# A real time digital receiver for correlation measurements in atomic systems

V. Mugundhan1, Maheswar Swar2, Subhajit Bhar3 and Saptarishi Chaudhuri4
Raman Research Institute,
C.V. Raman Avenue, Sadashivanagar,
Bangalore - 560080, India
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present the development and characterization of a generic, reconfigurable,
low-cost ($<$ 350 USD) software-defined digital receiver system (DRS) for
temporal correlation measurements in atomic spin ensembles. We demonstrate the
use of the DRS as a component of a high resolution magnetometer. Digital
receiver based fast Fourier transform spectrometers (FFTS) are generally
superior in performance in terms of signal-to-noise ratio (SNR) compared to
traditional swept-frequency spectrum analyzers (SFSA). In applications where
the signals being analyzed are very narrow band in frequency domain, recording
them at high speeds over a reduced bandwidth provides flexibility to study
them for longer periods. We have built the DRS on the STEMLab 125-14 FPGA
platform and it has two different modes of operation: FFT Spectrometer and
real time raw voltage recording mode. We evaluate its performance by using it
in atomic spin noise spectroscopy (SNS). We demonstrate that the SNR is
improved by more than one order of magnitude with the FFTS as compared to that
of the commercial SFSA. We also highlight that with this DRS operating in the
triggered data acquisition mode one can achieve spin noise (SN) signal with
high SNR in a recording time window as low as 100 msec. We make use of this
feature to perform time resolved high-resolution magnetometry. While the
receiver was initially developed for SNS experiments, it can be easily used
for other atomic, molecular and optical (AMO) physics experiments as well.

##  I Introduction

Digital receivers are a class of electronic systems where operations like
amplification, filtering, integration etc. are performed as a series of
mathematical operation on embedded components like FPGAs, Microprocessors or
GPUs. Compared to their analog counterparts, digital receivers are immune to
variations in gain and temperature. However, digital systems have quantization
noise, sampling rate and phase noise which can be minimized by choosing high
bit-width ADCs and low drift clock sources. These characteristics make digital
receivers an attractive option in applications where precision measurements
are required [1, 2]. AMO experiments are one such example.

Digital receivers are being used in AMO experiments for process control [3]
(e.g. temperature, current and wavelength control in lasers), synchronous
detection [4] and in compact magnetic resonance spectroscopy systems [5].
However, synchronous detection may not be feasible in many AMO experiments
including spin noise spectroscopy (SNS) [6, 7, 8]. The typical strength of the
raw SNS signal is $<100nV/{\sqrt{Hz}}$ which is far less compared to the
previous studies[3, 4, 5]. Therefore, the SNS signal has to be recorded
continuously (or on trigger) to improve the SNR.

In this paper, we discuss the development and utilization of a versatile
digital receiver to measure the spin noise (SN) in atomic vapor systems. We
have performed a comparative study with the results of our previous work in
[6]. We also demonstrate its utility in real-time precision magnetometry. This
digital receiver will be used as a component of a novel, miniaturized
magnetometer based on SNS techniques.

The paper is organized as follows: We introduce spin noise spectroscopy and
the digital receiver system developed for its measurement in section II. The
firmware architectures of the former is described in section III. The methods
adopted to mitigate the effects of electro-magnetic interference (EMI) in the
measurements are described in section IV. In section V, we present the SNS
data obtained by using our DRS as well as a comparison with the results
obtained from SFSA. Further, we demonstrate triggered data acquisition and
time resolved magnetometry using our DRS. We conclude in section VI after a
brief discussion on further applications and future scope of the developed
receiver.

##  II Digital Receivers for Spin noise spectroscopy

In this section, we describe the spin noise spectroscopy technique as well as
detection schemes by introducing our digital receiver.

###  II-A Spectroscopy Technique

Study of SN of an atomic ensemble has varied applications, ranging from
precision magnetometry, non-perturbative optical detection to metrology and
quantum sensing. The fluctuation in the spin population of an atomic system in
time leads to temporal changes in the refractive index of the medium. A
linearly polarized and far detuned probe laser beam can detect this temporal
variations of the refractive index in its polarization angle. We use a
polarimetric detection system where the polarization fluctuation of the probe
laser beam is detected in a balanced photodetector [9]. The power spectrum of
the polarization fluctuation (second order correlation function - g(2)) gives
the information about the spectral properties of the atomic spin ensemble. In
a typical experimental conditions we apply a uniform magnetic field
perpendicular to the propagation direction of the probe beam. The atomic spins
precess around the magnetic field with a Larmor frequency which is
proportional to the strength of the magnetic field. Therefore, the peak of the
spectrum appears at the Larmor frequency. Also, the width of the signal is
inversely proportional to the transverse spin relaxation time ($\sim$100 kHz
in our experimental conditions).

Our SNS experimental set-up is shown in Fig.1. A probe laser beam which is
detuned by $\approx$ 10 GHz from the strongest optical transition in neutral
rubidium (Rb) atoms, is used to probe the spin fluctuations of the atoms in
the vapor cell heated to temperatures ranging between 350 K and 400 K. The
spin fluctuations causes polarization fluctuations of the probe laser. The
polarimetric detection scheme employing a half wave plate (HWP), polarizing
beam splitter (PBS), and a balanced photo-detector (BPD) is capable of
measuring this polarization fluctuation. A uniform magnetic field
($B_{\bot}$), produced using a pair of magnetic coils in Helmholtz
configuration, is applied on the atomic vapor which is perpendicular to the
propagation direction of the probe laser beam. The output signal of the BPD is
recorded using the digital receiver described in this article and reported in
section V.

Figure 1: A typical spin noise spectroscopy (SNS) set-up of rubidium (Rb)
atomic vapor. L- plano-convex lens, VC- vapor cell (contains Rb atomic vapor),
M- dielectric mirror, HWP- half wave plate, PBS- polarizing beam-splitter,
BPD- balanced photodetector, $B_{\bot}$\- orthogonal magnetic field.

Previously we used a commercially available
SFSA111https://www.keysight.com/in/en/assets/7018-01953/data-
sheets/5989-9815.pdf to detect the SN signal from the balanced photo-detector.
It has a superheterodyne stage whose mixer provided a swept local oscillator
(LO) signal, such that the radio-frequency (RF) signal is translated to a
fixed intermediate frequency (IF). Since the LO has to be swept across a range
of frequencies, the sweep time increases with frequency span. This reduces the
dwell time at each frequency resulting in a decreased sensitivity. From the
experimental point of view, this leads to poor SNR. Moreover, if the signal is
expected to show variations smaller than the sweep-time, it cannot be
detected. SFSAs also have a low frequency cut-off (in our case 100 kHz) below
which the output amplitude and frequency measurements are not possible. We
have ongoing experiment to perform SNS on laser cooled atoms to investigate
spin dynamics in quantum regime, where a triggered data recording of a SN
signal over a short duration ($\sim$ few ms) is required. In this case a SFSA
can not be used.

There have been recent experiments reporting measurements of SNS in quantum
dots [10] as well as in atomic vapors [11], where the use of non-
reconfigurable and somewhat expensive digital receivers are reported. However,
the digital receiver described in this paper, allows us to overcome the
aforementioned limitations of SFSA, with certain trade-offs.

###  II-B Digital Receivers for SNS

We developed a digital receiver capable of operating in two modes, as listed
below.

  1. I

A fast Fourier transform spectrometer (FFTS) to probe the entire frequency
range of interest,

  2. II

A real time data recorder (RTDR) with trigger capabilities.

Both the aforementioned modes are implemented on the STEMLab
125-14222https://www.redpitaya.com/f130/STEMlab-board development board. This
board is selected for our application because it has two 14 bit analog-to-
digital converter (ADC) channels, with each channel providing a dynamic range
better than 80 dB. It has an analog bandwidth of 62.5 MHz, and is DC coupled.
The heart of the board is a Xilinx Zynq 7010 System on Chip (SoC), with
integrated programmable logic (PL) cells and ARM microprocessor based
processing system (PS). The signal processing algorithms in this work are
implemented on the PL side, while user control and data transmission programs
are implemented on the PS side.

##  III Firmware Description

In this section, we describe the firmware architecture of the two operational
modes of the DRS.

###  III-A Fast Fourier Transform Spectrometers (FFTS)

Fourier transform is used to find the spectral content of a time domain signal
[12]. Fast Fourier transform (FFT) is an algorithm which reduces the
complexity involved in calculating the Fourier transform from a $O(n^{2})$ to
$O(n\log{n})$ problem by using the periodicity and the symmetry properties of
the former [13, 14]. This in general reduces the number of operations required
to obtain the spectrum and results in resource savings when implemented in
embedded devices e.g. FPGA or in Microprocessors.

The SNR is proportional to $\sqrt{\beta\tau}$, where $\beta$ is the bandwidth,
and $\tau$ is the integration time [15]. For a conventional spectrum analyzer,
there are two time scales involved: $t_{s}$, the sweep time, and $t_{d}$, the
dead time. So, if a sweep contains $N_{s}$ points, the amount of time required
for obtaining the power at each frequency becomes ${t_{s}}/{N_{s}}$. In cases
where the data is to be acquired using interfaces such as GPIB, USB, or
ethernet, $t_{d}$ includes the time taken for the spectrum analyzer to
transfer the data to the DAQ system, during which time no new acquisition can
occur.

In case of an FFTS, the estimation of the power spectrum involves complex
weighting and summation of all time domain samples of the burst used to
perform the FFT. If a streaming algorithm is used, data acquisition,
performing FFT and data transfer can happen simultaneously resulting in zero
dead time. Thus for a single spectrum, with the same spectral resolution, an
FFTS ideally provides $\sqrt{N_{s}}$ improvement in the SNR as compared to
SFSA[16].

Figure 2: Top level block diagram of the FFTS. PS and related interfaces are
shown as green blocks. The red dashed lines indicate the flow of the master
clock at 125 MHz, which is derived from the ADC data clock. The data and clock
inputs from the ADC to the FPGA are Low Voltage Complementary Metal Oxide
Semiconductor (LVCMOS) signals.

The base firmware version of the Stem-Lab 125-14 provides a burst mode version
of the FFTS. However, we required capability to perform this operation in
streaming mode, and average the spectra on the FPGA itself. This allows us to
keep data rates below 30 MBps, beyond which loss-less data transfer via
ethernet becomes difficult, due to bottlenecks in communication between the PL
and the PS sides of the SoC. The block diagram of our FFTS implementation is
shown in Fig.2.

The analog signal is sampled by the on-board ADC at 125 MHz. The digitized
data is captured synchronously on the FPGA, converted to 2’s complement format
and passed on to the spectrometer block.

The spectrometer block is implemented using Simulink System Generator. The
signed data obtained so far is polyphase filtered using an 8 tap FIR
filter[17]. The output of the filterbank is an 18 bit fixed-point number, with
a binary point at the 17th bit. This is input to the biplex FFT block IP core
[18], available from the CASPER signal processing library [19, 20]. A 4096
points FFT is performed resulting in a spectral resolution of $\approx$ 30.5
kHz. To avoid overflow during subsequent stages of the FFT, the output of each
stage is scaled by a factor 2. As the data to the FFT block is real-valued,
the power spectrum is estimated by computing the magnitude of the positive
half of the spectrum. After the integration of a programmable number of
spectra, the dv signal is asserted high and the power spectrum is presented to
the subsequent blocks for transmission to the DAQ. For example, if 1000 such
spectra are summed, the resulting integration time is $\approx$ 32 ms.

The output spectra is recast as an Advanced eXtensible Interface (AXI) stream
and is written to a Block RAM (BRAM) through the AXIS BRAM WRITER IP block
[21]. Once the spectrum is written into the BRAM, a finished signal indicating
this is asserted and posted to the sts register.

The FPGA present on-board provides access only to the PS Ethernet, therefore
the transfer of data to the DAQ is mediated by executing a C code on the PS.
The sts, cfg and AXI BRAM READER provide memory mapped access to the PL. cfg
register is used to provide a master reset and configuration information to
the PL. sts register provides information on the BRAM address pointer and
holds the state of the finished signal. On the assertion of the finished
signal, the PS starts reading the contents of the BRAM. The BRAM data is
packetized as UDP packets with packet and spectrum count information and is
transmitted to the DAQ using linux socket functions.

