pdfborder = 0 0 0

# Accelerating End-host Congestion Response using P4 Programmable Switches

Nehal Baganal-Krishna 15, Tuan-Dat Tran5, Ralf Kundel3, Amr Rizk 5
0000-0002-9099-1754 0000-0002-9385-7729 1University of Ulm, Germany
5University of Duisburg-Essen, Germany  3 Techincal University of Darmstadt,
Germany

###### Abstract

Transport layer congestion control relies on feedback signals that travel from
the congested link to the receiver and back to the sender. This forward
congestion control loop, first, requires at least one rount-trip time (RTT) to
react to congestion and secondly, it depends on the downstream path after the
bottleneck. The former property leads to a reaction time in the order of RTT +
bottleneck queue delay, while the second may amplify the unfairness due to
heterogeneous RTT. In this paper, we present Reverse Path Congestion Marking
(RPM) to accelerate the reaction to network congestion events without changing
the end-host stack. RPM decouples the congestion signal from the downstream
path after the bottleneck while maintaining the stability of the congestion
control loop. We show that RPM improves throughput fairness for RTT-
heterogeneous TCP flows as well as the flow completion time, especially for
small Data Center TCP (DCTCP) flows. Finally, we show RPM evaluation results
in a testbed built around P4 programmable ASIC switches.

###### Index Terms:

Congestion Marking, AQM, P4, Data plane programming

##  I Introduction

Bufferbloat describes the excessive delays packets observe when traversing
links with deep queues [1]. Large buffer memory in combination with the
available bandwidth and congestion control mechanisms of most of the modern
(transport) protocols such as TCP (e.g. Cubic, BBR) and QUIC lead to large
packet delays. This is due to the available bandwidth estimation and
congestion control mechanisms actively inducing congestion signals at the
bottleneck to estimate the appropriate sending rate [2]. Active Queue
Management (AQM) tackles this problem by keeping the buffer filling small.
This is done by sending congestion signals, i.e., through packet drops or ECN
markings, when the delay due to buffering exceeds a target value. Prominent
AQM algorithms are Codel[3], PIE[4] and RED[5] and their variations.

Although AQM aims to manage the Bufferbloat problem, there still remain two
fundamental problems that are due to (i) the end-to-end construction of the
current congestion control mechanisms and (ii) traffic generation patterns of
modern applications, especially in data centers. Prevalent congestion control
mechanisms are self-clocked, i.e., they work on the time scale of roundtrip
time (RTT) with congestion signals created at bottleneck links having to
bounce back from the receiver to the sender. This binds the sender’s reaction
time to congestion to this time scale which is often used as a cornerstone to
showing the stability of congestion control [6]. However, it is also known
that this RTT time scale binding of congestion control leads to throughput
unfairness when multiple flows of different RTTs compete for the bandwidth at
the same bottleneck [7, 8, 9]. Note that the end-to-end argument in congestion
control also dictates that the excessive queueing time at the congested link
buffer becomes part of the time required for the congestion signal to travel
back to the sender.

The second problem arises especially in data centers where most traffic flows
are short, e.g., due to application request-response patterns [10, 11]. These
short flows, combined with the very high link capacities found in data center
fabrics [12], lead to so-called microbursts. These $\mu$bursts last shorter
than a single RTT [13], but most importantly, microbursts can lead to
excessive packet drops as data center buffers are typically very small [14].
It is evident that classical congestion control mechanisms struggle with such
phenomena leading to bloated flow completion times [15]. However, approaches
that change the congestion control stack tackle this problem at the cost of
special purpose end-host stacks [16].

In this paper, we propose an approach to reduce the congestion reaction time
as well as the flow completion time for short flows using in-network support
[17]. We denote this approach Reverse Path Congestion Marking (RPM)111RPM
stands on the shoulders of approaches for out of band congestion notification
using special packets, e.g. ICMP or others [18, 19].. RPM modifies the in-
flight ACKs for fast and stable congestion control in sub-RTT time scales. RPM
is compatible with TCP and DCTCP, hence no modifications to end-hosts are
required. Finally, we implement and show RPM in a programmable data-plane
using P4 on modern networking hardware (Intel Tofino switch).

To summarize, our contributions are as follows:

  * •

We illustrate an approach for in-band reverse path congestion marking without
additional control traffic.

  * •

We provide an analytical model of RPM, proving its stability.

  * •

We show an implementation of RPM in P4 on programmable networking hardware,
including results from a testbed deployment.

##  II Challenges for forward-path congestion control loops

Congestion control closed-loop mechanisms comprise two major steps, i.e. (i)
the congestion point signals congestion to the receiver using packet drops or
CE code points in the IP header, and (ii) the sender reduces the congestion
window conforming to the congestion notification received. The above-mentioned
control loop has two major challenges, i.e., congestion signal delay and path
dependency, which we discuss in the following.

###  II-A Delayed congestion control response

The congestion control loop requires at least one RTT or even more time to
ease the congestion at the bottleneck after congestion detection. To reach the
sender, the congestion signal has to traverse from the bottleneck to the
receiver and then from the receiver back to the sender. The propagation delay
of the forward loop from the bottleneck to the sender is
$\tau_{2}+\tau_{3}+\tau_{4}$, as depicted in Fig. 1. The congestion control
loop additionally requires the time $\tau_{1}$ to propagate the effect of a
reduced congestion window after the congestion signal arrives at the sender if
this reduction is executed immediately. Hence, the total congestion signal
propagation time elapsed from congestion occurrence in the bottleneck until
the bottleneck experiences a reduction in the sending rate is at least
$\sum_{i=1}^{4}\tau_{i}$ (cf. Fig. 1). The delay of the congestion control
loop leads to a delayed dynamical system at the output port queue experiencing
congestion.

###  II-B Downstream path dependent congestion response

The second challenge in classical forward-path congestion control loops is the
dependency of the reaction time of the sender node on the downstream path from
the bottleneck to the receiver. Considering a network where multiple flows
with different RTTs share one bottleneck, current congestion control
approaches lead to unfair bottleneck capacity distribution among these flows
[20, 21, 22, 23]. Hence, although the flows may experience congestion
simultaneously, the time of arrival of the congestion notification at the
respective sources varies and hence they react with different delays to the
congestion leading to throughput unfairness with sources having a smaller RTT
receiving more bandwidth.

From the two challenges above, we observe that end-to-end congestion control
without network support is difficult to configure. TCP friendliness [7, 8, 9]
and max-min fairness [6] are much harder to obtain if the congestion control
loops have heterogeneous delays and do not only depend on the shared
bottleneck but also on their own paths. To this end, we present a design of a
data-plane based support for congestion marking that aims to dispense (in
parts) with the two challenges above in the following sections.

Figure 1: Forward path ECN feedback loop.

##  III Design of Reverse Path Congestion Marking

In this section, we first present the design objectives behind Reverse Path
Congestion Marking (RPM), then present its design components before discussing
these.

###  III-A RPM Design objectives

To overcome the challenges discussed in the Sect. II, we first derive the
objectives for RPM that act as a framework for the subsequent implementation.

####  III-A1 Decouple congestion signals from the downstream path

In the current state-of-the-art congestion control loop, for the congestion
notification (either duplicate ACKs or ECE marked ACKs) to reach the sender,
the signal must traverse the path beyond the bottleneck to the receiver and
then take the path back to the sender. If the downstream network after the
bottleneck is dynamic, the time taken by the signal to reach the receiver
generally increases (on average) and its variability increases too.

Any additional time elapsed after the creation of a congestion signal leads to
additional buffer filling at the bottleneck and causes an increase in flow
packet drops as well as flow completion time. Therefore, it is necessary to
decouple congestion signals from the downstream path and its bottlenecks,
further disassociating the dependence of the sender’s reaction to the
congestion signal on the downstream path.

####  III-A2 Scalable data plane implementation

Common models for the analysis of AQM mechanisms, such as CoDel, consider a
segregation of flows that traverse the same bottleneck to correctly inform the
sender of congestion. As we consider scenarios where possibly hundreds or
thousands of flows traverse a bottleneck, and the storage of a per-flow AQM
state is impractical on modern data plane implementations, it is necessary to
obtain an RPM algorithm implementation of the data-plane that is flow-
agnostic.

####  III-A3 No modifications to the end-host stack

Transmission Control Protocol (TCP) and the Data Center TCP (DCTCP) are two of
the most widely used transport layer protocols. Modifications to the end-host
TCP or DCTCP congestion control stack are weary to standardize. To allow
seamless adoption, the RPM algorithm implemented in the network data plane
should adhere to TCP and DCTCP specifications without requiring changes to the
end-hosts.

###  III-B RPM Design Components

####  III-B1 Reverse path ECN-based Marking

The idea of reverse path congestion marking is to mimic the ECE flag marking
functionality of the receiver in the data plane. Replicating the receiver’s
marking scheme onto the switch improves the congestion indication flexibility,
as it can notify congestion by directly marking the ECE field of the in-flight
ACKs going back to the sender. The reverse marking feature shortens the
reaction time of the congestion control loop by decoupling the congestion
signal from the downstream bottleneck. Figure 2 illustrates the approach to
Reverse Path Congestion Marking on the data plane. Congestion detection and
marking are carried out by an AQM such as CoDel, e.g., using the data plane
implementation [24]. The reverse path marking on the data plane requires a
congestion state and a reverse marking module, which is explained below in
more detail.

In any ECN-enabled TCP congestion control loop, when the receiver accepts a
packet with marked CE code points, it sets ECE flags in all subsequent ACKs to
the sender to report congestion [25]. The receiver continues to mark ACKs
until it receives a packet with a congestion window reduced (CWR) flag set,
indicating a reduction in the congestion window at the sender. Even though the
sender receives multiple ACKs with ECE set, it halves its congestion window
only once per RTT. DCTCP clients apply similar functionality with an
additional single-bit state variable denoted DCTCP Congestion Encountered
(DCTCP.CE) to enable delayed ACKs [26]. When a DCTCP receiver receives a
packet with the CE code points set, DCTCP.CE is set to ”1” and transmits an
ACK with the ECE flag set to the sender. Here too, the receiver continues to
mark ACKs until it receives a packet with a CWR flag set thereby setting
DCTCP.CE to “0”. The DCTCP sender also reduces the congestion window once per
RTT but the reduction depends on the number of ACKs (with ECE flag) the sender
receives.

When the AQM on the switch data plane (see Fig. 2) detects congestion, the
marking mechanism of RPM sets the ECE bit in the in-flight ACKs of that flow,
to notify the sender. To identify this flow and the corresponding ACKs, a
reverse marking module is required to keep a minimum state mapping flows to a
binary congestion observed value. This can be realized on modern P4
programmable switches (such as the Intel Tofino ASIC) and will be discussed in
the next section. For every congestion detected by the AQM for a specific flow
at the bottleneck, one in-flight ACK of that flow on the reverse path will be
marked (by setting the ECE flag). The design of the marking operation at the
switch is the same for TCP and DCTCP, although the DCTCP receiver employs
specific state variables. Note that the data plane need not keep a state of
DCTCP Congestion Encountered, which renders RPM to be TCP and DCTCP compatible
simultaneously.

