# KnowGraph-PM: a Knowledge Graph based Pricing Model for Semiconductors
Supply Chains

Nour Ramzy  Leibniz Universität Hannover
Hannover, Germany
<EMAIL_ADDRESS>Sören Auer  TIB: Technische Informationsbibliothek
Hannover, Germany
<EMAIL_ADDRESS>Javad Chamanara  TIB: Technische Informationsbibliothek
Hannover, Germany
<EMAIL_ADDRESS>

Hans Ehm
Infineon Technologies AG
Munich, Germany
<EMAIL_ADDRESS>
###### Abstract

Semiconductor supply chains are described by significant demand fluctuation
that increases as one moves up the supply chain, the so-called bullwhip
effect. To counteract, semiconductor manufacturers aim to optimize capacity
utilization, to deliver with shorter lead times and exploit this to generate
revenue. Additionally, in a competitive market, firms seek to maintain
customer relationships while applying revenue management strategies such as
dynamic pricing. Price change potentially generates conflicts with customers.
In this paper, we present KnowGraph-PM, a knowledge graph-based dynamic
pricing model. The semantic model uses the potential of faster delivery and
shorter lead times to define premium prices, thus entail increased profits
based on the customer profile. The knowledge graph enables the integration of
customer-related information, e.g., customer class and location to customer
order data. The pricing algorithm is realized as a SPARQL query that relies on
customer profile and order behavior to determine the corresponding price
premium. We evaluate the approach by calculating the revenue generated after
applying the pricing algorithm. Based on competency questions that translate
to SPARQL queries, we validate the created knowledge graph. We demonstrate
that semantic data integration enables customer-tailored revenue management.

###### Index Terms:

knowledge graph, semiconductor, dynamic pricing, revenue management, customer
relationship management

##  I Introduction

The semiconductor industry is characterized by complex supply chain structures
due to its wide range of customers with fluctuating demands for products, long
production lead times, short product life cycles and rapidly changing
technologies [1]. This entails specific phenomena such as the bullwhip effect
that increases the variability of orders as they move up the supply chain from
retailers to wholesalers to manufacturers to suppliers [2]. Consequently,
matching the demand and capacity requires effective planning and optimal
utilization of production capacities. Moreover, manufacturing lead times are
longer than customer order lead times (the time interval between order entry
and requested delivery date), thus fulfilling order requests that go beyond
promised delivery time is costly [3]. Therefore, to keep utilizing capacity in
a profitable way and guarantee customer satisfaction, companies resort to
revenue management ideas such as dynamic pricing [4], i.e., exploiting faster
delivery to generate revenue. Potentially, conflicts with customers arise,
affecting their satisfaction and harming the firm’s long-term relationship
with the customer and ultimately its success [5].

In fact, to maintain advantage especially needed within the competitive
electronics market [6], semiconductor manufacturers strive to deliver and
maintain various customer relationships, and design customized portfolios with
distinct values to differentiate between customers’ requirements [7]. Yet,
revenue management and customer account management are often conducted
independently of each other [8], although, the flow of information in a supply
chain is crucial for carrying out effective exchanges between parties [9]. For
instance, centralized information and elaborate information integration among
supply chain partners can limit the increase in demand variability i.e.,
reducing the bullwhip effect [10].

In this article, we propose KnowGraph-PM a pricing solution to measure a price
change to each customer for an order. The knowledge graph, depicting customers
and orders, subsumes data about customer order behavior as well as customer
profile. The former describes order lead times for a customer, e.g.,
requested, confirmed, while the latter defines customer classes expressing the
relationship with the firm. For each customer, we calculate a price premium
based on lead time variations as well as the corresponding price adjustment
factor derived from the respective customer class. For each order placed by a
customer, we apply the customer price premium to get the new price for this
order. The approach enables, via a knowledge graph, semantic data integration
by combining customer demand and order-related data with customer portfolio
information. We apply the proposed approach to a use case of an international
semiconductor manufacturing firm; the results show that KnowGraph-PM approach
allows tailored revenue generation according to customers’ profile thus
reducing the risk of harming relationships.

The remainder of the paper is organized as follows: In Section II we introduce
the domain and the key terminology and motivation behind the work. We present
the approach and the KnowGraph-PM implementation in Section III. In Section IV
we evaluate the implemented solution and discuss the advantages and
limitations of our approach. Finally, we conclude and present an outlook for
further steps in Section V.

##  II Background and Motivation

###  II-A Background

####  II-A1 Lead Time

The order lead time (OLT) at each stage of the order, is the time frame
between the entry of an order to the time point it passes a designated stage
[11] as shown in Figure 1. A requested order lead time $OLT_{Requested}$ is
the difference between Customer Request Date for an order, i.e., when the
customer wishes to receive the order and the Order Date (when the customer
placed the order). A confirmed order lead time $OLT_{Confirmed}$ is the
difference between Customer Delivery Date i.e., the confirmed order delivery
date and the Order Date. SDT is defined as the difference between Standard
Delivery Date and Order Date. It is the lead time the customer can expect till
the order is delivered. In practice, customers request their orders earlier
than SDT, i.e., $SDT>OLT_{Requested}$. Moreover, due to the high-competitive
market of semiconductors, manufacturers usually sell their products with a
shorter $OLT_{Confirmed}$, closer to $OLT_{Requested}$. The potential of Lead
Time-Based Pricing (LTBP) is seen in the difference between Standard Delivery
Date and Customer Delivery Date. Namely, faster delivery, $SDT>$
$OLT_{Confirmed}$, is exploited by offering LTBP as part of dynamic pricing.

Figure 1: The definitions of lead times.

####  II-A2 Lead Time-based Pricing (LTBP)

Namely, dynamic pricing describes the firm’s practice to charge various
customers different prices for the same products [12]. For instance, [4]
proposes a supply chain planning framework for revenue management that
consists of solutions for demand steering and dynamic pricing where the
pricing algorithm relies on the data from order lead time measurement.
However, LTBP models to identify the price premium based on faster delivery
are still under development. In fact, some rely on simple mathematical
functions, e.g., linear, concave, convex. For instance, [13] introducing
Equation 1, explains the rationale behind a convex function; both opportunity
costs for the manufacturer and for the customer increase with decreasing lead
time.

$P_{Premium}=\alpha*\log(\frac{OLT_{Confirmed}}{SDT})$ (1)

where $\alpha$ is a factor obtained from simulating customer order behavior
for maximizing revenue for the company and $P_{Premium}$ is the price change
or the added portion to the original price of the order. In fact, revenue
management (RM) is built upon heterogeneity between customers; frequent buyers
need to be treated differently than occasional ones [14]. Typically, Customer
Relationship Management (CRM) enables customer profiling and delimits
characteristics for customers and how they can be used to determine the price
as part of RM [15].

####  II-A3 Customer Relationship Management (CRM)

CRM refers to building one‐to‐one relationships with customers that can drive
value for the firm [16]. Consequently, customization of requirements increases
maintenance of customers [17]. Namely, by segmenting customers into
portfolios, an organization can better understand the relative importance of
each customer to the company’s total profit [18]. Customer Portfolio
Management (CPM) is based on financial performance as well as strategic and
economic criteria, e.g., customer account types or classes, regional
importance. For each criterion, the company chooses measurable characteristics
to segment customers accordingly. For example, customer classes distinguish
customers according to their impact on business, based on the average revenue
for the current and previous fiscal year, the volume of purchases, potential
sales, the prestige of the account and market leadership.

Especially in semiconductor manufacturing, a competitive domain, meeting
customer-specific requirements incites maintaining close associations with
customers to identify their specific needs. In fact, CPM affects new product
development [19]. Especially with the short product life cycle trait for this
domain, it accentuates the need for manufacturers to act fast and maintain
good relations with the customer.

###  II-B Motivation

The bullwhip effect, a significant characteristic of the semiconductor supply
chain, entails high inventory capacity, possibly unnecessarily. However, to
keep utilizing capacity in a profitable way companies exploit revenue
management ideas such as dynamic pricing. Yet, LTBP strategies are myopic
because the decisions are optimized considering solely the profit that is
expected to be obtained from the currently arriving customer without any
foresight on the long-run impact of these decisions [11]. Companies realize
the need to keep a good relationship with the customer while generating
revenues.

Typically, interactions between revenue management and key account management
have been largely ignored as both are often conducted independently of each
other [8]. In addition, [20] explains how limited information sharing
increases the difficulty of reducing the bullwhip effect and leads to
inefficient supply chain management. Also, information integration increases
acquisition and maintenance of customers according to profitability [17].
Moreover, to the best of our knowledge, previously developed LTBP models
solely rely on customer order behavior while not including the customer
portfolio and specific characteristics. In this work, we create a knowledge
graph that semantically integrates data from different sources, i.e.,
customers, orders as well as customer account types. This solution suggests
information integration from various data sources to customer-specific revenue
management via LTBP. This improves revenue generation while maintaining
customer relationships.

##  III Implementation

This section describes the approach we implemented, integrating data from
customer relationship management and customer order data.

###  III-A Knowledge Graph (KG)

####  III-A1 Domain Ontology

We use the ontology-based data access (OBDA) approach for semantic data
integration. The schema is given in terms of an ontology representing the
formal and conceptual view of the domain [21]. Figure 2 shows the ontology
comprising the Order class, which enables the coupling of the remaining
classes of the domain, e.g., Product and Customer. The Order entity is
uniquely identified via an Order Number and is described via twofold data
properties. First, order information properties such as the order quantity and
the original price of the order. Second, an order is described with lead time
properties, e.g., order entry date, requested delivery date, and the standard
delivery time. Through its two object properties containsProduct and
wasPlacedBy, an order is linked to the Product and Customer entities
respectively. The Customer class describes a customer, by assigning each
customer a distinct customer code and by categorizing a customer into the
defined customer account type or class. This categorization into a customer
account type is then quantified with the data property hasAdjustmentFactor,
which assigns each customer a specific pre-defined value based on their
respective type. Finally, each product is uniquely identified with the help of
the product number. Also, a Product includes data properties that contain all
relevant product information such as the product basic type and the product
line.

####  III-A2 Data Sources: Use Case Description

The use case described here belongs to a international semiconductor
manufacturing firm. We utilize these two datasets: DS1 that contains data
about order details, e.g., lead times, products and corresponding customers.
We filter orders containing products within one product line. We process
roughly 65 thousand orders in the time span of 2016-10-04 to 2020-09-01. DS2
contains customer portfolio, i.e., customer account type, class and location.
It subsumes 177 customers and corresponding account types (regular, key and
others). Moreover, we use the corporate memory tool as a platform for data
integration111https://eccenca.com/products/enterprise-knowledge-graph-
platform-corporate-memory.

Figure 2: Domain knowledge graph comprising the order, customer and product
classes with associated properties.

###  III-B Lead Time-based Pricing Algorithm

The pricing algorithm is implemented as a SPARQL query that emulates the
equation developed by [22]. SPARQL222https://www.w3.org/TR/rdf-sparql-query/
is the query language standardized by the W3C for querying knowledge graphs.
Figure 3 describes how parts of the equation match to the nested query shown.
Steps [1-4] depict how the customer order behavior and customer account type
contribute to calculating the price premium $P_{premium}$ for a specific
customer. The last step is to calculate the new revenue management price
$P_{RM{j}}$ of a specific order j knowing the $P_{premium}$ of this customer.

Figure 3: Lead time-based pricing algorithm with the formula (top) and
corresponding SPARQL query implementation (bottom).

  1. 1.

We calculate the relative standard deviation (RSD) and the relative mean
deviation (RMD) for each customer individually. This is done only if revenue
management is allowed, i.e., $SDT>OLT_{Requested}$, when customers request
their orders earlier than Standard Delivery Date. This represents the customer
order behavior from previous orders. $\alpha$ and $\beta$ are coefficients to
offer the model user the chance to emphasize one parameter more than the
other. For instance, one could argue that the average deviation or mean is
less important. For simplicity, alpha and beta are chosen to be equal to 1.

  2. 2.

The sum of weighed RSD and RMD is then compared to a pre-determined threshold
$P_{max}$ which in this case is set to 2 as this represents the ceiling for
the highest premium that can be offered to a customer which is double the
original price. The sum is modified by the revenue adjustment parameter
$\rho$. As shown in Table I, for each customer account type or class there
exists a corresponding $\rho$ based on revenue ranges. Large revenue ranges,
entail more important customer accounts, thus have greater advantage
equivalent to higher $\rho$ leading to smaller premium price.