We characterize the developed spectrometer using continuous wave (CW) and SN
signals. For the CW tests, signals at different frequencies and different
powers are fed to the system and recorded. These tests are carried out to
estimate the SNR of the spectrometer at different frequencies and to identify
the linear regime. The power measured by the FFTS is found to linearly vary
with the input power.

###  III-B Real Time Data Recorder (RTDR)

TABLE I: The different options available in the RTDR firmware. Option | Description
---|---
baseband | Records signal in dc - 625 kHz base-band; LO is disabled
IF | Records signal in an IF band spanning from $f_{lo}$ to $f_{lo}+625$ kHz
triggered | Records a signal burst for a programmable predefined time on rising edge of trigger pulse
continuous | Records data continuously

The FFTS discussed thus far, provides a simple and compact measurement option
for SNS. In scenarios where the SN signal is short lived in time, a real time
triggered data acquisition and processing protocol is required.

We developed a low bandwidth, raw voltage recorder. In Fig.3, we show a top-
level block description of the same. Table I outlines the various options
available in the data recorder. Here, we use both the input channels, where
one is from the BPD and the other is a signal generator (BK PRECISION Model
no. 4040B), which is used as the local oscillator (LO). As in the FFTS
firmware, the signals from the ADC are converted to 2’s complement and
multiplied to obtain an intermediate frequency (IF) signal. The IF signal is
usually close to dc and tracking the variations in the former using the master
clock operating at 125 MHz would result in sub-optimal usage of resources.
Hence, we use a cascaded integrated comb (CIC) filter for down-sampling the
signal[22, 23]. When the factor by which the signal has to be down-sampled is
large, CIC filters, used as a front end for FIR filters result in decreased
number of filter taps required for anti-aliasing.

The response of the CIC filter is not uniform within the band of interest. To
compensate for this, a FIR filter with a complementary response is cascaded to
it. The FIR filter is implemented as a half-band filter, resulting in an
output data rate that is half of that of the input. A 256 tap Kaiser window,
with $\beta=10$, is used to shape the filter pass band. When both the filters
are cascaded, the resulting combination has an uniform amplitude response over
the band of operation and an out of band rejection $\geq$ 90 dB.

Figure 3: Firmware description of the triggered raw voltage recorder. The
dotted blue lines represent the flow of control signals and status signals to
and from the memory mapped AXI registers. The green blocks represent signal
processing elements of the design.

The data from the multiplier as it enters the CIC filter is at 125 MSPS. This
is decimated by 50, resulting in a data rate of 2.5 MSPS at the output of the
CIC filter and an aggregate data rate of 1.25 MSPS after the FIR filter,
resulting in a base-band data of 625 kHz.

We use four counters in the firmware for timekeeping purposes: a trigger
counter (TC), a free running counter (FRC) at 125 MHz rate, a over- flow
counter (OC) and a packet counter (PC). The TC keeps track of the triggers
received by the RTDR, while the FRC keeps track of the time since the power
on. The 32 bit OC counts the number of FRC overflows. These counters allow us
to obtain the time between two trigger events and also its occurance instances
since the start of acquisition. The PC helps us to ensure no data was missed
during packet framing and transmission.

Figure 4: Characterization of the DRS (in both RTDR and FFTS modes) with
signals of various power fed across the frequency range of operation. The ’+’
markers indicate the data and the solid lines represent the first order
polynomial fits. The bottom plot shows the residuals. The black dashed lines
encompass the linear range of the DRS.

The data, along with the counter values, is written to a BRAM and read out by
the PS and transmitted using ethernet.

CW signals at various frequencies within the 0-62.5 MHz band were injected at
different power levels from -90 to 10 dBm. As shown in Fig.4, the output power
scales linearly with the input power, irrespective of the frequency.

##  IV Electromagnetic Interference (EMI) and its mitigation

The ambient EMI can hinder the detection of weak signals. Some common sources
of unavoidable EMI are 50/60 Hz AC lines, switching regulators and LO
harmonics. Strong EMI affects the dynamic range of the receiver system at
frequencies $<$ 1 MHz. EMI can be mitigated during pre- and post-processing
stages. We describe the techniques adopted, during these stages of EMI
mitigation, in this section.

During our measurements, we found that the strong 50 Hz AC signal and its
harmonics were getting coupled into the system through the AC adapter of the
FPGA evaluation board. To overcome this, we replaced the power adapter with a
commercially available battery-bank of similar specifications.

Figure 5: The effect of EMI on the measured spectrum (green trace) and after
its mitigation (blue trace). The effect of EMI mitigation is clearly visible
through the absence of spikes and a reduction in the noise floor by $\approx$
20 dB in the blue trace.

A second prominent source of EMI were the switching circuits associated with
the magnetic coil current driver. This gave rise to strong peaks in the low
kHz frequencies. The SNS setup is located in a laboratory environment with
multiple sources of EMI. Therefore, we custom designed a mild-steel enclosure
for the DRS. In Fig.5 we show the significant mitigation of EMI by $\approx$
20 dB after adopting the aforementioned schemes. We performed further
processing on the archived data to excise low-level narrow-band and impulsive
broad-band EMI [24],[25]. For EMI that was stationary in frequency, we used a
combination of background and median subtraction. Impulsive, broadband EMI,
when present, was clipped from each channel when its value exceeded the
3$\sigma$ threshold.

##  V Results and Discussion

Our experimental set-up is shown schematically in Fig.1. A uniform transverse
magnetic field ($B_{\bot}$) was produced by the current flowing through a pair
of magnetic coils in Helmholtz configuration. The signal from the BPD333The
BPD used was a Newport 1807 model having a cut-off frequency of $\approx$ 80
MHz. was recorded either with our digital receiver system (DRS) or a
commercial spectrum analyzer (SFSA) for the purpose of performance comparison.

In Fig.6, we show the SN spectrum of Rb atomic vapor with $B_{\bot}$ $\approx$
5.1 gauss obtained using SFSA (top panel) and the FFTS (bottom panel). These
sets of data were recorded under similar experimental conditions. In each of
the panels, we observe two peaks appearing at $\approx$ 2.4 MHz and $\approx$
3.6 MHz corresponding to SN signal due to 85Rb and 87Rb, respectively. The RBW
of the SFSA was the same as the channel width of the FFTS which is 30 kHz. The
integration time for the SFSA to obtain the data presented in Fig.6 (top
panel) was 45 seconds whereas the same for the FFTS presented in Fig.6 (bottom
panel) was 10 seconds. In both the cases the spectrum was background
subtracted and normalized to it’s peak value. The SNR of the SN signal is
defined as the ratio of the strength of the strongest signal to the rms value
of the background signal. The background signal is obtained by recording the
SN signal at zero magnetic field. Comparing these two spectra, we note that
the SNR for the DRS is more than an order of magnitude better than that for
the SFSA for the same integration time. This improvement in the SNR along with
the fact that our DRS is light weight, portable, low-cost, low-power ($<$ 10
Watts) as compared to a commercial SFSA, makes it preferable for both
laboratory and field measurements.

Figure 6: Spin noise (SN) spectrum acquired from SFSA (top panel) and FFTS
(bottom panel). Note that the SNR is $\approx$ 50 for the FFTS for an
integration time 5x lower than SFSA. These two spectra were recorded under the
same experimental conditions, at $B_{\bot}\approx 5.1$ G.

Since the SN spectrum peak position is the Larmour frequency
$\nu_{L}(=g_{F}\mu_{B}B_{\bot}/h$, where $g_{F}$ is the g-factor of the
hyperfine levels, $\mu_{B}$ is the Bohr magneton, and $h$ is the Plank’s
constant$)$, by precisely measuring the peak position of the spectrum we can
estimate the strength of $B_{\bot}$. Therefore, this measurement technique can
be used as a precision magnetometry tool. Since the developed DRS described in
this article is easily field deployable, we would highlight the applicability
of this device to construct a robust miniaturized magnetometer. As an example,
the SN spectrum recorded at various magnetic field strengths are shown Fig.7.
By fitting a Lorentzian to the individual spectrum, we can estimate the
Larmour frequency and in turn the magnetic field.

Figure 7: Background subtracted spin noise (SN) spectra measured for different
magnetic field values. The error in the measurements are $\approx$ 0.02 G,
which is limited by the spectral resolution $\approx$ 30 kHz of the FFTS.

Another advantage of using this DRS is that the device is reconfigurable which
enables triggered real time measurements of the SN spectra. In Fig.8, we show
an example of real time data acquisition. In each of the four panels we show
SN spectrum obtained using our DRS with an integration time of as low as 100
ms after the TTL trigger pulse. The magnetic field strength was also changed
after each trigger pulse. Hence, we can sample the magnetic field with a time
resolution of 100 msec. The integration time of the data shown in Fig.8 was
100 ms and the corresponding spectral resolution was $\approx$ 610 Hz
resulting in an SNR $\approx$ 5\. Whereas for the data shown in Fig.7 the
integration time was 1 s and the spectral resolution was 30 kHz, hence an SNR
$\approx$ 15.

For testing and verification of the triggered mode acquisition, we generated
an external trigger TTL pulse from a function generator and fed this into a
SMA-to-GPIO board, which was connected to the Red-Pitaya using a ribbon cable
and connector assembly. The acquisition time was set to $\approx$ 100 ms. The
trigger rate was $\approx$ 10 per minute.

The output from the BPD was fed to the DRS. The triggered acquisition works as
follows: (a) The FPGA waits for a rising edge on the trigger input port, (b)
On the rising edge of the trigger input, a dv signal is asserted, upon which
the BRAM starts to store the data, (c) Once 256 such samples are written, the
data is packetized and transferred to the DAQ server. The trigger count and an
absolute time stamp is included in the header data of the Ethernet packet for
assisting in data analysis.

Figure 8: Spin noise (SN) spectrum at various magnetic field strengths from
the RTDR. These series of spectra has an SNR $\approx$ 5, and the peak
positions can be determined within an accuracy of $\approx$ 5$\%$.

Another application of using our device is the temporal and spatial
correlation measurements in a system with laser cooled atoms and ions.
Intrinsically the correlation signals are expected to be extremely narrow in
frequency domain and are promising candidates for various quantum technology
applications. However, the measurement duration is typically limited to few
milliseconds. Therefore, it is difficult to perform the SNS in cold atoms
using traditional SFSA and our DRS emerges as a promising candidate for this
purpose. In fact, we have already started using the DRS described in this
article in cold atom measurements. Also, since in the RTDR configuration, the
DRS records the voltage samples directly, it gives us the flexibility to
achieve high frequency resolutions, which is only limited by the timing jitter
of the on-board clock.

Figure 9: Waterfall plot of the DRS output when the magnetic field was swept
from $\sim$ 3.35 G to 3.85 G. We have synchronously varied the coherent drive
frequency (see text). The yellow patches represent the spin noise signal in
the time-frequency plane, the green dashed line is a fit to the centroids of
the yellow patches and its slope indicates the rate of variation of the
magnetic field in time.

Our DRS in the RTDR mode, allows for a minimum time resolution of 800 ns when
the data is treated in its raw form. When spectral analysis is carried out
with a N point fourier transform, the channel width and the time resolution is
$\frac{1.25MHz}{N}$ and $800ns\times N$, respectively. Therefore, when the
signal is inherently strong, higher time resolutions can be obtained at the
cost of reduced frequency resolution. However, for intrinsic SN signal from
atomic vapor presented in this article, the SNR deteriorates for shorter
integration times which affects the precision of the measurements. Therefore,
for the purpose of demonstration, we integrate the SN signal for 100 ms time
window and obtain a precision in measured magnetic field of the order of 800
$\mu$G.

To demonstrate the response of the detection system to fast varying magnetic
fields, we conducted an experiment [26], where we added a coherent drive [27]
using a pair of Raman beams, which enhanced the SN signal strength million
fold, improving the SNR. We then varied the magnetic field from 3.35 G to 3.85
G, synchronously with the coherent drive field frequency, and recorded the
signal using our DRS. The results from this experiment is shown in Fig. 9,
where each pixel along time and frequency axis is $\approx$ 800 $\mu s$ and
610 Hz, respectively. We see that the DRS simultaneously tracks the “step”
changes in the magnetic field, as well as its drifts in milli-second
timescales.

