#  : Text-to-Image Generation With Realistic Hand Appearances

Supreeth Narasimhaswamy∗1, Uttaran Bhattacharya2, Xiang Chen2,
Ishita Dasgupta2, Saayan Mitra2, and Minh Hoai1

1Stony Brook University, USA, 2Adobe Research, USA Work started when Supreeth
was an intern at Adobe Research

###### Abstract

Text-to-image generative models can generate high-quality humans, but realism
is lost when generating hands. Common artifacts include irregular hand poses,
shapes, incorrect numbers of fingers, and physically implausible finger
orientations. To generate images with realistic hands, we propose a novel
diffusion-based architecture called that achieves realism by injecting hand
embeddings in the generative process. consists of two components: a Text-to-
Hand-Params diffusion model to generate SMPL-Body and MANO-Hand parameters
from input text prompts, and a Text-Guided Hand-Params-to-Image diffusion
model to synthesize images by conditioning on the prompts and hand parameters
generated by the previous component. We incorporate multiple aspects of hand
representation, including 3D shapes and joint-level finger positions,
orientations and articulations, for robust learning and reliable performance
during inference. We conduct extensive quantitative and qualitative
experiments and perform user studies to demonstrate the efficacy of our method
in generating images with high-quality hands. Project page:
https://supreethn.github.io/research/handiffuser/index.html

##  1 Introduction

Figure 1: Generating realistic hands. Text-to-Image generative models, e.g.,
[56], often produce various hand artifacts (top row). We inject hand
embeddings, capturing hand shapes, poses, and articulations, in the generation
process to generate realistic hands (bottom row).

Text-to-Image (T2I) generative models have shown impressive advancement in
recent years. Generative models such as Stable Diffusion [56], Imagen [58],
and GLIDE [45] can generate high quality, photorealistic images. However,
these methods often struggle to synthesize high-quality and realistic hands.
The generated hands often have improbable hand poses, irregular hand shapes,
incorrect number of fingers, and poor hand-object interactions (Fig. 1).

Generating images with high-quality hands is a challenging problem since hands
often take up a small part of the image, but are highly articulate. They have
high degrees of freedom, with a wide variety of flexibility where fingers can
bend to various degrees relatively independently. Hands can also occur in
various shapes, sizes, and orientations and can be occluded with other human
body parts. Further, hands often interact with objects and can have a wide
range of grasps depending on the object’s size, shape, and affordance.
Therefore, capturing such a vast range of articulations and interactions
directly from text inputs remains challenging. Despite having billions of
parameters and several millions of trainable images, T2I models struggle to
generate realistic hands.

A central challenge in hand image generation is learning diverse hand poses
and configurations at scale. Existing hand representations based on keypoint
skeletons and shape formats [34, 57] are useful for generative tasks in pose
animation [66] and hand-object interactions [12]. These representations
provide a grounded understanding of plausible hand shapes and postures,
especially in relation to the rest of the body and different interacting
objects. However, the necessary steps to incorporate these hand
representations into T2I pipelines, in terms of both learning these
representations from text prompts and mapping these representations into the
pixel space of images, remain open problems. These problems are exacerbated
when we consider naturally constructed prompts, which often imply rather than
specify hand postures and articulations (e.g., all prompts in Fig. 1). Prompt
engineering [33, 16], focusing on hand descriptions, can potentially improve
the generation quality. But it comes with the cost of distilling and learning
appropriate prompts from large-scale data, and with the caveats of learning
spurious inter-relationships between the prompt and the hands or between the
hands and the rest of the images.

In this paper, we propose a learning-based model to generate images containing
realistic hands in an end-to-end fashion from text prompts. Our model, called
, consists of two key trainable components. The first component, Text-to-Hand-
Params (T2H), generates parameters of a hand model [34, 57] conditioned on the
input text prompts. The second component, Text-Guided Hand-Params-to-Image
(T-H2I), uses the hand parameters and the input text prompts as conditions to
generate images. By conditioning the image generation on accurate hand models,
can generate high-quality hands with plausible hand poses, shapes, and finger
articulations. Specifically, we consider three aspects of hand representation,
each serving a unique purpose. These include the spatial locations of hand
joints to capture the hand pose, the joint rotations to capture the finger
orientations and articulations, and the hand vertices to capture the overall
hand shape. We design a novel Text+Hand Encoder by extending the CLIP encoder
[54] to obtain joint embeddings for these three representations together with
the text. We use the proposed joint embeddings to condition the image
generation, allowing us to generate images by conditioning on both the hand
parameters and the text.

We train the two components of independently. We train T2H using around 450K
text and 3D human pairs and fine-tune T-H2I using around 900K text and image
pairs. Once trained, we use the two components end-to-end in a single
inference pipeline to generate images from text prompts. We conduct extensive
experiments and user studies to show the effectiveness of the in generating
images with high-quality hands.

In short, the contributions of our paper are:

  * •

, a generative model to synthesize images with high-quality hands by
conditioning on text and hand embeddings. It has two novel components: Text-
to-Hand-Params and Text-Guided Hand-Params-to-Image.

  * •

Text-to-Hand-Params, a diffusion model to generate SMPL-Body and MANO-Hand
parameters from text inputs. The generated MANO-Hands are used to further
condition the image generation.

  * •

Text-Guided Hand-Params-to-Image, a diffusion model to generate images with
high-quality hands by conditioning on hand and text embeddings. We design hand
embeddings to capture hand shape, pose, and finger orientations and
articulations.

Figure 2:  architecture. Our architecture consists of two components. The
first component, Text-to-Hand-Params (T2H), takes the text as input and
generates body and hand parameters. The second component, Text-Guided Hand-
Params-to-Image (T-H2I), uses the hand parameters from the first component and
the text to generate images with high-quality hands. The Text+Hand encoder
jointly encodes hand parameters and text, and captures hand pose,
articulation, and shape.

##  2 Related Work

We briefly summarize related work on text-to-image generation, concurrent work
on text-to-human generation, and commonly used hand representations.

Text-to-Image Generation. Text-guided image generation is a well-studied
problem, with modern approaches ranging from GANs [68, 78, 70, 65],
autoregressive generation approaches [37, 55], and VQ-VAE transformers [13] to
state-of-the-art diffusion models [61, 21, 11, 22]. Text-to-image generation
using diffusion models often bootstrap the generative pipeline with pre-
trained language models, such as BERT [10] or CLIP [54], to efficiently learn
from the text information [73, 38, 17, 45, 4]. More recently, Stable Diffusion
[56] performs diffusion in the latent image space to generate high-quality
images at low computational costs. Imagen [58], by contrast, diffuses the
pixels directly in a hierarchical fashion. ControlNet [74] provides additional
controllability in the image generation process in the form of conditioning
signals ranging from sketches to pose priors. Latest software products for
text-to-image generation include Midjourney [39], DALL-E 3 [46], and Firefly
[1]. While the advancements in this area have been rapid and significant,
generating highly articulate hands remains prone to unrealistic artifacts.

Text-to-Human Generation. Alongside image generation, there has also been
considerable progress in human pose and motion generation from text prompts.
Recent generative methods typically follow various skeletal joint formats,
such as OpenPose [5], or combined joint and mesh formats, such as SMPL [34],
to represent the human body. They train on various large-scale pose and motion
datasets, including KIT [51], AMASS [36], BABEL [53], and HumanML3D [18]. To
efficiently map text prompts to motion sequences, text-to-motion generation
methods learn combined representations of language and pose using techniques
ranging from recurrent neural networks [2, 30], hierarchical pose embeddings
[3, 14], and VQ-VAE transformers [49, 50] to motion diffusion models [75, 66,
28, 7]. Some approaches even generate 3D meshes on top of pose motion
synthesis to synthesize fully rendered humans [23, 71]. Separate from motion
synthesis, there are works on generating parametric pose models from text [8,
47, 9]. However, these methods focus on the human body and ignore the hand
regions. As a result, they cannot generate articulate hands. There is a method
[43] to generate plausible hands using ControlNet but it requires a hand
skeleton or mesh as an additional input. A concurrent work [35] proposes an
inpainting approach to refine hands. Given a generated image, the method
reconstructs 3D meshes for hands and further refines hand regions. As a
result, the quality of the reconstructed hand mesh, and consequently the final
refined hand, depends on the quality of the initially generated hand. On the
contrary, our method first generates hand mesh parameters from the prompt and
further conditions the image generation on such intermediate hand parameters.
Moreover, [35] ignores the hand-object interactions in the initial image and
might not preserve hand-object occlusions and interactions when refining
hands.

Hand Representations. Available datasets on hand configurations, gestures, and
hand-object interactions offer hand representations in a variety of formats,
including bounding boxes, silhouettes, depth maps [40, 29, 31, 42, 41, 25, 60,
44], keypoints and parametric models [57, 27, 26, 12]. These representations
are useful for multiple hand-centric tasks, including detection [67], gesture
and pose recognition [76], motion generation [77, 64], and hand-object
interactions [24, 63, 15]. Our work combines representations based on keypoint
and parametric models to efficiently encode diverse hand shapes and highly
articulated finger movements.

##  3

Fig. 2 illustrates the proposed architecture. Given a text input, first uses a
novel Text-to-Hand-Params diffusion model to generate the parameters of the
human body and hand models. The second component is the Text-Guided Hand-
Params-to-Image diffusion model that generates the output image by
conditioning on the hand model and the text. This section provides more
detailed insights into the Text-to-Hand-Params and Text-Guided Hand-Params-to-
Image models, following a brief introduction to the fundamentals of human
models and stable diffusion.

