# 3DGEN: A GAN-based approach for generating novel 3D models from image data

Antoine Schnepf
<EMAIL_ADDRESS>
Criteo AI Lab
&Flavian Vasile
<EMAIL_ADDRESS>
Criteo AI Lab
&Ugo Tanielian
<EMAIL_ADDRESS>
Criteo AI Lab

###### Abstract

The recent advances in text and image synthesis show a great promise for the
future of generative models in creative fields. However, a less explored area
is the one of 3D model generation, with a lot of potential applications to
game design, video production, and physical product design. In our paper, we
present 3DGEN, a model that leverages the recent work on both Neural Radiance
Fields for object reconstruction and GAN-based image generation. We show that
the proposed architecture can generate plausible meshes for objects of the
same category as the training images and compare the resulting meshes with the
state-of-the-art baselines, leading to visible uplifts in generation quality.

##  1 Introduction

Generative models such as StableDiffusion (Rombach et al., 2022) or DALLE2
(Ramesh et al., 2022) are rapidly changing the boundaries of machine-assisted
creativity, especially in the case of image synthesis. Researchers and
practitioners are inventing new ways to create and remix art, either by text-
conditioned image generation, image inpainting or outpainting, full video
generation. In the same time, the class of Neural Radiance Fields models
(NeRF, Mildenhall et al., 2021) are making rapid advances in photorealistic 3D
model/scene reconstruction from partial views. NeRF uses an implicit
volumetric representation to represent 3D scenes, making it possible to render
them at an arbitrary resolution with low memory costs.

Some of the existing work, such as Generative Radiance Fields (GRAF, Schwarz
et al., 2020), has been starting to bridge the gap between reconstruction and
generation. GRAF can generate new volumetric models from a set of views of
similar objects. However, a major limitation of the GRAF model is that this
volumetric representation is not adapted to produce plausible object meshes,
and is therefore not a good match for 3D-native creative environments such as
game design, virtual-reality (VR) world design, animation. On the other hand,
UNISURF (Oechsle et al., 2021) showed that radiance fields and implicit
surface representations can be unified, and proposed a joint optimization task
that both improves NERF and allows to extract 3D meshes.

In this paper, we propose a potential solution for the shortcomings of GRAF,
which we name 3DGEN. This solution builds on both GRAF (Schwarz et al., 2020)
and UNISURF (Oechsle et al., 2021), and can generate volumetric objects with a
corresponding implicit surface, hence making them easily exportable to 3D
meshes. In Figure 2, we showcase one potential way to control the object
generation: the interpolation in the latent space between two existing object
meshes leads to a set of plausible object meshes of the same type (in this
case cars and chairs).

##  2 Our approach

We begin by introducing a conditional version of UNISURF $g_{\theta}$ which
encodes an object conditionally to a shape code and an appearance code, in a
disentangled manner. Similarly to GRAF (Schwarz et al., 2020), we construct a
generator $G_{\theta}$ (sharing parameters with $g_{\theta}$) by stacking (i)
a module that casts rays, (ii) $g_{\theta}$ that conditionally computes the
emitted radiance and occupancy probabilities along the cast rays, and (iii) a
differentiable volumetric renderer to produce the output images (Mildenhall et
al., 2021). We introduce the discriminator $D_{\phi}$, a convolutionnal neural
network.

We train this setup with the non-saturating GAN objective (Goodfellow et al.,
2014) with R1-regularization (Mescheder et al., 2018), to which a smoothing
term for the implicit surface is added (Oechsle et al., 2021) (more details on
each term can be found in Appendix):

$\min_{\theta}\max_{\phi}\Bigl{(}\mathcal{L}_{\text{adv}}(\theta,\phi)-\lambda\mathcal{R}_{1}(\phi)+\gamma\mathcal{L}_{\text{smooth}}(\theta)\Bigr{)}$
(1)

#### Implementation details and surface extraction.

Our model is initialized such that the initial implicit surfaces are spheres
(Oechsle et al. (2021), Gropp et al. (2020)). As the training progresses, the
points are sampled along rays within a narrowing interval centered around the
first intersection with the implicit surface (Oechsle et al., 2021). By doing
so, in the early stages of the training, the formulation is similar to GRAF,
while in the later stages of the training, points are sampled close to the
implicit surface. It is therefore possible to extract a well defined surface
with the Marching Cube algorithm (Lorensen and Cline, 1987).

Experiments. We test our model on (i) a dataset of cars rendered from The
Carla Driving simulator (Dosovitskiy et al., 2017) and (ii) a dataset of
chairs, rendered from Photoshapes (Park et al., 2018). Figure 1 shows the
generated cars and chairs, as well as disentanglement of shape and appearance.
Figure 2 shows latent space interpolation and the exportation to meshes. In
the Appendix, 3DGEN is compared to the baseline GRAF for camera poses
interpolations in Figure 3 and meshes extraction in Figure 4. To evaluate both
methods, we report the Frechet Inception Distance (FID, Heusel et al., 2017).
GRAF / Ours: $71/97$ (Cars); $48/126$ (Chairs). For fairness, both methods are
trained for the same duration on similar hardware.

Figure 1: Left: 3DGEN with cars and chairs. Right: disentanglement of shape
and appearance.

Figure 2: Latent space interpolation and the corresponding meshes.

##  3 Conclusion and future work

This work present 3DGEN, a generative model that unifies radiance fields and
implicit surfaces. The model can learn an underlying distribution of radiance
fields and surfaces from a dataset composed only of 2D images of objects of a
similar class, and therefore _generate new objects_ from this class. During
inference, the model can both render views from any angle and easily export to
a mesh based representation, which makes it applicable to 3D content creation.
In our future work we intend to further improve the quality of the generated
objects by reducing artefacts in the shapes and producing more diverse
outputs. The ability to export to colored meshes would also be an useful
improvement.

## References

  * Dosovitskiy et al. (2017) Alexey Dosovitskiy, German Ros, Felipe Codevilla, Antonio Lopez, and Vladlen Koltun.  Carla: An open urban driving simulator, 2017.  URL https://arxiv.org/abs/1711.03938.
  * Goodfellow et al. (2014) Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.  Generative adversarial networks, 2014.  URL https://arxiv.org/abs/1406.2661.
  * Gropp et al. (2020) Amos Gropp, Lior Yariv, Niv Haim, Matan Atzmon, and Yaron Lipman.  Implicit geometric regularization for learning shapes, 2020.  URL https://arxiv.org/abs/2002.10099.
  * Heusel et al. (2017) M. Heusel, H. Ramsauer, T. Unterthiner, B. Nessler, and S. Hochreiter.  Gans trained by a two time-scale update rule converge to a local nash equilibrium.  In _Advances in Neural Information Processing Systems_ , pages 6626–6637, 2017.
  * Lorensen and Cline (1987) William E Lorensen and Harvey E Cline.  Marching cubes: A high resolution 3d surface construction algorithm.  _ACM siggraph computer graphics_ , 21(4):163–169, 1987.
  * Mescheder et al. (2018) Lars Mescheder, Andreas Geiger, and Sebastian Nowozin.  Which training methods for gans do actually converge?  2018\.  doi: 10.48550/ARXIV.1801.04406.  URL https://arxiv.org/abs/1801.04406.
  * Mildenhall et al. (2021) Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng.  Nerf: Representing scenes as neural radiance fields for view synthesis.  _Communications of the ACM_ , 65(1):99–106, 2021\.
  * Oechsle et al. (2021) Michael Oechsle, Songyou Peng, and Andreas Geiger.  Unisurf: Unifying neural implicit surfaces and radiance fields for multi-view reconstruction.  In _Proceedings of the IEEE/CVF International Conference on Computer Vision_ , pages 5589–5599, 2021.
  * Park et al. (2018) Keunhong Park, Konstantinos Rematas, Ali Farhadi, and Steven M. Seitz.  Photoshape.  _ACM Transactions on Graphics_ , 37(6):1–12, dec 2018.  doi: 10.1145/3272127.3275066.  URL https://doi.org/10.1145%2F3272127.3275066.
  * Ramesh et al. (2022) Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen.  Hierarchical text-conditional image generation with clip latents.  _arXiv preprint arXiv:2204.06125_ , 2022.
  * Rombach et al. (2022) Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer.  High-resolution image synthesis with latent diffusion models.  In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pages 10684–10695, 2022.
  * Schwarz et al. (2020) Katja Schwarz, Yiyi Liao, Michael Niemeyer, and Andreas Geiger.  Graf: Generative radiance fields for 3d-aware image synthesis.  _Advances in Neural Information Processing Systems_ , 33:20154–20166, 2020.

##  4 Appendix

###  4.1 Background

This section summarizes the mathematical formalism required for our proposed
model 3DGEN.

Neural Radiance Field (Mildenhall et al., 2021). Neural radiance fields are
neural networks parameterized by $\theta\in\Theta$ mapping a spatial location
$\mathbf{x}\in\mathds{R}^{3}$ and a viewing direction
$\mathbf{d}\in\mathds{R}^{3}$ to a view-dependant color
$c_{\theta}(\mathbf{x},\mathbf{d})$ and a volumetric opacity
$\sigma_{\theta}(\mathbf{x})\in\mathds{R}^{+}$:

$f_{\theta}:\mathbf{x},\mathbf{d}\longrightarrow
c_{\theta}(\mathbf{x},\mathbf{d}),\sigma_{\theta}(\mathbf{x})$ (2)

A neural radiance field encodes a 3D scene or object in a volumetric
representation.