####  III-B2 Per-Flow State Aggregation

When the flows pass through the switch and experience congestion, the switch
should notify congestion to the senders of each flow. Hence, the switch has to
remember the flow experiencing congestion and mark the ACK corresponding to
the stored flow. The Congestion State Register in the data plane holds a
memory unit for each flow traversing through the switch. When an AQM detects
congestion for a flow, the congestion state register is incremented at a
specific location pointed by the flow. Subsequently, when the corresponding
in-flight ACK of the previously congested flow enters the switch, the
congestion state register is decremented and ECE flag is marked. The per-flow
congestion state facilitates RPM to uniquely notify the sender of a flow
experiencing congestion and helps RPM achieve simultaneous compatibility with
both TCP and DCTCP.

###  III-C Discussion

RPM isolates the dependency of congestion notification on the downstream path
after the bottleneck, as the marking algorithm on the data plane marks the in-
flight ACKs going towards the sender instead of packets in the forward
direction, which may experience different and variable delays towards the
receiver and back. Hence, the time required for the congestion feedback to
reach the sender from the bottleneck is minimized.

One prerequisite of RPM is that the route taken by the data flow to reach the
receiver and the route taken by the ACK to reach the sender should be the
same, or at least the bottleneck switch should be on both paths forward and
reverse. Checking this condition is straightforward with an RPM data plane
implementation using P4 such that RPM can be turned off for flows that do not
adhere to this condition.

Also note that the size of the register present in the P4 programmable switch,
e.g., Intel Tofino, dictates the number of flows for which RPM is operational.
We report on the utilized on-switch memory in Sect. V.

The scarce memory availability on the data plane adds an upper threshold to
the number of flows passing through the single switch that RPM can handle at
any given time, as each flow requires one bit in the congestion state
register. Instead of storing information for every flow that passes through
the bottleneck, we can choose to store per port congestion information. All
flows routing through a single bottleneck switch port will be considered one
port group of flows for which RPM holds one state. We note that this marking
scheme might be too aggressive towards congestion window reduction.

RPM works with TLS but is difficult to implement with IPSec. In TLS, as the
TCP payload is encapsulated and RPM can mark the ECE field of the TCP header
in case of congestion; however, with IPSec the TCP header is encapsulated,
which means RPM cannot mark ACKs if RPM is not running at the IPSec tunnel
ingress.

Figure 2: Components of Reverse Path Congestion Marking.

##  IV Stability of Reverse Path Congestion Marking

In this section, we provide an analytical model of Reverse Path Congestion
Marking based on the sketch of the setup depicted in Fig. 3. We assume that
the TCP source on the sender (left-hand side) is in the congestion avoidance
state of an Additive increase and Multiplicative decrease (AIMD) congestion
control algorithm. We assume that the bottleneck switch signals congestion by
marking CE code points in the IP header and that the bottleneck link has a
capacity of $c$ bps. The following model extends the forward path congestion
control analysis in [27]. This extension is not trivial as the resulting
sending window ODE has multiple, different delay factors making the analysis
much harder than, e.g., in [27].

Figure 3: Deterministic dynamic system model of the congestion control loop of
a single connection over a bottleneck link.

We consider the scenario where an ACK packet traversing back to the sender
without congestion notification (marked ECE bits in the TCP header) causes the
time-dependent congestion window function $w(t)$ to increase by
$\frac{a}{w(t)}$, where $a$ relates to the additive increase factor. Now, we
consider an AQM that randomly marks packets with congestion bits given that
the queue at the switch output port experiences congestion, i.e., the queue
length is larger than an AQM target value of $x^{*}$. Precisely, the CE code
points are marked $1$ with probability $\eta[x(t)-x^{*}]^{+}$ and are marked
$0$ with $(1-\eta)[x(t)-x^{*}]^{+}$222We use $[x]^{+}$ to denote
$\max\\{x,0\\}.$. Note that the congestion marking probability depends on the
queue length exceeding the target $[x(t)-x^{*}]^{+}$ and the normalization
factor $\eta$ controls the “aggressiveness” (or slope) of the marking with the
grade congestion. As depicted in Fig. 3 we call the forward propagation delay
to the switch $\tau_{f}$ and propagation delay for the rest of the loop
$\tau_{r}=\sum_{i=1}^{3}\tau_{i}$. Note that the queueing delay at the switch
port is denoted $\tau_{q}$ such that we define $\tau_{f}+\tau_{r}:=\tau$ and
the round trip delay $d:=\tau+\tau_{q}$.

When RPM indicates congestion, the switch sets the ECE bit in the ACK of the
corresponding flow, which experiences a bottleneck. Hence, the congestion
signal (ECE bit) experiences a propagation delay back of approximately
$\tau_{rs}=\tau_{3}$ and the short control loop round trip time will be
$\tau_{s}=\tau_{f}+\tau_{rs}$. We note that at the time of congestion
experience there might not be an ACK travelling back, however, (i)
$\tau_{rs}<\tau_{r}$ always holds, and (ii) we approximate the time until an
ACK is seen on the reverse path to zero. Once the sender receives an ACK with
an ECE bit set, the sender’s congestion window decreases multiplicatively by a
factor of $b$. Hence, we obtain the following rate of change of the congestion
window $w(t)$ of the TCP sender

$\displaystyle\dot{w}(t)=$
$\displaystyle\frac{a}{w(t)}r(t-\tau-\tau_{q})(1-\eta\left[x(t-\tau_{r}-\tau_{q})-x^{*}\right]^{+}$
$\displaystyle-b\cdot
w(t)r(t-\tau_{s})\eta\left[x(t-\tau_{rs})-x^{*}\right]^{+}$ (1)

The first term on the right side is an additive increase rate each time the
sender receives an unmarked ACK. The rate of the arrival of unmarked ACKs at
the source is $r(t-\tau-\tau_{q})$, and a factor of
$(1-\eta(x(t-\tau_{r}-\tau_{q})-x^{*})$ ACKs are not marked by switch. The
second term represents the multiplicative decrease proportional to $b>0$ and
the current window rate $w(t)$. The arrival rate of congestion notifications
is $r(t-\tau_{s})$ as the propagation delay of ECE bit after congestion
indication at the switch is $\tau_{rs}$ and a factor of
$\eta(x(t-\tau_{rs})-x^{*})$ ACKs are marked by switch. Next, we use that
$d=\tau+\tau_{q}>0$ is the round trip delay for the connection and denote for
nomenclature homogeneity $d_{s}=\tau_{s}$ as the “short” round trip delay
between the sender and the bottleneck.

######  Lemma 1 (Stability of RPM).

Given a fluid AIMD TCP source with additive increase parameter $a$ (e.g. $1$)
and multiplicative decrease parameter $b$ (e.g. $\frac{1}{2}$) using a network
path utilizing Reverse Path Congestion Marking (RPM). Assume the path has a
forward path round-trip time $d$, a (reverse) short round trip time between
the sender and the bottleneck $d_{s}$, and a bottleneck link capacity $c$ and
define $\gamma:=\frac{abc}{a+bc^{2}d^{2}}$. The dynamical system describing
the congestion window of the AIMD source in (IV) is stable with a positive
normalization factor $\eta$ given by

$\normalsize\eta=\frac{(e^{-sd}-e^{-sd_{s}}-2)s\gamma-s^{2}}{\frac{e^{-sd}}{d^{2}}+\frac{c^{2}e^{-sd_{s}}}{2}}\,,$
(2)

for any $s<s^{*}<0$ with $s^{*}$ given in (6).

The proof of Lem. 1 is given in the appendix.

Based on Lem. 1 we can obtain the marking aggressiveness $\eta$ of the RPM. As
an exercise substituting common values for the system parameters one may find
that $\eta$ is typically small and positive.

By varying $d_{s}$ (the position of bottleneck) with a constant $d$ and $c$ in
(2), we find that $\eta$ is approximately constant. This indicates that the
marking factor $\eta$ need not be changed based on bottleneck location in the
network to maintain stability.

##  V Implementation & Evaluation

In this section, we discuss the implementation details of RPM. We further
indulge in discussing the testbed and evaluating RPM performance.

###  V-A Algorithmic implementation of RPM

As indicated in Fig. 2, the implementation of RPM has three components, (1)
congestion indication, (2) reverse path marking, and (3) congestion state
clearance. For congestion indication, the switch can use an arbitrary AQM; in
this paper, we employ CoDel in the data plane [24]. We modify this data-plane
implementation of CoDel to support RPM.

When CoDel detects a bottleneck, RPM increments a Congestion State Register
$\mathbf{R_{C}}$ at the location pointed by the flow’s 5-tuple Hash function
$h($IPsrc, IPdst, IPprot, psrc, p${}_{dst})$. The hash function $h$ uniquely
identifies each flow and maps the congestion state register to a unique per-
flow memory unit. To mark in-flight ACKs, RPM reads the congestion state
register at the location pointed by the flow’s 5-tuple Hash function $h$, but
the source-destination IP addresses IP(⋅) and L4-ports p(⋅) are swapped. If
the register $\mathbf{R_{C}}$ is greater than or equal to ”1” indicating
congestion, the ECE flag of the TCP header in the ACK packet is set and the
register is decremented by ”1”.

The packet entering the data plane goes through all the RPM components
mentioned above to seamlessly support TCP or DCTCP for the following reasons;
first, the data plane need not differentiate between the data packets
traveling from sender to receiver and ACKs traveling from receiver to sender.
In addition, TCP and DCTCP are bidirectional flows meaning both sender and
receiver can send data to each other while simultaneously acknowledging the
received data. So, a packet considered an ACK might carry data. Hence, flows
in the reverse direction, i.e., of the ACKs, can be marked. Note that a TCP
flow requires at least one ACK with ECE flag set to halve its congestion
window per RTT, while a DCTCP flow exactly needs the count of ACKs the sender
obtains to calculate congestion window per RTT [26, 10]. The marking scheme
employed by RPM satisfies the congestion window calculation of both TCP and
DCTCP. This is because every time a flow experiences congestion, in-flight
ACKs in the reverse direction of the flow will be marked. Algorithm 1
describes the core logic behind RPM when implemented in the data plane.

1: Run CoDel

// CoDel congestion state denoted $\text{drop}_{C}$

2: if $\text{drop}_{C}==\text{True}$ then

3: $pos\leftarrow h($IPsrc, IPdst, IPprot, psrc, p${}_{dst})$

4: $\mathbf{R_{C}}[pos]\leftarrow\mathbf{R_{C}}[pos]+1$

5: end if

// When an ACK arrives

6: $pos\leftarrow h($IPdst, IPsrc, IPprot, pdst, p${}_{src})$

7: if $\mathbf{R_{C}}[pos]\geq 1$ then

8: $\text{TCP.ECE}\leftarrow 1$

9: $\mathbf{R_{C}}[pos]\leftarrow\mathbf{R_{C}}[pos]-1$

10: end if

Algorithm 1 Pseudocode of the RPM Algorithm Figure 4: Testbed Topology.