###  3.1 Preliminaries

SMPL-H. Our Text-to-Hand-Params model generates parameters of human body and
hand models from text inputs. We use SMPL [34] and MANO [57] as our body and
hand model, respectively. The SMPL is a differentiable function
$\mathcal{M}_{b}(\theta_{b},\beta_{b})$ that takes a pose parameter
$\theta_{b}\in\mathbb{R}^{69}$ and shape parameter
$\beta_{b}\in\mathbb{R}^{10}$, and returns the body mesh
$\mathcal{M}_{b}\in\mathbb{R}^{6890\times 3}$ with 6890 vertices. Similarly,
MANO is a differentiable function $\mathcal{M}_{h}(\theta_{h},\beta_{h},s)$
that takes the hand pose parameter $\theta_{h}\in\mathbb{R}^{48}$, hand shape
parameter $\beta_{h}\in\mathbb{R}^{10}$, and the hand side
$s\in\\{\text{left},\text{right}\\}$, and returns hand mesh
$\mathcal{M}_{h}\in\mathbb{R}^{778\times 3}$ with 778 vertices. The 3D hand
joint locations $J_{h}\in\mathbb{R}^{k\times
3}=\mathcal{W}_{h}\mathcal{M}_{h}$ can be regressed from vertices using a pre-
trained linear regressor $\mathcal{W}_{h}$. The SMPL-H model combines the
body, left hand, and right hand model into a single differntiable function
$\mathcal{M}(\theta,\beta)$ with pose parameters
$\theta{=}(\theta_{b},\theta_{lh},\theta_{rh})$ and shape parameters $\beta$.
The pose parameters $\theta_{b}$, $\theta_{lh}$, and $\theta_{rh}$ captures
the root-relative joint rotations for body, left hand, and right hand,
respectively. The shape parameter $\beta$ captures the scale of the person.

Stable Diffusion. Our Text-Guided Hand-Params-to-Image model is built upon
Stable Diffusion [56]. Stable Diffusion is a latent diffusion model consisting
of an auto-encoder, a U-Net for noise estimation, and a CLIP text encoder. The
encoder $\mathcal{E}$ encodes an image $x$ into a latent representation
$z=\mathcal{E}(x)$ that the diffusion process operates on. The decoder
$\mathcal{D}$ reconstructs the image from $\hat{x}=\mathcal{D}(z)$ from the
latent $z$. The U-Net is conditioned on the denoising step $t$ and the text
$\tau_{text}(text)$, where $\tau_{text}(text)$ is a CLIP [54] text encoder
that projects a sequence of tokenized texts into an embedding space. To
jointly condition the image generation on hand parameters and the text, we
replace the text encoder $\tau_{text}(text)$ with a novel Text+Hand encoder
$\tau_{text+h}(text,hand)$ that jointly embeds the text and hand parameters
into a common embedding space.

###  3.2 Text-to-Hand-Params Diffusion

The Text-to-Hand-Params diffusion model takes a text as input and generates
the pose parameters $\theta=(\theta_{b},\theta_{lh},\theta_{rh})$ and shape
parameters $\beta$ for the SMPL-H model by conditioning on the text.

We define $x:=(\theta,\beta)$ and model the forward diffusion process by
iteratively adding Gaussian noise to $x$ for $T$ time steps:

$q\left(x_{t}|x_{t-1}\right)=\mathcal{N}\left(\sqrt{\alpha_{t}}x_{t-1},(1-\alpha_{t})I\right),$
(1)

where $\alpha_{t}\in(0,1)$ are constant hyper-parameters.

We model the text-conditioned SMPL-H generation distribution $p(x_{0}|c)$ as
the reverse diffusion process of gradually denoising $x_{T}$. Following [66],
we learn the denoising by directly predicting $\hat{x}_{0}=G(x_{t},t,c)$ using
a model $G$. We train the reverse diffusion using the training objective:

$\mathcal{L}_{1}=\mathbb{E}_{x_{0}\sim
q(x_{0}|c),t\sim[1,T]}||x_{0}-G(x_{t},t,c)||^{2}_{2}.$ (2)

We get the conditional text embeddings $c$ by encoding the text using CLIP
[54]. We implement $G$ using a transformer encoder-only architecture similar
to MDM [66].

Given a text during inference, we conditionally sample $x=(\theta,\beta)$. We
use the shape and pose parameters to obtain the joints $J_{lh},J_{rh}$ and
vertices $\mathcal{M}_{lh},\mathcal{M}_{rh}$ for left and right hands using
MANO-Hand model. We also choose camera parameters and project $J_{lh},J_{rh}$
into an image space and obtain the corresponding image-space joint locations
$J^{2D}_{lh},J^{2D}_{rh}$. We use the joint rotations
$\theta_{lh},\theta_{rh}$, hand vertices $\mathcal{M}_{lh},\mathcal{M}_{rh}$,
and spatial joint locations $J^{2D}_{lh},J^{2D}_{rh}$ to condition the image
generation in the next stage.

###  3.3 Text-Guided Hand-Params-to-Image Diffusion

The Text-Guided Hand-Params-to-Image diffusion model is built upon Stable
Diffusion [56] and conditions the image generation on hand parameters
generated from the Text-to-Hand-Params model and the text. Specifically, Text-
Guided Hand-Params-to-Image uses a novel Text+Hand Encoder $\tau_{text+h}$ to
first obtain joint embeddings for text and hand parameters. It then uses the
joint hand and text embeddings to condition the image generation. We provide
more details on this below.

Text+Hand Encoder. Given the provided text, along with the spatial joint
locations $J^{2D}_{h}$, vertices $\mathcal{M}_{h}$, and joint rotations
$\theta_{h}$ of the hand, our goal is to generate $D$-dimensional embeddings
to encode both the text and hand parameters. Here $D$ denotes the CLIP [54]
token embedding dimension. To encode hand joint locations in the image space,
we follow [6, 69] and introduce additional positional tokens. We quantize the
image height and width uniformly into $N_{bins}$ bins. This allows us to
approximate and tokenize any normalized spatial coordinate into one of
$N_{\text{bins}}$ tokens. We then encode the text tokens and the hand joint
spatial tokens into a $D$-dimensions using $f_{text+J^{2D}_{h}}$.
Specifically, we construct $f_{text+J^{2D}_{h}}$ by introducing an additional
$N_{\text{bins}}\times D$ embedding layer into an existing CLIP token embedder
and finetune it during training. To encode hand vertices, we transform them to
basis point set (BPS) [52] representations and pass through
$f_{\mathcal{M}_{h}}$, a Multi-Layer Perceptron (MLP) consisting of fully-
connected linear and ReLU layers. Similarly, we encode 6D hand joint rotations
$\theta_{h}$ using an MLP $f_{\theta_{h}}$ consisting of fully-connected
linear and ReLU layers. Finally, we concatenate embeddings from text, spatial
hand joints, hand vertices, and hand joint rotations to produce joint text and
hand embeddings.

Diffusion. We instantiate the Text-Guided Hand-Params-to-Image using Stable
Diffusion [56] and train using the following objective

$\mathcal{L}_{2}=\mathbb{E}_{\mathcal{E}(x),\epsilon\sim\mathcal{N}(0,1),t,y}||\epsilon-F\left(z_{t},t,\tau_{text+h}(y)\right)||^{2}_{2}.$
(3)

In the above equation, the condition
$y=(\text{text},J^{2D}_{h},\mathcal{M}_{h},\theta_{h})$ denotes the
combination of the text and the hand parameters, which include spatial joint
locations, vertices, and joint rotations. The function $F$ is a denoising
U-Net to predict the noise, $\tau_{text+h}$ is the trainable Text+Hand
encoder. We refer the readers to [56] for more details regarding Eq. (3).

Method | FID $\downarrow$ | KID $\downarrow$ | FID-H $\downarrow$ | KID-H $\downarrow$ | Hand Conf. $\uparrow$
---|---|---|---|---|---
Stable Diffusion | $29.005$ | $9.63{\times}{10}^{-3}$ | $34.372$ | $4.63{\times}{10}^{-2}$ | $0.887$
Stable Diffusion Fine-tuned | $20.056$ | $7.91{\times}{10}^{-3}$ | $31.219$ | $3.09{\times}{10}^{-2}$ | $0.913$
ControlNet | $18.694$ | $5.93{\times}{10}^{-3}$ | $28.091$ | $2.19{\times}{10}^{-2}$ | $0.969$
w/o 2D hand joints | $16.839$ | $5.21{\times}{10}^{-3}$ | $29.902$ | $2.46{\times}{10}^{-2}$ | $0.953$
w/o 3D joint rotation and vertices | $14.586$ | $4.14{\times}{10}^{-3}$ | $28.186$ | $2.21{\times}{10}^{-2}$ | $0.961$
(proposed) | $\mathbf{13.918}$ | $\mathbf{4.07{\times}{10}^{-3}}$ | $\mathbf{27.550}$ | $\mathbf{2.11{\times}{10}^{-2}}$ | $\mathbf{0.978}$

Table 1: Quantitative results. We report the scores of current baselines and
ablated versions of our method on multiple evaluation metrics. $\uparrow$
indicates higher values are better, $\downarrow$ indicates lower values are
better.