Differentiable volume rendering. Given the the camera origin
$\mathbf{o}\in\mathds{R}^{3}$ and a viewing direction $\mathbf{d}$, a ray
$\mathbf{r}=\\{\mathbf{o}+t\mathbf{d}|t\in\mathbb{R}^{+}\\}$ is cast through
the radiance field. Given N points $\\{\mathbf{x}_{i}\\}$ sampled along this
ray, differentiable volume rendering approximates the perceived color of the
ray as follows:

$\hat{C}(\mathbf{r})=\sum^{N}_{i=1}T_{i}\alpha_{i}c_{\theta}(\mathbf{x}_{i},\mathbf{d})$
(3) $\alpha_{i}=1-\exp(-\sigma_{\theta}(\mathbf{x}_{i})\delta_{i}))$ (4)
$T_{i}=\prod_{j<i}(1-\alpha_{j})$ (5)

where $T_{i}$ is the accumulated transmittance along the ray $\mathbf{r}$ and
$\delta_{i}=||\mathbf{x}_{i+1}-\mathbf{x}_{i}||_{2}$ is the distance between
two adjacent points.

By casting $K\times K$ carefully chosen rays and rendering their respective
colors, an image of the encoded object is obtained, where K is the desired
resolution.

UNISURF (Oechsle et al., 2021). Assuming solid non-transparent objects, i.e.
$\sigma_{\theta}(\mathbf{x}_{i})\in\\{0,+\infty\\}$, $\alpha_{i}$ can be
reinterpreted as the occupancy probability at position $\mathbf{x}_{i}$ from
equation 4. We can therefore derive an implicit surface
$\mathcal{S}_{\theta}$:

$\mathcal{S}_{\theta}=\\{\mathbf{x}\in\mathbb{R}^{3}|\alpha_{\theta}(\mathbf{x})=0.5\\}$
(6)

A regularization loss $\mathcal{L}_{\text{smooth}}$ on the implicit surface is
introduced:

$\mathcal{L}_{\text{smooth}}(\theta)=\sum_{\mathbf{x}\in\mathcal{S}_{\theta}}\left\|\mathbf{n}_{\theta}\left(\mathbf{x}\right)-\mathbf{n}_{\theta}\left(\mathbf{x}+\boldsymbol{\epsilon}\right)\right\|_{2}$
(7)

Here $\boldsymbol{\epsilon}$ is a small perturbation and
$\mathbf{n}(\mathbf{x}_{s})$ denotes the surface normal at position
$\mathbf{x}_{s}$. The surface normal can be computed using the formula:

$\mathbf{n_{\theta}}(\mathbf{x})=\frac{\nabla\alpha_{\theta}(\mathbf{x})}{\|\nabla\alpha_{\theta}(\mathbf{x})\|_{2}}$
(8)

We adapt this loss for adversarial training:

$\mathcal{L}_{\text{smooth}}(\theta)=\mathbb{E}_{\mathbf{z}_{s}\sim
p_{\text{latent}},\boldsymbol{\xi}\sim p_{\text{cam}}}\biggl{[}\
\sum_{\mathbf{x}\in\mathcal{S}_{\theta}(\mathbf{z}_{s})\cap\mathcal{R}(\boldsymbol{\xi})}\left\|\mathbf{n}_{\theta}\left(\mathbf{x},\mathbf{z}_{s}\right)-\mathbf{n}_{\theta}\left(\mathbf{x}+\boldsymbol{\epsilon},\mathbf{z}_{s}\right)\right\|_{2}\biggr{]}$
(9)

where the implicit surface $\mathcal{S}_{\theta}(\mathbf{z}_{s})$ and its
normal $\mathbf{n}_{\theta}(\mathbf{x},\mathbf{z}_{s})$ are now conditional to
a shape code $\mathbf{z}_{s}$. $\mathcal{R}(\boldsymbol{\xi})$ denotes the set
of rays cast in the radiance field under the camera parameters
$\boldsymbol{\xi}$.

GRAF (Schwarz et al., 2020). Built upon the original NeRF formulation, a
conditional neural radiance field (cNeRF) generates radiance fields
conditionally to an appearance code $z_{a}$ and a shape code $z_{s}$:

$g_{\theta}:\mathbf{x},\mathbf{d},\mathbf{z_{s}},\mathbf{z_{a}}\longrightarrow
c_{\theta}(\mathbf{x},\mathbf{d},\mathbf{z_{s}},\mathbf{z_{a}}),\sigma_{\theta}(\mathbf{x},\mathbf{z_{s}})$
(10)

The generator $G_{\theta}$ is composed of three modules:

  1. (i)

a module that sample the camera parameters $\boldsymbol{\xi}$ and subsequently
cast $K\times K$ rays

  2. (ii)

a cNeRF $g_{\theta}$ to conditionally compute radiances and volumetric
opacities along the sampled rays

  3. (iii)

a volume renderer to produce the output image

To train the discriminator $D_{\phi}$ and the generator $G_{\theta}$, the
adversarial loss $\mathcal{L}_{\text{adv}}(\theta,\phi)$ and the
regularization term $\mathcal{R}_{1}(\phi)$ are introduced:

$\mathcal{L}_{\text{adv}}(\theta,\phi)=\mathbb{E}_{\mathbf{I}\sim
p_{\text{data }},\boldsymbol{\nu}\sim
p_{\text{patch}}}\Bigl{[}f\left(-D_{\phi}(\Gamma(\mathbf{I},\boldsymbol{\nu}))\right)\Bigr{]}\\\
+\mathbb{E}_{\mathbf{z_{s}},\mathbf{z_{a}}\sim p_{\text{latent
}},\boldsymbol{\xi}\sim p_{\text{cam}},\boldsymbol{\nu}\sim
p_{\text{patch}}}\Bigl{[}f\left(D_{\phi}(G_{\theta}(\mathbf{z_{s}},\mathbf{z_{a}},\boldsymbol{\xi},\boldsymbol{\nu}))\right)\Bigr{]}$
(11)

$\mathcal{R}_{1}(\phi)=\mathbb{E}_{\mathbf{I}\sim p_{\text{data
}},\boldsymbol{\nu}\sim p_{\text{patch}}}\Bigl{[}\bigl{|}\bigl{|}\nabla
D_{\phi}\left(\Gamma(\mathbf{I},\boldsymbol{\nu})\right)\bigr{|}\bigr{|}_{2}^{2}\Bigr{]}$
(12)

with $f(x)=-\log(1+\exp(-x))$. Here, $\boldsymbol{\nu}$ denotes denotes the
patching strategy and $\Gamma$ the patching operator to transform the training
images into $K\times K$ patches. Note that during inference, the generator
$G_{\theta}$ can be used to generate images at any resolution. The patch
constraint only holds during training, to make this setup trainable in
practice. For more details, we refer the reader to Schwarz et al. (2020).

###  4.2 Qualitative comparison between 3DGEN and GRAF

Figure 3: Camera poses interpolations on both Cars and Chairs. 3DGEN (left)
and GRAF (right). Figure 4: Extracting surface meshes from GRAF at level set
$\sigma\in\\{1,10,50,100\\}$ (left) and from 3DGEN (right).


††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Quantum simulation of general semi-classical Rabi model beyond strong
driving regime

Kunzhe Dai  National Laboratory of Solid State Microstructures,
School of Physics, Nanjing University, Nanjing 210093, China  Haiteng Wu
National Laboratory of Solid State Microstructures,
School of Physics, Nanjing University, Nanjing 210093, China  Peng Zhao
National Laboratory of Solid State Microstructures,
School of Physics, Nanjing University, Nanjing 210093, China  Mengmeng Li
National Laboratory of Solid State Microstructures,
School of Physics, Nanjing University, Nanjing 210093, China  Qiang Liu
National Laboratory of Solid State Microstructures,
School of Physics, Nanjing University, Nanjing 210093, China  Guangming Xue
Synergetic Innovation Center of Quantum Information $\&$ Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Xinsheng Tan<EMAIL_ADDRESS>National Laboratory of Solid State
Microstructures,
School of Physics, Nanjing University, Nanjing 210093, China  Haifeng Yu
<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures,
School of Physics, Nanjing University, Nanjing 210093, China  Synergetic
Innovation Center of Quantum Information $\&$ Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Yang Yu  National Laboratory of Solid State Microstructures,
School of Physics, Nanjing University, Nanjing 210093, China  Synergetic
Innovation Center of Quantum Information $\&$ Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China

###### Abstract

We propose a scheme to simulate the interaction between a two-level system and
a classical light field. Under the transversal driving of two microwave tones,
the system Hamiltonian is identical to that of the general semi-classical Rabi
model. We experimentally realize this Hamiltonian with a superconducting
transmon qubit. By tuning the strength, phase and frequency of the two
microwave driving fields, we simulate the quantum dynamics from weak to
extremely strong driving regime. The resulting evolutions gradually deviate
from the normal sinusoidal Rabi oscillations with increasing driving strength,
in accordance with the predictions of the general semi-classical Rabi model
far beyond the weak driving limit. Our scheme provides an effective approach
to investigate the extremely strong interaction between a two-level system and
a classical light field. Such strong interactions are usually inaccessible in
experiments.

