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# Coupled states of cold $\mathbf{{}^{174}Yb}$ atoms in a high-finesse cavity

Saran Shaju  Experimentalphysik, Universität des Saarlandes, 66123
Saarbrücken, Germany  Dmitriy Sholokhov  Experimentalphysik, Universität des
Saarlandes, 66123 Saarbrücken, Germany  Simon B. Jäger  Physics Department and
Research Center OPTIMAS, University of Kaiserslautern-Landau, 67663
Kaiserslautern, Germany  Jürgen Eschner  Experimentalphysik, Universität des
Saarlandes, 66123 Saarbrücken, Germany

###### Abstract

We experimentally and theoretically study the formation of dressed states
emerging from strong collective coupling of the narrow intercombination line
of Yb atoms to a single mode of a high-finesse optical cavity. By permanently
trapping and cooling the Yb atoms during their interaction with the cavity, we
gain continuous experimental access to the dressed states. This allows us to
detect both their field and their atomic properties, by simultaneously
measuring the steady-state cavity transmission and free-space fluorescence. By
varying the cavity and probe frequencies, we observe coupled atom-cavity
states with atom number-dependent splitting, the hallmark of collective strong
coupling of the atoms with the single cavity mode. We find additional
fluorescence output at atomic resonance, which we explain by the effects of
dephasing and inhomogeneous broadening. We compare our experimental results
with a theoretical model and find good qualitative agreement.

cold atoms, cavity QED, collective coupling, ytterbium

##  I I. Introduction

Cold atomic ensembles in optical cavities are a versatile platform to study
collective effects such as self-organization [1, 2, 3, 4, 5, 6], super- and
subradiance [7, 8, 9, 10, 11, 12, 13, 14, 15], and in general long-range or
even arbitrary cavity-mediated interactions [16, 17, 18, 19, 20]. Based on
these effects various applications have been proposed reaching from quantum
simulators and computers [21, 22, 23] to ultra-precise lasers and active
optical clocks [24, 25, 26, 27, 28]. Trapping light in the confined geometry
of an optical cavity massively enhances the interaction between a photon and a
single atom. In the presence of many atoms, this interaction is in addition
collectively enhanced which allows for very strong light-matter interactions
and the achievement of very large collective linewidths even when working with
atoms that possess very narrow natural linewidths. With this one can work in a
regime where very strong non-linearities can be generated while unwanted
decoherence channels such as spontaneous emission are weak which is an ideal
ground for future quantum technologies.

However, this field faces a major challenge which arises from atomic motion
inducing line-broadening and also eventually the loss of atoms. Cooling and
trapping [29] of the atoms in a regime where the total broadening is smaller
than the collectively enhanced coupling strength is a minimum requirement. In
several experiments this is achieved in a sequential way where first the gas
is cooled and prepared in a trap and then one investigates collective effects
emerging from the atom-cavity coupling in a second step. In this second step
the atoms will heat which leads to the loss of atoms and sets the time limit
on which operations and simulations can be performed. This is detrimental and
only recently several groups in this field aim at operating such devices in a
continuous regime where atoms are constantly refilled [30, 31, 26, 32, 33, 34]
or where the atom-cavity interactions operate simultaneously with trapping and
cooling [35, 36, 37, 38, 39, 40, 41].

The latter is the situation that we investigate in this paper. We
experimentally and theoretically investigate the formation of dressed states
formed by strong collective coupling between the
1S${}_{0}\,\leftrightarrow\,^{3}$P1 intercombination line of 174Yb atoms and a
resonant optical cavity. Formation of dressed states has been studied in
several works reaching from single-atom experiments [42, 43, 44, 45, 46, 47,
48, 49] to atomic ensembles [50, 47, 51, 52]. More recently these studies have
also been extended to narrow intercombination lines [53, 33, 54]. In this work
we investigate a similar situation but observe both the atomic and cavity
degrees while operating a magneto-optical trap (MOT) on the
1S${}_{0}\,\leftrightarrow\,^{1}$P1 transition which permanently traps and
cools the atoms. This allows us to study their dynamics over long timescales
that are of the orders of several seconds. Over such long timescales we detect
the field as well as the atomic character of the dressed states by measuring
the cavity transmission and by detecting the fluorescence from the narrow
intercombination line, respectively. We compare these measurements to
theoretical predictions that include inhomogeneous broadening of the narrow
atomic line. We find qualitative good agreement for reasonable atom numbers
and broadening. Our work is a step towards exploring the behavior of narrow-
linewidth atoms strongly coupled to optical cavities, while they are
continuously trapped on a broader line, which a possible scenario for their
use in quantum technologies.