Generating SMPL-H vs Skeletons in Text-to-Hand-Params Diffusion. We design the
first component of to generate pose and shape parameters of SMPL-H instead of
keypoints or skeletons since SMPL-H encodes topological and geometric priors
about humans and encodes richer information than skeletons. Also, SMPL-H
parameters tend to be more robust to noise than skeletons; we can still get
plausible poses even with noisy SMPL-H parameters, whereas noisy joint
locations lead to implausible poses. Since we are generating parameters (51
joint rotations and 10 shape parameters) of SMPL-H mesh, the Text-to-Hand-
Params component is computationally lighter compared to the second component,
Text-Guided Hand-Params-to-Image .

Figure 3: Qualitative results. We compare the quality of hands in images
generated by different methods from the same text prompts. (Images are
generated at 512x512 resolution)

##  4 Experiments

This section describes the datasets used to train , implementation details,
and evaluation metrics used. We also present qualitative and quantitative
results, and user studies to show the efficacy of in generating images with
high-quality hands.

###  4.1 Datasets

We train the two components of using our own curated datasets. We start with
930K paired text and images, then curate it to remove inappropriate and
harmful content and validate the quality of the images through independent
content creators. We randomly split the dataset to obtain 900K train and 30K
test text-image pairs. We further preprocess the dataset to obtain SMPL-H
parameters. Specifically, we use [62] to obtain SMPL parameters for the body
and [57] to obtain MANO parameters for hands. We reject estimated SMPL body
and MANO hands that have low confidence scores. Finally, we curate two
datasets using the estimated hand and body parameters. The first dataset
consists of tuples of the form (text, SMPL-H). We keep such tuples only for
images where we can reliably estimate SMPL-H parameters. This dataset has 450K
tuples (text, SMPL-H) and is used to train the first component of the , Text-
to-Hand-Params. The second dataset consists of triplets of the form (text,
image, SMPL-H), and we keep all 930K triplets. We use this dataset to train
the second component of , Text-Guided Hand-Params-to-Image. During training,
we only conditioned the image generation on hand parameters when the SMPL-H
parameters were reliably estimated.

###  4.2 Implementation Details

We train the Text-to-Hand-Params diffusion to generate the SMPL-H pose
$\theta$ and shape $\beta$ by conditioning on the text. We encode the text
using a frozen CLIP-ViT-B/32 model [54]. We train the Text-to-Hand-Params
model using a classifier-free guidance [20] by randomly setting $10\%$ of the
text conditions to be empty. We train this model for 100 epochs on a single
A100 GPU using a batch size of 64. We use 1000 steps and a guidance scale of
$s=2.5$ during the inference. We fine-tune Text-Guided Hand-Params-to-Image
starting from the Stable Diffusion v1.4 checkpoint. To implement the Text+Hand
encoder $\tau_{text+h}$, we start with the CLIP ViT-L/14 model and introduce
additional $N_{bins}=1000$ positional tokens for spatial hand joints. We
choose simple three-layer MLPs $f_{\mathcal{M}_{h}}$ and $f_{\theta_{h}}$ to
encode hand vertices and joint rotations, respectively. We fine-tune Text-
Guided Hand-Params-to-Image, including the Text+Hand encoder $\tau_{text+h}$,
for 20 epochs on eight A100 GPUs using a batch size of 8 and AdamW optimizer
with a constant learning rate of $10^{-4}$. We perform inference with 50 PLMS
[32] steps using a classifier-free guidance [20] of 4.0.

Inference. Given a text input, we first sample SMPL-H parameters using our
trained Text-to-Hand-Params model. We then extract the MANO hand parameters
from SMPL-H and choose camera parameters randomly with some constraints to
make hands somewhat visible in the image and obtain spatial hand joint joints.
Finally, we use these spatial hand joints, MANO parameters, and the text to
conditionally sample an image from our trained Text-Guided Hand-Params-to-
Image model.

###  4.3 Evaluation Metrics

We access the quality of generated images from using the Frechet Inception
Distance (FID) and Kernel Inception Distance (KID) [19, 48]. Since FID and KID
measures the overall quality of the image, we also compute FID-H and KID-H to
measure the quality of images only in the hand regions. We perform this by
first extracting crops using hand bounding boxes and then computing FID and
KID using such hand crops. We also measure the quality of hands using average
hand detection confidence scores. Specifically, we run an off-the-shelf hand
detector [72] on generated images and compute the confidence scores for
detection. Higher confidence scores mean that the hand detector is more
confident of a region being a hand, indicating higher-quality hand
generations.

###  4.4 Quantitative Results and Ablation Studies

We compare the proposed with three different methods and report these results
in Table 1. First, we use an off-the-shelf Stable Diffusion [56] model pre-
trained on the LAION-5B [59] dataset. The LAION-5B dataset is a general-
purpose text and image pairs dataset and does not necessarily focus on humans.
Therefore, a Stable Diffusion model trained on the LAION-5B dataset does not
perform well on images solely focused on humans. Second, we fine-tuned Stable
Diffusion on our dataset and observed that it generated better images than the
pre-trained model. While this generates better performance than the pre-
trained model, the performance is still low compared to the proposed . Third,
we experiment with ControlNet [74], a popular latent diffusion model that uses
spatial control images to condition the image generation process. We train a
ControlNet architecture on our dataset using hand-pose skeleton images as
controls. However, unlike , which generates images directly from text input,
ControlNet requires an additional hand pose skeleton image as input during
inference. To address this, we directly use the ground-truth skeleton images
from the test data as control images. Despite employing these ground-truth
control images, ControlNet does not perform as well as .

It is important to note that the reported FID-H, KID-H, and hand confidence
scores for Stable Diffusion and Stable Diffusion Fine-tuned in Table 1 are
optimistic performance measures. To evaluate the performance of these two
methods, we first ran a hand detector [72] to obtain hand crops. However, this
approach is biased towards rejecting bad hand generations since the hand
detector cannot localize low-quality hand generations, leaving out
unrealistic-looking generated hands from evaluation. On the contrary, the
corresponding metrics for ControlNet and in Table 1 are the true performance
measures since both the methods generate images conditioned on hands, allowing
us to crop every generated hand precisely.

We also study the benefits of different hand representations that are used to
condition the hand generation in . First, we evaluate by omitting the spatial
hand joint locations $J^{2D}_{h}$ in hand embeddings. Second, we evaluate by
omitting the hand joint rotations $\theta_{h}$ and hand vertices
$\mathcal{M}_{h}$ in hand embeddings. We report these results in the fourth
and fifth row of Table 1. These results show that all three hand
representations help in generating quality hands.

Figure 4: Illustrative SMPL-H results, generated from our Text-to-Hand-Params
model.

###  4.5 Qualitative Results and Failure Cases

We report some good qualitative results in Fig. 3. We compare results from
Stable Diffusion, ControlNet [74], the proposed without hand joint rotations
and vertices embeddings, and . Stable Diffusion does not generate realistic
hands even after fine-tuning on human-centric datasets. It generates an
incorrect number of hand fingers, poor hand-object interactions, implausible
finger orientations, and hand shapes. ControlNet generates better-looking
results but requires hand skeleton control images as additional input. We can
see that generates hands with plausible hand poses by conditioning the image
generation on spatial hand joint locations. Further conditioning on hand joint
rotations and vertices enables to generate high-quality, detailed hands with
plausible orientations and shapes.

Fig. 4 shows a few SMPL-H results generated from text inputs using our Text-
to-Hand-Params. While we only use hand parameters from these SMPL-H outputs,
our Text-to-Hand-Params can be directly used in other applications that
require generating SMPL-H models from text inputs. We also show how Text-
Guided Hand-Params-to-Image maps these SMPL-H results to generated images in
Fig. 5. Fig. 6 shows some failure cases of .

###  4.6 User Studies

We evaluate the quality of both our generated images and the intermediate
outputs of our approach through two different user studies. We evaluate the
generated images in two aspects. The first is (A) plausibility, which
considers how natural the hands look, for example, in terms of hand shapes,
finger orientations, number of fingers, hands, and how clearly the hands are
in focus in the image. The second is  (B) relevance, which considers how
natural the hand poses or gestures appear given the prompt, for example,
holding objects or gesticulating conventionally (unless otherwise specified in
the prompt).

We evaluate the intermediate SMPL-H outputs for generating the images in three
aspects: (A) plausibility of the pose, (B) relevance to prompt, and (C)
consistency with the generated image.

Figure 5: Generating images from text via SMPL-H. The intermediate SMPL-H
representations are essential in generating realistic hand appearances. Figure
6:  failure cases. We note some failure cases when the given action may be
unclear (e.g., “adjusts” in the first image from left), the model does not
exactly follow the hand pose description (second image from left), the model
does not fully realize the finger dexterity when handling small and thin
objects (third image from left), and model does not strictly obey the intended
affordance of the object (fourth image from left).

(a) Results on the plausibility of generated images

(b) Results on the relevance of generated images to given prompts

Figure 7: User study results for generated images. We report the mean fraction
of responses for each point on the Likert scale.