###### pacs:

85.25.Cp, 42.50.Ct , 42.50.Hz

††preprint: AIP/123-QED

Light-matter interaction has been at the heart of important modern discoveries
of fundamental effects, both classical and quantum mechanical Cohen-Tannoudji
_et al._ (1992). As the simplest form of light-matter interaction, a two-level
atom interacting with a classical light field, which was introduced by I. I.
Rabi Rabi (1937), has been actively investigated to study and control various
quantum systems, including that of nuclear magnetic resonance Vandersypen and
Chuang (2005), cavity quantum electrodynamics (QED) Haroche and Raimond
(2006), and circuit QED Blais  _et al._ (2007); You and Nori (2003). Based on
this semi-classical model Braak  _et al._ (2016), Jaynes and Cummings
introduced a fully quantized version, i.e. quantum Rabi model Jaynes and
Cummings (1963); Tavis and Cummings (1968), which describes a two-level atom
interacting with a quantum field mode. This model is a cornerstone of various
areas of quantum physics such as quantum optics and quantum information
processing.

As per the usual description of the Rabi model (semi-classical or quantum),
the interaction between the atom and the field mode (classical or quantum) can
be decomposed into two parts, i.e. the rotating and the counter-rotating
terms. Usually, the coupling is significantly smaller than the atomic
transition frequency and field mode frequency. In this weak-coupling regime,
the dynamics of the atom-field system is well described by the Rabi model with
the usual rotating wave approximation (RWA) Jaynes and Cummings (1963); Tavis
and Cummings (1968), i.e. omitting the counter-rotating terms. However, when
the coupling is comparable to the atomic transition frequency and field mode
frequency, a limit which can be dubbed ultrastrong coupling (USC) regime, the
usual RWA breaks down Bloch and Siegert (1940), consequently the counter-
rotating terms manifest important impact on the dynamics of the atom-field
systemCao  _et al._ (2011); Ashhab and Nori (2010); Zueco  _et al._ (2009).
Xie et al. Xie  _et al._ (2014) have introduced an anisotropic version of the
quantum Rabi model thus generalized the quantum Rabi model to USC regime. The
experimental explorations of light-matter interaction in ultra-strong coupling
regime has long been restricted by the coupling strength between the atom and
field. Recently, a few experiments have reached USC regime in solid state
system Fuchs  _et al._ (2009); Laucht  _et al._ (2016); Rao and Suter (2017);
Forn-Díaz  _et al._ (2010); Tuorila  _et al._ (2010); Deng  _et al._ (2015);
Niemczyk  _et al._ (2010); Fedorov  _et al._ (2010); Todorov  _et al._ (2010);
Anappara  _et al._ (2009); Yoshihara  _et al._ (2017) and have demonstrated
complex dynamics unique to this regime, e.g. the Bloch-Siegert shift Forn-Díaz
_et al._ (2010); Tuorila  _et al._ (2010), the Floquet state Deng  _et al._
(2015), exotic behaviors distinct from the normal sinusoidal Rabi oscillation
Fuchs  _et al._ (2009); Laucht  _et al._ (2016); Rao and Suter (2017), and the
spectroscopic signatures of these driving regimes Niemczyk  _et al._ (2010);
Fedorov  _et al._ (2010); Todorov  _et al._ (2010); Anappara  _et al._ (2009);
Yoshihara  _et al._ (2017). However, it still remains an interesting but tough
object to achieve driving strength that is significantly larger than the
atomic transition frequency and field mode frequency (deep coupling regime
Casanova  _et al._ (2010)). As an alternative to access this fascinating
field, quantum simulation of light-matter interaction in deep strong coupling
(for quantum Rabi model) or extremely strong driving (for semi-classical Rabi
model) would be helpful for researching and/or understanding various
associated effects Ballester  _et al._ (2012); Grimsmo and Parkins (2013);
Pedernales  _et al._ (2015); Langford  _et al._ (2016); Fedortchenko  _et al._
(2017); Puebla  _et al._ (2017); Braumüller  _et al._ (2016).

In this paper, first we propose a scheme to simulate the general semi-
classical Rabi model, for which the rotating and counter-rotating terms have
two different driving strengths. By using bichromatic driving, we can engineer
the desired effective Hamiltonian describing the generalized semi-classical
Rabi model in various driving regimes. Then we experimentally realize the
effective Hamiltonian in a circuit-QED setup, where a superconducting transmon
qubit is driven by two phase-locked microwave tones simultaneously. We observe
the dynamics of this light-matter interaction system in different driving
regimes. The population evolutions gradually deviate from the normal
sinusoidal Rabi oscillations with increasing driving strength. From weak to
extremely strong driving regimes, the system’s behavior can be well described
by the general semi-classical Rabi model. Our scheme thus provides an
effective approach to explore the extremely strong interaction between a two-
level system and a classical light field, such strong interaction is usually
inaccessible in experiments.

The Hamiltonian of the generalized semi-classical Rabi model is given as Xie
_et al._ (2014)

$\displaystyle\hat{H}/\hbar$
$\displaystyle=\frac{\omega_{a}}{2}{\hat{\sigma}}_{z}+A_{d}(H_{r}+\lambda
H_{cr}),$ (1) $\displaystyle{\hat{H}}_{r}/\hbar$
$\displaystyle=e^{i\omega_{d}t}{\hat{\sigma}}_{-}+e^{-i\omega_{d}t}{\hat{\sigma}}_{+},$
$\displaystyle{\hat{H}}_{cr}/\hbar$
$\displaystyle=e^{-i\omega_{d}t}{\hat{\sigma}}_{-}+e^{i\omega_{d}t}{\hat{\sigma}}_{+}.$

where ${\hat{\sigma}}_{z}$ is the Pauli matrices,
${\hat{\sigma}}_{-}({\hat{\sigma}}_{+})$ are the ladder operators for the atom
of frequency $\omega_{a}$, $A_{d}$ is the driving strength of the rotating
interaction ${\hat{H}}_{r}$, and $\lambda$ is the relative strength between
the rotating terms ${\hat{H}}_{cr}$ and counter-rotating terms
${\hat{H}}_{r}$. When $\lambda=1$, the Hamiltonian reduces to that of the
usual semi-classical Rabi model. When $\lambda\neq 1$, this generalization can
be considered as a semi-classical version of the anisotropic Rabi model Xie
_et al._ (2014),or as the anisotropic generalization of the semi-classical
Rabi model.

In order to reach the strong driving regime, we employ a bichromatic driving,
rather than a single large amplitudeDeng  _et al._ (2015), to drive the two-
level system. The Hamiltonian under the bichromatic driving in the laboratory
frame with RWA can be written as

$\displaystyle\hat{H}/\hbar=$
$\displaystyle\frac{{\omega_{a}}}{2}{\hat{\sigma}_{z}}+\frac{{{\Omega_{1}}}}{2}\bigl{(}{e^{+i({\omega_{1}}t+{\varphi_{1}})}}{\hat{\sigma}_{-}}+\text{H.c.}\bigr{)}$
(2)
$\displaystyle+\frac{{{\Omega_{2}}}}{2}\bigl{(}{e^{+i({\omega_{2}}t+{\varphi_{2}})}}{\hat{\sigma}_{-}}+\text{H.c.}\bigr{)}.$

with ${\Omega_{i}}$ the driving strength and ${\omega_{i}}$ the frequency of
the $i\left({1,2}\right)$th drive. ${\varphi_{i}}$ denotes the relative phase
of the $i$th driving in the coordinate system of the qubit Bloch sphere in the
laboratory frame.

Then we transform from the laboratory frame to a reference frame that rotates
about the $z$ axis using the unitary transformation, $U_{1}=\exp[-i\omega
t\sigma_{z}/2]$, with ${\omega}=({\omega_{1}}+{\omega_{2}})/2$. Finally we
apply a time-independent rotation, $U_{2}=\exp[-i\phi\sigma_{z}/2]$, with
$\phi=({\varphi_{1}}+{\varphi_{2}})/2$. Since $H_{\text{eff}}=U^{-1}HU-
iU^{-1}\dot{U}$, after the above two steps the original Hamiltonian in Eq. (2)
becomes,

$\displaystyle\hat{H}_{\text{eff}}/\hbar=$
$\displaystyle\frac{{\omega_{a}^{*}}}{2}{\hat{\sigma}_{z}}+\frac{\Omega_{1}}{2}({e^{+i(\omega_{d}^{*}t+\varphi_{0}^{*})}}{\hat{\sigma}_{-}}+\text{H.c.})$
(3)
$\displaystyle+\frac{\Omega_{2}}{2}({e^{-i(\omega_{d}^{*}t+\varphi_{0}^{*})}}{\hat{\sigma}_{-}}+\text{H.c.}).$

Here we find that the effective qubit energy splitting
$\omega_{a}^{\ast}\equiv{\omega_{a}}-{\omega}$, the effective light field
frequency $\omega_{d}^{\ast}\equiv\frac{1}{2}({\omega_{1}}-{\omega_{2}}),$
$\omega_{1}>\omega_{2}$, and the initial phase
$\varphi_{0}^{\ast}=({\varphi_{1}}-{\varphi_{2}})/2$. The driving regime is
now completely determined by the tunable parameters $\omega_{a}^{\ast},$
$\omega_{d}^{\ast},$ ${\Omega_{1}}\mathrm{,}$ and ${\Omega_{2}.}$ Under the
condition ${\Omega_{1}=\Omega_{2}=\Omega^{\ast}},$ the effective Hamiltonian
can be reduced to

