# Open-Vocabulary Affordance Detection in 3D Point Clouds

Toan Nguyen1, Minh Nhat Vu2, An Vuong1, Dzung Nguyen1, Thieu Vo1, Ngan Le3,
Anh Nguyen4 1 FPT Software AI Center, Vietnam<EMAIL_ADDRESS>Automation & Control Institute (ACIN), TU Wien, Vienna, Austria
<EMAIL_ADDRESS>Department of Computer Science & Computer Engineering,
University of Arkansas<EMAIL_ADDRESS>Department of Computer Science,
University of Liverpool, UK<EMAIL_ADDRESS>
###### Abstract

Affordance detection is a challenging problem with a wide variety of robotic
applications. Traditional affordance detection methods are limited to a
predefined set of affordance labels, hence potentially restricting the
adaptability of intelligent robots in complex and dynamic environments. In
this paper, we present the Open-Vocabulary Affordance Detection (OpenAD)
method, which is capable of detecting an unbounded number of affordances in 3D
point clouds. By simultaneously learning the affordance text and the point
feature, OpenAD successfully exploits the semantic relationships between
affordances. Therefore, our proposed method enables zero-shot detection and
can detect previously unseen affordances without a single annotation example.
Intensive experimental results show that OpenAD works effectively on a wide
range of affordance detection setups and outperforms other baselines by a
large margin. Additionally, we demonstrate the practicality of the proposed
OpenAD in real-world robotic applications with a fast inference speed
($\approx$ $100\text{\,}\mathrm{ms}$).

##  I INTRODUCTION

The concept of affordance, proposed by the ecological psychologist James
Gibson [1], plays an important role in various robotic applications, such as
object recognition [2, 3, 4], action anticipation [5, 6, 7], agent’s activity
recognition [8, 9, 10], and object functionality understanding [11, 12, 13,
14]. In these applications, affordances are used to illustrate the potential
interactions between the robot and its surrounding environment. For instance,
with a general cutting task, the knife’s affordances can guide the robot to
use the knife’s blade to achieve requirements such as mincing meat or carving
wood. Detecting object affordances, however, is not a trivial task since the
robots need to understand in real-time the arbitrary correlations between
objects, actions, and effects in complex and dynamic environments [15].

Traditional methods for affordance detection utilize classical machine
learning methods on the images, such as Support Vector Machine (SVM) [16]
based affordance prediction [17], texture-based and object-level monocular
appearance cues [18], relational affordance model [19], and human-object
interactions [20]. With the rise of deep learning, several works have employed
Convolutional Neural Networks (CNN) [21, 22, 23] for different affordance-
related tasks, such as affordance reasoning [24, 25], pixel-based affordance
detection [26, 27, 28, 29, 30, 31], and functional scene understanding [32,
33, 34]. The key challenge in detecting object affordances via the imagery
data is that object affordances may differ in terms of visual information such
as shape, size, or geometry while being similar in object functionality [35].
In practice, object affordance detection from the images requires an
additional step to be applied to downstream robotic tasks, as we need to
transform the detected results from 2D to 3D using the depth information [36].

With the increasing availability of advanced depth cameras, 3D point cloud has
become a popular modality for robotic applications [37]. Compared to
conventional images, 3D point clouds directly provide the robot with 3D
information about surrounding objects and the environment [38]. Consequently,
several recent works directly utilize the 3D point clouds for affordance
detection [39, 40, 36, 41, 42, 43]. For instance, Kim et al. [39] detected
affordances by dividing point clouds into segments and classifying them using
logistic regression. The authors in [44] proposed a new grasp detection method
in point cloud data. More recently, Mo et al. [36] predicted affordance heat
maps from human-object interaction via the scene point cloud. In this work, we
address the task of affordance detection in 3D point clouds to directly apply
the results to robotic application tasks. More specifically, we consider the
affordances of point-level objects and propose a new method to generalize the
affordance understanding using the open-vocabulary setting.

Figure 1: The comparison between traditional affordance detection methods (a)
and our method (b). Traditional methods are restricted to predefined
affordance label sets, while our OpenAD enables open-set affordance labels.

While several works have been proposed for affordance understanding using 2D
images or 3D point clouds, they are mostly restricted to a  _predefined
affordance label set_. This limitation prevents robots from quickly adapting
to a wide range of real-world scenarios or responding to changes in the
operating environments. Recently, increasing the flexibility of affordance
labels has been studied in [45, 46, 47] as one-shot or few-shot learning
problems. However, the authors in [45, 46, 47] only consider the 2D images as
input and treat the problem as the classical pixel-wise affordance
segmentation task. In this work, we overcome the limitation of the fixed
affordance label set by addressing the affordance detection in 3D point clouds
under the  _open-vocabulary_ setting. Our key idea is to learn collaboratively
the mapping between the _language labels_ and the _visual features_ of the
point cloud. In contrast to traditional methods that are limited to a
predefined set of affordance labels, our approach allows the robot to utilize
an unrestricted number of natural language texts as input and, therefore, can
be used in a broader range of applications. Moreover, unlike [45, 46, 47], our
method does not require annotation examples for unseen affordances and can
also work directly with 3D data instead of 2D images. The main concept of our
approach is illustrated in Figure 1.

In this paper, we present Open-Vocabulary Affordance Detection (OpenAD). Our
main goal is to provide a framework that does not restrict the application to
a fixed affordance label set. Our method takes advantage of the recent large-
scale language models, i.e., CLIP [48], and enhances the generalizability of
the affordance detection task in 3D point clouds. Particularly, we propose a
simple, yet effective method for learning the affordance label and the visual
feature together in a shared space. Our method enables zero-shot learning with
the ability to process new open-language texts as query affordances.

Our contributions are summarized as follows:

  * •

We present OpenAD, a simple but effective method to tackle the task of open-
vocabulary affordance detection.

  * •

We conduct intensive experiments to validate our method and demonstrate the
usability of OpenAD in real-world robotic applications.

##  II Related Work

Pixel-Wise Affordance Detection. A large number of works consider affordance
detection as a pixel-wise labeling task, see, e.g., [26, 49, 50, 28, 51, 27,
29, 52, 53, 54, 55]. In [49], the authors predicted the ability to perform
actions in different locations using the concept of the action map. Nguyen et
al.[28] detected object affordances in real-world scenes using an object
detector and dense conditional random fields (CRF) [56]. The authors in [27]
proposed a two-branch framework that simultaneously recognizes multiple
objects and their affordances from RGB images. Chen et al. [54] presented a
multi-task dense prediction architecture and a tailored training framework to
address the problem of joint semantic, affordance, and attribute parsing. More
recently, Luo et al. [55] proposed a cross-view knowledge transfer framework
to extract invariant affordances from exocentric observations and transfer
them to egocentric views. Some other works designed different settings for
their task of affordance detection on images [20, 57]. In particular, Hassan
et al. [20] predicted high-level affordances by investigating mutual contexts
of humans, objects, and the ambient environment, while Chen et al. [57]
learned meaningful affordance indicators to predict actions for autonomous
driving.

Affordance Detection in 3D Point Clouds. Affordance detection is also studied
on 3D point cloud data [39, 40, 36, 41, 42, 43]. Particularly, Kim et al. [39]
presented a method to extract geometric features from point cloud segments and
classify affordances by logistic regression. Subsequently, Kim and Sukhame
[41] presented a technique that voxelizes point cloud objects and creates an
affordance map through interactive manipulation. Kokic et al. [40] proposed a
system for modeling relationships between task, object, and a grasp to address
the problem of task-specific robot grasping. Also focusing on detecting
affordances for the task of grasping, the authors in [42] localized grasping
affordances for industrial bin-picking applications. Lately, Mo et al. [43]
learned affordance heatmaps from object-object interaction. Regardless of
performing on images or 3D point clouds, none of the above methods addressed
the task of open-vocabulary affordance detection.

Language-Driven Segmentation. Language-driven segmentation has recently
attracted research interest in computer vision and machine learning. Promising
results of language-driven segmentation are achieved by large-scale language
models such as CLIP [48] or BERT [58]. Inspired by these works, several
researchers attempted to apply the same idea to other domains, including
semantic segmentation [59, 60, 61, 62, 63, 64, 65]. For instance, Li et al.
[59] aligned pixel-wise features and text features to tackle the task of zero-
shot semantic segmentation. With a similar focus, Xu et al. [60] proposed a
two-stage framework that first extracts mask proposals and then performs
classification on the masked image crops generated. Rozenberszki et al. [61]
proposed a language-driven contrastive pre-training method to address the task
of indoor semantic segmentation. Using the pre-trained text encoder from CLIP
for open-vocabulary 3D scene understanding, Peng et al. [62] proposed a
technique that fused 2D and 3D features and aligned the combination with text
embeddings. Despite recent development in the field of language-driven
segmentation, most recent works have focused only on 2D images, while the task
of open-vocabulary affordance detection in 3D point clouds remains an open
problem.

##  III Open-Vocabulary Affordance Detection

Figure 2: The overview of the proposed Open-Vocabulary Affordance Detection
(OpenAD) network. First, the input point cloud is fed into a point cloud
network to extract per-point embeddings. Second, the affordance labels are
passed into a text encoder to extract the text embeddings. Subsequently, the
correlation between the point-wise features and the corresponding text
embeddings is computed using the cosine similarity function. Finally, a
softmax layer is employed to predict language-driven affordances.

###  III-A Problem Formulation

We consider the task of open-vocabulary affordance detection in 3D point
clouds by training a combined vision-language model. Specially, we take into
account an input point cloud
$\mathcal{C}=\left\\{\mathbf{p}_{1},\mathbf{p}_{2},...,\mathbf{p}_{n}\right\\}$
of $n$ unordered points, $\mathbf{p}_{i}\in\mathbb{R}^{3}$, $i=1,...,n$. Each
point is represented by its coordinate in Euclidean space. The affordance
labels are presented in a natural language form, i.e.,
$\mathcal{L}=\left\\{\mathbf{l}_{1},\mathbf{l}_{2},...,\mathbf{l}_{m}\right\\}$.
Note that, in our open-vocabulary label setting, the number of labels $m$ can
ideally be unlimited. The testing label set can differ from the training label
set and can contain unseen affordance labels.

Our goal is to jointly learn the affordance labels and the input point cloud.
We first extract the point cloud feature using a deep point cloud network. The
affordance label is encoded by a text encoder. We then introduce a simple,
however, effective correlation metric to jointly learn the point cloud visual
feature and the text embedding features. In this manner, our method leverages
the similarities of text embeddings of unseen affordances that are
semantically related to the ones seen in the training process. The overall
framework of our method is depicted in Figure 2.

###  III-B Open-Vocabulary Affordance Detection

Text encoder. The text encoder $f_{\rm{text}}\left(\cdot\right)$ embeds the
set of potential affordance labels into an embedding space $\mathbb{R}^{D}$.
Similar to other works [59, 60], the text encoder can be an arbitrary network.
In this work, we employ the state-of-the-art pre-trained ViT-B/32 text encoder
from CLIP [48]. The text encoder produces $m$ word embeddings
$\mathbf{T}_{1},\mathbf{T}_{2},...,\mathbf{T}_{m}\in\mathbb{R}^{D}$ as the
presentation of the input affordance labels. Note that we only use the CLIP
text encoder to extract the text features and freeze
$f_{\rm{text}}\left(\cdot\right)$ during both training and testing.

