# Learning Optical Flow on ImageNet

Adrian Wälchli and Paolo Favaro
University of Bern, Switzerland
{adrian.waelchli<EMAIL_ADDRESS>
# Cheap Optical Flow Ground-Truth

Adrian Wälchli and Paolo Favaro
University of Bern, Switzerland
{adrian.waelchli<EMAIL_ADDRESS>
# Inexpensive Method to Obtain Ground-Truth Optical Flow

Adrian Wälchli and Paolo Favaro
University of Bern, Switzerland
{adrian.waelchli<EMAIL_ADDRESS>
# Ground-Truth Is All You Need: An Inexpensive Method to Obtain Ground-Truth
Optical Flow From Single Images

Adrian Wälchli and Paolo Favaro
University of Bern, Switzerland
{adrian.waelchli<EMAIL_ADDRESS>
# Optical Flow Ground-Truth Is All You Need

Adrian Wälchli and Paolo Favaro
University of Bern, Switzerland
{adrian.waelchli<EMAIL_ADDRESS>
# Learning Optical Flow from Single Images

Adrian Wälchli and Paolo Favaro
University of Bern, Switzerland
{adrian.waelchli<EMAIL_ADDRESS>
# On Building Optical Flow Datasets from Single Images

Adrian Wälchli and Paolo Favaro
University of Bern, Switzerland
{adrian.waelchli<EMAIL_ADDRESS>
# An Approach to Build Optical Flow Datasets from Single Images

Adrian Wälchli and Paolo Favaro
University of Bern, Switzerland
{adrian.waelchli<EMAIL_ADDRESS>
# Optical Flow Dataset Synthesis from Unpaired Images

Adrian Wälchli and Paolo Favaro
University of Bern, Switzerland
{adrian.waelchli<EMAIL_ADDRESS>
###### Abstract

The estimation of optical flow is an ambiguous task due to the lack of
correspondence at occlusions, shadows, reflections, lack of texture and
changes in illumination over time. Thus, unsupervised methods face major
challenges as they need to tune complex cost functions with several terms
designed to handle each of these sources of ambiguity. In contrast, supervised
methods avoid these challenges altogether by relying on explicit ground truth
optical flow obtained directly from synthetic or real data. In the case of
synthetic data, the ground truth provides an exact and explicit description of
what optical flow to assign to a given scene. However, the domain gap between
synthetic data and real data often limits the ability of a trained network to
generalize. In the case of real data, the ground truth is obtained through
multiple sensors and additional data processing, which might introduce
persistent errors and contaminate it. As a solution to these issues, we
introduce a novel method to build a training set of pseudo-real images that
can be used to train optical flow in a supervised manner. Our dataset uses two
unpaired frames from real data and creates pairs of frames by simulating
random warps, occlusions with super-pixels, shadows and illumination changes,
and associates them to their corresponding exact optical flow. We thus obtain
the benefit of directly training on real data while having access to an exact
ground truth. Training with our datasets on the Sintel and KITTI benchmarks is
straightforward and yields models on par or with state of the art performance
compared to much more sophisticated training approaches.

##  1 Introduction

Optical flow estimation is the task of associating pixel locations across two
consecutive frames in a video. More specifically, given two frames, the
objective is to assign a shift at each pixel of the first frame, which
corresponds to the location that that pixel has in the second frame. However,
determining such correspondence is challenging due to ambiguities caused, for
example, by occlusions, shadows, reflections, lack of texture and changes in
illumination over time.

This task has been originally formulated as a per-image-pair energy
optimization [12] and much of the effort has been devoted to designing
regularization terms that would handle the ambiguities mentioned above and
also capture some prior knowledge about the optical flow (_e.g_., smoothness
and discontinuities). More recently, deep learning methods have allowed to
train a single neural network model to estimate optical flow for any input
image pair, with a remarkable improvement in terms of accuracy and estimation
efficiency. The natural evolution of the original approach by Horn and Schunk
[12] has led to the unsupervised methods for optical flow [17, 18, 21, 22, 25,
33]. In fact, also these methods formulate a loss function for training that
consists of a range of terms, each addressing one of the key ambiguities in
the optical flow estimation task. The main difficulty in using and building on
these methods is that they require a careful simultaneous tuning of several
hyperparameters, and the number of these parameters grows with the number of
challenges that we desire to address.

Figure 1:  From a single image, we create a pair of frames and the
corresponding exact optical flow ground truth by warping and displacing
superpixels.

An alternative approach is supervised learning, which consists of training a
network on data where the optical flow corresponding to image pairs is
explicitly given. This solution moves the problem of designing and tuning
multiple energy terms to the design of training data. Thanks to supervised
learning the field of Computer Vision has made great progress in image
classification, segmentation and object detection. However, while the manual
annotation for ImageNet [6] was expensive and time-consuming, but feasible,
the manual definition of accurate ground truth optical flow for pairs of video
frames is unmanageable and error-prone. One effort to automate the definition
of ground truth optical flow is to use alternative sensors to measure it [11].
However, real measures are often noisy and may be sparse. Thus, post-
processing of measured optical flow is needed, but often introduces error
bias. To avoid these limitations, synthetic datasets have been introduced,
such as FlyingChairs [7], which consists of 3D chairs against a flat
background texture, FlyingThings [24], which uses a wider range of object
classes than just chairs, and MPI-Sintel [4], which is a 3D animated short
film that comes with optical flow annotations directly extracted from the
rendering pipeline. Because these datasets are synthetic, the optical flow
they provide is exact. However, synthetic and real data have also a domain gap
and a network trained on one dataset might yield unpredictable outcomes on the
other dataset.

In this work, we present a simple, flexible and inexpensive pipeline to
generate as much optical flow ground truth data as needed and on the image
domain the system is targeted for. Our data preparation requires no per-sample
manual intervention and it only requires a set of (even non consecutive)
images taken, for example, from a video or an unordered photo collection such
as ImageNet [6]. The key idea is to take a single frame and to create two
consecutive frames such that they share the texture of the given frame (and
possibly some texture from another image), which provides both the foreground
and the background. We first segment the single frame in superpixels, randomly
choose a cluster of superpixels, apply a global translation and warping and
then superimpose them to another part of the image. This operation is also
repeated multiple times to simulate multiple occlusions. Moreover, the
background (the unselected superpixels) are also warped in a random fashion.
We show that this simple method for augmenting single images with optical flow
ground truth is powerful enough to compete with state of the art methods that
involve training on several synthetic and real datasets (see Figure 1).

Our contributions are summarized as follows:

  * •

We introduce a general data pre-processing pipeline that is able to generate
unlimited, diverse image pairs with ground truth optical flow in an
unsupervised, inexpensive way;

  * •

The data we generate is composed of realistic texture and occlusion
boundaries, while the motions are synthetic;

  * •

We show the role of texture in optical flow estimation and we find that our
data generalizes best on the datasets used to build our synthetic training
set.

##  2 Prior Work

###  2.1 Supervised Flow Estimation

The first end-to-end learning-based optical flow works [7, 13, 16] were
inspired by the success of convolutional neural networks (CNNs) in per-pixel
prediction tasks such as semantic segmentation [23, 29] or depth estimation
[9]. FlowNet [7] and all subsequent works take two or more consecutive video
frames as input and estimate the dense forward flow between them. These models
are trained end-to-end with supervision through synthetic data and are fine-
tuned on sparsely annotated real world videos [11]. Several works have made
architectural improvements to boost performance, _e.g_., coarse-to-fine [27,
30], efficient cost volume computation [30, 34] and warping in feature space
[13]. Recently RAFT [31] achieved state of the art performance with impressive
numbers on all popular benchmarks. They introduce a new architecture based on
GRU [5] that performs progressive updates to a single flow estimate at
multiple scales, and, in addition, the iterative approach enables an efficient
correlation cost volume computation at full spatial resolution between all
pairs of pixels. Despite the impressive advancements in the supervised domain,
these methods still require extensive pretraining with synthetic data [7, 24]
and need to be fine-tuned on sparse real-world data [11], which is not readily
available.

###  2.2 Unsupervised Flow Estimation

Unsupervised methods try to avoid altogether the use of ground truth training
data. Early works focused on minimizing the photometric error between the
first frame and the second frame warped by the estimated flow [18]. Together
with a smoothing regularizer for the flow, this method is very effective in
learning accurate predictions in non-occluded regions, but fails when the
brightness-constancy constraint is not satisfied, _e.g_., at occlusion
boundaries across specular surfaces. Subsequent works improved these
shortcomings by excluding the pixels in occluded regions from the loss using a
mask obtained by forward warping [33] or a forward-backward consistency check
[25]. Janai _et al_. [17] include multiple frames for occlusion reasoning to
obtain sharper flow at boundaries. Instead of excluding occluded pixels from
the optimization, Liu _et al_. [21] simulate the optical flow at image
boundaries by cropping the images and imposing the flow predicted by a teacher
model onto the student model. [22] take this idea a step further and
hallucinate occlusions anywhere in the image with superpixels [1]. The
advantage of these distillation methods is that the flow estimate for occluded
target pixels can be included in the optimization objective, but they rely on
a strong teacher model. Inspired by the distillation methods, ARFlow [20]
transforms (augments) a secondary pair of frames with a transformation and
enforces the estimated optical flows to be consistent (equivariant) with the
transformation. With only a handful of basic transformations such as cropping,
zoom, affine and thin plate spline warps, and a model that has less than 2.5M
trainable parameters, they achieve the state of the art performance among
unsupervised methods.

