# A practical guide to Digital Micro-mirror Devices (DMDs) for wavefront
shaping

Sébastien M. Popoff  0000-0002-7199-9814 Institut Langevin, ESPCI Paris, PSL
University, CNRS, France<EMAIL_ADDRESS>Yaron Bromberg
0000-0003-2565-7394 Racah Institute of Physics, The Hebrew University of
Jerusalem, Israel  Maxime W. Matthès  Institut Langevin, ESPCI Paris, PSL
University, CNRS, France  Rodrigo Gutiérrez-Cuevas  0000-0002-3451-6684
Institut Langevin, ESPCI Paris, PSL University, CNRS, France
<EMAIL_ADDRESS>
###### Abstract

Digital micromirror devices have gained popularity in wavefront shaping,
offering a high frame rate alternative to liquid crystal spatial light
modulators. They are relatively inexpensive, offer high resolution, are easy
to operate, and a single device can be used in a broad optical bandwidth.
However, some technical drawbacks must be considered to achieve optimal
performance. These issues, often undocumented by manufacturers, mostly stem
from the device’s original design for video projection applications. Herein,
we present a guide to characterize and mitigate these effects. Our focus is on
providing simple and practical solutions that can be easily incorporated into
a typical wavefront shaping setup.

  * September 2023

##  1 Introduction

Since the advent of adaptive optics, various technologies have been employed
to modulate the amplitude and/or phase of light. Early adaptive optics
devices, utilized in fields like microscopy and astronomy, offer rapid
modulation capable of compensating for the aberrations of optical systems in
real-time. However, these devices are constrained by a limited number of
actuators, restricting their utility in complex media where a large number of
degrees of freedom is essential. Liquid Crystal Spatial Light Modulators (LC-
SLMs), which allow for the control of light phase across typically more than a
million pixels, have emerged as powerful tools for wavefront shaping in
complex media since the seminal work of A. Most and I. Vellekoop in the
mid-2000s [1]. Nonetheless, LC-SLMs are hampered by their slow response time,
permitting only a modulation speed ranging from a few Hz to about 100 Hz.

Digital Micromirror Devices (DMDs) have emerged as a technology bridging the
gap between these two types of systems; they offer a large number of pixels
(similar to LC-SLMs) and fast modulation speeds (typically up to several tens
of kHz). Their high speed capabilities made them attractive for real-time
applications, in particular for high-resolution imaging microscopy requiring
fast scanning or illumination shaping [2, 3], biolithography [4], and optical
tweezers [5]. However, DMDs are restricted to hardware binary amplitude
modulation and are not optimized for coherent light applications. Utilizing
DMDs for coherent control of light in complex media is therefore non-trivial
and necessitates specific adaptations for efficient use.

To comprehend both the capabilities and limitations of DMD technology for
coherent wavefront shaping, it is crucial to understand the device’s operating
principles and its original design intentions. Investigated and developed by
Texas Instruments since the 1980s, DMDs gained prominence in the 1990s for
video projection applications since the 1990s under the commercial name of
Digital Light Processing (DLP) [6, 7]. The technology enables high-resolution,
high-speed, and high-contrast-ratio modulation of light. DMDs operate by
toggling the state of small mirrors between two distinct angles, denoted as
$\pm\theta_{\text{DMD}}$. The device is originally engineered for amplitude
modulation in video projection applications. In this configuration, one mirror
angle directs light into the projection lens, while the alternate angle
results in the light path being blocked (see Fig. 1). Given that projectors
utilize incoherent light and that the DMD plane is optically conjugated with
the projection screen, aberrations within the DMD plane are generally not
problematic. Similarly, phase fluctuations induced by temperature variations,
as well as minor vibrations from the cooling hardware, are inconsequential in
this context. The DMD is designed to produce binary on/off modulation, which
is then leveraged to generate grayscale images via pulse-width modulation.
Color modulation is accomplished through the use of a color wheel in
conjunction with a bright white light source.

Third-party companies have developed kits tailored for research applications,
which include a DMD, a control board, and a software interface. Specifically,
Vialux devices [8] offer an FPGA board that enables high-speed modulation by
allowing frames to be stored in the device’s memory [9]. However, standard
Texas Instruments video projector evaluation modules can also be repurposed
into wavefront shaping devices [10], though at a compromised modulation speed.
These systems can further be converted into phase or complex field modulators.
This is typically achieved by encoding the optical phase into the spatial
displacement of binary spatial fringes displayed on the DMD, followed by
filtering the high spatial frequencies in the Fourier plane [11]. Such a
configuration permits multi-level complex modulation but sacrifices spatial
resolution. The implementation and performance of such systems are explored in
a separate tutorial [12] and will not be elaborated upon here. For the
remainder of this paper, it will be assumed that the DMD is used for complex
modulation via such a method.

While other articles exist describing the various aspects of DMDs [13, 14, 10,
15], this tutorial aims to provide a guide for easily setting up a DMD for
wavefront shaping applications in complex media. In particular, we provide
characterization and validation procedures that requires minimal changes
compared to typical wavefront shaping setups.
We first introduce the diffraction properties of a DMD and elaborate on how
these could impact the system’s efficiency. We also furnish a straightforward
criterion for selecting the appropriate DMD parameters for a specified
excitation wavelength. In the next section, we delve into the aberration
impacts brought about by the non-flatness of the DMD surface. We demonstrate a
simple process to characterize this effect and provide compensation solutions.
In the third segment, we detail the influence of mechanical vibrations that
are induced by the DMD’s cooling system. Lastly, we discuss how the
thermalization of the DMD chip can potentially result in variations to the DMD
response over time.

Figure 1:  Principle of operation of a DMD in a digital projector. Left,
incident light can be reflected towards the projection lens (state on), or
onto a beam dump (state off). Right, zoom on the pixels. Image adapted from
[16].

##  2 Choosing the right DMD: Diffraction effects

###  2.1 A 1D model

A significant distinction between liquid crystal modulators and DMDs lies in
the geometry of the pixel surface. This difference gives rise to diffraction
effects that can adversely affect both the modulation quality and system
efficiency. The impact of these diffraction effects is highly contingent on
several factors: the wavelength of the illumination, the pixel pitch, and both
the incident and outgoing angles. Therefore, alongside selecting an
appropriate anti-reflection coating, it is crucial to ensure that the pixel
pitch is compatible with the specific configuration being used. Texas
Instruments offers chips with a variety of pixel pitches $d$, ranging
approximately from $5$ to $\sim 25$ µm [17].

To achieve a qualitative understanding of this issue, we consider a 1D array
of pixels as illustrated in Fig. 2. Initially, let’s assume that all pixels
are in the same state and are illuminated by a plane wave originating from the
far field. Under these conditions, the pixelated modulator essentially
functions as a grating, with a period $d$ that is equivalent to the pixel
pitch. It is important to underscore that these modulators possess a hardware-
limited fill factor, typically around 90%. This translates to an effective
active pixel size of $d^{\prime}<d$.

In general, a grating gives rise to various diffraction orders with differing
intensities and angles $\theta_{p}$, as dictated by the grating equation:
$\text{sin}(\theta_{p})=p\lambda/d=p\,\text{sin}(\theta_{D}),$ where $\lambda$
is the wavelength of the light, $\theta_{D}$ is the angle of the first-order
diffraction, and $p$ is an integer value denoting the orders of diffraction.
The intensity of the individual diffraction orders is influenced by the
response of a single pixel, constituting the unit cell of the grating, and
that is governed by $d$, $d^{\prime}$, and its rotation angle in the case of a
DMD. Importantly, we can decouple the effects of the grating’s periodicity,
which influences the angles of the diffraction orders, from the effects of the
response of a single pixel, which shapes the envelope of the angular response.