###  V-B Evaluation Results

Next, we compare RPM with CoDel-Forward Congestion Marking (CoDel-FWD). We use
the topology in Fig. 4 to study the performance of RPM. First, we study the
throughput fairness when the flows in the network have different RTTs but
share the same bottleneck. The hosts $H_{1}$ to $H_{10}$ act as senders and
hosts $H_{11}$ to $H_{20}$ act as receivers, such that $H_{i}$ sends its
traffic only to $H_{i+10}$. Host links are set to $10\ Gbps$. The hosts may be
run using high performance traffic generation as in [28]. In CoDel-FWD, after
congestion identification by the CoDel algorithm, the marking scheme directly
sets the CE code points in IP header, indicating congestion to the receiver.
The CoDel parameters, i.e., the target delay and the CoDel interval, are set
to $1\ ms$ and $20\ ms$. As the Tofino switch has finite memory, the buffer
has an upper limit on the number of packets it can store, which restricts the
delay in this configuration to be below $2\ ms$.

Figure 5: Average flow completion time for short flows. The shorter the flows
the more RPM helps decreasing the flow completion time due to shorter reaction
time compared to CoDel on the forward path.

##### Fairness Experiment

For the fairness experiments we set the delay of each sender-receiver pair
link using the Traffic Control (tc) linux command on both senders and
receivers. The link delays of the links $H_{i}-S_{1}$ are all set to $10\ ms$
and the link delays of the links $S_{2}-H_{j}$ for each experiment conducted
are given in Tab. I. The P4 switch $R_{1}$ (in which AQM is deployed) is
connected to $S_{1}$ with a $100\ Gbps$ link and to $S_{2}$ with a $10\ Gbps$
link leading to congestion at the output port of $R_{1}$. We deploy AQM on
$R_{1}$. We measure Jain’s fairness index333Jain’s fairness index defined as
$J=\frac{E[X]^{2}}{E[X^{2}]}$ [29]. of the flow throughput for long TCP Cubic
flows.

TABLE I: Experiment configurations of receiver link delays.

Experiment | $S_{2}$-$H_{11/12}$ | $S_{2}$-$H_{13/14}$ | $S_{2}$-$H_{15/16}$ | $S_{2}$-$H_{17/18}$ | $S_{2}$-$H_{19/20}$
---|---|---|---|---|---
$1$ | $10\ ms$ | $10\ ms$ | $10\ ms$ | $10\ ms$ | $10\ ms$
$2$ | $10\ ms$ | $20\ ms$ | $30\ ms$ | $40\ ms$ | $50\ ms$
$3$ | $20\ ms$ | $40\ ms$ | $60\ ms$ | $80\ ms$ | $100\ ms$

TABLE II: Jain’s Fairness index comparison. $J\rightarrow 1$ indicates
complete fairness. RPM increases throughput fairness for heterogeneous flow
RTTs (Experiments 2/3).

Experiment | J(CoDel-FWD) | J(RPM)
---|---|---
$1$ | $0.89$ | $0.89$
$2$ | $0.74$ | $0.81$
$3$ | $0.82$ | $0.86$

Table II shows the results of our fairness experiments. Experiment 1, that is
used for calibration, sets homogeneous link delays of 10ms on all links. We
observe that both RPM and CoDel-FWD obtain the same fairness index of the flow
throughput. Experiments 2 and 3 show that when the RTTs are heterogeneous, RPM
improves the throughput fairness as all the flows have the same RPM reaction
time of $20$ms. Note that RPM cannot improve the RTT unfairness due to
differing RTTs on the first link, i.e. $H_{i}-S_{1}$. Also, RTT unfairness in
the additive part of AIMD remains unchanged.

##### Flow Completion Time Experiment

Here, we evaluate the average flow completion time, especially of short flows,
as we expect RPM to perform best with short flows as it shortens the reaction
time. We deploy RPM/CoDel_FWD on switch $S_{1}$. We set the link capacity of
all links to $10\ Gbps$ and remove the link delays introduced in the fairness
experiment.

Figure 5 shows the average flow completion time for short flows given in the
number of MSS packets. As RPM helps to decrease the reaction time of the
congestion control loop, it is especially beneficial for short flows if they
suffer from congestion losses.

##  VI Related Work

Established AQM schemes, such as Random Early Detection (RED) [5], CoDel[3],
and PIE[4], send forward path congestion notifications such that the sender
adjusts its sending rate. The congestion signal is either packet drop or
setting ECN flelds in IP header. when CE code points of IP header are set by
the router in case of congestion [30] the receiver reflects this information
in the ECN-Echo bit of the TCP header of the ACK to notify the sender. Once
the sender receives this ACK with ECN-Echo bit set, it reduces its sending
rate. The minimum time required to observe a reduced sending rate by a
congested port, i.e, the reaction time, is at least one RTT.

To decrease this reaction time, Backward ECN (BECN) for network congestion was
originally proposed for ATM networks [19], wherein the router informs the
sender about the congestion by sending out-of-band ICMP Source Quench (ISQ)
messages without the intervention of the receiver. This reduces the reaction
time but generates additional traffic in the feedback loop and loss of the
feedback ISQ packets would not be noted by the sender.

EXplicit Congestion Protocol (XCP) by Zhang and Henderson [16] proposed a
constant exchange of information between router and end hosts. The routers
notify the receiver about the state of congestion in the network by marking it
in the XCP header. The sender receives congestion notifications from the
receiver through ACKs. This process takes around one RTT and involves
modifying the end-hosts.

Similar to BECN, the work in [18] proposed a network-assisted congestion
feedback using NACK packets. Instead of sending ISQ packets, this approach
sends NACKs generated by programmable switches to notify the sender regarding
congestion. Here too, the sender stack has to be modified to understand the
NACKs and the additional out-of-band backward traffic remains.

One approach to use in-flight ACKs on their way to the sender is found in
different forms in [31] and [32]. In both works the ACKs are delayed in the
router. This expires the TCP Retransmission Timeout at the sender, forcing the
sender to reduce its sending rate. Both works [31] and [32] estimate the
latency between the access point and the receiver by taking the queue delay as
well as transmission delay into account. The work in [32] is only confined to
wireless communication with last-mile access points, whereas [31] mainly
tackles incast problems in multi-tenant data centers. In contrast, RPM is a
congestion marking scheme that can be combined with any ECN-enabled congestion
control at the receiver and any AQM algorithm deployed on the switch to mark
in-flight ACKs to improve the responsiveness of the congestion control loop.

##  VII Conclusion

In this paper, we presented Reverse Path Congestion Marking (RPM), i.e., an
in-network method to improve the congestion control reaction time without
changing the end-host stack. With RPM, we can send congestion signals directly
back to the sender, decoupling congestion signaling from the downstream path
past the bottleneck. We mathematically prove that RPM is stable. Further,
using a testbed built around P4 programmable ASIC switches, we showed that it
improves throughput fairness for RTT-heterogeneous TCP flows as well as the
flow completion time, especially for small DCTCP flows.

##  VIII Appendix

###### Proof of Lemma 1.

By Little’s theorem we know that the average congestion window equals the
average sending rate times the average delay. With this in mind we approximate
the sending window, hence, in the case of unmarked ACKs $w(t)=r(t)d$ and in
the case of marked ACKs we have $w(t)=r(t)d_{s}$. we can rewrite (IV) as

$\displaystyle\Large\dot{r}(t)=$
$\displaystyle\left[\frac{a}{r(t)d^{2}}r(t-d)\right](1-\eta\left[(x(t-d+\tau_{f})-x^{*}\right]^{+})$
$\displaystyle-\left[b\cdot
r(t)r(t-d_{s})\right]\eta\left[x(t-d_{s}+\tau_{f})-x^{*}\right]^{+}$ (3)

By linearizing (VIII), substituting the dynamics of $x(t)$, and using $r(t)$
again, we obtain the following delay differential equation

$\displaystyle\Large\ddot{r}(t)$
$\displaystyle+\gamma\left[2\dot{r}(t)-\dot{r}(t-d)+\dot{r}(t-d_{s}\right]$
$\displaystyle+\alpha r(t-d)+\Omega r(t-d_{s})=0$ (4)

where $\gamma=\frac{abc}{a+bc^{2}d^{2}}$, $\alpha=\frac{\eta a}{d^{2}}$ and
$\Omega=bc^{2}\eta$. To study the stability of the model, we substitute
$r(t)=e^{st}$ in (VIII) and solve for $s$. We obtain a characteristic equation
as follows

$\displaystyle\Large s^{2}+(2-e^{-sd}+e^{-sd_{s}})\gamma s+\alpha
e^{-sd}+\Omega e^{-sd_{s}}=0$ (5)

For this dynamical system to be stable, the roots of (5) should have a
negative real term. Note that this equation will have infinite roots, and
hence, we seek conditions, i.e., TCP AIMD and scenario parameters, under which
the system is always stable.

Classically, we consider AIMD with $a=1$ and $b=\frac{1}{2}$, hence,
$\gamma,\alpha$ and $\Omega$ are, $\frac{c}{2+c^{2}d^{2}}$,
$\frac{\eta}{d^{2}}$ and $\frac{\eta c^{2}}{2}$ respectively. Substituting in
(5) and reorganizing we obtain $\eta$ in (2). Now, for the dynamical system to
be stable, $\eta$ should be greater than 0 under the condition that $s$ has a
negative real part. For simplicity, assume that $s$ is real; hence for
$\eta>0$ we obtain the condition $(e^{-sd}-e^{sd_{s}}-2)s\gamma-s^{2}$ $<\ 0$.
After some algebraic manipulation, and using a power series expansion and
solving for $s$ we obtain

$\displaystyle\Large
s^{*}=\frac{-\sqrt{5d^{2}\gamma^{2}-2d\gamma^{2}d_{s}+2d\gamma-3\gamma^{2}d_{s}^{2}-2\gamma
d_{s}+1}-d\gamma+\gamma d_{s}-1}{\gamma(d^{2}-d_{s}^{2})}$ (6)