$\displaystyle\hat{H}_{\text{eff}}/\hbar$
$\displaystyle=\frac{{\omega_{a}^{*}}}{2}{\hat{\sigma}_{z}}+{\Omega^{*}}\cos(\omega_{d}^{*}t+\varphi_{0}^{*}){\sigma_{x}}.$
(4)

Eq. (4) is identical to the semi-classical Rabi Hamiltonian. It is interesting
to observe the close resemblance between the formalism of the effective
Hamiltonian in Eq. (3) and that of the Hamiltonian given in Eq. (1). This
implies that we can simulate the generalized semi-classical Rabi model with
controllable parameters. For a qubit with fixed ${\omega_{a},}$ we can tune
$\omega$ and $\Omega_{1,2},$ thus changing the system from weak driving, to
ultrastrong driving $\Omega_{1,2}>0.1\omega_{a}^{\ast}$, then deep strong
driving $\Omega_{1,2}>\omega_{a}^{\ast}$, and even extremely strong driving
$\Omega_{1,2}\gg\omega_{a}^{\ast}.$ By choosing different parameters, we can
investigate the behaviors in anisotropic Rabi model, phase sensitivity, and
bias-modulated dynamics Lü  _et al._ (2016); Oliver  _et al._ (2005); Ashhab
_et al._ (2007) in various driving regimes.

Figure 1: (Color Online) The schematic of the bichromatic driving scheme,
capable of generating two distinct microwave tones, each with its own tunable
frequency, strength, and phase, as illustrated by different colors in the
figure.
Figure 2: (Color Online) Qubit evolution in strong driving regime. (a), (b),
and (c) are measured qubit population as a function of time for tuning
different parameters. (d), (e), and (f) are the corresponding numerical
simulations with Hamiltonian in Eq. (3).

We use a 3D transmon qubit to experimentally implement the quantum simulation.
A 3D transmon qubit comprises a superconducting transmon qubit and an aluminum
holder in which the qubit sits, forming a typical circuit QED system. By our
design, the detuning between the qubit frequency and cavity frequency is much
larger than the coupling strength between the qubit and the cavity, thus the
system works in the dispersive region. The main purpose of the cavity in our
experiments is to serve as a convenient tool to manipulate and measure the
qubit. The sample is cooled in a cryogen-free dilution refrigerator to a base
temperature of about $20~{}\mathrm{mK}$. The details of qubit control and
measurement can be found elsewhereReed  _et al._ (2010). From spectroscopy
measurement, we obtain the qubit transition frequency at $\sim
7.173~{}\mathrm{GHz}$, the frequency difference between the ground state and
the first excited state of the qubit. The resonant frequency of the 3D cavity
is $\sim 9.052~{}\mathrm{GHz}$. The coupling strength between the qubit and
the cavity is about $50~{}\mathrm{MHz}$. The energy relaxation time of the
qubit is about $10~{}\mathrm{\mu s}$ and the decoherence time measured from
Ramsey experiment is about $10~{}\mathrm{\mu s}$.

In order to ensure phase control of the driving microwaves with respect to the
qubit Bloch sphere coordinate system, we use a single microwave source
together with two digital to analog converter(DAC) channels of an Arbitrary
Wave Generator(AWG) of the model Tektronix 5014c to generate two microwave
tones. Our bichromatic driving scheme (shown in Fig. 1) is different from that
typical used in strong driving experiments Braumüller  _et al._ (2016). We use
only one microwave source for Local Oscillator(LO) input, with frequency equal
to the mean frequency of the two tones and amplitude fixed. The two DAC
channels synchronously generate a cosine and a sine waveform of different
amplitudes for heterodyne IQ (In-phase and Quadrature) mixing. Except for the
difference of a quadrature, these two waveforms have the same frequency and
initial phase. Using this scheme, we can obtain two separate microwave tones
with fully controllable frequency, strength and phase, which have simple
relations to the input waveforms and can be readily deduced. The power of
microwave required in this experiment is less than $20~{}\mathrm{dBm}$
(corresponding to driving strength at $2\pi\times 50~{}\mathrm{MHz}$ under our
experimental setup), and the frequency applied is comparable to qubit
transition frequency. These requirements are met by most microwave sources
from common brands.

Setting | $\omega_{1}-\omega_{a}$ | $\omega_{2}-\omega_{a}$ | $\Omega_{1}$ | $\Omega_{2}$ | $\varphi_{1}-\varphi_{2}$ | $\omega_{a}^{*}$ | $\omega_{d}^{*}$ | $\varphi_{0}^{*}$
---|---|---|---|---|---|---|---|---
a | 0 | 0$\sim$20 | 20 | 20 | 0 | 0$\sim$10 | 0$\sim$10 | 0
b | 0 | 10 | 20 | 1$\sim$20 | 0 | 5 | 5 | 0
c | 0 | 10 | 20 | 20 | 0$\sim$$2\pi$ | 5 | 5 | 0$\sim$$\pi$
Table 1: The parameters used for quantum simulations in strong driving regime.
The frequency and the driving strength are in the unit of
2$\pi~{}\mathrm{MHz}$, while the phase in radians. We used the same parameters
for numerical simulations.

According to Eq. (3), we can individually tune multiple parameters to simulate
quantum dynamics in strong driving regime. In our experiments, we choose three
representative settings to fulfill the requirement of strong driving. Table I
lists the parameters used for different settings. At first, we keep the
driving strength constant, i.e.
$\Omega_{1}/2\pi=\Omega_{2}/2\pi=20~{}\mathrm{MHz}$, the model reduces to Rabi
model as described in Eq. (4). The initial phases of both microwave tones are
the same, rendering $\varphi^{\ast}=0$. As we keep the first microwave tone in
resonance with the qubit and change the frequency of the second tone, we
obtain different values for the effective transition frequency
$\omega_{a}^{\ast}$, hence also for the ratio
$\Omega^{\ast}/\omega_{a}^{\ast}$, which scales the relative driving strength.
Shown in Fig. 2(a) is the qubit population at the first excited state as a
function of time for different driving strengths with $\omega_{a}^{\ast}/2\pi$
adjusted from $10~{}\mathrm{MHz}$ to $0$, corresponding to the ratio
$\Omega^{\ast}/\omega_{a}^{\ast}$ increasing from $2$ to infinity. The system
in the process reaches and even goes beyond the deep strong driving regime.
The resulting oscillations are obviously anharmonic and nonlinear, exhibiting
the dynamics of semi-classical Rabi model. In the second setting, we only
change the driving strength of the second microwave tone $\Omega_{2}$ and keep
the other parameters fixed. From Eq. (3) we can see that $\Omega_{2}$ defines
the strength of the counter rotating field. This setting can be used to
investigate the anisotropic Rabi model. We measure the qubit population
evolution under various values for $\Omega_{2}$, as shown in Fig. 2(b). When
$\Omega_{2}$ is small, corresponding to the bottom part of Fig. 2(b),
distinctly sinusoidal oscillations are observed because the the effects of the
counter-rotating terms are negligible. With the increase of $\Omega_{2}$, the
sinusoidal fringes deform and transition into an irregular pattern, exhibiting
the influence of the counter-rotating terms under large $\Omega_{2}$. In the
third setting, we only change the phase difference of the two tones, which
defines the initiate phase $\varphi^{\ast}$ in the effective Hamiltonian in
Eq. (3). In the weak driving regime this initial phase has negligible effects
on the dynamics of the systemLondon  _et al._ (2014); Rao and Suter (2017).
However, this is not the case when we set the parameters for strong driving
regime and measure the corresponding qubit evolution, as shown in Fig. 2(c),
in which the phase-sensitivity of the evolution dynamics is evident. We note
that the period of phase $\varphi^{*}_{0}$ for this pattern is $\pi$ instead
of $2\pi$. In addition, we provide numerical simulation results by solving the
master equations. The computed qubit evolutions agree with the experimental
results.

Figure 3: (Color Online) The time evolution dynamics in Bloch sphere
representation (left) and the corresponding $\langle\sigma_{z}\rangle$
component (right) of the state vector with different initial phases. (a) and
(b) [(c) and (d)] are evolutions in weak (deep strong) driving limit, with
initial phase $\varphi_{0}^{*}=0$ and $\pi/2$, respectively. Solid lines and
symbols correspond to theoretical calculation and experimental data.