Point cloud network. The second component of our method is the point cloud
network. The $n$ input points are plugged into the point cloud network $f_{\rm
pc}\left(\cdot\right)$ producing an embedding vector for every input point.
Similar to the text encoder, the architecture of the point cloud network can
be various. In this work, we use the state-of-the-art point cloud model, i.e.,
PointNet++ [66] as the underlying architecture. Furthermore, we append to the
end of the backbone with a convolutional layer with $D$ output units shared
across all points, followed by a batch norm layer. More specifically, the
point cloud network $f_{\rm pc}\left(\cdot\right)$ produces a set of $n$
vectors $\mathbf{P}_{1},\mathbf{P}_{2},...,\mathbf{P}_{n}\in\mathbb{R}^{D}$.
In contrast to the text encoder, the weights of the point cloud network
$f_{\rm pc}\left(\cdot\right)$ are updated during the training.

Learning text-point correlation. To enable open-vocabulary affordance
detection, the semantic relationships between the point cloud affordance and
its potential labels have to be computed. Particularly, we correlate point-
wise embeddings of the input point cloud $\mathbf{P}_{i}$ and text embeddings
of affordance labels $\mathbf{T}_{j}$ using the cosine similarity function.
The correlation value $F_{i,j}$, which is an element of the $i$-th row and
$j$-th column of the correlation matrix $\mathbf{F}\in\mathbb{R}^{n\times m}$,
is computed as

${F}_{i,j}=\frac{{\mathbf{P}^{\top}_{i}}\mathbf{T}_{j}}{\left\|\mathbf{P}_{i}\right\|\left\|\mathbf{T}_{j}\right\|}\>\>.$
(1)

The point-wise softmax output of a single point $i$ is then computed in the
form

${S}_{i,j}=\dfrac{\exp\left({F}_{i,j}/\tau\right)}{\sum_{k=1}^{m}\exp({{F}_{i,k}}/\tau)}\>\>,$
(2)

where $\tau$ is a learnable temperature parameter [67]. This computation is
applied for every point in the point cloud.

During the training, we encourage the point cloud network $f_{\rm
pc}\left(\cdot\right)$ to provide point embeddings that are close to the text
embeddings. These text embeddings are produced by the text encoder $f_{\rm
text}\left(\cdot\right)$ of the corresponding ground-truth classes.
Specifically, given the embedding $\mathbf{P}_{i}\in\mathbb{R}^{D}$ of point
$i$, we aim to maximize the value of the entry ${F}_{i,j}$ that is the
similarity of $\mathbf{P}_{i}$ and the text embedding $\mathbf{T}_{j}$
corresponding to the ground-truth label $j=y_{i}$. This can be accomplished by
optimizing the weighted negative log-likelihood loss of the point-wise softmax
output over the entire point cloud in the form

${L=-\sum_{i=1}^{n}w_{y_{i}}\log{S}_{i,{y_{i}}}}\>\>,$ (3)

where $w_{y_{i}}$ is the weighting parameter to the imbalance problem of the
label classes during the training. Inspired by [66], we define this weight as

$w_{j}=\left(\frac{\mathrm{max}\left\\{c_{1},c_{2},...c_{m}\right\\}}{c_{j}}\right)^{1/3},$
(4)

where $c_{j}$ is the number of points of the class $j$ of the training set.

###  III-C Training and Inference

During the training, we fix the text encoder and train the rest of the network
end-to-end. Similar to [36], we fix the number of points in a point cloud to
$n=2048$. We set $D$ to $512$. We train our network using the Adam optimizer
[68] with the learning rate $\alpha=10^{-3}$ and the weight decay
$\gamma=10^{-4}$. The proposed framework is trained over $200$ epochs on a
24GB-RAM NVIDIA Geforce RTX 3090 Ti with a batch size of $16$. We initialize
the value of $\ln{(1/0.07)}$ for $\tau$. During the inference, we feed the
point cloud and any text as the input label to detect the wanted affordance
from the point cloud. The inference process takes approximately
$100\text{\,}\mathrm{ms}$ on average with our proposed network.

##  IV Experiments

In this section, we perform several experiments to validate the effectiveness
of our OpenAD. We start with a zero-shot detection setting to verify the
ability of OpenAD to generalize to previously unseen affordances. Secondly, we
present OpenAD’s notable qualitative results together with visualizations.
Finally, we conduct additional ablation studies to further investigate other
aspects of OpenAD.

###  IV-A Zero-Shot Open-Vocabulary Affordance Detection

Dataset. We use the 3D AffordanceNet dataset [36] in our experiments. 3D
AffordanceNet dataset is currently the largest dataset for affordance
understanding using 3D point cloud data with $22,949$ instances from $23$
object categories and $18$ affordance labels. As in other zero-shot setups
[69, 70, 71], we need more label classes to verify the robustness of the
methods, therefore, we re-label the 3D AffordanceNet with the extra $18$
affordance classes and also consider the background as a class, hence making
the total number of affordance labels $37$. Following [36], we benchmark on
the two tasks: the full-shape and partial-view tasks. The partial-view setup
is more useful in robotics as the robot usually can only observe a partial-
view of the object’s point cloud.

Baselines and Evaluation Metrics. We compare our method with the following
recent methods for zero-shot learning in 3D point clouds: ZSLPC [69], TZSLPC
[70], and 3DGenZ [71]. Note that, since these baselines used GloVe [72] or
Word2Vec [73] for word embedding, which are less powerful models than CLIP, we
replace their original text encoders with CLIP for a fair comparison. For
ZSLPC [69] and TZSLPC [70], we change their classification heads to the
segmentation task. As in [66, 74, 75], we use three metrics to evaluate the
results: mIoU (mean IoU over all classes), Acc (overall accuracy over all
points), and mAcc (mean accuracy over all classes).

Resutls. Table I shows that our OpenAD achieves the best results on both tasks
and all three metrics. In particular, on the full-shape task, OpenAD
significantly surpasses the runner-up model (ZSLPC) by 4.40% on mIoU. OpenAD
also outperforms others on Acc (0.84% over 3DGenZ) and on mAcc (0.81% over
ZSLPC). Similarly, for the partial-view task, our method has the highest mIoU
(3.98% higher than the second-best ZSLPC), and also the highest Acc and mAcc.

TABLE I: Zero-shot Open-vocabulary detection results
Task | Method | mIoU | Acc | mAcc
---|---|---|---|---
Full-shape | TZSLPC[70] | 3.86 | 42.97 | 10.37
| 3DGenZ [71] | 6.46 | 45.47 | 18.33
| ZSLPC [69] | 9.97 | 40.13 | 18.70
| OpenAD (ours) | 14.37 | 46.31 | 19.51
Partial-view | TZSLPC [70] | 4.14 | 42.76 | 8.49
| 3DGenZ [71] | 6.03 | 45.24 | 15.86
| ZSLPC [69] | 9.52 | 40.91 | 17.16
| OpenAD (ours) | 12.50 | 45.25 | 17.37

###  IV-B Qualitative Results

We present several examples to demonstrate the generality and flexibility of
OpenAD. Primarily, we use objects from the 3D AffordanceNet [36] for our
visualizations. We also select objects from the ShapeNetCore dataset [76] to
analyze the capability of OpenAD to generalize to unseen object categories and
new affordance labels.

Generalization to New Affordance Labels. We illustrate several examples
showing the ability of OpenAD to generalize to unseen affordance classes in
Figure 3. In the upper row of Figure 3, we present the detection results of
OpenAD for nine seen affordances on appropriate objects. As trained on these
affordances, OpenAD produces good detection results. Next, for each object, we
feed new affordance labels to the models while keeping the same corresponding
object, and present the detection results in the two rows below. The
visualization shows that OpenAD successfully detects the associated regions
for the queried new affordance labels, even though the labels are not included
in the training set.

Figure 3: The visualization of our OpenAD’s capability to detect seen and
unseen affordances. The upper row: Detection results of OpenAD on seen
affordances. The middle and lower rows: Detection results of OpenAD on unseen
affordances that do not exist in the training set.

Generalization to Unseen Object Categories. In this work, OpenAD is trained on
3D AffordanceNet dataset [36], which covers 23 object categories. To verify
the generalization ability of our method on new unseen objects, we select new
novel objects from the ShapeNetCore dataset [76] and test them in both cases:
seen and open-vocabulary affordance labels. We use the farthest point sampling
algorithm to uniformly sample $2,048$ points from the surface of each object.
All points are then centered and scaled before being fed into OpenAD. Figure 4
summarises the results. This figure shows that OpenAD is able to detect the
affordance classes on new object categories. This confirms the generalization
of our OpenAD for downstream robotic tasks.

Figure 4: Results on unseen object categories. OpenAD gives reasonable outputs
when detecting affordances on object categories that are not in the training
set. Upper row: Seen affordances. Lower row: Unseen affordances.

Multi-Affordance Detection. In OpenAD, the number of affordances in the label
set, $m$, can vary. This flexible design allows OpenAD to detect multiple
affordances at once. Figure 5 presents the detection results of two objects
given label sets with different numbers of affordances. Moreover, we observe a
notable ability of OpenAD that it does not fix the label for a particular
point but finds the most suitable label in a specific label set. Concretely,
given an object, OpenAD first detects affordances in a particular label set.
By maintaining the earlier affordances, once a new affordance label is added
to the label set, there are certain points will be labeled by new affordances
over the previous affordance classes. For instance, in the left column of
Figure 5, the points in the upper body of the bottle are labeled as contain in
the first run, then they are re-labeled as wrap-grasp in the second run and
finally as grasp in the last run. This OpenAD’s ability can also be observed
in the case of the knife object in the right column of Figure 5. It further
demonstrates our OpenAD’s flexibility.

Figure 5: Multi-affordance detection using our method.

###  IV-C Ablation Study

Will language-driven architecture affect the detection results? In this work,
we mainly focus on jointly learning a vision-language model to improve the
generalization in downstream robotic tasks, and do not aim for improving the
accuracy of the traditional affordance detection tasks. This leads to the
question that whether our language-driven model affects the accuracy of the
traditional affordance detection task. To verify this, we train our OpenAD on
the original 3D AffordanceNet with its original label set and training split,
and compare our result with other state-of-the-art methods, including
PointNet++ [66], Dynamic Graph CNN (DGCNN) [74] and Point Transformer [75].
For PointNet++ and DGCNN, we follow similar designs in [36] and change the
final classifier to a linear layer detecting the affordance classes. For Point
Transformer, we apply the same architecture in [75]. Table II presents the
results of all methods on 3D AffordanceNet [36]. From this table, we find that
OpenAD performs competitively when compared to other methods. Therefore, we
can conclude that while our method is designed for a different purpose, it
still can be used as a strong benchmark for closed-set affordance detection.

