa)}{4\Omega_{r}^{2}+\kappa^{2}}$
(32) $\displaystyle\langle{a^{\dagger}}^{2}\rangle_{\infty}$
$\displaystyle=\frac{\lambda(2\tilde{\omega}_{r}-i\kappa)}{4\Omega_{r}^{2}+\kappa^{2}}.$
(33)

Plugging these results in Eq. 17, we obtain an analytic expression for the
dispersive shift of the spin,

$\displaystyle\delta\tilde{\omega}_{s}=\frac{g^{2}}{\tilde{\omega}_{s}^{2}-\Omega_{r}^{2}}\left(\frac{4\lambda^{2}\tilde{\omega}_{s}}{4\Omega_{r}^{2}+\kappa^{2}}+\tilde{\omega}_{r}+\tilde{\omega}_{s}\right)$
(34)

Figure 4: (a) Dispersive shift and (b) dissipation rate of the spin, as a
function of the squeezing parameter $r$. The continuous lines are our analytic
expressions for the dispersive shift, Eq. 34, and for the total dissipation
$\Gamma_{\rm tot}=\Gamma_{\rm Purcell}/2+\Gamma_{\phi}^{\rm photon}$, Eqs. 22
and VI. This approximation fits very well to the exact numerical results
(dots), except for the range$|\Delta|\ll g$ (the gray shaded area marks where
$|\Delta|<0.02g$).

For a quantum system described by a Markovian master equation, the two-time
correlator of Eq. 27 can be computed using the quantum regression theorem [19]
(see also Appendix C for a simple proof). According to this theorem, given a
set of operators $\vec{O}$ that satisfy Eq. (29), the two-time correlation
functions satisfy the differential equation

(a) Numerics |  (b) Analytic
---|---

Figure 5: Ratio between the dispersive shift of a spin and its spectral width,
$\delta\tilde{\omega}_{s}/\Gamma$, as a function or the squeezing parameter
$r$ and the loss-rate of the resonator $\kappa$. The ratio has a local maximum
for $r=1.1\pm 0.1$, where one obtains an optimal value for the effective
coupling $\chi$, see Fig. 3.

$\displaystyle\frac{d}{d\tau}\left<O_{i}(t+\tau)O_{k}(t)\right>=\sum_{j}G_{i,j}\left<O_{j}(t+\tau)O_{k}(t)\right>.$
(35)

Using this theorem, we can compute the two-time correlation function of the
squeezed resonator (see Appendix D for details). Plugging this result into Eq.
27 we obtain the dephasing rate due to photon noise in the squeezed resonator:

$\displaystyle\Gamma_{\phi}^{\rm photon}=$
$\displaystyle\frac{2\chi_{r}^{2}\lambda^{2}}{\kappa\left(\Omega_{r}+\tilde{\omega}_{s}\right)^{2}}\left[\frac{\left(2\Omega_{r}^{2}+\kappa^{2}\right)}{\left(4\Omega_{r}^{2}+\kappa^{2}\right)}+4\tilde{\omega}_{r}\tilde{\omega}_{s}\frac{\left(4\Omega_{r}^{2}+3\kappa^{2}\right)}{\left(4\Omega_{r}^{2}+\kappa^{2}\right)^{2}}\right.$
$\displaystyle\left.+2\tilde{\omega}_{s}^{2}\frac{\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}\right)\left(4\Omega_{r}^{2}+5\kappa^{2}\right)}{\left(4\Omega_{r}^{2}+\kappa^{2}\right)^{3}}\right].$
(36)

Equation (Section VI) is proportional to $\lambda^{2}$: The photon noise is
associated with the occupation of the resonator, induced by squeezing, and
vanishes for $\lambda=0$. See also Appendix D for the cases of a resonator in
a coherent state and in a thermal state. In all cases, the resulting
expression is inversely proportional to $\kappa$: if the photons are allowed
to escape rapidly from the resonator, they have a smaller effect on the
dephasing of the spin.