In strong driving regime, a remarkable feature is the presence of counter-
rotating evolution, which interestingly shows the initial phase dependence
Laucht  _et al._ (2016); London  _et al._ (2014). One of the advantages of our
scheme is precise control of initial phase. As a result, we can investigate
the dynamics with a specific initial phase instead of using an averaged phase
Laucht  _et al._ (2016); London  _et al._ (2014). Fig. 3 shows the state
evolution on Bloch sphere (left) and its corresponding
$\langle\sigma_{z}\rangle$ component (right) as a function of time. For
simplicity, we choose on-resonance condition
$\omega_{d}^{\ast}=\omega_{a}^{\ast}.$ By changing the values of
$\omega_{a}^{\ast}$ while keeping $\Omega^{\ast}/2\pi=20~{}\mathrm{MHz}$, we
simulate the dynamics from weak driving to deep strong driving regime. For (a)
[(c)] and (b) [(d)], $\varphi_{0}^{\ast}=0$ and $\pi/2,$ respectively. In Fig.
3(a), (b), $\omega_{a}^{\ast}/2\pi=2~{}\mathrm{GHz}$, which is much larger
than $\Omega^{\ast}/2\pi$. The system is in weak driving regime, where RWA is
valid. The qubit state vector rotates normally in the $y$–$z$ plane on the
surface of the Bloch sphere and the trajectories of both
$\varphi_{0}^{\ast}$’s coincide. In Fig. 3(c), (d),
$\omega_{a}^{\ast}/2\pi=5~{}\mathrm{MHz}$, and
$\Omega^{\ast}>\omega_{a}^{\ast}$, corresponding to the deep strong driving
regime. Two trajectories completely deviate from the $y$–$z$ plane, the
$\langle\sigma_{z}\rangle$ component of state vector shows complicated
oscillations. The trajectories of state vector for two initial phases are
completely different. The dynamics exhibits strong sensitivity to initial
phase. These phenomena result from the significant contribution of counter-
rotating terms. In addition, the oscillations of $\langle\sigma_{z}\rangle$
component agree with the numerical simulations, indicating precise control of
the initial phase realized by our scheme.

In summary, we propose a bichromatic driving method to simulate the general
Rabi model in strong driving regime. This method allows us to reach the strong
driving regime with easily attainable experimental conditions. We demonstrate
the dynamics of general Rabi model in various driving regimes. When the
driving strength is higher than the effective energy splitting
$\omega_{a}^{*}$, the model exhibits highly anharmonic and nonlinear
behaviors, clearly deviating from the usual sinusoidal Rabi oscillations. Such
effects manifest the influence of the counter-rotating terms upon the
breakdown of RWA. Moreover, these sophisticated patterns and trajectories,
obtained experimentally, are in agreement with the computed results. This
quantum simulation scheme provides a useful test bed for studying general Rabi
model in regimes defined by arbitrarily strong driving, and for exploring
optimal quantum gate operations.

This work was partly supported by the the NKRDP of China (Grant No.
2016YFA0301802), NSFC (Grant No. 11504165, No. 11474152, No. 61521001).

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# Age analysis of extrasolar planets: Insight from stellar isochrone models

C. Swastik Indian Institute of Astrophysics, Koramangala 2nd Block, Bangalore
560034, India  Pondicherry University, R.V. Nagar, Kalapet, 605014,
Puducherry, India  Ravinder K. Banyal Indian Institute of Astrophysics,
Koramangala 2nd Block, Bangalore 560034, India  Mayank Narang Department of
Astronomy and Astrophysics, Tata Institute of Fundamental Research Homi Bhabha
Road, Colaba, Mumbai 400005, India  Academia Sinica Institute of Astronomy
$\&$ Astrophysics, 11F of Astro-Math Bldg., No. 1, Sec. 4, Roosevelt Road,
Taipei 10617, Taiwan, Republic of China  Athira Unni Indian Institute of
Astrophysics, Koramangala 2nd Block, Bangalore 560034, India  Department of
Physics and Astronomy, University of California, Irvine, 4129 Frederick Reines
Hall, Irvine, CA 92697, USA  T. Sivarani Indian Institute of Astrophysics,
Koramangala 2nd Block, Bangalore 560034, India

(Received April 18, 2022; Revised April 18, 2022)

###### Abstract

There is growing evidence from stellar kinematics and galactic chemical
evolution (GCE) suggesting that giant planets (M${}_{P}\geq$0.3$M_{J}$) are
relatively young compared to the most commonly occurring population of small
planets (M${}_{P}<$0.3$M_{J}$). To further test the validity of these results,
we analyzed the ages for a large number of 2336 exoplanet hosting stars
determined using three different but well-established isochrone fitting
models, namely, PARSEC, MIST, and Yonsei Yale (YY). As input parameters, we
used Gaia DR3 parallaxes, magnitudes, and photometric temperature, as well as
spectroscopically determined more accurate temperatures and metallicities from
the Sweet Catalog. Our analysis suggests that $\sim$ 50$\%$ to 70$\%$ of stars
with planets are younger than the sun. We also find that, among the confirmed
exoplanetary systems, stars hosting giant planets are even younger compared to
small planet hosts. The median age of $\sim$ 2.61 to 3.48 Gyr estimated for
the giant planet-hosting stars (depending on the model input parameters)
suggests that the later chemical enrichment of the galaxy by the iron-peak
elements, largely produced from Type Ia supernovae, may have paved the way for
the formation of gas giants. Furthermore, within the giant planet population
itself, stars hosting hot Jupiters (orbital period $\leq$10 days) are found to
be younger compared to the stellar hosts of cool and warm Jupiters (orbital
period $>$10 days), implying that hot Jupiters could be the youngest systems
to emerge in the progression of planet formation.

Stellar ages(1581), Exoplanets(498), Planet hosting stars(1242), Exoplanet
formation(492), Extrasolar gaseous giant planets(509), Hot Jupiters(753)

††journal: AJ††software: Numpy (Harris et al., 2020), Topcat (Taylor, 2005),
Astropy (Astropy Collaboration et al., 2013), Scikit-learn (Pedregosa et al.,
2011), Matplotlib (Hunter, 2007), Scipy (Virtanen et al., 2020)

.

##  1 Introduction

The field of exoplanets has seen a rapid rise in the past three decades after
the discovery of the first planet by Mayor & Queloz (1995). We realized that
planets are ubiquitous and their architecture and properties are considerably
more diverse and complex (Carter et al., 2012; Gaudi et al., 2017; Gillon et
al., 2017; Zhu & Dong, 2021). Apart from the strange planet discovered around
the pulsar: PSR1257+12 (Wolszczan & Frail, 1992), the earliest exoplanets
discovered were all giants, equivalent to Jupiter in mass and size. Among
those initial discoveries were hot Jupiters which orbit their stars with
periods of a few days as they were easier to detect compared to smaller
planets(Mayor & Queloz, 1995; Charbonneau et al., 2000; Henry et al., 2000).
Thus, the first exoplanet detection, is itself a surprise to the astronomy
community since no such planet exists in our solar system. Numerous
revelations have emerged over the years since the first discovery. To date,
more than 5000 exoplanets have been discovered, which span a wide range of
population. While some planets orbit in close proximity to their parent star
(Mills et al., 2016; MacDonald et al., 2016; Smith et al., 2018; Barragán et
al., 2018; Malavolta et al., 2018; Lam et al., 2021), others have incredibly
extended orbits (Naef et al., 2001; Tamuz et al., 2008; Chauvin, 2018; Vigan
et al., 2021; Wahhaj et al., 2021; Currie et al., 2022; Benisty et al., 2021;
Ren et al., 2023; Wahhaj et al., 2024).

Several correlations connecting the stellar and planetary properties have
emerged in the past decade (Santos et al., 2004; Fischer & Valenti, 2005;
Fischer et al., 2014; Narang et al., 2018; Sousa et al., 2021; Swastik et al.,
2021; Unni et al., 2022). One such correlation is the stellar age-planetary
mass correlation. The ages of Stars With Planets (SWP) are crucial for
investigating numerous aspects of planetary system evolution, such as
dynamical interactions among planets (Laughlin & Chambers, 2002) and tidal
effects generated by SWP (Pätzold et al., 2004; Barker & Ogilvie, 2009). In
fact, understanding the ages of stars holds significant importance in the
process of selecting stellar candidates for planet detection (Bonfanti et al.,
2015) and assessing their potential habitability. The rotation and activity
levels of a star which serve as indicators of stellar age, play a crucial role
in determining the habitability of planets orbiting around them.

Figure 1: Stellar isochrone models generated using MIST. The red symbols
represent planet-hosting stars. The isochrones are drawn for the solar-scaled
abundances. Both X and Y scales are in logarithmic units.

The majority of planets have been detected around main-sequence FGK stars. Due
to the degeneracy of parameters and the slow evolution of stars in main-
sequence, it is difficult to accurately constrain the ages of these stars.
Owing to the ambiguities inherent in estimating age, greater precision is
required for these investigations. Unlike other stellar properties, such as
$T_{eff}$, log g and [Fe/H], ages cannot be directly observed or measured. To
estimate stellar ages, one uses an indirect model-dependent technique such as
isochrone fitting (Valls-Gabaud, 2014). Other approaches, such as
gyrochronology and activity index, are also used in addition to isochrone
fitting from stellar evolutionary models. Chemical analysis (also known as
chemiochronology, (Delgado Mena et al., 2019; Swastik et al., 2022)) and
stellar kinematics (Binney et al., 2000; Wu et al., 2021) can be used to
estimate the ages of an ensemble of stars but cannot be used for individual
cases. Asteroseismology stands as the sole method capable of ascertaining the
age of a star with an impressive level of precision, reaching uncertainties as
low as $\sim$ 11$\%$ (Bellinger et al., 2019). However, it requires longer
time-series data, which is only accessible for a limited sample of stars.
Additionally, it only applies to stars hotter than about spectral type K, as
cooler stars do not typically exhibit the oscillations required for estimating
ages using asteroseismology (Silva Aguirre et al., 2015; Christensen-Dalsgaard
& Aguirre, 2018). Each of the aforementioned models requires input parameters
derived from various sources. Each input parameter is accompanied by its own
uncertainty, which ultimately propagates into the age estimation.

Table 1:  Sample distribution of stars hosting small and giant planets used in
this paper.