Substituting any value of s less than $s^{*}$ in (2) insures stability. ∎

## References

  * [1] J. Gettys, “Bufferbloat: Dark buffers in the internet,” _IEEE Internet Computing_ , vol. 15, no. 3, pp. 96–96, 2011.
  * [2] T.-Y. Huang, N. Handigol, B. Heller, N. McKeown, and R. Johari, “Confused, timid, and unstable: picking a video streaming rate is hard,” in _Proc. of Internet Measurement Conference (IMC)_ , 2012, pp. 225–238.
  * [3] K. Nichols and V. Jacobson, “Controlling queue delay,” _Communications of the ACM_ , vol. 55, no. 7, pp. 42–50, 2012.
  * [4] R. Pan, P. Natarajan, C. Piglione, M. S. Prabhu, V. Subramanian, F. Baker, and B. VerSteeg, “PIE: A lightweight control scheme to address the bufferbloat problem,” in _Proc. of IEEE International conference on high performance switching and routing (HPSR)_ , 2013, pp. 148–155.
  * [5] S. Floyd and V. Jacobson, “Random early detection gateways for congestion avoidance,” _IEEE/ACM Transactions on networking_ , vol. 1, no. 4, pp. 397–413, 1993.
  * [6] R. Srikant and L. Ying, _Communication Networks: An Optimization, Control and Stochastic Networks Perspective_. Cambridge University Press, 2014.
  * [7] S. Floyd, “Connections with multiple congested gateways in packet-switched networks part 1: One-way traffic,” _ACM SIGCOMM Computer Communication Review_ , vol. 21, no. 5, pp. 30–47, 1991.
  * [8] C. Barakat, E. Altman, and W. Dabbous, “On TCP performance in a heterogeneous network: a survey,” _IEEE Communications Magazine_ , vol. 38, no. 1, pp. 40–46, 2000.
  * [9] E. Altman, C. Barakat, E. Laborde, P. Brown, and D. Collange, “Fairness analysis of TCP/IP,” in _Proc. of IEEE conference on decision and control_ , vol. 1, 2000, pp. 61–66.
  * [10] M. Alizadeh, A. Greenberg, D. A. Maltz, J. Padhye, P. Patel, B. Prabhakar, S. Sengupta, and M. Sridharan, “Data Center Tcp (DCTDC),” in _Proc. of ACM SIGCOMM_ , 2010, pp. 63–74.
  * [11] I. Cho, K. Jang, and D. Han, “Credit-scheduled delay-bounded congestion control for datacenters,” in _Proc. of ACM SIGCOMM_ , 2017, pp. 239–252.
  * [12] F. Uyeda, L. Foschini, F. Baker, S. Suri, and G. Varghese, “Efficiently measuring bandwidth at all time scales,” in _Proc. of USENIX NSDI_ , 2011\.
  * [13] M. Alizadeh, A. Kabbani, T. Edsall, B. Prabhakar, A. Vahdat, and M. Yasuda, “Less Is More: Trading a Little Bandwidth for $\\{$Ultra-Low$\\}$ Latency in the Data Center,” in _Proc. of USENIX NSDI_ , 2012, pp. 253–266.
  * [14] Y. Ren, Y. Zhao, P. Liu, K. Dou, and J. Li, “A survey on TCP Incast in data center networks,” _International Journal of Communication Systems_ , vol. 27, no. 8, pp. 1160–1172, 2014.
  * [15] D. Zats, T. Das, P. Mohan, D. Borthakur, and R. Katz, “DeTail: Reducing the flow completion time tail in datacenter networks,” in _Proc. ACM SIGCOMM_ , 2012, pp. 139–150.
  * [16] Y. Zhang and T. R. Henderson, “An implementation and experimental study of the explicit control protocol (XCP),” in _Proc. of IEEE INFOCOM_ , 2005, pp. 1037–1048.
  * [17] R. Kundel, N. B. Krishna, C. Gärtner, T. Meuser, and A. Rizk, “Poster: Reverse-path congestion notification: Accelerating the congestion control feedback loop,” in _2021 IEEE 29th International Conference on Network Protocols (ICNP)_ , 2021, pp. 1–2.
  * [18] A. Feldmann, B. Chandrasekaran, S. Fathalli, and E. N. Weyulu, “P4-enabled network-assisted congestion feedback: a case for NACKs,” in _Proc. of the 2019 Workshop on Buffer Sizing_ , 2019, pp. 1–7.
  * [19] J. H. Salim, B. Nandy, and N. Seddigh, “A proposal for backward ECN for the internet protocol (IPv4/IPv6),” _RFC proposal_ , 1998.
  * [20] T. Henderson and S. Floyd, “The NewReno Modification to TCP’s Fast Recovery Algorithm,” _RFC_ , no. 2582, Apr. 1999.
  * [21] P. Brown, “Resource sharing of TCP connections with different round trip times,” in _Proc. IEEE INFOCOM_ , vol. 3, 2000, pp. 1734–1741.
  * [22] S. Ha, I. Rhee, and L. Xu, “CUBIC: a new TCP-friendly high-speed TCP variant,” _ACM SIGOPS operating systems review_ , vol. 42, no. 5, pp. 64–74, 2008.
  * [23] T. Lakshman and U. Madhow, “The performance of TCP/IP for networks with high bandwidth-delay products and random loss,” _IEEE/ACM Transactions on Networking_ , vol. 5, no. 3, pp. 336–350, 1997.
  * [24] R. Kundel, J. Blendin, T. Viernickel, B. Koldehofe, and R. Steinmetz, “P4-codel: Active queue management in programmable data planes,” in _Proc. of IEEE NFV-SDN_ , 2018, pp. 1–4.
  * [25] K. Ramakrishnan, S. Floyd, and D. Black, “The addition of explicit congestion notification (ECN) to IP,” Internet Requests for Comments, RFC 3168, 2001.
  * [26] S. Bensley, D. Thaler, P. Balasubramanian, L. Eggert, and G. Judd, “Data center TCP (DCTCP): TCP congestion control for data centers,” Internet Requests for Comments, RFC 8257, 2017.
  * [27] A. Kumar, D. Manjunath, and J. Kuri, _Communication networking: an analytical approach_. Academic Press, 2004\.
  * [28] R. Kundel, F. Siegmund, R. Hark, A. Rizk, and B. Koldehofe, “Network testing utilizing programmable network hardware,” _IEEE Commun. Mag._ , vol. 60, no. 2, pp. 12–17, 2022.
  * [29] R. Jain, _The art of computer systems performance analysis - techniques for experimental design, measurement, simulation, and modeling._ Wiley, 1991.
  * [30] K. Ramakrishnan, S. Floyd, and D. Black, “The Addition of Explicit Congestion Notification (ECN) to IP,” Internet Requests for Comments, RFC 3168, September 2001.
  * [31] X. Du, K. Xu, L. Xu, K. Zheng, M. Shen, B. Wu, and T. Li, “R-AQM: Reverse ACK Active Queue Management in Multitenant Data Centers,” _IEEE/ACM Transactions on Networking_ , 2022.
  * [32] Z. Meng, Y. Guo, C. Sun, B. Wang, J. Sherry, H. H. Liu, and M. Xu, “Achieving consistent low latency for wireless real-time communications with the shortest control loop,” in _Proc. of ACM SIGCOMM_ , 2022, pp. 193–206.


# Performance of an optical single-sideband laser system for atom
interferometry

Clemens Rammeloo1  Lingxiao Zhu1,2  Yu-Hung Lien1,$\ast$ Kai Bongs1  Michael
Holynski1,${\dagger}$ 1School of Physics and Astronomy, University of
Birmingham, Edgbaston, Birmingham, B15 2TT, UK
2College of Intelligence Science and Technology, National University of
Defense Technology, Changsha, 410073, China<EMAIL_ADDRESS><EMAIL_ADDRESS>
††journal: osajournal

This paper reports on a detailed performance characterization of a recently
developed optical single-sideband (OSSB) laser system based on an IQ modulator
and second-harmonic generation for rubidium atom interferometry experiments.
The measured performance is used to evaluate the noise contributions of this
OSSB laser system when it is applied to drive stimulated Raman transitions in
87Rb for precision measurements of gravitational acceleration. The laser
system suppresses unwanted sideband components, but additional phase shift
compensation needs to be applied when performing frequency chirps with such an
OSSB laser system. The total phase noise contribution of the OSSB laser system
in the current experiment is 72 mrad for a single atom-interferometry sequence
with interrogation times of $T=120$ ms, which corresponds to a relative
precision of 32 n$g$ per shot. The dominant noise sources are found in the
relative intensity fluctuations between sideband and carrier components and
the phase noise of the microwave source.

##  1 Introduction

Atom interferometry is increasingly being applied outside laboratory
environments, for instance, in surveys of precision measurements of
gravitational acceleration [1, 2, 3, 4] and on mobile platforms [5, 6, 7].
This drives a demand for compact and efficient laser systems that address
transitions in alkali atoms. Such precision measurements rely on two laser
beams having both a stable phase-relation and a controlled frequency
difference, for which several compact laser systems have been developed. These
laser systems can generally be categorized in either consisting of two lasers
that are phase stabilized with respect to each other [8, 2] or a single laser
with a phase modulation to generate multiple frequency components [9, 10, 11,
12, 13, 14]. The latter requires only a single seed-laser, thus reducing
complexity and potentially enabling more compact designs. However, most laser
systems that are based on direct phase modulation apply an electro-optic
modulator (EOM) which also generates undesired frequency components. This
method is relatively inefficient as it wastes the available optical power and
could drive parasitic transitions that perturb measurements of gravitational
acceleration [15, 16].

A new method that suppresses undesirable sideband components is based on an IQ
modulator (IQM) that generates an optical single-sideband (OSSB) for tunable
laser frequency components with a single seed-laser [16]. This laser system
design relies on components from the telecommunication industry operating at
$1560\text{\,}\mathrm{nm}\text{/}$ wavelengths and second-harmonic generation
to $780\text{\,}\mathrm{nm}\text{/}$ wavelengths in order to drive transitions
in Rb atoms. We have previously demonstrated the applicability of an OSSB
laser system in measurements of gravitational acceleration [16], including the
suppression of systematic phase shifts from undesirable sidebands. In general,
such an IQM-based laser system can be used in all stages of a typical atom-
interferometry experiment that include laser cooling and detection of rubidium
atoms [17]. The application of an IQM instead of an EOM provides additional
controls over the sideband to carrier ratios. Due to this, however, an OSSB
laser system has the potential to introduce significant intensity changes in
sideband and carrier components.

In this work we discuss a detailed performance characterization of an OSSB
laser system, with a focus on the purpose of driving stimulated Raman
transitions between the hyperfine-split ground-states in 87Rb. These
transitions are applied in Mach–Zehnder (MZ) type pulse sequences for
precision measurements of gravitational acceleration [18]. Here we evaluate
for the parameters of our atom-interferometry experiment, the impact of the
noise contributions from the OSSB laser system. This paper first describes the
OSSB laser system setup and its wavelength conversion performance. This is
followed by measurements of the output power, intensity noise and stability of
the sideband component. We also identify additional phase shifts that need to
be considered when doing frequency chirps with an IQM. Finally, we report on
laser phase and frequency noise measurements, and compare the expected phase
shift contributions from all these effects for our MZ type atom-
interferometer.

##  2 Laser system setup

The OSSB laser system applied in this work is based on second harmonic
generation (SHG) of light from components operating at
$1560\text{\,}\mathrm{nm}\text{/}$ wavelengths as shown in the diagram in Fig.
1. The use of fiber-coupled components from the C-band telecommunication
industry enables a relatively efficient and compact design. Additionally, to
the best of our knowledge there are currently no IQ modulators commercially
available for wavelengths outside telecommunication bands.

The seed-laser is an erbium-doped fiber laser (NKT Photonics, Koheras BASIK
E15) and the single-sideband frequency component is generated via an IQ
modulator (Photline, MXIQ-LN-40). An erbium-doped fiber amplifier (Orion
Laser, YEDFA-PM) boosts the power to about $1.5\text{\,}\mathrm{W}\text{/}$
before an SHG module (NTT Electronics, WH-0780-000-F-B-C) which converts the
light to $780\text{\,}\mathrm{nm}\text{/}$ wavelengths. The latter consists of
a periodically-poled lithium niobate ridge-waveguide (PPLN RW) that is
temperature stabilized and has its input and output ports pigtailed with
polarization maintaining (PM) fibers. The output from the SHG fiber is
collimated and passed through a polarizing beam splitter (PBS) to filter
polarization fluctuations. A half-wave plate before the PBS is set such that
about $1\text{\,}\mathrm{\char 37\relax}\text{/}$ of the light is reflected by
the PBS towards a scanning Fabry–Pérot cavity for recording the relative
powers between the laser frequency components. The transmitted beam after the
PBS is switched by an acousto-optic modulator (AOM) to create the pulse
sequences for the Raman beam. The first-order diffracted output from the AOM
is coupled into a PM fiber to the Raman beam collimator of the atom
interferometer. Details of the atom-interferometer experiment can be found in
[19].