Figure 2:  1D grating geometry. Schematic representation of the geometry of
two types of modulators: (a) the liquid crystal modulator, equivalent to a
flat grating, and (b) the DMD geometry, equivalent to a blazed grating.
$\alpha$ denotes the incident angle relative to the normal of the array plane,
$\theta_{0}$ refers to the angle of the zeroth diffraction order, and
$\theta_{B}$ is the tilt angle of the mirrors.

For a case of normal incidence with an LC-SLM (Liquid Crystal Spatial Light
Modulator), we can assume that the response of a single pixel is uniform over
its surface. Similarly, in the scenario of a blazed grating, such as a DMD, a
linear phase slope is present on each pixel. This is due to the tilt angle
$\theta_{B}$ of the mirrors. For an arbitrarily selected incidence angle
$\alpha$, a global phase slope is introduced. This results in a trivial shift
of the angular diffraction pattern by an angle $\alpha$. In essence, the
incident angle $\alpha$ serves to shift the entire angular diffraction pattern
relative to the case of normal incidence, while the blazed angle $\theta_{B}$
—the tilt angle of the mirror in the DMD projected onto the axis we consider—
only shifts the envelope by an angle of $2\theta_{B}$. Whenever the fill
factor approaches 100%, i.e. when $d\approx d^{\prime}$, the envelope for a
flat grating achieves its maximum at $\theta_{0}=-\alpha$; this corresponds to
the angle of the zeroth-order diffraction, and the intensity nears zero for
all other orders. In these specific conditions, a singular diffraction order
is visually perceived, corresponding to the optimum scenario. The addition of
a blazed angle $\theta_{B}$ results in both a shift in the envelope and in the
position of its maximum, now indicated by $\theta_{0}=2\theta_{B}-\alpha$. In
the general case, this position may not align with a diffraction order anymore
[13]. We provide in Appendix A. a 1D computation of this effect. A more
accurate computation of the far field can be found in [15].

We represent in Fig. 3 the angular response of a flat grating and a blazed
grating for a 1D filling fraction of 95% (correponding to a 2D filling
fraction of $\approx 90$%). For a flat grating, the zero-th order contains
most of the intensity, the other orders being negligible in comparison. For
the blazed grating example shown, we are in a situation close to the worst
case scenario: Two diffraction orders have a significant and comparable
intensity, and other orders also have a non-negligible contributions. In the
optimal scenario, where the peak of the envelope corresponds to a diffraction
order, it results in a single diffraction order carrying the majority of the
energy. This state is achieved when the conditions of the blazed grating
equation are fulfilled [18]:

$\text{sin}(2\theta_{B}-\alpha)+\text{sin}(\alpha)=2\,\text{sin}(\theta_{B})\,\text{cos}(\theta_{B}-\alpha)=p\frac{\lambda}{d}\,.$
(1)

We note that the incident angle $\alpha$ olso affects the diffraction
efficiency.

Figure 3:  Flat grating vs blazed grating. Far field diffraction patterns for
a 1D flat grating (left) and a 1D blazed grating (right) for an input angle of
$\alpha=-20^{\circ}$, a filling fraction of 95% (corresponding to a 2D filling
fraction of $\approx 90$%), a pixel tilt angle of $\theta_{B}=5^{\circ}$, and
a wavelength to pixel pitch ratio $0.05$. Vertical lines represent the angles
of the diffraction orders and the black dashed curve represent the amplitude
of the field.

###  2.2 The 2D case

To analyze more precisely the effect of diffraction in a DMD, one needs to
consider the 2D surface of the modulator. We can establish a Cartesian
coordinate system on the plane of the DMD, with axes $x$ and $y$ aligned with
the pixel sides (refer to Fig. 4a). Pixels are uniformly repeated along these
axes. However, a technical challenge arises in that the axis of rotation of
the pixels aligns with the pixel diagonals, resulting in a rotation by 45
degrees with respect to the $x$ and $y$ axes. For the convenience of alignment
and manipulation of the optical setup, it is preferable to work with the
incident and outgoing beams which have the optical axis contained in the
horizontal plane, i.e. a plane parallel to the table surface. A
straightforward and prevalent solution is to rotate the chip by 45 degrees
relative to the horizontal plane, which aligns the pixel’s axis of rotation to
be vertical. This configuration is depicted in Fig. 4b.

Figure 4:  2D grating geometry. (a) Schematic representation of the geometry
of the DMD. The incident and reflection angles $\alpha^{\prime}$ and
$\theta_{0}^{\prime}$ are situated within the horizontally plane, illustrated
in yellow. (b) Photograph of the DMD chip oriented so that the rotation axis
of the pixels is aligned vertically.

The 2D array can be seen as 2 orthogonal 1D gratings in the $x$ and $y$
directions, both having the same properties. A more comprehensive depiction of
the 2D system can be found in [14]. In order to utilize Eq.(1), one first
needs to project the different angles of the problem onto the incident planes
of the two 1D gratings. This yields
$\alpha=\text{arctan}\left(\text{tan}(\alpha^{\prime})/\sqrt{2}\right)$,
$\theta=\text{arctan}\left(\text{tan}(\theta^{\prime})/\sqrt{2}\right)$, and
$\theta_{B}=\text{arctan}\left(\text{tan}(\theta_{\text{DMD}})/\sqrt{2}\right)$,
where $\theta_{\text{DMD}}$ represents the true rotation angle of the mirrors
relative to the diagonal of the pixels, and $\alpha^{\prime}$ and
$\theta^{\prime}$ denote the incident and outgoing angles in the horizontal
plane. We can quantify how close we are to the ideal case, i.e., when
satisfying the blazed equation outlined by Eq.(1), by defining a blazed number
$\mu$ as introduced in [19]:

$\mu=\left|\lfloor
4\frac{d}{\lambda}\left[\text{sin}(\theta_{B})\text{cos}(\theta_{B}-\alpha)\right]\rfloor\mod{2}-1\right|\,,$
(2)

with $\lfloor.\rfloor$ representing the floor function, and $\mod{2}$ the
modulo 2 operation. $\mu$ is maximal and equals $1$ when the blazing equation
is satisfied, i.e. when one order of diffraction contains most of the energy,
and minimal when we are in the worst-case scenario, i.e. when four orders of
diffraction have a significant and equal intensity.

To demonstrate the effect, we conduct a simulation of a DMD using Python
(refer to tutorial and code in [19]) with two pixel pitches of $d=7.6$µm and
$d=10.8$µm, under a coherent excitation at $\lambda=633$nm. Fig. 5 shows the
estimated blazed number $\mu$ as a function of the angle of incidence
$\alpha^{\prime}$ in the horizontal plane, along with the far field
diffraction pattern for two distinct incident angles. It should be noted that
the efficiency of diffraction can be altered by adjusting the angle of
incidence. However, its impact is relatively confined within an acceptable
angular range that aligns with experimental limitations (i.e. for angles far
from $\pm 90^{\circ}$). Far field patterns are centered around the maximum of
the envelope $\theta_{\text{max}}=2\theta_{B}-\alpha$ (marked by a yellow
cross). We see that for small values of $\alpha$, the pixel pitch of
$d=10.8$µm leads to a blazed number $\mu$ close $1$. It corresponds in the far
field to having one bright order of diffraction close to the maximum of the
envelope.