##  3 Generating an Optical Flow Dataset

(a)

(b)

(c)

(d)

Figure 2:  Superpixel extraction for occlusion simulation. (a) template image;
(b) auxilliary image for background replacement; (c) superpixel segmentation
of template; (d) selected superpixels for foreground motion.

Since our method follows the training procedure of supervised optical flow
approaches, we present and discuss in detail only the preparation of the
pseudo-synthetic dataset. For training, we use the state of the art neural
network RAFT [31].

We illustrate our data preparation pipeline in Figure 2, 3, 4 and 5 using
images from the Sintel [4] movie for clarity, though our method applies to all
kind of image data. In the following paragraphs we describe our method of
synthesizing motion using two random unrelated images. The idea is to segment
the image into parts and then shift and deform them independently on new
layers to simulate occlusions and their corresponding exact optical flow. It
is similar to what FlyingChairs [7] and FlyingThings [24] do but less
expensive and textures are specific to the testing domain. In our method all
textures for the foreground come from a single image and no additional assets
like 3D geometry or shaders are required.

Given an image $I\in\mathbb{R}^{H\times W\times C}$, we compute superpixel
segmentation maps $S_{k}\in\mathbb{N}^{H\times W}$ for $k=1,\dots,K$ from
coarse to fine, where $S_{k}(x)=i$ if the pixel $x$ belongs to superpixel $i$.
We use SLIC [1] over other segmentation methods, because it is general,
applicable to all domains and does not require pretraining. Because all
superpixels from a single scale $k$ are roughly the same size, we collect
neighboring superpixels and group them together until a target size $m$ is
reached. We denote the pixels that belong to the chosen group with a mask
$M\in\\{0,1\\}^{H\times W}$. The superpixel segmentation and occluder mask $M$
is illustrated in Figure 2.

(a)

(b)

(c)

(d)

Figure 3:  Background motion through warping. (a) Superpixel source texture
replaced with auxiliary image; (b-c) Background motion through thin plate
warping and corresponding forward flow (d).

Copying the pixels in $M$ to a different location in the same image would
introduce an ambiguity in the motion between the new frames due to the
duplication of texture. We prevent this by painting the source region from
where the superpixel was extracted with texture from an auxiliary image
$A\in\mathbb{R}^{H\times W\times C}$ (see Figure 2(b)), _i.e_.,

$I^{\prime}=(1-M)\odot I+M\odot A$ (1)

where $\odot$ is the element-wise multiplication. The result $I^{\prime}$ is
shown in Figure 3(a). Next, the single image $I^{\prime}$ is transformed twice
with a Thin Plate Spline (TPS) warping [3, 8] to simulate a smooth background
motion between two frames. By applying the TPS transformation twice we make
sure the two resulting images have an equal amount of distortion and
interpolation artifacts. The extent of the deformation is limited by the
_control points_ , which in our case are sampled by constructing a regular
grid of size $L\times L$ and adding uniform noise to each grid point. Applying
the spline once to the regular grid of pixel coordinates
$\\{x_{i}\\}\subset\mathbb{R}^{H\times W}$ yields a deformed grid
$\\{\phi(x_{i})\\}$ from which we can derive the per-pixel motion as
$\phi(x_{i})-x_{i}$. Thus, we perform two image warpings with $\phi_{1}$ and
$\phi_{0}$ in sequence, add a global shift $d$ to all coordinates and obtain a
final background image pair

$\displaystyle B_{1}(x)$ $\displaystyle=I^{\prime}(\phi_{1}(x)),$ (2)
$\displaystyle B_{0}(x)$ $\displaystyle=B_{1}(\phi_{0}(x)+d).$ (3)

We use bilinear interpolation at non-integer locations. The optical flow from
image $B_{0}$ to image $B_{1}$ is

$\omega^{B}(x)=\phi_{0}(x)-x+d.$ (4)

The pair $(B_{0},B_{1})$ and corresponding optical flow $\omega^{B}$ is shown
in Figure 3.

In the next stage, we extract the foreground texture $F=M\odot I$ from the
original, undistorted image and apply two new TPS warps $\psi_{1}$ and
$\psi_{0}$ analogous to eq. (2) and (3)

$\displaystyle F_{1}(x)$ $\displaystyle=F(\psi_{1}(x))$ (5) $\displaystyle
F_{0}(x)$ $\displaystyle=F_{1}(\psi_{0}(x))$ (6)

with the new foreground flow

$\omega^{F}(x)=\psi_{0}(x)-x.$ (7)

$F_{0}$ and $F_{1}$ with the corresponding optical flow is shown in Figure 4.
The same warps $\psi_{0}$ and $\psi_{1}$ are applied to the mask $M$ and we
get two new masks $M_{0}$ and $M_{1}$.

(a)

(b)

(c)

(d)

Figure 4:  Foreground motion through warping. (a) Selected superpixel from
original image in Figure 2. (b-c) Foreground deformation through thin plate
spline and corresponding optical flow (d).

Finally, we combine the foreground with the background by placing the
foreground $F_{0}$ and $F_{1}$ at positions $p_{0}$ and $p_{1}$, respectively,
_i.e_.,

$I_{i}(x)=M_{i}(x-p_{i})F_{i}(x-p_{i})+(1-M_{i}(x-p_{i}))B_{i}(x),$ (8)

for $i=1,2$ and $\delta=p_{1}-p_{0}$ is the relative global motion of the
foreground from the first to the second image. The optical flow is composed by

$\omega(x)=M_{0}(x-p_{0})\omega^{F}(x-p_{0})+(1-M_{0}(x-p_{0}))\omega^{B}(x).$
(9)

In Figure 5 we show an example with $p_{0}=p_{1}$, _i.e_. no global foreground
motion ($|\delta|=0$) and an example with $|\delta|>0$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5:  Composition of background and foreground motion. Left column
(a,c,e): Background from Figure 3 with foreground from Figure 4 combined.
Right column: Additional global shift added to foreground.

The above procedure can easily be adapted to generate a pair of frames with
several (overlapping) occlusions. Instead of choosing a single group of
superpixels, we sample $N$ distinct groups and each time draw $k$ and $m$ from
a uniform distribution, yielding individual mask $M_{1},\dots,M_{N}$. All
masked regions get inpainted collectively to form a single background and are
then warped using eq. (2), (3) and (4). The different foreground superpixels
and their motions get added in order of depth following eq. (5) - (9), with
the last layer on top. Example data is shown in Figure 6.

With our simple occlusion synthesis, we are able to impose the exact optical
flow in ambiguous regions where a high-level of reasoning based on the
environment is required to recover the motion, _e.g_., at the occlusion
boundaries or on surfaces with uniform or repeated textures.

###  3.1 Additional Augmentations

We add additional augmentations as a means to increase the amount of data, and
also for the purpose of robustness. With probability $p_{\text{s}}$, we
convert the foreground layer to a shadow, which we model as a semi-transparent
black texture with the shape of the superpixel. All warpings and translations
remain the same, except that we set $\omega^{F}\coloneqq\omega^{B}$ in eq.
(9), _i.e_., we impose invariance to transparent objects and their motion.
Following prior works [16, 31], we add color jittering to simulate global
illumination changes, scaling transforms to both views, random horizontal and
vertical flips, random crop, and random erasing in a rectangle in the second
view. The optical flow is transformed as well according to the augmentation
performed on the images.

Figure 6:  Examples of our training data (first frame, second frame and
optical flow) built on either Sintel [4] (top) or KITTI [11] (bottom). More
examples can be found in the supplementary material.

##  4 Experiments

We conduct all experiments on the standard benchmark datasets Sintel and KITTI
and take the raw, unlabelled data for our unsupervised data preprocessing.
Quantitative performance is measured by the average End-Point Error (EPE,
Euclidean distance between predicted- and ground truth flow) and the F1-all
outlier ratio (percentage of EPE that are larger than 3 pixels or not within
5% of the ground truth). We use the state of the art RAFT [31] as a baseline
and train with their code and exact settings for fair comparison. The only
difference is the data.