  3. 3.

In order to ensure that no customer receives a premium price smaller than the
original we select the max between our first part of the formula and 1.

  4. 4.

We assign $P_{premium}$ to each customer i where $1<P_{premium}<2$

  5. 5.

Upon a new order j, we apply Equation 2 to get the revenue management price
$P_{RM{j}}$ from the original price $P_{O{j}}$ of the order and customer-
specific $P_{premium}$ assigned in the previous step. As mentioned, the
difference between SDT and $OLT_{Confirmed}$, is exploited for revenue
management.

$P_{RM{j}}=P_{O{j}}+P_{O{j}}[(1-\frac{OLT_{Confirmed_{j}}}{SDT{j}})*(P_{premium}-1)]$

(2)

TABLE I: Exemplary values of price adjustment factor

Revenue (r) Range | Customer Class | Price Adjustment Factor $\rho$
---|---|---
10$<$r$<$100m | Key | 0.1
5$<$r$<$10m | Regular | 0.05
0$<$r$<$5m | Others | 0.025

##  IV Evaluation

The implemented approach combines data from customer relationship management
with customer order behavior. We use OBDA to represent the domain and create a
knowledge graph. We evaluate KnowGraph-PM twofold. First, we check if the
created knowledge graph covers the domain in question. Second, we calculate
the total revenue generated using static and dynamic pricing algorithms and
compare them with our KnowGraph-PM model.

###  IV-A Competency Questions and KG Evaluation

We translate the listed competency questions, defined by domain experts, to
SPARQL queries. We execute them on the described knowledge graph in Section
III. The results are extracted and represented in Figure 4.

Figure 4: Chart visualization of SPARQL query results.

CQ1: Top 20 most profitable customers: This competency question retrieves the
top 20 customers that generate the highest revenues who are assigned a price
premium and who buy the order with the price change. The results shown in
Figure 4(a) give a valuable insight into the most influential customers for a
firm. This can be extended by examining these customer classes and accordingly
tailor marketing strategies.

CQ2: Occurrence based customer ranking: We can examine the overall customer
order behavior of the different customer classes. For this, we rank customer
types based on the highest percentage of occurrences of $SDT>OLT_{Requested}$
(RM is possible). We analyze, which customer classes are more likely to
request a delivery date that is earlier than the Standard Delivery Date. This
makes them eligible for revenue management with the adjustment factor $\rho$
in Table I suitable to their respective customer class. The results in Figure
4(b) show that Key account customers are least likely to request an early
delivery. This is an expected outcome. In fact, key customers have strong
relationship established with the company. Thus, they are more likely to enter
long-term agreements and contracts entailing stable order behavior i.e.,
limited variation in lead times. Additionally, because of the frequency of
their past orders, they can accurately forecast their needs and how a firm can
fulfill them. Accordingly, this constrains dynamic pricing and revenue
management potential.

CQ3: Per customer class price premium estimation: This question provides
information about the customer class that is most likely to be the most
profitable upon applying revenue management. The information contains the
highest, the lowest and the average Price Premium that will be paid by
customers of each customer class. Also, it gives insights into the customer
order behavior of the different customer classes. We observe in the results
shown in Figure 4(c) that the Regular customer type has the highest maximum,
average, and minimum price premiums. This means that this customer class would
yield the highest average revenue when the dynamic pricing model is applied.

CQ4: Initial customer and product selection: This CQ determines the
combination of customers and products that should be released with the first
practical implementation of the LTBP model in order to maximize the initial
potential revenue increase using the adjusted prices. Results in Figure 4(d)
provide the combination of customers and products in terms of profitability
and revenue generation in the case where the lead time-based pricing is
applied.

###  IV-B Revenue Management (RM)

Dynamic pricing as part of RM is about generating revenue for the company.
Existing dynamic pricing algorithms, e.g., convex model consider customer
behavior while ignoring customer profiles. KnowGraph-PM customizes the RM
price based on customer order behavior and customer relationship with the
company. To evaluate this, we execute the query in Listing 1 where we sum the
order prices with and without applying revenue management. The results show
that $TotalOrginalPrice<TotalRMPrice<TotalConvexPrice$. In that,
TotalOrginalPrice is for the case that no RM is applied, which indeed is the
summation of original prices for each order $P_{O{j}}$ as provided in Equation
2, TotalRMPrice is the sum of RM price, i.e., $P_{RM{j}}$ as of Equation 2,
and TotalConvexPrice is the sum of order prices if Equation 1 is applied.

These results indicate that by applying KnowGraph-PM, a company can generate
revenue while taking into consideration the customer portfolio. Other models
such as the convex generate higher revenue but can entail a disturbed
relationship with the customer. The implemented approach integrates data from
customer relationship management as well as customer order data to generate a
price premium that fits the customer portfolio. The proposed solution is
portable and can be applied to other domains with the right mapping between
data and the ontology. Additionally, the two evaluation techniques show that
it is up to the company to adapt pricing strategies based on customer
portfolio. SPARQL queries can be tweaked to show details about a specific
customer and consequently tailor marketing strategies.

Listing 1: Evaluation query

⬇

SELECT

(sum(?rmprice) as ?TotalRMPrice)

(sum(?orignalprice) as ?TotalOrginalPrice)

(sum(?convexprice) as ?TotalConvexPrice)

FROM <http://xyz.com/LTBP/>

WHERE {

?order :hasRMPrice ?rmprice.

?order :hasOriginalPrice ?orignalprice.

?order :hasConvexPrice ?convexprice.

}

###  IV-C Discussion

The implemented approach integrates data from customer relationship management
as well as customer order data to generate a price premium that fits the
customer portfolio. The proposed solution is portable and can be applied to
other domains with the right mapping between data and the ontology.
Additionally, the two evaluation techniques show that it is up to the company
to adapt pricing strategies based on customer portfolio. SPARQL queries can be
tweaked to show details about a specific customer and consequently tailor
marketing strategies.

However, the approach has some limitations. First, we identify a limitation in
the evaluation as we compare our model to a convex model with $\alpha$ chosen
to be -0.5 in Equation 1. This choice is not accurate as the parameter is set
after using simulation models to optimize the revenue for a specific set of
customers. Second, for the evaluation, we calculate the total revenue with an
assumption that all customers accept the price change. However, in practice,
customers can refute the premium price and stick to the original.

##  V Conclusion and Outlook

Semiconductor supply chains are strongly affected by the bullwhip effect,
i.e., increasing demand fluctuation. Thus, in a competitive market,
semiconductor manufacturers seek to optimize capacity utilization, deliver
with shorter lead times than the agreed contractual times and guarantee
customer satisfaction and loyalty. Firms exploit faster delivery by resorting
to revenue management ideas such as dynamic pricing. However, this potentially
affects relationships with customers.

In this work, we propose KnowGraph-PM, a knowledge graph lead time-based
pricing approach allowing tailored revenue generation according to customers’
profile thus reducing the risk of harming relationships. We present dynamic
pricing as part of revenue management taking into consideration customer
relationship with the customer. Using a pricing algorithm, implemented as a
SPARQL query, we assign a price premium to each customer based on their order
behavior and changes in the order lead time. Consequently, we calculate a
price change for a customer’s order. We apply the proposed approach to a use
case of an international semiconductor manufacturing firm and proved that
semantic data integration enables customer profiling aware lead time-based
pricing.

As future work, we aim to integrate into KnowGraph-PM an acceptance rate
distribution to model customer’s behavior towards dynamic pricing. This would
lead to more realistic figures in the evaluation thus overcoming the mentioned
limitation. Moreover, we propose extending the knowledge-graph with more
contextual data about the customer. For instance, Customer Master Data
(address data, contract data) is an enabler for all customer-driven
operations. Thus, it can be used to determine the strategic internal decision
to treat the customer in a strategic manner, e.g., customer prioritization in
times of allocation. Consequently, further pricing models that incorporate
more factors about a customer can be explored.

## Acknowledgment

This work has received funding from the EU Electronic Component Systems for
European Leadership (ECSEL) Joint Undertaking within Integrated Development
4.0 project (idev40).

## References

  * [1] B. A. Mousavi, R. Azzouz, C. Heavey, and H. Ehm, “Simulation-based analysis of the nervousness within semiconductors supply chain planning: insight from a case study,” in 2019 Winter Simulation Conference (WSC), pp. 2396–2407, IEEE, 2019.
  * [2] E. Sucky, “The bullwhip effect in supply chains—an overestimated problem?,” International Journal of Production Economics, vol. 118, no. 1, pp. 311–322, 2009.
  * [3] H. Ott, S. Heilmayer, C. Sng, et al., “Granularity dependency of forecast accuracy in semiconductor industry,” Research in Logistics & Production, vol. 3, 2013.
  * [4] A. Seitz, H. Ehm, R. Akkerman, and S. Osman, “A robust supply chain planning framework for revenue management in the semiconductor industry,” Journal of Revenue and Pricing Management, vol. 15, no. 6, pp. 523–533, 2016\.
  * [5] J. Wirtz, S. E. Kimes, J. H. P. Theng, and P. Patterson, “Revenue management: resolving potential customer conflicts,” Journal of Revenue and Pricing Management, vol. 2, no. 3, pp. 216–226, 2003.
  * [6] L. Mönch, R. Uzsoy, and J. W. Fowler, “A survey of semiconductor supply chain models part i: semiconductor supply chains, strategic network design, and supply chain simulation,” International Journal of Production Research, vol. 56, no. 13, pp. 4524–4545, 2018.
  * [7] C.-M. Chou, “How does manufacturing service perceived value influence customer satisfaction? an investigation of global semiconductor industry,” International Journal of Production Research, vol. 52, no. 17, pp. 5041–5054, 2014.
  * [8] X. L. Wang and R. Brennan, “A framework for key account management and revenue management integration,” Industrial Marketing Management, vol. 43, no. 7, pp. 1172–1181, 2014.
  * [9] G. Stefansson, “Business-to-business data sharing: A source for integration of supply chains,” International journal of production economics, vol. 75, no. 1-2, pp. 135–146, 2002.
  * [10] F. Chen, Z. Drezner, J. K. Ryan, and D. Simchi-Levi, “Quantifying the bullwhip effect in a simple supply chain: The impact of forecasting, lead times, and information,” Management science, vol. 46, no. 3, pp. 436–443, 2000.
  * [11] M. Öner-Közen and H. Ehm, “Dynamic price and lead time quotation under semiconductor industry related challenges,” in 2018 Winter Simulation Conference (WSC), pp. 3386–3396, IEEE, 2018.
  * [12] M. D. Wittman and P. P. Belobaba, “Customized dynamic pricing of airline fare products,” Journal of Revenue and Pricing Management, vol. 17, no. 2, pp. 78–90, 2018.
  * [13] N. Berger and A. Strauss, “Order prioritization in the high-tech industry: A lead-time based pricing approach.” 2021.
  * [14] J. Hailwood and D. Gottlieb, “Customer relationship management,” Apr. 10 2003\.  US Patent App. 09/972,277.
  * [15] M. Zhang and P. Bell, “Price fencing in the practice of revenue management: An overview and taxonomy,” Journal of Revenue and Pricing Management, vol. 11, no. 2, pp. 146–159, 2012.
  * [16] V. Kumar, “Customer relationship management,” Wiley international encyclopedia of marketing, 2010.
  * [17] D. S. Johnson, B. H. Clark, and G. Barczak, “Customer relationship management processes: How faithful are business-to-business firms to customer profitability?,” Industrial Marketing Management, vol. 41, no. 7, pp. 1094–1105, 2012.
  * [18] R. Thakur and L. Workman, “Customer portfolio management (cpm) for improved customer relationship management (crm): Are your customers platinum, gold, silver, or bronze?,” Journal of Business Research, vol. 69, no. 10, pp. 4095–4102, 2016.
  * [19] H. Yli-Renko and R. Janakiraman, “How customer portfolio affects new product development in technology-based entrepreneurial firms,” Journal of marketing, vol. 72, no. 5, pp. 131–148, 2008.
  * [20] C. Li, “Controlling the bullwhip effect in a supply chain system with constrained information flows,” Applied Mathematical Modelling, vol. 37, no. 4, pp. 1897–1909, 2013.
  * [21] G. De Giacomo, D. Lembo, M. Lenzerini, A. Poggi, and R. Rosati, “Using ontologies for semantic data integration,” in A Comprehensive Guide Through the Italian Database Research Over the Last 25 Years, pp. 187–202, Springer, 2018.
  * [22] N. Berger, “Revenue management in the semiconductor industry.” 2017.