Figure 5:  Blazed number and far field diffraction patterns, extreme cases.
Blazing number $\mu$ (Eq.(2)) as a function of the incident angle
$\alpha^{\prime}$ (top) for a pixel pitch of $d=7.6$µm (left) and $d=10.8$µm
(right). Corresponding far field diffraction pattern (bottom) for two incident
angles $\alpha^{\prime}=0^{\circ}$ and $\alpha^{\prime}=25^{\circ}$. The
yellow cross indicates the maximum of the envelope
$\theta_{\text{max}}=2\theta_{B}-\alpha$.

###  2.3 Modulation cross-talk

In practice, we place a pinhole or iris to select one order of diffraction,
corresponding to the on state. Having a small value for the blaze number $\mu$
not only restricts the amount of light modulation due to the diminished
diffraction efficiency, it also influences the modulation quality by inducing
cross-talk between the two states of the DMD pixels. Until now, we have
assumed that all the pixels are in the same state. In actual usage of the DMD,
it becomes necessary to modulate the state of each pixel individually. When
$\mu$ approaches zero, higher orders of diffraction still possess a
significant intensity, as demonstrated in the 1D case in Fig 3. One adverse
implication is that pixels in the off state may contain orders of diffraction
that are not blocked by the pinhole, and therefore, will contribute as an
interference to the modulated wavefront. We show in Fig. 6 the normalized
amplitude of the diffraction patterns corresponding to all the pixels in the
on (blue curve) and off (orange curve) states for the same experimental
conditions but with two different pixel pitches leading to situations close to
the worst (a) and best (b) case scenarios. We observe the presence of a non-
negligible contribution of the off state at the main diffraction order of the
on state in the first case. While this contribution might appear weak, it does
affect the quality of the modulation since the modulation scheme typically
necessitates about half the pixels to be in the off state for phase modulation
[11], and even more so for elaborate modulation schemes [20, 12].

Figure 6:  Cross-talk between on and off states. We show the computed
normalized amplitude of the diffraction patterns corresponding to all the
pixels in the on state (blue) and off state (orange) for two different pixel
pitches, $d=7.6$µm (a.) and $d=10.8$µm (b.) with the same experimental
conditions. In the first case, $\mu$ is close to $0$, we observe a non-
negligible contribution of the off state at the main diffraction order of the
on state, thus creating unwanted cross-talk. In the second case, $\mu$ is
close to $1$, the contribution of the off state is negligible.

###  2.4 Dispersion

DMDs are composed of metallic small mirrors, the response of which is
minimally affected by wavelength changes. This is particularly advantageous
for broadband applications requiring amplitude modulation and operating on a
plane conjugated to the DMD’s surface. This is the case for the originally
intended application of video projection. However, for wavefront-shaping
applications, it is typically required to select a specific diffraction order
to acheive phase or complex modulation [11, 12]. Under such circumstances, the
wavelength-dependency of the diffraction effect becomes important. The blazed
number, denoted by $\mu$ (according to Eq.(2)), scales inversely with the
wavelength. Fig. 7 shows the blazed number $\mu$ as a function of the
wavelength for two pixel pitches $d=7.6$µm and $d=10.8$µm, with an incident
angle of $\alpha^{\prime}=20^{\circ}$. Within the visible spectrum, $\mu$
fluctuates between $0$ to $1$ over a typical range of roughly 100 nm.

Figure 7:  Dispersion of the diffraction effect. Blazing number $\mu$ (Eq.(2))
as a function of the incident wavelength for an incident angle
$\alpha^{\prime}=20^{\circ}$ and a pixel pitch of $d=7.6$µm (blue curve) and
$d=10.8$µm (dashed black curve).

TL;DR:
DMDs act as blazed gratings. For a given operation wavelength, we need to find
the right pixel pitch to have a good modulation quality and diffraction
efficiency. It can be done by estimating the blazed number $\mu$ introduced in
eq. (2) or directly using our custom online tool [21].

###  2.5 Python code example

We provide in the paper repository [22] a Python code to simulate the
diffraction effect of a DMD by computing the far field pattern for a set of
realistic parameters. It provides a simple way to estimate the blazed number
$\mu$ introduced in eq. (2) to assess the quality of the modulation at the
desired wavelength for a given pixel pitch. We also propose an online tool
accessible at https://www.wavefrontshaping.net/post/id/49.

##  3 Characterizing aberration effects

###  3.1 Presentation of the problem

While only capable of providing hardware binary amplitude modulation, DMDs
serve as a potent tool for wavefront shaping and sensing. These applications
critically require characterization and correction of aberrations caused by
the non-flatness of the DMD surface. For LC-SLMs, the manufacturer typically
characterizes the surface’s inhomogeneities within the plane of the modulator
and provides a spatial phase profile of the introduced aberrations based on
the operational wavelength. It is noteworthy that when utilized for intensity
modulation on a plane conjugated to the DMD plane, as it is done in digital
projectors, the system becomes insensitive to the aberrations caused by the
DMD surface. Consequently, these effects are commonly overlooked and rarely
documented in the information provided by the manufacturers. Nonetheless, they
are non-negligible, with deviations from a flat surface that typically spans
several wavelengths [23].

###  3.2 Finding the correction pattern

In the literature, various methods have been proposed to characterize the
phase pattern of the DMD aberrations in the plane of the modulator. Typically,
this involves using a model for the aberrations, tweaking the parameters to
align with the measurements [24, 14, 23]. An alternative approach entails
direct measurement of the distorted wavefront, either employing a wavefront
sensor such as a Shack-Hartmann [25], or via interferometry. While such
methods yield accurate results, they necessitate adaptation to the particular
setup conditions, and frequently require supplementary optical components,
meticulous alignment, and custom software. Furthermore, they also demand a
meticulously calibrated and stable setup. In this section, we introduce a
straightforward method to characterize the aberrations using a lens and a
camera. This technique can be employed for any system that offers phase
modulation, such as LC-SLMs or deformable mirrors.

Figure 8:  Setup for aberration correction. (a) A schematic representation of
the DMD array divided into macropixels, of which the phase can be controlled.
The precise shape of the macropixels and the area controlled are not crucial
for the characterization of aberrations to be effective, provided that the
spatial sampling is sufficient to capture the highest spatial frequencies of
the aberrations. (b) A schematic representation of the setup employed to
characterize the aberrations. A camera is used to image the far field on the
DMD surface using a lens. (c) shows the ideal intensity pattern corresponding
to the numerical aperture of the illumination setup, i.e. its theoretical
ideal point spread function. (d) show the actual recorded intensity pattern
corresponding to a distorted point spread function due to the DMD aberrations.

We assume the DMD is configured to deliver phase modulation [11, 12]. This
implies that the modulator can be divided into $N$ sections, which we
designate as macropixels, where the phase can be controlled independently. We
use a lens and a camera in its Fourier plane, illuminating the modulator with
a collimated beam that extends over the entire area of the modulator intended
for use. In scenarios where there are minimal or no aberrations, the intensity
pattern observed would mimic the PSF of the lens, such as an Airy disk
depicted in Fig. 8.c. However, in practice, we encounter a substantially
distorted pattern, like the one represented in Fig. 8.d. A more detailed
depiction of the setup for aberration characterization, in the context of a
wavefront shaping application in complex media is presented in Fig. 9.a. In
the case where the medium is a multimode fiber, one could leverage the
reflection from the input surface to directly visualize the intensity pattern
of the input plane, as shown in Fig. 9.b.