#### Implementation.

We precompute two superpixel segmentation maps ($K=2$) per image with 100 and
1000 components. In each image, we select between $N=8$ and $N=14$ superpixel
groups for foreground motion. Each superpixel group is chosen according to the
maximum size $M\sim\mathcal{U}\\{6000,50000\\}$ and for each deformation
(foreground and background) we sample the grid size of TPS control points
$L\sim\mathcal{U}\\{3,5\\}$. The auxiliary image is randomly picked from the
whole dataset. For the foreground and background motion, we limit the extent
of the warping by sampling the displacement of the individual control points
from the normal distribution $\mathcal{N}(0,25^{2})$. Similarly, the global
background translations $d$ and the foreground translations $\delta$ are
sampled from $\mathcal{N}(0,30^{2})$. The probability of a foreground becoming
a shadow is $p_{\text{s}}=0.2$ and the transparency is also chosen randomly
according to $\mathcal{U}(0.4,0.6)$. Our method is implemented in PyTorch [10,
26]. Code and data are available online.111Code will be released upon
publication.

Data | Method | SINTEL (train) | SINTEL (test) | KITTI 12 | KITTI 15 |
---|---|---|---|---|---|---
clean | final | clean | final | train | test | train | test (F1) |
SY / Kraw | UnFlow-CSS [25] | - | (7.91) | - | 10.22 | 3.29 | - | 8.10 | - |
C + S / K | OccAwareFlow [33] | (4.03) | (5.95) | 7.95 | 9.15 | - | - | - | - |
R + Sraw/ Kraw | MFOccFlow$\dagger$ [17] | (3.89) | (5.52) | 7.23 | 8.81 | - | - | - | - |
Kraw | BridgeDepthFlow§ [19] | - | - | - | - | 2.56 | - | 7.02 | - |
Kraw | CCFlow§ [28] | - | - | - | - | - | - | 5.66 | 25.27% |
Kraw | UnOS-stereo§ [32] | - | - | - | - | 1.64 | 1.80 | 5.58 | 18.00% |
C + S / K | DDFlow [21] | (2.92) | (3.98) | 6.18 | 7.40 | 2.35 | 3.00 | 5.72 | 14.29% |
S + Sraw/ K | SelFlow$\dagger$ [22] | (2.88) | (3.87) | 6.56 | 6.57 | 1.69 | 2.20 | 4.84 | 14.19% |
Sraw/ Kraw | ARFlow [20] | 2.79 | 3.73 | 4.78 | 5.89 | 1.44 | 1.80 | 2.85 | 11.80% |
Sraw/ Kraw |  RAFT-Ours “C” | 2.00 | 3.49 | - | - | 2.42 | - | 6.17 | - |
Sraw/ Kraw |  RAFT-Ours “C + T” | 1.93 | 3.47 | 4.06 | 5.97 | 2.39 | 2.90 | 5.99 | 15.75% |
Kraw/ Sraw |  RAFT-Ours “C” | 2.60 | 4.15 | - | - | 2.76 | - | 6.41 | - |
Kraw/ Sraw |  RAFT-Ours “C + T” | 2.14 | 3.82 | - | - | 2.57 | - | 6.66 | - |
Table 1:  Comparison with unsupervised methods on MPI-Sintel [4] and KITTI
[11] benchmarks. All numbers are AEPE (average end-point-error) except for the
KITTI test set where outlier ratio F1 is reported. Numbers enclosed by
parentheses mean that the training is performed on the same dataset. The
results for entries marked as (-) were not reported by the original authors.
Methods marked with $\dagger$ are using more than two frames. Methods trained
with geometrical constraints are marked with §.

#### Sintel.

This is an animated movie with $\sim$21k video frames from which a subset of
$\sim$1k pairs are annotated with ground truth optical flow and occlusion
masks. There exist two versions, a clean and final pass, where the latter
contains more realistic motion blur and illumination. The clean and final
subsets are only used for evaluation. We download the raw version of the movie
rendered at a resolution of $1280\times 544$ pixels and manually discard
roughly 1.6k frames that belong to scene transitions, intro and credits roll.
We label this dataset as Sraw in our tables.

#### KITTI.

The KITTI dataset contains a large amount of raw stereo image data. There are
a total of $\sim$49k images with dimensions $1242\times 375$ pixels. We take
all images from the left view and apply our unsupervised data augmentation to
all video frames. We refer to this dataset as Kraw in our tables. The
evaluation is done on the official training and test splits of the KITTI data
(denoted as K), where the training split has sparse optical flow labels.
Samples of our KITTI data are shown in Figure 6.

#### Comparison with Unsupervised Methods.

Since our data is generated in an unsupervised way, we compare against other
unsupervised methods in Table 1. RAFT [31] was originally published as a
supervised method trained in stages on FlyingChairs [7], FlyingThings [24],
Sintel [4] and KITTI [11] in this order for 100k iterations per stage. We
retrain their model with our data following the same training schedules and
augmentation settings and we evaluate after each stage. The results are shown
in Table 1. We label our models evaluated after the FlyingChairs stage as “C”
and the models evaluated at the FlyingThings stage with “C + T”. The results
show that RAFT trained with our Sintel raw data (Sraw) outperforms or is on
par with prior unsupervised works when compared on the Sintel [4] datasets.
When trained with KITTI raw data (Kraw), we are worse than the state of the
art and compete with SelFlow [22] and DDFlow [21].

#### Comparison with Supervised Methods.

Since our dataset is synthetic, we can compete and compare with supervised
methods trained on Chairs [7] and Things [24] and evaluate the capability to
generalize on unseen data from Sintel [4] and KITTI [11]. We show RAFT trained
with our data in Table 2. Following Teed _et al_. [31], we group methods by
the dataset(s) they were trained on. We find that RAFT trained on the Chairs
and Things (C + T) datasets achieves a better performance than us on Sintel
and KITTI except for the outlier ratio F1-all on the KITTI training split
where we achieve 16.03%. We find that the additional 100k iterations of
training from our “C” training to our “C + T“ training does not result in a
significant reduction in EPE, hence this shows that RAFT is fitting our data
already well in the “C” stage and does not benefit as much when training for
longer. While our dataset combined with the state of the art RAFT yields
competitive results, the variability of our data seems to be the major factor
of influence.

We continue the training with the ground truth image- and optical flow data
from Sintel [4] and KITTI [11] to compare with other methods in the finetuned
regime. We again follow the training schedule and settings from RAFT [31] with
the exception that we do not include the Things dataset and only finetune on
either the Sintel or the KITTI dataset. To prevent overfitting, we only train
for 50k iterations in contrast to RAFT-ft which was finetuned for 100k
iterations. Results are shown in Table 2.

Data | Method | SINTEL (train) | SINTEL (test) | KITTI 15 (train) | KITTI 15 (test) |
---|---|---|---|---|---|---
clean | final | clean | final | EPE | F1 [%] | F1 [%] |
C | RAFT [31] | 2.26 | 4.51 | - | - | 9.85 | 37.56 | - |
C | RAFT [31] _reproduced_ | 2.32 | 4.68 | - | - | 10.95 | 40.12 | - |
Sraw/ Kraw |  RAFT-Ours “C” | 2.00 | 3.49 | - | - | 6.17 | 16.77 | - |
Kraw/ Sraw |  RAFT-Ours “C” | 2.60 | 4.15 | - | - | 6.41 | 17.22 | - |
C + T | LiteFlowNet [13] | 2.48 | 4.04 | - | - | 10.39 | 28.5 | - |
C + T | PWC-Net [30] | 2.55 | 3.93 | - | - | 10.35 | 33.7 | - |
C + T | LiteFlowNet2 [14] | 2.24 | 3.78 | - | - | 8.97 | 25.9 | - |
C + T | VCN [34] | 2.21 | 3.68 | - | - | 8.36 | 25.1 | - |
C + T | MaskFlowNet [36] | 2.25 | 3.61 | - | - | - | 23.1 | - |
C + T | FlowNet2 [16] | 2.02 | $3.54^{1}$ | 3.96 | 6.02 | 10.08 | 30.0 | - |
C + T | RAFT [31] | 1.43 | 2.71 | - | - | 5.04 | 17.4 | - |
C + T | RAFT [31] _reproduced_ | 1.52 | 2.68 | - | - | 5.08 | 17.58 | - |
Sraw/ Kraw |  RAFT-Ours “C + T” | 1.93 | 3.47 | 4.06 | 5.97 | 5.99 | 16.03 | 15.75 |
Kraw/ Sraw |  RAFT-Ours “C + T” | 2.14 | 3.82 | - | - | 6.66 | 17.05 | - |
C + T + S/K | FlowNet2-ft [16] | (1.45) | (2.01) | 4.16 | 5.74 | (2.30) | (6.8) | 11.48 |
C + T + S/K | HD3 [35] | (1.87) | (1.17) | 4.79 | 4.67 | (1.31) | (4.1) | 6.55 |
C + T + S/K | IRR-PWC [15] | (1.92) | (2.51) | 3.84 | 4.58 | (1.63) | (5.3) | 7.65 |
C + T + S/K | ScopeFlow [2] | - | - | 3.59 | 4.10 | - | - | 6.82 |
C + T + S/K | RAFT-ft [31] | (0.77) | (1.20) | 2.08 | 3.41 | (0.64) | (1.5) | 5.27 |
Sraw\+ S / Kraw\+ K | RAFT-Ours-ft | (0.75) | (1.30) | 2.43 | 3.29 | (0.69) | (1.84) | - |
Table 2:  Comparison with supervised methods on MPI-Sintel [4] and KITTI [11]
benchmarks. All numbers are AEPE (average end-point-error) except for KITTI
where both EPE and outlier ratio F1 is reported. Numbers enclosed by
parentheses mean that the training is performed on the same dataset. The
results for entires marked as (-) were not reported by the original authors.