TABLE II: Closed-set detection results
Task | Method | mIoU | Acc | mAcc
---|---|---|---|---
Full-shape | Point Transformer [75] | 41.26 | 68.67 | 67.03
| PointNet++ [66] | 41.26 | 68.51 | 68.14
| DGCNN [74] | 42.09 | 68.47 | 61.47
| OpenAD (ours) | 42.00 | 69.03 | 67.31
Partial-view | Point Transformer [75] | 40.51 | 69.22 | 65.34
| PointNet++ [66] | 41.10 | 69.40 | 66.74
| DGCNN [74] | 41.93 | 69.38 | 63.12
| OpenAD (ours) | 41.87 | 69.91 | 66.34

Backbones and Text Encoders. In table III, we conduct an ablation study on two
different point cloud backbones, i.e., PointNet++ [66] and DGCNN [74], and two
different pretrained text encoders, i.e., CLIP ViT-B/32 [48] and BERT [58].
Note that with BERT, the parameter $D$ is set to $768$. We observe that
different combinations of backbones and text encoders perform equivalently on
the closed-set tasks. Meanwhile, on the open-vocabulary tasks, PointNet++
performs better than DGCNN, and CLIP ViT-B/32 performs better than BERT. The
gap in the performance of frameworks using CLIP text encoder compared to those
using BERT is significant, demonstrating the superiority of CLIP in semantic
language-vision understanding.

TABLE III: Ablation study on point cloud backbone and text encoder
Task & setting |  Point cloud backbone |  Text encoder | mIoU | Acc | mAcc
---|---|---|---|---|---
Full-shape & | PointNet++ | CLIP | 14.37 | 46.31 | 19.51
Open-vocabulary | PointNet++ | BERT | 10.55 | 46.81 | 16.02
| DGCNN | CLIP | 10.88 | 45.21 | 15.40
| DGCNN | BERT | 9.43 | 46.37 | 14.61
Partial-view & | PointNet++ | CLIP | 12.50 | 45.25 | 17.37
Open-vocabulary | PointNet++ | BERT | 9.33 | 45.71 | 14.45
| DGCNN | CLIP | 11.19 | 44.21 | 16.43
| DGCNN | BERT | 7.00 | 44.93 | 10.97
Full-shape & | PointNet++ | CLIP | 42.00 | 69.03 | 67.31
Closed-set | PointNet++ | BERT | 41.04 | 68.79 | 67.22
| DGCNN | CLIP | 42.11 | 68.59 | 61.31
| DGCNN | BERT | 42.06 | 68.31 | 61.26
Partial-view & | PointNet++ | CLIP | 41.87 | 69.91 | 66.34
Closed-set | PointNet++ | BERT | 41.50 | 69.77 | 65.55
| DGCNN | CLIP | 41.20 | 69.16 | 64.16
| DGCNN | BERT | 41.27 | 69.62 | 58.09

###  IV-D Robotic Demonstration

The experiment setup, shown in Fig. 6, comprises five main components, i.e.
the robot KUKA LBR iiwa R820, the PC1 running the real-time automation
software Beckhoff TwinCAT, the Intel RealSense D435i camera, the Robotiq 2F-85
gripper, and the PC2 running Robot Operating System (ROS) Noetic 20.04. PC1
communicates with the robot via a network interface card (NIC) using the
EtherCAT protocol, marked by the blue region in Fig. 6. Note that the robot
control is implemented in a C++ module in PC1. The sampling time is set to
$125\text{\,}\mathrm{\SIUnitSymbolMicro s}$ for the robot sensors and
actuators. PC2 controls the gripper and the camera via the USB protocol in the
ROS environment. Additionally, the two PCs communicate with each other via an
Ethernet connection. After receiving point cloud data of the environment from
the RealSense D435i camera, we utilize the state-of-the-art object
localization method [77] to identify the object, then perform point sampling
to get $2048$ points. We then feed this point cloud with a natural language
affordance command. Note that, using our OpenAD, we can have a general input
command and are not restricted to a predefined affordance label set. Our
OpenAD returns the affordance region, which can be used for the grasp pose
detection module [44], the analytical inverse kinematics module [78], and the
trajectory optimization module [79]. Several demonstrations, such as holding
and raising a bag, wrapping a bottle, and pushing an earphone, can be found in
our supplementary material.

USB
---
USB
---
ROS
---
NIC
---
TwinCAT
---
RealSense D435i
---
Robotiq 2F-85
---
Object
---
(a)
---
Object
---
Points
---
OpenAD
---
(b)
---
Localisation
---
Sampling
---
Figure 6: Example of a robot demonstration. (a) Experimental setup. (b) Result
from OpenAD.

###  IV-E Discussion

Despite achieving promising results, OpenAD has its limitation. Our method is
still far from being able to detect completely unseen affordances. The upper
row of Figure 7 shows cases when OpenAD fails to detect unseen affordances on
seen objects. Moreover, we present false-positive predictions of OpenAD in the
lower row of Figure 7. In these cases, OpenAD detects affordances that the
objects do not provide, i.e., display for the bag, support for the microwave,
and openable for the hat. From our intensive experiments, we see several
improvement points for future work: i) learning the visual-language
correlation plays an important role in this task and can be further improved
by using more complicated techniques such that cross-attention mechanism [80]
, ii) applying a stronger point cloud backbone would likely improve the
result, and iii) having a large-scale dataset with several affordance classes
would be beneficial for benchmarking and real-world robotic applications.
Finally, we will also release our source code to encourage further study.

Figure 7: Failure cases of OpenAD. Upper row: Cases when OpenAD fails to
detect unseen affordances. Lower row: OpenAD detects affordances that are not
furnished by the objects.

##  V Conclusions

We proposed OpenAD, a simple yet effective method for open-vocabulary
affordance detection in 3D point clouds. Different from traditional
approaches, OpenAD, with its capability of semantic understanding, can
effectively detect unseen affordances without requiring annotated examples.
Empirical results show that OpenAD outperforms other methods by a large
margin. We further verified the capability of OpenAD to detect unseen
affordances on both known and unseen objects. Additionally, we demonstrated
OpenAD’s usefulness in real-world robotic applications.

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# Quantum nondemolition measurements in the relativistic Dirac oscillator

Wei Sun  Quantum Institute for Light and Atoms, State Key Laboratory of
Precision Spectroscopy, School of Physics and Electronic Science, East China
Normal University, Shanghai 200241, China  Keye Zhang<EMAIL_ADDRESS>Quantum Institute for Light and Atoms, State Key Laboratory of Precision
Spectroscopy, School of Physics and Electronic Science, East China Normal
University, Shanghai 200241, China  Pierre Meystre  J. Wyant College of
Optical Sciences and Department of Physics, University of Arizona, Tucson,
Arizona 85721, USA.  Weiping Zhang<EMAIL_ADDRESS>School of Physics and
Astronomy, and Tsung-Dao Lee Institute, Shanghai Jiao Tong University,
Shanghai 200240, China  Collaborative Innovation Center of Extreme Optics,
Shanxi University, Taiyuan, Shanxi 030006, China  Shanghai Research Center for
Quantum Sciences, Shanghai 201315, China

###### Abstract

We investigate the feasibility of performing quantum nondemolition (QND)
measurements in the one-dimensional Dirac oscillator, a frequently used toy
model to investigate relativistic extensions of nonrelativistic quantum
effects. We derive general expressions for its QND observables and find that
they are in general complex combinations of the position, momentum, and spin
operators, which makes them challenging to realize experimentally. However,
the situation is considerably simplified in both the weakly and strongly
relativistic limits, where they should be amenable to an experimental
demonstration. This suggests exploiting continuous measurements of small
departures from the QND character of these variables in the intermediate
regime as probes of relativistic effects.

††preprint: APS/123-QED

##  I Introduction

Quantum nondemolition (QND) measurements were first proposed by Braginsky
Braginsky (1967); Braginsky and Nazarenko (1969) as a way to circumvent the
Standard Quantum Limit of measurements, that is, the backaction on a second
measurement of a quantum observable $\hat{a}(t)$ from the uncertainty imposed
on its conjugate variable. Under normal circumstances the uncertainty in a
second measurement of an observable $\hat{A}$ is limited by the Heisenberg
Uncertainty Principle. QND variables are variables that commute with
themselves at the times of $t_{0}$ and $t_{1}$ of these measurements,
$[\hat{A}(t_{0}),\hat{A}(t_{1})]=0$. If this condition is satisfied at all
times, $d\hat{A}/dt=0$, $\hat{A}(t)$ is called a continuous QND observable,
else it is a stroboscopic QND observable see e.g. Refs. Braginsky (1967);
Braginsky and Nazarenko (1969); Braginsky  _et al._ (1975, 1980); Thorne  _et
al._ (1978).

Reference. Thorne  _et al._ (1978), see also the reviews Caves  _et al._
(1980) and Bocko and Onofrio (1996), showed that in the case of a simple
harmonic oscillator of mass $m$, frequency $\omega$ and Hamiltonian
$\hat{H}_{0}=\hbar\omega\hat{a}^{\dagger}\hat{a}$, either of the two
quadratures

$\displaystyle\hat{Y}_{1}$ $\displaystyle=$ $\displaystyle\hat{x}\cos\omega
t-(\hat{p}/m\omega)\sin\omega t,$ $\displaystyle\hat{Y}_{2}$ $\displaystyle=$
$\displaystyle\hat{x}\sin\omega t+(\hat{p}/m\omega)\cos\omega t,\,$ (1)

with $[\hat{Y}_{1},\hat{Y}_{2}]=i\hbar/m\omega$, $\Delta Y_{1}\Delta
Y_{2}\geq\hbar/2m\omega$, and $d\hat{Y}_{\ell}/dt=0$, can be chosen as a QND
observable (but of course not both simultaneously). This property was
exploited in particular in optical and atomic interferometry Chen  _et al._
(2017), weak force and gravitational wave detection Chen  _et al._ (2011); Suh
_et al._ (2014), and the preparation of squeezed states Vasilakis  _et al._
(2015).

The goal of this paper is to extend these considerations to the case of a
Dirac oscillator, an extension of the free particle Dirac equation obtained by
the addition of a term linear in position and governed by the equation

$i\hbar\partial_{t}\psi=\left[c{\bm{\alpha}}\cdot\left({\bf p}-im\omega{\bf
r}\beta\right)+\beta mc^{2}\right]\psi.$ (2)

Here ${\bf p}=-i\hbar{\bm{\nabla}}$ and

$\bm{\alpha}=\begin{pmatrix}0&{\bm{\sigma}}\\\ {\bm{\sigma}}&0\\\
\end{pmatrix}\,\,\,\,\,;\,\,\,\,\,\beta=\begin{pmatrix}I&0\\\ 0&-I\\\
\end{pmatrix}$ (3)

with $\bm{\sigma}$ the vector Pauli spin matrix, so that $\alpha_{i}$ and
$\beta$ satisfy the anticommutation relations ($i\neq j=x,y,z$)

$\displaystyle\alpha_{i}\alpha_{j}$ $\displaystyle=-\alpha_{j}\alpha_{i}\,,$
(4) $\displaystyle\alpha_{i}\beta$ $\displaystyle=-\beta\alpha_{i}\,,$

and $\alpha_{i}^{2}=\beta^{2}=1$.