We highlight that a time resolved measurement of magnetic field has
applications ranging from geophysics [28] to physiology[29, 30]. The time
stamp contained in the received data (see Section III B), can be used to
determine the absolute time variation of the magnetic field strength.

##  VI Conclusions

We presented the development of a software defined digital receiver system
(DRS) with two operating modes for spin noise spectroscopy (SNS) experiments.
The highlight of this work is that we show the applicability of SNS in
precision magnetometry and measure fast temporal variations in the magnetic
field. This receiver is fully re-configurable and had a short development time
at a low cost.

The FFTS mode allows for user programmable integration times. The RTDR mode
was specifically developed for high spectral and temporal resolution studies
of spin noise (SN) signals from both hot and cold atoms. Such a mode, where
high time resolution voltage data can be recorded, does not exist in SFSA. The
FFTS can also be used to supplement the RTDR mode as follows: using the FFTS,
the user will be able to explore the spectrum over a wide range of frequencies
and once the peak location is known, the RTDR can be used to obtain a time-
frequency resolved picture of the signal of interest.

Future directions for the system development include using a DDS core which
can internally generate the LO signals to facilitate a two channel
implementation, replace the DAQ computer with a ARM based microprocessor
system viz. Raspberry Pi for system miniaturization.

The receiver described here was developed to be a part of a compact, portable
SNS magnetometer. While we have demonstrated the application of the digital
receiver using the SNS experiments, it can be easily adapted for use in other
experiments requiring correlation measurements.

## Acknowledgments

The authors acknowledge partial support provided by Ministry of Electronics
and Information Technology (MeitY), Govt. of India under grant for “Center for
Excellence in Quantum Technology” with Ref. No. 4(7)/2020-ITEA. Authors also
acknowledge Department of Science and Technology (DST), Govt. of India, Prof.
Hema Ramachandran, Prof. Dibyendu Roy, Electronics Engineering Group (EEG) and
central workshop facility, Raman Research Institute. We extend our thanks to
the CASPER signal processing community and Dr. Pavel Demin for maintaining
many useful open-source IP cores. We are grateful to Xilinx for donation of
Vivado Design Suite, through their University Program.

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# On the structure factor of jammed particle configurations on the one-
dimensional lattice

Jean-Marc Luck  Université Paris-Saclay, CNRS, CEA, Institut de Physique
Théorique, 91191 Gif-sur-Yvette, France<EMAIL_ADDRESS>
###### Abstract

A broad class of blocked or jammed configurations of particles on the one-
dimensional lattice can be characterized in terms of local rules involving
only the lengths of clusters of particles (occupied sites) and of holes (empty
sites). Examples of physical relevance include the metastable states reached
by kinetically constrained spin chains, the attractors of totally irreversible
processes such as random sequential adsorption, and arrays of Rydberg atoms in
the blockade regime. The configurational entropy of ensembles of such blocked
configurations has been investigated recently by means of an approach inspired
from the theory of stochastic renewal processes. This approach provides a
valuable alternative to the more traditional transfer-matrix formalism. We
show that the renewal approach is also an efficient tool to investigate a
range of observables in uniform ensembles of blocked configurations, besides
their configurational entropy. The main emphasis is on their structure factor
and correlation function.

##  1 Introduction

Blocked or jammed configurations are met in many guises in the statistical
physics of complex systems, including first and foremost glassy and granular
materials (see [1, 2] for reviews). In this work we focus our attention onto
blocked configurations of particles on the one-dimensional lattice. Various
situations of physical significance involving such arrays of particles are
recalled below.

In the one-dimensional geometry, metastability can only occur exactly at zero
temperature. Moreover, metastable states typically consist of a single
configuration. For instance, a variety of kinetic spin models are known to
have an exponentially large number of such metastable configurations, each of
them being an attractor of the underlying zero-temperature dynamics. This
class includes pristine models, such as the Ising chain with Kawasaki dynamics
[3, 4], disordered models, such as the Ising spin glass [5, 6], and a breadth
of kinetically constrained spin models [7, 8, 9, 10, 11, 12, 13, 14, 15].
Several lattice gas models share the same phenomenology [16, 17, 18, 19, 20].
The zero-temperature dynamics of the above models is irreversible and strongly
non-ergodic. Some of these kinetic models can be exactly mapped onto the
dynamics of the deposition of hard objects on the one-dimensional lattice.
Prototypical examples are RSA (random sequential adsorption) or CSA
(cooperative sequential adsorption), where particles or clusters are deposited
irreversibly and sequentially on an initially empty substrate [21, 22, 23].
The process stops when no further object can be inserted into the system,
which is left in a blocked or jammed configuration with a non-trivial density
or coverage. Another more recent motivation to investigate blocked
configurations of particles defined by local constraints comes from an
entirely different area, namely the physics of ultracold atoms. Trapped
Rydberg atoms appear as a promising benchmark for what concerns quantum
computation, simulation and information processing [24]. Their large size and
strong interactions may give rise to a blockade, preventing the excitation of
Rydberg atoms in the vicinity of an already existing one [25, 26, 27, 28, 29,
30]. In the simple setting of a one-dimensional optical lattice, each site
occupied by a Rydberg atom must have at least $b$ empty sites on either side,
where $b\geq 1$ is referred to as the blockade range [31, 32]. To close this
panorama with an example outside the realm of physics, we mention the Riviera
model, where houses are sequentially built on an infinite array of pre-drawn
plots along a beach, with the constraint that every house should enjoy the
sunlight from at least one of the side directions. New houses are successively
introduced until a blocked configuration is reached [33, 34, 35, 36].

A prominent quantity characterizing statistical ensembles of blocked
configurations is their configurational entropy $S_{\star}$, describing the
exponential growth of the number of blocked configurations,

${\cal N}_{N}\sim\exp(NS_{\star}),$ (1.1)

with the system size $N$. In many situations, the configurational entropy has
been determined either by combinatorial means [10, 32, 37, 38, 39, 40, 41] or
by the transfer-matrix approach [13, 15]. We have proposed a novel method to
evaluate $S_{\star}$ in a recent paper in collaboration with Krapivsky [42].
This approach, inspired from the theory of renewal processes, works whenever
the rules defining blocked configurations involve only the lengths of clusters
of particles and holes. Within this scope, which embraces most situations of
physical significance, the renewal approach is more systematic and easier to
implement than the more traditional transfer-matrix approach.

The goal of the present work is to show that the same approach can be
efficiently used to investigate a range of observables, besides the
configurational entropy. of uniform ensembles of blocked configurations. The
setup of this paper is as follows. Section 2 is devoted to generalities and to
simple observables. We first recall the definition of the statistical
ensembles introduced in [42] (section 2.1) and the determination of their
configurational entropy by the renewal approach (section 2.2). In section 2.3
we use the same approach to investigate simple local observables, such as
boundary probabilities and the length distributions of clusters of particles
and holes. Section 3 is devoted to the main subject of this work, namely the
structure factor and the correlation function of uniform ensembles. After
recalling some definitions (section 3.1), we successively evaluate Fourier
amplitudes (section 3.2), structure factors (section 3.3), and correlation
functions (section 3.4), describing in some detail the general properties of
these quantities. The next sections concern examples of ensembles of blocked
configurations. Explicit results are derived in section 4 for the three
simplest statistical ensembles already considered in [42]. We then consider
the ensemble of attractors of the $k$-mer deposition model (section 5) and of
arrays of Rydberg atoms with blockade range $b$ (section 6). Single
configurations pertaining to both models can be mapped onto each other, with
the correspondence $k=b+1$. A brief discussion is given in section 7. An
appendix is devoted to the Hendricks-Teller model [43], a simple example of a
random structural model whose structure factor can be studied by means of the
renewal approach.

##  2 Model and simple observables

###  2.1 The model

Let us first recall the statistical model of blocked configurations introduced
in [42]. Configurations are semi-infinite sequences of particles (or occupied
sites, noted ${\bullet}$) and of holes (or empty sites, noted ${\circ}$). Such
a configuration ${\cal C}$ is described by the lengths
$i_{1},j_{1},i_{2},j_{2},\dots$ of the successive clusters of particles and
holes. It is therefore either of the form

${\cal
C}_{I}=\underbrace{{\bullet}\cdots{\bullet}}_{i_{1}}\underbrace{{\circ}\cdots{\circ}}_{j_{1}}\underbrace{{\bullet}\cdots{\bullet}}_{i_{2}}\underbrace{{\circ}\cdots{\circ}}_{j_{2}}\cdots$
(2.1)

if the first site is occupied, or of the form

${\cal
C}_{J}=\underbrace{{\circ}\cdots{\circ}}_{j_{1}}\underbrace{{\bullet}\cdots{\bullet}}_{i_{1}}\underbrace{{\circ}\cdots{\circ}}_{j_{2}}\underbrace{{\bullet}\cdots{\bullet}}_{i_{2}}\cdots$
(2.2)

if the first site is empty.

Blocked configurations are defined by the following local rules:

  * •

All lengths $i_{1},i_{2},\dots$ of particle clusters belong to some set ${\cal
I}$ of positive integers.

  * •

All lengths $j_{1},j_{2},\dots$ of hole clusters belong to some set ${\cal J}$
of positive integers.

In the above, ${\cal I}$ and ${\cal J}$ are two prescribed finite or infinite
subsets of the positive integers, which entirely define the configurational
model. Hereafter we always deal with the uniform statistical ensemble obtained
by attributing equal weights to all blocked configurations ${\cal C}$
constructed as above. We assume that the model is non-trivial, namely that
both sets ${\cal I}$ and ${\cal J}$ are non-empty and at least one of them
contains more than one integer.

###  2.2 Configurational entropy

To set the stage, we begin by recalling the analysis of the configurational
entropy. This quantity, denoted by $S_{\star}$, describes the exponential
growth (see (1.1))

${\cal N}_{N}\sim\exp(NS_{\star})$ (2.3)

of the number ${\cal N}_{N}$ of configurations ${\cal C}_{N}$ on a finite
system of length $N$. No overhang is allowed in the construction, so that the
rightmost cluster ends exactly at site $N$. We also introduce the numbers
${\cal N}_{N,I}$ and ${\cal N}_{N,J}$ of finite configurations ${\cal
C}_{N,I}$ and ${\cal C}_{N,J}$ beginning with a cluster of particles or of
holes, in correspondence with (2.1) and (2.2).