##  VII Numeric simulation of a spin in a squeezed resonator

To test the validity of the expressions obtained for the dispersive shift of a
spin, Eq. 34, and the dissipation rate of the spin, Sections VI and 22, we
compute numerically the power spectrum of of the spin
$S_{s}\left[\omega\right]$, defined as

$\displaystyle
S_{s}[\omega]=\int_{-\infty}^{\infty}\langle\sigma_{x}(\tau)\sigma_{x}(0)\rangle_{\infty}e^{-i\omega\tau}d\tau$
(37)

The dispersive shift is the distance between the position of the maximum of
$S_{s}\left[\omega\right]$ and the bare frequency $\tilde{\omega}_{s}$. This
quantity is plotted in Fig. 4(a) as a function of $r$. The dispersive shift of
the spin is initially negative (because
$\tilde{\omega}_{s}<\tilde{\omega}_{r}$) and its absolute value increases as a
function of $r$ for all $r<r_{c}$. At $r=r_{c}$, the dispersive shift changes
sign and starts to decrease. At large $r\gtrsim 1$, the dispersive shift
increases again, signaling an enhancement of the coupling between the spin and
the resonator. Our numerical findings are in excellent agreement with our
analytical result, Eq. (34) , except for a narrow region around $r\approx
r_{c}$, where the analytical curve diverges, while the numerical one remains
finite. From the width of the numerical spectrum, we estimate the dephasing
rate $\Gamma$, see Fig. 4(b). We find a quantitative agreement with the
analytical prediction $\Gamma=\Gamma_{\rm Purcell}/2+\Gamma_{\phi}^{\rm
photon}$, where $\Gamma_{\rm Purcell}$ and $\Gamma_{\phi}^{\rm photon}$ are
given in Sections VI and 22.

From the computed values of the dispersive shift and decoherence rate, we can
estimate the relative strength of these two effects. The ratio between these
two quantities is shown in Fig. 5 and, again, shows an excellent agreement
between analytics and numerics. At a fixed $\kappa$, this ratio follows a non-
monotonic behavior, with local minima at resonance, for $r\approx r_{c}$, and
at large squeezing, for $r\gg 1$. In these regions, the ratio between the
dispersive shift and the decoherence rate tends to zero, indicating that the
dephasing effect of squeezing dominates over the enhanced coupling between the
spin and the resonator. This ratio obtains an optimal value for $r\approx
1.1$, in correspondence to the local maximum of the effective coupling
parameter $\chi$ found in Fig. 3(a).

##  VIII Discussion and Conclusion

In conclusion, we studied the coherence properties of a spin embedded in a
resonator under parametric drive. We found out that such a drive gives rise to
an enhanced effective coupling between the spin and the resonator but also
increases relaxation and decoherence rates of both systems. These non-unitary
effects are related to the relatively large number of photons in the squeezed
resonator. In particular, the number of photons in a squeezed resonator
modifies the relaxation rate of the spin. This effect limits the relative
enhancement of the coupling by squeezing to approximately $20\%$. This
situation is in contrast to what usually occurs for coherent or thermal
states, where the relaxation of the spin via the resonator - the so-called
Purcell rate - is fixed by the intrinsic properties of the system and is
independent of the photon occupation. Here, the squeezing terms change both
the coupling between the two systems and their loss rates. The theoretical
methods developed in this work enabled us to consider these two effects on
equal footing and could be extended further to more ‘exotic’ quantum states
like cat states [20] or Gottesman-Kitaev-Preskill (GKP) states [21, 22]. Our
analytical approach based on generalized Schrieffer Wolf transformations and
quantum regression theorem are valid for squeezed cavities on a large
parameter scale and were found to be in quantitative agreement with the
numerical calculations.

###### Acknowledgements.

This work was supported by the Israel Science Foundation, Grants No. 426/15,
151/19, 154/19 and 898/19. I. S. acknowledges support from the Institute for
Nanotechnology and Advanced Materials at Bar-Ilan University. We wish to thank
T. Kontos, D. Vion and L. Bello for inspiring and fruitful discussions.

## Appendix

##  Appendix A Comparison of the two decay mechanisms

Figure 6: Spectrum of the spin, in the absence of squeezing (blue) and in the
presence of squeezing (red). The upper and lower panel differ in the nature of
the Lindblad operator used in the calculation: in (a) we use
$L^{\prime}=\sqrt{\kappa}\gamma$, which corresponds to a resonator that is
immersed in a squeezed vacuum. In (b) we use $L=\sqrt{\kappa}a$, which
corresponds to the physical situation of an unsqueezed vacuum. In the presence
of squeezing, a splitting is observed for (a) but not for (b).