Count | Stellar-hosts | Planets
---|---|---
Total | 2336 | 3034
Small ($M_{P}<0.3M_{J}$) | 1834 | 2509
Giant ($0.3M_{J}\leq M_{P}\leq 5M_{J}$) | 502 | 526

Note – The above values are listed after the sample curation as described in
section 2.

The ages of the individual planet-hosting stars are often determined using
different models and input parameters. Due to the inherent uncertainties
associated with each model and its input parameters, carrying out a meaningful
statistical comparison becomes challenging. There have been limited
homogeneous studies for the ages of the planet-hosting stars. Early studies
such as Saffe et al. (2005) have estimated the stellar ages of the 49 planets
hosting stars. Since the sample was very small, it was not possible to draw
any robust conclusions about different populations of planets. Recent studies
of 326 planets hosting stars by Bonfanti et al. (2015) using PARSEC isochrones
have found that $\sim$6$\%$ of stars have ages lower than 0.5 Gyr, while
$\sim$7$\%$ of stars are older than 11 Gyr. Additionally, their results showed
that the majority of their planet-hosting stars fall within the age range of
1.5 to 2 Gyr, indicating a prevalence of younger systems. Using
astroseismology data for 33 Kepler exoplanet host stars, Silva Aguirre et al.
(2015) have found that the majority of these Kepler host stars are older than
the Sun. Further study of 335 transiting planets hosting stars by Bonfanti et
al. (2016) has shown that the median age of the sample is $\sim$5 Gyr, which
is similar to the solar age. These studies motivated us to look for possible
correlations between stellar ages and planet mass. However, these studies are
based on a limited sample of stars, and the ages derived in these papers have
strong dependence on the models and input parameters. As a result, the
estimates become less reliable for studying any statistical correlation
between the stellar ages and the properties of their planetary companions.

Figure 2: Comparison of stellar ages computed from photometric
$T_{\textrm{eff}}$ from Gaia DR3 and spectroscopic $T_{\textrm{eff}}$ from
sweet-cat. The color coding represents the density of points.

In order to understand how stellar ages are correlated with planetary
properties, we need to analyse a large number of planet-hosting stars with
minimal errors in their ages. Because of the difficulties in getting precise
age estimates from isochrone models due to high uncertainties, we confirmed
these estimates using established isochrone fitting models and parameters from
photometric and spectroscopic data. Our findings indicate that, while slight
variations in age estimates may occur based on the specific input parameters
or models employed, the overall statistical trends for a large sample of
planet hosting stars remain unchanged. In this study, our specific objective
is to determine the ages of stars that host planets and examine the
relationships between stellar ages and planetary properties, namely, the
orbital period and mass of the exoplanets. We used combinations of input
parameters from photometry and spectroscopy and three isochrone fitting models
(MESA Isochrones & Stellar Tracks (MIST) (Dotter, 2016; Choi et al., 2016),
PAdova and tRieste Stellar Evolutionary Code (PARSEC) (Bressan et al., 2012)
and q2-Yonsei-Yale (YY) (Han et al., 2009; Ramírez et al., 2014)) for the
analysis. The paper is organized as follows. In section 2 we describe our
sample of stars hosting planets. Section 3 discusses the methodology of the
age determination using isochrones. In section 4 and section 5 we discuss the
trends for stellar ages as a function of planet mass and interpret the results
in terms of the chemical evolution of our galaxy. Finally, we summarise and
conclude our findings in section 6.

Figure 3: Comparison of stellar ages from MIST model using V-band (Johnson)
and G-band (Gaia) mag. The color coding represents the density of stars.
Figure 4: Comparison of stellar ages from MIST, PARSEC and Yonsei-Yale
isochrone fitting models. The color coding represents the density of stars.

##  2 Sample selection

To study the dependence of stellar ages on planetary properties such as planet
mass ($M_{P}$) and orbital period, we selected an initial sample of 3775
planet-hosting stars from the NASA exoplanet archive (Akeson et al., 2013;
NASA Exoplanet Science Institute, 2020). To calculate stellar ages using
isochrone fitting techniques, we need accurate stellar parameters from
spectroscopy (such as T${}_{\textrm{eff}}$, log g and [Fe/H]), photometry
(like G-band magnitude), and astrometry (like parallax, proper-motion etc).
The photometric and astrometric data was obtained from GAIA observations by
cross-matched 111Initially, we used a wider search radius, but in this
instance, a 3” search radius was adequate to retrieve all planet-hosting
stars. We also used other sources, such as the SIMBAD, to confirm that they
are actual planet-hosting stars. our sample with GAIA DR3 (Gaia Collaboration
et al., 2022), while for the spectroscopic data was obtained from the sweet-
cat (Andreasen et al., 2017; Sousa et al., 2018, 2021) which is a catalogue of
stellar parameters for SWP determined homogeneously from spectroscopy. We
narrowed down our analysis to main sequence stars due to the complexities
associated with accounting for evolutionary effects, such as photospheric
mixing, which can introduce variations in photospheric metallicities
(Bergemann et al., 2011; Swastik et al., 2022). Subsequently, we excluded
super-Jupiters ($M_{P}\geq 5M_{J}$) and multi-planetary systems hosting at
least one small planet ($M_{P}<0.3M_{J}$) and a giant planet ($M_{P}\geq
0.3M_{J}$), as including such multi-planetary systems would make it difficult
to discern differences in stellar populations between small and giant planets
as multi-planetary systems show properties similar to giant-planets and are
statistically younger (see Appendix B of Swastik et al. (2022)). The
distinction between small and giant planets is taken at the mass of Saturn,
drawing on the mass-density rationale presented in Hatzes & Rauer (2015). This
boundary is based on the observed shift in the slope within the mass-density
relationship for exoplanets.Further, Bashi et al. (2017) used Mass-Radius
(M-R) relations and arrived at similar conclusions. They further suggested
that the location of the breakpoint is linked to the onset of electron
degeneracy in hydrogen, and therefore to the planetary bulk composition. For
further analysis, we retained only those stars that have age uncertainties
smaller than their main-sequence lifetime, as recommended by Pont & Eyer
(2004). Additionally, we excluded lower main-sequence stars
(T${}_{\textrm{eff}}<$4400K) from our sample since the main-sequence lifetime
for such stars can exceed the age of the universe, and therefore, the current
stellar isochrone models cannot reliably estimate their age. Consequently, our
final dataset comprised 2336 stars hosting 3034 planets (see table 1).

##  3 Stellar age determination: Isochrones

To determine the ages from isochrone, one places the star on the
Hertzsprung$-$Russell diagram (HRD) with $T_{eff}$ on the x-axis and
luminosity $L$ on the y-axis (Figure 1). The $T_{eff}$ and $L$ can be obtained
by several techniques. In the case of $T_{eff}$, it can be determined both by
spectroscopy and photometry (colour-index), while the $L$ is computed from the
observed total flux, which is obtained from the photometric magnitude (Rodrigo
et al., 2017), and distance from the parallax ($\pi$). These observables have
their own intrinsic errors and systematics based on the techniques used to
obtain them. Thus, the isochrone placement technique becomes challenging to
determine the ages of a star with high accuracy. Further, for the lower main-
sequence stars (T${}_{eff}<$4400K), the isochrone ages are not reliable as the
evolution timescales for these stars are $>$ age of the universe, and thus it
becomes challenging to model the evolution of such systems.

Another factor that influences the determination of isochrone ages is the
selection of models/grids. While different isochrone models share a common
goal of estimating stellar ages, they diverge in their underlying assumptions.
For instance, the equation of state (EOS) employed by the PARSEC models
primarily relies on the Free EOS tool 222https://freeeos.sourceforge.net/. In
contrast, MIST and YY isochrones predominantly utilize the OPAL (Opacity
Project at Livermore) (Iglesias & Rogers, 1996) and the SCVH (Stewart,
Colwell, Vasil, and Helfand) equation of state (Saumon et al., 1995),
respectively. Furthermore, variations in solar abundances among the different
isochrone models contribute to discrepancies in their results. For example,
YY-isochrones adopt solar abundances from (Grevesse & Sauval, 1998), while
MIST isochrones employ the values from Asplund et al. (2009). Additionally,
the choice of atmospheric models, such as ATLAS12, PHOENIX (BT-Settl), SYNTHE,
MARCS, and others, in conjunction with the EOS, opacity values and solar
abundances, further contributes to systematic differences in the estimation of
stellar ages. These differences are critical factors that must be taken into
account when determining isochrone ages. They highlight the complexities and
uncertainties involved in age estimation and demonstrate the need for careful
consideration and comparison of multiple isochrone models to mitigate
potential biases.

We used isoclassify333https://github.com/danxhuber/isoclassify(Huber et al.,
2017; Berger et al., 2020, 2023), a robust tool designed to estimate stellar
ages using MIST isochrone grids. For the estimation of stellar ages using
PARSEC isochrones, we used PARAM-1.5 444http://stev.oapd.inaf.it/cgi-
bin/param, while for the Yonsei Yale (YY) isochrones, we used q2 tool
555https://github.com/astroChasqui/q2. To obtain the stellar ages, we used the
observed stellar parameters such as effective temperature ($T_{\text{eff}}$),
luminosity ($L$), and metallicity ([Fe/H]) as inputs into these codes, which
then matches these observations with theoretical isochrones to estimate
stellar ages. To ensure the robustness of our age determinations, we conducted
a comparative analysis using the age estimates derived from each of the three
isochrone sets. This approach allowed us to evaluate the consistency of age
estimations across different stellar models and to identify any systematic
discrepancies that may arise due to the underlying assumptions of each
isochrone model.