The key observation made in [42] is the existence of a formal analogy between
the present setting and the theory of stochastic renewal processes [44, 45,
46] (see [47, 48] for presentations in the physics literature). This analogy
suggests to introduce the generating series of the numbers of configurations

$\displaystyle{\cal N}_{I}(z)=\sum_{N\geq 1}{\cal N}_{N,I}z^{N},\qquad{\cal
N}_{J}(z)=\sum_{N\geq 1}{\cal N}_{N,J}z^{N},$ $\displaystyle{\cal
N}(z)=\sum_{N\geq 0}{\cal N}_{N}z^{N}.$ (2.4)

We have

${\cal N}(z)=1+{\cal N}_{I}(z)+{\cal N}_{J}(z),$ (2.5)

where the initial term ${\cal N}_{0}=1$ is conventional. The series ${\cal
N}_{I}(z)$ and $N_{J}(z)$ obey linear renewal equations of the form

${\cal N}_{I}(z)=I(z)(1+{\cal N}_{J}(z)),\qquad{\cal N}_{J}(z)=J(z)(1+{\cal
N}_{I}(z)),$ (2.6)

where

$I(z)=\sum_{i\in{\cal I}}z^{i},\qquad J(z)=\sum_{j\in{\cal J}}z^{j}$ (2.7)

are the series associated with the sets ${\cal I}$ and ${\cal J}$ defining the
statistical ensemble. Solving the linear equations (2.6), we obtain the
results

${\cal N}_{I}(z)=\frac{I(z)(1+J(z))}{1-I(z)J(z)},\qquad{\cal
N}_{J}(z)=\frac{J(z)(1+I(z))}{1-I(z)J(z)}$ (2.8)

and

${\cal N}(z)=\frac{(1+I(z))(1+J(z))}{1-I(z)J(z)}.$ (2.9)

As $z$ increases from 0 to 1, the product $I(z)J(z)$ increases from
$I(0)J(0)=0$ to $I(1)J(1)={\left|{\cal I}\right|}{\left|{\cal J}\right|}\geq
2$ (possibly infinite). There is therefore a single value $z_{\star}$ of $z$
in the range $0<z_{\star}<1$ such that the denominator of (2.9) vanishes,
i.e.,

$I(z_{\star})J(z_{\star})=1.$ (2.10)

The configurational entropy entering the exponential growth law (1.1) reads

$S_{\star}=-\ln z_{\star}.$ (2.11)

Throughout the following, along the lines of [42], we restrict this study to
the rational class of models where both generating series

$I(z)=\frac{A_{I}(z)}{B_{I}(z)},\qquad J(z)=\frac{A_{J}(z)}{B_{J}(z)}$ (2.12)

are rational functions, i.e., ratios of polynomials in $z$. This class
encompasses all examples considered below, and virtually all situations of
physical significance (see [42] for more details). The formula (2.9) then
takes the form

${\cal N}(z)=\frac{C(z)}{D(z)},$ (2.13)

where $C(z)$ and $D(z)$ are polynomials in $z$. The rational fraction (2.13)
is assumed to be irreducible. The degree of $D(z)$ is denoted by $\Delta$. Its
smallest zero $z_{\star}$ is real, positive, and simple. The degree $\Delta$
provides a natural measure of the complexity of configurational models in the
rational class.

###  2.3 Local observables

Our main goal is to show that the renewal method recalled above provides an
efficient tool to evaluate a good deal of observables in the uniform ensemble,
besides its configurational entropy. The case of the Hendricks-Teller model,
considered in A, serves as a warming up for what follows. The remainder of
this section is devoted to local observables such as boundary probabilities
and the length distribution of particle and hole clusters. Section 3 will be
about the main subject of this work, namely the structure factor and the
correlation function.

#### Boundary probabilities.

Our first observables are the boundary probabilities $W_{I}$ (resp. $W_{J}$)
that an infinitely long configuration ${\cal C}$ begins with a cluster of
particles (resp. a cluster of holes), i.e.,

$W_{I}=\lim_{N\to\infty}\frac{{\cal N}_{N,I}}{{\cal N}_{N}},\qquad
W_{J}=\lim_{N\to\infty}\frac{{\cal N}_{N,J}}{{\cal N}_{N}}.$ (2.14)

We have ${\cal N}_{N}={\cal N}_{N,I}+{\cal N}_{N,J}$ for $N\geq 1$, and so
$W_{I}+W_{J}=1$, as expected. All generating series entering (2.8) and (2.9)
have poles at $z=z_{\star}$, so that all numbers of configurations grow as
$\exp(NS_{\star})$. We have therefore

$W_{I}=\lim_{z\to z_{\star}}\frac{{\cal N}_{I}(z)}{{\cal N}(z)},\qquad
W_{J}=\lim_{z\to z_{\star}}\frac{{\cal N}_{J}(z)}{{\cal N}(z)}.$ (2.15)

The formulas (2.8) and (2.9) yield

$\displaystyle
W_{I}=\frac{I(z_{\star})}{1+I(z_{\star})}=\frac{1}{1+J(z_{\star})},$
$\displaystyle
W_{J}=\frac{J(z_{\star})}{1+J(z_{\star})}=\frac{1}{1+I(z_{\star})}.$ (2.16)

#### Distributions of cluster lengths and their moments.

We now turn to the length distributions of the clusters of particles and of
holes, i.e.,

$p_{I,k}={\mathop{\rm Prob}}(i=k),\qquad p_{J,k}={\mathop{\rm Prob}}(j=k),$
(2.17)

where $i$ (resp. $j$) stands for the length of any particle cluster (resp. any
hole cluster) in an infinitely long configuration. Consider particle clusters
for definiteness. All such clusters are interchangeable, so that we have

$p_{I,k}={\mathop{\rm Prob}}(i_{1}=k)=\lim_{N\to\infty}\frac{{\cal
N}_{N,I,k}}{{\cal N}_{N,I}},$ (2.18)

where ${\cal N}_{N,I,k}$ is the number of configurations of length $N$
beginning with a cluster of particles of length $i_{1}=k$. The corresponding
generating series reads

${\cal N}_{I,k}(z)=\sum_{N\geq 1}{\cal
N}_{N,I,k}z^{N}=z^{k}\,\chi_{I}(k)(1+{\cal N}_{J}(z)),$ (2.19)

where

$\chi_{I}(k)=\left\\{\begin{array}[]{ll}1&(k\in{\cal I}),\\\ 0&(k\notin{\cal
I})\end{array}\right.$ (2.20)

is the indicator function of the set ${\cal I}$. In line with (2.15), we have

$p_{I,k}=\lim_{z\to z_{\star}}\frac{{\cal N}_{I,k}(z)}{{\cal N}_{I}(z)}.$
(2.21)

We thus obtain

$p_{I,k}=\frac{z_{\star}^{k}}{I(z_{\star})}\,\chi_{I}(k),$ (2.22)

and similarly

$p_{J,k}=\frac{z_{\star}^{k}}{J(z_{\star})}\,\chi_{J}(k).$ (2.23)

In particular, the mean lengths of particle and hole clusters read

$\displaystyle{\langle i\rangle}=\sum_{k\geq 1}k\,p_{I,k}=\sum_{k\in{\cal
I}}\frac{kz_{\star}^{k}}{I(z_{\star})}=\frac{z_{\star}I^{\prime}(z_{\star})}{I(z_{\star})}=z_{\star}I^{\prime}(z_{\star})J(z_{\star}),$
$\displaystyle{\langle j\rangle}=z_{\star}I(z_{\star})J^{\prime}(z_{\star}).$
(2.24)

Here and throughout the following, ${\langle.\rangle}$ denotes an average over
the uniform ensemble of blocked configurations. The mean distance between
consecutive particle clusters (or, equivalently, between consecutive hole
clusters) is therefore

$\ell={\langle i\rangle}+{\langle
j\rangle}=z_{\star}(I^{\prime}(z_{\star})J(z_{\star})+I(z_{\star})J^{\prime}(z_{\star})).$
(2.25)

The particle density can be obtained as

$\rho=\frac{{\langle
i\rangle}}{\ell}=\frac{I^{\prime}(z_{\star})J(z_{\star})}{I^{\prime}(z_{\star})J(z_{\star})+I(z_{\star})J^{\prime}(z_{\star})}.$
(2.26)

An equivalent expression was derived by another route in [42]. Higher moments
of the cluster lengths can be determined along the same lines:

$\displaystyle{\langle i^{p}\rangle}=J(z_{\star})\Bigl{(}z\frac{{\rm d}}{{\rm
d}z}\Bigr{)}^{p}I(z)\Big{|}_{z=z_{\star}},$ $\displaystyle{\langle
j^{p}\rangle}=I(z_{\star})\Bigl{(}z\frac{{\rm d}}{{\rm
d}z}\Bigr{)}^{p}J(z)\Big{|}_{z=z_{\star}}.$ (2.27)

We have especially

$\displaystyle{\langle
i^{2}\rangle}=z_{\star}J(z_{\star})(I^{\prime}(z_{\star})+z_{\star}I^{\prime\prime}(z_{\star})),$
$\displaystyle{\langle
j^{2}\rangle}=z_{\star}I(z_{\star})(J^{\prime}(z_{\star})+z_{\star}J^{\prime\prime}(z_{\star})).$
(2.28)

##  3 General results on structure factor and correlation function

###  3.1 Definitions

Each configuration ${\cal C}$ can be alternatively described by the occupation
numbers

$\eta_{m}=\left\\{\begin{array}[]{ll}1&(\hbox{site $m$ is occupied}),\\\
0&(\hbox{site $m$ is empty}).\end{array}\right.$ (3.1)

The correlation functions of the model are ${\langle\eta_{m}\rangle}$,
${\langle\eta_{m}\eta_{l}\rangle}$, and so on, where averages are taken over
the uniform ensemble. Translation invariance and clustering of correlations
hold in the bulk of the system, i.e., far from its boundary. We have in
particular

${\langle\eta_{m}\rangle}\to\rho,\qquad{\langle\eta_{m}\eta_{l}\rangle}-{\langle\eta_{m}\rangle}{\langle\eta_{l}\rangle}\to
C_{m-l}$ (3.2)

as $m$ and $l$ are simultaneously large, the difference $m-l$ being kept
fixed. The particle density $\rho$ is given by (2.26). The connected pair
correlation function $C_{n}$ is an even function of $n$ which falls off to
zero as the distance ${\left|n\right|}$ between both sites gets large. It is
invariant under the exchange of particles and holes
($\eta_{m}\longleftrightarrow 1-\eta_{m}$).

The (random) Fourier amplitude $G_{N}(q)$ and intensity $S_{N}(q)$ associated
with the first $N$ sites read

$G_{N}(q)=\sum_{n=1}^{N}{\rm e}^{-{\rm i}mq}\eta_{m},\qquad
S_{N}(q)=\frac{{\left|G_{N}(q)\right|}^{2}}{N}.$ (3.3)

Our main interest is in the structure factor

$S(q)=\lim_{N\to\infty}{\langle S_{N}(q)\rangle}.$ (3.4)

This self-averaging quantity identifies with the Fourier transform of the
correlation function $C_{n}$:

$S(q)=\sum_{n=-\infty}^{\infty}C_{n}{\rm e}^{-{\rm i}nq}.$ (3.5)

###  3.2 Fourier amplitudes

In order to determine the Fourier amplitudes, it is advantageous to split the
definition (3.3), denoting by $G_{N,I}(q)$ (resp. $G_{N,J}(q)$) the Fourier
amplitude of a finite configuration of type ${\cal C}_{N,I}$ (resp. ${\cal
C}_{N,J}$). These amplitudes obey the renewal equations

$\displaystyle G_{N,I}(q)=a(i_{1},q)+{\rm e}^{-{\rm
i}qi_{1}}G_{N-i_{1},J}(q),$ $\displaystyle G_{N,J}(q)={\rm e}^{-{\rm
i}qj_{1}}G_{N-j_{1},I}(q),$ (3.6)

with

$a(i_{1},q)=\sum_{m=1}^{i_{1}}{\rm e}^{-{\rm i}mq}=\frac{{{\rm e}^{-{\rm
i}q}}}{1-{{\rm e}^{-{\rm i}q}}}\,\left(1-{\rm e}^{-{\rm i}qi_{1}}\right).$
(3.7)

In (3.6), the amplitude denoted by $G_{N-i_{1},J}(q)$ is associated with the
$N-i_{1}$ atoms of the configuration ${\cal C}_{I}$ numbered from $i_{1}+1$ to
$N$. It is a probabilistic copy of the amplitude $G_{N-i_{1},J}(q)$.
Similarly, the amplitude denoted by $G_{N-j_{1},I}(q)$ is associated with the
$N-j_{1}$ atoms of the configuration ${\cal C}_{J}$ numbered from $j_{1}+1$ to
$N$. It is a probabilistic copy of the amplitude $G_{N-j_{1},I}(q)$.