In this section we compare the effects of Lindblad operators coupled to the
physical operator $a$, or to the squeezed resonator operator $\gamma$. Fig. 6
shows the spectrum of the spin, defined in Eq. 37, for the two cases. Fig.
6(a) reproduces Fig. 1 of Ref. [10] and shows that in the presence of
squeezing, the system reaches the strong coupling regime. Fig. 6(b) describes
the physical situation, where the resonator is immersed in the regular vacuum.
In this case, no level splitting is observed.

##  Appendix B Numerical calibration of the truncation parameter $N$

In this appendix we study the effect of the truncation parameter $N$, and
compare two different frames. The first frame, which we denote as the original
frame, corresponds to the Hamiltonian in Eq. 6 and $L=\sqrt{\kappa}a$. The
second frame, which we denote as the squeezed frame, corresponds to the
transformed Hamiltonian Eq. 9 and Lindblad operator in the form of Eq. 12. In
each frame, we truncate the matrix representing the annihilation and creation
operators, respectively $a,a^{\dagger}$ and $\gamma,\gamma^{\dagger}$, and
vary the maximal number of photon occupation $N$.

For concreteness, we consider the integrated two-time correlation

$\displaystyle\eta=\int_{-\infty}^{\infty}\langle(n(\tau)-\bar{n})(n(0)-\bar{n})\rangle
d\tau$ (38)

for a squeezed resonator, whose analytical expression is computed in Appendix
D, Eq. 64. In the original frame, Fig. 7(a), the results converge slowly to
the analytic solution as we increase $N$. For intermediate amount of
squeezing, $r\approx 1.5$, the numerical solution requires extremely large
values of the truncation parameter $N\gtrsim 150$ to converge to the analytic
solution. In contrast, the numerical results in the squeezed frame, Fig. 7(b),
are less sensitive to the truncation of the matrix size, thus converging much
faster to the analytic result. In this article we are interested in squeezing
parameters $r\lesssim 1.5$ and, hence, we work in the squeezed frame and use
$N=30$.

Figure 7: Integrated two-time correlation function $\eta$, defined in Eq. 64,
for different values of the numerical truncation parameter $N$, (a) in the
original frame and (b) in the squeezed frame.

##  Appendix C Quantum Regression Theorem

In this appendix we offer a simple derivation of the quantum regression
theorem for Markovian master equations of the form

$\displaystyle\frac{d}{dt}\rho=\mathcal{L}\rho,$ (39)

where $\mathcal{L}$ is the Liouvillian superoperator and $\rho$ is the density
matrix. For time-independent $\mathcal{L}$, the time evolution of $\rho$ is
given by

$\displaystyle\rho(t+\tau)=e^{\mathcal{L}\tau}\rho(t).$ (40)

The expectation value of an operator $A$ is defined as

$\displaystyle\langle A(t)\rangle={\rm Tr}\left[A\rho(t)\right].$ (41)

Its time derivative is given by

$\displaystyle\frac{d}{dt}\langle A(t)\rangle=\frac{d}{dt}{\rm
Tr}\left[A\rho(t)\right]={\rm Tr}\left[A\frac{d}{dt}\rho(t)\right].$ (42)

Plugging-in Eq. 39 we obtain

$\displaystyle\frac{d}{dt}\langle A(t)\rangle={\rm
Tr}\left[A\mathcal{L}\rho(t)\right].$ (43)

The quantum regression theorem allows one to compute the two-time correlation
function

$\displaystyle\langle A(t+\tau)O(t)\rangle={\rm
Tr}\left[Ae^{\mathcal{L}\tau}O\rho(t)\right]$ (44)

for systems where the time derivative of $d\langle A(t)\rangle/dt$ has a
linear dependence on the system’s operators $B_{j}$,

$\displaystyle\frac{d}{dt}\langle A(t)\rangle=\sum_{j}G_{j}\langle
B_{j}(t)\rangle.$ (45)

Because the trace of sum of matrices is the sum of their traces, we can
rewrite Eq. 45 as

$\displaystyle\frac{d}{dt}\langle A(t)\rangle=\sum_{j}G_{j}{\rm
Tr}\left[B_{j}\rho(t)\right]=\sum_{j}{\rm Tr}\left[G_{j}B_{j}\rho(t)\right]$
(46)