Figure 9:  Detailed aberration characterization setup. (a) A laser beam is
expanded and collimated using a telescope (lenses L and L’). The DMD is
represented in transmission for simplicity. Light reflected from the DMD is
focused by a lens L${}_{\text{1}}$ onto the complex medium to study, namely a
scattering medium (SC) or a multimode fiber (MMF). A beamsplitter is used to
image the far field of the DMD onto a camera using a lens (L${}_{\text{2}}$).
The intensity pattern, up to a homotetic transformation, represent the input
excitation on the medium which also corresponds to the PSF of the illumination
setup. (b) In the case of a multimode fiber, one can take advantage of the
reflection of the input facet to directly image the intensity pattern of the
input plane.

We hypothesize that the aberrations brought about by the DMD are smooth, and
can be depicted by a phase pattern $\phi^{\text{aber}}$ in the plane of the
DMD array. This could be feasibly approximated by a finite number of Zernike
polynomials $Z_{n}(r,\theta)$ [26], as follows:

$\phi^{\text{aberr}}(r,\theta)\approx\sum_{n=0}^{N}a_{n}Z_{n}(r,\theta)\,.$
(3)

The goal is to find and display the phase value $\phi_{i}^{\text{corr}}$ on
each macropixel $i$ that best compensates for the aberrations, i.e.
$\phi_{i}^{\text{corr}}=-\phi^{\text{aber}}(r_{i},\theta_{i})$. We create this
pattern in the basis of Zernike polynomials

$\phi_{i}^{\text{corr}}=\sum_{n=0}^{N}a^{\prime}_{n}Z_{n}(r,\theta)\,,$ (4)

the best correction is obtained for

$a^{\prime}_{n}=-a_{n}\,\,\forall n\in[0..N]\,.$ (5)

Figure 10:  Correction phase map. (a) Schematic representation of the function
we want to optimize. We both want to maximize the intensity of the main lobe
of the point spread function and minimize its side lobes. (b) Values of the
experimentally obtained optimal coefficients in the Zernike polynomials basis.
(c) Resulting optimal phase pattern in the chosen illumination area.

We perform a sequential optimization of parameters $a_{n}$ to maximize a
specific function designed to be maximal for the ideal correction of optical
aberrations. We first generate a mask, represented by a disk, centered around
the point of maximum intensity of the original image (refer to Fig. 8.d) with
a radius equivalent to a single speckle grain. The exact size of this radius
for a successful optimization is not critical and can be determined by
approximating the dimensions of the ideal point spread function, expressed as
$r_{0}\approx M\frac{\lambda}{2NA}$, where $NA$ is the numerical aperture of
the optical system and $M$ refers to its magnification. For a given output
intensity pattern, we compute an element-wise product between this image and
the created mask, followed by a summation. This calculated sum is then divided
by an analogous product, but with the complementary mask substituted in place
of the original in order to minimize the side lobes of the point spread
function. We set initial parameter values as $a^{\prime}_{n}=0\,\,\forall
n\in[0..N]$. For each parameter, we test different values of $a^{\prime}_{n}$,
construct the phases for every micropixel according to
$\phi_{i}^{\text{corr}}=\sum_{n=0}^{N}a^{\prime}_{n}Z_{n}(r_{i},\theta_{i})$,
record the resulting intensity profile, and evaluate the corresponding cost
function. For each parameter, the value that results in the maximum output is
retained. To mitigate potential noise or instability, this complete process is
reiterated $3$ times for each parameter.

To estimate the quality of the correction, we compute the Strehl ratio of the
point spread function. It is defined as the maximum of the measured PSF
divided by the maximum of the ideal one. This optimal PSF is the squared
modulus of the Fourier transform of a circular aperture [27]:

$PSF_{\text{ideal}}\propto\left[J_{1}\left(ka\frac{R}{\sqrt{R^{2}+f^{2}}}\right)\right]^{2}\,,$
(6)

with $J_{1}$ the Bessel function of the first kind of order $1$,
$k=2\pi/\lambda$ the wavenumber, $a$ the radius of the aperture, $R$ the
radial coordinate in the Fourier plane, and $f$ the focal length of the lens.

As an illustration, we conduct an optimization procedure using 11 Zernike
polynomials. We use a V-9501 Vialux DMD with a DLP9500 TI chip of resolution
1920 by 1200 pixels and a pixel pitch of 10.8 µm. The optimization is
performed on a disk of radius $340$ pixels, The illumination is done using an
expanded laser beam at 633 nm, corresponding to the aperture of our optical
setup. We exclude the first three Zernike polynomials in the optimization
process, starting from the radial degree $2$. Indeed, the initial one, known
as the piston, does not influence the PSF quality. The subsequent ones, the
tip and the tilt, cause the PSF to shift. Our procedure relies on optimizing
the maximum of the PSF, wherever that is, rendering us indifferent to these
two parameters. After optimization, it is possible to generate the correction
pattern using a selected number of Zernike polynomials in order to investigate
their impact on the PSF quality. We see here that using about 10 Zernike
polynomials is sufficient to obtain a Strehl ratio $>0.99$. Fig. 11
demonstrates the Strehl ratio and the intensity profiles of the PSF for
different counts of the utilized Zernike polynomials. It is important to note
that we do not use the full surface of the DMD. Using a larger area may lead
to stronger deformations of the PSF, requiring a larger number of Zernike
polynomials to be corrected accurately. The experimental data, in addition to
the Python code used to generate the figures, can be accessed in the dedicated
repository [22].

Figure 11:  Effect of Zernike polynomials on the optimization of the PSF. (a)
Strehl ratio of the point spread function as a function of the number of
Zernike polynomials used for the compensation of the aberrations. (b)
Intensity profile of the point spread function for different number of Zernike
polynomials in linear and logarithmic scale. To generate this data, we take
the final solution of the optimization experimental procedure, i.e. the
optimal coefficients $a^{\prime}_{n}$, and generate the corrected PSF with an
increasing number of Zernike polynomials.

###  3.3 Python code example

We provide in the paper repository [22] a Python code example to simulate the
effect of aberration on a DMD and then perform a sequential optimization as
previously proposed to learn the characterize pattern. We make use of the
aotools package [28] for generating Zernike polynomials.

TL;DR:
DMDs are not flat and can introduce aberrations; these are typically much
stronger than those commonly observed with liquid crystal SLMs. This can be
counteracted in situ using a standard setup and a straightforward optimization
procedure to maximize the intensity at the central position of the point
spread function.

##  4 Mechanical and thermal stability

Unlike the original purpose of the DMD, i.e. amplitude modulation for video
projectors, typical scientific applications require a high stability of the
generated wavefront. This is particularly true for applications in complex
media, such as strongly scattering media or multimode fiber, where a small
change in the phase front can lead to a large change in the output intensity
profile. While LC-SLMs design has been improved and adapted to scientific
applications over the last decades, DMD are still relatively new tools for
wavefront shaping and sensing and are prone to instabilities that need to be
addressed by the user. In this section, we present the effect of mechanical
and thermal instabilities and how to limit their impact on the wavefront
quality with simple solutions.

###  4.1 Mechanical stability

Most DMD kits consist of two primary components, the chip itself and the
electronic board that controls it. This could be the standard electronics
board typically used for video projectors, as seen in TI evaluation kits, or
an FPGA specifically designed for rapid scientific usage, as offered by Vialux
[8] for instance. Integral to these electronics is a fan designed to cool both
the chip and the electronic board. However, due to the use of a rigid flat
cable for connection between the chip and the electronics board, these parts
are not mechanically independent. As such, vibrations originating from the
board are partially transmitted to the chip, resulting in minute rotations of
the mirror surface. Although this perturbation is inconsequential for video
projection, they can have significant impacts on applications involving
complex media given their high sensitivity to phase front variations.