Let us introduce the generating series

$\displaystyle\Gamma_{I}(z,q)=\sum_{N\geq 1}\sum_{{\cal
C}_{N,I}}G_{N,I}(q)z^{N},$ $\displaystyle\Gamma_{J}(z,q)=\sum_{N\geq
1}\sum_{{\cal C}_{N,J}}G_{N,J}(q)z^{N},$
$\displaystyle\Gamma(z,q)=\Gamma_{I}(z,q)+\Gamma_{J}(z,q).$ (3.8)

The renewal equations (3.6) entail that these quantities obey

$\displaystyle\Gamma_{I}(z,q)=\gamma(z,q)+I(z{{\rm e}^{-{\rm
i}q}})\Gamma_{J}(z,q),$ $\displaystyle\Gamma_{J}(z,q)=J(z{{\rm e}^{-{\rm
i}q}})\Gamma_{I}(z,q),$ (3.9)

with

$\gamma(z,q)=\frac{{{\rm e}^{-{\rm i}q}}}{1-{{\rm e}^{-{\rm
i}q}}}\left(I(z)-I(z{{\rm e}^{-{\rm i}q}})\right)\left(1+{\cal
N}_{J}(z)\right).$ (3.10)

Solving the linear equations (3.9), we obtain

$\displaystyle\Gamma_{I}(z,q)=\frac{\gamma(z,q)}{1-I(z{{\rm e}^{-{\rm
i}q}})J(z{{\rm e}^{-{\rm i}q}})},$
$\displaystyle\Gamma_{J}(z,q)=\frac{J(z{{\rm e}^{-{\rm
i}q}})\gamma(z,q)}{1-I(z{{\rm e}^{-{\rm i}q}})J(z{{\rm e}^{-{\rm i}q}})},$
$\displaystyle\Gamma(z,q)=\frac{(1+J(z{{\rm e}^{-{\rm
i}q}}))\gamma(z,q)}{1-I(z{{\rm e}^{-{\rm i}q}})J(z{{\rm e}^{-{\rm i}q}})}.$
(3.11)

In the limit of an infinitely long sample, in line with (2.15), the mean
Fourier amplitude is given by

${\langle G(q)\rangle}=\lim_{z\to z_{\star}}\frac{\Gamma(z,q)}{{\cal N}(z)}.$
(3.12)

Using (2.9), (3.10) and (3.11), we are left with

${\langle G(q)\rangle}=\frac{{{\rm e}^{-{\rm i}q}}}{1-{{\rm e}^{-{\rm
i}q}}}\,\frac{1+J(z_{\star}{{\rm e}^{-{\rm
i}q}})}{1+I(z_{\star})}\,\frac{I(z_{\star})-I(z_{\star}{{\rm e}^{-{\rm
i}q}})}{1-I(z_{\star}{{\rm e}^{-{\rm i}q}})J(z_{\star}{{\rm e}^{-{\rm
i}q}})}.$ (3.13)

This result diverges at small wavevectors as

${\langle G(q)\rangle}\approx\frac{\rho}{{\rm i}q}\qquad(q\to 0),$ (3.14)

where the particle density $\rho$ is given by (2.26).

###  3.3 Structure factor

The structure factor $S(q)$ encodes bulk properties of the model. Boundary
conditions are therefore irrelevant, so that it is sufficient to consider
configurations of type ${\cal C}_{I}$. The simplest route to evaluate $S(q)$
consists in eliminating amplitudes of the form $G_{N,J}(q)$ from the renewal
equations (3.6), obtaining thus

$G_{N,I}(q)=a(i_{1},q)+{\rm e}^{-{\rm
i}q(i_{1}+j_{1})}G_{N-i_{1}-j_{1},I}(q),$ (3.15)

with the same convention as in (3.6), and so

$\displaystyle G_{N,I}(q){\overline{G}}_{N,I}(q)$
$\displaystyle=a(i_{1},q){\overline{a}}(i_{1},q)$ (3.16)
$\displaystyle+a(i_{1},q){\rm e}^{{\rm
i}q(i_{1}+j_{1})}{\overline{G}}_{N-i_{1}-j_{1},I}(q)$
$\displaystyle+{\overline{a}}(i_{1},q){\rm e}^{-{\rm
i}q(i_{1}+j_{1})}G_{N-i_{1}-j_{1},I}(q)$
$\displaystyle+G_{N-i_{1}-j_{1},I}(q){\overline{G}}_{N-i_{1}-j_{1},I}(q).$

Here and throughout the following, a bar denotes complex conjugation.

Let us introduce the generating series

$\Sigma(z,q)=\sum_{N\geq 1}\sum_{{\cal
C}_{N,I}}G_{N,I}(q){\overline{G}}_{N,I}(q)z^{N}.$ (3.17)

This quantity is given by the sum of four terms, in correspondence with the
four lines of the right-hand side of (3.16), namely

$\displaystyle\Sigma_{1}(z,q)=\frac{2I(z)-I(z{{\rm e}^{{\rm i}q}})-I(z{{\rm
e}^{-{\rm i}q}})}{2(1-\cos q)}\,\left(1+{\cal N}_{J}(z)\right),$
$\displaystyle\Sigma_{2}(z,q)=\frac{{{\rm e}^{-{\rm i}q}}}{1-{{\rm e}^{-{\rm
i}q}}}(I(z{{\rm e}^{{\rm i}q}})-I(z))J(z{{\rm e}^{{\rm
i}q}}){\overline{\Gamma}}_{I}(z,q),$ $\displaystyle\Sigma_{3}(z,q)=\frac{{{\rm
e}^{{\rm i}q}}}{1-{{\rm e}^{{\rm i}q}}}(I(z{{\rm e}^{-{\rm
i}q}})-I(z))J(z{{\rm e}^{-{\rm i}q}})\Gamma_{I}(z,q),$
$\displaystyle\Sigma_{4}(z,q)=I(z)J(z)\Sigma(z,q).$ (3.18)

We have therefore

$\Sigma(z,q)=\frac{\Sigma_{1}(z,q)+\Sigma_{2}(z,q)+\Sigma_{3}(z,q)}{1-I(z)J(z)}.$
(3.19)

In the limit of an infinitely long sample, in line with (2.15), the structure
factor is given by

$S(q)=\lim_{z\to z_{\star}}\frac{\Sigma(z,q)}{z{\cal N}_{I}^{\prime}(z)},$
(3.20)

where the denominator

$z{\cal N}_{I}^{\prime}(z)=\sum_{N\geq 1}N{\cal N}_{N,I}z^{N}$ (3.21)

takes care of the factor $N$ in the definition (3.3) of $S_{N}(q)$. This
quantity has a double pole at $z=z_{\star}$, of the form

$z{\cal
N}_{I}^{\prime}(z)\approx\frac{\ell\,I(z_{\star})(1+J(z_{\star}))}{(1-I(z)J(z))^{2}},$
(3.22)

where the length $\ell$ is given by (2.25).

Some algebra using (3.18), (3.19), (3.22) leads us to the following expression
for the structure factor:

$S(q)=\frac{\Phi({{\rm e}^{{\rm i}q}})+\Phi({{\rm e}^{-{\rm
i}q}})}{2\ell\,(1-\cos q)},$ (3.23)

with

$\Phi({{\rm e}^{{\rm i}q}})=\frac{(I(z_{\star})-I(z_{\star}{{\rm e}^{{\rm
i}q}}))(J(z_{\star})-J(z_{\star}{{\rm e}^{{\rm i}q}}))}{1-I(z_{\star}{{\rm
e}^{{\rm i}q}})J(z_{\star}{{\rm e}^{{\rm i}q}})}.$ (3.24)

The formula (3.23) is the key result of this work. First of all, this
expression is invariant under the interchange of particles and holes, as it
should be. For all models in the rational class under consideration, the
structure factor $S(q)$ can be reduced to a rational function of $\cos q$.
Numerous explicit examples will be given hereafter (see (4.4), (4.10), (4.15),
(5.7), (5.11), (6.6), (6.10)). Its denominator is a polynomial in $\cos q$
whose degree is generically $\Delta-1$, where $\Delta$ is the degree of the
polynomial $D(z)$ introduced in (2.13). The structure factor has poles at the
complex values $\pm q_{m}$ of the wavevector, such that

${\rm e}^{{\rm i}q_{m}}=\frac{z_{\star}}{z_{m}},$ (3.25)

where the $z_{m}$ are the zeros of $D(z)$, except the smallest one
$z_{\star}$. There are $\Delta-1$ such zeros, counted with their
multiplicities. The $z_{m}$ are either real, or they occur in complex
conjugate pairs. They are assumed to be ordered such that

$z_{\star}<{\left|z_{1}\right|}\leq{\left|z_{2}\right|}\leq\dots,\qquad
0<{\mathop{\rm Im}\,}q_{1}\leq{\mathop{\rm Im}\,}q_{2}\leq\dots$ (3.26)

The structure factor at zero wavevector,

$S(0)=\sum_{n=-\infty}^{\infty}C_{n},$ (3.27)

is nothing but the compressibility of the model. This quantity describes the
linear growth of the variance of the total particle number,

$M=\sum_{m=1}^{N}\eta_{m},$ (3.28)

in a finite system of size $N$:

${\mathop{\rm var}\,}M={\langle M^{2}\rangle}-{\langle M\rangle}^{2}\approx
S(0)\,N.$ (3.29)

Expanding (3.23) in $q$, and using (2.24) and (2.28), we obtain

$S(0)=\frac{\bigl{(}{\langle i^{2}\rangle}-{\langle
i\rangle}^{2}\bigr{)}{\langle j\rangle}^{2}+\bigl{(}{\langle
j^{2}\rangle}-{\langle j\rangle}^{2}\bigr{)}{\langle
i\rangle}^{2}}{\ell^{3}}.$ (3.30)

An equivalent formula was derived by another route in [42], where the
compressibility is denoted by $c_{2}$.

The structure factor at wavevector $\pi$,

$S(\pi)=\sum_{n=-\infty}^{\infty}(-1)^{n}C_{n},$ (3.31)

can be referred to as the staggered compressibility. It reads

$S(\pi)=\frac{(I(z_{\star})-I(-z_{\star}))(J(z_{\star})-J(-z_{\star}))}{2\ell\,(1-I(-z_{\star})J(-z_{\star}))}.$
(3.32)

The structure factor exhibits systematic extinctions in cases where the
lengths of all clusters of particles (or of holes) are multiples of some fixed
integer $K=2,3,\dots$ This phenomenon is to be expected. Consider for
definiteness the case of particle clusters. If all their lengths are multiples
of $K$, $I(z)$ is a rational function of $z^{K}$, so that the expression
(3.23) of $S(q)$ vanishes whenever ${\rm e}^{{\rm i}qK}=1$, but ${\rm e}^{{\rm
i}q}\neq 1$, i.e., for all wavevectors of the form

$q=\frac{2\pi m}{K}\qquad(m\neq 0\;\hbox{mod}\;K).$ (3.33)

In the particular case of a symmetric ensemble, where particles and holes play
similar roles, such that ${\cal I}={\cal J}$, $I(z_{\star})=J(z_{\star})=1$,
and $\rho=1/2$, (3.23) and (3.30) boil down to

$S(q)=\frac{1}{\ell(1-\cos q)}\,\frac{1-I(z_{\star}{{\rm e}^{{\rm
i}q}})I(z_{\star}{{\rm e}^{-{\rm i}q}})}{(1+I(z_{\star}{{\rm e}^{{\rm
i}q}}))(1+I(z_{\star}{{\rm e}^{-{\rm i}q}}))},$ (3.34) $S(0)=\frac{{\langle
i^{2}\rangle}-{\langle i\rangle}^{2}}{4{\langle i\rangle}}.$ (3.35)

The above expressions bear a striking resemblance with those of the Hendricks-
Teller model (see (1.11), (1.13)).

###  3.4 Correlation function

The correlation function is given by (see (3.5))

$C_{n}=\int_{0}^{2\pi}\frac{{\rm d}q}{2\pi}\,{\rm e}^{{\rm i}nq}S(q).$ (3.36)

Setting $y={\rm e}^{{\rm i}q}$ and using (3.23), this reads

$C_{n}=-\frac{1}{\ell}\oint\frac{{\rm d}y}{2\pi{\rm
i}}\,\frac{y^{n}}{(y-1)^{2}}\bigl{(}\Phi(y)+\Phi(1/y)\bigr{)}.$ (3.37)

The integration contour in (3.37) is the unit circle, along which the
integrand is regular. The differential element ${\rm d}y/(y-1)^{2}$ is
invariant if $y$ is changed into $1/y$. This ensures the expected symmetry
$C_{n}=C_{-n}$.

Henceforth we consider $n\geq 0$ for definiteness. The integral in (3.37)
receives contributions from the poles of $\Phi(1/y)$ at $y_{m}={\rm e}^{{\rm
i}q_{m}}=z_{\star}/z_{m}$ (see (3.25)).