#### Cross Dataset Evaluation.

While the motions in our data are synthetic, the textures come from the
dataset we evaluate on, _i.e_., so far either Sintel or KITTI. To study the
impact of texture in optical flow performance, we take the model trained with
our KITTI and evaluate on the Sintel benchmark. We do the same also in
reverse, _i.e_., we take the model trained with our Sintel data and evaluate
it on the KITTI benchmark dataset. The results of these ablations are listed
in the rows below our best results in both Table 1 and 2. We observe that the
EPE in this cross-evaluation is consistently higher. Since the distribution of
motions is the same between the two datasets and the only real difference is
the texture in the data, these results show, to a limited degree, that the
texture does in fact have an influence on the ability to generalize on unseen
data.

Figure 7:  Qualitative results. We evaluate RAFT [31] trained on C + T
(middle) and RAFT trained on our data (right) on the Sintel [4] (top two rows)
and KITTI [11] test set (bottom two rows).

#### Qualitative Results.

In addition to the quantitative evaluations, we also show qualitative flow
predictions of RAFT trained with our data vs. RAFT trained on Chairs and
Things in Figure 7. The input images are from the KITTI [11] and Sintel [4]
test split and we only show the first image. Most notably, the flow
predictions from our method and also the predictions of RAFT [31] contain the
shadow as part of the moving car. By the strict definition of optical flow and
photometric consistency, this estimate for shadows is correct. However, the
ground truth data in these regions does not agree and instead assigns the
motion of the ground plane to the region where we predict the motion of the
shadow. It is clear that this discrepancy between real optical flow and sensor
data from KITTI [11] is a major problem for evaluating the performance of
optical flow models as this systematic error cannot be reduced even if a
method is qualitatively better in these ambiguous regions.

#### Ablation Study.

To study the effect of the various hyperparameters in the flow synthesis, we
generate several smaller datasets of 5k samples each using the MPI-Sintel
images. in Table 3 shows the effect on EPE for different settings of $K$
(number of coarse-to-fine segmentation maps to choose from), $L$ (grid size of
TPS warping), $M$ (max. size of a superpixel group in pixels), and $N$ (number
of occlusions). We find that a greater value for $N$ has the highest impact.
Choosing a larger grid size $L$ and using two segmentation maps is beneficial
too, however adding a third segmentation map does not improve the EPE further.

##  5 Conclusions and Future Work

We have presented a novel, unsupervised method to synthesize an unlimited
amount of exact ground truth data for optical flow. Our approach effectively
leverages all available texture in a single image to create realistic
occlusions and deformations across several layers of motion. We compared
against supervised as well as unsupervised methods and showed how our dataset
can reduce the reliance on multiple datasets and schedules for pretraining
such as FlyingChairs and FlyingThings. While we made specific choices in
regard to the segmentation method or the warping model, our formulation is
generic and is compatible with arbitrarily complex segmentation- and motion
models. We believe that introducing even more variability to the motions in
our dataset is a promising direction for future work that could address the
generalization gap where we currently fall short or are on par with state of
the art methods.

K | L | N | M | clean | final
---|---|---|---|---|---
1 | 4 | 10 | 25k | 2.77 | 4.10
1 | 4 | 15 | 25k | 2.39 | 3.57
1 | 4 | 5 | 25k | 2.61 | 3.85
1 | 6 | 10 | 25k | 2.58 | 4.00
1 | 2 | 10 | 25k | 2.65 | 4.31
1 | 4 | 10 | 50k | 2.47 | 3.83
1 | 4 | 10 | 5k | 3.18 | 4.35
2 | 4 | 10 | 25k | 2.42 | 3.72
3 | 4 | 10 | 25k | 2.44 | 3.72
Table 3:  Impact of dataset hyperparameters. All experiments use textures from
MPI-Sintel [4]. All numbers are AEPE (average end-point-error). Parameters in
bold mark the difference to the baseline (first row).

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# Preparation of thermal coherent environments for quantum coherence
protection

Asghar Ullah<EMAIL_ADDRESS>Department of Physics, Koç University, 34450
Sarıyer, Istanbul, Türkiye  M. Tahir Naseem<EMAIL_ADDRESS>Department of
Physics, Koç University, 34450 Sarıyer, Istanbul, Türkiye  Faculty of
Engineering Science, Ghulam Ishaq Khan Institute of Engineering Sciences and
Technology,
Topi 23640, Khyber Pakhtunkhwa, Pakistan  Özgür E. Müstecaplıoğlu
<EMAIL_ADDRESS>Department of Physics, Koç University, 34450 Sarıyer,
Istanbul, Türkiye  TÜBITAK Research Institute for Fundamental Sciences, 41470
Gebze, Türkiye

###### Abstract

The unavoidable interaction between thermal environments and quantum systems
leads to the degradation of the quantum features, which can be fought against
by engineered environments. In particular, preparing a thermal coherent
environment can be promising for prolonging quantum properties relative to
incoherent baths. We propose that a thermal coherent state (TCS) can be
realized by using an ancilla qubit to thermally and longitudinally driven
resonator modes. Using the master equation approach to describe the open
system dynamics, we obtain the steady-state solution of the master equation
for the qubit and resonator. Remarkably, the state of the resonator is a TCS,
while the ancilla qubit remains thermal. Furthermore, we study the second-
order correlation coefficient and photon number statistics to validate its
quantum properties. To sum up, we also investigate a mechanism for generating
quantum coherence based on a hybrid system composed of two-level systems and
resonator to claim that an ancilla-assisted engineered thermal coherent bath
prolongs the coherence lifetimes of qubits. Our results may provide a
promising direction for preparing and practically implementing TCSs and
environments for quantum science and technology.

##  I Introduction

Quantum coherence is a fundamental concept in quantum mechanics related to the
superposition of quantum states. Quantum coherence is a crucial resource that
underpins many potential advantages of quantum technologies over classical
Madsen  _et al._ (2012); Filip (2009); Marek  _et al._ (2011); Grangier  _et
al._ (1987); Takeda  _et al._ (2013); Yokoyama  _et al._ (2013); Ourjoumtsev
_et al._ (2006). Quantum coherence is utilized as a resource in quantum
thermodynamics to enhance the performance of several tasks, such as quantum
information processing Bouwmeester and Zeilinger (2000); Braunstein and van
Loock (2005a), quantum sensing and metrology Agarwal (2012); TAY (2016); Ou
(2007); Pezzè  _et al._ (2018); Podhora  _et al._ (2022); Ullah  _et al._
(2023), quantum computing Steane (1998), and energy harvesting Caruso  _et
al._ (2010); Mohseni  _et al._ (2008). In the context of thermodynamics,
quantum coherence was shown to be a promising candidate for the optimal
charging of quantum batteries Gyhm  _et al._ (2022); Seah  _et al._ (2021);
Kamin  _et al._ (2020); Andolina  _et al._ (2019); Ferraro  _et al._ (2018);
Campaioli  _et al._ (2017); Alicki and Fannes (2013), enhancing the efficiency
of heat engines and thermal machines Ptaszyński (2018); Dorfman  _et al._
(2018); Dağ  _et al._ (2016); Çakmak  _et al._ (2017); Hammam  _et al._
(2022).

Non-classical states of light are the cornerstone of quantum mechanics, which
provide important resources for quantum information sciences, such as fast
quantum computation, secure quantum communication, and precise quantum
metrology Pan  _et al._ (2012); Braunstein and van Loock (2005b); Reid  _et
al._ (2009). Several techniques have been proposed for the preparation of non-
classical states of light. For example, multiphoton non-classical states can
be generated by single and two-photon cavity-induced transparency Zhao  _et
al._ (2019). Squeezed states can be generated by using single-photon source
Filip and Zapletal (2014), in hyperradiance regime, when two qubits are placed
in a cavity with opposite coupling coefficients to the cavity Li  _et al._
(2022), oscillator-qubit gates coupled through rabi interactions Hastrup  _et
al._ (2021), and cavity quantum electrodynamics (QED) Eickbusch  _et al._
(2022); Hastrup and Andersen (2022); Huang  _et al._ (2020). Manipulating the
motion of trapped ions also results in the generation of quantum coherent
states (QCSs) Obšil  _et al._ (2018); Zhong  _et al._ (2018); Zou  _et al._
(2002). Preparing QCSs in bosonic systems is done through quantum state
engineering, which involves designing the interactions between the bath and
quantum systems so that they exhibit the desired quantum properties Guarnieri
_et al._ (2018).