# Fast spin squeezing by distance-selective long-range interactions with
Rydberg molecule dressing

Huaizhi Wu  Fujian Key Laboratory of Quantum Information and Quantum Optics
and Department of Physics, Fuzhou University, Fuzhou 350116, People’s Republic
of China  Xin-Yu Lin  Fujian Key Laboratory of Quantum Information and Quantum
Optics and Department of Physics, Fuzhou University, Fuzhou 350116, People’s
Republic of China  Zong-Xing Ding  Fujian Key Laboratory of Quantum
Information and Quantum Optics and Department of Physics, Fuzhou University,
Fuzhou 350116, People’s Republic of China  Shi-Biao Zheng  Fujian Key
Laboratory of Quantum Information and Quantum Optics and Department of
Physics, Fuzhou University, Fuzhou 350116, People’s Republic of China  Igor
Lesanovsky  School of Physics and Astronomy, University of Nottingham,
Nottingham NG7 2RD, United Kingdom  Centre for the Mathematics and Theoretical
Physics of Quantum Non-equilibrium Systems, University of Nottingham,
Nottingham NG7 2RD, United Kingdom  Weibin Li  School of Physics and
Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom
Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium
Systems, University of Nottingham, Nottingham NG7 2RD, United Kingdom

###### Abstract

We propose a Rydberg molecule dressing scheme to create strong and long-ranged
interactions at selective distances. This is achieved through laser coupling
ground-state atoms off-resonantly to an attractive molecular curve of two
interacting Rydberg atoms. Although dephasing due to Rydberg state decay
occurs in all dressing schemes, an advantage of the molecule dressing is that
a large ratio of dressed interaction to dephasing rate can be realized at
large atomic separations. In an optical lattice or tweezer setting, we show
that the strong interaction permits the fast generation of spin squeezing for
several tens of dressed atoms. The proposed setting offers a new route to
study complex many-body dynamics and to realize quantum information processing
with non-convex long-range interactions.

Introduction.–The long lifetime and strong interaction emerging between atoms
excited to Rydberg states offer an effective way for the implementation of
quantum computation protocols Jaksch  _et al._ (2000); Lukin  _et al._ (2001);
Wilk  _et al._ (2010); Zhang  _et al._ (2010); Isenhower  _et al._ (2010);
Keating  _et al._ (2013); Hankin  _et al._ (2014); Keating  _et al._ (2015);
Li and Kuzmich (2016); Levine  _et al._ (2018), and many-body quantum
simulation Honer  _et al._ (2010); Mattioli  _et al._ (2013); Glaetzle  _et
al._ (2014, 2015); Schauß  _et al._ (2015); de Léséleuc  _et al._ (2017); van
Bijnen and Pohl (2015); Li and Sarma (2015); Labuhn  _et al._ (2016a); de
Léséleuc  _et al._ (2018). One approach to utilize the Rydberg interaction is
through the Rydberg dressing Santos  _et al._ (2000); Pupillo  _et al._
(2010); Henkel  _et al._ (2010); Johnson and Rolston (2010); Balewski  _et
al._ (2014); Jau  _et al._ (2015); Zeiher  _et al._ (2016); Mukherjee  _et
al._ (2016); Buchmann  _et al._ (2017); Płodzień  _et al._ (2017); Chai  _et
al._ (2017); Borish  _et al._ (2020), where ground state atoms are off-
resonantly coupled to a Rydberg state. As direct Rydberg excitation is
avoided, the dressed interaction - although weakened in strength - comes in
principle with the benefit of prolonged coherence times. For Rydberg atoms
interacting with a van der Waals potential, the dressing leads to a soft-core
interaction (with characteristic soft-core radius $R_{c}$) between the dressed
atoms. This interaction is nearly a constant when the distance $R$ between the
atoms is much smaller than $R_{c}$ and it decays rapidly when $R>R_{c}$. The
strength and $R_{c}$ of the dressed interaction can be controlled by the
Rydberg dressing laser (i.e. detuning, Rabi frequency and the coupled Rydberg
state). Such tunability Carroll  _et al._ (2004); Barredo  _et al._ (2014) has
stimulated the study of exotic topological states Glaetzle  _et al._ (2014,
2015); van Bijnen and Pohl (2015); Shi and Kennedy (2018), density-wave orders
Khasseh  _et al._ (2017); Li  _et al._ (2018, 2020), correlated spin dynamics
Zeiher  _et al._ (2017); Guardado-Sanchez  _et al._ (2021), and quantum
sensing Bouchoule and Mølmer (2002); Gil  _et al._ (2014); Arias  _et al._
(2019a) with Rydberg dressed interactions.

Figure 1:  Rydberg molecule dressing. (a) Level scheme. The ground states
$|0\rangle$ and $|1\rangle$ are coupled via optically Raman transitions. The
ground state $|1\rangle$ is optically dressed to a Rydberg state $|r\rangle$.
Rydberg state $|nS\rangle$ and $|n^{\prime}P\rangle$ are coupled by a
microwave field (Rabi frequency $\Omega$ and detuning $\Delta$), forming a
superposition state $|r\rangle$. (b) Rydberg molecule dressing. Two-photon
transition from the ground state $|11\rangle$ to the Rydberg molecular dimer
state $|rr\rangle$ is detuned by $2\Delta+U(R)$, with $U(R)$ being the
attractive interaction potential between atoms in Rydberg state $|r\rangle$.
(c) RMD induced interaction $V(R)$, nonlocal two-body decay $\gamma^{(2)}$,
and coherence strength $\mathcal{C}$ as a function of the interatomic distance
$R$. For comparison, $V(R)$, $\mathcal{C}$, and $\gamma^{(2)}$ are shown in
units of $V_{0}$, $\gamma^{(1)}$ and $\mathcal{C}_{s}$, respectively.
Parameters are $(\Omega,\Delta,\gamma)/2\pi=(1,10,0.01)$ MHz,
$\Delta_{2}(R_{c})=2\Omega$, and lattice spacing $a=1\,\mu$m. See text for
details.

Dissipation Weber  _et al._ (2012); Malossi  _et al._ (2014); Schempp  _et
al._ (2014); Goldschmidt  _et al._ (2016); Boulier  _et al._ (2017), such as
spontaneous decay due to the finite lifetime of Rydberg states Barredo  _et
al._ (2020) and motional dephasing Li  _et al._ ; Gambetta  _et al._ (2020);
Lorenz  _et al._ (2021), is ubiquitous in any Rydberg dressing scheme. In the
weak Rydberg dressing scheme (i.e. $|\Delta|\gg\Omega$ with $\Delta$ and
$\Omega$ the detuning and Rabi frequency of the dressing laser) Henkel  _et
al._ (2010); Pupillo  _et al._ (2010); Keating  _et al._ (2013); Zeiher  _et
al._ (2016), the effective decay rate of the dressed state scales with
$\sim\Omega^{2}\gamma/\Delta^{2}$ with $\gamma$ to be the spontaneous decay
rate in Rydberg states. As a result, the effective lifetime of the dressed
atom is increased by a factor $\Delta^{2}/\Omega^{2}$. The strength of the
dressing interaction scales as $V_{0}\sim\Omega^{4}\Delta^{-3}$. Its coherence
strength is given by the ratio between $V_{0}$ and the effective decay rate,
$\mathcal{C}\propto\Omega^{2}/(\Delta\gamma)$ Keating  _et al._ (2016); Zeiher
_et al._ (2016); Lee  _et al._ (2017). These estimations, however, are based
on a simple two-level description. There are other detrimental processes such
as level crossings at short distances that have to be taken care of in an
experimental implementation Zeiher  _et al._ (2016). To effectively implement
quantum simulation and computation with the dressed long-range interactions
Gross and Bloch (2017); Schauss (2018); Wu  _et al._ (2021), it is worth to
developing new Rydberg dressing schemes where such unwanted effects can be
suppressed, and which offers a fundamentally better scaling of the coherence
strength $\mathcal{C}$.

In this work, we propose a Rydberg dressing scheme where ground state atoms
are coupled off-resonantly to macroscopic Rydberg molecular dimers which form
attractive potentials at micrometer separations Petrosyan and Mølmer (2014).
This results in a long-range dressed interaction that is strongly attractive
only at selective distances, while saturating to $V_{0}$ at short distances
($R\ll R_{c}$). The selective distance locates at the minimal of the molecular
potential, whose depth and therefore the dressed interaction can be modulated
by external microwave fields. Though laser dressing to Rydberg states
generally leads to single- and two-body dephasing, we show that the Rydberg
molecule dressing (RMD), however, greatly enhances $\mathcal{C}$. As an
application, we study spin squeezing of the Rydberg dressed atom in an one
dimensional optical lattice. We show that this allows to implement stronger
spin squeezing within a much shorter time in comparison to ”traditional” soft-
core Rydberg dressing (SRD) Henkel  _et al._ (2010); Pupillo  _et al._ (2010),
provided that the atoms are positioned at certain preselected distances. Our
study reveals insights on the coherent and dissipative features in Rydberg
gases, which are important for implementing many-body physics and quantum
information processing with Rydberg dressed interactions Honer  _et al._
(2010); Mattioli  _et al._ (2013); Glaetzle  _et al._ (2014, 2015); van Bijnen
and Pohl (2015); Li and Sarma (2015).

Rydberg molecule dressing.–In our setting, each atom is modeled by two
hyperfine ground states $|0\rangle$ and $|1\rangle$, coupled through a Raman
process with strength $g$, as shown in Fig. 1(a). A microwave field with the
Rabi frequency $\Omega_{m}$ and the detuning $\Delta_{m}$ couples Rydberg $nS$
and $n^{\prime}P$ states (where $n$ and $n^{\prime}$ are principal quantum
numbers) Arias  _et al._ (2019b). This leads to a Rydberg superposition state
$|r\rangle$. The dispersion coefficients of the two Rydberg states are assumed
to have opposite signs. In this case, two atoms in the superposition state
$|r\rangle$ experience an interaction potential $U(R)$ with an attractive tail
and a repulsive core, depicted in Fig. 1(b). More details can be found in the
supplementary material (SM) SM . The minimum (located at $R_{c}$) of $U(R)$
can be a few micrometers, supporting exotic many-body states and macroscopic
dimers Petrosyan and Mølmer (2014).

In the proposed RMD scheme, the hyperfine state $|1\rangle$ is coupled to
state $|r\rangle$ where the laser is red-detuned with respect to the potential
minimum, as illustrated in Fig. 1(b). When $\Delta\gg\Omega$ and the two-
photon detuning $\Delta_{2}(R)=U(R)+2\Delta$ is much larger than the effective
Rabi frequency $\Omega_{2}\equiv\left(\sqrt{2}\Omega\right)^{2}/2\Delta$ of
the collective transition $|11\rangle\leftrightarrow|rr\rangle$ (via the
intermediate state $|1r\rangle_{s}\equiv(|1r\rangle+|r1\rangle)/\sqrt{2}$),
i.e. $|\Delta_{2}(R_{c})|\gg\Omega_{2}$, we obtain an effective interaction
potential between two atoms in the dressed $|1\rangle$ state,

$\displaystyle
V(R)=\frac{V_{0}\Delta_{2}(R)}{\Delta_{2}^{2}(R)+\gamma^{2}}U(R),$ (1)

with $V_{0}\equiv\Omega^{4}/8\Delta{}^{3}$ and $\gamma$ being the spontaneous
decay rate in state $|r\rangle$ SM . As $|\Delta_{2}(R)|\gg\gamma$, the
dressed potential can be further approximated to be $V(R)\approx
V_{0}U(R)/\Delta_{2}(R)$. The dressed potential is repulsive and attractive at
short and large distances [see Fig. 1(c)], respectively. Its depth is largest
around the distance $R_{c}$ determined by $U(R_{c})/\Delta_{2}(R_{c})$. Note
that the interaction strength at $R_{c}$ can be several times larger than the
saturated value ($\sim V_{0}$) at short distances. Such profile is different
from the soft-core potential Jau  _et al._ (2015); Keating  _et al._ (2016);
Zeiher  _et al._ (2016); Lee  _et al._ (2017).