Let us assume for a while that all the zeros $z_{m}$ are simple and that
$\Phi(\infty)$ is finite. In this situation, the correlation function is a
finite sum of decaying exponentials:

$C_{n}=\sum_{m=1}^{\Delta-1}A_{m}\Bigl{(}\frac{z_{\star}}{z_{m}}\Bigr{)}^{n}\qquad(n\geq
0),$ (3.38)

with amplitudes

$A_{m}=\frac{z_{\star}}{\ell}\,\frac{2-I(z_{\star})J(z_{m})-I(z_{m})J(z_{\star})}{(z_{\star}-z_{m})^{2}(I^{\prime}(z_{m})J(z_{m})+I(z_{m})J^{\prime}(z_{m}))}.$
(3.39)

The correlation therefore decays exponentially with distance, as

$C_{n}\sim{\rm e}^{-n/\xi},$ (3.40)

where the correlation length reads

$\xi=\frac{1}{\ln{\displaystyle{\displaystyle{\left|z_{1}\right|}\over\displaystyle
z_{\star}}}}=\frac{1}{{\mathop{\rm Im}\,}q_{1}}.$ (3.41)

Let us now relax the two assumptions above. If the zero $z_{m}$ is not simple,
but has multiplicity $m\geq 2$, the corresponding amplitude $A_{m}$ in (3.38)
is replaced by a polynomial in $n$ of degree at most $m-1$. If $\Phi(\infty)$
is not finite, then $\Phi(y)$ grows as $y^{\mu}$ at large $y$, where $\mu$ is
some positive integer. As a consequence, the formula (3.38) for $C_{n}$ might
be modified for the first few values of the distance $n$ ($n=0,\dots,\mu-1$).
In all cases, the correlation length is given by (3.41).

The correlation function admits an appealing alternative expression, which
holds in full generality. Consider the function $\Phi({{\rm e}^{{\rm i}q}})$
defined in (3.24). This function equals unity in the limit ${{\rm e}^{{\rm
i}q}}\to 0$, whereas it vanishes for $q\to 0$ as

$\Phi({{\rm e}^{{\rm i}q}})\approx-{\rm i}qR,$ (3.42)

with

$R=\frac{z_{\star}I^{\prime}(z_{\star})J^{\prime}(z_{\star})}{I^{\prime}(z_{\star})J(z_{\star})+I(z_{\star})J^{\prime}(z_{\star})}=\frac{{\langle
i\rangle}{\langle j\rangle}}{\ell}=\rho(1-\rho)\ell.$ (3.43)

Moreover, the function $\Phi({{\rm e}^{{\rm i}q}})$ is regular at least for
${\left|{{\rm e}^{{\rm i}q}}\right|}<1$. Its poles are indeed situated at
${\rm e}^{-{\rm i}q_{m}}$ (see (3.25)). It can therefore be expanded as

$\Phi({{\rm e}^{{\rm i}q}})=\sum_{a=1}^{\infty}r_{a}(1-{\rm e}^{{\rm i}aq}).$
(3.44)

The amplitudes $r_{a}$ are real, but not positive in general. They obey the
sum rules

$\sum_{a=1}^{\infty}r_{a}=1,\qquad\sum_{a=1}^{\infty}ar_{a}=R.$ (3.45)

Moreover, the rational expression (3.24) of $\Phi({{\rm e}^{{\rm i}q}})$
implies that the amplitudes $r_{a}$ obey a linear recursion of the form

$\lambda_{0}r_{a}+\lambda_{1}r_{a+1}+\lambda_{2}r_{a+2}+\cdots=0,$ (3.46)

except for finitely many initial values of $a$. The number of terms involved
in this recursion is generically $\Delta$, the degree of the polynomial $D(z)$
introduced in (2.13). The coefficients
$\lambda_{0},\,\lambda_{1},\,\lambda_{2}\,\dots$ are constants dictated by the
model.

In terms of the amplitudes $r_{a}$, the formula (3.23) for the structure
factor reads

$S(q)=\frac{1}{\ell}\sum_{a=1}^{\infty}r_{a}\,\frac{1-\cos aq}{1-\cos q}.$
(3.47)

Inserting this expression into (3.36), we obtain

$C_{n}=\frac{1}{\ell}\sum_{a=1}^{\infty}r_{a}I_{a,n},$ (3.48)

with

$I_{a,n}=\int_{0}^{2\pi}\frac{{\rm d}q}{2\pi}\,{\rm e}^{{\rm
i}nq}\,\frac{1-\cos aq}{1-\cos
q}=\left\\{\begin{array}[]{cl}a-{\left|n\right|}&(a>{\left|n\right|}),\\\
0&(a\leq{\left|n\right|}).\end{array}\right.$ (3.49)

Let us henceforth consider $n\geq 0$. The correlation function has the
following expression in terms of the amplitudes $r_{a}$:

$C_{n}=\frac{1}{\ell}\sum_{a=n+1}^{\infty}(a-n)r_{a}.$ (3.50)

Using (3.43) and (3.45), this formula can be recast as

$C_{n}=\rho(1-\rho)-\frac{1}{\ell}\biggl{(}n-\sum_{a=1}^{n-1}(n-a)r_{a}\biggr{)}.$
(3.51)

The correlation function therefore obeys the three-term recursion

$2C_{n}-C_{n-1}-C_{n+1}=-\frac{r_{n}}{\ell}\qquad(n\geq 1).$ (3.52)

As a consequence, the $C_{n}$ also obey the linear recursion relation (3.46),
i.e.,

$\lambda_{0}C_{n}+\lambda_{1}C_{n+1}+\lambda_{2}C_{n+2}+\cdots=0,$ (3.53)

except for finitely many initial values of $n$, and $C_{n}$ and $r_{n}$ share
the same exponential decay (3.40).

The expression (3.51) yields in particular

$C_{0}=\rho(1-\rho),\qquad C_{1}=\rho(1-\rho)-\frac{1}{\ell}.$ (3.54)

These two results are expected. The first one is obvious from the definition
of $C_{0}$. The second one expresses that the densities of ${\bullet}{\circ}$
and ${\circ}{\bullet}$ cluster interfaces have the expected value, namely

$\lim_{m\to\infty}{\langle\eta_{m}(1-\eta_{m+1})\rangle}=\rho-\rho^{2}-C_{1}=\frac{1}{\ell}.$
(3.55)

At larger distances ($n\geq 2$), the correlation function is not universal, in
the sense that it depends on the individual amplitudes $r_{a}$. We have

$C_{2}=\rho(1-\rho)-\frac{2-r_{1}}{\ell},\qquad
C_{3}=\rho(1-\rho)-\frac{3-2r_{1}-r_{2}}{\ell},$ (3.56)

and so on.

##  4 Three simple examples

In this section we illustrate the above general results on the structure
factor $S(q)$ and on the correlation function $C_{n}$ by considering three
simple examples of statistical ensembles, already studied in [42], which
virtually exhaust all cases with $\Delta=1$ or $\Delta=2$.

###  4.1 Flat ensemble

The simplest statistical ensemble of all is the flat one, where all cluster
lengths are permitted. This corresponds to ${\cal I}={\cal
J}=\\{1,2,3,\dots\\}$, hence

$I(z)=J(z)=\frac{z}{1-z},$ (4.1)

so that

$D(z)=1-2z,$ (4.2)

and so $\Delta=1$. We have

$z_{\star}=\frac{1}{2},\qquad\rho=\frac{1}{2},\qquad\ell=4.$ (4.3)

Equation (3.23) reads

$S(q)=\frac{1}{4},$ (4.4)

and so

$C_{n}=\frac{\delta_{n0}}{4},$ (4.5)

where $\delta_{ab}$ is the Kronecker symbol.

The absence of correlations between the occupancies of different sites
reflects the property that each site of the lattice is independently either
occupied or empty. Concomitantly, the sum entering (3.38) is empty for
$\Delta=1$.

###  4.2 Isolated empty sites

Our second example is defined by the condition that empty sites (i.e., holes)
are isolated. The ensemble where occupied sites (i.e., particles) are isolated
is related to the present one by exchanging the roles of particles and holes.
Both models share the same structure factor and correlation function. These
ensembles have been considered in several contexts, including the attractors
of repulsion processes [49] and packings of disks in narrow channels [50, 51].

This example corresponds to ${\cal I}=\\{1,2,3,\dots\\}$ and ${\cal
J}=\\{1\\}$, hence

$I(z)=\frac{z}{1-z},\qquad J(z)=z,$ (4.6)

so that

$D(z)=1-z-z^{2},$ (4.7)

and so $\Delta=2$. We have

$\displaystyle z_{\star}$ $\displaystyle=\frac{\sqrt{5}-1}{2}\approx
0.618033,$ (4.8) $\displaystyle\rho$
$\displaystyle=\frac{5+\sqrt{5}}{10}\approx
0.723606,\qquad\ell=\frac{5+\sqrt{5}}{2}\approx 3.618033.$ (4.9)

Equation (3.23) reads

$S(q)=\frac{1}{\sqrt{5}\,(3+2\cos q)},$ (4.10)

and so

$C_{n}=\frac{1}{5}\left(-\frac{3-\sqrt{5}}{2}\right)^{n}\qquad(n\geq 0).$
(4.11)

The correlation function is a pure decaying exponential, in agreement with
(3.38) for $\Delta=2$.

###  4.3 Even particle clusters

Our third example is defined by the condition that clusters of particles have
even lengths. This corresponds to ${\cal I}=\\{2,4,6,\dots\\}$ and ${\cal
J}=\\{1,2,3,\dots\\}$, i.e.,

$I(z)=\frac{z^{2}}{1-z^{2}},\qquad J(z)=\frac{z}{1-z}.$ (4.12)

Here again,

$D(z)=1-z-z^{2}$ (4.13)

and $\Delta=2$, so that $z_{\star}$ is given by (4.8). We have

$\rho=\frac{5-\sqrt{5}}{5}\approx
0.552786,\qquad\ell=\frac{5+3\sqrt{5}}{2}\approx 5.854101.$ (4.14)

Equation (3.23) reads

$S(q)=\frac{2\,(1+\cos q)}{\sqrt{5}\,(3+2\cos q)}.$ (4.15)

The structure factor vanishes for $q=\pi$. This is an extinction of the form
(3.33) with $K=2$. The correlation function reads

$C_{n}=\frac{\delta_{n0}}{\sqrt{5}}-\frac{1}{5}\left(-\frac{3-\sqrt{5}}{2}\right)^{n}\qquad(n\geq
0).$ (4.16)

The expression for $n=0$ is modified with respect to the decaying exponential
(3.38). This is an instance of the situation described below (3.41), with
$\mu=1$.

Figure 1 shows plots of the structure factor $S(q)$ against $q/\pi$ for the
three ensembles investigated in this section.

Figure 1: Structure factor $S(q)$ of the three ensembles investigated in
section 4, plotted against $q/\pi$. Red curve (1): flat ensemble (see (4.4)).
Green curve (2): isolated empty sites (see (4.10)). Blue curve (3): even
particle clusters (see (4.15)).

##  5 The $k$-mer deposition model

The sequential deposition of $k$-mers is a fully irreversible process where
$k$-mers (clusters of $k$ particles) are deposited at random positions on an
initially empty one-dimensional lattice. The process stops when the system
reaches a blocked (or jammed) configuration where no further $k$-mer can be
inserted any more [52, 53, 54]. The integer $k\geq 2$ is the only parameter of
the model. The sequential deposition of dimers ($k=2$) is equivalent to a
dimerization model investigated long ago by Flory [55].

The statistical ensemble of blocked configurations of $k$-mers has been
studied by various approaches [39, 40, 32, 42, 41]. In [42] it has been shown
that this ensemble can be described in terms of independent clusters, and
therefore studied within the present renewal formalism, with ${\cal
I}=\\{k,2k,3k,\dots\\}$ and ${\cal J}=\\{1,2,\dots,k-1\\}$, so that

$I(z)=\frac{z^{k}}{1-z^{k}},\qquad J(z)=\frac{z-z^{k}}{1-z}.$ (5.1)

The polynomial $D(z)$ can be reduced to

$D(z)=1-(z^{k}+z^{k+1}+\cdots+z^{2k-1}),$ (5.2)

and so

$\Delta=2k-1.$ (5.3)

For all values of the integer $k$, the structure factor is expected to exhibit
systematic extinctions at wavevectors that are multiples of $2\pi/k$ (see
(3.33)).