Coherent states are minimal uncertain states that describe the maximum
coherence and classical behavior Glauber (1963) have played an important role
in various fields of physics Klauder and Skagerstam (1985); Zhang  _et al._
(1990). Superpositions of coherent states can be obtained in the resonant
Jynes-Cummings model (JCM) if the cavity field is initially coherent with a
large mean photon number Gea-Banacloche (1990). Thermal coherent state (TCS)
generalizes the notion of coherent state to finite temperatures. By
definition, it is a state that is produced by a thermalization operation
applied to a displaced (coherent) state Oz-Vogt  _et al._ (1991); Du  _et al._
(2017); Bishop and Vourdas (1987); Marian and Marian (2007); guo Meng  _et
al._ (2018); Vourdas and Bishop (1994). A few physical models have been
proposed to prepare TCSs; for example, a superposition of displaced thermal
states can be obtained via the resonant interaction of a two-level atom with a
cavity field Zheng (2007). The TCS can be generated by quantum state
engineering using an on-off detector scheme Magaña-Loaiza  _et al._ (2019) by
pumping a nonlinear crystal with the help of a pump beam. An incident signal
is a thermal field mixed with a vacuum field that results in conditional TCS.
A probabilistic quantum amplifier can also produce a TCS using a thermal noise
source and a photon number subtraction scheme Usuga  _et al._ (2010); Marek
and Filip (2010). TCSs have been extensively studied in thermofield theory
(TFD). The formalism of TFD permits a more comprehensive formulation of the
uncertainty relations, which include the effects of both quantum and thermal
fluctuations Bishop and Vourdas (2007); Mann  _et al._ (1989). The TCSs can be
particularly interesting as they saturate the generalized uncertainty
relations, as regular coherent states do for the Heisenberg uncertainty
relations. Recently, displaced thermal bath has been found to play a
significant role in studying steady-state entanglement between two nitrogen-
vacancy centers in a diamond host on an ultrathin yttrium iron garnet
stripUllah  _et al._ (2022).

In this paper, we propose a theoretical model for preparing a TCS. The
proposed model consists of a two-level system (qubit) coupled longitudinally
to a resonator similar to what can be found in electromechanical setups and
circuit QED Abari  _et al._ (2022); Chu  _et al._ (2017); Sletten  _et al._
(2019); Rouxinol  _et al._ (2016); Santos  _et al._ (2019); Wollack  _et al._
(2022). In addition; the qubit is also coupled to an external thermal bath.
Using the master equation approach, we derive the combined state of the qubit
and resonator. Notably, while the qubit remains in a thermal state, the final
state of the resonator is a TCS. To validate the coherence and thermal nature
of the state, we analyze the photon statistics of the state, including the
second-order correlation coefficient and photon number distribution. Finally,
we validate that an ancilla-assisted engineered thermal coherent bath prolongs
the coherence lifetimes of qubits. The TCS is prepared in a weak coupling
regime and at low temperatures, making it feasible and appealing for practical
quantum technologies.

The rest of the paper is organized as follows: In Sec. II, we present the
model system, and after diagonalizing the Hamiltonian, we write the master
equation of the system and obtain its steady-state solution for the qubit and
resonator. The results of TCS are presented in Sec. III. In Sec. IV, we
present a possible application of a thermal coherent bath that can enhance
qubits’ coherence lifetime with the help of an ancilla qubit. Finally, we
conclude the paper in Sec. V.

##  II Model description

We consider a two-level system (TLS), such as a qubit interacting
longitudinally with a single-mode bosonic resonator field. In addition, the
qubit is coupled to a thermal bath of temperature $T$ as depicted in Fig. 1.
The Hamiltonian of the system of interest is given by (we work in the units
where $\hbar=1=k_{B}$ throughout this study) LaHaye  _et al._ (2009); Naseem
and Müstecaplıoğlu (2022); Richer and DiVincenzo (2016); Didier  _et al._
(2015)

$\hat{H}=\frac{\omega_{q}}{2}\hat{\sigma}_{z}+\omega_{R}\hat{a}^{\dagger}\hat{a}+g\hat{\sigma}_{z}(\hat{a}+\hat{a}^{\dagger}),$
(1)

where g is the coupling strength between the qubit and resonator, $\omega_{q}$
and $\omega_{R}$ are the frequencies of the qubit and bosonic field,
respectively, and $\hat{\sigma}_{i}(i=1,2,3)$ are the Pauli operators of the
qubit. The creation(annihilation) operator of the bosonic field is represented
by $\hat{a}^{\dagger}(\hat{a})$. The longitudinal coupling between the
resonator and the qubit has been implemented in QED circuits to address
scalability problems and to achieve fast and high fidelity quantum
nondemolition qubit readout Richer and DiVincenzo (2016); Didier  _et al._
(2015). This coupling has also been realized for a superconducting flux qubit
coupled off resonantly to a transmission line resonator, offering advantages
such as enhanced qubit lifetimes, high-fidelity measurements, entanglement
over longer distances, and the generation of important microwave photon states
for quantum communication Blais  _et al._ (2004); Xiang  _et al._ (2013).
Another mechanism utilizes strain coupling between nitrogen-vacancy centers
and a diamond mechanical nanoresonator to enable long-range spin-spin
interactions, generating squeezed states and suppressing spin dephasing and
relaxation Bennett  _et al._ (2013). It is also utilized for reversible
information transfer between superconducting qubits and ultracold atoms using
a microwave resonator, enabling rapid quantum logic gates and long-term
storage with collective metastable states Petrosyan  _et al._ (2009). This
model has also been used to predict and describe phonon blockade in a
nanomechanical resonator coupled to a Cooper-pair box (i.e., a charge qubit)
for observing strong antibunching and sub-Poissonian phonon-number statistics
Wang  _et al._ (2016); LaHaye  _et al._ (2009).

In our setup, the qubit is coupled to a multimode bosonic bath which is
modeled as ensembles of harmonic oscillators and described by the Hamiltonian

$\hat{H}_{B}=\sum_{k}\omega_{k}\hat{b}^{\dagger}_{k}\hat{b}_{k},$ (2)

and the system bath interaction has the following form

$\hat{H}_{I}=\sum_{k}\gamma_{k}\hat{\sigma}_{z}(\hat{b}_{k}+\hat{b}^{\dagger}_{k}),$
(3)

where $\hat{b}(\hat{b}^{\dagger})$ is the annihilation (creation) bosonic
operator of the k-th bath modes with
$[\hat{b}_{k},\hat{b}^{\dagger}_{k^{\prime}}]=\delta_{k,k^{\prime}}$, and
$\gamma_{k}$ refers to the coupling strength between the k-th bath and qubit.
To derive the Master equation, we first diagonalize the Hamiltonian given in
Eq. (1) with the aid of the following unitary transformation

$\hat{\text{U}}=e^{-\alpha\hat{\sigma}_{z}(\hat{a}^{\dagger}-\hat{a})},$ (4)

where $\alpha:=g/\omega_{R}$. After performing the unitary transformation, the
diagonalized Hamiltonian takes the following form

$\tilde{H}=\hat{\text{U}}\hat{H}\hat{\text{U}}^{\dagger}=\frac{\omega_{q}}{2}\tilde{\sigma}^{+}\tilde{\sigma}^{-}+\omega_{R}\tilde{a}^{\dagger}\tilde{a}-\frac{g^{2}}{\omega_{R}}.$
(5)

Remarkably, the frequencies of the qubit and bosonic field modes are
unaffected by the longitudinal coupling. The transformed operators are given
as

$\tilde{\sigma}^{-}=\hat{\text{U}}\hat{\sigma}^{-}\hat{\text{U}}^{\dagger}=\hat{\sigma}^{-}e^{-\alpha(\hat{a}^{\dagger}-\hat{a})},$
(6)
$\tilde{a}=\hat{\text{U}}\hat{a}\hat{\text{U}}^{\dagger}=\hat{a}-\alpha\hat{\sigma}_{z}.$
(7)

Figure 1: Model description. We consider a TLS, for example, in the current
study, a qubit of frequency $\omega_{q}$ is coupled longitudinally to a
resonator of frequency $\omega_{R}$. In addition, the qubit is also coupled to
a thermal bath of temperature $T$. The system bath coupling is represented by
$g$. For simplicity, we assume that the resonator is at zero temperature, such
that $T_{R}=0$. We will observe the thermal effects of the bath in the
resonator via qubit.