Because Eq. (46) and Eq. (43) are satisfied for any $\rho(t)$, we obtain that

$\displaystyle A\mathcal{L}=\sum_{j}G_{j}B_{j}$ (47)

This result allows us to compute the time derivative of Eq. 44,

$\displaystyle\frac{d}{d\tau}\langle A(t+\tau)O(t)\rangle$
$\displaystyle=\frac{d}{d\tau}{\rm
Tr}\left[Ae^{\mathcal{L}\tau}O\rho(t)\right]$ (48) $\displaystyle={\rm
Tr}\left[A\mathcal{L}e^{\mathcal{L}\tau}O\rho(t)\right].$ (49)

Using Eq. 47 we obtain

$\displaystyle\frac{d}{d\tau}\langle A(t+\tau)O(t)\rangle$ $\displaystyle={\rm
Tr}\left[\sum_{j}G_{j}B_{j}e^{\mathcal{L}\tau}O\rho(t)\right]$ (50)
$\displaystyle=\sum_{j}G_{j}{\rm
Tr}\left[B_{j}e^{\mathcal{L}\tau}O\rho(t)\right]$ (51)
$\displaystyle=\sum_{j}G_{j}\langle B_{j}(t+\tau)O(t)\rangle$ (52)

Eq. (52) is the quantum regression theorem used in Eq. (35).

##  Appendix D Two-time correlation functions of the resonator

In this appendix we compute the two-time correlation function of the number of
photons in a squeezed resonator coupled to a thermal bath and in a coherent
state. The Hamiltonian of a squeezed resonator is given by Eq. 6 with $g=0$

$\displaystyle
H/\hbar=\tilde{\omega}_{r}a^{\dagger}a-\frac{\lambda}{2}\left(a^{2}+{a^{\dagger}}^{2}\right).$
(53)

For a system described by the Lindblad master-equation Eq. 10, the time
evolution of an arbitrary operator $O_{i}(t)$ is given by

$\displaystyle\begin{split}&\frac{d}{dt}\left<O_{i}(t)\right>=\frac{i}{\hbar}\left<\left[H,O_{i}(t)\right]\right>+\\\
&\sum_{m}\left(\left<L_{m}^{\dagger}O_{i}(t)L_{m}\right>-\frac{1}{2}\left<L^{\dagger}_{m}L_{m}O_{i}(t)\right>-\frac{1}{2}\left<O_{i}(t)L^{\dagger}_{m}L_{m}\right>\right),\end{split}$
(54)

where $L_{m}$ are the Lindblad superoperators and $m$ is a positive integer
$\\{m=1,2,..\\}$. If the resonator is coupled to a thermal bath, the Lindblad
operators are

$\displaystyle L_{1}=\sqrt{\kappa\left(\bar{n}+1\right)}a$ (55) $\displaystyle
L_{2}=\sqrt{\kappa\bar{n}}a^{\dagger}$ (56)

The time derivative of $\vec{O}(t)$, where $\vec{O}(t)=\begin{pmatrix}n(t),\
a^{2}(t),\ {a^{\dagger}}^{2}(t),\ \mathds{1}\end{pmatrix}^{T}$, gives rise to
a set of coupled differential equations that can be written in the form of Eq.
29, with

$\displaystyle
G=\left(\begin{matrix}-\kappa&-i\lambda&i\lambda&\kappa\bar{n}\\\
i2\lambda&-i2\tilde{\omega}_{r}-\kappa&0&i\lambda\\\
-i2\lambda&0&i2\tilde{\omega}_{r}-\kappa&-i\lambda\\\
0&0&0&0\end{matrix}\right).$ (57)

###  D.1 Squeezed resonator

For a squeezed resonator coupled to a zero-temperature bath, $\bar{n}=0$, Eq.
57 becomes

$\displaystyle G=\begin{pmatrix}-\kappa&-i\lambda&i\lambda&0\\\
i2\lambda&-i2\tilde{\omega}_{r}-\kappa&0&i\lambda\\\
-i2\lambda&0&i2\tilde{\omega}_{r}-\kappa&-i\lambda\\\ 0&0&0&0\end{pmatrix}.$
(58)

To obtain the two-time correlation function $\langle n(\tau)n(0)\rangle$, we
use the quantum regression theory, see Appendix C, to obtain the relation