Figure 1: Diagram of the optical single-sideband laser system, see text for
details. EDFA: erbium-doped fiber amplifier, PPLN RW: periodically-poled
lithium niobate ridge-waveguide, AFG: arbitrary function generator, BPF:
bandpass filter, FC: fiber coupler, PBS: polarizing beamsplitter, AOM:
acousto-optic modulator. The plot is a measured transmission spectrum of a
Fabry–Pérot cavity scan showing the laser spectrum after the PPLN RW. The
carrier $\omega_{0}$ and first-order sideband $\omega_{1}$ are the laser
frequency components used to drive the two-photon Raman transition, while the
other $\omega_{-1}$ sideband is suppressed through tuning of the RF phase
shift and bias voltages on the IQ modulator.

The laser frequency components that drive the two-photon Raman transition are
the carrier $\omega_{0}$ and +1st order sideband $\omega_{1}$ generated by the
IQ modulator and subsequent wavelength conversion. The tunable RF signal for
the IQ modulation is generated by mixing a $7\text{\,}\mathrm{GHz}\text{/}$
signal from a signal generator (Keysight, E8267D) with the output of an
arbitrary function generator (Tektronix, AFG3252C) via an RF mixer (Mini-
Circuits, ZMX-10G+). The down converted frequency component is around
$6.835\text{\,}\mathrm{GHz}\text{/}$, to be resonant with the hyperfine-
splitting frequency of the ${{}^{5}S}_{1/2}$ state in 87Rb, while other RF
components are suppressed by more than $50\text{\,}\mathrm{dB}\text{/}$ by a
cavity bandpass filter (ELHYTE, BP6834-70/T-5CS). After amplification (Nextec,
NBL00426), the RF signal is split for the I and Q ports. A tunable RF phase
shift is introduced before the Q port by either a manual phase shifter
(Fairview Microwave, SMP0820) or a voltage controlled phase shifter (RF-
Lambda, RVPT0408GBC).

The optical output spectrum of the OSSB laser system shown in Fig. 1, is
measured by the scanning Fabry–Pérot cavity. These recordings are used to
monitor and to set the +1st order sideband / carrier power ratio to
${\sim}{1/2}$ such that the first-order AC Stark shifts are canceled in our
experiment. The -1st order sideband $\omega_{-1}$ is suppressed below
$-23\text{\,}\mathrm{dB}\text{/}$ with respect to the carrier by tuning the RF
phase shifter and bias voltages on the IQ modulator [16].

##  3 Wavelength conversion and output power

For an efficient second-harmonic generation of
$780\text{\,}\mathrm{nm}\text{/}$ wavelength light the quasi-phase-matching
condition in the PPLN RW needs to be maintained, which depends on both the
laser frequency and the temperature of the PPLN RW [20]. The frequency
bandwidth of the quasi-phase-matching condition is measured by tuning the
fiber laser frequency and simultaneously recording both frequency and power of
the $780\text{\,}\mathrm{nm}\text{/}$ wavelength light at the PPLN RW output.
The thus measured FWHM bandwidth is $80\pm 1.\text{\,}\mathrm{GHz}\text{/}$
and is much larger than both the sideband separation as well as the frequency
tuning range required in our atom-interferometry experiment.

Figure 2: SHG output power of the carrier $\omega_{0}$ and $\pm 1$st order
sidebands $\omega_{\pm 1}$ at $780\text{\,}\mathrm{nm}\text{/}$ as a function
of the temperature of the PPLN RW. The $\omega_{-1}$ sideband is not
suppressed in this measurement to show the shift in the optimum PPLN RW
temperature.

The temperature for optimum SHG is around
$53\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$ for the particular PPLN RW
used in this laser system as shown in Fig. 2. The PPLN RW temperature is
stabilized by a compact temperature controller (Meerstetter Engineering,
TEC-1091) that keeps the temperature fluctuations within
${\pm$0.005\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$}$. The temperature
range over which the quasi-phase-matching condition of this PPLN RW is
maintained has a HWHM of $0.9\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$
around the optimum temperature. However, the optimum phase-matching
temperatures of the sideband frequency components are offset from the
carrier’s optimum by about
$0.15\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{/}$ as seen in Fig. 2. This
offset can be exploited for increased sideband suppression. For instance, if
the PPLN RW temperature is tuned to a slightly lower temperature, the phase-
matching condition for the $\omega_{1}$ sideband is enhanced while
simultaneously reducing that of the $\omega_{-1}$ sideband. This can result in
an additional $2\text{\,}\mathrm{dB}\text{/}$ in sideband suppression in the
$780\text{\,}\mathrm{nm}\text{/}$ wavelength output, but this method has a
trade-off with a reduction in total output power.

The wavelength conversion efficiency is characterized by measuring the
$780\text{\,}\mathrm{nm}\text{/}$ wavelength power after the PPLN RW output
fiber as a function of the $1560\text{\,}\mathrm{nm}\text{/}$ wavelength power
supplied by the EDFA, see Fig. 3. The relation between the output power of a
PPLN medium $P_{2\omega}$ and the input power $P_{\omega}$ is described by the
equation [21]

$P_{2\omega}=\epsilon P_{\omega}\tanh^{2}{\\!\left(\sqrt{\eta\epsilon
P_{\omega}}\right)}.$ (1)

Here is $\eta$ the SHG efficiency and $\epsilon$ the light coupling
efficiency. In our laser system, $\epsilon$ represents the accumulated
coupling efficiencies between the ridge-waveguide and its input and output
fibers, as well as the fiber-to-fiber coupling efficiency from the EDFA to the
SHG module.

In the results of Fig. 3 an almost linear relation is observed for optical
powers above about $200\text{\,}\mathrm{mW}\text{/}$ where the wavelength
conversion is limited by the light coupling efficiency. Fitting the
measurements with (1) shows a coupling efficiency of $\epsilon=$64\pm
1.\text{\,}\mathrm{\char 37\relax}\text{/}$$ and an SHG efficiency
$\eta=$(1.3\pm 0.3)\text{\times}{10}^{3}\text{\,}\mathrm{\char
37\relax}\text{/}\mathrm{W}$$. The SHG efficiency $\eta$ is higher compared to
bulk PPLN crystals [22, 23], but the output power of these PPLN RW modules
reaches saturation for input powers around $2\text{\,}\mathrm{W}\text{/}$
[24]. The output power of the OSSB laser system presented here, enables a
$\pi/2$-pulse duration of
$\tau_{\mathrm{R}}=$25\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ for the
current Raman beam size in our atom-interferometry experiment at a red-
detuning of $2\text{\,}\mathrm{GHz}\text{/}$ with respect to the
${{}^{5}S}_{1/2}\leftrightarrow{{}^{5}P}_{3/2}$ transition in 87Rb.

Figure 3: The optical power at the output of the PPLN RW
($780\text{\,}\mathrm{nm}\text{/}$ wavelength) as a function of the EDFA
output ($1560\text{\,}\mathrm{nm}\text{/}$ wavelength) and the conversion
efficiency as the ratio between both powers. From the fit with (1) a second-
harmonic generation efficiency of $\eta=$(1.3\pm
0.3)\text{\times}{10}^{3}\text{\,}\mathrm{\char 37\relax}\text{/}\mathrm{W}$$
and a coupling efficiency of $\epsilon=$64\pm 1.\text{\,}\mathrm{\char
37\relax}\text{/}$$ are extracted.

##  4 Intensity noise and sideband/carrier ratio

Since the OSSB laser system is applied to drive stimulated Raman transitions
between atomic ground states, laser power fluctuations cause variations in the
Rabi frequency and thus reduce the signal to noise ratio in the measurements
of the atomic state population [25]. Besides this effect, the influence of
short-term intensity noise during an interferometry sequence can cause phase
shifts at the atom-interferometer output. These phase shifts are usually
suppressed through tuning of the sideband/carrier intensity ratio. However, it
will be shown that an IQM-based laser system can experience significant
variations in the sideband/carrier ratio. In the following sections we
investigate the impact from the measured laser power fluctuations and the
changes in the sideband/carrier ratio.

###  4.1 Power stability and RIN

Measurements of both the long-term stability of the laser system’s output
power over several hours, as well as the short-term relative intensity noise
(RIN) spectrum are presented in Fig. 4. At the start of the output power
recording in Fig. 4a a peak is observed during the warm-up period of the laser
system, but after an hour the long-term relative drift in output power reduces
to below $0.5\text{\,}\mathrm{\char 37\relax}\text{/}$. When comparing the
power fluctuations before and after the PBS, it is seen that one of the main
causes of the power drifts originate from the changing polarization at the
output fiber of the PPLN RW.

The short-term power fluctuations are measured with a
$125\text{\,}\mathrm{MHz}\text{/}$-bandwidth photodiode (TTi, TIA-525)
connected to a spectrum analyzer (Tektronix, RSA5115B). This allows the
recording of the RIN spectrum of the laser system, shown in Fig. 4b. When
integrating the RIN spectrum over the sensitive frequency range of a MZ pulse
sequence with $\tau_{\mathrm{R}}=$25\text{\,}\mathrm{\SIUnitSymbolMicro
s}\text{/}$$ and an interrogation time $T=$120\text{\,}\mathrm{ms}\text{/}$$
between the pulses, the integrated RIN is estimated to be
$-65\text{\,}\mathrm{dB}\text{/}$c for a single MZ pulse sequence.

Figure 4: (a) Relative power of the $780\text{\,}\mathrm{nm}\text{/}$
wavelength output of the laser system with respect to an average output power
of $0.4\text{\,}\mathrm{W}\text{/}$ as recorded before and after the
polarizing beam splitter (PBS). (b) Power spectral density (PSD) of the
relative intensity noise measured at the output fiber (solid line) with
respect to the measurement noise floor (dashed line). The PSD is averaged for
frequencies above $20\text{\,}\mathrm{Hz}\text{/}$ for clarity.

To evaluate the effect of these power fluctuations on the atom interferometer,
repeated MZ pulse sequences are produced with the AOM while the pulse powers
are measured with the high-bandwidth photodiode at the output fiber of the
OSSB laser system. Each pulse sequence is integrated and normalized to the
average total Raman beam power to create the plot in Fig. 5a. The variation in
the integrated Raman beam power over a complete MZ pulse sequence is smaller
in comparison to a single $\pi/2$-pulse, due to the averaging over the three
pulses. From the relative power fluctuations in Fig. 5a we estimate its
contribution to the atom-interferometer signal-to-noise ratio to about
$50\text{/}1$. This effect could in principle be mitigated through power
stabilization with the AOM, or reduced in post-process when the Raman beam
power is monitored for each MZ pulse sequence.

Figure 5: Simultaneous measurements of (a) the relative change in total beam
power for repeated Mach–Zehnder pulse sequences in comparison with single
$\pi/2$-pulses, and (b) the sideband/carrier ratio $R$ in comparison to the
ratio $-\beta/\alpha$ where the light shift is canceled. (c) The phase shift
$\Delta\Phi_{I}$ in the atom interferometer calculated via (3) with the
measured intensity and ratio $R$.