This paper is structured as follows. In Sec. II we describe the experimental
setup while Sec. III introduces our theoretical tools. In Sec. IV we report
the measurements of the cavity field transmission and the free-space
fluorescence and compare them to theory. In Sec. V we summarize our results
and conclude.

##  II Experimental Setup and Measurement Protocol

###  II.1 Lasers

Two lasers are employed. One is used for cooling and trapping the Yb atoms in
a conventional 6-beam magneto-optical trap (MOT) on the
1S${}_{0}\,\leftrightarrow\,^{1}$P1 dipole-allowed transition at
$\lambda_{\mathrm{MOT}}=\quantity{399}{nm}$ (”blue”, linewidth
$\gamma_{\mathrm{MOT}}=2\pi\times\quantity{29.1}{MHz}$). The other laser is
near-resonant with the 1S${}_{0}\,\leftrightarrow\,^{3}$P1 intercombination
line at $\lambda=\quantity{556}{nm}$ (”green”, linewidth
$\gamma=2\pi\times\quantity{182.4}{kHz}$), it is used as probe to drive the
atom-cavity system through a cavity mirror. The blue laser is frequency-
stabilized via a transfer locking scheme [55], the green laser is stabilized
to the green transition by atomic-beam spectroscopy. Both lasers are tuned
with acousto-optical modulators (AOMs). The relevant transitions are displayed
in Fig. 1(b).

###  II.2 Atoms

In the experiment we work with 174Yb atoms that are evaporated from an oven
operating at $\quantity{500}{}$. The collimated atoms are guided to a magneto-
optical trap (MOT) and slowed down by a Zeeman slower operating on the
1S${}_{0}\,\leftrightarrow\,^{1}$P1 transition. The MOT, typically operating
at $\Delta_{\text{MOT}}=-2\pi\times\quantity{30}{MHz}$, traps and cools the
atoms in a $\sim\quantity{1}{mm}$-size spatial region around the center of our
optical cavity, see Fig. 1(a). In the overlap region between MOT and cavity
mode, about $10^{4}$ to $10^{5}$ atoms are held at 5 to $\quantity{10}{mK}$
temperature.

###  II.3 Cavity

The high-finesse optical cavity has a length of $L=\quantity{4,78}{cm}$ in
Fabry-Perot configuration and $\quantity{87}{\micro m}$ waist radius of the
TEM00 mode. It couples to the 1S${}_{0}\,\leftrightarrow\,^{3}$P1 atomic
transition at $\lambda=\quantity{556}{nm}$, where it has a loss rate of
$\kappa=2\pi\times\quantity{70}{kHz}$ (finesse $\mathcal{F}=$ {45000}). A
single atom in this setup couples to the cavity with a vacuum Rabi frequency
$g_{0}=2\pi\times\quantity{66}{kHz}$, corresponding to a single-atom
cooperativity $C_{1}=g_{0}^{2}/(\kappa\gamma)=0.34$ [56]. By trapping a large
number $N$ of atoms in the cavity mode, collective strong coupling
$C=NC_{1}\gg 1$ is achieved, which leads to the formation of atom-cavity
coupled states in the system that we investigate. Nevertheless, since the beam
waist of the cavity is small compared to the size of the cloud itself, only a
small percentage of the total number of atoms interact with the cavity.