$\displaystyle\frac{\partial}{\partial\tau}\left<\vec{O}(\tau)n(0)\right>=G\left<\vec{O}(\tau)n(0)\right>.$
(59)

We solve this equation by diagonalizing $G$ and using the initial conditions
for the steady-state solution of $\left<\vec{O}(\tau)n(0)\right>$. To obtain
the latter, we compute the equations of motion of the expectation values of
the operator

$\displaystyle\vec{\psi}=\begin{pmatrix}a^{\dagger}a,\ a^{2},\
{a^{\dagger}}^{2},\ {a^{\dagger}}^{2}a^{2},\ {a^{\dagger}}^{3}a,\
a^{\dagger}a^{3},\ a^{4},\ {a^{\dagger}}^{4},\ \mathds{1}\end{pmatrix}^{T}$
(60)

and write them as
$\frac{d}{dt}\langle\vec{\psi}(t)\rangle=M\langle\vec{\psi}(t)\rangle$, where

$\displaystyle M=\left(\begin{matrix}-\kappa&-i\lambda&i\lambda&0&0&0&0&0&0\\\
2i\lambda&-2i\tilde{\omega}_{r}-\kappa&0&0&0&0&0&0&i\lambda\\\
-2i\lambda&0&2i\tilde{\omega}_{r}-\kappa&0&0&0&0&0&-i\lambda\\\
0&-i\lambda&i\lambda&-2\kappa&2i\lambda&-2i\lambda&0&0&0\\\
-3i\lambda&0&0&-3i\lambda&2i\tilde{\omega}_{r}-2\kappa&0&0&i\lambda&0\\\
3i\lambda&0&0&3i\lambda&0&-2i\tilde{\omega}_{r}-2\kappa&-i\lambda&0&0\\\
0&6i\lambda&0&0&0&4i\lambda&-2\kappa-4i\tilde{\omega}_{r}&0&0\\\
0&0&-6i\lambda&0&-4i\lambda&0&0&-2\kappa+4i\tilde{\omega}_{r}&0\\\
0&0&0&0&0&0&0&0&0\end{matrix}\right)$ (61)

The steady-state solution is given by the null eigenvector of $M$, defined by
$M\langle\vec{\psi}\rangle_{\infty}=0$, and corresponds to

$\displaystyle\langle\vec{\psi}\rangle_{\infty}=\begin{pmatrix}\frac{2\lambda^{2}}{4\tilde{\omega}_{r}^{2}+\kappa^{2}-4\lambda^{2}}\\\
\\\
-\frac{\lambda(2\tilde{\omega}_{r}+i\kappa)}{-4\tilde{\omega}_{r}^{2}-\kappa^{2}+4\lambda^{2}}\\\
\\\
-\frac{-2\tilde{\omega}_{r}\lambda+i\kappa\lambda}{4\tilde{\omega}_{r}^{2}+\kappa^{2}-4\lambda^{2}}\\\
\\\
-\frac{-4\tilde{\omega}_{r}^{2}\lambda^{2}-\kappa^{2}\lambda^{2}-8\lambda^{4}}{\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}-4\lambda^{2}\right)^{2}}\\\
\\\
\frac{6\left(2\tilde{\omega}_{r}\lambda^{3}-i\kappa\lambda^{3}\right)}{\left(-4\tilde{\omega}_{r}^{2}-\kappa^{2}+4\lambda^{2}\right)^{2}}\\\
\\\
\frac{6\left(2\tilde{\omega}_{r}\lambda^{3}+i\kappa\lambda^{3}\right)}{\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}-4\lambda^{2}\right)^{2}}\\\
\\\
\frac{3\left(4\tilde{\omega}_{r}^{2}\lambda^{2}+4i\tilde{\omega}_{r}\kappa\lambda^{2}-\kappa^{2}\lambda^{2}\right)}{\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}-4\lambda^{2}\right)^{2}}\\\
\\\
\frac{3\left(4\tilde{\omega}_{r}^{2}\lambda^{2}-4i\tilde{\omega}_{r}\kappa\lambda^{2}-\kappa^{2}\lambda^{2}\right)}{\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}-4\lambda^{2}\right)^{2}}\\\
\\\ 1\end{pmatrix},$ (62)

From this expression, and using the canonical commutation relations, we obtain
the initial conditions of Eq. 59 for $\tau=0$, namely