Due to its high sensitivity in complex media, it is convenient to characterize
this effect directly on the system’s response, rather than constructing a
distinct setup to analyze the wavefront itself. An example of such a setup is
demonstrated in Fig. 12, although a similar approach can be employed with a
scattering medium. We enlarge a laser beam onto the DMD and transmit the
incoming light through a multimode fiber. Additional elements are required in
the setup to fulfill the requirement for complex modulation [12]. For the sake
of clarity, we present a simplified version of the setup where those elements
are not present. The output from the fiber is then made to interfere with a
reference arm in an off-axis configuration [29], allowing us to detect changes
in the output complex field by recording the interference pattern using a
camera.

Figure 12:  Measuring stability at the output of a multimode fiber. Schematic
representation of the setup used to measure the phase fluctuations through a
multimode fiber. An equivalent setup can be used with a scattering medium.

In the supplementary materials [30], we present an animation illustrating the
dynamic pattern. In off-axis holography, the local transverse displacement of
the fringes is directly proportional to the phase, with a displacement
equivalent to the period of the fringes corresponding to $2\pi$ [29]. This
permits us to estimate the fluctuation in phase over time at a given position
of the output plane. As illustrated in Fig. 13 (depicted by the red curve),
the phase varies rapidly over time, a fluctuation attributed to the rapid
mechanical vibrations transmitted by the board.

Figure 13:  Vibration induced phase fluctuations. Phase fluctuations measured
over time at a specific position on a plane located at the distal end of a
multimode fiber. We employ off-axis holography using the setup depicted in
Fig. 12. The red and blue curves correspond to the phase fluctuations measured
respectively without and with vibration damping by securing the flat cable
with foam, as illustrated in Fig. 14.

A simple yet effective solution consists in dampening the vibrations at the
flat cable’s level by clamping it with a soft material, as depicted in Fig.
14. This can be achieved using commonly available materials. In this context,
we utilize simple foam, typically used for packaging, and secure it to the
cable with two metallic plates, screws, and nuts. We observe a significant
decrease in the phase fluctuations, as demonstrated in Fig. 13 (blue curve).

Figure 14:  Dumping mechanical vibrations. Pictures of experimental setups
where a damping of the vibrations is implemented. (a) The flat cable is
secured with foam and metallic plates. (b) The flat cable is in tension
against a foam material supported by a metallic plate.

TL;DR:
The functioning of DMDs can be perturbed by vibrations transmitted from the
electronics board via the rigid flat cable that links it to the DMD chip. This
adverse effect can be minimized by securing the cable with a soft material
that serves to dampen these vibrations.

###  4.2 Thermal stability

Electronics utilized to control the DMD chip experience thermal variations
during operation. The dynamics of this effect are dependent on the frame rate.
Specifically, the chips heats up more quickly when increasing the frame rate.
The increase of temperature can reach more than $15$ degrees Celsius when
running a sequence at maximum speed (20-30 kHz) [31]. Notably, this effect is
less pronounced when the device is on but not running a sequence. Temperature
fluctuations can cause deformations on the chip’s surface and can modify the
phase response of the glass protective window. This creates low order
aberrations, which degrade the quality of the wavefront. While this issue is
comparatively less critical than static aberrations and mechanical
instabilities previously detailed, it nonetheless has a substantial impact on
the complex medium’s response when a DMD is employed to modulate the input
wavefront.

Before initiating a wavefront shaping experiment, it is important to
characterize the influence of temperature to assess the extent to which it
impacts the results. Although the exact effect on the wavefront distortion can
be directly quantified [31], it is typically more convenient to directly
measure the effect on the studied system’s response. To do so, we use a setup
similar to the one presented in the previous section and depicted in Fig. 12.
We can then estimate the field or intensity decorrelation over time. We
present here results with intensity correlation, as it does not require a
reference arm. The measured output pattern typically take the form of a
seemingly random speckle pattern, that is sensitive to minute changes in the
input wavefront. The correlation estimation is obtained by comparing the
output intensity pattern $I(\vec{r},t)$ at a given time $t$ to the one at
$t=0$. We use the following expression for the correlation:

$C(t)=\frac{\left\langle\bar{I}(\vec{r},t)\bar{I}(\vec{r},t=0)\right\rangle_{\vec{r}}}{\sqrt{\left\langle\bar{I}(\vec{r},t)^{2}\right\rangle_{\vec{r}}\left\langle\bar{I}(\vec{r},t=0)^{2}\right\rangle_{\vec{r}}}}\,,$
(7)

with $\bar{I}(\vec{r},t)=I(\vec{r},t)-\left\langle
I(\vec{r},t)\right\rangle_{t}$ and $\left\langle.\right\rangle_{\vec{r}}$.
$\left\langle.\right\rangle_{t}$ represent the spatial averaging over the
region of interest of the output plane (i.e. the plane of the camera sensor)
and the temporal averaging over the measured frames. Fig. 15 shows the
measured decorrelation over time when running an sequence in an infinite loop.
$t=0$ correspond in the first case (a) to the start of a sequence and is set
in the second one to 4.2 hours post the after of the sequence (b). For the
scenario (a), it is to note that before the sequence commenced, the DMD was
active but remained in the idle state, meaning that it was not executing a
sequence. We observe a non-negligible decorrelation over time after the
sequence started that slows down after about 1 hour. When running the same
experiment 4.2 hours after starting the sequence, the correlation is now
stable.

Figure 15:  Effect of temperature. Temporal decorrelation of a speckle pattern
measured at the output of a multimode fiber when running a sequence in an
infinite loop. We use Eq.(7) to estimate the correlation between the intensity
pattern at a given time $t$ and the one at $t=0$. $t=0$ corresponds
respectively to the start of the sequence (a) and to 4.2 hours after the start
of the sequence (b).

The more efficient way to counteract this effect is to use a closed-loop
system to stabilize the temperature of the DMD chip. This can be achieved by
using a thermoelectric cooler as demonstrated in Ref. [31] and depicted in
Fig. 16.

Figure 16:  Thermal stabilization of the DMD chip. Schematic representation of
the setup utilized to regulate the temperature of the DMD chip. A
thermoelectric cooler and a heat sink is employed to control the temperature,
while a thermistor is used to measure deviations from the temperature
setpoint. A closed-loop system is used to regulate the temperature. Image
adapted from [31].

TL;DR:
DMDs take about an hour to thermaly stabilize when a sequence is running. It
can countered by using a thermoelectric cooler and a temperature sensor. It
can also be simply mitigated by letting a sequence run for few hours before
starting the experiment.

##  5 Conclusion

DMDs are powerful tools for wavefront shaping applications in complex media
due to their high pixel count, relative low cost, and high refresh rate.
However, mostly due to their original purpose of amplitude modulation of
incoherent light for video projection, several effects need to be taken into
account when using DMDs for wavefront shaping. First, the choice of its pixel
pitch must be made carefully by considering the wavelength of operation to
ensure a good modulation quality and diffraction efficiency. Furhtermore, the
DMD surface is not flat and can introduce aberrations which can be corrected
using a simple optimization procedure. Finally, the DMD is sensitive to
mechanical vibrations and thermal variations, which can be mitigated ensuring
a good mechanical and thermal isolation of the device.