The Dirac oscillator is characterized by the presence of a spin-orbit coupling
term, as becomes particularly apparent when expressing the Dirac wave function
in terms of its large and small components $\psi_{1}$ and $\psi_{2}$ as
Moshinsky and Szczepaniak (1989)

$\psi=\begin{pmatrix}\psi_{1}\\\ \psi_{2}\\\ \end{pmatrix}e^{-iEt/\hbar}$ (5)

with

$\displaystyle(E-mc^{2})\psi_{1}$ $\displaystyle=$ $\displaystyle
c{\bm{\sigma}}\cdot({\bf p}+im\omega{\bf r})\psi_{2}$
$\displaystyle(E+mc^{2})\psi_{2}$ $\displaystyle=$ $\displaystyle
c{\bm{\sigma}}\cdot({\bf p}-im\omega{\bf r})\psi_{1}\,$ (6)

exhibiting the familiar spin-orbit coupling of the Dirac equation.
Reexpressing the energy $E$ as $E=mc^{2}+{\cal E}$, one finds that in the
weakly relativistic limit ${\cal E}\ll mc^{2}$ and when expressed in terms of
the large component $\psi_{1}$ only, the energy of the system takes the form

${\cal E}=\frac{\bf p^{2}}{2m}+\frac{m\omega^{2}\bf
r^{2}}{2}-\frac{2\omega}{\hbar}{\bf L}\cdot{\bf S}-\frac{3\hbar\omega}{2},$
(7)

where ${\bf L}={\bf r}\times{\bf p}$ and ${\bf S}=(\hbar/2){\bm{\sigma}}$
Moshinsky and Szczepaniak (1989). That is, in addition to the familiar energy
of a simple harmonic oscillator, $\cal E$ comprises a spin-orbit coupling
component of strength $-2\omega/\hbar$. It is this property that justifies
calling this system a Dirac oscillator.

The Dirac oscillator is frequently used as a toy model to investigate
relativistic extensions of nonrelativistic quantum effects. These include in
particular the generation of Dirac cat states Bermudez  _et al._ (2007), the
study of a relativistically deformed uncertainty principle in (1+1) dimensions
Benzair  _et al._ (2012), and a relativistic quantum heat engine using a
single Dirac particle trapped in an infinite one-dimensional potential as its
working substance Muñoz and Peña (2012). It has also found wide applications
in nuclear physics, condensed matter physics and quantum optics, see e.g. the
review Quesne (2017).

The spin-orbit coupling characteristic of the Dirac equation is also central
to on-going efforts to control the spin and electronic ground state topology
in quantum systems, which requires its detailed characterization Fasth  _et
al._ (2007), and is also becoming an efficient mechanism of quantum sensing in
hybrid optomechanical systems Abdi  _et al._ (2017); He  _et al._ (2016). For
this reason, extending the familiar physics associated with QND measurements
in the simple harmonic oscillator to this system can help clarify the
difficulties associated with this additional interaction and suggest ways to
exploit this complication in diagnostic tools.

Because of its nonequidistant excitation spectrum and the presence of spin-
orbit coupling, the dynamics of the Dirac oscillator differs significantly
from that of the simple harmonic oscillator. For example, we showed in earlier
work Zhang  _et al._ (2018) that the interference between positive and
negative energy states prevents the realization of quantum-mechanics free
subsystems that avoid quantum measurement backaction. But whether a more
direct generation of QND observables is possible remains an open question.

##  II Model

We consider for simplicity the (1+1)-dimensional version of the Dirac
oscillator. In that case the Dirac operators $\bm{\alpha}$ and $\beta$, reduce
to $2\times 2$ matrices that can be taken as any two of the three Pauli
matrices, for example,

$\alpha_{x}\to\hat{\sigma}_{x}\;\;;\;\;\beta\to\hat{\sigma}_{z}.$ (8)

and the Dirac oscillator is governed by the equation

$i\hbar\frac{\partial\psi}{\partial
t}=\left[c\hat{\sigma}_{x}\hat{p}-mc\omega\hat{\sigma}_{y}\hat{x}+mc^{2}\hat{\sigma}_{z}\right]\psi\equiv\hat{H}_{\rm
D}\psi.$ (9)

We introduce the bosonic annihilation and creation operators $\hat{a}$ and
$\hat{a}^{\dagger}$ and the excitation number operator
$\hat{n}=\hat{a}^{\dagger}\hat{a}$ in the usual way via $\hat{x}=x_{\rm
zpt}(\hat{a}+\hat{a}^{\dagger})$ and $\hat{p}=-ip_{\rm
zpt}(\hat{a}-\hat{a}^{\dagger})$, where $x_{\text{zpt}}=\sqrt{\hbar/2m\omega}$
and $p_{\text{zpt}}=\sqrt{m\hbar\omega/2}$ are the zero-point position and
momentum. $\hat{H}_{D}$ takes then the form of the anti-Jaynes-Cumming
Hamiltonian

$\hat{H}_{\text{D}}=\eta\hat{\sigma}_{-}\hat{a}+\eta^{\ast}\hat{\sigma}_{+}\hat{a}^{\dagger}+mc^{2}\hat{\sigma}_{z},$
(10)

where

$\eta=-imc^{2}\sqrt{2\epsilon}\;\;\;\;;\;\;\;\;\epsilon=\hbar\omega/mc^{2}.$
(11)

As in (3+1) dimensions, the oscillator wave functions $\psi$ are spinors with
components $\psi_{1}$ and $\psi_{2}$. With

$\psi=\begin{pmatrix}\psi_{1}\\\ \psi_{2}\\\ \end{pmatrix}e^{-iEt/\hbar},$
(12)

Eq. (9) takes the form of the coupled equations

$\displaystyle(E-mc^{2})\psi_{1}$
$\displaystyle=c(\hat{p}+im\omega\hat{x})\psi_{2},$ (13)
$\displaystyle(E+mc^{2})\psi_{2}$
$\displaystyle=c(\hat{p}-im\omega\hat{x})\psi_{1}.$

or, in terms of $\hat{a}$ and $\hat{a}^{\dagger}$,

$\displaystyle(E-mc^{2})\psi_{1}$
$\displaystyle=\eta^{*}\hat{a}^{\dagger}\psi_{2},$ (14)
$\displaystyle(E+mc^{2})\psi_{2}$ $\displaystyle=\eta\hat{a}\,\psi_{1},$

which explicitly show that the frequency $\omega$ is a measure of the spin-
orbit coupling between $\psi_{1}$ and $\psi_{2}$ already encountered in Eq.
(7).

The eigenstates of the Hamiltonian (10) are the dressed states

$\displaystyle|E_{n}^{+}\rangle$ $\displaystyle=$ $\displaystyle
A_{n}|n,\uparrow\rangle-iB_{n}|n-1,\downarrow\rangle,$
$\displaystyle|E_{n}^{-}\rangle$ $\displaystyle=$ $\displaystyle
B_{n+1}|n+1,\uparrow\rangle+iA_{n+1}|n,\downarrow\rangle\,,$ (15)

with associated positive and negative eigenenergies

$E_{n}^{-}=-E_{n+1}^{+}=-mc^{2}\sqrt{1+2(n+1)\epsilon}\,.$ (16)

Here $\left|\uparrow\right\rangle,\,\left|\downarrow\right\rangle$ are Pauli
spinors, the states $\left|n\right\rangle$ are eigenstates of the
nonrelativistic harmonic oscillator, and

$\displaystyle A_{n}$ $\displaystyle=$
$\displaystyle\sqrt{(E_{n}^{+}+mc^{2})/2E_{n}^{+}},$ $\displaystyle B_{n}$
$\displaystyle=$ $\displaystyle\sqrt{(E_{n}^{+}-mc^{2})/2E_{n}^{+}}.$ (17)

The asymmetry between the positive and negative energy branches and the
property that the eigenstates are linear superpositions if motional states of
opposite spins are direct consequences of the relativistic spin-coupling
component of $\hat{H}_{\rm D}$.

##  III QND observables

This section derives the general form of a pair of QND variables of the
Hamiltonian $\hat{H}_{\rm D}$ and determines their dependence on the
relativistic level of the oscillator, as characterized by the parameter

$\epsilon=\hbar\omega/mc^{2},$ (18)

with $\omega$ the oscillator frequency and $m$ its rest mass. We consider
first the two limiting cases of the weakly relativistic regime, $\epsilon\ll
1$, and the extreme relativistic regime, $\epsilon\gg 1$, and show that the
corresponding QND variables reduce then to the familiar quadrature operators
$\hat{Y}_{1,2}$ of the simple harmonic oscillator, albeit with spin-dependent
frequencies. In the intermediate regime, however, they take the form of
intricate spin-orbit coupling operators. The following section will give
estimates of experimental parameters achievable in QND measurements of
implementations of the Dirac oscillator with either electrons or cold atoms.

###  III.1 Weakly relativistic limit

In the weakly relativistic limit $\epsilon\ll 1$ we have $E_{n}^{+}\approx
mc^{2}+n\hbar\omega$, $A_{n}\approx 1$ and $B_{n}\approx 0$, so that the
superpositions of opposite spin states disappear, the eigenstates reducing to
$\left|n,\uparrow\right\rangle$ and $i\left|n,\downarrow\right\rangle$,
respectively. This can be seen more explicitly when eliminating $\psi_{2}$ or
$\psi_{1}$ from these two equations by multiplying them by $(E+mc^{2})$ and
$(E-mc^{2})$, respectively, resulting in the Klein-Gordon-like equations

$\displaystyle(E^{2}-m^{2}c^{4})\psi_{1}$
$\displaystyle=2mc^{2}\left(\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}-\frac{\hbar\omega}{2}\right)\psi_{1},$
(19) $\displaystyle(E^{2}-m^{2}c^{4})\psi_{2}$
$\displaystyle=2mc^{2}\left(\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}+\frac{\hbar\omega}{2}\right)\psi_{2}.$

Consider first the component $\psi_{1}$ of the Dirac spinor. With
$E=mc^{2}+{\cal E}$, the weakly relativistic regime corresponds to ${\cal
E}\ll mc^{2}$, so that

$(E^{2}-m^{2}c^{4})\approx 2mc^{2}{\cal E}+O({\cal E}^{2})$ (20)

so that

${\cal
E}\psi_{1}=\left(\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}-\frac{\hbar\omega}{2}\right)\psi_{1}.$
(21)

with corresponding effective positive energy Hamiltonian

$\hat{H}_{\rm{eff}}^{+}=mc^{2}+\left(\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}-\frac{\hbar\omega}{2}\right).$
(22)

Similarly, for the component $\psi_{2}$ we set $E=-mc^{2}+{\cal E}$. The
weakly relativistic limit $|{\cal E}|\ll mc^{2}$ yields then

$(E^{2}-m^{2}c^{4})\approx-2mc^{2}{\cal E}+O({\cal E}^{2}),$ (23)

so that

${\cal
E}\psi_{2}=-\left(\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}+\frac{\hbar\omega}{2}\right)\psi_{2}$
(24)

with corresponding negative energy Hamiltonian

$\hat{H}_{\rm{eff}}^{-}=-mc^{2}-\left(\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}+\frac{\hbar\omega}{2}\right).$
(25)

Combining Eqs. (22) and (25) gives finally the effective Hamiltonian
characterizing the weakly relativistic limit of the (1+1)-dimensional Dirac
oscillator as

$\displaystyle\hat{H}_{\rm nr}$ $\displaystyle=$
$\displaystyle\left[mc^{2}+\frac{\hat{p}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}\right]\hat{\sigma}_{z}-\frac{1}{2}\hbar\omega$
(26) $\displaystyle=$ $\displaystyle
mc^{2}\hat{\sigma}_{z}+\hbar\omega\left(\hat{a}^{\dagger}\hat{a}+\frac{1}{2}\right)\hat{\sigma}_{z}-\frac{1}{2}\hbar\omega.$

In that limit the two components of the spinor $\psi$ become therefore
approximately decoupled and $\hat{\sigma}_{z}$ is a constant of motion, hence
a QND observable. In addition, it also follows that the system possesses an
additional pair of QND observables $\hat{X}_{\ell,\text{nr}}$, $\ell=(1,2)$,
of precisely the same form as the quadrature operators $\hat{Y}_{\ell}$ of
Eqs. (1) but with the important difference that the frequency $\omega$ becomes
now a spin-dependent frequency operator

$\omega\to\hat{\omega}_{\rm{nr}}\equiv\omega\hat{\sigma}_{z},$ (27)

so that

$\displaystyle\hat{X}_{1,\text{nr}}$ $\displaystyle=$ $\displaystyle
x_{\text{zpt}}(\hat{a}e^{i\hat{\omega}_{\text{nr}}t}+{\rm h.c.})\,,$
$\displaystyle\hat{X}_{2,\text{nr}}$ $\displaystyle=$ $\displaystyle
ix_{\text{zpt}}(e^{-i\hat{\omega}_{\text{nr}}t}\hat{a}^{\dagger}-{\rm h.c.}).$
(28)

Since $[\hat{a},\hat{\omega}_{\text{nr}}]=0$ these operators obey the same
commutation relations as the harmonic oscillator quadratures $\hat{Y}_{1,2}$.