$\displaystyle\langle\vec{O}(0)n(0)\rangle_{\infty}=\begin{pmatrix}\frac{3\lambda^{2}\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}\right)}{\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}-4\lambda^{2}\right)^{2}}\\\
\\\
\frac{2\lambda(2\tilde{\omega}_{r}+i\kappa)\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}-\lambda^{2}\right)}{\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}-4\lambda^{2}\right)^{2}}\\\
\\\
\frac{6\lambda^{3}(2\tilde{\omega}_{r}-i\kappa)}{\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}-4\lambda^{2}\right)^{2}}\\\
\\\
\frac{2\lambda^{2}}{4\tilde{\omega}_{r}^{2}+\kappa^{2}-4\lambda^{2}}\end{pmatrix}.$
(63)

Next, we solve Eq. 59 by diagonalizing $M$ and derive an analytic expression
for $\langle n(t)n(0)\rangle$, see Fig. 8. Using this expression, we can
compute the integrated two-time correlation defined in Eq. (64) and used in
Fig 7,

$\displaystyle\eta$
$\displaystyle=\frac{2\lambda^{2}\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}\right)\left(4\tilde{\omega}_{r}^{2}+5\kappa^{2}-4\lambda^{2}\right)}{\kappa\left(4\tilde{\omega}_{r}^{2}+\kappa^{2}-4\lambda^{2}\right)^{3}}.$
(64)

To evaluate the dephasing rate of a squeezed resonator, we need to compute the
two-time correlation of $\tilde{n}$, see Eqs. 18 and 27. In order to find this
quantity, we compute all permutations obtained from the multiplication between
$\tilde{n}(\tau)$ and $\tilde{n}((0))$, i.e. all the matrix elements of the
$9\times 9$ matrix $\langle\vec{O}^{T}(\tau)\vec{O}(0)\rangle$. We refer the
reader to the Wolfram Mathematica code in the online repository for all the
details
222https://github.com/InbarShani2610/Coherence_Properties_of_a_Spin_in_a_Squeezed_Resonator/tree/Mathematica_code.
This calculation enables us to compute the integrated two-time correlation
function $\tilde{\eta}$ and infer the decoherence rate of a spin coupled to a
squeezed resonator, Section VI.

###  D.2 Thermal bath

We now consider a thermal bath in the absence of squeezing ($\lambda=0$). In
this case, it is sufficient to compute the expectation value of
$\vec{O}=\begin{pmatrix}n(t),\ \mathds{1}\end{pmatrix}^{T}$, whose time
derivative takes the form of Eq. 30 with

$\displaystyle G=\left(\begin{matrix}-\kappa&\kappa\bar{n}\\\
0&0\end{matrix}\right).$ (65)

Figure 8: The two-time correlation for the photon-number operator vs $\tau$,
in a squeezed state for different values of r (solid blue and dashed purple).
This correlation provide an indication for the effective temperature of the
system, where the fluctuations increase for larger squeezing parameter $r$.

The steady-state expectation values are obtained by the null eigenstate of
$G$, defined by $G\langle\vec{O}\rangle=0$, leading to $\langle
n(t)\rangle_{\infty}=\bar{n}$. To obtain the two-time correlation $\langle
n(t+\tau)n(t)\rangle_{\infty}$, we use the quantum regression theorem Eq. 35
and obtain

$\displaystyle\frac{d}{d\tau}\left<\begin{matrix}n(t+\tau)n(t)\\\
\mathds{1}\;n(t)\end{matrix}\right>=\left(\begin{matrix}-\kappa&\kappa\bar{n}\\\
0&0\end{matrix}\right)\left<\begin{matrix}n(t+\tau)n(t)\\\
\mathds{1}\;n(t)\end{matrix}\right>.$ (66)

Next, we compute the initial conditions (as done previously), and find
$\langle n(0)n(0)\rangle=\bar{n}(2\bar{n}+1)$. Finally, by diagonalizing $G$,
we obtain

$\displaystyle\langle
n(\tau)n(0)\rangle_{\infty}=\bar{n}^{2}+\left(\bar{n}^{2}+\bar{n}\right)e^{-\kappa\left|\tau\right|},$
(67)

Here, one can check that at $\tau\rightarrow\infty$ the operator $n(\tau)$ is
not correlated to the operator $n(0)$ and $\lim_{\tau\to\infty}\langle
n(\tau)n(0)\rangle_{\infty}=\langle n(\tau)\rangle_{\infty}\langle
n(0)\rangle_{\infty}=\bar{n}^{2}$. Using Eq. 67, we can compute