Statistical inferences drawn for a sample of stars become inherently
unreliable due to the dependence of individual star age determinations on both
the input parameters and the models utilized. Therefore, we estimate the ages
of the stars using various models and different combinations of input
parameters. Our objective is to assess whether consistent statistical
conclusions could be drawn across different combinations of models and input
parameters. Here, we vary certain input parameters (for instance, we use Teff
form photometry and spectroscopy) while keeping the other parameters constant
(for instance using G-band magnitude and MIST isochrone models) to demonstrate
how the age varies from one case to another and how the overall stellar age
for the population of exoplanets varies statistically.

Figure 5: Histogram for ages of the planets hosting stars for small and giant
planets. The label is indicated in the following format: isochrone model-
photometric band-temperature from spectroscopy or photometry. The dashed lines
represent the median ages corresponding to their color labels in the
histograms.  Figure 6: Cumulative distribution for the stellar ages of small
(red) and giant (blue) planet-hosting stars obtained using different isochrone
models and input parameters. Figure 7: Planet mass as a function of stellar
age. The colorbar on the right represents the stellar metallicity values
obtained from the literature. The dashed red line (MP = 0.3MJ) separates the
small and giant planet-hosting stars.  Table 2: Comparison of stellar ages for
the small and giant planet-hosting stars using different models and input
parameters

| Giant-planet | Small-planet | p-value
---|---|---|---
| (Gyr) | (Gyr) |
MIST G Spec | 3.00$\pm$2.56 | 4.69$\pm$2.47 | $10^{-26}$
MIST G Phot | 3.34$\pm$3.12 | 5.08$\pm$2.95 | $10^{-16}$
MIST V Spec | 2.38$\pm$2.89 | 4.58$\pm$2.37 | $10^{-16}$
MIST V Phot | 3.42$\pm$3.34 | 5.03$\pm$2.83 | $10^{-29}$
Yonsei Y G Spec | 3.45$\pm$3.15 | 5.05$\pm$2.48 | $10^{-36}$
PARSEC G Spec | 2.63$\pm$2.21 | 3.78$\pm$2.01 | $10^{-13}$

Note – The errors quoted are the median absolute deviation (MAD) in the above
distributions. Here the p-value obtained from the KS - test represents the
probability that the two samples belong to the same distribution.

###  3.1 Choice of stellar temperatures

The temperature of a star is mostly obtained by spectroscopy or photometric
measurements. Both of these techniques have certain assumptions while
obtaining the estimate of the $T_{eff}$ and this leads to a systematic
difference in the estimation of ages (Wing & Yorka, 1979). To verify how the
input ages affect the stellar age estimates from isochrone fitting we took the
$T_{eff}$ from two sources: sweet-cat (Sousa et al., 2021) (spectroscopic) and
GAIA DR3 (Gaia Collaboration et al., 2022) (photometric). Figure 2 shows the
spread in age using photometric and spectroscopic temperatures using Yonsei
Yale isochrone models (Han et al., 2009) and GAIA parallaxes (Lindegren et
al., 2021). We find that $\sim$85$\%$ of the stars have an age difference $<$
0.5 Gyrs. Similarly, when employing MIST and PARSEC isochrone models, we
observe that 72$\%$ and 75$\%$ of the stars, respectively show an age
difference of less than 0.5 Gyrs, indicating that the scatter is small with no
significant systematic in the age estimates from spectroscopic and photometric
temperatures.

###  3.2 Choice of photometric band magnitude

The luminosity estimate in our model relies on the total flux ($f$), which is
determined from the band magnitude provided. Given this, we explored whether
the selection of photometric band magnitudes has any noteworthy impact on the
process of estimating stellar ages. We decided to use the Johnsons V-band and
Gaia G-band magnitudes to estimate the ages of the exoplanet host star. Stars
for which V-band magnitude was not available, we used the relationship as
described in GAIA archive 666https://www.cosmos.esa.int/web/gaia-
users/archive/gdr3-documentation and used G, Gbp and Grp magnitudes to obtain
the V-band magnitude. We also tested the above relation for the stars whose
both G-band and V-band were available and we found the V-band obtained from
the empirical relation matches with the observed ones. Figure 3 shows the
correlation between the ages determined from V- and G-band magnitudes using
spectroscopic temperatures and MIST. We find that the ages obtained from
G-band and V-band are strongly correlated and do not show any significant
systematic differences. We also performed this analysis with several other
combinations of input parameters (for instance, using photometric and
spectroscopic $T_{eff}$ ) and did not find any considerable dispersion in any
case.

###  3.3 Choice of models

Most of the stellar ages obtained using isochrone use a standard stellar
evolutionary isochrone fitting model to estimate the age of stars. However,
the choice of the model plays an equally important role just as input
parameters (Delgado Mena et al., 2019). Figure 4 shows the scatter plot in the
stellar ages obtained using various isochrone fitting models using
spectroscopic temperatures and G magnitude. We find that the ages estimated
using MIST and YY are well in agreement, with a moderate spread but no
significant systematic differences. For the stellar ages obtained using PARSEC
models, we find a large scatter for stars with ages $>$6 Gyr, when comparing
with the ages obtained using MIST and YY. We also find a notable systematic
difference in the stellar ages obtained from PARSEC models when comparing the
ages with MIST and YY.

Table 3: Distribution of Jupiter hosting stars in terms of their stellar ages
and orbital period.

| Young hot Jupiters | Old hot Jupiters | Young cool Jupiters | Old Cool Jupiters | Hot Jupiters | Cool Jupiters | p-value
---|---|---|---|---|---|---|---
| (Count) | (Count) | (Count) | (Count) | (Gyr) | (Gyr) |
MIST | 207 | 60 | 115 | 46 | 2.43 | 3.50 | $10^{-3}$
Yonsei Y | 191 | 76 | 97 | 67 | 3.08 | 4.35 | $10^{-5}$
PARSEC | 289 | 48 | 132 | 56 | 2.15 | 3.38 | $10^{-9}$

The standard error of the mean is typically 0.01 Gyr for all the cases. The
p-value is obtained in the same way as in table 2.

##  4 Results

###  4.1 Ages of the planet-hosting stars

We compute the ages of the planet-hosting stars from isochrone fitting methods
using different models and input parameters. Figure 5 shows the age histograms
of stars hosting small planets and giant planets. Note that, even though the
distribution of stellar ages computed using different models and input
parameters has noticeable differences, the relative offset (median age
difference) between the ages of small and giant planet-hosting stars, is
always positive. In other words, the median ages of the stars hosting small
planets are higher compared to stars hosting giant planets in all cases (see
Table 3). We also find that the median age of stars obtained from PARSEC
models is slightly lower when compared with the MIST and YY isochrone models,
and this is due to the systematics of the stellar ages as obtained in Section
3.

Further, the cumulative age fraction shown in figure 6 implies that stars
hosting small and giant planets belong to different populations. We also
performed the Kolmogorov Smirnov (K-S) test on the sample and found that the
sample of stars hosting small and giant planets fall into distinct age groups
(table 2). This result clearly suggests that small planets are common around
both young and older stars, whereas giant planets are more prevalent around
younger stars.

It is important to note that due to high stellar activity and RV jitter777RV
jitter is the intrinsic noise in radial velocity measurements of a star,
caused by factors such as stellar activity, granulation, oscillations, and
instrumental limitations. It poses challenges in detecting planets, raising
the detection threshold, and introducing false positives/negatives., the
detection of small and low-mass planets around young stars is far more
challenging than the detection of giant planets. Therefore, in the current
exoplanet census, there is a strong possibility that some small planets might
not have been detected around younger stars. However, the lack of giant
planets around older stars is not likely caused by any detection or selection
bias, as detecting giant planets is relatively easier than detecting small
planets, regardless of the method used or the age of the star.

###  4.2 Planet mass as a function of stellar age

Figure 7 shows the planet-mass distribution as a function of stellar age with
$\sim$ 70%-85% of stars in the sample having an age below 7 Gyr. We also note
that the population of giant planets began to rise about 4-5 Gyr ago,
indicating that the formation epoch of giant planets is relatively recent
compared to the population of small planets, with the formation onset
occurring as early as 7-8 Gyr ago. Moreover, the colorbar in Figure 7, which
signifies stellar metallicity (taken from Sweet-Cat Sousa et al. (2021)),
reveals a metallicity gradient. This suggests that young stars that host
planets are statistically richer in metals. This is also in line with the
previous studies that have shown giant planet-hosting stars are metal-rich
(Fischer & Valenti, 2005; Sousa et al., 2019; Narang et al., 2018; Swastik et
al., 2022, 2023), which supports the core-accretion model for planet
formation.

Figure 8: Fraction of stars hosting giant planets to small planets as a
function of stellar age using different models and input parameters. The data
is binned at an interval of 1 Gyr. Figure 9: Distribution of giant planets as
a function of stellar age. The colour bars represent planet mass and labels in
each plot window represent the isochrone models used. For all three cases,
spectroscopic $T_{eff}$ and G magnitude were used as input parameters.