###  III.2 Extreme relativistic limit

In the other extreme limit, $\epsilon\gg 1$, the spin-orbit coupling term
$\eta\hat{\sigma}_{-}\hat{a}+\eta^{\ast}\hat{\sigma}_{+}\hat{a}^{\dagger}$
dominates the Hamiltonian so that $E_{n}^{+}\approx mc^{2}\sqrt{2n\epsilon}$
and $A_{n}\approx B_{n}\approx 1/\sqrt{2}$. Moreover, the energy difference
between neighboring motional states $\left|n\right\rangle$ and
$\left|n-1\right\rangle$ scales as $1/\sqrt{n}$ and approaches zero for large
excitations $n\gg 1$. The eigenstates of $\hat{H}_{\rm D}$ can then be
approximated by the factorized motional and spin states
$\left|E_{n}^{\pm}\right\rangle\approx|n\rangle(\left|\uparrow\right\rangle\mp
i\left|\downarrow\right\rangle)/\sqrt{2}$, corresponding to the effective
Hamiltonian

$\hat{H}_{\rm r}=-\hat{\sigma}_{y}\sqrt{2mc^{2}\hbar\omega\hat{n}}\,.$ (29)

In contrast to $\hat{H}_{\rm nr}$ the eigenenergy spectrum of $\hat{H}_{\rm
r}$ are not equidistant, and $\hat{\sigma}_{y}$ rather than $\hat{\sigma}_{z}$
is now a constant of motion, hence a QND observable. Like in the weakly
relativistic case $\hat{H}_{\rm r}$ also possesses a pair of QND observables
$\hat{X}_{\ell,\rm r}$ that are formally identical with the quadrature
operators (28), but now with the replacement of $\omega$ by a frequency
operator $\hat{\omega}_{\rm r}$ with both a motional and spin dependence,

$\omega\to\hat{\omega}_{\rm
r}\equiv\omega\sqrt{2/\varepsilon}(\sqrt{\hat{n}-1}-\sqrt{\hat{n}})\hat{\sigma}_{y}\,.$
(30)

With the equality $\hat{a}f(\hat{n})=f(\hat{n}+1)\hat{a}$ one can readily show
that the dynamical evolution and uncertainty relation of $\hat{X}_{\ell,\rm
r}$ are the same as those of the quadrature operators $\hat{Y}_{\ell}$. (Note
that since $[\hat{a},\hat{\omega}_{\text{r}}]\neq 0$ the operator ordering in
$\hat{X}_{\ell,\rm r}$ cannot be arbitrarily changed.)

###  III.3 General case

It is easily verified that the formal similarity between the QND observables
$\hat{X}_{\ell}$ of the Dirac oscillator in the limits $\epsilon\ll 1$ and
$\epsilon\gg 1$ and the quadratures $\hat{Y}_{\ell}$ of the simple harmonic
oscillator follows from the fact that in both limits $\hat{n}$ is a conserved
quantity (approximately so for the case $\epsilon\gg 1$), in which case QND
observables in that general form, with an appropriately chosen frequency
operator $\hat{\omega}$, can always be found. However, in the intermediate
regime of the Dirac oscillator $\hat{n}$ is no longer a conserved quantity, a
consequence of the spin-orbit induced exchange of energy between the motional
and spin degrees of freedom.

The effects of this spin-orbit coupling can be formally eliminated in the
Foldy-Wouthuysen (FW) representation Moreno and Zentella (1989), as we now
show. For the one-dimensional Dirac oscillator, the FW and the original Dirac
representation are related by the unitary transformations Verlan and Razumova
(2001),

$\hat{U}=\begin{bmatrix}\hat{A}_{n}&i\frac{\hat{B}_{n}}{\sqrt{\hat{n}}}\hat{a}^{\dagger}\\\
\hat{a}\frac{\hat{B}_{n}}{\sqrt{\hat{n}}}&-i\hat{A}_{n+1}\end{bmatrix},$ (31)

where the operators $\hat{A}_{n}$ and $\hat{B}_{n}$ are obtained by replacing
$n$ by $\hat{n}$ in the expressions for $A_{n}$ and $B_{n}$ of Eqs. (17). The
transformed Hamiltonian is then the sum of $\hat{n}$-conserving Hamiltonians
$\hat{H}_{\text{F}}(\hat{n})$ of the form

$\hat{H}_{\text{F}}(\hat{n})=\hat{U}\hat{H}_{\text{D}}\hat{U}^{\dagger}=\hat{\sigma}_{z}mc^{2}\sqrt{1+(2\hat{n}-\hat{\sigma}_{z}+1)\varepsilon}\,,$
(32)

with eigenstates $\left|n,\uparrow\right\rangle$ and
$\left|n,\downarrow\right\rangle$ and associated positive and negative
eigenenergies $E_{n}^{+}$ and $E_{n}^{-}$, respectively. Importantly, the FW
eigenstates do not involve superpositions of the two spin states, that is, the
positive and negative energy eigenstates are uniquely associated with spin up
and spin down, respectively, as was the case in the weakly relativistic limit
$\epsilon\ll 1$. Following a similar approach we can therefore again construct
a pair of QND observables

$\displaystyle\hat{X}_{1,\rm F}$ $\displaystyle=$ $\displaystyle x_{\rm
zpt}(\hat{a}e^{i\hat{\omega}_{\epsilon}t}+{\rm h.c.})\,,$
$\displaystyle\hat{X}_{2,\rm F}$ $\displaystyle=$ $\displaystyle ix_{\rm
zpt}(e^{-i\hat{\omega}_{\epsilon}t}\hat{a}^{\dagger}-{\rm h.c.})\,,$ (33)

where the frequency operator is now

$\hat{\omega}_{\epsilon}\equiv\left[\hat{H}_{\text{F}}(\hat{n})-\hat{H}_{\text{F}}(\hat{n}-1)\right]/\hbar,$
(34)

and accounts for the unequally spaced energy spectrum. As in the previous
cases these quadratures obey the commutation relation
$[\hat{X}_{1,\text{F}},\hat{X}_{2,\text{F}}]=i\hbar/m\omega$.

From Eq. (34), the eigenvalues $\omega_{n}^{\pm}$ of
$\hat{\omega}_{\epsilon}$, given by
$\hat{\omega}_{\epsilon}|n,\uparrow\rangle=\omega_{n}^{+}|n,\uparrow\rangle$
($n\geq 1$) and
$\hat{\omega}_{\epsilon}|n,\downarrow\rangle=\omega_{n}^{-}|n,\downarrow\rangle$
($n\geq 0$), are simply the differences in frequency of neighboring
eigenstates of the FW-transformed Dirac oscillator. In the weakly relativistic
limit $\epsilon\ll 1$ they approach the constant value $\omega$ characteristic
of a harmonic oscillator, while for a system deep in the relativistic regime
we have $\left|\omega_{n}^{\pm}\right|/\omega\to 0$ for large $n$, a signature
of the increasing anharmonicity of the system.

Figure 1: (Color online) (a) Time evolution of the mean values (color lines)
and uncertainties (background shadows) of $\hat{X}_{1,\text{F}}$ (blue) and
$\hat{X}_{2,\text{F}}$ (red) under a continuous QND measurement described by
the Hamiltonian (35); (b) Corresponding evolution of $\hat{x}$ (blue) and
$\hat{p}$ (red). Here the Dirac oscillator is initially in a coherent state
with $\alpha=0.5$, $c_{1}=c_{2}=1/\sqrt{2}$, and $\epsilon=0.1$; the probe
optical field is in a coherent state $|\beta=2\rangle$ and $g=0.1$. Position
in units of $x_{\text{zpt}}=\sqrt{\hbar/2m\omega}$ and momentum in units of
$p_{\text{zpt}}=\sqrt{m\hbar\omega/2}$.

If one were to ignore the important point that $\hat{X}_{1,\rm F}$ needs to be
transformed back to the Dirac representation to assess possible practical
measurement schemes, one might argue that it could formally be monitored e.g.
by continuous QND measurements described by the total Hamiltonian

$\hat{H}_{\rm{tot}}=\hat{H}_{\rm
F}+\hbar\omega_{b}\hat{b}^{\dagger}\hat{b}+g\hat{X}_{1,\rm
F}\hat{b}^{\dagger}\hat{b},$ (35)

where $\hat{b}$ represents the annihilation operator of an optical mode of
frequency $\omega_{b}$, a form reminiscent of the optomechanical Hamiltonian
of quantum optics. The associated Heisenberg equations of motion
$d\hat{X}_{1,\rm F}/dt=0$, $d\hat{X}_{2,\rm
F}/dt=-g\hat{b}^{\dagger}\hat{b}/m\omega$ and
$d\hat{b}/dt=-i(\omega_{b}+g\hat{X}_{1,\rm F}/\hbar)\hat{b}$ show that the QND
variable $\hat{X}_{1,\rm F}$ is imprinted on the phase of the optical mode
$\hat{b}$, while $\hat{X}_{2,\rm F}$ is subject to the the measurement
backaction from the radiation pressure force $g\hat{b}^{\dagger}\hat{b}$. For
example, for a Dirac oscillator initially in the state $|\phi\rangle_{\rm
F}=|\alpha\rangle\otimes(c_{1}\left|\uparrow\right\rangle+c_{2}\left|\downarrow\right\rangle)$,
with $|\alpha\rangle$ a coherent state and $c_{1}$ and $c_{2}$ arbitrary
probability amplitudes, we have that $\Delta\hat{X}_{1,\rm
F}^{2}(0)=\Delta\hat{X}_{2,\rm F}^{2}(0)=\hbar/2m\omega$. In that case, as
shown in Fig. 1(a), the mean value and the uncertainty in the measured
observable $\hat{X}_{1,\rm F}$ remain constant in time as expected of QND
measurements, while the mean value and variance of $\hat{X}_{2,\rm F}$
become$\langle\hat{X}_{2,\rm F}(t)\rangle=\langle\hat{X}_{2,\rm
F}(0)\rangle-(g/m\omega)\langle\hat{n}_{b}\rangle t$ and $\Delta\hat{X}_{2,\rm
F}^{2}(t)=\hbar/2m\omega+(g/m\omega)^{2}\Delta\hat{n}_{b}^{2}t^{2}$.