The eigenenergies of the Hamiltonian in Eq. (5) are
$E_{n_{q},n_{R}}=\omega_{q}n_{q}/2+\omega_{R}n_{R}-g^{2}/\omega_{R}$, where
$n_{q}$ and $n_{R}$ are the photon numbers of qubit and resonator,
respectively. The master equation can be derived by transforming these
operators into the interaction picture followed by the standard Born-Markov,
and secular approximations Naseem  _et al._ (2018); Gelbwaser-Klimovsky and
Kurizki (2014), the resulting form of the master equation is written by

$\displaystyle\frac{d\tilde{\rho}}{dt}=$
$\displaystyle-i[\tilde{H},\tilde{\rho}]+G(\omega_{q})\mathcal{D}[\tilde{\sigma}^{-}]+G(-\omega_{q})\mathcal{D}[\tilde{\sigma}^{+}]$
(8)
$\displaystyle+\alpha^{2}\\{G(\omega_{q}-\omega_{R})\mathcal{D}[\tilde{\sigma}^{-}\tilde{a}^{\dagger}]$
$\displaystyle+G(-\omega_{q}+\omega_{R})\mathcal{D}[\tilde{\sigma}^{+}\tilde{a}]$
$\displaystyle+G(\omega_{q}+\omega_{R})\mathcal{D}[\tilde{\sigma}^{-}\tilde{a}]$
$\displaystyle
G(-\omega_{q}-\omega_{R})\mathcal{D}[\tilde{\sigma}^{+}\tilde{a}^{\dagger}]\\}.$

Here $\mathcal{D}[\hat{x}]$
($\hat{x}\equiv\hat{a},\hat{a}^{\dagger},\sigma^{\pm}$) is the Lindblad
dissipator which is defined as

$\mathcal{D}[\hat{x}]=\hat{x}\hat{\rho}\hat{x}^{\dagger}-\\{\hat{\rho},\hat{x}^{\dagger}\hat{x}\\}/2,$
(9)

where $\\{\hat{A},\hat{B}\\}=\hat{A}\hat{B}+\hat{B}\hat{A}$ represents the
anti-commutation relation between two quantum operators. The bath spectral
response function in Eq. (8) is given by

$G(\omega)=\begin{cases}\gamma(\omega)(1+\bar{n}(\omega)),&\text{if
$\omega>0$},\\\ \gamma(|\omega|)\bar{n}(|\omega|),&\text{if
$\omega<0$},\end{cases}$ (10)

where $\gamma(\omega)$ is the coupling strength between system and bath and
$\bar{n}(\omega):=1/(e^{\omega/T}-1)$ is average photon number of the bath.

The steady-state solution of the global Master Eq. (8) gives the global
thermal state and is given by

$\tilde{\rho}_{G}=\tilde{\rho}^{(q)}_{th}\otimes\tilde{\rho}^{R}_{th},$ (11)

where $\tilde{\rho}^{(q)}_{th}$ and $\tilde{\rho}^{(R)}_{th}$ are the qubit
and resonator thermal density matrices, respectively. We can transform the
global thermal state in local basis by applying back transformation, such as

$\hat{\rho}=\hat{\text{U}}[\tilde{\rho}^{(q)}_{th}\otimes\tilde{\rho}^{R}_{th}]\hat{\text{U}}^{\dagger}.$
(12)

To find the reduced state for the resonator in local basis, one can take a
trace over the qubit state, which gives the state of the resonator

$\hat{\rho}^{(R)}=Tr_{q}[\tilde{\rho}^{(q)}_{th}\otimes\hat{\text{U}}\tilde{\rho}^{(R)}_{th}\hat{\text{U}}^{\dagger}]=Tr_{q}[\hat{\text{U}}\tilde{\rho}^{(R)}_{th}\hat{\text{U}}^{\dagger}].$
(13)

The state of the resonator is finally given as

$\hat{\rho}^{(R)}=e^{-\alpha^{\prime}(\hat{a}^{\dagger}-\hat{a})}\tilde{\rho}^{(R)}_{th}e^{+\alpha^{\prime}(\hat{a}^{\dagger}-\hat{a})},$
(14)

where $\rho^{(R)}_{th}$ is the thermal state of the resonator in Fock basis,
i.e.,
$\tilde{\rho}^{(R)}_{th}=\mathcal{Z}^{-1}e^{-\beta\omega_{R}\hat{a}^{\dagger}\hat{a}}|n\rangle\langle
n|$ with $\mathcal{Z}$ being the partition function and $\beta:=1/k_{B}T$
being the inverse temperature. In the unitary operator
$\hat{\text{U}}=e^{-\alpha^{\prime}(\hat{a}-\hat{a}^{\dagger})}$, the
parameter $\alpha^{\prime}$ is real and is given by

$\alpha^{\prime}=\frac{g}{\omega_{R}}\langle\hat{\sigma}_{z}\rangle=-\frac{g}{2}\Omega\tanh{(\frac{\omega_{q}}{2T})},$
(15)

where we define $\Omega:=\omega_{q}/\omega_{R}$. The density matrix obtained
for the resonator in local basis (see Eq. (14)) has the form of a thermal
coherent state $D(\alpha)\rho_{th}D^{\dagger}(\alpha)$ with displacement
operator defined as
$D(\alpha)=e^{-(\alpha^{*}\hat{a}-\alpha\hat{a}^{\dagger})}$. Generally, a
thermal coherent state is obtained if a thermal state is displaced by an
amount of $\alpha$ in phase space. We can obtain the normally ordered form of
the thermal coherent state of the resonator such that

$\hat{\rho}^{(R)}=\frac{1}{\sqrt{\mu}}e^{-w[(\hat{a}^{\dagger}+\alpha)(\hat{a}+\alpha)]},$
(16)

where $\mu:=(\bar{n}+1)^{2}$ and $w:=1/\bar{n}$.

##  III Results

In the next section, we discuss two examples that are complementary for the
investigation of the thermal coherent state of the resonator. The first one
shows the non-classicality witnessed by the second-order correlation function
$g^{2}(0)$. In contrast, the second one is the occupation probability (for
example, Eq. (25)) of TCS to perceive the coherence.

###  III.1 Photon statistics

The second-order correlation coefficient measures the quantum state’s degree
of photon bunching or anti-bunching. It is a fundamental tool for
characterizing the quantum properties of optical fields. We analyze the
second-order correlation coefficient $g^{2}(0)$ that gives a direct measure of
the degree of coherence of photon states in terms of the variance of the
photon statistics and is defined for at zero delay time as Tatsi  _et al._
(2022)

$g^{2}(0)=\frac{\langle\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}\rangle}{(\langle\hat{a}^{\dagger}\hat{a}\rangle)^{2}}.$
(17)

After calculating the average photon number
$\langle\hat{n}\rangle=\langle\hat{a}^{\dagger}\hat{a}\rangle$ and
$\langle\hat{a}^{\dagger}\hat{a}^{\dagger}\hat{a}\hat{a}\rangle$, the second
order correlation coefficient can be written in the following form

$g^{2}(0)=\frac{\alpha^{4}+2\bar{n}^{2}+4\alpha^{2}\bar{n}}{(\bar{n}+\alpha^{2})^{2}}.$
(18)

Figure 2: (a). The second-order correlation coefficient $g^{2}(0)$ for the
steady state of the resonator as a function of bath temperature $T$. The solid
red and cyan dashed curve is for $g=0.01$ and $g=0.04$, respectively. (b). The
absolute value of $\alpha^{\prime}=g\langle\hat{\sigma}^{z}\rangle/\omega_{R}$
is plotted against $T$. The other parameters are $g=0.01$, $\omega_{q}=1$, and
$\omega_{R}=1$.

In the case of a Poisson distribution, $g^{2}(0)=1$, which means that photons
in coherent states are uncorrelated, while for $g^{2}(0)=2$, the photons are
correlated or bunched. If the value of $g^{2}(0)$ is less than 1, we describe
the photon field’s statistics as sub-Poissonian, meaning that the state is
nonclassical. On the other hand, if $g2(0)$ is greater than 1, we refer to the
statistics as super-Poissonian, which indicates a classical state of the
photon field Mandel and Wolf (1995); Filip and Marek (2014).

We plot $g^{2}(0)$ and the absolute value of $\alpha$ as a function of bath
temperature $T$ for different values of the coupling strength $g$ in Fig. 2.
The results are shown for the thermal coherent state of the resonator obtained
after solving the full master equation (see Eq. (8)). This indicates that the
thermal coherent state of the resonator at lower temperatures follows the
Poissonian statistics. However, the photon field attains super-Poissonian
statics as the temperature increases. Another factor affecting the
distribution is the coupling strength between the resonator and qubit; strong
coupling results in a slow variation in the photon field distribution.