###  4.2 Sideband/carrier ratio

Intensity fluctuations in the Raman laser beams during the pulse sequence
cause varying light shifts on the atomic states[26]. The differential light
shift contribute to phase shifts in the atom interferometer, but can be
suppressed through optimization of the intensity ratio between sideband and
carrier components. However, the sideband/carrier ratio in the OSSB laser
system exhibits a drift due to instabilities in the IQ modulator biases [27].
This is clearly observed in the measurements from the scanning Fabry–Pérot
cavity plotted in Fig. 5b. These measurements were performed simultaneously
with the total power fluctuations recorded for Fig. 5a. This enables us to
evaluate the effects on the atom-interferometer phase shift from the varying
sideband/carrier ratio both independently from, and in combination with the
measured intensity noise. The phase shift contribution from the resulting
differential light shift could be reduced through the application of an active
stabilization of the IQ modulator biases, or via the subtraction of the phase
shift obtained with the following method in post-process.

As each of the Raman laser beams are detuned with respect to the electronic
transition ${{}^{5}S_{1/2}}\leftrightarrow{{}^{5}P_{3/2}}$ in 87Rb, they
induce a light shift on the two-photon Raman transition. The detuning
$\delta\nu$ due to this light shift can be expressed as a linear combination
of the laser intensities [28]: $\delta\nu=\alpha I_{1}+\beta I_{0}$. Here are
$I_{1}$ and $I_{0}$ the intensities of the +1st-order sideband component
$\omega_{1}$ and the carrier $\omega_{0}$, respectively. The $\alpha$ and
$\beta$ components are calculated using the formalism from [29] and for Raman
beams with a red-detuning of $2\text{\,}\mathrm{GHz}\text{/}$, we find
$\alpha=$5.43\text{\,}\mathrm{kHz}\text{\,}{\mathrm{cm}}^{2}\text{/}\mathrm{mW}$$
and
$\beta=-$2.76\text{\,}\mathrm{kHz}\text{\,}{\mathrm{cm}}^{2}\text{/}\mathrm{mW}$$.
Because we measure the ratio $R=I_{1}/I_{0}$ with the scanning Fabry–Pérot
cavity and the total Raman beam intensity $I_{\mathrm{tot}}=I_{1}+I_{0}$ with
a fast photodiode, the detuning from the differential light shift is expressed
here as

$\delta\nu=\alpha I_{\mathrm{tot}}\left(\frac{R+\beta/\alpha}{R+1}\right).$
(2)

This light shift is canceled by controlling the intensity ratio between
sideband and carrier such that $R=-\beta/\alpha$, which is in our experiment
approximately $0.51$. In the current laser system, however, the ratio $R$
drifts in the range of $15\text{\,}\mathrm{\char
37\relax}\text{/}25\text{\,}\mathrm{\char 37\relax}\text{/}$ relative to the
$-\beta/\alpha$ value as observed in Fig. 5b. Depending on $R$, the stimulated
Raman transitions become sensitive to temporal fluctuations in intensity
during the interferometry sequence, which leads to a phase shift
$\Delta\Phi_{I}$ in the atom interferometer given by [26]

$\Delta\Phi_{I}=\int_{-\infty}^{+\infty}g(t)2\pi\delta\nu(t)\mathrm{d}t.$ (3)

In this equation is $g(t)$ the sensitivity function of a MZ type pulse
sequence as derived by [30].

The expected phase shift $\Delta\Phi_{I}$ with the current OSSB laser system
is determined in two cases. First, the contribution from the fluctuating total
intensity noise $I_{\mathrm{tot}}(t)$ is evaluated while assuming a constant
ratio $R$ in (2) during each MZ pulse sequence. Secondly, the effect from a
varying sideband/carrier ratio $R(t)$ during the pulse sequence is determined
by setting the total intensity $I_{\mathrm{tot}}$ equal to the average
intensity of the Raman laser beams.

The phase noise contribution from the total intensity noise is evaluated by
performing the integration of (3) over each measured pulse sequence. Fast
changes in the sideband/carrier ratio during an interferometry sequence cannot
be measured with the scanning Fabry–Pérot cavity due to its limited scan
speed. Therefore, the ratio $R$ is assumed constant during a MZ single
sequence when evaluating the integration of (3) over the measured intensity
$I_{\mathrm{tot}}(t)$. The resulting phase shifts for each pulse sequence are
plotted in Fig. 5c and reach about $30\text{\,}\mathrm{mrad}\text{/}$ per shot
with a standard deviation of $6\text{\,}\mathrm{mrad}\text{/}$ over an hour of
repeated pulse sequences. These phase shifts are higher than those reported in
other works [26], but can be subtracted in post-process. However, this method
is currently limited by the measurement precision of the ratio $R$.

Fast changes of the carrier and sideband ratio $R(t)$ on time scales down to
$10\text{\,}\mathrm{ms}\text{/}$ are measured via a beat signal spectrum
generated in combination with a second laser operating at
$780\text{\,}\mathrm{nm}\text{/}$ wavelengths. A spectrogram of the beat
signal with both carrier and sideband components of the OSSB laser system is
recorded by a real-time spectrum analyzer (Tektronix, RSA5115). The
sideband/carrier ratio in these measurements shows RMS fluctuations below
$0.9\text{\,}\mathrm{\char 37\relax}\text{/}$ on time-scales of a single
interferometry sequence. We evaluate the effect of this changing
sideband/carrier ratio $R(t)$ during a MZ pulse sequence again via (2) and
(3). For our atom-interferometry experiment with an average intensity
$I_{\mathrm{tot}}=$34\text{\,}\mathrm{mW}\text{/}{\mathrm{cm}}^{2}$$, the
light shift sensitivity is $12\text{\,}\mathrm{kHz}\text{/}\mathrm{\char
37\relax}$ change in the sideband/carrier ratio. Thus the measured variation
in the ratio $R$ between pulses corresponds here to a contribution of
$56\text{\,}\mathrm{mrad}\text{/}$ in the atom-interferometer phase shift. The
effect on the determination of the gravitational acceleration $g$ using MZ
pulse sequences with $T=$120\text{\,}\mathrm{ms}\text{/}$$ is then at a level
of ${$25$~{}\mathrm{n}g}$ per shot.

The changing intensity $I_{\mathrm{tot}}(t)$ not only originate from the laser
system’s RIN, but also from the movement of the atoms in the Raman laser beam.
The effect of the latter depends on the experimental conditions, for example
the atom-cloud temperature, but can be suppressed via differential
measurements with reversed frequency chirps [31]. Such methods do not,
however, eliminate uncorrelated fluctuations in intensity or sideband/carrier
ratio. Recording the sideband/carrier ratio during MZ pulse sequences, or
stabilizing $R$ via the IQM bias voltages [32], is thus necessary when using
this OSSB laser systems for precision measurements of gravity.

##  5 Frequency chirping

In order to keep the Raman laser on resonance with the accelerating atoms in
the atom interferometer, the RF signal applied to the IQ modulator is chirped.
These frequency chirps can introduce variations in the sideband/carrier ratio
due to a changing RF phase difference between the I- and Q-ports. This
frequency dependence is attributed to the microwave components and dispersion
in the RF cables, the result of which could cause a bias in the atom-
interferometer phase shift.

The RF phase imbalance is measured by combining the signals for the I- and
Q-ports with an RF mixer (Mini-Circuits, ZMX-10G+) and recording the DC
voltage output while chirping the RF signal. The resulting phase imbalance,
within the microwave frequency range applicable in our atom-interferometer, is
shown in Fig. 6a for two different configurations. In case a manual RF phase
shifter is used, there is no active compensation of the RF imbalance and we
observe a variation of about $11\text{\,}\mathrm{mrad}\text{/}\mathrm{MHz}$.
On the other hand, when a voltage controlled phase shifter is applied to
compensate the frequency dependent RF phase shifts, this variation is reduced
below $3\text{\,}\mathrm{mrad}\text{/}\mathrm{MHz}$.

The effect of this RF phase compensation on the power ratio between +1st-order
sideband and carrier components is clearly visible in Fig. 6b. The variation
in the ratio $R$ as measured with the Fabry–Pérot cavity, is about
$4\text{\,}\mathrm{\char 37\relax}\text{/}$ when the RF phase compensation is
applied. However, this measurement is limited by the temporal variations in
$R(t)$ which were already discussed in section 4.2, thus the frequency
chirping effect on the sideband/carrier ratio is likely smaller. Without this
RF phase shift compensation the changes in the ratio $R$ are an order of
magnitude larger. It should be noted that with the applied RF phase
compensation, the -1st-order sideband stays suppressed to a level
$-23\text{\,}\mathrm{dB}\text{/}$ with respect to the carrier.

Figure 6: The effect of a changing microwave frequency for the single-sideband
generation on (a) the phase difference between the RF signals at the I- and
Q-ports of the IQ modulator, and (b) the power ratio between +1st-order
sideband and carrier components.

The contribution to the atom-interferometry phase shift can be considered as a
bias when the same frequency chirp is applied during each measurement cycle.
We estimate from the measurements of Fig. 6b and (3), that the RF phase
compensation can reduce this frequency chirping bias to below
$82\text{\,}\mathrm{mrad}\text{/}$ for MZ pulse sequences with
$T=$120\text{\,}\mathrm{ms}\text{/}$$. This bias could in principle be
corrected either in post-process with a characterization of the microwave
components and IQ modulator, or via feedback from the Fabry–Pérot cavity
measurements to the voltage controlled phase shifter. Foremost, this effect
highlights the need for careful RF design when applying IQM-based laser
systems to perform accurate measurements using atom interferometry.

##  6 Laser phase noise

The main noise source in many atom-interferometers for precision measurements
of gravitational acceleration, is the phase noise between the two Raman laser
beams. The phase noise between two retro-reflected Raman beams is usually
dominated by vibrations of the retro-reflecting mirror. These perturbing
accelerations are filtered through vibration isolation [25, 1, 3] or
compensated using recordings from classical accelerometers [33, 34]. The
remaining limit of the Raman beam phase noise is then governed by phase
fluctuations between the two frequency components of the Raman laser system.

Because the two frequency components share the same optical path in the OSSB
laser system, phase noise from perturbations in the optical path up to the
atom interferometer is common-mode and cancels out in the atomic state
population. The phase noise at the fiber output is therefore limited by the
phase noise of the RF source as observed in the measurements in Fig. 7. The
phase noise power spectral density (PSD) of Fig. 7a is measured by a signal
source analyzer (Agilent, N9030B). It is observed that the phase noise of the
optical beat signal between carrier and sideband of the Raman laser is limited
by the $7\text{\,}\mathrm{GHz}\text{/}$ signal generator source at frequencies
below $500\text{\,}\mathrm{Hz}\text{/}$. Between
$500\text{\,}\mathrm{Hz}\text{/}$ and about $100\text{\,}\mathrm{kHz}\text{/}$
the phase noise contribution from the AFG is dominating. At higher frequencies
the measured phase noise of the laser beat signal is higher than that of the
RF modulation signal as a result of intensity noise of the fiber laser.

Figure 7: (a) The power spectral density (PSD) of the phase noise between the
Raman laser frequency components measured in the optical beat signal. The
measured microwave signal used to create the sideband components and the
$7\text{\,}\mathrm{GHz}\text{/}$ microwave source are also plotted in
comparison to the spectrum analyzer’s measurement noise floor. (b) The
cumulative phase noise in an atom interferometer as calculated with (4) and
the laser phase noise PSD of (a) weighted by (5) for a MZ pulse sequence with
$\tau_{\mathrm{R}}=$25\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ and
$T=$120\text{\,}\mathrm{ms}\text{/}$$.