###  5.1 The first few values of $k$

It is interesting to start by considering the first few values of $k$, where
closed-form formulas can be derived.

#### Dimers ($k=2$).

This case is the simplest of all. We have

$D(z)=1-z^{2}-z^{3},$ (5.4)

and so $\Delta=3$. We thus recover that

$z_{\star}=\frac{1}{6}\bigl{(}(100+12\sqrt{69})^{1/3}+(100-12\sqrt{69})^{1/3}-2\bigr{)}\approx
0.754877$ (5.5)

is the reciprocal of the so-called plastic number [42], whereas

$\rho=\frac{2(7+2z_{\star}+3z_{\star}^{2})}{23}\approx 0.822991.$ (5.6)

Even in this simple case, (3.23) yields a rather lengthy expression for the
structure factor. Repeated use of $D(z_{\star})=0$ reduces this expression to
a minimal form where the highest power of $z_{\star}$ is $\Delta-1$ (here, 2).
We shall do this reduction throughout the following. We thus obtain

$S(q)=\frac{2z_{\star}(1-z_{\star})(1+\cos q)}{A_{2}(\cos q)},$ (5.7)

with

$\displaystyle A_{2}(\cos q)$ $\displaystyle=$ $\displaystyle
3-z_{\star}^{2}-2(4-5z_{\star}-5z_{\star}^{2})\cos q$ (5.8) $\displaystyle-$
$\displaystyle 4(3-2z_{\star}-4z_{\star}^{2})\cos^{2}q.$

The extinction at $q=\pi$ is clearly visible on (5.7).

#### Trimers ($k=3$).

In this case we have $\Delta=5$, but $D(z)$ factors as

$D(z)=(1+z^{2})(1-z^{2}-z^{3}),$ (5.9)

so that $z_{\star}$ is again given by (5.5), whereas

$\rho=\frac{3(26+7z_{\star}-2z_{\star}^{2})}{115}\approx 0.786377.$ (5.10)

Equation (3.23) yields

$S(q)=\frac{(1+2(1-z_{\star})\cos q)(1+2\cos q)^{2}}{A_{3}(\cos q)},$ (5.11)

with

$\displaystyle A_{3}(\cos q)$ $\displaystyle=$
$\displaystyle-(1-2z_{\star})(2+3z_{\star})+2(9+z_{\star}-13z_{\star}^{2})\cos
q$ (5.12) $\displaystyle+$ $\displaystyle
4(4-3z_{\star})(2+3z_{\star})\cos^{2}q-8(3-6z_{\star}-10z_{\star}^{2})\cos^{3}q$
$\displaystyle-$ $\displaystyle 16(1-2z_{\star})(2+3z_{\star})\cos^{4}q.$

The extinction at $q=2\pi/3$ is clearly visible on (5.11).

The formulas (5.7) and (5.11) show that the complexity of the analytical
expression of $S(q)$ increases fast with the integer $k$. The structure factor
indeed exhibits more and more detailed structures as $k$ increases, as
testified by figure 2.

Figure 2: Structure factor $S(q)$ of the $k$-mer deposition model, plotted
against $q/\pi$, for $k$ ranging from 2 to 6 (see legend).

###  5.2 Scaling behavior at large $k$

Static observables of the $k$-mer deposition model have been shown in [42] to
exhibit an unusual behavior at large $k$, where logarithmic corrections to
scaling are ubiquitous. These corrections to scaling already show up in
$z_{\star}$. Setting

$z_{\star}=\exp\left(-\frac{u_{\star}}{k}\right),$ (5.13)

we find that $u_{\star}$ obeys the transcendental equation

$u_{\star}{\rm e}^{u_{\star}}\approx k.$ (5.14)

This equation holds up to corrections of relative order $1/k$, which will be
consistently neglected throughout the following. The solution to (5.14) reads

$u_{\star}\approx W(k),$ (5.15)

in terms of the Lambert $W$ function. To leading order, this reads

$u_{\star}\approx\ln k.$ (5.16)

More precisely, setting

$\lambda=\ln k,\qquad\mu=\ln\lambda=\ln\ln k,$ (5.17)

we obtain the full asymptotic series

$u_{\star}=\lambda-\mu+\frac{\mu}{\lambda}+\frac{\mu(\mu-2)}{2\lambda^{2}}+\frac{\mu(2\mu^{2}-9\mu+6)}{6\lambda^{3}}+\cdots$
(5.18)

The main outcomes of the scaling analysis performed in [42] which are relevant
for the present purpose are as follows:

$\rho\approx\frac{u_{\star}}{u_{\star}+1},\qquad\ell\approx\frac{u_{\star}+1}{u_{\star}}\,k.$
(5.19)

The scaling analysis recalled above can be extended to the structure factor
all over its scaling regime where $q$ scales as the inverse of $k$.
Introducing the rescaled wavevector

$Q=kq,$ (5.20)

we have the estimates

$\displaystyle I(z_{\star})\approx\frac{u_{\star}}{k},\qquad I(z_{\star}{\rm
e}^{\pm{\rm i}q})\approx\frac{u_{\star}\,{\rm e}^{\pm{\rm i}Q}}{k},$
$\displaystyle J(z_{\star})\approx\frac{k}{u_{\star}},\qquad J(z_{\star}{\rm
e}^{\pm{\rm i}q})\approx\frac{k}{u_{\star}\mp{\rm i}Q},$ (5.21)

so that (3.23) evaluates to

$S(q)\approx k\,\frac{u_{\star}}{u_{\star}+1}\,\frac{2(1-\cos
Q)}{2u_{\star}^{2}(1-\cos Q)+2u_{\star}Q\sin Q+Q^{2}}.$ (5.22)

The second factor of this expression is equal to the particle density $\rho$
(see (5.19)). The third factor mainly depends on the rescaled wavevector $Q$.
The presence of powers of $u_{\star}$ however generates logarithmic
corrections to this main scaling behavior. We have in particular

$S(0)\approx\frac{ku_{\star}}{(u_{\star}+1)^{3}}.$ (5.23)

An equivalent formula was derived by another route in [42]. The expression

$C_{0}=\rho(1-\rho)\approx\frac{u_{\star}}{(u_{\star}+1)^{2}}$ (5.24)

translates to the non-trivial identity

$\int_{-\infty}^{\infty}{\rm d}Q\,\frac{1-\cos Q}{2u_{\star}^{2}(1-\cos
Q)+2u_{\star}Q\sin Q+Q^{2}}=\frac{\pi}{u_{\star}+1}.$ (5.25)

The scaling form (5.22), with its fast decay in $1/Q^{2}$, corroborates the
most salient feature of figure 2, namely that most of the intensity
concentrates onto smaller and smaller values of $q$ as $k$ is increased.

Let us focus our attention onto the leading peak of the structure factor,
located slightly before the first extinction at $q=2\pi/k$, whose height grows
rather fast with $k$. The scaling behavior of the position $Q_{\rm
max}=kq_{\rm max}$ of this peak and of its height $S_{\rm max}$ can be
extracted from the scaling result (5.22). The peak position obeys the
transcendental equation

$u_{\star}=\frac{Q_{\rm max}(2(1-\cos Q_{\rm max})-Q_{\rm max}\sin Q_{\rm
max})}{2(1-\cos Q_{\rm max})(Q_{\rm max}-\sin Q_{\rm max})}.$ (5.26)

Some algebra yields the following asymptotic series in inverse powers of
$u_{\star}$:

$\displaystyle Q_{\rm max}$ $\displaystyle=$ $\displaystyle
2\pi\left(1-\frac{1}{u_{\star}}+\frac{1}{u_{\star}^{2}}+\frac{\pi^{2}-3}{3u_{\star}^{3}}+\cdots\right),$
(5.27) $\displaystyle S_{\rm max}$ $\displaystyle=$
$\displaystyle\frac{k}{\pi^{2}}\left(1+\frac{1}{u_{\star}}-\frac{2\pi^{2}}{3u_{\star}^{3}}+\cdots\right).$
(5.28)

The peak height $S_{\rm max}$ therefore grows linearly with $k$, i.e., faster
than $S(0)\approx k/(\ln k)^{2}$ (see (5.23)) by a factor growing as $(\ln
k)^{2}$. This is illustrated in figure 3, showing plots of $S(0)$ and $S_{\rm
max}$ against $k$. Dashed lines with corresponding colors show appropriate
approximations (see caption).

Figure 3: Plots of $S(0)$ and $S_{\rm max}$ against $k$, for the $k$-mer
deposition model. Full blue curve: exact values of $S(0)$ for finite $k$ (see
(3.30)). Dashed blue curve: scaling prediction (see (5.13), (5.23)). Full red
curve: numerically exact maximal values of $S(q)$ for finite $k$. Dashed red
curve: leading linear growth in $k/\pi^{2}$ (see (5.28)).

The peak position $Q_{\rm max}$ is very near the first pair of complex poles
of the structure factor, i.e., $Q_{1}=kq_{1}$ and
${\overline{Q}}_{1}=k{\overline{q}}_{1}$, where

$Q_{1}=2\pi\left(1-\frac{1}{u_{\star}}+\frac{1+{\rm
i}\pi}{u_{\star}^{2}}-\frac{4\pi^{2}+3+9{\rm
i}\pi}{3u_{\star}^{3}}+\cdots\right).$ (5.29)

Taking the imaginary part of this expression to leading order in $1/u_{\star}$
and using (3.41), we obtain that the correlation length scales as

$\xi\approx\frac{ku_{\star}^{2}}{2\pi^{2}}\approx\frac{k(\ln
k)^{2}}{2\pi^{2}}.$ (5.30)

##  6 Arrays of Rydberg atoms

This section is devoted to blocked configurations of arrays of Rydberg atoms
on a one-dimensional optical lattice. We consider the simple setting recalled
in section 1, where each lattice site occupied by a Rydberg atom must have at
least $b$ empty sites on either side. The integer $b\geq 1$, referred to as
the blockade range, is the only parameter of the model.

Our aim is to investigate the uniform ensemble of all blocked configurations,
where no further atom can be inserted into the system. Along the lines of
earlier works [32, 41, 42], blocked configurations consist of isolated
occupied sites (the Rydberg atoms) separated by clusters of holes whose length
is at least $b$, in order to obey the blockade constraint, and at most $2b$,
since an extra Rydberg atom could be inserted in the middle of an empty range
of size $2b+1$.

Within the present formalism, clusters of particles and of holes respectively
correspond to ${\cal I}=\\{1\\}$ and ${\cal J}=\\{b,b+1,\dots,2b\\}$, and so

$I(z)=z,\qquad J(z)=\frac{z^{b}(1-z^{b+1})}{1-z}.$ (6.1)

The polynomial $D(z)$ can be reduced to

$D(z)=1-(z^{b+1}+z^{b+2}+\cdots+z^{2b+1}).$ (6.2)

This expression coincides with (5.2), with the identification

$k=b+1.$ (6.3)

We have in particular

$\Delta=2b+1.$ (6.4)

There is indeed an equivalence, at the level of single configurations, between
blocked configurations of $k$-mers and of Rydberg atoms. A Rydberg atom
followed by $b$ empty sites (to its right) can be mapped onto a $k$-mer, where
$b$ and $k$ are related by (6.3). The mapping between configurations of
$k$-mers and of Rydberg atoms is however not unique [41, 42]. Many observables
pertaining to both models are simply related to each other. For instance, the
density of the $k$-mer deposition model is $k$ times larger than the
corresponding density of Rydberg atoms.

As a consequence of the above correspondence, the structure factors of
$k$-mers and Rydberg atoms with $b=k-1$ share the same denominator. The latter
is indeed a polynomial in $\cos q$ that is essentially dictated by $D(z)$.
Both structure factors have different numerators, if only because there are no
extinctions in the case of Rydberg atoms.