The Rydberg dressing induces not only the dispersive interaction, but also
dissipation, whose origin lies in the finite lifetimes in Rydberg states. In
our system, the Rydberg dressing laser induces a single-body dephasing (SBD)
with the rate $\gamma^{(1)}=\Omega^{2}\gamma/4\Delta^{2}$ and a distance
dependent two-body dephasing (TBD) with a rate given by

$\displaystyle\gamma^{(2)}(R)=\frac{\Omega^{4}\gamma}{2\Delta^{2}[\Delta_{2}^{2}(R)+\gamma^{2}]}.$
(2)

This two-body dephasing rate reaches the maximal value at $R=R_{c}$.

Dephasing is ubiquitous in any Rydberg dressing scheme. However, the advantage
of the RMD is that the interaction strength $V(R)$ can be enhanced by an order
of magnitude with the dephasing only being increased by about a factor of two.
To illustrate this, we define the interaction coherence strength $\mathcal{C}$

$\mathcal{C}\equiv\frac{V(R)}{2\gamma^{(1)}+\gamma^{(2)}(R)}\approx\frac{\Omega^{2}}{4\Delta\gamma}\cdot\frac{\Delta_{2}(R)U(R)}{\Omega^{2}+\Delta_{2}^{2}(R)},$
(3)

where we have taken into account the SBD and TBD. The value of $\mathcal{C}$
strongly depends on the laser parameters. When
$\left|\Delta_{2}(R)\right|\sim\left|\Delta\right|\gg\Omega$, one finds the
familiar scaling for the SRD
$\mathcal{C}_{s}=V_{0}/2\gamma^{(1)}=\Omega^{2}/4|\Delta|\gamma$. For
$\left|\Delta\right|\gg\left|\Delta_{2}(R)\right|\sim\Omega$, the ratio
becomes $|\mathcal{C}|=\left|\Delta_{2}(R)\right|/4\gamma$, which can be
large. For example, we can choose $|\Delta_{2}(R)|=\Omega$, such that
$\mathcal{C}_{m}=\Omega/4\gamma\sim|\Delta|/\Omega\mathcal{C}_{s}$. As
typically $|\Delta|\gg\Omega$, one can gain an order of magnitude increasing
in $\mathcal{C}$ at $R=R_{c}$, depicted in Fig. 1(c).

Figure 2:  Driven-dissipative dynamics. (a)-(c) Population of the two atom
states $|0_{1}\rangle|0_{2}\rangle$,
$(|0_{1}\rangle|1_{2}\rangle+|1_{1}\rangle|0_{2}\rangle)/\sqrt{2}$, and
$|1_{1}\rangle|1_{2}\rangle$ versus the scaled time $V_{0}t$ for $N=2$.
(d)-(f) Time-dependence of the mean spin components $\langle J_{x}\rangle$,
$\langle J_{z}\rangle$, and the covariance $\langle(\Delta J_{z})^{2}\rangle$
for $N=7$. Atoms are initially in the state
$\otimes_{k=1}^{N}(|0_{k}\rangle+|1_{k}\rangle)/\sqrt{2}$. We show numerical
data with (red dash) and without (green solid) TBD, as well as the coherent
dynamics (blue solid) by turning off both the SBD and TBD. Other parameters
are $\Omega=1$, $(\Delta,\gamma)/\Omega=(5.5,0.005)$, $R_{c}=a$,
$g/V_{0}=0.2$, $n=50$, and $n^{\prime}=58$.

Effective many-body spin model of dressed Rydberg atoms.–We now consider the
many-body dynamics of the atoms trapped in a deep one-dimensional (1D) optical
lattice Zhang  _et al._ (2011); Li  _et al._ (2013). The dynamics with the
dressed ground state manifold is then described by Hamiltonian
$\mathcal{H}_{s}=\sum_{k=1}^{N}[gJ_{x}^{(k)}+\delta^{(k)}J_{z}^{(k)}]+\sum_{k<l}V(R_{kl})J_{z}^{(k)}J_{z}^{(l)}$
with the spin-$1/2$ Pauli operators $J_{x}^{(k)}=\left(|1_{k}\rangle\langle
0_{k}|+|0_{k}\rangle\langle 1_{k}|\right)/2$,
$J_{y}^{(k)}=i\left(|0_{k}\rangle\langle 1_{k}|-|1_{k}\rangle\langle
0_{k}|\right)/2$, and $J_{z}^{(k)}=\left(|1_{k}\rangle\langle
1_{k}|-|0_{k}\rangle\langle 0_{k}|\right)/2$. The dressing-induced interaction
between atoms located on lattice sites $k$ and $l$ at a distance
$R_{kl}/a=|k-l|$ is given by $V(R_{jk})$. Taking into account of the single
and two-body dephasing, the dynamics of the many-body two-level system is
governed by the master equation,

$\stackrel{{\scriptstyle.}}{{\rho}}=-i[\mathcal{H}_{s},\rho]+\sum_{k=1}^{N}\mathcal{L}[o_{k}]\rho+\sum_{k<l}\mathcal{L}[o_{kl}]\rho,$
(4)

where the Lindblad operator $\mathcal{L}[o]\rho=o\rho
o^{\dagger}-\frac{1}{2}\\{o^{\dagger}o,\rho\\}$ with
$o_{k}=\sqrt{\gamma^{(1)}}|1_{k}\rangle\langle 1_{k}|$, and
$o_{kl}=\sqrt{\gamma^{(2)}(R_{kl})}|1_{k}1_{l}\rangle\langle 1_{k}1_{l}|$
being the so-called jump operator corresponding to single- and two-body
dephasing, respectively.

We first explore the competition between the dephasing and dressed two-body
interaction with an example of two atoms, shown in Fig. 2(a)-(c). During the
early stage of dynamical evolution $0<V_{0}t<10$, both the SBD and TBD hardly
affect the coherent dynamics. This is seen from the population dynamics, where
the master equation simulation agrees with the coherent dynamics. They only
become important in the long-time dynamics, where the many-body coherence is
damped. In this region, the dominant contribution results from the SBD, as
found from our numerical simulation.

The short-time dynamics is largely coherent even for a moderate number of
spins. In Fig. 2, the expectation values of the collective spin operator
$J_{x}=\sum_{k}J_{x}^{(k)}$ and $J_{z}=\sum_{k}J_{z}^{(k)}$, and the quantum
fluctuation $\langle(\Delta J_{z})^{2}\rangle$ of the operator $J_{z}$ for a
chain of $N=7$ atoms are shown. The numerical simulations show that the
dephasing plays visible roles when $V_{0}t>4$. When $V_{0}t<4$, the high
coherence strength guarantees that the dispersive interaction dominates in the
short-time dynamics.

Fast spin squeezing.–We will show that the large $\mathcal{C}$ in the RMD
allows to achieve rapid spin squeezing in the 1D lattice Ma  _et al._ (2011).
The degree of spin squeezing can be quantified through the squeezing parameter
(SP) Wineland  _et al._ (1994),

$\xi^{2}=\frac{N\left(\Delta
J_{\vec{n}_{\bot}}\right)^{2}}{|\langle\vec{J}\rangle|^{2}},$ (5)

where $J_{\vec{n}_{\bot}}$ is the spin component perpendicular to the
direction of the mean total spin $\langle\vec{J}\rangle$ with
$\vec{J}=\sum_{k}\bm{J}{}^{(k)}$. The parameter $\xi^{2}$ is then defined by
choosing the direction $\vec{n}_{\bot}$ at which $\Delta J_{n_{\bot}}$ is
minimized. For the state with $\xi^{2}<1$, its phase sensitivity is improved
over the standard quantum limit ($\sim 1/\sqrt{N}$) given by the coherent spin
state Qin  _et al._ (2020).

To achieve spin squeezing, initially the atoms are prepared in the state
$|0\rangle$ such that the mean spin direction is along the $z$-direction, and
thus the perpendicular spin component is simply
$J_{\vec{n}_{\bot}}\left(\theta\right)=\textrm{cos}\left(\theta\right)J_{x}+\textrm{sin}\left(\theta\right)J_{y}$
with $\theta$ being the angle with respect to the $x$ axis, which immediately
implies $\left(\Delta
J_{\vec{n}_{\bot}}\right)^{2}=\textrm{cos}^{2}\left(\theta\right)\langle
J_{x}^{2}\rangle+\textrm{si$n^{2}$}\left(\theta\right)\langle
J_{y}^{2}\rangle+\text{sin}\left(2\theta\right)\langle
J_{x}J_{y}+J_{y}J_{x}\rangle/2$. We then implement the spin-echo type
squeezing protocol Gil  _et al._ (2014), which consists of a $\pi/2$ pulse of
length $t_{\pi/2}=\pi/\left(2g\right)$ to rotate the spin along the $x$ axis,
RMD of duration $\tau/2$, a $\pi$ pulse to rotate the spin, RMD of duration
$\tau/2$, and then a $\pi/2$ pulse. During the spin rotation, the Rydberg
dressing laser is turned off. At the end of the sequence, the mean spin is
along the $z$-direction again with the noise uncertainty for
$J_{\vec{n}_{\bot}}$ along one of the perpendicular direction being squeezed.

The SP resulting from the above protocol can be analytical calculated in the
absence of dissipation Gil  _et al._ (2014). In the presence of dissipation,
it is not possible to carry out analytical calculations for the dressed spin
model. For small systems with a few atoms we can study the driven-dissipative
dynamics by numerically solving the master equation and evaluate the SP.
Approximations have to be taken when there are tens and even hundreds of
lattice sites due to the large Hilbert space. Since the $\pi/2$ and $\pi$
rotational pulses are much shorter than the typical time scale of the SBD and
TBD, dissipation is therefore not important during the dynamical evolution.
This allows us to simulate the dynamics approximately with a non-Hermitian
Hamiltonian
$\mathcal{H}_{c}=\mathcal{H}_{s}-i(\gamma^{(1)}/2)\sum_{k=1}^{N}|1_{k}\rangle\langle
1_{k}|-\left(i/2\right)\sum_{j<k}\gamma^{(2)}(R_{jk})|1_{j}1_{k}\rangle\langle
1_{j}1_{k}|$, where the single- and two-body jump processes have been
neglected. Furthermore, we apply the mean-field approximation to the
decoherence part induced by the TBD, and finally arrive at the non-Hermitian
Hamiltonian

$\displaystyle\mathcal{H}_{c}$ $\displaystyle\approx$
$\displaystyle\frac{1}{2}\sum_{j=1}^{N}\left(\sum_{k\neq
j}V(R_{jk})-i\Gamma_{z}\right)J_{z}^{(j)}$ (6)
$\displaystyle+\sum_{j<k}V(R_{jk})J_{z}^{(j)}J_{z}^{(k)}-i\frac{\bar{\Gamma}}{2},$

where the mean-field decay rate
$\bar{\Gamma}=\frac{N}{2}[\gamma^{(1)}+\Gamma_{0}(\frac{1}{4}-\langle{J}_{z}\rangle{}^{2}/N^{2})]$
and
$\Gamma_{z}=\gamma^{(1)}+{\Gamma}_{0}(\frac{1}{2}+\langle{J}_{z}\rangle/N)$
with $\Gamma_{0}=\sum_{|k-j|=1}^{N-1}\gamma^{(2)}(R_{jk})$. We then evaluate
the time-dependent many-body state $|\psi(t)\rangle\approx
e^{-i\mathcal{H}_{c}t}|\psi(0)\rangle$ where the initial state
$|\psi(0)\rangle$ corresponds to all atoms in state $|0\rangle$.

Figure 3: The squeezing parameter as a function of the dressing time.
Numerical results obtained from the master equation (4), the conditional
Hamiltonian with and without the two-body decay, and the coherent dynamics,
are shown. We consider two different lattice constants (a) $R_{c}=a$ and (b)
$R_{c}=3a$. The coordinates for the optimal coherent and dissipative squeezing
(i.e. the minimum) are labeled. Here $N=10$ and other parameters are the same
as in Fig. 2.