## Data and code availability

Data and code examples are available in the dedicated repository [22].

## Appendix A: 1D calculation of the DMD diffraction effect

Assuming the effect of the device’s finite size and illumination to be
negligible, and situating ourselves within the context of the small-angle
approximation, we can represent the field reflected from the device under the
influence of plane wave illumination across two systems as follows:

$\displaystyle R_{\text{flat}}(x)$
$\displaystyle\propto\left[\Pi\left(x/d^{\prime}\right)\otimes_{x}\sum_{k}\delta(x-kd)\right]e^{-j\frac{2\pi}{\lambda}\text{sin}(\alpha)x}$
(8) $\displaystyle R_{\text{blazed}}(x)$
$\displaystyle\propto\underbrace{\left(\underbrace{\Pi\left(x/d^{\prime}\right)}_{\text{pixel
size}}\times\underbrace{e^{j\frac{2\pi}{\lambda}2\text{sin}(\theta_{B}-\alpha)x}}_{\begin{subarray}{c}\text{blazed
angle}\\\ \text{+ angle of incidence}\end{subarray}}\right)}_{\text{pixel
response}}\otimes_{x}\left[\underbrace{\sum_{k}\delta(x-kd)}_{\text{periodicitiy}}.\underbrace{e^{-j\frac{2\pi}{\lambda}\text{sin}(\alpha)kd}}_{\text{angle
of incidence}}\right]\,,$

with $\Pi(x)$ the rectangular function, representing the finite size of the
pixel, defined as:

$\Pi(x)=\begin{cases}1,&\text{if }-\frac{1}{2}<x<\frac{1}{2},\\\
0,&\text{otherwise}.\end{cases}$ (9)

The intensity as a function of the angle in the far-field is given, up to a
homotetic transformation, by the absolute value squared of

The Fourier transform of Eq. 8 can be written as:

$\displaystyle
I_{\text{flat}}(\theta)\propto\sum_{p}\delta(\text{sin}(\theta)+\text{sin}(\alpha))\times\text{sinc}^{2}\left(\left[\text{sin}(\theta)+\text{sin}(\alpha)\right]\frac{\lambda}{d^{\prime}}\right)$
(10) $\displaystyle
I_{\text{blazed}}(\theta)\propto\underbrace{\sum_{p}\delta(\text{sin}(\theta)+\text{sin}(\alpha)-p\,\theta_{D})}_{\text{orders
of
diffraction}}\times\underbrace{\text{sinc}^{2}\left(\left[\text{sin}(\theta)+\text{sin}(\alpha)\right]\frac{\lambda}{d^{\prime}}\right)}_{\text{envelope}}\,.$

We observe that the envelope (right hand term) is maximal for
$\text{sin}(\theta_{max})=\text{sin}(\alpha-2\theta_{B})$ while the effect of
the periodicity (left hand term) is maximal for
$\text{sin}(\theta_{p})+\text{sin}(\alpha)=p\lambda/d$, representing the
orders of diffraction.

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# Quantum beat spectroscopy of repulsive Bose polarons

A. M. Morgen  Center for Complex Quantum Systems, Department of Physics and
Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark  S.
S. Balling  Center for Complex Quantum Systems, Department of Physics and
Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark  K.
Knakkergaard Nielsen  Max-Planck Institute for Quantum Optics, Hans-
Kopfermann-Str. 1, D-85748 Garching, Germany  T. Pohl  Center for Complex
Quantum Systems, Department of Physics and Astronomy, Aarhus University, Ny
Munkegade 120, DK-8000 Aarhus C, Denmark  Institute for Theoretical Physics,
Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Vienna,
Austria  G. M. Bruun  Center for Complex Quantum Systems, Department of
Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C,
Denmark  J. J. Arlt  Center for Complex Quantum Systems, Department of Physics
and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark

###### Abstract

The physics of impurities in a bosonic quantum environment is a paradigmatic
and challenging many-body problem that remains to be understood in its full
complexity. Here, this problem is investigated for impurities with strong
repulsive interactions based on Ramsey interferometry in a quantum degenerate
gas of 39K atoms. We observe an oscillatory signal that is consistent with a
quantum beat between two co-existing coherent quasiparticle states: the
attractive and repulsive polarons. The interferometric signal allows us to
extract the polaron energies for a wide range of interaction strengths,
complimenting earlier spectroscopic measurements. We furthermore identify
several dynamical regimes towards the formation of the Bose polaron in good
agreement with theory. Our results improve the understanding of quantum
impurities interacting strongly with a bosonic environment, and demonstrate
how quasiparticles as well as short-lived non-equilibrium many-body states can
be probed using Ramsey interferometry.

As famously argued by Landau, a bare impurity particle smoothly evolves into a
quasiparticle as the interaction with a surrounding quantum environment is
adiabatically increased. Whereas the concept of quasiparticles was originally
developed to understand the motion of electrons in solids [1], it now has a
much broader scope and forms an incredibly successful platform for
understanding strongly interacting quantum many-body systems [2, 3, 4, 5, 6].

Ultracold quantum gas experiments have in recent years improved our
understanding of quasiparticles, as they allow one to perform Landau’s
gedanken experiment by tuning the interaction strength between an impurity
atom and a surrounding quantum gas. This has led to the observation of the
Fermi polaron [7, 8, 9], which is now well understood theoretically even for
strong interactions [10]. Furthermore, its non-equilibrium formation dynamics
[11, 12] and mediated interactions between two Fermi polarons [13] have been
observed. There is also strong experimental evidence for the Bose polaron with
an energy [14, 15, 16, 17] and formation dynamics [18, 19] that matches
theoretical predictions well for attractive interactions. For repulsive
interactions, the spectroscopic measurements are generally complicated with
broader signals due to damping, the possible presence of few-body bound
states, and a many-body continuum. Except in the perturbative regime [20, 21],
there are indeed many different theoretical predictions, reflecting the
richness of the problem [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].

Figure 1: Illustration of the employed Ramsey interferometry method and a
simplified energy landscape of the system as a function of inverse interaction
strength $1/(k_{n}a)$. A first radio frequency (rf) pulse (black arrow) is
applied to a Bose-Einstein condensate (BEC) in the initial electronic state
$\ket{1}$ (blue dots), which generates a small admixture in the impurity state
$\ket{2}$ (red dots). Its interaction with the background BEC results in
attractive and repulsive Bose polaron branches (green lines). A second rf
pulse closes the interferometric sequence. The presence of two polaronic
branches leads to the observation of a beat signal in the amplitude of the
oscillatory interferometer output.

In this Letter, we use Ramsey interferometry to probe the non-equilibrium
dynamics of impurities in a weakly interacting Bose-Einstein condensate (BEC)
for repulsive interactions. The underlying method is illustrated in Fig. 1. We
observe a fast initial decay and a revival in the amplitude of the oscillatory
interferometric signal. This revival corresponds to a quantum beat, indicating
the presence of two states, which we refer to as the repulsive and attractive
branches. The interferometric measurements allow for a characterization of the
energy of these branches as a function of interaction strength. Based on the
initial decay, several non-equilibrium dynamical regimes are identified and
explained theoretically. These interferometric results provide valuable
information on impurity dynamics and the energy of the Bose polaron for strong
and repulsive interactions, complimenting previous spectroscopic measurements
[14, 17].