Setup. We compare three methods in the user study to evaluate the generated
images: fine-tuned Stable Diffusion (SD-FT), trained with only 2D hand joints
(-2D), and trained with all its model components. We show participants 20 sets
of images. Each set consists of a unique prompt randomly selected from the
test partition of LAION-5B [59] and the images generated by the three methods
given that prompt. We arrange the three images within each set in a random
order not known to the participants. For each image in each set, we ask them
to respond to two questions: “How is the visual quality of the hands?” (image
plausibility) and “How well do the hands follow the prompt?” (image
relevance). We collect responses to the two questions on a 5-point Likert
scale, consisting of the following choices: “Poor (e.g., too many or severe
mistakes)”, “Bad (e.g., some aspects reasonable but still many or severe
mistakes)”, “Fair (e.g., some aspects are plausible but some mistakes
visible)”, “Good (e.g., most aspects are plausible but a few mistakes
visible)”, and “Excellent (e.g., everything looks good, no visible mistakes)”.
Note that we perform the user study on methods that only require text prompts
to generate image outputs at test time. Therefore, we exclude methods such as
ControlNet [74], which additionally requires pose information to generate
similar images. Moreover, the overall performance of ControlNet is also at the
same level as -2D (Table 1 rows 3 and 4) even if we manually provide the
ground-truth poses, leading to no meaningful differences between their
responses in a pilot study.

Image Aspect | SD-FT |  -2D |  (proposed)
---|---|---|---
Plausibility $\uparrow$ | $2.74\pm 0.08$ | $3.30\pm 0.11$ | $\mathbf{3.51\pm 0.11}$
Relevance $\uparrow$ | $3.83\pm 0.12$ | $4.11\pm 0.17$ | $\mathbf{4.23\pm 0.18}$

Table 2: User study for generated images score summary. We compute the mean
and standard deviation of the Likert-scale scores of the three evaluated
methods across all the 700 responses of 35 participants. Higher scores are
better.

For the user study to evaluate the intermediate SMPL-H outputs, we show
participants 9 random triplets of (prompt, SMPL-H poses, generated image),
where the prompts are randomly selected from the test partition of LAION-5B
[59] and the SMPL-H poses and generated images come from our approach. For
each triplet, we ask the participants to respond to three questions: “How
plausible is the pose?” (SMPL-H plausibility), “How relevant are the hands in
the pose given the prompt?” (SMPL-H relevance), and “How consistent are the
hands in the pose with the hands in the image?” (SMPL-H consistency). We
collect responses to the three questions on the same 5-point Likert scale as
above. ask them to evaluate the poses on 5-point Likert scales for each of the
three aspects of plausibility, relevance, and consistency. To evaluate
consistency, we additionally ask participants to focus primarily on the hand
configurations and gestures described or implied in the text. We ask them to
ignore distractors, such as the quality of any facial expressions (or lack
thereof), any component of the 3D pose that is not visible in the image, and
the differences in orientations of the body between the pose and the image.

Figure 8: User study results for SMPL-H poses. We report the mean fraction of
responses for each point on the Likert scale.

Results. Our user study to evaluate the generated images was completed by 35
participants, resulting in a total of 700 responses over the 20 image sets. We
did not observe any notable response differences across genders and age
groups. We report the distribution of scores for the two aspects across all
the responses in Fig. 7. We also summarize the mean and the standard deviation
of the scores of the two aspects for each of the three methods in Table 2. To
compute these values, we assign numbers $1$ through $5$ to the response
choices Poor through Excellent. Consequently, higher scores indicate better
performance. outperforms the other methods in both aspects for generated
images. Looking at the distribution of image plausibility scores (Fig. 7(a)),
we observe the mode of SD-FT on “Fair”, while the modes of both versions are a
point higher, on “Good”. Overall, $55\%$ of scores are “Good” or better,
compared to $47\%$ of -2D scores and $27\%$ of SD-FT scores. Looking at the
distribution of image relevance scores (Fig. 7(b)), we observe the modes of
all the three methods on “Excellent”, indicating their efficacy in generating
hand appearances aligned with text prompts. Among the three methods, we note a
relatively higher distribution of good responses for HandDiffuser variants.
Specifically, $78\%$ of scores are “Good” or better, compared to $75\%$ of -2D
scores and $65\%$ of SD-FT scores. Looking at the mean scores across all the
700 responses (Table 2), we note marked improvements for . Its image
plausibility scores are $0.77$ points (or $15\%$ on the 5-point scale) higher
than SD-FT and $0.21$ points (or $4\%$) higher than -2D. Correspondingly, its
image relevance scores are $0.40$ points (or $8\%$) higher than SD-FT and
$0.12$ points (or $2\%$) higher than -2D.

Our user study to evaluate the intermediate SMPL-H poses was completed by 18
participants, resulting in a total of 171 responses over the 9 triplets. We
did not observe any notable response differences across genders and age
groups. We report the distribution of scores for the two aspects across all
the responses in Fig. 8. We also summarize the mean and the standard deviation
of the scores of the two aspects for each of the three methods in Table 3. To
compute these values, we assign numbers $1$ through $5$ to the response
choices Poor through Excellent. Consequently, higher scores indicate better
performance.

Plausibility $\uparrow$ | Relevance $\uparrow$ | Consistency $\uparrow$
---|---|---
$4.00\pm 0.81$ | $3.83\pm 0.98$ | $3.67\pm 1.04$
Table 3: User study for SMPL-H score summary of . We compute the mean and
standard deviation of the Likert-scale scores for the three evaluation aspects
across all 162 responses of 18 participants. Higher scores are better.

##  5 Conclusions and Limitations

We have presented , an end-to-end model to generate images with realistic hand
appearances from text prompts. Our model explicitly learns hand embeddings
based on hand shapes, poses and finger-level articulations, and combines them
with text embeddings to generate images with high-quality hands. We
demonstrate the state-of-the-art performance of our method on the benchmark
T2I dataset both quantitatively, through multiple evaluation metrics, and
qualitatively, through a user study.

In the future, we plan to extend our model to more unexplored territories of
hand generation. These include images consisting of multiple people, complex
hand-object interactions, prompts describing highly specialized hand
activities (e.g., origami), the same person handling multiple objects
simultaneously, hand-hand interactions of two or more people, and non-
anthropomorphic hands (e.g., a dog a using a computer). A concurrent future
direction is to make the hand generation pipeline style- and shape-aware, such
that it can consistently generate the same hands when asked to generate the
same person in different images.

Acknowledgements. This project was partially supported by US National Science
Foundation Award NSDF DUE-2055406.

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##  6 Additional Results

We show additional qualitative comparisons with the Stable Diffusion [56]
baseline and the proposed in Fig. 9 and Fig. 10.

We also show additional qualitative results and intermediate outputs from the
two components of in Fig. 11 and Fig. 12.

Figure 9: Qualitative Comparison. We compare the quality of hands in images
generated from Stable Diffusion and the proposed .

Figure 10: Qualitative Comparison. We compare the quality of hands in images
generated from Stable Diffusion and the proposed .

Figure 11:  Qualitative Results. Given a text input, Text-to-Hand-Params (T2H)
generates SMPL-H [34, 57] parameters. We extract the MANO-Hand from SMPL-H and
use some camera parameters to obtain 2D hand poses. The text, MANO-Hand and 2D
hand poses are used to generate the final image using Text-Guided Hand-Params-
to-Image .

Figure 12:  Qualitative Results. Given a text input, Text-to-Hand-Params (T2H)
generates SMPL-H [34, 57] parameters. We extract the MANO-Hand from SMPL-H and
use some camera parameters to obtain 2D hand poses. The text, MANO-Hand and 2D
hand poses are used to generate the final image using Text-Guided Hand-Params-
to-Image .


# Observability of the scalar Aharonov-Bohm effect inside a 3D Faraday cage
with time-varying exterior charges and masses

RY Chiao  University of California, Merced, Schools of Natural Sciences and
Engineering, P.O. Box 2039, Merced, CA 95344, USA  XH Deng  University of
California, Merced, School of Natural Sciences, P.O. Box 2039, Merced, CA
95344, USA  KM Sundqvist  Electrical and Computer Engineering, Texas AM
University, College Station, TX 77843, USA  NA Inan  University of California,
Merced, School of Natural Sciences, P.O. Box 2039, Merced, CA 95344, USA  GA
Munoz  California State University, Fresno, CA 93740, USA  DA Singleton
California State University, Fresno, CA 93740, USA  BS Kang  University of
California, Merced, School of Natural Sciences, P.O. Box 2039, Merced, CA
95344, USA  LA Martinez  University of California, Merced, School of Natural
Sciences, P.O. Box 2039, Merced, CA 95344, USA

([; date; date; date; date)

###### Abstract

In this paper we investigate the scalar Aharonov-Bohm (AB) effect in two of
its forms, i.e., its electric form and its gravitational form. The standard
form of the electric AB effect involves having particles (such as electrons)
move in regions with zero electric field but different electric potentials.
When a particle is recombined with itself, it will have a different phase,
which can show up as a change in the way the single particle interferes with
itself when it is recombined with itself. In the case where one has quasi-
static fields and potentials, the particle will invariably encounter fringing
fields, which makes the theoretical and experimental status of the electric AB
effect much less clear than that of the magnetic (or vector) AB effect. Here
we propose using time varying fields $outside$ of a spherical shell, and
potentials $inside$ a spherical shell to experimentally test the scalar AB
effect. In our proposal a quantum system will always be in a field–free region
but subjected to a non-zero time-varying potentials. Furthermore, our system
will not be spatially split and brought back together as in the magnetic AB
experiment. Therefore there is no $spatial$ interference and hence no shift in
a $spatial$ interference pattern to observe. Rather, there arises purely
$temporal$ interference phenomena. As in the magnetic AB experiments, these
effects are non-classical. We present two versions of this idea: (i) a
Josephson temporal interferometry experiment inside a superconducting
spherical shell with a time-varying surface charge; (ii) a two-level atom
experiment in which the atomic spectrum acquires FM sidebands when it is
placed inside a spherical shell whose exterior mass is sinusoidally varying
with time. The former leads to a time-varying internal magnetic field, and the
latter leads to a time-varying gravitational redshift.

one two three

###### pacs:

PACS number

††preprint: HEP/123-qed

year number number identifier Date text]date

LABEL:FirstPage1 LABEL:LastPage#1

##  I Introduction

In the 19th century, Faraday showed that when the exterior of a large,
enclosed cubical metallic cage (i.e., a “Faraday cage”) is electrified at such
a high voltage that sparks started to dart from the corners of the cage, he
could still safely conduct many sensitive electrical experiments within the
cage, such as sensitive electroscope measurements of the charge residing on
the interior surface of the cage. He found the complete absence of any charges
residing on the interior surface. Therefore in the special case of a spherical
“Faraday cage” configuration, such as the one depicted in Figure 1, one would
never expect any kind of electrical effects to be detectable inside the hollow
spherical cavity which is carved out of this metallic sphere.