Eq. (6) enables us to find an analytical solution in the presence of the
dephasing and for large systems [see SM SM ]. To verify the accuracy of the
analytical solution against the exact result, we first examine a situation
with $N=10$ atoms with $(\Delta,\gamma)/\Omega=(10,0.005)$ (corresponding to
$\gamma^{(1)}/\gamma\approx 8.3\times 10^{-3}$), depicted in Fig. 3. For
$R_{c}=a$, both the effective master equation Civitarese  _et al._ (2009);
Kominis (2008) and the non-Hermitian Hamiltonian give the similarly optimal SP
(i.e. the minimum) $\xi_{\text{min}}^{2}\sim 0.6$ around the dressing time
$V_{0}\tau_{\text{min}}\approx 0.17$, which is much shorter than that by the
traditional SRD scheme Gil  _et al._ (2014), and see Fig. 4(a). The
dissipative SP is slightly weaker than the coherent counterpart $\xi^{2}\sim
0.58$ without including the decoherence. For $R_{c}=3a$, because the repulsive
interactions among the nearest and the second-nearest neighboring sites
counteract the effect of the strong attractive interactions, the optimal SP
reduces to $\xi_{\text{min}}^{2}\sim 0.83$. On the the hand, in comparison
with the individually spontaneous decay, the TBD does not cause significant
effects on the rapid spin squeezing at the time scale
$V_{0}\tau_{\text{min}}$. We find that the dissipative dynamics is well
captured by the conditional Hamiltonian $\mathcal{H}_{c}$ in general, which
motivates us to investigate larger systems with $\mathcal{H}_{c}$ (see SM SM
).

Figure 4: Spin squeezing with RMD and SRD. The optimal spin squeezing
$\xi_{\text{min}}^{2}$ [(a1)-(f1)] and the required dressing time
$V_{0}\tau_{\text{min}}$ [(a2)-(f2)] of the RMD and SRD schemes are shown. For
$V_{0}\tau_{\text{min}}=0$ in (a2)-(f2), the squeezing parameter $\xi^{2}$
remains larger than 1 for all interaction time $\tau$. The atomic decay rates
are $\gamma/\Omega=\text{ }0.01$ in (a)-(c), and $\gamma/\Omega=\text{ }0.005$
in (d)-(f). Other parameter are the same as in Fig. 3.

Based on Eq. (6), it is not difficult to find that the time-dependent squeeze
parameter $\xi^{2}(\tau)$ exhibits an exponential decay as the dressing time
$\tau$ increases, namely, $\xi^{2}(\tau)\propto e^{-\bar{\Gamma}\tau}$, with
$\bar{\Gamma}$ depending on the effective single- and two-body dephasing
rates. We show in Fig. 4 the optimally dissipative SP $\xi_{\text{min}}^{2}$
and the corresponding dressing time $V_{0}\tau_{\text{min}}$ versus the number
of the lattice sites with different lattice constant $R_{c}=ka$ ($k=1,2,3$).
For comparison, we also show the SP influenced only by the SRD, where the TBD
is negligible Gil  _et al._ (2014). In both cases, the detrimental effects of
the dephasing are to diminish the squeezing when the number of atoms
increases. In particular, the spin squeezing is more robust in case of RMD,
where the SP is smaller than 1 for $N$ up to 200 [see the first and third row
in Fig. 4]. In contrast the SP reaches 1 typically at smaller $N$ with the
SRD. On the other hand, for the RMD scheme with $R_{c}=a$, where the dressed
atoms next to each other get the strongest interaction coherence strength
$\mathcal{C}$, the optimal SP exists for $N>150$ and the dressing times are
all within $V_{0}\tau_{\text{min}}<0.17$; the SRD scheme however allows for
$\xi_{\text{min}}^{2}<1$ when the system contains only a few tens of atoms and
requires a much longer dressing time. As the lattice constant decreases from
$a=R_{c}/2$ to $R_{c}/3$, the SP for the RMD scheme decreases. This reduction
results from the fact that the phase of the many-body state $|\psi(t)\rangle$
due to repulsive [$V(R)>0$] and attractive [$V(R)<0$] interactions cancels. In
contrast, the SRD scheme becomes more robust to the dephasing for $a=R_{c}/3$
due to the constructive phase accumulation by all repulsive interactions (i.e.
the interaction potential is positive for all realized interatomic distances).
In general, a compromise could be found in between getting a large interaction
coherence strength and keeping all negative (or positive) phase terms. In this
regard, the SP under the RMD can be better than that by the SRD for $N>40$ and
$R_{c}=2a$ , but the SRD scheme is able to achieve strong SP, i.e.,
$\xi_{\text{min}}^{2}<0.5$ for $R_{c}\geq 3a$ and $N<30$.

Conclusion.–We have proposed a Rydberg molecule dressing scheme, where the
dressing laser is tuned off-resonantly to be the attractive molecular
potential of two Rydberg atoms, which can be created by coupling two different
Rydberg states with MW fields. The Rydberg molecule dressing leads to
distance-selective attractive interaction potential of the Rydberg dressed
ground-state atoms. We have shown that enhanced Rydberg dressed interactions
can be achieved, while the respective dissipation is still weak. This allows
us to generate entangling phases for fast spin squeezing. Our work is relevant
to the study of quantum computation and simulation with finite systems of
Rydberg atoms trapped in optical arrays Labuhn  _et al._ (2016b); Bernien  _et
al._ (2017); Browaeys and Lahaye (2020) and trapped Rydberg ions Zhang  _et
al._ (2020). In the future, it is worth to explore squeezing in higher
dimensional settings and investigate whether the distance selective two-body
interaction also here lead to enhanced squeezing. The controllable interaction
and dephasing furthermore opens opportunities to quantum simulate interesting
many-body dynamics, complementary to other studies that aim at the creation of
exotic phases and many-body states Glaetzle  _et al._ (2017a, b); Sándor  _et
al._ (2017); Thomas  _et al._ (2018). We note that a recent experiment has
demonstrated the distance selection interaction in an atom array Hollerith
_et al._ (2021), where the ground state atom is dressed to Rydberg macrodimers
Boisseau  _et al._ (2002). Our scheme offers the flexibility to engineer the
dressed interaction through tuning the MW field Petrosyan and Mølmer (2014).

###### Acknowledgements.

Acknowledgments. IL acknowledges support from the Baden-Württemberg
Foundation, through the project BWST$\\_$ISF2019-23 (“Internationale
Spitzenforschung”) and the Deutsche Forschungsgemeinschaft through SPP 1929
GiRyd, Grant No. 428276754. W. L. acknowledges support from the EPSRC Grant
No. EP/R04340X/1 via the QuantERA project “ERyQSenS”, and the Royal Society
through the International Exchanges Cost Share award No.
IEC$\backslash$NSFC$\backslash$181078\. H.W. and S.B.Z. acknowledge support
from the National Natural Science Foundation of China under Grants No.
11774058, No. 11874114, and No. 12174058.