The cavity length is stabilized by locking it to a reference derived from the
green laser. In our setup, we use a reference beam which is
$\quantity{0.4}{GHz}$ shifted from the green transition frequency and resonant
to the $15^{\text{th}}$ transverse mode of the cavity. Feedback is applied to
piezo-electric actuators attached to the cavity mirrors. The uncertainty of
the cavity frequency is on the order of the cavity linewidth. The cavity
frequency is then controlled by shifting the frequency of the reference beam
with an AOM.

Figure 1: (a) Schematic of the experimental setup for probing the atom-cavity
coupled states. Atoms are held by a conventional 3D MOT on the
1S${}_{0}\,\leftrightarrow\,^{1}$P1 blue line, and a laser drive is provided
along the cavity axis. The cavity transmission is detected using a photo-
multiplier tube (PMT) and an avalanche photo detector (APD) is used for
counting the fluorescence photons emitted by the atoms into free space. (b)
The atomic level scheme shows the relevant transitions for the experiment,
where the atoms are held using the broad blue transition and the atom-cavity
interaction is on the narrow 1S${}_{0}\,\leftrightarrow\,^{3}$P1
intercombination line.

###  II.4 Detection

Detection in our setup happens in two channels: first, detection of the cavity
transmission, and second, detection of the atomic fluorescence emitted into
free space. The transmitted light from the cavity is collected to a single
mode fiber and detected with a photo-multiplier tube (PMT). In the figures
shown below, 1 mV of PMT signal translates to about {70}pW of cavity output,
or $\sim 4\times 10^{4}$ photons inside the cavity. Fluorescence is recorded
using a telescope imaging scheme with a pair of 1-inch lenses to collect and
collimate the photons, which are then fiber-coupled to an avalanche photo
detector (APD) in counting mode. A color filter rejects the blue MOT
fluorescence.

###  II.5 Measurement protocol

The measurement begins with loading the atoms into the MOT, tracking the
fluorescence at $\quantity{399}{nm}$ as a measure of number of atoms trapped.
The atomic flux to the MOT is regulated using a motorized mechanical shutter.
Using this we control the number of atoms that interact with the cavity. The
cloud position is adjusted by the MOT beam directions and powers, and
monitored with CCD cameras to ensure the geometrical overlap with the cavity
mode. Once good overlap is attained that supports the formation of the atom-
cavity states, the cavity is driven with the probe laser.

With the atoms inside the cavity, we set the detuning of the cavity to a fixed
value, $\Delta_{\text{c}}$, and scan the frequency of the probe laser,
$\Delta_{\text{p}}$. For each parameter set
($\Delta_{\text{c}},\Delta_{\text{p}}$) we measure the cavity transmission and
free-space fluorescence using the PMT and APD, respectively, for 5 to 10
seconds to obtain a good signal-to-noise ratio.

We emphasize some aspects in which our experimental situation is peculiar, and
regarding which our results are novel. First, the atom-cavity states are
subject to dephasing induced by (i) the MOT beams and (ii) by the strong
saturation of the green transition inside the cavity. Second, on the scale of
the cavity and atomic linewidth, there is significant inhomogeneous (Doppler)
broadening due to the MOT temperature. To overcome these effects we trap a
large number of atoms and produce a Rabi splitting nearly 2 orders of
magnitude larger than the linewidth. Finally, by measuring the fluorescence
when the strong probe is inside the cavity, we do not only observe the cavity
transmission but also have simultaneous access to the atomic response of the
coupled system.

##  III Theoretical description

###  III.1 Mean-field description

We use a mean-field description for the dynamics of $N$ two-level atoms with
internal ground (excited) state $\ket{g}$ ($\ket{e}$) interacting with the
driven cavity mode. This means we replace all quantum operators by their
expectation values. The equations of motion describing the coupled atom-cavity
dynamics are given by