But what is impossible classically is sometimes possible quantum mechanically.
For example, a 2D, cylindrical (i.e., tubular) Faraday cage was used in
Aharonov and Bohm’s original paper AB effect , in which they first proposed
the electric (or “scalar”) Aharonov-Bohm (AB) effect.111We do not use the term
“scalar” here to refer to neutron interferometry experiments that have been
conducted in a uniform magnetic field, but reserve it to refer to the electric
and gravitational AB effects. A metallic tube shielded an electron passing
through the tube from any exterior electric fields. However, if a voltage
pulse were to be applied to the $exterior$ of the tube only when the electron
wavepacket were to be deep in the $interior$ of the tube, then the electron
could not feel any forces during its passage through the tube. Nevertheless,
the electron would pick up the scalar AB phase shift

$\varphi(t)=\frac{e}{\hbar}\int_{0}^{t}V\left(t^{\prime}\right)dt^{\prime}$
(1)

caused by the voltage pulse $V\left(t\right)$ applied to the tube whilst the
electron was deep in the interior of the Faraday cage.

The scalar (electric) AB effect is less known than the vector (magnetic)
version of the AB effect since it is harder to achieve the situation required
for the electric AB effect where fields are vanishing while the potentials are
non-zero. If one considers the fields and potentials for the cylindrical
Faraday cage used in Aharonov and Bohm’s original paper, then the electron
will invariably pass through some region with a non-zero electric field,
although the field may be extremely small. To have the electron pass only
through regions where there is no field, one needs to switch the fields and
potentials on and off completely, i.e., it is necessary to consider the time-
dependent fields and potentials described by equation (1), where $V(t)$ is a
function with compact support.

The existence of the scalar AB effect has been questioned (see for example the
paper by Walstad Walstad ) exactly on the basis that some experimental
confirmations of the effect Oudenaarden  have the interfering electron passing
through regions where the electric field is non-zero, and thus (potentially)
one could explain the shift in an interference pattern in terms of classical
forces rather than as a quantum phase effect.

In this paper we are proposing two variants of the scalar AB effect as two
“thought experiments” which address these questions with setups where a
quantum system that is influenced by potentials is always deep inside a field-
free region of space. However, since both of these “thought experiments”
involve time-dependent potentials which have no spatial gradients, the
resulting effect on the system is not a $spatial$ interference phenomenon, as
in the vector (magnetic) AB effect, but rather a $temporal$ interference
phenomenon.

From equation (1) it can be seen that the phase in the case of the electric AB
effect involves just an open time integration as opposed to the usual closed-
path spatial line integral of the vector (magnetic) AB effect. This opens the
possibility of setting up a scalar AB experiment where one does not split the
system along different spatial paths, as in the vector AB setup, but instead
the quantum system stays at a single location while the potential will be
varying in time. Since we will not be spatially splitting and recombining our
system, no $spatial$ interference pattern will result, and therefore no shift
in the $spatial$ fringes as in the vector (magnetic) AB effect. However, we
shall see below that there can still be a shift in the fringes of a purely
$temporal$ interference pattern, or a change of the frequency spectrum of the
system.

Our proposal directly tests the scalar AB effect without any “loopholes” that
would allow for the effect to be explained any other way. The basic setup
involves a spherical metallic shell (i.e., a Faraday cage) that has an
oscillating charge $Q\left(t\right)$ or mass $M\left(t\right)$ deposited on
it. Inside the shell, the potential is spatially uniform, but is time-varying.
The two systems that are placed inside this shell are a Josephson-circuit
setup and a two-level atom. In both cases, the interior system does not move
spatially and there is a complete absence of any field (electric or
gravitational), but the system will experience a time-varying potential energy
$U\left(t\right)$ which creates an observable AB effect.

##  II An electric scalar AB effect via Josephson interferometry

We present here a method to observe the electric scalar AB effect by a
superconducting artificial atom Clarke Review ; Makhlin Review . In order to
shield the “atom” from external magnetic fields, as shown in Figure 1, it is
confined within a superconducting Faraday cage. Internal scalar potential
differences arise by sending rf-signals to the surface of the cage. Due to the
shielding of the cage, communication between the exterior and interior can
only be by way of the scalar AB phase of a Cooper pair, in the limit of
electrostatics equilibrium established instantly all over the whole Faraday
cage. This assumption satisfies the fact that the skin effect on the surface
of the cage doesn’t affect the interior. It may be possible that indications
of this internal effect can also be measured by the external circuit.

Figure 1: In the spherical superconducting (SC) shell, the SC artificial atom
is formed with the hollow cavity and a superconducting wire on the horizontal
axis. A SC island is connected to the cavity with a SC wire on one end and via
a large Josephson junction on the other end. The effective inductance between
the wire and cavity can be enhanced by increasing the number of switchbacks
depicted as a solenoid. The spherical SC shell is made thick enough to prevent
magnetic flux penetration into interior.

We demonstrate this system by considering the superconducting model circuit of
Figure 2. An AC voltage is introduced onto the sphere by charging its self-
capacitance. For a superconductor subject to a voltage $V\left(t\right)$, a
phase factor $\varphi(t)=\tfrac{2e}{\hbar}\int_{0}^{t}V\left(t\right)dt$
develops within the order parameter of the Cooper pair condensate,
$\Psi=\sqrt{\rho}e^{i\varphi}$. Furthermore, a phase difference between the
superconducting banks of the Josephson junction also gives rise to a
supercurrent according to the inverse AC Josephson effect, i.e. the Levinsen
effect Inverse ac JosephsonE . It is therefore possible that the scalar AB
phase allows the fully enclosed superconducting circuit to be driven by the
external signal generator.

Figure 2: A simplified circuit includes three circuit loops that connected on
node 1. In the leftmost loop, a signal generator drives the system. The next
loop with the capacitor $C_{sphere}$ represents the spherical SC shell.
$C^{\prime}$ and $L$ are the effective capacitance and inductance of the SC
island, respectively. The superconducting phase $\varphi_{2}$ on the central
superconductor (node 2) and $\varphi_{1}$ on the sphere (node 1) are separated
by a Josephson junction modeled with nonlinear inductance $L_{J}$ and
effective capacitance $C_{J}$. The right loop demonstrates a simple example of
measurement, an rf-SQUID (light blue), although many possible variations of
external low-noise readout exist to detect the change of $\varphi_{1}$ in
time.

Neglecting possible readout schemes for our model circuit, we construct the
Lagrangian using node potentials and phases
$\\{V_{1},\varphi_{1};V_{2},\varphi_{2}\\}$ in the standard way QuantumCircuit
; Devoret . Node phases are related to fluxes by
$\varphi_{1,2}=\dfrac{2\pi}{\Phi_{0}}$
$\Phi_{1,2}=\dfrac{2\pi}{\Phi_{0}}\int\mathbf{A}_{1,2}\cdot d\mathbf{x,}$
where flux quanta are $\Phi_{0}=h/2e$. Omitting bias conditions, we find

$\displaystyle\mathcal{L}\left(\varphi_{2},t\right)$
$\displaystyle=\frac{C_{sphere}}{2}V_{1}^{2}$ $\displaystyle\hskip
14.45377pt+\frac{C_{\Sigma}}{2}\left(V_{1}-V_{2}\right)^{2}$
$\displaystyle\hskip
14.45377pt-\frac{1}{2L}\left(\Phi_{1}-\Phi_{2}\right)^{2}$
$\displaystyle\hskip 14.45377pt+E_{J}\cos\alpha\left(\Phi_{1}-\Phi_{2}\right)$
(2)
$\displaystyle=\left(\frac{\hbar}{2e}\right)^{2}\left[\tfrac{1}{2}C_{sphere}\dot{\varphi}_{1}^{2}\right.$
$\displaystyle\hskip
14.45377pt+\frac{C_{\Sigma}}{2}\left(\dot{\varphi}_{2}-\dot{\varphi}_{1}\right)^{2}$
$\displaystyle\hskip
14.45377pt-\frac{1}{2L}\left(\varphi_{2}-\varphi_{1}\right)^{2}$
$\displaystyle\hskip
14.45377pt\left.+E_{J}\cos\left(\varphi_{2}-\varphi_{1}\right)\right]$ (3)

Considering $V_{1}$ is driven $\left(V_{1}=V_{0}\sin\omega t\right)$, the
interior phase $\Delta\varphi=\varphi_{2}-\varphi_{1}$ and
$\Delta\dot{\varphi}=\dot{\varphi}_{2}-\dot{\varphi}_{1}$ become the
independent variables in the Lagrangian. This allows us to get the following
equation of motion with driving on $\dot{\varphi}_{1}$.