However, when expressed in the Dirac representation the observables
$\hat{X}_{\ell}=\hat{U}^{\dagger}\hat{X}_{\ell,{\rm F}}\hat{U}$ become in
general complex combinations of $\hat{x}$, $\hat{p}$, and $\bm{\sigma}$, so
that realizing the system-probe interaction (35) is anything but obvious in
general. That this is the case is also apparent from Fig. 1(b), which shows
the evolution of $\langle\hat{x}\rangle$ and $\langle\hat{p}\rangle$ under the
measurement scheme and the same parameters as Fig 1(a). In addition to slow
oscillations at a frequency of order $\omega$, the evolution of these
observables is characterized by high-frequency Zitterbewegung oscillations
induced by the spin-orbit coupling induced interferences between the positive
and negative energy states Zhang  _et al._ (2018), as well as by significant
measurement backaction. This situation is considerably more complex than for
the simple harmonic oscillator, where the QND nature quadratures (1) follows
readily and intuitively from the simple oscillatory evolution of $\hat{x}$ and
$\hat{p}$.

##  IV Implementation

The explicit expressions of $\hat{X}_{1,2}$ in terms of $\hat{x}$, $\hat{p}$,
and $\bm{\sigma}$ in the Dirac representation are in general quite cumbersome,
as they include several spin-orbit-coupling composite observables (see
Appendix A for their explicit form). Their measurement represents therefore a
considerable challenge for arbitrary $\epsilon$, and while possible in
principle, they are likely to remain unrealistic in practice. The situation is
considerably simplified in both the weakly relativistic and extreme
relativistic limits, as we now show.

###  IV.1 Weakly relativistic limit

For $\epsilon\ll 1$ we have

$\hat{X}_{1,\rm nr}=\hat{x}\cos\omega
t-(\hat{p}\hat{\sigma}_{z}/m\omega)\sin\omega t,$ (36)

so that in addition to the measurement of $\hat{x}$ the composite observable
$\hat{p}\hat{\sigma}_{z}$ requires only spin-sensitive momentum measurements.
This is because in this limit the spin-orbit coupling is so weak as not to
couple the components $\psi_{1}$ and $\psi_{2}$ of the spinor, as we have
seen. $\hat{\sigma}_{z}$ is then a constant of motion and commutes with
$\hat{p}$. Consequently, if the initial state of the Dirac oscillator is known
not to exhibit spin-motion correlations, these measurements can be performed
by measuring the spin and the momentum independently.

###  IV.2 Extreme relativistic limit

The situation is somewhat more complex, but still manageable in principle in
the extreme relativistic limit $\epsilon\gg 1$ where

$\hat{X}_{1,\rm
r}=\frac{1}{2}(\hat{x}+i\hat{p}/m\omega)e^{i\hat{\omega}_{r}t}+{\rm h.c.}$
(37)

As we have seen, that regime is characterized by a relativistically induced
anharmonicity of the oscillator which results in an excitation frequency that
is no longer a constant, but rather a quantum observable $\hat{\omega}_{\rm
r}$ that does not commute with $\hat{x}$ and $\hat{p}$, see Eq. (30). In that
case the measurement of the composite observables that comprise
$\hat{X}_{1,\rm r}$ requires a coupling between the probe and $\hat{x}$ and
$\hat{p}$ that is sensitive to the excitation frequency. However, as we shall
see the time evolution of $\hat{\omega}_{\rm r}$ is far slower than the other
observables, so that this coupling can be approximated by introducing an
additional probe of $\hat{\omega}_{\rm r}$ and then combining with a probe of
the motional observables.

To see more concretely how this scheme can be implemented we consider a probe
with internal degree of freedom described by the Pauli spin operators
$\hat{s}_{x,y,z}$, and external degree of freedom by the bosonic operators
$\hat{b}$ and $\hat{b}^{\dagger}$. The measurement scheme is described by the
total Hamiltonian

$\hat{H}_{\rm{tot}}=\hat{H}_{\rm D}+\hbar\omega_{\rm
b}\hat{b}^{\dagger}\hat{b}+\hbar\omega_{\rm{s}}\hat{s}_{z}+\hat{V}_{\rm m},\\\
$ (38)

where $\omega_{b}$ and $\omega_{s}$ are the external and the internal
characteristic frequencies of the probe. in the extreme relativistic limit
considered here $\hat{H}_{\rm D}\approx\hat{H}_{\rm r}$. Finally
$\hat{V}_{\rm{m}}$ describes the measurement interaction. Its explicit form is

$\hat{V}_{\rm m}=\frac{\hbar\hat{\omega}_{\rm r}}{2}\hat{s}_{x}+gx_{\rm
zpt}\big{[}\hat{a}(\hat{s}_{z}-i\hat{s}_{y})+{\rm
h.c.}\big{]}(\hat{b}^{\dagger}+\hat{b}),$ (39)

where the first term accounts for the rotation of the probe internal spin due
to $\hat{\omega}_{r}$, and the second describes the Dirac oscillator driven
modification of the spin-orbit coupling of the probe. The Heisenberg equations
of motion ot the oscillator-probe system are

$\displaystyle\frac{d\hat{s}_{x}}{dt}$ $\displaystyle=$
$\displaystyle-2\omega_{\rm{s}}\hat{s}_{y}-i\frac{g}{p_{\rm
zpt}}[\hat{a}(\hat{s}_{z}-i\hat{s}_{y})-{\rm
h.c.}](\hat{b}^{\dagger}+\hat{b}),$ (40)
$\displaystyle\frac{d\hat{s}_{y}}{dt}$ $\displaystyle=$
$\displaystyle-\hat{\omega}_{\rm{r}}\hat{s}_{z}+2\omega_{\rm{s}}\hat{s}_{x}+\frac{g}{p_{\rm
zpt}}(\hat{a}\hat{s}_{x}+{\rm h.c.})(\hat{b}^{\dagger}+\hat{b}),$ (41)
$\displaystyle\frac{d\hat{s}_{z}}{dt}$ $\displaystyle=$
$\displaystyle\hat{\omega}_{\rm{r}}\hat{s}_{y}+i\frac{g}{p_{\rm
zpt}}(\hat{a}\hat{s}_{x}-{\rm h.c.})(\hat{b}^{\dagger}+\hat{b}),$ (42)
$\displaystyle\frac{d\hat{\omega}_{\rm{r}}}{dt}$ $\displaystyle=$
$\displaystyle\frac{i\hat{\sigma}_{y}}{\hbar}cg(2\sqrt{\hat{n}}-\sqrt{\hat{n}+1}-\sqrt{\hat{n}-1})\hat{a}(\hat{s}_{z}-i\hat{s}_{y})(\hat{b}^{\dagger}+\hat{b})+{\rm
h.c.},$ (43) $\displaystyle\frac{d\hat{a}}{dt}$ $\displaystyle=$
$\displaystyle-i\hat{a}{\hat{\omega}_{\rm{r}}}-\frac{i\hat{\sigma}_{y}}{\hbar}cp_{\rm
zpt}(2\sqrt{\hat{n}}-\sqrt{\hat{n}+1}-\sqrt{\hat{n}-1})\hat{a}\hat{s}_{x}-\frac{ig}{2p_{\rm
zpt}}(\hat{s}_{z}+i\hat{s}_{y})(\hat{b}^{\dagger}+\hat{b}),$ (44)
$\displaystyle\frac{d\hat{b}}{dt}$ $\displaystyle=$
$\displaystyle-i\omega_{b}\hat{b}-i\frac{g}{2p_{\rm
zpt}}[\hat{a}(\hat{s}_{z}-i\hat{s}_{y})+{\rm h.c.}],$ (45)

where the coupling between Eqs. (43) and (44) is due to the fact
$[\hat{a},\hat{\omega}_{\text{r}}]\neq 0$, which leads to the difficulty of
measuring $\hat{x}$, $\hat{p}$, and $\hat{\omega}_{\rm r}$ independently, in
contrast to the weakly relativistic limit.

Assuming that $\langle\hat{a}\rangle$ scales as $\sqrt{n}$, Eq. (43) shows
that $\langle\dot{\hat{\omega}}_{\rm r}\rangle$ scales then as $1/n$, so that
$\langle\dot{\hat{\omega}}_{\rm r}\rangle/\langle\hat{\omega}_{\rm r}\rangle$
scales as $g/\sqrt{n\hbar\omega m}$, where we have used Eq. (30). For weak
enough measurement strength $g$, large enough oscillator excitation $n$, and
assuming further that $\langle\hat{\omega}_{r}\rangle\gg\omega_{s}$, so that
it dominates the evolution of the probe field, see Eqs. (40), (41), and (42),
the Heisenberg equations for $\hat{s}_{x,y,z}$ can be approximated as

$\frac{d\hat{s}_{x}}{dt}\approx 0,\ \
\frac{d\hat{s}_{y}}{dt}\approx-\hat{\omega}_{\rm r}\hat{s}_{z},\ \
\frac{d\hat{s}_{z}}{dt}\approx\hat{\omega}_{\rm r}\hat{s}_{y},$ (46)

so that

$\displaystyle\hat{s}_{x}(t)\approx\hat{s}_{x}(0),$ (47)
$\displaystyle\hat{s}_{y}(t)\approx\hat{s}_{y}(0)\cos{\hat{\omega}_{\rm
r}t}-\hat{s}_{z}(0)\sin{\hat{\omega}_{\rm r}t},$
$\displaystyle\hat{s}_{z}(t)\approx\hat{s}_{y}(0)\sin{\hat{\omega}_{\rm
r}t}+\hat{s}_{z}(0)\cos{\hat{\omega}_{\rm r}t}.$

Substituting these expressions into $\hat{V}_{\rm m}$, and for the probe
initially in the internal state $\left|\uparrow\right\rangle$ so that
$\langle\hat{s}_{x,y}(0)\rangle=0$ and $\langle\hat{s}_{z}(0)\rangle=1$, the
effective form of the measurement interaction becomes

$\hat{V}_{\rm m}\approx g\hat{X}_{\rm{1,r}}(\hat{b}^{\dagger}+\hat{b}),$ (48)

so that this model detection scheme performs indeed approximate QND
measurements on that observable.

###  IV.3 Near QND measurements as probes of relativistic effects

The fact that in the weakly and extreme relativistic limits $\epsilon\ll 1$
and $\epsilon\gg 1$ the QND variables $\hat{X}_{1,\rm nr}$ and $\hat{X}_{1,\rm
r}$ are amenable to direct detection suggests exploiting continuous
measurement departures from their QND character as probes of relativistic
effects, that is, of spin-orbit coupling.