The effect of the laser phase noise on the atom-interferometer phase noise
$\Delta\Phi_{\phi}$ can be evaluated with the equation [30]:

$\Delta\Phi_{\phi}=\sqrt{\int^{\infty}_{0}S_{\phi}(f)|H(2\pi
f)|^{2}\mathrm{d}f}.$ (4)

Here is $S_{\phi}(f)$ the phase noise PSD and $H(2\pi f)=H(\omega)$ is the
spectral weighting function of a Mach–Zehnder type pulse sequence according to
the formulation of [30]:

$|H(\omega)|^{2}=\left|\frac{4\Omega_{\mathrm{R}}\omega}{\omega^{2}-\Omega_{\mathrm{R}}^{2}}\right|^{2}\sin^{2}\\!\left(\frac{\omega(T+2\tau_{\mathrm{R}})}{2}\right)\times\left[\cos\\!\left(\frac{\omega(T+2\tau_{\mathrm{R}})}{2}\right)+\frac{\Omega_{\mathrm{R}}}{\omega}\sin\\!\left(\frac{\omega
T}{2}\right)\right]^{2},$ (5)

where $\Omega_{\mathrm{R}}=\pi/(2\tau_{\mathrm{R}})$ is the Rabi frequency of
the stimulated Raman transition. Applying the measured phase noise PSD of the
laser beat signal in (4), we calculate the cumulative phase noise for a MZ
pulse sequence with $\tau_{\mathrm{R}}=$25\text{\,}\mathrm{\SIUnitSymbolMicro
s}\text{/}$$ and $T=$120\text{\,}\mathrm{ms}\text{/}$$. The cumulative phase
noise introduced in the atom interferometer is evaluated by taking the
integration in (4) up to the frequency indicated in Fig. 7b. This shows that
the total phase noise contribution $\Delta\Phi_{\phi}$ of
$35\text{\,}\mathrm{mrad}\text{/}$ is limited by phase noise from the
microwave source below $10\text{\,}\mathrm{Hz}\text{/}$. Bespoke microwave
sources can reduce this phase noise contribution as demonstrated by other
research groups [8, 35], but the characterization here is currently limited by
the noise floor of the spectrum analyzer.

##  7 Laser frequency noise

Frequency noise from the fiber laser introduces phase noise in stimulated
Raman transitions with counter-propagating beams, due to a propagation delay
of the retro-reflected Raman beam [36]. The phase noise contribution
$\Delta\Phi_{\nu}$ from this effect is estimated via

$\Delta\Phi_{\nu}=2\pi
t_{\mathrm{d}}\sqrt{\int^{\infty}_{0}S_{\nu}(f)\left|H(2\pi
f)\right|^{2}\mathrm{d}f}.$ (6)

Here is $S_{\nu}(f)$ the power spectral density of the laser frequency noise
and $t_{\mathrm{d}}$ is the propagation delay time of the retro-reflected
Raman beam. In our atom-interferometer setup the average distance between
mirror and atom cloud is $75\text{\,}\mathrm{cm}\text{/}$, resulting in a
propagation delay of $t_{\mathrm{d}}=$5\text{\,}\mathrm{ns}\text{/}$$.

Figure 8: (a) The power spectral density (PSD) of the laser frequency noise
determined from an error signal of a rubidium modulation transfer spectroscopy
setup. The PSD is averaged for frequencies above
$10\text{\,}\mathrm{kHz}\text{/}$ for clarity. The white noise floor level
(dashed line) for frequencies above $50\text{\,}\mathrm{kHz}\text{/}$ is
determined from the linewidth of the beat signal spectrum with a similar
laser. (b) The cumulative phase noise from the weighted PSD of (a) for a
propagation delay of $5\text{\,}\mathrm{ns}\text{/}$ and a MZ type pulse
sequence with $\tau_{\mathrm{R}}=$25\text{\,}\mathrm{\SIUnitSymbolMicro
s}\text{/}$$, $T=$120\text{\,}\mathrm{ms}\text{/}$$.

The PSD $S_{\nu}(f)$ of the laser frequency noise is determined by taking the
Fourier transform of the error signal from a rubidium spectroscopy setup. A
modulation transfer spectroscopy setup [37] is used to convert the optical
frequency at the $780\text{\,}\mathrm{nm}\text{/}$ wavelength output of the
laser to an error signal voltage, as described in [38]. The fiber laser is
tuned via the internal piezo element such that the laser frequency is within
the linear range of the error signal around the
$\ket{{}^{5}S_{1/2},F=2}\leftrightarrow\ket{{}^{5}P_{3/2},F=3}$ transition in
87Rb. In this setup no active frequency stabilization is applied, i.e. the
fiber laser is unlocked. The resulting PSD of the error signal voltage is
converted to laser frequency noise and plotted in Fig. 8a. The two distinct
peaks visible at $50\text{\,}\mathrm{Hz}\text{/}$ and
$150\text{\,}\mathrm{Hz}\text{/}$ in the PSD are from mains-power pick-up. The
demodulation circuit of the spectroscopy setup has a cut-off frequency at
$50\text{\,}\mathrm{kHz}\text{/}$, where nearby a small bump is visible in the
spectrum from the resonance of the piezo tuning structure in the fiber laser.

For frequencies above $50\text{\,}\mathrm{kHz}\text{/}$ the laser frequency
noise is estimated by a white noise floor level of
$S_{\mathrm{L}}=$6.7\text{\times}{10}^{3}\text{\,}{\mathrm{Hz}}^{2}\text{/}\mathrm{Hz}$$,
as indicated by the dashed line in Fig. 8a. This white noise amplitude is
calculated from the laser linewidth, which in turn is measured by creating a
beat signal with a second laser of the same model. The recorded spectrum of
the beat signal results in a Lorenztian laser linewidth of $\gamma=$21\pm
2.\text{\,}\mathrm{kHz}\text{/}$$ (FWHM) and is used to estimate the laser
frequency noise level via $S_{\mathrm{L}}=\gamma/\pi$ [39].

The cumulative phase noise in a MZ pulse sequence, plotted in Fig. 8b, is
calculated from the measured PSD of Fig. 8a using (5) and (6). The noise
contribution in the atom-interferometry phase is found to be
$\Delta\Phi_{\nu}=$5.5\text{\,}\mathrm{mrad}\text{/}$$ and is mostly due to
the frequency noise below $1\text{\,}\mathrm{kHz}\text{/}$ in the absence of
active frequency stabilization. This phase noise contribution from propagation
delay is similar to those reported in other laser systems [36, 10] and could
also be reduced by decreasing the distance between the retro-reflecting mirror
and the atom cloud.

##  8 Concluding remarks

The evaluated contributions in the atom-interferometry phase noise from the
current OSSB laser system are summarized in Table 1, together with the
contribution in the relative precision or measurement bias in $g$. For
comparison with other gravimeter experiments, these contributions are also
evaluated for Mach–Zehnder type pulse sequences at different interrogation
times, $T=$60\text{\,}\mathrm{ms}\text{/}$$ and
$240\text{\,}\mathrm{ms}\text{/}$. The most significant phase shifts come from
differential light shifts due to a changing ratio between sideband and carrier
intensity. Thus, for improved measurement precision and accuracy a
characterization of RF phase shifts is necessary. This becomes even more
crucial for increased interrogation times $T$, as seen in Table 1, since a
larger frequency chirp would be required. In comparison, the laser intensity
noise does not pose a significant limit to the atom-interferometer measurement
precision, because the resulting reduction in signal-to-noise ratio and
additional phase shift can in principle be corrected for in post-process when
monitoring the Raman beam powers and sideband/carrier ratio. We estimate that
when sideband/carrier ratio fluctuations in the order of
$0.1\text{\,}\mathrm{\char 37\relax}\text{/}$ can be achieved, the phase noise
contribution $\Delta\Phi_{I}$ reduces to the same level as $\Delta\Phi_{\nu}$.
Such a measurement precision of the sideband/carrier ratio is reachable with
the installed Fabry–Pérot cavity, however the limiting factor here is its
measurement speed.

The current OSSB laser system impacts our atom-interferometry experiment with
$T=$120\text{\,}\mathrm{ms}\text{/}$$ to the level of
$72\text{\,}\mathrm{mrad}\text{/}$ per shot, corresponding to a relative
measurement precision in the local gravitational acceleration of
$32~{}\mathrm{n}g$. This study thus shows that when the sideband/carrier ratio
is monitored, the performance of an OSSB laser system in atom-interferometry
mainly lies in the RF design and choice of microwave components that drive the
IQ modulator. The presented IQM-based laser system thus provides a tunable,
phase-stable and efficient light source for high-precision atom-interferometry
experiments.

Table 1: RMS phase noise and frequency chirping bias contributions from the OSSB laser system in our atom-interferometry experiment using a MZ type pulse sequences with $\tau_{\mathrm{R}}=$25\text{\,}\mathrm{\SIUnitSymbolMicro s}\text{/}$$ and different interrogation times $T$. In brackets are the corresponding values relative to the local gravitational acceleration. Interrogation time $T$ |  $60\text{\,}\mathrm{ms}\text{/}$ |  $120\text{\,}\mathrm{ms}\text{/}$ |  $240\text{\,}\mathrm{ms}\text{/}$
---|---|---|---
Laser phase noise $\Delta\Phi_{\phi}$ |  $30\text{\,}\mathrm{mrad}\text{/}$ $($52$~{}\mathrm{n}g)$ |  $35\text{\,}\mathrm{mrad}\text{/}$ $($16$~{}\mathrm{n}g)$ |  $37\text{\,}\mathrm{mrad}\text{/}$ $($4.0$~{}\mathrm{n}g)$
Frequency noise $\Delta\Phi_{\nu}$ |  $5.0\text{\,}\mathrm{mrad}\text{/}$ $($8.8$~{}\mathrm{n}g)$ |  $5.5\text{\,}\mathrm{mrad}\text{/}$ $($2.4$~{}\mathrm{n}g)$ |  $6.1\text{\,}\mathrm{mrad}\text{/}$ $($0.7$~{}\mathrm{n}g)$
Intensity noise $\Delta\Phi_{I}$ from: |  |  |
\- total intensity $I_{\mathrm{tot}}(t)$ |  - |  $6\text{\,}\mathrm{mrad}\text{/}$ $($2.6$~{}\mathrm{n}g)$ |  -
\- intensity ratio $R(t)$ |  - |  $56\text{\,}\mathrm{mrad}\text{/}$ $($25$~{}\mathrm{n}g)$ |  -
Frequency chirping bias |  $14\text{\,}\mathrm{mrad}\text{/}$ $($25$~{}\mathrm{n}g)$ |  $82\text{\,}\mathrm{mrad}\text{/}$ $($36$~{}\mathrm{n}g)$ |  $175\text{\,}\mathrm{mrad}\text{/}$ $($19$~{}\mathrm{n}g)$

##  9 Funding

Engineering and Physical Sciences Research Council (EP/M013294); Defence
Science and Technology Laboratory (DSTLX-1000095040).

##  10 Acknowledgments

The authors thank A. Kaushik, M. Ali Khan and C. McRae for useful discussions
and their help with the laser system components.