$\displaystyle\frac{d\alpha}{dt}=$
$\displaystyle\left[i(\Delta_{p}-\Delta_{c})-\frac{\kappa}{2}\right]\alpha-i\frac{\eta}{2}-i\frac{g_{0}}{2}\sum_{j}s_{j}$
(1) $\displaystyle\frac{ds_{j}}{dt}=$
$\displaystyle\left[i(\Delta_{p}-\omega_{j})-\frac{\gamma+\gamma_{d}}{2}\right]s_{j}+i\frac{g_{0}}{2}\alpha
z_{j}$ (2) $\displaystyle\frac{dz_{j}}{dt}=$
$\displaystyle-\gamma(1+z_{j})+ig_{0}(\alpha^{*}s_{j}-s_{j}^{*}\alpha),$ (3)

where $\alpha=\langle\hat{a}\rangle$,
$s_{j}=\langle\hat{\sigma}_{j}^{-}\rangle$, and
$z_{j}=\langle\hat{\sigma}_{j}^{z}\rangle$ are the mean values of the cavity
field annihilation operator $\hat{a}$, the lowering operator
$\hat{\sigma}_{j}^{-}=\ket{g}_{j}\bra{e}$, and the inversion operator
$\hat{\sigma}_{j}^{z}=\ket{e}_{j}\bra{e}-\ket{g}_{j}\bra{g}$, respectively.
This description is derived in a frame rotating with the laser frequency
$\omega_{L}$, and we have introduced the respective detunings between cavity
and atom, $\Delta_{c}=\omega_{c}-\omega_{a}$, and between laser and atom,
$\Delta_{p}=\omega_{p}-\omega_{a}$. In Eq. (1) we have introduced the cavity
linewidth $\kappa$, the cavity-field driving amplitude $\eta$, and the vacuum
Rabi frequency $g_{0}$. The frequencies governing the dynamics of the atomic
degrees of freedom, Eqs. (2)-(3), are the natural atomic linewidth $\gamma$,
homogeneous dephasing $\gamma_{d}$, and an atom-dependent frequency
$\omega_{j}$ which is drawn from a distribution function

$\displaystyle
f(\omega)=\frac{1}{\sqrt{2\pi\Delta\omega^{2}}}e^{-\frac{\omega^{2}}{2\Delta\omega^{2}}}.$
(4)

Inhomogeneous broadening described by $\Delta\omega$ and homogeneous dephasing
described by $\gamma_{\text{d}}$ is introduced to model several effects in our
experiment including Doppler broadening, magnetic shifts, and random motion
introduced by the MOT.

Equations (1)-(3) are the basis of our theoretical analysis, and in the
following we will derive the stationary state which is used for comparison to
the experimental results.

###  III.2 Stationary state

We will now derive the stationary state of Eqs. (1)-(3). We are interested in
both the cavity field and in the excited state population, which are accessed
in the experiment by measuring the cavity transmission and the atomic
fluorescence, respectively.

We find the stationary solution of Eq. (2), which reads

$\displaystyle
s_{j}=\frac{-\frac{g_{0}}{2}z_{j}}{(\Delta_{p}-\omega_{j})+i\frac{\gamma+\gamma_{d}}{2}}\alpha.$
(5)

Using this result in Eq. (3) and imposing $dz_{j}/dt=0$, we obtain the steady
state expression for $z_{j}$ which takes the form

$\displaystyle
z_{j}=-\frac{1}{1+\frac{\frac{g_{0}^{2}|\alpha|^{2}(\gamma+\gamma_{d})}{2\gamma}}{(\Delta_{p}-\omega_{j})^{2}+\left(\frac{\gamma+\gamma_{d}}{2}\right)^{2}}}.$
(6)

With this solution we find $s_{j}$ [Eq. (5)] which is then used to solve
$d\alpha/(dt)=0$ in Eq. (1), such that

$\displaystyle\alpha=\frac{\frac{\eta}{2}}{(\Delta_{p}-\Delta_{c})+i\frac{\kappa}{2}+i\frac{\Gamma}{2}}.$
(7)