Figure 2: (color online) Upper row: Time evolution of the normalized mean
values (colored lines) and corresponding uncertainties (background shadows of
the same colors) of $\hat{X}_{1,{\rm nr}}$ (blue solid) and
$\hat{X}_{2,\rm{nr}}$ (red dashed) under continuous measurements for
increasing relativistic parameters, (a) $\epsilon=0.001$, (b) $0.02$, and (c)
$0.1$. Middle row: Corresponding time evolution of $\hat{x}$ (blue solid) and
$\hat{p}$ (red dashed) for (d) $\epsilon=0.001$, (e) $0.02$, and (f) $0.1$.
Lower row: Dependence on $\epsilon$ of $\left\langle X_{1,{\rm
nr}}\right\rangle$ (blue solid) and $\left\langle X_{2,{\rm nr}}\right\rangle$
(red dashed) and their variances at time $t=4\pi/\omega$. The Dirac oscillator
is initially in a coherent state with $\alpha=1$ and $c_{1}=c_{2}=1/\sqrt{2}$,
the optical field is in the coherent state $\beta=1$ and the measurement
strength is $g=1$. The normalized units are same as in Fig. 1.

This is illustrated for the example of $\hat{X}_{1,\rm nr}$ in Fig. 2, which
shows both its evolution and the evolution of its conjugate variable
$\hat{X}_{2,\rm nr}$ for three values of $\epsilon$. In the weakly
relativistic regime $\epsilon=10^{-3}$ of Figs. 2(a) and 2(d), $\hat{X}_{1,\rm
nr}$ behaves as expected as a QND observable, but it starts departing from
this behavior as $\epsilon$ increases. This is apparent also from the temporal
increase in its variance, as well as by the onset of Zitterbewegung superposed
to damped oscillations at frequency $2\omega$ for large enough $\epsilon$.
Fig. 2(g) shows the dependence on $\epsilon$ of $X_{1,{\rm nr}}$ and
$X_{2,{\rm nr}}$ at a fixed time $t=4\pi/\omega$, for the same parameters,
illustrating the sensitivity of both $\langle\hat{X}_{1,\rm nr}\rangle$ and
its variance on $\epsilon$.

More quantitatively, on the weakly relativistic side, and keeping terms up to
the first order of $\epsilon$ in $\hat{H}_{\rm D}$, the Heisenberg equations
for $\hat{X}_{\ell,\rm{nr}}$ become

$\displaystyle\frac{d\hat{X}_{1,\rm{nr}}}{dt}$ $\displaystyle\approx$
$\displaystyle-\frac{\hat{\omega}_{\rm{nr}}\epsilon}{2}[i\hat{X}_{1,\rm{nr}}+(2\hat{n}-\hat{\sigma}_{z}+1)\hat{X}_{2,\rm{nr}}],$
(49) $\displaystyle\frac{d\hat{X}_{2,\rm{nr}}}{dt}$ $\displaystyle\approx$
$\displaystyle-\frac{\hat{\omega}_{\rm{nr}}\epsilon}{2}[i\hat{X}_{2,\rm{nr}}-(2\hat{n}-\hat{\sigma}_{z}+1)\hat{X}_{1,\rm{nr}}]-\frac{g\hat{b}^{\dagger}\hat{b}}{m\omega}.$

which illustrates the $\epsilon$-dependence of measurement induced backaction
on $\hat{X}_{1,\rm{nr}}$. (The results of Fig. 2 were obtained numerically,
keeping $\epsilon$ to all orders.)

Similarly, slightly departing from the extreme relativistic limit the
Heisenberg equations for $\hat{X}_{\ell,\rm{r}}$ are

$\displaystyle\frac{d\hat{X}_{1,\rm{r}}}{dt}$ $\displaystyle\approx$
$\displaystyle\frac{1}{8\epsilon}[(i\hat{X}_{1,\rm{r}}-\hat{X}_{2,\rm
r})\frac{\hat{\omega}_{\rm r}}{\hat{n}}-{\rm h.c.}],$ (50)
$\displaystyle\frac{d\hat{X}_{2,\rm{r}}}{dt}$ $\displaystyle\approx$
$\displaystyle\frac{1}{8\epsilon}[(\hat{X}_{1,\rm
r}+i\hat{X}_{2,\rm{r}})\frac{\hat{\omega}_{\rm r}}{\hat{n}}+{\rm
h.c.}]-\frac{g\hat{b}^{\dagger}\hat{b}}{m\omega}.$

Turning finally to order-of-magnitude estimates of relevance for a potential
laboratory implementation of this system, we note first that for a sample of
electrons with mass $m_{e}\sim 10^{-31}$kg in the relativistic limit with
$\epsilon\sim 10^{3}$ and in highly excited states $n\sim 10^{4}$, the
frequency of the Dirac oscillator is $\omega\sim 10^{23}$Hz, and the
uncertainty $\Delta X_{1}\sim 10^{-13}$m, This corresponds to electron
energies of the order of $E_{e}\sim\langle\hat{H}_{\rm r}\rangle\sim 10$MeV,
rendering an experimental implementation challenging. On the other hand, these
numbers are significantly improved in tabletop simulations of relativistic
quantum systems. For example, for a Dirac oscillator simulated by a spin-orbit
coupled ultracold atomic sample Zhang  _et al._ (2018) with effective mass
$m\sim 10^{-27}$kg and effective velocity of light $c\sim 10^{-2}$m/s, and for
a same relativistic limit condition, the oscillating frequency $\omega\sim
10^{6}$Hz and $\Delta X_{1}\sim 10^{-7}$m, with characteristic energy levels
decreased to a manageable $10^{-9}$eV.

##  V Conclusion and outlook

We have established that despite the existence of anharmonicity and spin-orbit
coupling resulting from relativistic effects one can identify a pair of
conjugate QND observables of the one-dimensional Dirac oscillator. They can be
monitored relatively simply in the weakly and strongly relativistic limits of
the oscillator, and may be used for instance as probes of modulated spin-orbit
coupling by relativistic effect. A laboratory implementation should be
possible in artificial Dirac systems. Although a single-particle theory, the
existence of these observables may provide a simple model to investigate
aspects of relativistic quantum measurements Marolf and Rovelli (2002) and the
generalized relativistic uncertainty principle Rosen and Vallarta (1932); Al-
Hashimi and Wiese (2009); Todorinov  _et al._ (2019), and may also find
applications in the development of alternative detection technique for
topological insulators and other quantum systems with genuine or synthetic
spin–orbit coupling Cooper  _et al._ (2019).

###### Acknowledgements.

We acknowledge enlightening discussions with Baiqiang Zhu. This work was
supported by the National Natural Science Foundation of China (Grants No.
11974116 and No. 11654005), the National Key Research and Development Program
of China (Grant No. 2016YFA0302001), the Fundamental Research Funds for the
Central Universities, the Shanghai Municipal Science and Technology Major
Project under Grant No. 2019SHZDZX01, the Chinese National Youth Talent
Support Program and and the Shanghai Talent Program.

## Appendices

###  V.1 Transforms between the Dirac representation and the Foldy–Wouthuysen
representation

In the Dirac representation the Hamiltonian of a (1+1)-dimensional Dirac
oscillator can be written in the form of anti-Jaynes-Cummings model,

$\hat{H}_{\rm{D}}=\eta\hat{\sigma}_{-}\hat{a}+\eta^{\ast}\hat{\sigma}_{+}\hat{a}^{\dagger}+mc^{2}\hat{\sigma}_{z},$
(51)

where the matrix forms of the Pauli operators are

$\hat{\sigma}_{z}=\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix},\
\hat{\sigma}_{+}=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix},\
\hat{\sigma}_{-}=\begin{bmatrix}0&0\\\ 1&0\end{bmatrix}.$ (52)

Then its eigenstates are in the form of dressed states as shown in Eq. (15),

$\displaystyle|E^{+}_{n}\rangle=A_{n}|n,\uparrow\rangle-
iB_{n}|n-1,\downarrow\rangle,$ (53)
$\displaystyle|E^{-}_{n}\rangle=B_{n+1}|n+1,\uparrow\rangle+iA_{n+1}|n,\downarrow\rangle.$

In the Foldy-Wouthuysen (FW) representation, the complications associated with
the spin-orbit coupling are formally eliminated. The form of the Hamiltonian
becomes into a diagonal $2\times 2$ matrix

$\displaystyle\hat{H}_{\rm{F}}(\hat{n})$
$\displaystyle=\hat{U}\hat{H}_{\rm{D}}\hat{U}^{\dagger}$ (54)
$\displaystyle=mc^{2}\begin{bmatrix}\sqrt{1+2\hat{n}\epsilon}&0\\\
0&-\sqrt{1+2(\hat{n}+1)\epsilon}\end{bmatrix}.$

where the unitary transformation operators $\hat{U}$ is given in Eq. (31) and
we have used the equality

$\hat{a}f(\hat{n})=f(\hat{n}+1)\hat{a}.$ (55)

The eigenstates become $|n,\uparrow\rangle$ and $|n,\downarrow\rangle$
corresponding to the positive eigenenergies $E^{+}_{n}$ and the negative
eigenenergies $E^{-}_{n}$, respectively. The eigenstates in the FW and the
Dirac representations are related by the unitary transformation

$\displaystyle|n,\uparrow\rangle=\hat{U}|E^{+}_{n}\rangle,$ (56)
$\displaystyle|n,\downarrow\rangle=\hat{U}|E^{-}_{n}\rangle.$

Then the QND observables for a (1+1)-dimensional Dirac oscillator can be
concisely expressed in the FW representation,

$\displaystyle\hat{X}_{1,\rm{F}}=\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}e^{i\hat{\omega}_{\epsilon}t}+e^{-i\hat{\omega}_{\epsilon}t}\hat{a}^{\dagger}),$
(57)
$\displaystyle\hat{X}_{2,\rm{F}}=i\sqrt{\frac{\hbar}{2m\omega}}(e^{-i\hat{\omega}_{\epsilon}t}\hat{a}^{\dagger}-\hat{a}e^{i\hat{\omega}_{\epsilon}t}),$

where we defined a frequency operator

$\hbar\hat{\omega}_{\epsilon}=\hat{H}_{\rm F}(\hat{n})-\hat{H}_{\rm
F}(\hat{n}-1).$ (58)

We have the corresponding minimum uncertainty state

$\left|\phi\right\rangle_{\rm
F}=\left|\alpha\right\rangle\otimes(c_{1}\left|\uparrow\right\rangle+c_{2}\left|\downarrow\right\rangle).$
(59)

If we go back to the Dirac representation, the QND observables become

$\hat{X}_{\ell,\rm
D}=\hat{U}^{\dagger}\hat{X}_{\ell,\rm{F}}\hat{U},\,\,\ell=1,2,$ (60)

where the operator $\hat{a}$ and $\hat{a}^{\dagger}$ are transformed into the
forms with complex spin-orbit coupling,