## References

  * Jaksch  _et al._ (2000) D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Phys. Rev. Lett. 85, 2208 (2000).
  * Lukin  _et al._ (2001) M. D. Lukin, M. Fleischhauer, R. Cote, L. M. Duan, D. Jaksch, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 87, 037901 (2001).
  * Wilk  _et al._ (2010) T. Wilk, A. Gaëtan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, Phys. Rev. Lett. 104, 010502 (2010).
  * Zhang  _et al._ (2010) X. L. Zhang, L. Isenhower, A. T. Gill, T. G. Walker, and M. Saffman, Phys. Rev. A 82, 030306 (2010).
  * Isenhower  _et al._ (2010) L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson, T. G. Walker, and M. Saffman, Phys. Rev. Lett. 104, 010503 (2010).
  * Keating  _et al._ (2013) T. Keating, K. Goyal, Y.-Y. Jau, G. W. Biedermann, A. J. Landahl, and I. H. Deutsch, Phys. Rev. A 87, 052314 (2013).
  * Hankin  _et al._ (2014) A. M. Hankin, Y.-Y. Jau, L. P. Parazzoli, C. W. Chou, D. J. Armstrong, A. J. Landahl, and G. W. Biedermann, Phys. Rev. A 89, 033416 (2014).
  * Keating  _et al._ (2015) T. Keating, R. L. Cook, A. M. Hankin, Y.-Y. Jau, G. W. Biedermann, and I. H. Deutsch, Phys. Rev. A 91, 012337 (2015).
  * Li and Kuzmich (2016) L. Li and A. Kuzmich, Nat. Commun. 7, 13618 (2016).
  * Levine  _et al._ (2018) H. Levine, A. Keesling, A. Omran, H. Bernien, S. Schwartz, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Phys. Rev. Lett. 121, 123603 (2018).
  * Honer  _et al._ (2010) J. Honer, H. Weimer, T. Pfau, and H. P. Büchler, Phys. Rev. Lett. 105, 160404 (2010).
  * Mattioli  _et al._ (2013) M. Mattioli, M. Dalmonte, W. Lechner, and G. Pupillo, Phys. Rev. Lett. 111, 165302 (2013).
  * Glaetzle  _et al._ (2014) A. W. Glaetzle, M. Dalmonte, R. Nath, I. Rousochatzakis, R. Moessner, and P. Zoller, Phys. Rev. X 4, 041037 (2014).
  * Glaetzle  _et al._ (2015) A. W. Glaetzle, M. Dalmonte, R. Nath, C. Gross, I. Bloch, and P. Zoller, Phys. Rev. Lett. 114, 173002 (2015).
  * Schauß  _et al._ (2015) P. Schauß, J. Zeiher, T. Fukuhara, S. Hild, M. Cheneau, T. Macrì, T. Pohl, I. Bloch, and C. Gross, Science 347, 1455 (2015).
  * de Léséleuc  _et al._ (2017) S. de Léséleuc, D. Barredo, V. Lienhard, A. Browaeys, and T. Lahaye, Phys. Rev. Lett. 119, 053202 (2017).
  * van Bijnen and Pohl (2015) R. M. W. van Bijnen and T. Pohl, Phys. Rev. Lett. 114, 243002 (2015).
  * Li and Sarma (2015) X. Li and S. D. Sarma, Nat. Commun. 6, 7137 (2015).
  * Labuhn  _et al._ (2016a) H. Labuhn, D. Barredo, S. Ravets, S. de Leseleuc, T. Macri, T. Lahaye, and A. Browaeys, Nature 534, 667 (2016a).
  * de Léséleuc  _et al._ (2018) S. de Léséleuc, S. Weber, V. Lienhard, D. Barredo, H. P. Büchler, T. Lahaye, and A. Browaeys, Phys. Rev. Lett. 120, 113602 (2018).
  * Santos  _et al._ (2000) L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, Phys. Rev. Lett. 85, 1791 (2000).
  * Pupillo  _et al._ (2010) G. Pupillo, A. Micheli, M. Boninsegni, I. Lesanovsky, and P. Zoller, Phys. Rev. Lett. 104, 223002 (2010).
  * Henkel  _et al._ (2010) N. Henkel, R. Nath, and T. Pohl, Phys. Rev. Lett. 104, 195302 (2010).
  * Johnson and Rolston (2010) J. E. Johnson and S. L. Rolston, Phys. Rev. A 82, 033412 (2010).
  * Balewski  _et al._ (2014) J. B. Balewski, A. T. Krupp, A. Gaj, S. Hofferberth, R. Löw, and T. Pfau, New J. Phys. 16, 063012 (2014).
  * Jau  _et al._ (2015) Y.-Y. Jau, A. M. Hankin, T. Keating, I. H. Deutsch, and G. W. Biedermann, Nat. Phys. 12, 71 (2015).
  * Zeiher  _et al._ (2016) J. Zeiher, R. van Bijnen, P. Schauß, S. Hild, J.-y. Choi, T. Pohl, I. Bloch, and C. Gross, Nat. Phys. 12, 1095 (2016).
  * Mukherjee  _et al._ (2016) R. Mukherjee, T. C. Killian, and K. R. A. Hazzard, Phys. Rev. A 94, 053422 (2016).
  * Buchmann  _et al._ (2017) L. F. Buchmann, K. Mølmer, and D. Petrosyan, Phys. Rev. A 95, 013403 (2017).
  * Płodzień  _et al._ (2017) M. Płodzień, G. Lochead, J. de Hond, N. J. van Druten, and S. Kokkelmans, Phys. Rev. A 95, 043606 (2017).
  * Chai  _et al._ (2017) X. Chai, L. Zhang, D. Ma, L. Yan, H. Bao, and J. Qian, Phys. Rev. A 96, 053417 (2017).
  * Borish  _et al._ (2020) V. Borish, O. Marković, J. A. Hines, S. V. Rajagopal, and M. Schleier-Smith, Phys. Rev. Lett. 124, 063601 (2020).
  * Carroll  _et al._ (2004) T. J. Carroll, K. Claringbould, A. Goodsell, M. J. Lim, and M. W. Noel, Phys. Rev. Lett. 93, 153001 (2004).
  * Barredo  _et al._ (2014) D. Barredo, S. Ravets, H. Labuhn, L. Béguin, A. Vernier, F. Nogrette, T. Lahaye, and A. Browaeys, Phys. Rev. Lett. 112, 183002 (2014).
  * Shi and Kennedy (2018) X.-F. Shi and T. A. B. Kennedy, Phys. Rev. A 97, 033414 (2018).
  * Khasseh  _et al._ (2017) R. Khasseh, S. H. Abedinpour, and B. Tanatar, Phys. Rev. A 96, 053611 (2017).
  * Li  _et al._ (2018) Y. Li, A. Geißler, W. Hofstetter, and W. Li, Phys. Rev. A 97, 023619 (2018).
  * Li  _et al._ (2020) Y. Li, H. Cai, D.-w. Wang, L. Li, J. Yuan, and W. Li, Phys. Rev. Lett. 124, 140401 (2020).
  * Zeiher  _et al._ (2017) J. Zeiher, J.-y. Choi, A. Rubio-Abadal, T. Pohl, R. van Bijnen, I. Bloch, and C. Gross, Phys. Rev. X 7, 041063 (2017).
  * Guardado-Sanchez  _et al._ (2021) E. Guardado-Sanchez, B. M. Spar, P. Schauss, R. Belyansky, J. T. Young, P. Bienias, A. V. Gorshkov, T. Iadecola, and W. S. Bakr, Phys. Rev. X 11, 021036 (2021).
  * Bouchoule and Mølmer (2002) I. Bouchoule and K. Mølmer, Phys. Rev. A 65, 041803 (2002).
  * Gil  _et al._ (2014) L. I. R. Gil, R. Mukherjee, E. M. Bridge, M. P. A. Jones, and T. Pohl, Phys. Rev. Lett. 112, 103601 (2014).
  * Arias  _et al._ (2019a) A. Arias, G. Lochead, T. M. Wintermantel, S. Helmrich, and S. Whitlock, Phys. Rev. Lett. 122, 053601 (2019a).
  * Weber  _et al._ (2012) T. M. Weber, T. Niederprüm, T. Manthey, P. Langer, V. Guarrera, G. Barontini, and H. Ott, Phys. Rev. A 86, 020702 (2012).
  * Malossi  _et al._ (2014) N. Malossi, M. M. Valado, S. Scotto, P. Huillery, P. Pillet, D. Ciampini, E. Arimondo, and O. Morsch, Phys. Rev. Lett. 113, 023006 (2014).
  * Schempp  _et al._ (2014) H. Schempp, G. Günter, M. Robert-de Saint-Vincent, C. S. Hofmann, D. Breyel, A. Komnik, D. W. Schönleber, M. Gärttner, J. Evers, S. Whitlock, and M. Weidemüller, Phys. Rev. Lett. 112, 013002 (2014).
  * Goldschmidt  _et al._ (2016) E. A. Goldschmidt, T. Boulier, R. C. Brown, S. B. Koller, J. T. Young, A. V. Gorshkov, S. L. Rolston, and J. V. Porto, Phys. Rev. Lett. 116, 113001 (2016).
  * Boulier  _et al._ (2017) T. Boulier, E. Magnan, C. Bracamontes, J. Maslek, E. A. Goldschmidt, J. T. Young, A. V. Gorshkov, S. L. Rolston, and J. V. Porto, Phys. Rev. A 96, 053409 (2017).
  * Barredo  _et al._ (2020) D. Barredo, V. Lienhard, P. Scholl, S. de Léséleuc, T. Boulier, A. Browaeys, and T. Lahaye, Phys. Rev. Lett. 124, 023201 (2020).
  * (50) W. Li, C. Ates, and I. Lesanovsky, Phys. Rev. Lett. 110, 213005.
  * Gambetta  _et al._ (2020) F. M. Gambetta, W. Li, F. Schmidt-Kaler, and I. Lesanovsky, Phys. Rev. Lett. 124, 043402 (2020).
  * Lorenz  _et al._ (2021) N. Lorenz, L. Festa, L.-M. Steinert, and C. Gross, SciPost Phys. 10, 052 (2021).
  * Keating  _et al._ (2016) T. Keating, C. H. Baldwin, Y.-Y. Jau, J. Lee, G. W. Biedermann, and I. H. Deutsch, Phys. Rev. Lett. 117, 213601 (2016).
  * Lee  _et al._ (2017) J. Lee, M. J. Martin, Y.-Y. Jau, T. Keating, I. H. Deutsch, and G. W. Biedermann, Phys. Rev. A 95, 041801 (2017).
  * Gross and Bloch (2017) C. Gross and I. Bloch, Science 357, 995 (2017).
  * Schauss (2018) P. Schauss, Quantum Sci. Technol. 3, 023001 (2018).
  * Wu  _et al._ (2021) X. Wu, X. Liang, Y. Tian, F. Yang, C. Chen, Y.-C. Liu, M. K. Tey, and L. You, Chin. Phys. B 30, 020305 (2021).
  * Petrosyan and Mølmer (2014) D. Petrosyan and K. Mølmer, Phys. Rev. Lett. 113, 123003 (2014).
  * Arias  _et al._ (2019b) A. Arias, G. Lochead, T. M. Wintermantel, S. Helmrich, and S. Whitlock, Phys. Rev. Lett. 122, 053601 (2019b).
  * (60) See SupplementalMaterial for details about the dressed potential, and discussions on numerical calculations, which includes Refs. Gil  _et al._ , 2014 and Wineland  _et al._ , 1994.
  * Zhang  _et al._ (2011) S. Zhang, F. Robicheaux, and M. Saffman, Phys. Rev. A 84, 043408 (2011).
  * Li  _et al._ (2013) L. Li, Y. O. Dudin, and A. Kuzmich, Nature 498, 466 (2013).
  * Ma  _et al._ (2011) J. Ma, X. Wang, C. Sun, and F. Nori, Phys. Rep. 509, 89 (2011).
  * Wineland  _et al._ (1994) D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, Phys. Rev. A 50, 67 (1994).
  * Qin  _et al._ (2020) W. Qin, Y.-H. Chen, X. Wang, A. Miranowicz, and F. Nori, Nanophotonics 9, 4853 (2020).
  * Civitarese  _et al._ (2009) O. Civitarese, M. Reboiro, L. Rebon, and D. Tielas, Phys. Lett. A 373, 754 (2009).
  * Kominis (2008) I. K. Kominis, Phys. Rev. Lett. 100, 073002 (2008).
  * Labuhn  _et al._ (2016b) H. Labuhn, D. Barredo, S. Ravets, S. de Léséleuc, T. Macrì, T. Lahaye, and A. Browaeys, Nature 534, 667 (2016b).
  * Bernien  _et al._ (2017) H. Bernien, S. Schwartz, A. Keesling, H. Levine, A. Omran, H. Pichler, S. Choi, A. S. Zibrov, M. Endres, M. Greiner, V. Vuletić, and M. D. Lukin, Nature 551, 579 (2017).
  * Browaeys and Lahaye (2020) A. Browaeys and T. Lahaye, Nat. Phys. 16, 132 (2020).
  * Zhang  _et al._ (2020) C. Zhang, F. Pokorny, W. Li, G. Higgins, A. Pöschl, I. Lesanovsky, and M. Hennrich, Nature 580, 345 (2020).
  * Glaetzle  _et al._ (2017a) A. W. Glaetzle, R. M. W. van Bijnen, P. Zoller, and W. Lechner, Nat. Commun. 8, 15813 (2017a).
  * Glaetzle  _et al._ (2017b) A. W. Glaetzle, K. Ender, D. S. Wild, S. Choi, H. Pichler, M. D. Lukin, and P. Zoller, Phys. Rev. X 7, 031049 (2017b).
  * Sándor  _et al._ (2017) N. Sándor, R. González-Férez, P. S. Julienne, and G. Pupillo, Phys. Rev. A 96, 032719 (2017).
  * Thomas  _et al._ (2018) O. Thomas, C. Lippe, T. Eichert, and H. Ott, Nat. Commun. 9, 2238 (2018).
  * Hollerith  _et al._ (2021) S. Hollerith, K. Srakaew, D. Wei, A. Rubio-Abadal, D. Adler, P. Weckesser, A. Kruckenhauser, V. Walther, R. van Bijnen, J. Rui, C. Gross, I. Bloch, and J. Zeiher, arXiv:2110.10125 (2021).
  * Boisseau  _et al._ (2002) C. Boisseau, I. Simbotin, and R. Côté, Phys. Rev. Lett. 88, 133004 (2002).

Supplementary material for: “Fast spin squeezing by distance-selective long-
range interactions with Rydberg molecule dressing”

This supplementary material contains additional details on the analysis in the
main text.

##  I microwave modulated Rydberg interaction energies and Rydberg molecular
dressing

We consider two interacting Rydberg atoms separated by $R$, where two Rydberg
states $|np\rangle$ and $|n^{\prime}s\rangle$ are continuously driven by a
microwave (MW) field with Rabi frequency $\Omega_{mw}$ and detuning
$\Delta_{mw}$. The Rydberg interaction between the states $|np\rangle$ and
$|n^{\prime}s\rangle$ is of the dipole nature and scales as $R^{-3}$, while
the van der Waals interaction scaling as $R^{-6}$ takes over if the atoms
simultaneously populate the same Rydberg level $|n^{\prime}s\rangle$ (or
$|np\rangle)$. The Hamiltonian for the coupled Rydberg pair states
$\\{|n^{\prime}s,n^{\prime}s^{\prime}\rangle,(|n^{\prime}s,np\rangle+|np,n^{\prime}s\rangle)/\sqrt{2},|np,np\rangle\\}$
reads

$H_{sp}(R)=\left[\begin{array}[]{ccc}-\frac{C_{6}^{(s,s)}}{R^{6}}&\frac{\Omega_{mw}}{\sqrt{2}}&0\\\
\frac{\Omega_{mw}}{\sqrt{2}}&\Delta_{mw}+\frac{C_{3}^{(s,p)}}{R^{3}}&\frac{\Omega_{mw}}{\sqrt{2}}\\\
0&\frac{\Omega_{mw}}{\sqrt{2}}&2\Delta_{mw}-\frac{C_{6}^{(p,p)}}{R^{6}}\end{array}\right].$
(S1)

The eigenenergies $E_{n}^{(2)}(R)$ ($n=1,2,3$) of the Hamiltonian (S1) give
the modified level configuration (e.g. the eigenstates
$|\phi_{n}^{(2)}\rangle$) for the Rydberg pair states, which strongly depend
on the coupling strength and detuning between the MW field and the Rydberg
states. As a specific example, we show, in Fig. S1(a), the distance-dependent
interactions between the Rydberg states $|58p\rangle$ and $|50s\rangle$ for Rb
atoms. The Rydberg molecular potential appears at the upper (black solid)
branch corresponding to the pair state $|50s\rangle+|50s\rangle$ at large
distances. It shows an attractive character when the interatomic separation
$R$ is larger than 2 $\mu\text{m}$ and is repulsive at short distances.

The ground state atoms are coupled from the ground state $|5s\rangle$ to the
MW coupled Rydberg states
$|E_{+}^{(1)}\rangle=\text{sin}\left(\phi/2\right)|n^{\prime}s\rangle+\text{\text{cos}$\left(\phi/2\right)$}|np\rangle$
($|E_{-}^{(1)}\rangle=-\text{sin}\left(\phi/2\right)|np\rangle+\text{cos}\left(\phi/2\right)|n^{\prime}s\rangle$)
with the laser detunings $\Delta_{+}$($\Delta_{-}$) and coupling strengths
$\Omega_{+}=$$\text{
cos}\left(\phi/2\right)|\vec{d}_{5n}\cdot\vec{E}_{0}|/\hbar$
($\Omega_{-}=$$\text{
sin}\left(\phi/2\right)|\vec{d}_{5n}\cdot\vec{E}_{0}|/\hbar$), where
$\phi=\text{tan}^{-1}(2\Omega_{mw}/\Delta_{mw})$, $\vec{d}_{5n}$ is the dipole
moment of the transition $|5s\rangle\leftrightarrow|np\rangle$, and
$\vec{E}_{0}$ is the electric field amplitude. We are interested in the RMD
based on two-photon dispersive interactions, that is, the doubly-excitation
states are far-off resonant from the two-photon transition intermediated by
$|E_{+}^{(1)}\rangle$. At large distances, the effective Rabi frequencies for
the transition channel
$|5s\rangle|5s\rangle\rightarrow|5s\rangle|E_{+}^{(1)}\rangle\rightarrow|\phi_{1}^{(2)}\rangle$
($\sim|E_{+}^{(1)}\rangle|E_{+}^{(1)}\rangle$) is approximately given by
$\Omega_{2}\approx\Omega_{+}^{2}/\Delta_{+}$. While at short distances,
$\Omega_{2}(R)$ becomes distance-dependent. The parameter regime for RMD is
thus given by $|\Delta_{+}|\gg\Omega_{+}$ and
$|2\Delta_{+}+E_{1}^{(2)}(\infty)-E_{1}^{(2)}(R)|\gg\Omega_{2},\gamma$, from
which we can then find the dressed interaction potential via the perturbation
theory (see the following section). The perturbation result of the dressed
interaction potential agrees with the full numerical calculation, shown in
Fig. S1(b).