$\displaystyle 0$
$\displaystyle=\Delta\ddot{\varphi}+\omega_{C}^{2}\Delta\varphi+\left(\frac{2e}{\hbar}\right)^{2}\frac{E_{J}}{C_{\Sigma}}\sin\Delta\varphi$
$\displaystyle\hskip
14.45377pt+\left(\frac{2e}{\hbar}\right)^{2}\frac{C_{g}V_{0}\omega}{C_{\Sigma}}\cos\omega
t$ (4)

where $\Delta\varphi=\left(\varphi_{2}-\varphi_{1}\right)$,
$\omega_{C}=\dfrac{1}{LC^{\prime}}$. Here $C_{g}$ is the effective gate
capacitance of the SC island and $C_{\Sigma}$ is the total capacitance of the
system. We adopt the following initial conditions, which assume an initial
steady state throughout the system.

$\left\\{\begin{array}[c]{c}\varphi_{2}\left(t=0\right)=\varphi_{1}\left(t=0\right)\\\
\dot{\varphi}_{2}\left(t=0\right)=\dot{\varphi}_{1}\left(t=0\right)\end{array}\right.$
(5)

Figure 3 depicts the calculated dynamics of $\Delta\varphi$. When the floating
part node 2 is initially grounded and the internal circuit is cooled down to
ground state, a signal of $V_{1}$ drives $\Delta\varphi$ starting at some
moment according to the equation of motion (4). Hence an interior supercurrent
is being driven after the signal is turned off at some point and causes a
nonvanishing $\Delta\varphi$ because of the Josephson effect, which reflects
on changing $\varphi_{1}$ from the internal circuit. An rf-SQUID (light blue
in Figure 2) at the read out port picks up the time dependent phase
$\varphi_{1}$ and converts it into time dependent magnetic flux which can be
detected from an exterior circuit. We have therefore demonstrated that the
external generator affects the fully enclosed superconducting circuit.

Figure 3: Phase difference across JJ $\Delta\varphi$ in terms of tme, solved
from the Lagrangian equation of motion, when $\omega_{0}\approx
8.5GHz,\alpha\approx 2\times 10^{-15}Wb,V_{0}=1\mu V,\omega=150MHz.$

Also notice that the internal Josephson junction circuit, with the possible
addition of current or flux biasing, shares a similar circuit topology to
phase or flux qubits, respectively Phase qubit ; Flux qubit . In any case,
when $E_{J}\gg E_{C}=\tfrac{\left(2e\right)^{2}}{2C_{\Sigma}}$, phase is a
good quantum number. The anharmonic potential energy,
$U\left({\Delta\varphi}\right)=\left({\tfrac{\hbar}{{2e}}}\right)^{2}\tfrac{1}{{2L}}\left({\Delta\varphi}\right)^{2}-E_{J}\cos\left({\Delta\varphi}\right)$
is depicted as in Figure 4 and gives quantized energy levels similar to an
atom occupying the local minimas of this periodic potential. This artificial
atomic system, as we have just determined, is affected by the external
modulating voltage. Note that the electric potential becomes
$\partial/\partial\varphi$ which is the conjugate variable of
${\Delta\varphi}$ in the Hamiltonian. Driving on the potential could displace
the phase from ${\Delta\varphi=0}$ to ${\Delta\varphi=2\pi}$. A $2\pi$ phase
in the internal circuit corresponds to a flux quantum in the cavity generated
by the persistent current going through the central SC wire. After releasing
the driving, the system may stay in the ${\Delta\varphi=2\pi}$ state until
there is a relaxation back to ${\Delta\varphi=0}$, which can be detected via
measurement of the rf-SQUID (rightmost loop in Figure 2). If the driving
finishes one cycle from ${\Delta\varphi=0}$ to ${\Delta\varphi=0}$, the
magnetic flux quantum comes back to zero in the cavity. Therefore, the
internal magnetic flux becomes the consequence of a temporal scalar AB effect.

Figure 4: A plot of potential energy for $E_{L}=1GHz\times h,E_{J}=25GHz\times
h.$ Blue is the ground state for each well while red is the first excited
state. Inter-well tunneling may occur, in which microwave photons may be
emitted or absorbed.

Here we make the following observations:

(1) The supercurrent is created due interference between two paths for the
propagation of the SC phase to node-2. One of the paths is via the JJ and the
other is via the solenoid.

(2) The scalar AB phase discussed here is only for a single Cooper pair
instead of a bulk system. As pointed out earlier, the phase difference drives
the supercurrent that in principle can be detected. Considering the bulk
system, the AB phase factor should include the phase from all Cooper pairs,
$\varphi_{C.P.}\left(t\right)=\frac{2e}{\hbar}\int_{0}^{t}N_{C.P.}\left(t\right)V\left(t\right)dt$,
and electrons,$\
\varphi_{els}\left(t\right)=\frac{e}{\hbar}\int_{0}^{t}N_{els}\left(t\right)V\left(t\right)dt$.
Here $N_{C.P.}\left(t\right)$ is the total number of Cooper pairs (which is
not conserved in principle) and $N_{els}\left(t\right)$ is the total number of
electrons of the whole system. Also, AB phase factor from ionic lattice should
be included as well
$\varphi_{ion}\left(t\right)=\frac{-e}{\hbar}\int_{0}^{t}N_{ion}\left(t\right)V\left(t\right)dt$.
For the case of a 3D rf-SQUID confined within the Faraday cage (see Figure 1),
there will arise time-varying charge imbalances between the charge of the
fixed ionic lattice and the charge of the mobile Cooper pairs on the SC
island, which will give rise to a nonzero total AB phase developed in the
internal circuit. The AB effect here merely drives the Cooper pair condensate
and generates a magnetic field in the internal cavity. This is the physical
reaction caused by the scalar AB phase that we expect here by theoretical
analysis.

(3) Driving the SC atom away from ${\Delta\varphi=0}$ to
${\Delta\varphi=2}\pi$ becomes analogous to the case of the ionization of an
atom. Consequently, we can expect that treating this perturbation results in a
nonlocal photoelectric effect.

##  III A gravitational AB phase shift observable as a time-dependent
gravitational redshift

Here we consider the problem of a two-level atom that is undergoing a time-
dependent gravitational redshift when the atom is placed inside a time-varying
spherical mass shell $M(t)$ (see Figure 5) Again, as in the electric case, the
gravitational scalar potential will be uniform everywhere within the interior
of a mass shell, so that no gravitational force will be experienced by the
atom. Nevertheless, there can in general arise a scalar AB phase Hohensee ,
which arises from the Newtonian gravitational scalar potential $\Phi$

$\varphi\left(t\right)=\frac{1}{\hbar}\int_{0}^{t}m\left(t^{\prime}\right)\Phi\left(t^{\prime}\right)dt^{\prime}$
(6)

where $m\left(t^{\prime}\right)$ is the time-varying rest mass of the quantum
system that is acquiring this phase shift. The Newtonian gravitational scalar
potential for a time-varying mass shell $M\left(t\right)$, such as that
associated with the sinusoidally time-varying charge $Q\left(t\right)$ on the
surface of the shell depicted in Figure 1, is given by

$\displaystyle\Phi\left(t\right)$
$\displaystyle=-\frac{GM\left(t\right)}{r_{0}}$
$\displaystyle=-\frac{G\left(M_{0}+M_{1}\cos\omega t\right)}{r_{0}}$ (7)

where $G$ is Newton’s constant, $r_{0}$ is the radius of the mass shell,
$M_{0}$ is the DC component of the mass shell, and $M_{1}$ is the amplitude of
the AC component of the time-varying mass shell. It goes without saying that
the mass-to-charge ratio of the electron is so tiny that the AC component
associated with $M_{1}$ will be extremely small, so that there would be no
hope for any practical laboratory experiment in connection with the Faraday
cage configuration shown in Figure 5. However, we are concerned here with the
problem of whether _in principle_ the gravitational AB phase exists or not, so
that a “thought experiment” would suffice here.

Nevertheless, there exist astrophysical situations, such as in the case of an
exploding mass shell of a supernova, where the radius $r_{0}\left(t\right)$ is
the time-dependent quantity rather than the mass $M\left(t\right)$. Then the
gravitational AB phase shift may be large enough to be seen in practice. In
any case, although the gravitational scalar potential may vary with time,
nevertheless it must be independent of the position of the field point within
the interior of the mass shell. This follows from Gauss’s theorem. Therefore
the atom within the mass shell experiences no classical forces. However, it
can experience a nonzero quantum phase shift arising from the gravitational AB
effect.

Figure 5: Two-level atom inside a spherical mass shell with a time-dependent
mass $M(t)$ that arises from a time-dependent charge $Q(t)$. The mass $M(t)$
results a time-dependent gravitational redshift that leads to observable FM
sidebands in the spectrum of the atom.