## References

  * [1] T. Farah, C. Guerlin, A. Landragin, P. Bouyer, S. Gaffet, F. Pereira Dos Santos, and S. Merlet, “Underground operation at best sensitivity of the mobile LNE-SYRTE cold atom gravimeter,” Gyroscopy and Navigation 5, 266–274 (2014). 10.1134/S2075108714040051.
  * [2] B. Wu, Z. Wang, B. Cheng, Q. Wang, A. Xu, and Q. Lin, “The investigation of a $\upmu$Gal-level cold atom gravimeter for field applications,” Metrologia 51, 452–458 (2014). 10.1088/0026-1394/51/5/452.
  * [3] C. Freier, M. Hauth, V. Schkolnik, B. Leykauf, M. Schilling, H. Wziontek, H.-G. Scherneck, J. Müller, and A. Peters, “Mobile quantum gravity sensor with unprecedented stability,” Journal of Physics: Conference Series 723, 012050 (2016). 10.1088/1742-6596/723/1/012050.
  * [4] V. Ménoret, P. Vermeulen, N. L. Moigne, S. Bonvalot, P. Bouyer, A. Landragin, and B. Desruelle, “Gravity measurements below 10-9 g with a transportable absolute quantum gravimeter,” Scientific Reports 8, 12300 (2018). 10.1038/s41598-018-30608-1.
  * [5] B. Barrett, L. Antoni-Micollier, L. Chichet, B. Battelier, T. Lévèque, A. Landragin, and P. Bouyer, “Dual matter-wave inertial sensors in weightlessness,” Nature Communications 7, 13786 (2016). 10.1038/ncomms13786.
  * [6] Y. Bidel, N. Zahzam, C. Blanchard, A. Bonnin, M. Cadoret, A. Bresson, D. Rouxel, and M. F. Lequentrec-Lalancette, “Absolute marine gravimetry with matter-wave interferometry,” Nature Communications 9, 627 (2018). 10.1038/s41467-018-03040-2.
  * [7] D. Becker, M. D. Lachmann, S. T. Seidel, H. Ahlers, A. N. Dinkelaker, J. Grosse, O. Hellmig, H. Müntinga, V. Schkolnik, T. Wendrich, A. Wenzlawski, B. Weps, R. Corgier, T. Franz, N. Gaaloul, W. Herr, D. Lüdtke, M. Popp, S. Amri, H. Duncker, M. Erbe, A. Kohfeldt, A. Kubelka-Lange, C. Braxmaier, E. Charron, W. Ertmer, M. Krutzik, C. Lämmerzahl, A. Peters, W. P. Schleich, K. Sengstock, R. Walser, A. Wicht, P. Windpassinger, and E. M. Rasel, “Space-borne Bose–Einstein condensation for precision interferometry,” Nature 562, 391–395 (2018). 10.1038/s41586-018-0605-1.
  * [8] M. Schmidt, M. Prevedelli, A. Giorgini, G. M. Tino, and A. Peters, “A portable laser system for high-precision atom interferometry experiments,” Applied Physics B 102, 11–18 (2010). 10.1007/s00340-010-4263-8.
  * [9] O. Carraz, F. Lienhart, R. Charrière, M. Cadoret, N. Zahzam, Y. Bidel, and A. Bresson, “Compact and robust laser system for onboard atom interferometry,” Applied Physics B 97, 405–411 (2009). 10.1007/s00340-009-3675-9.
  * [10] F. Theron, O. Carraz, G. Renon, N. Zahzam, Y. Bidel, M. Cadoret, and A. Bresson, “Narrow linewidth single laser source system for onboard atom interferometry,” Applied Physics B 118, 1–5 (2014). 10.1007/s00340-014-5975-y.
  * [11] X. Wu, F. Zi, J. Dudley, R. J. Bilotta, P. Canoza, and H. Müller, “Multiaxis atom interferometry with a single-diode laser and a pyramidal magneto-optical trap,” Optica 4, 1545 (2017). 10.1364/optica.4.001545.
  * [12] C. Diboune, N. Zahzam, Y. Bidel, M. Cadoret, and A. Bresson, “Multi-line fiber laser system for cesium and rubidium atom interferometry,” Optics Express 25, 16898 (2017). 10.1364/OE.25.016898.
  * [13] W. Li, X. Pan, N. Song, X. Xu, and X. Lu, “A phase-locked laser system based on double direct modulation technique for atom interferometry,” Applied Physics B 123, 54 (2017). 10.1007/s00340-016-6630-6.
  * [14] J. Fang, J. Hu, X. Chen, H. Zhu, L. Zhou, J. Zhong, J. Wang, and M. Zhan, “Realization of a compact one-seed laser system for atom interferometer-based gravimeters,” Optics Express 26, 1586 (2018). 10.1364/oe.26.001586.
  * [15] O. Carraz, R. Charrière, M. Cadoret, N. Zahzam, Y. Bidel, and A. Bresson, “Phase shift in an atom interferometer induced by the additional laser lines of a raman laser generated by modulation,” Physical Review A 86, 033605 (2012). 10.1103/PhysRevA.86.033605.
  * [16] L. Zhu, Y.-H. Lien, A. Hinton, A. Niggebaum, C. Rammeloo, K. Bongs, and M. Holynski, “Application of optical single-sideband laser in Raman atom interferometry,” Optics Express 26, 6542 (2018). 10.1364/OE.26.006542.
  * [17] H. Gouraud and B. Battelier, “ModBox CS-SSB: the agile iXblue laser source for cold atoms accelerometers,” Photoniques Special EOS 3, 37 (2019). www.photoniques.com/articles/photon/abs/2019/02/contents/contents.html.
  * [18] M. A. Kasevich and S. Chu, “Atomic interferometry using stimulated Raman transitions,” Physical Review Letters 67, 181–184 (1991). 10.1103/PhysRevLett.67.181.
  * [19] L. Zhu, “A cold atoms gravimeter for use in absolute gravity comparisons,” Ph.D. thesis, University of Birmingham (2018). etheses.bham.ac.uk/8152.
  * [20] D. Hum and M. Fejer, “Quasi-phasematching,” Comptes Rendus Physique 8, 180–198 (2007). 10.1016/j.crhy.2006.10.022.
  * [21] N. Chiodo, F. Du Burck, J. Hrabina, Y. Candela, J. P. Wallerand, and O. Acef, “CW frequency doubling of 1029nm radiation using single pass bulk and waveguide PPLN crystals,” Optics Communications 311, 239–244 (2013). 10.1016/j.optcom.2013.08.020.
  * [22] Q. Wang, Z. Wang, Z. Fu, W. Liu, and Q. Lin, “A compact laser system for the cold atom gravimeter,” Optics Communications 358, 82 – 87 (2016). 10.1016/j.optcom.2015.09.001.
  * [23] F. Theron, Y. Bidel, E. Dieu, N. Zahzam, M. Cadoret, and A. Bresson, “Frequency-doubled telecom fiber laser for a cold atom interferometer using optical lattices,” Optics Communications 393, 152 – 155 (2017). 10.1016/j.optcom.2017.02.013.
  * [24] T. Lévèque, L. Antoni-Micollier, B. Faure, and J. Berthon, “A laser setup for rubidium cooling dedicated to space applications,” Applied Physics B 116, 997–1004 (2014). 10.1007/s00340-014-5788-z.
  * [25] A. Peters, K. Y. Chung, and S. Chu, “High-precision gravity measurements using atom interferometry,” Metrologia 38, 25 (2001). 10.1088/0026-1394/38/1/4.
  * [26] J. Le Gouët, T. Mehlstäubler, J. Kim, S. Merlet, A. Clairon, A. Landragin, and F. Pereira Dos Santos, “Limits to the sensitivity of a low noise compact atomic gravimeter,” Applied Physics B 92, 133–144 (2008). 10.1007/s00340-008-3088-1.
  * [27] J. P. Salvestrini, L. Guilbert, M. Fontana, M. Abarkan, and S. Gille, “Analysis and control of the DC drift in LiNbO3-based Mach–Zehnder modulators,” Journal of Lightwave Technology 29, 1522–1534 (2011). 10.1109/jlt.2011.2136322.
  * [28] J. Bateman, A. Xuereb, and T. Freegarde, “Stimulated raman transitions via multiple atomic levels,” Physical Review A 81, 043808 (2010). 10.1103/physreva.81.043808.
  * [29] P. Cheinet, “Conception et réalisation d’un gravimètre à atomes froids,” Ph.D. thesis, Université Paris (2006). tel.archives-ouvertes.fr/tel-00070861.
  * [30] P. Cheinet, B. Canuel, F. P. D. Santos, A. Gauguet, F. Yver-Leduc, and A. Landragin, “Measurement of the sensitivity function in a time-domain atomic interferometer,” IEEE Transactions on Instrumentation and Measurement 57, 1141–1148 (2008). 10.1109/TIM.2007.915148.
  * [31] A. Louchet-Chauvet, T. Farah, Q. Bodart, A. Clairon, A. Landragin, S. Merlet, F. Pereira, and D. Santos, “The influence of transverse motion within an atomic gravimeter,” New Journal of Physics 13, 065025 (2011). 10.1088/1367-2630/13/6/065025.
  * [32] X. Li, L. Deng, X. Chen, M. Cheng, S. Fu, M. Tang, and D. Liu, “Modulation-format-free and automatic bias control for optical IQ modulators based on dither-correlation detection,” Opt. Express 25, 9333–9345 (2017). 10.1364/OE.25.009333.
  * [33] J. Lautier, L. Volodimer, T. Hardin, S. Merlet, M. Lours, F. Pereira Dos Santos, and A. Landragin, “Hybridizing matter-wave and classical accelerometers,” Applied Physics Letters 105, 144102 (2014). 10.1063/1.4897358.
  * [34] P. Cheiney, L. Fouché, S. Templier, F. Napolitano, B. Battelier, P. Bouyer, and B. Barrett, “Navigation-compatible hybrid quantum accelerometer using a kalman filter,” Physical Review Applied 10, 034030 (2018). 10.1103/physrevapplied.10.034030.
  * [35] J. Lautier, M. Lours, and A. Landragin, “A compact micro-wave synthesizer for transportable cold-atom interferometers,” Review of Scientific Instruments 85, 063114 (2014). 10.1063/1.4884338.
  * [36] J. Le Gouët, P. Cheinet, J. Kim, D. Holleville, A. Clairon, A. Landragin, and F. Pereira Dos Santos, “Influence of lasers propagation delay on the sensitivity of atom interferometers,” The European Physical Journal D 44, 419–425 (2007). 10.1140/epjd/e2007-00218-2.
  * [37] D. McCarron, S. King, and S. Cornish, “Modulation transfer spectroscopy in atomic rubidium,” Measurement Science and Technology 19, 105601 (2008). 10.1088/0957-0233/19/10/105601.
  * [38] C. Rammeloo, “Optimization of a compact cold-atoms interferometer for gravimetry,” Ph.D. thesis, University of Birmingham (2018). etheses.bham.ac.uk/8146.
  * [39] G. D. Domenico, S. Schilt, and P. Thomann, “Simple approach to the relation between laser frequency noise and laser line shape,” Applied Optics 49, 4801 (2010). 10.1364/ao.49.004801.