###  6.1 The first few values of $b$

In is again interesting to first look at a few smallest values of the blockade
range $b$.

####  $b=1$.

This case is the simplest of all. It corresponds to $k=2$, so that (5.4) and
(5.5) still hold, whereas

$\rho=\frac{7+2z_{\star}+3z_{\star}^{2}}{23}\approx 0.411495.$ (6.5)

Equation (3.23) yields

$S(q)=\frac{1}{B_{1}(\cos q)},$ (6.6)

with

$\displaystyle B_{1}(\cos q)$ $\displaystyle=$ $\displaystyle
5+7z_{\star}+5z_{\star}^{2}+2(7+8z_{\star}+2z_{\star}^{2})\cos q$ (6.7)
$\displaystyle+$ $\displaystyle 4(2+3z_{\star})\cos^{2}q.$

We have (see (5.8))

$B_{1}(\cos q)=(2+3z_{\star}+2z_{\star}^{2})A_{2}(\cos q).$ (6.8)

It is indeed expected from the correspondence between $k$-mers and Rydberg
atoms that the denominators of (5.7) and (6.6) should coincide, up to some
$q$-independent multiplicative constant.

####  $b=2$.

This case corresponds to $k=3$, so that (5.5) and (5.9) still hold, whereas

$\rho=\frac{26+7z_{\star}-2z_{\star}^{2}}{115}\approx 0.262125.$ (6.9)

Equation (3.23) yields

$S(q)=\frac{(1+z_{\star})^{2}+2z_{\star}\cos q}{B_{2}(\cos q)},$ (6.10)

with

$\displaystyle B_{2}(\cos q)$ $\displaystyle=$ $\displaystyle
5+3z_{\star}-z_{\star}^{2}-2(3-6z_{\star}-10z_{\star}^{2})\cos q$ (6.11)
$\displaystyle+$ $\displaystyle
4(5+13z_{\star}+14z_{\star}^{2})\cos^{2}q+8(13+10z_{\star}+3z_{\star}^{2})\cos^{3}q$
$\displaystyle+$ $\displaystyle 16(5+3z_{\star}-z_{\star}^{2})\cos^{4}q.$

We have (see (5.12))

$B_{2}(\cos q)=(1+z_{\star})^{2}A_{3}(\cos q).$ (6.12)

A relationship of this kind was again expected from the correspondence between
$k$-mers and Rydberg atoms.

Equations (6.6) and (6.10) demonstrate that the complexity of the structure
factor increases fast with the blockade range $b$. Concomitantly, $S(q)$
exhibits more and more detailed structures as $b$ increases (see figure 4).

Figure 4: Structure factor $S(q)$ of the blocked arrays of Rydberg atoms,
plotted against $q/\pi$, for blockade ranges $b$ running from 1 to 5 (see
legend).

###  6.2 Scaling behavior at large $b$

The scaling analysis of the $k$-mer deposition model at large $k$ summarized
in section 5.2 applies, mutatis mutandis, to the blocked configurations of
Rydberg atoms in the regime where the blockade range $b$ becomes large [42].
Here, too, logarithmic corrections to scaling are ubiquitous. Setting

$z_{\star}=\exp\left(-\frac{u_{\star}}{b}\right),$ (6.13)

we find that $u_{\star}$ again obeys the transcendental equation (5.14), and
we have

$\rho\approx\frac{u_{\star}}{b(u_{\star}+1)},\qquad\ell\approx\frac{u_{\star}+1}{u_{\star}}\,b.$
(6.14)

The scaling analysis again extends to the structure factor all over the regime
where $q$ scales as the inverse of the blockade range $b$. Introducing the
rescaled wavevector

$Q=bq,$ (6.15)

and skipping every detail, we find that (3.23) translates to

$S(q)\approx\frac{u_{\star}}{b(u_{\star}+1)}\,\frac{Q^{2}}{2u_{\star}^{2}(1-\cos
Q)+2u_{\star}Q\sin Q+Q^{2}}.$ (6.16)

The first factor is equal to the density $\rho$ of Rydberg atoms (see (6.14)).
The second factor mainly depends on the rescaled wavevector $Q$. The
denominators of (5.22) and (6.16) coincide, as expected from the
correspondence recalled above between $k$-mers and Rydberg atoms. The
numerators are however different. The presence of powers of $u_{\star}$ in
(6.16) again generates logarithmic corrections of various kinds. We have in
particular

$S(0)\approx\frac{u_{\star}}{b(u_{\star}+1)^{3}}.$ (6.17)

The correlation length is again described by the scaling form (5.30), up to
the replacement of $k$ by $b$, i.e.,

$\xi\approx\frac{bu_{\star}^{2}}{2\pi^{2}}\approx\frac{b(\ln
b)^{2}}{2\pi^{2}}.$ (6.18)

At variance with the scaling form (5.22), which falls off as $1/Q^{2}$ for $Q$
large, the expression (6.16) predicts that

$S(q)\approx\rho$ (6.19)

is roughly constant and equal to the density of Rydberg atoms for
${\left|Q\right|}\gg 1/b$, i.e., in practice, for $b$ large and
${\left|q\right|}\gg 1/b$. This density is larger than $S(0)$ by a factor
growing as $(\ln b)^{2}$. The structure factor exhibits oscillations around
$\rho$ virtually all over the range of wavevectors, whose amplitude falls off
rather rapidly as the blockade range $b$ is increased (see figure 5).
Integrating (6.19) over $q$ yields $C_{0}\approx\rho$, in agreement with
(3.54) since $\rho$ is small at large $b$.

Figure 5: Full lines: structure factor $S(q)$ of Rydberg atoms, multiplied by
$b$ and plotted against $q/\pi$, for $b=10$ (blue) and $b=20$ (red). Dashed
lines with corresponding colors: prediction (6.19).

##  7 Discussion

This work follows on from a previous paper in collaboration with Krapivsky
[42]. There, we have put forward an alternative approach, inspired from the
theory of renewal processes, to study statistical ensembles of constrained
configurations of particles on a one-dimensional lattice, and especially to
determine their configurational entropies. Here, we demonstrate that a range
of observables pertaining to such ensembles, including chiefly their structure
factor and correlation function, can also be investigated by means of the
renewal approach. It is worth recalling that the scope of this approach is
restricted to local constraints which are expressible in terms of the lengths
of clusters of occupied and empty sites.

We have emphasized the pivotal importance of the class of rational models,
which encompasses all examples considered in this paper, and virtually all
situations of physical significance. Within this rational class, the
complexity of a statistical ensemble is measured by the degree $\Delta$ of the
polynomial $D(z)$ introduced in (2.13). The integer $\Delta$ is fully
analogous to the dimension of the transfer matrix in the more traditional
transfer-matrix approach. To mention just one example, the decomposition
(3.38) of the correlation function as a sum of $\Delta-1$ decaying
exponentials has a perfect analogue in terms of the subleading eigenvalues of
the transfer matrix.

To close, we briefly tackle the situation of ensembles of blocked
configurations beyond the rational class. To take a striking example, if
${\cal I}=\\{1,2,4,8,\dots\\}$, i.e., the allowed lengths of particle clusters
consist of the powers of two, the corresponding generating series

$I(z)=\sum_{k=0}^{\infty}z^{2^{k}}$ (7.1)

is a prototypical lacunary series, having the full unit circle of the complex
$z$-plane as a natural boundary. As a consequence, for a generic set ${\cal
J}$, the structure factor $S(q)$ has a natural boundary along some curve in
the complex $q$-plane. The structure of the correlation function is therefore
exceedingly more complex than the finite sum (3.38). It would be desirable to
have even one physically motivated model of this kind.

It is a pleasure to thank Paul Krapivsky for extensive discussions which
motivated this work.

## Data availability statement

Data sharing not applicable to this article as no datasets were generated or
analyzed.

## Conflict of interest

The author declares no conflict of interest.

## Orcid id

Jean-Marc Luck https://orcid.org/0000-0003-2151-5057

##  Appendix A The Hendricks-Teller model

In this appendix we illustrate the renewal approach used in the body of this
paper on a simpler structural model, namely the Hendricks-Teller model [43],
defined as follows. Identical pointlike atoms are put along an infinite half-
line at random positions $x_{n}$ such that $x_{0}=0$, and the distances

$\ell_{n}=x_{n}-x_{n-1}$ (1.1)

are independent and identically distributed with some continuous distribution
$\rho(\ell)$ such that ${\langle\ell^{2}\rangle}$ is convergent.

The (random) Fourier amplitude $G_{N}(q)$ and Fourier intensity $S_{N}(q)$ of
the first $N$ atoms read

$G_{N}(q)=\sum_{n=1}^{N}{\rm e}^{-{\rm i}qx_{n}},\qquad
S_{N}(q)=\frac{{\left|G_{N}(q)\right|}^{2}}{N}.$ (1.2)

Our main interest is in the structure factor

$S(q)=\lim_{N\to\infty}{\langle S_{N}(q)\rangle},$ (1.3)

which is a self-averaging quantity.

The Fourier amplitudes obey the renewal equation

$G_{N}(q)={\rm e}^{-{\rm i}q\ell_{1}}+{\rm e}^{-{\rm i}q\ell_{1}}G_{N-1}(q).$
(1.4)

The mean Fourier amplitude can be derived by averaging (1.4) over the atomic
positions. This yields

${\langle G_{N}(q)\rangle}=g(q)+g(q){\langle G_{N-1}(q)\rangle},$ (1.5)

where

$g(q)=\int_{0}^{\infty}{\rm e}^{-{\rm i}q\ell}\rho(\ell)\,{\rm d}\ell$ (1.6)

is the Fourier transform of the distribution of interatomic distances. In the
limit of an infinitely long sample, the mean amplitude therefore reads

${\langle G(q)\rangle}=\frac{g(q)}{1-g(q)}.$ (1.7)

This result diverges at small wavevectors as

${\langle G(q)\rangle}\approx\frac{1}{{\rm i}q{\langle\ell\rangle}}\qquad(q\to
0).$ (1.8)

The structure factor can be derived by multiplying (1.4) by its complex
conjugate, using (1.2), and averaging over the atomic positions. This yields

$N{\langle S_{N}(q)\rangle}=1+{\langle
G_{N-1}(q)\rangle}+{\langle{\overline{G_{N-1}}}(q)\rangle}+(N-1){\langle
S_{N-1}(q)\rangle}.$ (1.9)

In the $N\to\infty$ limit, the structure factor therefore reads

$S(q)=1+{\langle G(q)\rangle}+{\langle{\overline{G}}(q)\rangle}.$ (1.10)

Using (1.7), we thus recover the main result of [43] for the present model,
i.e.,

$S(q)=\frac{1-g(q){\overline{g}}(q)}{(1-g(q))(1-{\overline{g}}(q))}.$ (1.11)

Figure 6: Structure factor (1.14) of the Hendricks-Teller model with a uniform
distribution of interatomic distances over the interval $[0,a]$, plotted
against $qa/(2\pi)$.

Let us mention a few properties of the above formula. Whenever the distance
distribution $\rho(\ell)$ is continuous, its Fourier transform $g(q)$ falls
off to zero at large $q$, and so the structure factor goes to the limit

$S(\infty)=1.$ (1.12)

Its value at $q=0$,

$S(0)=\frac{{\langle\ell^{2}\rangle}-{\langle\ell\rangle}^{2}}{{\langle\ell\rangle}^{2}},$
(1.13)

is nothing but the reduced variance of interatomic distances. This quantity
may be either smaller or larger than unity.

The exponential distance distribution $\rho(\ell)=(1/a)\exp(-\ell/a)$ yields a
Poissonian distribution of atoms, for which we have $S(q)=1$ for all $q$, as
expected.

A simple yet non-trivial example is provided by a uniform distribution of
interatomic distances over some interval $[0,a]$. The ensuing structure factor

$S(q)=\frac{(qa)^{2}+2(\cos qa-1)}{(qa)^{2}+2(1-\cos qa-qa\sin qa)}$ (1.14)

is a function of the product $qa$, starting from $S(0)=1/3$ and exhibiting
slowly damped oscillations with period $2\pi$ around its limit $S(\infty)=1$
(see figure 6).

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