$\displaystyle\eta=\int_{-\infty}^{\infty}\langle
n(\tau)-\bar{n}\rangle\langle n(0)-\bar{n}\rangle
d\tau=\frac{2\left(\bar{n}^{2}+\bar{n}\right)}{\kappa}.$ (68)

leading to the known result [24]

$\displaystyle\Gamma_{\phi}^{\rm
photon}=\frac{4\chi_{0}^{2}\left(\bar{n}^{2}+\bar{n}\right)}{\kappa}.$ (69)

###  D.3 Coherent state

To consider the dephasing of a spin coupled to a resonator in a coherent
state, we consider the following Hamiltonian and Lindblad operator

$\displaystyle H$ $\displaystyle=\tilde{\omega}_{r}a^{\dagger}a$ (70)
$\displaystyle L$ $\displaystyle=\sqrt{\kappa}\left(a-\alpha
e^{i\tilde{\omega}_{r}t}\right).$ (71)

The steady state of this system corresponds to a pure state
$\rho(t)=|\psi(t)\rangle\langle\psi(t)|$, where the coherent state
$|\psi(t)\rangle=|\alpha e^{i\tilde{\omega}_{r}t}\rangle$ satisfies
$L|\psi(t)\rangle=0$. Applying the transformation
$U=e^{-i\tilde{\omega}_{r}a^{\dagger}at}$ on the Hamiltonian and the Lindblad
operator above we obtain

$\displaystyle\tilde{H}$ $\displaystyle=0$ (72) $\displaystyle\tilde{L}$
$\displaystyle=\sqrt{\kappa}\left(a-\alpha\right)e^{i\tilde{\omega}_{r}t}.$
(73)

In the new frame, we can write the coupled equations of motion for $\vec{O}$,
where $\vec{O}=\begin{pmatrix}n(t),\ a(t),\ {a^{\dagger}}(t),\
\mathds{1}\end{pmatrix}^{T}$, in the form of Eq. 29 with

$\displaystyle
G=\left(\begin{matrix}-\kappa&\frac{1}{2}\alpha^{*}\kappa&\frac{1}{2}\alpha\kappa&0\\\
0&-\frac{1}{2}\kappa&0&\frac{1}{2}\alpha\kappa\\\
0&0&-\frac{1}{2}\kappa&\frac{1}{2}\alpha^{*}\kappa\\\ 0&0&0&0\\\
\end{matrix}\right).$ (74)

The steady state solutions result in

$\displaystyle\langle\vec{O}\rangle_{\infty}=\left(\left|\alpha\right|^{2},\leavevmode\nobreak\
\alpha,\leavevmode\nobreak\ \alpha^{*},\leavevmode\nobreak\ 1\right)^{T}.$
(75)

In order to find $\langle n(\tau)n(0)\rangle$ we diagonalize the system and
plug in the initial conditions in the steady-state solution for $\tau=0$ (as
done previously for Eq. 62). Thus, we obtain

$\displaystyle\langle
n(\tau)n(0)\rangle_{\infty}=\left|\alpha\right|^{4}+\left|\alpha\right|^{2}e^{-\frac{1}{2}\kappa\left|\tau\right|}.$
(76)

We can, again, check that at $\tau\rightarrow\infty$, the operator $n(\tau)$
does not correlate with the operator $n(0)$ such that
$\lim_{\tau\to\infty}\langle n(\tau)n(0)\rangle_{\infty}=\langle
n(\tau)\rangle_{\infty}\langle n(0)\rangle_{\infty}=\left|\alpha\right|^{4}$.
In this case, we obtain

$\displaystyle\eta=\int_{-\infty}^{\infty}\langle
n(\tau)-\bar{n}\rangle\langle n(0)-\bar{n}\rangle
d\tau=\frac{4\left|\alpha\right|^{2}}{\kappa}.$ (77)

By changing the notation $\left|\alpha\right|^{2}$ to $\bar{n}$, we obtain the
dephasing rate due to photon noise in a coherent-state [25]

$\displaystyle\Gamma_{\phi}^{\rm photon}=\frac{8\chi^{2}\bar{n}}{\kappa}.$
(78)

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