The rest mass of the excited state of an atom or of a nucleus will be larger
than the rest mass of the ground state of this atom or nucleus. This follows
from Einstein’s equation

$E=mc^{2}$ (8)

In quantum mechanics, Einstein’s equation becomes

$mc^{2}=\left\langle\psi\left(t\right)\right|i\hbar\frac{\partial}{\partial
t}\left|\psi\left(t\right)\right\rangle$ (9)

Thus the rest mass $m$ in relativity is the expectation value of the energy
operator $i\hbar\partial/\partial t$ in quantum mechanics. For example, when
an atom is in a stationary state
$\left|\psi\left(t\right)\right\rangle=\left|\Psi_{E}\right\rangle\exp\left(-iEt/\hbar\right)$,
it follows from (9) that

$mc^{2}=E\left\langle\Psi_{E}\right.\left|\Psi_{E}\right\rangle=E$ (10)

and thus we recover Einstein’s equation (8). We shall assume that (9) holds in
general for open quantum systems, such as that of an atom inside a time-
varying mass shell depicted in Figure 5.

Now from the expression for the gravitational AB phase (6), we expect that the
atom in a stationary state with an energy $E$ inside the mass shell will pick
up an AB phase factor, so that

$\left|\psi\left(t\right)\right\rangle=\left|\Psi_{E}\right\rangle\exp\left(-iEt/\hbar\right)\exp\left(-i\varphi\left(t\right)\right)$
(11)

Substituting this into (9), we find

$mc^{2}=\left\langle\Psi_{E}\right|\\{E+\hbar\dot{\varphi}\\}\left|\Psi_{E}\right\rangle=E+\hbar\dot{\varphi}$
(12)

which is an application of (9) to the case of a time-dependent environment,
such as the time-varying mass shell. To find $\dot{\varphi}$, let us take the
time derivative of the expression for the AB phase (6). Then one obtains the
relationship

$\dot{\varphi}=\frac{1}{\hbar}m\left(t\right)\Phi\left(t\right)$ (13)

The physical meaning of this relationship is that the “instantaneous”
frequency $\dot{\varphi}$ associated with the modulation of the phase of the
atomic wavefunction due to an “instantaneous” change in the gravitational
potential energy $m\left(t\right)\Phi\left(t\right)$ of the atom inside the
mass shell, leads to an ”instantaneous” energy change of the energy level of
the atom given by

$\delta E=\hbar\dot{\varphi}=m\left(t\right)\Phi\left(t\right)$ (14)

Upon substitution of $\dot{\varphi}$ from (13) back into (12), it follows that

$mc^{2}=E+m\left(t\right)\Phi\left(t\right)$ (15)

from which we infer that

$E=mc^{2}\left(1-\frac{\Phi}{c^{2}}\right)$ (16)

which leads to the conclusion that the rest mass of quantum mechanical systems
may be increased due to its external gravitational environment.

If we now take the difference in the upper and lower energy levels of the two-
level atom and set it equal to the frequency of an emitted photon $f_{0}$
times Planck’s constant $h$, we will find that

$hf_{0}=hf\left(1-\frac{\Phi}{c^{2}}\right)$ (17)

Since $\Phi$ is a negative quantity, this implies that the photon of energy
$hf$ as seen by an observer at infinity will be smaller in energy than the
photon of energy $hf_{0}$ as seen by an observer near to the atom. Thus we
have recovered Einstein’s gravitational redshift starting from the
gravitational AB phase shift.

Now if the potential $\Phi$ were to be time varying due to changes in the mass
shell, the gravitational red shift would be changed by the time variation of
$\Phi$. If the time variation were to be sinusoidal, then we would expect the
emission and absorption spectrum of the atom to undergo FM modulation. To see
this, let us assume that the states of a two-level atom are represented by
$\left|i\right\rangle$ for the initial state and by $\left|f\right\rangle$ for
the final state. Then Fermi’s Golden Rule states that the rate of transitions
$w_{f\leftarrow i}$ between these two states will be given by the absolute
square of the transition matrix element connecting the initial and final
states, i.e.,

$w_{f\leftarrow i}\propto\left|\left\langle
f\right|H^{\prime}\left|i\right\rangle\right|^{2}$ (18)

where $H^{\prime}$ is the time-dependent perturbation that causes the
transitions to occur. (For the present purposes, we ignore the proportionality
constant and the density of final states in Fermi’s Golden Rule.)

Let us first consider the effect of the $electric$ scalar AB effect on the
transitions between the initial and final states of a $charged$ atom, i.e., an
ion, within the spherical shell of the Faraday cage. The electric AB phase is
given by

$\varphi_{q}\left(t\right)=\frac{q}{\hbar}\int_{0}^{t}V\left(t^{\prime}\right)dt^{\prime}$
(19)

where $q$ is the charge of the ion. Since the initial and final states of the
transition must have the same charge $q$ (which follows from Wigner’s charge
superselection rule), it follows that the electric AB phase factors in the
transition matrix elements in Fermi’s Golden Rule must cancel out, i.e.,

$\displaystyle w_{f\leftarrow i}$ $\displaystyle\propto\left|\left\langle
f\right|e^{+i\varphi_{q}(t)}H^{\prime}e^{-i\varphi_{q}(t)}\left|i\right\rangle\right|^{2}$
$\displaystyle=\left|\left\langle
f\right|H^{\prime}\left|i\right\rangle\right|^{2}$ (20)

since $[H^{\prime},V\left(t\right)]=0$. From (20), we conclude that the
electric AB effect cannot be observed in the spectroscopy of any charged
atomic system.

However, this is $not$ true for the gravitational AB effect. This is because
of the fact that the $rest$ $mass$ of an excited atom will be greater than the
$rest$ $mass$ of the unexcited atom. (Recall that there exists no
superselection rule for mass, unlike for the case of charge.)

Now from Einstein’s equation (8), it follows that there exists a rest mass
difference between the final and the initial states of the two-level atom,
which is given by

$\Delta m=\frac{\Delta E}{c^{2}}=\frac{E_{f}-E_{i}}{c^{2}}$ (21)

where $E_{f}$ is the energy level of the final state and $E_{i}$ is the energy
level of the initial state. Therefore from (6), we see that the difference in
the gravitational AB phase picked up by the final state and the phase picked
up the initial state will no longer vanish, but will differ by the amount

$\displaystyle\Delta\varphi\left(t\right)$ $\displaystyle=\frac{\Delta
m}{\hbar}\int_{0}^{t}\Phi\left(t^{\prime}\right)dt^{\prime}$
$\displaystyle=\frac{E_{f}-E_{i}}{\hbar
c^{2}}\int_{0}^{t}\Phi\left(t^{\prime}\right)dt^{\prime}$ (22)

where, to a a first approximation, the energy difference $E_{f}-E_{i}$ is
independent of $\Phi$.

As a simple example of how the difference in the gravitational AB phases in
the initial and final states can lead to an observable AB interference effect,
let us consider a superposition of the initial and final states which is
initially given by

$\left|\Psi\left(t=0\right)\right\rangle\propto\left|i\right\rangle+\left|f\right\rangle$
(23)

After a time $t=T$, this superposition will evolve to pick up phase factors,
viz.,

$\displaystyle\left|\Psi\left(t=T\right)\right\rangle$
$\displaystyle\propto\left|i\right\rangle
e^{-\frac{i}{\hbar}E_{i}T}e^{-i\varphi_{i}T}$
$\displaystyle+\left|f\right\rangle
e^{-\frac{i}{\hbar}E_{f}T}e^{-i\varphi_{f}T}$ (24)

where

$\varphi_{i}\left(T\right)=\frac{E_{i}}{\hbar
c^{2}}\int_{0}^{T}\Phi\left(t^{\prime}\right)dt^{\prime}$ (25)
$\varphi_{f}\left(T\right)=\frac{E_{f}}{\hbar
c^{2}}\int_{0}^{T}\Phi\left(t^{\prime}\right)dt^{\prime}$ (26)

are the gravitational AB phases picked up by the initial and final states,
respectively. From (22) and (24) it follows that

$\left|\Psi\left(t=T\right)\right\rangle\propto\left|i\right\rangle+\left|f\right\rangle
e^{-\frac{i}{\hbar}\left(E_{f}-E_{i}\right)T}e^{-i\Delta\varphi T}$ (27)

where
$\Delta\varphi\left(T\right)=\varphi_{f}\left(T\right)-\varphi_{i}\left(T\right)$
is the difference of the two AB phases. From (6) and (7), we see that

$\varphi_{i}\left(T\right)=\alpha_{i}\sin\omega t$ (28)
$\varphi_{f}\left(T\right)=\alpha_{f}\sin\omega t$ (29)

where $\alpha_{i}=GM_{1}m_{i}/\hbar\omega r_{0}$ and
$\alpha_{f}=GM_{1}m_{f}/\hbar\omega r_{0}$ are the FM modulation parameters
for the initial and final wavefunctions, respectively, of the two-level atom.
The sinusoidal modulations of the phases given by (28) and (29) will lead to
many FM harmonics of the frequency $\omega$ via the Jacobi-Anger expansion of
the wavefunctions of the two-level atom (see Appendix A). For large values of
$\alpha_{i}$ and $\alpha_{f}$, the dominant upper and lower FM sidebands of
the FM-modulated wavefunctions of the atom will occur at the frequency shifts
$\pm\alpha_{i}\omega$ and $\pm\alpha_{f}\omega$ away from their usual
frequencies of $E_{i}/\hbar$ and $E_{f}/\hbar$.