###  4.3 Planet fraction vs stellar age

In order to understand the progression of planet formation in the galaxy, we
calculated the ratio of stars hosting giant planets to the stars hosting small
planets as a function of the stellar age. In all cases that we analyzed in
this paper, the ratio falls as a function of stellar age, as shown in figure
8. However, It is possible that the ratio is higher for younger stars ($<$5
Gyrs) due to selection biases and sensitivity limitations of current detection
techniques. However, for stars, $>$5 Gyrs, the finding is certainly not due to
any selection or observation biases. This is because regardless of stellar
type, the giant planets are easier to detect with any technique compared to
the small planets. This finding possibly suggests that it is more likely that
young stars have a higher ratio of giant planets to small planets when
compared to older stars. A small scatter at the age bin 10-12 Gyr in figure 8
is due to the small number of stars in that bin.

###  4.4 Hot Jupiters are younger

For a subsample of the giant planet population, we also looked at the
distribution of their orbital period as a function of the ages of their parent
stars obtained from different models using spectroscopic $T_{eff}$ and G-band
magnitude. A closer look at figure 9 shows that the majority of these stars
are young ($\leq$ 5 Gyrs) and they are hosting hot-Jupiters with orbital
periods of fewer than 10 days. This is evident from the clustering of points
around the lower-left sides of each plot in figure 9. For comparison, the
median age of hot-Jupiters (orbital period $\leq$ 10 days) and warm/cool
Jupiters (orbital period $>$ 10 days) is listed in table 3. On average, the
hot Jupiter hosting stars are $\sim$ 1.2 Gyr younger than the stars harboring
warm/cool Jupiters.

Specifically, we note that $\sim$ 70% of giant planets have age 5 Gyr or less,
indicating that overall the giant planetary systems are younger. Although the
high number of young hot Jupiter (lower-left region of each plot in figure 9)
can be possibly due to detection bias as hot Jupiters are easier to detect
(Kipping & Sandford, 2016), we find that the number of young warm/cool
Jupiters (top-left region of each plot in figure 9) is notably higher than old
hot-Jupiters (bottom-right region of each plot in figure 9), which is unlikely
due to any observational bias. In fact, it again points to the scenario of the
late onset of giant planet formation in the galaxy.

###  4.5 Isochrone vs. Asteroseismology and Chemical Clock Ages

We compared the stellar ages derived from MIST isochrone models with those
obtained using Asteroseismology and from $\alpha$ abundances (also known as
chemical clocks). For Asteroseismology we used the ages from Silva Aguirre et
al. (2015) and our analysis reveals that isochrone-based ages are slightly
overestimated when compared to asteroseismology-derived ages as shown in
Figure 10. Conversely, when comparing the ages obtained using the $\alpha$
abundances using the relationship in Delgado Mena et al. (2019), we find that
the ages are over-estimated for the younger stars ($\leq$ 4 Gyrs) while they
slightly underestimated for the older stars ($>$ 4 Gyrs) as shown in Figure
10.

The discrepancies observed between the ages derived from MIST isochrone
models, asteroseismology, and $\alpha$ abundances underscore the need for
methodological refinement across age-determination techniques. Specifically,
the tendency of asteroseismology to underestimate ages, compared to isochrone
models, and the accuracy of chemical clocks for stars of different ages
highlights the importance of cross-validating stellar ages to identify and
correct systematic biases.

##  5 Discussion

The results from section 4 indicate that the ages for the majority of stars
hosting planets are around 6 to 7 Gyrs depending on the model. Further, within
the planet’s population, the giant planet-hosting stars are younger when
compared to the stars hosting small planets. The reason why younger stars host
more giant planets compared to older stars could be understood from the
knowledge of the chemical composition and dust-to-gas ratio of their
circumstellar environment. A higher dust-to-gas ratio favors the formation of
giant planets (Kama et al., 2015) as: a) the solid dust grains act as the
building blocks for planetesimals and planetary cores. When the dust-to-gas
ratio is higher, it means there is a larger amount of solid material available
compared to the surrounding gas. This increased availability of material
provides a larger reservoir for the growth and accumulation of solid cores. b)
In a higher dust-to-gas ratio environment, collisions between dust grains
become more frequent. These collisions can lead to sticking and aggregation,
allowing the particles to grow in size. With a greater number of collisions
occurring, the growth process can proceed more rapidly, enabling the formation
of larger planetesimals and planetary cores over shorter timescales ($\sim$
5-10 Myrs). c) Larger solid cores have stronger gravitational forces, enabling
them to attract and capture more surrounding gas. When the dust-to-gas ratio
is higher, there is a denser population of dust grains that can coalesce into
larger planetesimals and cores. These more massive cores can then more
effectively accrete gas from the protoplanetary disk, rapidly increasing their
size and leading to the formation of gas giant planets (Emsenhuber et al.,
2021; Drazkowska et al., 2022).

Figure 10: Top: Comparison of ages obtained from stellar isochrones and from
$[\alpha$/Fe] abundances. Bottom: Comparison of ages obtained from stellar
isochrones and from Asteroseismology. The colour represents the effective
temperature of the star.

Together with the dust-to-gas ratio, the grain composition also plays a key
role in the formation of the planetary core (Dorschner et al., 1995; Fabian et
al., 2001; Draine, 2003). During the early stages of galaxy evolution, the
circumstellar disk is devoid of sufficient grains as the ISM is mostly
enriched with Type II supernovae. The dust grains consisted of mostly
silicates, magnesium, and other $\alpha$ but lacked iron or other heavier
elements, thus resulting in a lower dust-to-gas ratio ($<<$0.01). As the
galaxy evolved, the ISM was enriched with Fe-peak elements from Type Ia
supernovae, which in turn increased the dust-to-gas ratio and thus favouring
the formation of both small and giant planets (Nissen, 2015; Bedell et al.,
2018; Anders et al., 2018; Feuillet et al., 2018; Buder et al., 2019; Delgado
Mena et al., 2019). The presence of giant planets around metal-rich and young
stars also correlates with higher dust-to-gas ratio ($\sim$ 0.01) in young and
metal-rich protoplanetary disks which makes them conducive to the formation of
giant planets. This is also consistent with the core-accretion process
(Pollack et al., 1996; Matsuo et al., 2007) leading to the formation of giant
planet, where a solid core of $\sim$ 10-15 M⊕ needs to form quickly ($\sim$ 10
Myr) before the gas in the disk dissipates (Haisch et al., 2001; Kraus et al.,
2012), otherwise the resulting planet would end up rocky in nature. The
presence of a higher dust-to-gas ratio, thus promotes faster core formation,
thereby facilitating the formation of giant planets.

##  6 Summary and conclusions

The characteristics of exoplanets and their host stars exhibit a close
interdependence. In the present study, we focused on estimating the stellar
ages of planets orbiting main-sequence stars. To accomplish this, we used the
isochrone fitting technique to estimate the stellar ages of the planet-hosting
stars. Furthermore, we conducted an extensive analysis, exploring possible
correlations between stellar and planetary properties to gain insights into
their formation mechanisms. Our sample consisted of 2336 stars known to host
planets, detected through both transit and radial velocity (RV) methods. In
conclusions:

  1. 1.

We computed the stellar ages for the main sequence planets hosting stars using
the isochrone fitting technique. Since isochrone ages are highly model- and
input-parameter-dependent, we used several models and input parameters in
order to understand the systematic age differences obtained from different
isochrone models.

  2. 2.

We find that, even though individual age estimates and their distributions
vary depending on the choice of model or input parameters, the underlying
statistical trends remain unaffected.

  3. 3.

Our findings suggest that 70% to 85% of planets have stellar ages $<$ 7Gyrs
and most of the planets started forming after the ISM was enriched
sufficiently to form the cores of the planets.

  4. 4.

Our analysis reveals a distinct divergence in the ages of stars hosting small
planets compared to those hosting giant planets. Specifically, we observe a
statistically significant age difference, with stars hosting giant planets
being notably younger than those hosting small planets. This disparity
suggests that the formation of Jupiter-sized planets occurred at a later stage
in the galaxy’s evolution, specifically when the necessary dust-to-gas ratio
had reached a threshold, enabling the formation of a significant number of
giant planets. These findings corroborate the core-accretion theory of planet
formation.

  5. 5.

Among the giant-planet population, we find that the hot Jupiters are the
youngest, and they are the most recently formed systems in the context of
planet formation.

In conclusion, we have analyzed the stellar ages for a large number of
exoplanet-hosting stars, connecting the planet formation process to the ages
of their hosts. The fact that stars hosting giant planets are younger is
largely consistent with the chemical evolution of the galaxy. From the
observed trends between stellar ages and planet masses, we conclude that the
small planet formation started to rise after the ISM was sufficiently enriched
($\sim$ 6-7 Gyr), while the giant planet formation is much younger and has
started to form in large numbers only in the past $\sim$ 4-5 Gyr.

## ACKNOWLEDGMENTS

This work has made use of (a) the NASA Exoplanet Database, which is run by the
California Institute of Technology under an Exoplanet Exploration Program
contract with the National Aeronautics and Space Administration and (b) the
European Space Agency (ESA) space mission Gaia, the data from which were
processed by the Gaia Data Processing and Analysis Consortium (DPAC) (c) the
exoplanet.eu database maintained developed and maintained by the exoplanet
TEAM. C Swastik would also like to thank Athul Ratnakar for the insightful
discussion.

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