###  III.2 Multimode resonator

To investigate further, we consider a multimode resonator coupled to a qubit.
The Hamiltonian of which is given by,

$\hat{H}=\frac{\omega_{q}}{2}\hat{\sigma}^{z}+\sum_{i}\omega_{i}\hat{a}^{\dagger}_{i}\hat{a}_{i}+\sum_{i}g_{i}\hat{\sigma}_{z}(\hat{a}^{\dagger}_{i}+\hat{a}_{i}),$
(19)

Figure 3: Second order correlation coefficient $g^{2}(0)$ as function of bath
temperature $T$ for different values of frequencies $\omega_{i}$:
$\omega_{1}=0.3$ (solid magenta line), $\omega_{2}=0.4$ (dashed red line),
$\omega_{3}=0.5$ (dotted blue line), $\omega_{4}=0.6$ (dot-dashed green line).
The other parameters are $\omega_{q}=1$ and $g_{1}=g_{2}=g_{3}=g_{4}=0.01$.

where $g_{k}$ is the coupling strength between the qubit and each resonator
mode of frequency $\omega_{k}$. The case of identical resonator modes is
straightforward, where all the modes have the same frequency $\omega_{i}$ and
coupling strengths $g_{i}$. Hence, the thermal coherent states for all modes
have the same second-order correlation coefficient such that
$g^{2}_{1}(0)=g^{2}_{2}(0)=g^{2}_{3}(0)=g^{2}_{4}(0)=g^{2}_{i}(0)$ and so does
the states. We now assume that all the modes have different frequencies and
coupling strengths. Therefore, the thermal coherent state of the resonator
takes the following form

$\hat{\rho}^{(R)}=\sum_{i}\frac{1}{\sqrt{\mu_{i}}}exp[-w_{i}\\{(\hat{a}_{i}+\alpha_{i})(\hat{a}^{\dagger}_{i}+\alpha_{i})\\}],$
(20)

where $\mu_{i}$ ad $w_{i}$ are defined in Sec. II. The second-order
correlation coefficient for the resonator having i-th modes can be written as

$g^{2}(0)=\sum_{i}\frac{\alpha^{4}_{i}+2\bar{n}^{2}_{i}+4\bar{n}_{i}\alpha^{2}_{i}}{(\bar{n}_{i}+\alpha^{2}_{i})^{2}},$
(21)

where $\alpha^{\prime}_{i}=g_{i}\langle\hat{\sigma}_{z}\rangle/\omega_{i}$. We
plot the second-order correlation coefficient $g^{2}(0)$ for various resonator
modes for the bath temperature. We assume that the coupling strengths for
different modes are the same but different frequencies, as shown in Fig. 3.

Every optical field possesses a crucial characteristic known as the photon
number distribution, which describes the probability of finding the specific
number of photons within the field. The PND provides a fundamental
understanding of the nature of the optical field, including its coherence,
statistical properties, and quantum behavior. To calculate the photon number
distribution of a thermal coherent state, it is convenient to write the
P-function representation of a thermal state

${\rho}_{th}=\frac{1}{\pi\bar{n}}\int
e^{-|\alpha|^{2}/\bar{n}}|\alpha\rangle\langle\alpha|d^{2}\alpha,$ (22)

where $\bar{n}=1/(e^{\omega_{R}/T}-1)$ is the mean photon number of the
thermal field. Indeed, this is a thermal state if one checks the matrix
elements in the Fock basis. In the Fock basis, the thermal coherent state
reads as

$\displaystyle\rho^{(R)}=$ $\displaystyle\frac{1}{\pi\bar{n}}\int
e^{-|\alpha|^{2}/\bar{n}}|\alpha+\alpha^{\prime}\rangle\langle\alpha+\alpha^{\prime}|$
(23) $\displaystyle=\frac{1}{\pi\bar{n}}\sum_{n,m}\int d^{2}\alpha
e^{-|\alpha|^{2}/\bar{n}}e^{-|\alpha+\alpha^{\prime}|^{2}}\frac{(\alpha+\alpha^{\prime})^{n}(\alpha+\alpha^{\prime})^{m}}{\sqrt{m!}\sqrt{n!}}$
$\displaystyle\times|n\rangle\langle m|d^{2}\alpha.$

The probability of finding $n$ photons in a quantum state described by the
density operator $\rho$ can be calculated by taking the projection over the
thermal coherent state of the resonator and reads as

$P(n)=Tr(\rho^{(R)}|n\rangle\langle n|).$ (24)

For the state of the resonator, it takes the following form

$P^{(R)}(n)=\frac{1}{\pi\bar{n}}\sum_{n}\int d^{2}\alpha
e^{-|\alpha|^{2}/\bar{n}}e^{-|\alpha+\alpha^{\prime}|^{2}}\frac{(\alpha+\alpha^{\prime})^{2n}}{n!}.$
(25)

To visualize the coherence in the steady state of the resonator, we plotted
Eq. (25), the PND or occupation probability $P(n)=Tr(\rho_{R}|n\rangle\langle
n|)$ of the thermal coherent state of the resonator in Fig. 4. This indicates
that as we increase the coupling strength $g$, the probability distribution
changes from thermal to coherent, as indicated in Fig. 4.

(a)  (b)

(c)  (d)

Figure 4: [(a)–(d)] The occupation probability (OP) of the resonator state
with respect to the occupation number of the Fock states $n$ for
$\omega_{q}=1$, $\omega_{c}=0.4$, and $T=0.1$. The distributions are for
different values of coupling strengths between qubit and resonator such as:
(a),$g=0.04$ (b), $g=0.2$ (c), $g=0.4$ and (d), $g=0.6$.

##  IV Coherence longevity

Here we provide a more intuitive understanding of coherence protection and
longevity based on thermal coherent environments. To begin with, we present a
possible application of ancilla-assisted engineered thermal coherent
environments that can enhance the coherence lifetimes of qubits. The scheme is
schematically depicted in Fig. 5. For this purpose, we consider a simple case
where a target qubit is coupled to a transmission line resonator, assume that
there is no ancilla qubit (for example, $g_{2}=0$), and observe the coherence
dynamics of TLR with the help of a target qubit. The interaction between the
target qubit of frequency $\omega_{T}$ and the resonator is given by the
Hamiltonian

$\hat{H}_{TR}=\hat{\sigma}_{x}\otimes(\hat{a}+\hat{a}^{\dagger}).$ (26)

The combined initial state of the coupled system supposed to be the product
state and reads as

$\rho_{i}(0)=\rho_{q}(0)\otimes\rho_{R}(0).$ (27)

We assume that the target qubit is initially prepared in a state
$\rho_{q}=|\psi_{q}(0)\rangle\langle\psi_{q}(0)|$, where
$|\psi_{q}(0)\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ is the coherent
superposition state and $|0\rangle[|1\rangle]$ being the ground[excited] state
of qubit. The initial state of TLR is a thermal state such that
$\rho_{R}(0)=e^{-\beta\hat{H}_{R}}/\mathcal{Z}$ with $\beta$ is the inverse
temperature of the resonator and, $\hat{H}=\omega_{R}\hat{a}^{\dagger}\hat{a}$
is the Hamiltonian of the resonator and
$\mathcal{Z}=\text{Tr}[e^{-\beta\hat{H}_{R}}]$ is the canonical partition
function. The evolved state of the system can be found by using Vonn Neumann
equation

$\dot{\rho}_{f}(t)=-i[\hat{H},\rho_{i}(0)].$ (28)

Figure 5: A transmission line resonator (TLR) of frequency $\omega_{R}$ is
coupled to two qubits.The ancilla qubit is coupled via coupling strength
$g_{1}$ while the other qubit is the target qubit coupled with strength
$g_{2}$. In addition, the ancilla qubit is also attached to a thermal bath of
temperature $T$. The purpose of the ancilla qubit is to build coherence in the
resonator modes fast enough to slow down the decay of the initial coherence of
the target qubit. The ancilla qubit is prepared coherently, and the target
qubit is prepared in the ground state. The coherence of the system is probed
via the target qubit.