It follows that the usual energy-conservation-enforcing delta function in the
Fermi Golden rule will be modified from the usual two-level atom resonance
condition not only by the usual gravitational red shift stemming from the DC
component of the mass shell $M_{0}$, but it will also be modified due to the
FM sidebands that arise from the AC component of the time-varying mass shell
$M\left(t\right)$. The bottom line of this analysis is that the usual
absorption or emission line of the two-level atom will be split into upper and
lower FM sideband frequencies occurring on either side of the unsplit line of
the atom with frequency shifts of $\pm(\Delta\alpha)\omega$, where

$\displaystyle\Delta\alpha$ $\displaystyle=\left(\alpha_{f}-\alpha_{i}\right)$
$\displaystyle=\frac{GM_{1}\left(m_{f}-m_{i}\right)}{\hbar\omega r_{0}}$ (30)

is the difference between the FM modulation parameters of the final state and
the initial state that stems from the difference in their rest masses,
$m_{f}-m_{i}$.

##  IV Conclusions

We conclude from the above two “counter-examples” that the claims of the non-
existence of the scalar AB effect are false. Although these two “counter-
examples” are by nature merely “thought experiments,” they do establish the
$existence$ of the electric AB effect and the $existence$ of the gravitational
AB effect _in principle_. However, they may ultimately lead to actual
experiments in the laboratory in the Josephson interferometry case, and to
actual observational evidence in astrophysical settings in the gravitational
redshift case.

Finally, we note that the magnetic (vector) AB effect as observed by Tonomura
Tonomura  using ferromagnetic toroids in electron interference experiments,
are obviously topological in nature. However, the electric and gravitational
(scalar) AB effects that are predicted to occur here inside the metallic
shells of Figures 1 and 5, are obviously non-topological in nature.

##  V Appendix A: Bloch’s theorem for scalar potentials that are periodic in
time

Consider the general case in which the potential energy inside the Faraday
cages depicted in Figures 1 and 5 satisfy the periodicity condition in time

$U\left(t+T\right)=U\left(t\right)$ (31)

where the period $T=2\pi/\omega$ is that of an arbitrary periodic waveform
generator that replaces the sine-wave generators in Figures 1 and 5. Then it
is apparent that this temporal periodicity condition is mathematically
identical to the spatial periodicity condition

$U\left(x+a\right)=U\left(x\right)$ (32)

that applies to a 1D crystalline lattice with a lattice constant $a$.

Bloch’s theorem Kittel  then tells us that the wavefunction inside the 1D
crystalline lattice is given by

$\psi\left(x\right)=e^{ipx/\hbar}u_{p}\left(x\right)$ (33)

where $p$ is the “crystal momentum” or “quasi-momentum”, and where

$u_{p}\left(x+a\right)=u_{p}\left(x\right)$ (34)

is a periodic function of $x$ within the spatial crystalline lattice.

Similarly, the temporal version of Bloch’s theorem (also known as “Floquet’s
theorem”) is given by

$\psi\left(t\right)=e^{-iEt/\hbar}u_{E}\left(t\right)$ (35)

where $E$ is the “crystal energy” or “quasi-energy” Zeldovich Silveri , and
where

$u_{E}\left(t+T\right)=u_{E}\left(t\right)$ (36)

is a periodic function of $t$ within a certain “temporal crystalline lattice.”
Wilczek  (We have suppressed the spatial dependence of the wavefunction
$\psi\left(t\right)$ and of the periodic function $u_{E}\left(t\right)$ as
being understood in (35). This also applies to all of the following
expressions for $\psi\left(t\right)$ and $u_{E}\left(t\right)$). Both the
“crystal momentum” and the “crystal energy” are _physically observable
quantities_ that obey _conservation laws_ , because of the discrete
translational symmetry of the crystalline systems in $x$ and in $t$,
respectively, which follow from the translational symmetry of (31) and (32).

Since $u_{E}\left(t\right)$ is a periodic function of time with a period $T$,
it can be expanded by Fourier’s theorem into a Fourier series expansion

$u_{E}\left(t\right)=\sum_{n=-\infty}^{+\infty}c_{n}\exp\left(-in\omega
t\right)$ (37)

where $c_{n}$ are the Fourier coefficients of $u_{E}\left(t\right)$, and where
$\omega=2\pi/T$ is the frequency of periodic charge waveform $Q\left(t\right)$
that is being injected onto the surface of the Faraday cage in Figures 1 and 5
by the arbitrary periodic waveform generator. Substituting (37) into (35), one
finds that

$\displaystyle\psi\left(t\right)$
$\displaystyle=\sum_{n=-\infty}^{+\infty}c_{n}\exp\left(-i\left(E+n\hbar\omega\right)t/\hbar\right)$
$\displaystyle=\sum_{n=-\infty}^{+\infty}c_{n}\exp\left(-iE_{n}t/\hbar\right)$
(38)

so that we conclude that

$E_{n}=E+n\hbar\omega$ (39)

which describes the “quasi-energy” levels Zeldovich  of any charged quantum
system inside the cavity of a Faraday cage which is being driven by an
arbitrary periodic waveform.

Note that this derivation of the spectrum of quasi-energy levels (39) applies
to the case of $any$ periodic potential energy function $U\left(t\right)$ .
However, let us now consider the important special case of a $sinusoidal$ time
variation of $U\left(t\right)$.

The wavefunction of a quantum system inside the Faraday cage such as the ones
depicted in Figures 1 and 5, will be phase modulated by the time-varying
potential energy in accordance with the time-dependent Schrödinger equation

$i\hbar\frac{\partial\psi}{\partial t}=H\psi=(H_{0}+U(t))\psi$ (40)

where the $H$ is the total Hamiltonian, $H_{0}$ is the unperturbed
Hamiltonian, and $U\left(t\right)$ is the potential energy of the quantum
system inside the spherical shell, which results, for example, from the
injection of the charge $Q\left(t\right)$ onto the surface of the spherical
metallic shell (i.e., Faraday cage) in Figure 1. Note that $U\left(t\right)$
will be independent of the position of any field point in the volume within
the shell. Thus

$U\left(t\right)=U_{0}\cos\omega t$ (41)

for the case of an oscillating charge $Q\left(t\right)=Q_{0}\cos\omega t$
which is $exterior$ to the Faraday cage. However, any field arising from the
spatial gradients of $U\left(t\right)$ $interior$ to the space containing the
quantum system is zero.

Now we shall assume that the quantum system is initially in an unperturbed
eigenstate of the unperturbed Hamiltonian $H_{0}$, i.e.,

$H_{0}\psi=E\psi$ (42)

where $E$ is the unperturbed energy level of the system. Since
$\left[H_{0},U(t)\right]=0$, it follows that the solution to the time-
dependent Schrödinger equation is

$\displaystyle\psi\left(t\right)$
$\displaystyle=\psi\left(0\right)e^{-\frac{i}{\hbar}\left(Et+\int_{0}^{t}U(t^{\prime})dt^{\prime}\right)}$
$\displaystyle=\psi\left(0\right)e^{-\frac{i}{\hbar}Et}e^{-i\varphi\left(t\right)}$
(43)

where $\varphi(t)$ is the phase shift of the wavefunction of the system which
is associated with the scalar AB effect, i.e.,

$\varphi\left(t\right)=\frac{1}{\hbar}\int_{0}^{t}U\left(t^{\prime}\right)dt^{\prime}$
(44)

in agreement with (1) and (6). Using the explicit functional form of (41) in
order to evaluate this integral, we find that

$\varphi\left(t\right)=\alpha\sin\omega t$ (45)

where the “FM depth of modulation” $\alpha$ parameter is defined as follows:

$\alpha=\frac{U_{0}}{\hbar\omega}$ (46)

Thus we find that the wavefunction of the system in the presence of the
$interior$ potential energy $U\left(t\right)$, which is caused, for example,
by the $exterior$ charge $Q\left(t\right)$, will have the form

$\psi\left(t\right)=\psi\left(0\right)e^{-\frac{i}{\hbar}Et}e^{-i\alpha\sin\omega
t}$ (47)

Now from the generating function for Bessel functions, one obtains the Jacobi-
Anger expansion Abramowitz

$e^{-i\alpha\sin\omega
t}=\sum_{n=-\infty}^{\infty}J_{n}\left(\alpha\right)e^{-in\omega t}$ (48)

where $J_{n}\left(\alpha\right)$ is the $n^{th}$ order Bessel function of the
argument $\alpha$. The meaning of the index $n$ is that it denotes the
$n^{th}$ harmonic sideband of the phase modulated wavefunction, which will end
up modifying the quasi-energy level structure of the quantum system. Positive
values of $n$ will correspond to upshifted-frequency sidebands, and negative
values of $n$ to downshifted-frequency sidebands in the quasi-energy spectrum.

Substituting the Jacobi-Anger expansion into the wavefunction (47), we
conclude that

$\displaystyle\psi\left(t\right)$
$\displaystyle=\psi\left(0\right)e^{-\frac{i}{\hbar}Et}\sum_{n=-\infty}^{\infty}J_{n}\left(\alpha\right)e^{-in\omega
t}$
$\displaystyle=\psi\left(0\right)\sum_{n=-\infty}^{\infty}J_{n}\left(\alpha\right)e^{-\frac{i}{\hbar}\left(E+n\hbar\omega\right)t}$
$\displaystyle=\psi\left(0\right)\sum_{n=-\infty}^{\infty}J_{n}\left(\alpha\right)e^{-\frac{i}{\hbar}E_{n}t}$
(49)

where the quasi-energy levels $E_{n}$ are once again given by the expression

$\displaystyle E_{n}$ $\displaystyle=E+n\hbar\omega$ (50)

##  VI Acknowlegements

We thank Profs. Dan Stamper-Kurn and Chih-Chun Chien for helpful discussions.

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