Here, we introduced

$\displaystyle\Gamma=i$
$\displaystyle\sum_{j}\frac{\frac{\frac{g_{0}^{2}}{2}}{(\Delta_{p}-\omega_{j})+i\frac{\gamma+\gamma_{d}}{2}}}{1+\frac{\frac{g^{2}_{0}|\alpha|^{2}(\gamma+\gamma_{d})}{2\gamma}}{(\Delta_{p}-\omega_{j})^{2}+\left(\frac{\gamma+\gamma_{d}}{2}\right)^{2}}}$
$\displaystyle=i$
$\displaystyle\int_{-\infty}^{\infty}d\omega\,F(\omega)\frac{\frac{\frac{g_{0}^{2}}{2}}{(\Delta_{p}-\omega)+i\frac{\gamma+\gamma_{d}}{2}}}{1+\frac{\frac{g_{0}^{2}|\alpha|^{2}(\gamma+\gamma_{d})}{2\gamma}}{(\Delta_{p}-\omega)^{2}+\left(\frac{\gamma+\gamma_{d}}{2}\right)^{2}}}$

using the function

$\displaystyle\tilde{f}(\omega)=\sum_{j}\delta(\omega-\omega_{j}).$ (8)

For very large atom numbers $N$ this function approaches $\tilde{f}(\omega)\to
Nf(\omega)$ with the distribution function introduced in Eq. (4). We get then

$\displaystyle\Gamma=i\int_{-\infty}^{\infty}d\omega\,f(\omega)\frac{\frac{\frac{Ng_{0}^{2}}{2}}{(\Delta_{p}-\omega)+i\frac{\gamma+\gamma_{d}}{2}}}{1+\frac{\frac{g^{2}_{0}|\alpha|^{2}(\gamma+\gamma_{d})}{2\gamma}}{(\Delta_{p}-\omega)^{2}+\left(\frac{\gamma+\gamma_{d}}{2}\right)^{2}}}.$
(9)

Note that for $\Delta_{\text{p}}=0=\Delta\omega$ and zero dephasing
$\gamma_{d}=0$ we have $\Gamma=NC\kappa$, with the cooperativity
$C=g_{0}^{2}/(\kappa\gamma)$. More generally, $\Gamma$ represents the line
broadening and dispersive shift of the cavity due the the presence of the
atoms.

At last we calculate the average photon number leaving the cavity per unit
time

$\displaystyle
T=\kappa|\alpha|^{2}=\frac{\frac{\eta^{2}}{4}}{\left|\Delta_{p}-\Delta_{c}+i\frac{\kappa}{2}+i\frac{\Gamma}{2}\right|^{2}}$
(10)

and fluorescent photon number per unit time

$\displaystyle
F=\gamma\sum_{j}\frac{1+z_{j}}{2}=\int_{-\infty}^{\infty}d\omega\frac{\frac{\frac{g_{0}^{2}|\alpha|^{2}(\gamma+\gamma_{d})}{4}}{(\Delta_{p}-\omega)^{2}+\left(\frac{\gamma+\gamma_{d}}{2}\right)^{2}}}{1+\frac{\frac{g_{0}^{2}|\alpha|^{2}(\gamma+\gamma_{d})}{2\gamma}}{(\Delta_{p}-\omega)^{2}+\left(\frac{\gamma+\gamma_{d}}{2}\right)^{2}}}.$
(11)

We solve self-consistently Eq. (9) and Eq. (10) for $|\alpha|^{2}$ which is
then used to calculate the expressions $T$ and $F$. In the figures shown later
we normalize the quantities $T$ and $F$ by the empty cavity transmission
$T_{0}=\eta^{2}/\kappa$ and the collective spontaneous decay rate $N\gamma$.

In the next section we use the results of this model and compare them to our
experimental data.

##  IV Collective light-matter states

In this section we present the main results which include the measured cavity
transmission and fluorescence and its comparison to the theory described in
the previous section.