We numerically simulate the above equation with the help of QuTiP packages
Johansson  _et al._ (2012, 2013) and find the final state $\rho_{q}(t)$ of the
target qubit from which we can investigate the coherence dynamics of the
target qubit in the absence of ancilla. For the enhancement of the coherence
lifetime of the target qubit, we attach an ancilla qubit to the TLR (such that
$g_{2}\neq 0$ in this case). This time, the TLR is attached to two qubits. An
ancilla qubit is also attached to a thermal bath of temperature $T$ and is
responsible for injecting the coherence in TLR. The coherences in the TLR can
be read out with the help of a second qubit called the target qubit. The total
Hamiltonian of this coupled system, as shown in Fig. 5 can be written as

$\hat{H}_{total}=\hat{H}_{0}+\hat{H}_{I},$ (29)

where the Hamiltonian of individual systems and interaction terms are given by

$\hat{H}_{0}=\frac{\omega_{2}}{2}\hat{\sigma}_{z}+\frac{\omega_{2}}{2}\hat{\sigma}_{z}+\omega_{R}\hat{a}^{\dagger}\hat{a}.$
(30)
$\hat{H}_{I}=g_{1}\hat{\sigma}_{x}\otimes(\hat{a}+\hat{a}^{\dagger})+g_{2}\hat{\sigma}_{z}\otimes(\hat{a}+\hat{a}^{\dagger}).$
(31)

The ancilla qubit interacts with the TLR via
$\hat{\sigma_{z}}(\hat{a}+\hat{a}^{\dagger})$ interaction so that it can
convert thermal energy into coherence in the resonator and put it in a thermal
coherent state. In Sec. II, we rigorously derived the global master equation
and presented analytical results for preparing the TCS of the resonator.
However, in this section, we rely on the local master equation for two
reasons; first, the ancilla-assisted system has become complicated for
deriving the global master equation; second, we prove that the TCS can also be
prepared by employing a local master equation. To obtain a more realistic
understanding of the system’s dynamics, it is necessary to consider the
influences of dissipation and decoherence effects. For an open quantum system
that interacts with an environment, we can write the final density matrix of
the system in the following form

$\frac{\partial\hat{\rho}_{f}}{\partial
t}=-i[\hat{H}_{total},\hat{\rho}_{i}(0)]+\mathcal{L}(\hat{\rho})$ (32)

where

$\displaystyle\mathcal{L}(\hat{\rho})=$
$\displaystyle\gamma_{q}(\bar{n}_{q}+1)\Big{(}\hat{\sigma}_{2}^{-}\hat{\rho}\hat{\sigma}_{2}^{+}-\frac{1}{2}\hat{\sigma}_{2}^{+}\hat{\sigma}_{2}^{-}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{\sigma}_{2}^{+}\hat{\sigma}_{2}^{-}\Big{)}$
(33)
$\displaystyle+\gamma_{q}\bar{n}_{q}\Big{(}\hat{\sigma}_{2}^{+}\hat{\rho}\hat{\sigma}_{2}^{-}-\frac{1}{2}\hat{\sigma}_{2}^{-}\hat{\sigma}_{2}^{+}\hat{\rho}-\frac{1}{2}\hat{\rho}\hat{\sigma}_{2}^{-}\hat{\sigma}_{2}^{+}\Big{)},$

where $\gamma_{q}$ denotes the relaxation rate of the ancilla qubit, which
accounts for its interaction with the bath, and the mean photon number in the
thermal bath of the ancilla qubit is given by

$\bar{n}_{q}=\frac{1}{e^{\hbar\omega/k_{b}T_{q}}-1}$ (34)

Let’s assume that the ancilla qubit is weakly coupled to the environment and
resonator. Therefore, we start with the initial state of the compound system
as the product state, which reads as

$\hat{\rho}_{i}(0)=\hat{\rho}_{R}(0)\otimes\hat{\rho}_{A}(0)\otimes\hat{\rho}_{T}(0).$
(35)

The initial state of an ancilla qubit is the superposition state; the target
qubit has a ground state $|0\rangle$, while the resonator has a thermal state.
The final state of the target qubit is obtained by applying partial trace such
that $\hat{\rho}_{T}(t)=\text{Tr}_{R,A}[\hat{\rho}_{f}(t)]$ from which one can
easily calculate the $l_{1}$-norm of coherence of qubit.

Figure 6: Coherence dynamics of the target qubit as a function of time. The
black dashed curve is for the case when there is no ancilla qubit coupled to
the resonator, while the blue dot-dashed curve represents the coherence
dynamics of the target qubit in the presence of an ancilla qubit. The inset
shows the coherence dynamics of the resonator. The target qubit is initially
in a coherent state while the resonator is in the thermal state with
temperature $T_{R}=0.5$ and other parameters are $\omega_{T}=1$,
$\omega_{A}=0.1$, $\omega_{R}=0.01$,$g_{1}=0.03$ and $g_{2}=0.03$. The initial
temperature of the resonator in the presence of ancilla is $T_{R}=10^{-3}$.
The coupling of qubit to the thermal bath is $\gamma_{q}=10^{-3}$.

By solving the Eqs. (28) and (32) numerically Johansson  _et al._ (2012,
2013), we investigate the coherence dynamics of target qubit in the case when
there is an ancilla qubit attached to the resonator, and when there is not. In
order to see the coherence dynamics of the target qubit coupled to a
resonator, we simulate the system driven by the Hamiltonian (26) and ignore
the coupling to any external environment (in the absence of ancilla qubit).
The initial state of the resonator is the thermal state, and due to finite
modes, the resonator acts as an approximate thermal coherent bath for the
target qubit. We probe the coherence dynamics with the target qubit in the
weak coupling regime such that $g=0.03\omega_{T}$. The decay in the coherence
of the target qubit can be seen in Fig. 6 (black dashed curve). The coherence
decays rapidly with time and becomes zero over a longer time. To prolong the
coherence time of the target qubit, we introduced an ancilla qubit to the
system, as shown in Fig. 6. When the ancilla qubit is there, it will inject
coherence into the TLR, making the dephasing of the initially coherent ancilla
qubit slower and having a relatively longer-lived coherence. The ancilla qubit
is longitudinally coupled to the TLR. The coherence dynamics of the target
qubit can be seen in Fig. 6 (blue dot-dashed curve). This time, it decays
slower as the bath of resonator modes has some coherence due to the ancilla
qubit presence. Basically, now it behaves as a decay of a two-level system
into a coherent thermal bath. As a result, the ancilla qubit builds coherence
in resonator modes (as shown in the inset of Fig. 6 fast enough to slow down
the decay of the initial coherence of the target qubit. The interaction
$g\sigma^{A}_{z}(\hat{a}+\hat{a}^{\dagger})$ between ancilla and resonator
plays a vital role in the emergence of coherence by generating coherent
thermal photons probed by the target qubit. In the presence of an ancilla
qubit, the coherence of the target qubit strongly depends on the initial
temperature of the resonator modes. In our case, the initial temperature of
the resonator is very low such as $T_{R}=10^{-3}$. As we increase the
temperature, the coherence decays faster. The other parameter significantly
affecting the results is the coupling strength $g_{2}$ between the ancilla
qubit and resonator. We presented the weak and similar coupling result, i.e.,
$g_{1}=g_{2}=0.03$ of the ancilla and target qubit. The strong coupling
$g_{2}$ doesn’t build coherence so fast in the resonator, and eventually, it
decays faster in the target qubit. However, this coherence decay can be made
slower by considering weak coupling $g_{2}$ for the ancilla and resonator.
Thus, quantum coherence can be protected using ancilla-assisted engineered
environments capable of generating thermal coherent baths, making them a
promising candidate for precise quantum sensing and metrology Degen  _et al._
(2017); Giovannetti  _et al._ (2011).

##  V Conclusion

We investigated the preparation of a thermal coherent state of light in a
coupled qubit-resonator system. We considered a qubit coupled longitudinally
to a resonator; such systems can also be found in many electromechanical
setups. The qubit is excited by attaching a thermal bath of temperature $T$ to
it. With the help of unitary transformations, we diagonalized the Hamiltonian
of the given system and solved the master equation. The steady-state solution
of the master equation gives the state of the resonator in the local basis,
which is, interestingly, a thermal coherent state. To quantify its properties,
we studied the photon statistics of the state at low temperatures and in a
weak coupling regime. The results show that the TCS follows a Poissonian
distribution ($g^{2}(0)=1$) at low temperatures, indicating the state’s
coherent properties. However, the photon statistics become super-Poissonian
($g^{2}(0)>1$) as the temperature of the bath increases, which confirms the
thermal properties of a coherent state. At zero temperature ($T=0$), the state
of the resonator is purely coherent. Furthermore, we examine the application
of thermal coherence environments to protect and increase the coherence
lifetime of qubits. With the help of an ancilla qubit initially prepared in
the coherent superposition state and in addition, attached to a thermal bath
can inject coherence in the resonator, which is probed by coupling a target
qubit to the resonator. This ancilla qubit has a positive effect on generating
coherence in the resonator. An improvement can be observed in terms of slowing
the decay of coherence in the target qubit. Moreover, the coherence dynamics
strongly depend on the initial temperature of the resonator and the coupling
strength with the ancilla qubit. Our results can pave the way for the
preparation of substantial thermal coherent and squeezed states of light via
inter-system interaction, such as qubit-resonator coupling, which is
practically realizable in optomechanical systems. Since the thermal bath may
destroy the quantumness of quantum states due to thermal fluctuations,
therefore, our results can be of fundamental and practical interest where the
quantum states are desirable in the presence of thermal environments in
quantum optical setups such as coherent thermal magnon baths that can be used
to entangle two distant nitrogen-vacancy centers Ullah  _et al._ (2022).

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