Figure S1: (Color online) (a) Potential energy $E_{n}^{(2)}(R)$ for the
coupled Rydberg pair states $|58p\rangle$ and $|50s\rangle$. The depth of the
Rydberg dressed potential can be modulated by the MW field driving strength
and detuning. (b) Interaction potential for two ground state atoms dressed to
the Rydberg state that corresponds to the $E_{2}^{(2)}$ potential. The dressed
interaction potential with full numerical simulation (red solid) agrees well
with the perturbation result (blue dashed). The avoided crossing at the very
short distance (e.g. smaller than the lattice spacing) is due to the Rydberg
antiblockade, where $|E_{n}^{(2)}(R)-2\Delta_{+}|=0$. Parameters are
$\Omega_{mw}/2\pi=134$ MHz, $\Delta_{mw}=0$.

##  II Derivation of the effective master equation

We consider two hyperfine ground states $|0\rangle$ and $|1\rangle$ where
state $|1\rangle$ is weakly coupled to the Rydberg state $|r\rangle=|+\rangle$
by the dressing laser (Rabi frequency $\Omega=\Omega_{+}$ and detuning
$\Delta=\Delta_{+}$). The Hamiltonian of the atoms is given by
$H_{o}=H_{a}+H_{i}$, where the atomic Hamiltonian for the spin states
$|0\rangle$, $|1\rangle$ and Rydberg state $|r\rangle$ is

$H_{a}=\sum_{j}\left[\Delta\sigma_{rr}^{(j)}+\Delta_{0}\sigma_{00}^{(j)}+\left(\frac{g}{2}\sigma_{01}^{(j)}+\frac{\Omega}{2}\sigma_{1r}^{(j)}+\text{H.c.}\right)\right]$
(S2)

where $\Delta_{0}$ and $g$ is the detuning and Rabi frequency between the two
hyperfine states (e.g. coupled through a MW field or a Raman process). The
two-body interaction in the Rydberg state is

$H_{i}=\frac{1}{2}\sum_{jk}U(R_{jk})\sigma_{rr}^{(j)}\sigma_{rr}^{(k)},$ (S3)

where $U(R)=E_{2}^{(2)}$ is the molecular interaction potential of the MW
coupled Rydberg state. Taking into account of the spontaneous decay in the
Rydberg state, the dynamics is governed by a master equation

$\dot{\rho}=-[H_{o},\rho]-\gamma\sum_{j}\left(\sigma_{1r}^{(j)}\rho\sigma_{r1}^{(j)}-\frac{1}{2}\left\\{\sigma_{r1}^{(j)}\sigma_{1r}^{(j)},\rho\right\\}\right).$
(S4)

We first derive the effective master equation of two atoms. This is done in a
regime where the detuning of the dressing laser is large, i.e.
$\Delta>\Omega\gg\gamma\gg g\approx\Delta_{0}$. We will then eliminate
dynamics in the Rydberg state to obtain the effective dynamics by separating
different time scales. As the dynamics due to the dressing laser is fast, this
allows us to split the Hilbert space into two parts, $P$ and $Q$ where $P$\-
and $Q$-subspace define the slowly and rapidly varying quantities with
$P+Q=\mathbb{1}$. The effective Hamiltonian for the slow dynamics can be
obtained through

$H_{eff}=PHP-PHQ\frac{1}{QHQ}QHP.$ (S5)

In the two atom basis
$\\{|00\rangle,\,|01\rangle,\,|10\rangle,\,|11\rangle,\,|rr\rangle\\}$, the
matrix form of the Hamiltonian is expressed as

$\displaystyle H_{eff}=\begin{pmatrix}2\Delta_{0}&0&0&0&0\\\
0&\Delta_{0}-\frac{\Omega^{2}}{4\Delta}&0&0&0\\\
0&0&\Delta_{0}-\frac{\Omega^{2}}{4\Delta}&0&0\\\
0&0&0&-\frac{\Omega^{2}}{2\Delta}&-\frac{\Omega^{2}}{2\Delta}\\\
0&0&0&-\frac{\Omega^{2}}{2\Delta}&U+2\Delta-\frac{\Omega^{2}}{2\Delta}\end{pmatrix}$

To study the dissipative process, we assume decay of Rydberg states is fast
than the coherent couplings. To consider different time scales, the master
equation $\dot{\hat{\rho}}=(\mathcal{L}_{0}+\mathcal{L}_{1})\hat{\rho}$ is
split into the fast (denoted by $\mathcal{L}_{0}\hat{\rho}$) and slow (denoted
by $\mathcal{L}_{1}\hat{\rho}$) parts, where

$\displaystyle\frac{\mathcal{L}_{0}\hat{\rho}}{\gamma}$ $\displaystyle=$
$\displaystyle\sum_{j=1,2}\left(\hat{\sigma}_{1r}^{j}\hat{\rho}\hat{\sigma}_{r1}^{j}-\frac{1}{2}\\{\hat{\rho},\hat{\sigma}_{r1}^{j}\hat{\sigma}_{1r}^{j}\\}\right),$
$\displaystyle\mathcal{L}_{1}\hat{\rho}$ $\displaystyle=$
$\displaystyle-i[\hat{H}_{eff},\hat{\rho}].$ (S6)

We will trace the fast dynamics and derive an effective master equation for
the slow dynamics via the second order perturbation calculation.

Here we define a projection operator
$\mathcal{P}_{0}=\lim_{t\to\infty}e^{t\mathcal{L}_{0}}$, which projects the
density matrix to the subspace corresponding to the relatively slow dynamics,
i.e. $\hat{\rho}_{0}=\mathcal{P}_{0}\hat{\rho}$. The first order correction to
the slowly varying dynamics vanishes
$\mathcal{P}_{0}\mathcal{L}_{1}\mathcal{P}_{0}\hat{\rho}=0$. We then calculate
the second order correction
$-\mathcal{P}_{0}\mathcal{L}_{1}(\mathcal{I}-\mathcal{P}_{0})\mathcal{L}_{1}\mathcal{P}_{0}\hat{\rho}$.
A tedious but straightforward calculation yields an effective master equation
depending on the two-atom dephasing,

$\dot{\hat{\rho}}_{e}\approx-i[\widetilde{V}_{12},\rho_{e}]+\text{$\gamma$}_{12}\left(\hat{\sigma}_{11}^{1}\hat{\sigma}_{11}^{2}\hat{\rho}_{e}\hat{\sigma}_{11}^{2}\hat{\sigma}_{11}^{1}-\frac{1}{2}\\{\hat{\sigma}_{11}^{2}\hat{\sigma}_{11}^{1},\hat{\rho}_{e}\\}\right).$
(S7)

where
$\gamma_{12}=\Omega^{4}\gamma/2\Delta^{2}[(U(R_{12})+2\Delta)^{2}+\gamma^{2}]$
and the dressed interaction
$V_{12}=\Omega^{4}[U(R_{12})+2\Delta]/2\Delta^{2}[(U(R_{12})+2\Delta)^{2}+\gamma^{2}]$
without including the correction induced by the single-photon Stark shifts.

To illustrate the accuracy of the perturbative calculation, we consider the
system dynamics under the Rydberg molecule dressing with
$(U(R_{12}),\Delta)/\Omega=(21,10)$ (left panel) and
$(U(R_{12}),\Delta)/\Omega=(40,10)$ (right panel), respectively, shown in Fig.
S2. The former leads to the dressed interaction $V_{12}/\Omega\approx
2.5\times 10^{-3}$ and the two-body decay rate
$\gamma_{12}\sim\Omega^{2}\gamma/2\Delta^{2}=5\times 10^{-3}\gamma$, which is
comparable to the effective single-photon decay
$\gamma_{1}\sim\Omega^{2}\gamma/4\Delta^{2}$; while for the latter, the
dressed energy and the effective decay rate are $V_{12}/\Omega\approx
2.5\times 10^{-4}$ and $\gamma_{12}\sim 10^{-3}\gamma_{1}$, respectively. The
dissipative dynamics of the two-atom system is then determined by the ratios
with respect to the driving strength: $g/V_{12}$ and
$g/(2\gamma_{1}+\gamma_{12})$. For the limit $V_{12}/g\gg 1$, the system can
only transit between the states $|00\rangle$ and $|10\rangle$ ($|01\rangle$),
corresponding to the two-photon blockade, and the two-body decay accelerates
the convergence to the steady state. In contrast, for $g/V_{12}\gg 1$, the
two-body decay effect can be largely neglected.

Figure S2:  Dynamics of two interacting atoms under RMD. Population of states
$|00\rangle$, $\left(|01\rangle+|10\rangle\right)/\sqrt{2}$, and $|11\rangle$
with the full three-level Hamiltonian (S4) (solid) and the effective two-level
model (S7) (dash) for $U(R_{12})/\Omega=21$ (left panel) and
$U(R_{12})/\Omega=40$ (right panel), respectively. Note that the difference is
barely seen in the numerical simulation. Other parameters are $\Omega=1$,
$\Delta/\Omega=10$, $\gamma=0.1$, $g=2.5\times 10^{-3}$, and
$\Delta_{0}=\Omega^{2}/4\Delta$.

##  III Spin squeezing of the Rydberg dressed atoms

Consider the effective decay described by the non-Hermitian (conditional)
Hamiltonian

$\displaystyle H_{c}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\left[\frac{1}{2}\sum_{j\neq
i}\overline{\kappa}_{ij}-i\frac{\gamma^{(1)}}{2}\right]J_{z}^{(i)}+\sum_{i<j}\overline{\kappa}_{ij}J_{z}^{(i)}J_{z}^{(j)}-i\frac{\Gamma_{G}}{2},$
(S8)
$H_{m}=\sum_{i=1}^{N}\left(-\frac{\Delta_{m}}{2}J_{z}^{(i)}+\Omega_{m}J_{x}^{(i)}\right)+\text{H.c.},$
(S9)

where

$\overline{\kappa}_{ij}=V(R_{ij})-i\frac{\gamma_{ij}^{(2)}}{2},$
$\Gamma_{G}=\frac{N\gamma^{(1)}}{2}+\frac{1}{2}\sum_{i<j}\frac{\gamma_{ij}^{(2)}}{2},$

with

$\gamma_{ij}^{(2)}=\frac{\Omega^{4}\gamma}{2\Delta^{2}[(U(R_{ij})+2\Delta)^{2}+\gamma^{2}]}$

being the distance-dependent two-atom collective decay rate.