$\displaystyle\hat{U}^{\dagger}\hat{a}\hat{U}=\begin{bmatrix}(\hat{A}_{n}\hat{A}_{n+1}+\frac{\sqrt{\hat{n}}}{\sqrt{\hat{n}+1}}\hat{B}_{n}\hat{B}_{n+1})\hat{a}&i(\frac{\hat{A}_{n}\hat{B}_{n+1}}{\sqrt{\hat{n}+1}}-\frac{\hat{A}_{n+1}\hat{B}_{n}}{\sqrt{\hat{n}}})\hat{a}^{\dagger}\hat{a}+i\frac{\hat{A}_{n}\hat{B}_{n+1}}{\sqrt{\hat{n}+1}}\\\
i\hat{a}(\frac{\hat{A}_{n}\hat{B}_{n+1}}{\sqrt{\hat{n}+1}}-\frac{\hat{A}_{n+1}\hat{B}_{n}}{\sqrt{\hat{n}}})\hat{a}&(\hat{A}_{n+1}\hat{A}_{n+2}+\frac{\sqrt{\hat{n}+2}}{\sqrt{\hat{n}+1}}\hat{B}_{n+1}\hat{B}_{n+2})\hat{a}\end{bmatrix},$
(61)
$\displaystyle\hat{U}^{\dagger}\hat{a}^{\dagger}\hat{U}=\begin{bmatrix}\hat{a}^{\dagger}(\hat{A}_{n}\hat{A}_{n+1}+\frac{\sqrt{\hat{n}}}{\sqrt{\hat{n}+1}}\hat{B}_{n}\hat{B}_{n+1})&-i\hat{a}^{\dagger}(\frac{\hat{A}_{n}\hat{B}_{n+1}}{\sqrt{\hat{n}+1}}-\frac{\hat{A}_{n+1}\hat{B}_{n}}{\sqrt{\hat{n}}})\hat{a}^{\dagger}\\\
-i\hat{a}^{\dagger}\hat{a}(\frac{\hat{A}_{n}\hat{B}_{n+1}}{\sqrt{\hat{n}+1}}-\frac{\hat{A}_{n+1}\hat{B}_{n}}{\sqrt{\hat{n}}})-i\frac{\hat{A}_{n}\hat{B}_{n+1}}{\sqrt{\hat{n}+1}}&\hat{a}^{\dagger}(\hat{A}_{n+1}\hat{A}_{n+2}+\frac{\sqrt{\hat{n}+2}}{\sqrt{\hat{n}+1}}\hat{B}_{n+1}\hat{B}_{n+2})\end{bmatrix},$

and the transformed frequency operator $\hat{\omega}_{\epsilon}$ is no longer
diagonal in the matrix form

$\displaystyle\frac{\hat{U}^{\dagger}\hat{\omega}_{\epsilon}\hat{U}}{\omega}=\begin{bmatrix}\chi_{\epsilon}(\hat{n})&i\sqrt{2\epsilon}\chi_{\epsilon}(\hat{n})\hat{a}^{\dagger}\\\
-i\hat{a}\sqrt{2\epsilon}\chi_{\epsilon}(\hat{n})&-\chi_{\epsilon}(\hat{n}+1)\end{bmatrix},$
(62)

where

$\chi_{\epsilon}(\hat{n})=\frac{1}{\epsilon}(1-\sqrt{1-\frac{2\epsilon}{1+2\epsilon\hat{n}}}).$
(63)

The minimum uncertainty state in the Dirac representation becomes

$|\phi\rangle_{\rm D}=\hat{U}^{\dagger}|\phi\rangle_{\rm
F}=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}(c_{1}|E_{n}^{+}\rangle+c_{2}|E_{n}^{-}\rangle).$
(64)

###  V.2 QND observables, weakly relativistic limit

In the weakly relativistic limit $\epsilon\sim 0$, we have

$\hat{A}_{n}\approx 1-\hat{n}\epsilon/4,\ \
\hat{B}_{n}\approx\sqrt{\hat{n}\epsilon}/\sqrt{2}.$ (65)

To the first order of $\epsilon$, the transforms (61) of the operators
$\hat{a}$ and $\hat{a}^{\dagger}$ are approximated as

$\hat{U}^{\dagger}\hat{a}\hat{U}\approx\hat{a}-\frac{\epsilon}{4}\hat{\sigma}_{z}\hat{a}+i\sqrt{\frac{\epsilon}{2}}\hat{\sigma}_{+},$
(66)

and because

$\chi_{\epsilon}(\hat{n})\approx 1+(\frac{1}{2}-2\hat{n})\epsilon,$ (67)

the transformed frequency operator is approximated as

$\frac{\hat{U}^{\dagger}\hat{\omega}_{\epsilon}\hat{U}}{\omega}\approx\hat{\sigma}_{z}+\epsilon-(2\hat{n}+\frac{1}{2})\epsilon\hat{\sigma}_{z}+i\sqrt{2\epsilon}(\hat{a}^{\dagger}\hat{\sigma}_{+}-\hat{a}\hat{\sigma}_{-}).$
(68)

So in the Dirac representation for a vanishing $\epsilon$, retaining the
leading order terms the QND observables (57) are still in the form of
quadrature operators

$\displaystyle\hat{X}_{1}\approx\hat{X}_{1,\rm{nr}}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}e^{i\omega\hat{{\sigma}}_{z}t}+\hat{a}^{\dagger}e^{-i\omega\hat{{\sigma}}_{z}t}),$
$\displaystyle=$ $\displaystyle\hat{x}\cos{\omega
t}-\frac{\hat{p}\hat{\sigma}_{z}}{m\omega}\sin{\omega t},$
$\displaystyle\hat{X}_{2}\approx\hat{X}_{2,\rm{nr}}$ $\displaystyle=$
$\displaystyle-i\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}e^{i\omega\hat{{\sigma}}_{z}t}-\hat{a}^{\dagger}e^{-i\omega\hat{{\sigma}}_{z}t}),$
(69) $\displaystyle=$ $\displaystyle\hat{x}\hat{\sigma}_{z}\sin{\omega
t}+\frac{\hat{p}}{m\omega}\cos{\omega t},$

associated with a weakly relativistic minimum uncertainty state (64),

$|\phi\rangle_{\rm
D}\approx|\alpha\rangle\otimes(c_{1}|\uparrow\rangle+ic_{2}|\downarrow\rangle).$
(70)

When the Dirac oscillator departs from the weakly relativistic limit, its
Hamiltonian, retaining the small terms of next order, becomes

$\hat{H}_{\rm{D}}\approx\hat{H}_{\rm{nr}}-\frac{\hbar\omega}{8}\hat{\sigma}_{z}(2\hat{n}-\hat{\sigma}_{z}+1)^{2}\epsilon.$
(71)

To the present relativistic level, the QND observables should also include the
small terms of next order,

$\displaystyle\hat{X}_{1}$
$\displaystyle\approx\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}e^{i\omega\hat{\sigma}_{z}t}+i\sqrt{\frac{\varepsilon}{2}}\hat{\sigma}_{+}e^{i\omega\hat{\sigma}_{z}t}+h.c)$
(72)
$\displaystyle+\sqrt{\frac{\hbar}{2m\omega}}\sqrt{2\varepsilon}[(\hat{a}^{\dagger}\hat{\sigma}_{+}-\hat{a}\hat{\sigma}_{-})\hat{a}^{\dagger}+h.c]\sin\omega
t,$ $\displaystyle\hat{X}_{2}$
$\displaystyle\approx-i\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}e^{i\omega\hat{\sigma}_{z}t}+i\sqrt{\frac{\varepsilon}{2}}\hat{\sigma}_{+}e^{i\omega\hat{\sigma}_{z}t}-h.c)$
$\displaystyle-i\sqrt{\frac{\hbar}{2m\omega}}\sqrt{2\varepsilon}[(\hat{a}^{\dagger}\hat{\sigma}_{+}-\hat{a}\hat{\sigma}_{-})\hat{a}^{\dagger}-h.c]\sin\omega
t,$

in which except the terms in Eq. (V.2) there are additional terms involving
complex spin-orbit coupling. As the Dirac oscillator slides deeper into the
relativistic regime, the QND observables have increasingly complex forms,
which means their measurements are more challenging.

###  V.3 QND observables, extreme relativistic limit

In the extreme relativistic limit $\epsilon\gg 1$, we have

$\hat{A}_{n}\approx\hat{B}_{n}\approx 1/\sqrt{2},$ (73)

and then the transforms on $\hat{a}$ becomes

$\displaystyle\hat{a}_{\rm r}$ $\displaystyle=\hat{U}^{\dagger}\hat{a}\hat{U}$
(74)
$\displaystyle\approx\frac{1}{2}(1+\frac{\sqrt{\hat{n}}}{\sqrt{\hat{n}+1}})\hat{a}+(\frac{\sqrt{\hat{n}+2}-\sqrt{\hat{n}}}{\sqrt{\hat{n}+1}})\frac{1-\hat{\sigma}_{z}}{4}\hat{a}$
$\displaystyle+\frac{i}{2}[(\frac{1}{\sqrt{\hat{n}+1}}-\frac{1}{\hat{n}})\hat{a}^{\dagger}\hat{\sigma}_{+}+\hat{\sigma}_{-}\hat{a}(\frac{1}{\sqrt{\hat{n}+1}}-\frac{1}{\hat{n}})]\hat{a}$
$\displaystyle+\frac{i}{2}\frac{\hat{\sigma}_{+}}{\sqrt{\hat{n}+1}},$

and the operator $\chi_{\epsilon}(\hat{n})$ is approximated as

$\chi_{\epsilon}(\hat{n})\approx\frac{1}{\epsilon}(1-\sqrt{1-\frac{1}{\hat{n}}}),$
(75)

such that the transformed frequency operator $\hat{\omega}_{\epsilon}$ becomes

$\displaystyle\frac{\hat{\omega}_{\rm
r}}{\omega}=\frac{\hat{U}^{\dagger}\hat{\omega}_{\epsilon}\hat{U}}{\omega}\approx\sqrt{\frac{2}{\epsilon}}\begin{bmatrix}0&i(1-\sqrt{1-\frac{1}{\hat{n}}})\hat{a}^{\dagger}\\\
-i\hat{a}(1-\sqrt{1-\frac{1}{\hat{n}}})&0\end{bmatrix}.$ (76)

Then the QND observables are in the forms

$\displaystyle\hat{X}_{1,\rm r}$
$\displaystyle\approx\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}_{\rm
r}e^{i\hat{\omega}_{\rm r}t}+e^{-i\hat{\omega}_{\rm r}t}\hat{a}_{\rm
r}^{\dagger}),$ (77) $\displaystyle\hat{X}_{2,\rm r}$
$\displaystyle\approx-i\sqrt{\frac{\hbar}{2m\omega}}(\hat{a}_{\rm
r}e^{i\hat{\omega}_{\rm r}t}-e^{-i\hat{\omega}_{\rm r}t}\hat{a}_{\rm
r}^{\dagger}),$

In the high excited level $n\gg 1$, the operators $\hat{a}_{\rm r}$ and
$\hat{a}_{\rm r}^{\dagger}$ can be approximated as $\hat{a}$ and
$\hat{a}^{\dagger}$, respectively, and then

$\displaystyle\hat{X}_{1,\rm{r}}\approx\frac{1}{2}(\hat{x}+\frac{i\hat{p}}{m\omega})e^{i\hat{\omega}_{\rm{r}}t}+h.c.,$
(78)
$\displaystyle\hat{X}_{2,\rm{r}}\approx-\frac{i}{2}((\hat{x}+\frac{i\hat{p}}{m\omega})e^{i\hat{\omega}_{\rm{r}}t}-h.c.),$

where

$\hat{\omega}_{\rm
r}\approx\omega\sqrt{2/\varepsilon}(\sqrt{\hat{n}-1}-\sqrt{\hat{n}})\hat{\sigma}_{y},$
(79)

associated with the approximate minimum uncertainty state

$|\phi\rangle_{\rm
D}\approx|\alpha\rangle\otimes\frac{(c_{1}+c_{2})\left|\uparrow\right\rangle-i(c_{1}-c_{2})\left|\downarrow\right\rangle}{\sqrt{2}}.$
(80)

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