The experiment is performed with ${}^{39}\text{K}$ atoms in an optical dipole
trap [34] and follows the same procedure as the investigations for attractive
impurity-medium interactions [18]. The medium is a BEC in the
$\ket{F=1,m_{F}=-1}\equiv\ket{1}$ hyperfine state, where $F$ and $m_{F}$
denote the total angular momentum quantum number and its projection,
respectively. The $\ket{F=1,m_{F}=0}\equiv\ket{2}$ hyperfine state constitutes
the impurity state. The interaction between atoms in these two states is
characterized by the scattering length, $a$, which can be controlled with a
magnetic Feshbach resonance at $113.8G$ [35, 36]. As a result, the interaction
strength $k_{n}a$ can be tuned at will from attractive, $k_{n}a<0$, to
repulsive, $k_{n}a>0$, interactions, where $k_{n}=(6\pi^{2}n_{B})^{1/3}$ is
determined by the average density, $n_{B}$, of the BEC. The typical energy
scale of the system is given by $E_{n}=\hslash^{2}k_{n}^{2}/(2m)$, with the
mass $m$ of ${}^{39}K$, which also provides the characteristic timescale
$t_{n}=\hslash/E_{n}$.

The interferometric sequence is initiated by a radio frequency (rf) pulse on
resonance with the atomic transition between the $\ket{1}$ and $\ket{2}$
states [37], which produces a small $5\%$ coherent admixture of the impurity
state. Subsequently, the system evolves at the chosen interaction strength
$k_{n}a$ for a variable time, $t$, until a second rf pulse with a variable
phase $\phi$ between $-\pi$ and $\pi$ is applied. We extract the signal from
the loss of atoms due to three-body recombination, involving two condensate
atoms and one impurity. The final number of atoms in the BEC has a sinusoidal
dependence on the phase of the second rf pulse, which we parametrize as
$N(\phi)=N_{0}-\mathcal{A}\cos(\phi-\phi_{c})$. The normalized coherence
function is then given by,
$C(t)=\left|\mathcal{A}(t)/\mathcal{A}(0)\right|e^{i\phi_{c}(t)}$, which
corresponds to the time-dependent Green’s function of the impurity,
$G(t)=-iC(t)=-i\bra{\Psi_{\text{BEC}}}\hat{c}(t)\hat{c}^{\dagger}(0)\ket{\Psi_{\text{BEC}}}$,
with the state of the BEC $\ket{\Psi_{\text{BEC}}}$, and the creation operator
for the impurity, $\hat{c}^{\dagger}$ [12, 18].

Figure 2: Impurity dynamics at three characteristic interaction strengths. The
coherence function is given for (a,b) strong $1/(k_{n}a)=0.4$, (c,d)
intermediate $1/(k_{n}a)=0.7$, and (c,f) weak $1/(k_{n}a)=1.5$ interaction
strengths. For comparison, theoretical results for the unitarity-limited case
in Eq. (1) (blue dashed line) and for weak two-body interaction Eq. (2) (green
dash-dotted line) are shown, without any free fitting parameters. Additional
decoherence effects in the system due to three-body losses and the
inhomogeneous density distribution, beyond Eqs. (1) and (2), are accounted for
in the theoretical description [38].

The observed coherence amplitude and phase evolution are shown in Fig. 2 for
three interaction strengths on the repulsive side of the resonance, displaying
a number of striking features. The coherence amplitude shows a fast initial
decay and a clear revival for intermediate and weak interaction strengths in
Figs. 2 (c) and 2 (e). The phase initially increases and then reverses on a
characteristic timescale, in Fig. 2 (d), indicating the crossover between
different dynamical regimes.

Importantly, the dynamics entails the population of a continuum of states,
which together with decoherence effects in the system, leads to a decrease in
the coherence amplitude. On top of this, the energy difference between the
attractive and repulsive branches (see Fig. 1) leads to a quantum beat signal.
In the following, the results shown in Fig. 2 are discussed in detail, and the
energies of the attractive and repulsive branches are inferred from the
signal.

Theoretically, the coherence function can be obtained from a Fourier transform
of the spectral function. At short times, an expansion in orders of $t/t_{n}$
yields

$\displaystyle C(t)$
$\displaystyle=1-(1-i)k_{3/2}\left(\frac{t}{t_{n}}\right)^{3/2}-k_{2}\left(\frac{t}{t_{n}}\right)^{2},$
(1)

valid for times $t\ll t_{a}=ma^{2}/\hslash$. The second term is a consequence
of unitarity-limited two-body interactions [39] with the universal constant
$k_{3/2}=16/(9\pi^{3/2})$ [18, 40]. The third term is an approximate second-
order correction with $k_{2}=\left(Z_{g}\left(E_{g}/E_{n}\right)^{2}/2-4/(3\pi
k_{n}|a|)\right)$, which takes the attractive polaron with energy $E_{g}$ and
residue $Z_{g}$, as well as the high-frequency tail of the spectral function
[19] into account.

The measured coherence function in Figs. 2 (a) and 2 (b) for strong repulsive
interactions $1/(k_{n}a)=0.4$ indeed agrees well with Eq. (1) at short times.
We are, thus, able to quantitatively account for both the fast initial decay
of the coherence amplitude as well as the initial positive phase evolution
[41]. At intermediate interaction strengths of $1/(k_{n}a)=0.7$ in Figs. 2 (c)
and 2 (d), similarly good agreement is obtained for short times, showing the
presence of a universal regime, where Eq. (1) is valid, also in this case. For
weak interactions at $1/(k_{n}a)=1.5$, shown in Figs. 2 (e) and 2 (f), the
universal regime contains only a few data points, since $t_{a}=0.2t_{n}$.

For intermediate to weak interactions, the attractive polaron state lies close
to the molecular state, whose energy to first order is given by
$-\hslash^{2}/ma^{2}$. Therefore, the quantum beat signal between the two
branches enters the dynamics on the same timescale, $t_{a}=ma^{2}/\hslash$, as
the system exits the unitarity-limited regime [38]. As a result, we expect a
dynamical regime described by weak coupling dynamics superimposed with quantum
beats after $t=t_{a}$, which is qualitatively different from the case of
attractive interactions, $1/(k_{n}a)<0$, where no beating is observed. The
weak coupling dynamics is to a good approximation described by

$\displaystyle\begin{split}C(t)&=e^{-iE_{\text{mf}}t/\hslash}e^{-(1+i)(t/t_{w})^{1/2}}.\end{split}$
(2)

In this case the dynamics is governed by the mean-field phase evolution with
$E_{\rm mf}=\frac{4\pi\hslash^{2}an_{B}}{m}$, and the coherence decays on an
interaction strength dependent timescale $t_{w}=\frac{m}{32\pi\hslash
n_{B}^{2}a^{4}}$ [42], following a stretched-exponential form.

For longer times, the measured coherence amplitude and phase in Figs. 2 (c)
and 2 (d) both agree well with this prediction, without, however, capturing
the crossover from the initial dynamics or the beat signal. This is expected
since Eq. (2) does not include the low-lying attractive polaron branch.

Figure 3: Dynamics of impurity evolution for repulsive interactions. The
characteristic dynamical regimes are indicated as a function of inverse
interaction strength $1/(k_{n}a)$ and evolution time $t/t_{n}$. The
experimentally extracted crossover times (blue dots) indicate the transition
from universal (blue region) to weak coupling and beating dynamics (orange
region). For strong interactions and long times, the repulsive polaron decays,
and the system is expected to transition into another dynamical regime (yellow
region), in which the attractive polaron dominates the dynamics.