Under the mean-field approximation, the collective decay can be rewritten in
the following form,

$\displaystyle\sum_{j<k}\gamma_{jk}^{(2)}\hat{\sigma}_{n}^{(j)}\hat{\sigma}_{n}^{(k)}\approx$
$\displaystyle\frac{1}{2}\sum_{j\neq
k}\gamma_{jk}^{(2)}\left(\hat{\sigma}_{n}^{(j)}\langle\hat{\sigma}_{n}^{(k)}\rangle+\langle\hat{\sigma}_{n}^{(j)}\rangle\hat{\sigma}_{n}^{(k)}-\langle\hat{\sigma}_{n}^{(j)}\rangle\langle\hat{\sigma}_{n}^{(k)}\rangle\right)$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{j}\hat{\sigma}_{n}^{(j)}\left(\sum_{k\neq
j}\gamma_{jk}^{(2)}\langle\hat{\sigma}_{n}^{(k)}\rangle\right)+\frac{1}{2}\sum_{j}\langle\hat{\sigma}_{n}^{(j)}\rangle\left(\sum_{k\neq
j}\gamma_{jk}^{(2)}\hat{\sigma}_{n}^{(k)}\right)$
$\displaystyle-\frac{1}{2}\sum_{j}\langle\hat{\sigma}_{n}^{(j)}\rangle\left(\sum_{k\neq
j}\gamma_{jk}^{(2)}\langle\hat{\sigma}_{n}^{(k)}\rangle\right)$
$\displaystyle=$
$\displaystyle\hat{\Gamma}(0)\langle\hat{\sigma}_{n}\rangle\sum_{j}^{N}\hat{\sigma}_{n}^{(j)}-\frac{N}{2}\hat{\Gamma}(0)\langle\hat{\sigma}_{n}\rangle^{2},$
(S10)

where $\hat{\Gamma}(\bm{q})$ is the Fourier transform of the dissipation-rate
matrix $\gamma_{jk}^{(2)}=\gamma^{(2)}(|\bm{R}_{j}-\bm{R}_{k}|)$:

$\hat{\Gamma}(\bm{q})=\sum_{\bm{R}}\gamma^{(2)}(\bm{R})e^{-i\bm{q}\cdot\bm{R}}$
(S11)

and therefore $\hat{\Gamma}(0)=\sum_{|k-j|=1}^{N-1}\gamma_{jk}^{(2)}$, which
is denoted as $\Gamma_{0}$ later for conciseness. For nearest neighbor
interactions only, one has $\Gamma_{0}=Z\gamma_{12}^{(2)}$, where $Z$ is the
lattice coordination number, i.e. the number of nearest neighbors of any given
site. For the one dimensional case, we thus have

$\displaystyle H_{c}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\left[-i\frac{\gamma^{(1)}}{2}+\frac{1}{2}\sum_{j\neq
i}\left(V_{ij}-i\frac{\gamma_{ij}^{(2)}}{2}\right)\right]J_{z}^{(i)}+\sum_{i<j}\left(V_{ij}-i\frac{\gamma_{ij}^{(2)}}{2}\right)J_{z}^{(i)}J_{z}^{(j)}-i\frac{\Gamma_{G}}{2},$
(S12) $\displaystyle=$
$\displaystyle\sum_{i=1}^{N}\left(\frac{1}{2}\sum_{j\neq
i}V_{ij}\right)J_{z}^{(i)}+\sum_{i<j}V_{ij}J_{z}^{(i)}J_{z}^{(j)}-i\left[\frac{\gamma^{(1)}}{2}+\frac{\Gamma_{0}}{4}\right]\sum_{i=1}^{N}J_{z}^{(i)}$
$\displaystyle-\frac{i}{2}\left[\Gamma_{0}\langle
J_{z}^{(i)}\rangle\sum_{j}^{N}J_{z}^{(j)}-\frac{N}{2}\Gamma_{0}\langle
J_{z}^{(i)}\rangle^{2}\right]-i\frac{\Gamma_{G}}{2}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{N}\left(\frac{1}{2}\sum_{k\neq
j}V_{jk}\right)J_{z}^{(j)}+\sum_{j<k}V_{jk}J_{z}^{(j)}J_{z}^{(k)}-i\frac{N}{4}\left[\gamma^{(1)}+\Gamma_{0}\left(\frac{1}{4}-\bar{\langle
J_{z}\rangle}{}^{2}\right)\right]$
$\displaystyle-\frac{i}{2}\left[\gamma^{(1)}+\Gamma_{0}\left(\frac{1}{2}+\bar{\langle
J_{z}\rangle}\right)\right]\sum_{j=1}^{N}J_{z}^{(j)},$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{j=1}^{N}\left(\sum_{k\neq
j}V_{jk}-i\Gamma_{z}\right)J_{z}^{(j)}+\sum_{j<k}V_{jk}J_{z}^{(j)}J_{z}^{(k)}-i\frac{\text{$\bar{\Gamma}$}}{2},$

where

$\text{$\bar{\Gamma}$}=\frac{N}{2}\left[\gamma^{(1)}+\Gamma_{0}\left(\frac{1}{4}-\bar{\langle
J_{z}\rangle}{}^{2}\right)\right],$
$\Gamma_{z}=\gamma^{(1)}+\Gamma_{0}\left(\frac{1}{2}+\bar{\langle
J_{z}\rangle}\right),$

and $\bar{\langle J_{z}\rangle}$ is the mean value of the $z$-component of the
angular momentum for the individual atom. For numeric, we evaluate
$\bar{\langle J_{z}\rangle}$ by the weighted mean for all spins and the time
average for the implementation period $\tau$, namely,

$\overline{\langle
J_{z}(\tau)\rangle}=\frac{1}{N\tau}\int_{0}^{\tau}d\tau\langle
J_{z}(\tau)\rangle,$

which will be analytically calculated in the Heisenberg picture and should be
effective in the mean-field regime.

We measure the degree of spin squeezing by the parameter proposed by Wineland
et al. Wineland  _et al._ (1994),

$\xi^{2}=\frac{N\left(\Delta
J_{\vec{n}_{\bot}}\right)^{2}}{|\langle\vec{J}\rangle|^{2}},$ (S13)

where

$\langle\vec{J}\rangle=\langle\sum_{k=1}^{N}J^{(k)}\rangle=\sqrt{\langle
J_{x}\rangle^{2}+\langle J_{y}\rangle^{2}+\langle
J_{z}\rangle^{2}}\hat{\varepsilon}_{z}$ (S14)

is the mean collective angular momentum of all spins with its direction
$\hat{\varepsilon}_{z}$ being predefined along the $z$-axis, corresponding to
the initial unitary population of the ground state for all atoms, and

$J_{\vec{n}_{\bot}}\left(\theta\right)=\textrm{cos}\left(\theta\right)J_{x}+\textrm{sin}\left(\theta\right)J_{y}$
(S15)

is the perpendicular spin component with $\theta$ being the angle with respect
to the $x$ axis. Thus, the fluctuations in
$J_{\vec{n}_{\bot}}\left(\theta\right)$ can be calculated by

$\displaystyle\left(\Delta J_{\vec{n}_{\bot}}\right)^{2}=$
$\displaystyle\langle J_{\vec{n}_{\bot}}^{2}\left(\theta\right)\rangle-\langle
J_{\vec{n}_{\bot}}\left(\theta\right)\rangle^{2}$ $\displaystyle=$
$\displaystyle\left(\textrm{cos}^{2}\left(\theta\right)\langle
J_{x}^{2}\rangle+\textrm{sin}^{2}\left(\theta\right)\langle
J_{y}^{2}\rangle+\text{sin}\left(\theta\right)\textrm{cos}\left(\theta\right)\langle
J_{x}J_{y}+J_{y}J_{x}\rangle\right)$
$\displaystyle-\left(\textrm{cos}^{2}\left(\theta\right)\langle
J_{x}\rangle^{2}+\textrm{sin}\left(\theta\right)\langle
J_{y}\rangle^{2}+2\text{sin}\left(\theta\right)\textrm{cos}\left(\theta\right)\langle
J_{x}\rangle\langle J_{y}\rangle\right),$ $\displaystyle=$
$\displaystyle\textrm{cos}^{2}\left(\theta\right)\left(\Delta
J_{x}\right)^{2}+\textrm{sin}^{2}\left(\theta\right)\left(\Delta
J_{y}\right)^{2}$
$\displaystyle+\text{sin}\left(\theta\right)\textrm{cos}\left(\theta\right)\left(\langle
J_{x}J_{y}+J_{y}J_{x}\rangle-2\langle J_{x}\rangle\langle
J_{y}\rangle\right).$ (S16)

Without considering the effect of the atomic spontaneous emission (i.e.,
$\gamma^{(1)}=0$ and $\gamma_{ij}^{(2)}=0$), and based on the spin-echo
protocol, one can find mean values of the spin components analytically Gil
_et al._ (2014):

$\langle J_{x}\rangle=\langle J_{y}\rangle=0,\text{ }\langle
J_{z}\rangle=\langle\sum_{i=1}^{N}J_{z}^{(i)}\rangle=-\frac{1}{2}\sum_{i}^{N}\prod_{j\neq
i}^{N}\textrm{cos}(\varphi_{ij}),\text{ }$ (S17) $\displaystyle\langle
J_{x}^{2}\rangle=$
$\displaystyle\langle\sum_{k=1}^{N}J_{x}^{(k)}\cdot\sum_{l=1}^{N}J_{x}^{(l)}\rangle=\frac{N}{4}+2\sum_{k<l}^{N}\langle
J_{x}^{(k)}J_{x}^{(l)}\rangle$ $\displaystyle=$
$\displaystyle\frac{N}{4}+\frac{1}{4}\sum_{i<j}^{N}\left[\prod_{k\neq
i,j}^{N}\textrm{cos}(\varphi_{ijk}^{-})-\prod_{k\neq
i,j}^{N}\textrm{cos}(\varphi_{ijk}^{+})\right],$ (S18)

$\langle J_{y}^{2}\rangle=\frac{N}{4}+\underbrace{2\sum_{i<j}^{N}\langle
J_{y}^{(i)}\cdot J_{y}^{(j)}\rangle}_{0}=\frac{N}{4},$ (S19)

and

$\langle
J_{x}J_{y}+J_{y}J_{x}\rangle=-\sum_{i<j}^{N}\textrm{sin}(\varphi_{ij})\prod_{k\neq
i,j}^{N}\textrm{cos}(\varphi_{ik}),$ (S20)

where $\varphi_{ij}=V_{ij}\tau/2$ and
$\varphi_{ijk}^{\pm}=\varphi_{ik}\pm\varphi_{jk}.$

When we take into account the non-Hermitian Hamiltonian (S12), where the
decoherence is estimated by the mean-field single-body decay, and redo the
spin-echo protocol by using the following transformations

$\displaystyle e^{i\alpha^{*}J_{z}^{(k)}}J_{x}^{(k)}e^{-i\alpha J_{z}^{(k)}}=$
$\displaystyle\text{cos}(\text{Re}\alpha)J_{x}^{(k)}-\text{\text{sin}(\text{Re}$\alpha$)}J_{y}^{(k)},$
(S21) $\displaystyle e^{i\alpha^{*}J_{z}^{(k)}}J_{y}^{(k)}e^{-i\alpha
J_{z}^{(k)}}=$
$\displaystyle\text{cos}(\text{Re}\alpha)J_{y}^{(k)}+\text{\text{sin}(\text{Re}$\alpha$)}J_{x}^{(k)},$
(S22) $\displaystyle e^{i\alpha^{*}J_{z}^{(k)}}J_{z}^{(k)}e^{-i\alpha
J_{z}^{(k)}}=$ $\displaystyle e^{2\text{Im}\alpha J_{z}^{(k)}}J_{z}^{(k)},$
(S23)

the mean values and the quantum fluctuations of the spin angular momentum
operators are alternatively given by

$\langle J_{m}\rangle\rightarrow e^{-\text{$\bar{\Gamma}$}\tau}\langle
J_{m}\rangle,\text{ }\langle J_{m}J_{n}\rangle\rightarrow
e^{-\text{$\bar{\Gamma}$}\tau}\langle J_{m}J_{n}\rangle,$ (S24)

and

$\displaystyle\left(\Delta J_{\vec{n}_{\bot}}\right)^{2}=$ $\displaystyle
e^{-\text{$\bar{\Gamma}$}\tau}\left[\textrm{cos}^{2}\left(\theta\right)\langle
J_{x}^{2}\rangle+\textrm{si$n^{2}$}\left(\theta\right)\langle
J_{y}^{2}\rangle\right]$
$\displaystyle+\frac{1}{2}\text{sin}\left(2\theta\right)e^{-\text{$\bar{\Gamma}$}\tau}\langle
J_{x}J_{y}+J_{y}J_{x}\rangle,$ (S25)

where $m,\text{ }n=x,$ $y$, or $z$, $\tau$ is the total optical dressing time,
and we have assumed the rotation pulses is fast enough, such that the
dissipation is negligible during the operation. Moreover, the dissipative
effects in the $z$-component of the individual spin angular momentum
[$J_{z}^{(k)}(\tau)\sim e^{-\Gamma_{z}\tau}$] during the two halves of optical
dressing time counteract each other due to spin-flip induced by the rotation
$\pi$ pulse. Using the above results, we then calculate the squeezing
parameter for different number of sites and interactions (RMD and SRD).