To examine the crossover in detail, the associated crossover time is extracted
from the coherence phase [43]. We pinpoint the time at which the
experimentally extracted coherence phase becomes closer to the result of Eq.
(2) rather than Eq. (1). The crossover time is identified with the time
between this and the previous data point. The extracted times are presented in
Fig. 3, along with the expected dynamical regimes. The extracted crossover
timescale is in very good agreement with the transition between unitarity-
limited dynamics (blue region) and weak-coupling beating dynamics (orange
region), set by the timescale $t_{a}=ma^{2}/\hslash$ for $1/(k_{n}a)>0.5$.
This shows the existence of universal behavior at short times for all
interaction strengths, extending the result in Ref. [18] to repulsive
interactions.

For strong interactions, there is no regime where Eq. (2) is accurate.
Instead, the short-time unitarity-limited dynamics given by Eq. (1) is
expected to transition directly to many-body dynamics for later times,
consistent with the case of strong attractive interactions [18].
Theoretically, it is expected that the repulsive polaron eventually dampens
out, resulting in the attractive polaron dominating the dynamics, at which
point the system enters a new dynamical regime (Fig. 3 yellow region).
Experimentally, however, we did not observe signatures of this crossover. This
is expected, since we theoretically observe that trap-induced density
inhomogeneities significantly delay the crossover [38].

We now turn to analyze the quantum beat signal, clearly visible in Figs. 2 (c)
and 2 (e), and present throughout the orange region in Fig. 3. Importantly,
such a signal is not present for attractive interactions of $1/(k_{n}a)<0$,
but a similar feature was observed in interferometric investigations of the
Fermi polaron [12]. In close analogy, the signal stems from the quantum
interference of the repulsive and attractive branches, as demonstrated in the
following. We apply a phenomenological model for the coherence function,

$\displaystyle
C(t)=Z_{g}\,e^{-iE_{g}t/\hslash}+Z_{r}\,e^{-iE_{r}t/\hslash}e^{-\Gamma_{r}t/2}.$
(3)

This describes two states; a ground state at energy $E_{g}$, and a damped
excited state with energy $E_{r}>E_{g}$ and damping rate $\Gamma_{r}$,
corresponding to the attractive and the repulsive polarons with residues
$Z_{g}$ and $Z_{r}$, respectively. This implies two main features of the
coherence function for weak-coupling beating dynamics. The slope of the phase
corresponds to the energy of the repulsive branch since it is expected to have
the largest residue, i.e. $\phi_{C}\approx-E_{r}t/\hslash$. In addition, the
squared amplitude of the coherence function contains a sinusoidal oscillation
at a frequency corresponding to the energy difference between the two
branches.

Based on this approach, the experimental coherence amplitude squared and phase
are fitted with the functions

$\displaystyle|\mathcal{C}(t)|^{2}$
$\displaystyle=A\,e^{-t/\tau_{1}}\cos\omega t+B\,e^{-t/\tau_{2}},$ (4)
$\displaystyle\varphi_{c}$ $\displaystyle=A_{\varphi}-B_{\varphi}\,t.$ (5)

This corresponds to the modulus squared of Eq. (3) with an additional
exponential decay of the oscillation amplitude to model three-body decay and
other loss mechanisms. The energy difference can be extracted from the fitting
parameters as $E_{r}-E_{g}=\hslash\omega$. An example of the coherence
amplitude fit is shown in the inset of Fig. 2 (e). Generally, a single beat is
resolved, due to the decoherence effects and losses in the experiment
discussed above [38]. Furthermore, the repulsive branch is expected to be
damped, such that the beat signal, even in an ideal setting, attenuates in
time.

Additionally, the fit directly yields the energy of the repulsive branch,
$E_{r}=\hslash B_{\varphi}$. This technique, however, can only be used when
the phase evolution shows a linear evolution for long times. This is the case
for the phase evolution in Fig. 2 (f) but not for Fig. 2 (d), where the last
data point shows an increase. This limits our analysis to $1/(k_{n}a)>0.85$.
To avoid the influence of the unitary regime, the linear fit is, additionally,
restricted to $t>t_{a}$. Figure 4 (upper panel) shows the obtained energies
for the repulsive branch for this range of interaction strengths. These
energies lie consistently higher than the simple mean-field expectation and
agree with a Monte-Carlo prediction [17]. Moreover, they lie systematically
higher than our diagrammatic calculation based on the ladder approximation
[38].

Figure 4: Energies of the repulsive and attractive branches as a function of
the inverse interaction strength. The measured energies are shown as gray
points and the result of our diagrammatic calculation is indicated by green
dashed lines [38]. (top panel) The repulsive branch energies lie clearly above
the simple mean-field expectation (black dash-dotted line) and agree with a
quantum Monte-Carlo prediction [17] (yellow dashed line, diamonds). (bottom
panel) The attractive branch energies do not allow for a distinction between
the diagrammatic result and the molecular energy (red line).

Based on these results, the energy of the attractive branch is obtained by
subtracting the extracted energy difference from the repulsive branch energy
$E_{g}=E_{r}-\hslash\omega$ as shown in Fig. 4 (lower panel). The energies
agree with both the attractive polaron energy, obtained from diagrammatic
calculations, and the molecular energy with an effective range given by the
Van der Waals length of our medium, $R^{*}=60a_{0}$ [14, 38]. Experimentally,
we cannot distinguish between these two results. However, the rf response of
the molecular state is expected to be very small compared to the attractive
polaron state, as has been previously measured for the Fermi polaron [8]. In
particular, since the molecular state does not have a plane wave component, we
expect such a signal to scale with the density of generated impurities. This
is indeed much weaker than the response from the attractive polaron state
scaling with the density of the entire BEC. With this in mind, the above
agreement confirms our interpretation of the beat signal arising from an
interference between the two polaron states. Furthermore, the simple linear
fitting for the repulsive branch energy yields a robust measurement beyond the
accuracy obtained spectroscopically [14]. Note that we see no evidence of the
recently predicted presence of meta-stable states at intermediate energies
between the two branches [33].

In conclusion, we have performed Ramsey interferometry on a BEC of 39K to
investigate the physics of Bose polarons in a wide range of repulsive
interactions. The observed quantum beat signal is consistent with a repulsive
and attractive quasiparticle branch that form and exist in a coherent
superposition. From the beat frequency and the slope of the coherence phase,
we determine the energy of these branches. The repulsive branch lies
systematically above the simple mean-field expectation and agrees well with
quantum Monte-Carlo calculations [17]. Furthermore, the initial dynamics is
seen to agree very well with the exact unitarity-limited behavior with the
non-analytical time-dependence of $t^{3/2}$ [18, 40], and the crossover to a
weak-coupling regime closely follows the expected $t=ma^{2}/\hslash$ behavior
for $1/(k_{n}a)>0.5$. At even stronger interactions of $1/(k_{n}a)<0.5$, the
dynamics follows the unitarity-limited behavior for long times. This, along
with the quantitative discrepancy from the diagrammatic calculations, calls
for further theoretical analyses, especially at intermediate to strong
repulsive interactions of $0<1/(k_{n}a)<1$.

###### Acknowledgements.

The authors thank Ragheed Alhyder for valuable discussions and Adam Chatterley
for the careful review of the manuscript. This work was supported by the
Danish National Research Foundation through the Center of Excellence “CCQ”
(Grant no. DNRF156), by the Novo Nordisk Foundation NERD grant (Grant no.
NNF22OC0075986), and by the Independent Research Fund Denmark (Grant no.
0135-00205B). KKN acknowledges support from the Carlsberg Foundation through a
Carlsberg Internationalisation Fellowship.

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