###  IV.1 Measured cavity transmission and fluorescence

Figure 2:  Atom-cavity coupled states. Measurement (top row) and theory
(bottom row) for cavity transmission (left) and fluorescence (right). (a) The
cavity transmission detected with a PMT when an effective atom number of
25,000 is present in the cavity mode. The white dashed line marks the
resonance position according to the simple dressed-state formula, Eq. (12). A
background originating mainly from detector noise has been subtracted. (b)
Simultaneously measured atomic fluorescence photon counts in 10 seconds. Atom-
cavity coupled states are observed as in the case of cavity transmission. The
peculiar feature which appears to be an additional broad resonance is visible
around $\Delta_{\text{p}}\approx 0$. (c) and (d) show the calculated
normalized cavity transmission $T/T_{0}$, Eq. (10), and fluorescence
$F/(N\gamma)$, Eq. (11), respectively, when the cavity is driven with
$\eta=2\pi\times\quantity{87}{MHz}$ and an inhomogeneous broadening of
$\Delta\omega=2\pi\times\quantity{0.9}{MHz}$ and dephasing of
$\gamma_{\text{d}}=2\pi\times\quantity{1}{kHz}$ are used.

In Fig. 2 we show (a) the measured cavity transmission in units of the voltage
detected by the PMT and (b) the measured fluorescence in units of photon
counts measured by the APD, as functions of the cavity-atom detuning
$\Delta_{\text{c}}$ and probe-atom detuning $\Delta_{\text{p}}$. Both
detunings are varied in a range
$2\pi\times(\quantity{-15}{MHz}\dots\quantity{15}{MHz})$. In the cavity
signal, Fig. 2(a), we observe transmission for large detunings
$|\Delta_{\text{c}}|,|\Delta_{\text{p}}|\geq 2\pi\times\quantity{10}{MHz}$,
and when $\Delta_{\text{c}}\approx\Delta_{\text{p}}$. Closer to atomic
resonance, the transmission peaks show an avoided crossing and their magnitude
tends to zero. The fluorescence signal, displayed in Fig. 2(b), shows a
similar behavior as the transmission, but additionally it exhibits a bright
central feature close to resonance $\Delta_{\text{c}}\approx 0$ across nearly
the whole range of $\Delta_{\text{p}}$. This feature does not appear in the
cavity transmission. To understand this behavior better we apply our
theoretical model and show the corresponding numerical results in Fig. 2(c)
and (d). The cavity transmission displayed in Fig. (2)(c) is in good agreement
with the experimental data in Fig. 2(a). The physical origin of the avoided
crossing is the formation of coupled eigenstates between the atoms and the
cavity. The coupled state resonances are approximately described by

$\displaystyle\Delta^{\prime}_{\text{c}\pm}=\frac{\Delta_{\text{c}}}{2}\pm\frac{1}{2}\sqrt{\Delta_{\text{c}}^{2}+Ng_{0}^{2}}.$
(12)

We show the result of this formula as a guide to the eye in Fig. 2(a)-(d) for
$N=$ {25000}, which is thereby used to estimate the effective number of atoms
participating in the atom-cavity interaction.

The coupled states are also visible in the fluorescence, Fig. 2(b) and (d), as
their atomic contribution undergoes spontaneous decay into free space.
Importantly, in the theory prediction in Fig. 2(d) we also find the central
feature close to atom-probe resonance, $\Delta_{\text{p}}\approx 0$, which is
not part of the formation of atom-cavity coupled states. It will be
investigated in more detail in section IV.3.

###  IV.2 Atom number dependence

Controlling the atomic flux into the trapping volume allows us to extend our
experiments and include variation of the atom number. We focus here on showing
the fluorescence excited via the probe, measured as in the previous section,
but for different numbers of atoms in the cavity mode. Figure 3 shows the
results when $\Delta_{\text{c}}$ and $\Delta_{\text{p}}$ are varied in the
same range as in Fig. 2. As we go from Fig. 3(a) to (d), the atom number is
increased and correspondingly more splitting is visible. In Fig. 3(a),
corresponding to the lowest number of atoms, the fluorescence appears very
close to the diagonal, $\Delta_{\text{c}}=\Delta_{\text{p}}$; for the next
higher atom number one can observe the beginning of splitting close to the
atomic resonance. As we continue to increase the atom number, in Fig. 3(c) a
clear splitting is observed, that becomes noticeably wider as we go to Fig.
3(d). In the latter two maps, the splitting is accompanied by the already
observed central fluorescence maximum; for the smaller values of atom number,
that feature masks the splitting to some extent.

Figure 3:  Fluorescence from coupled atom-cavity states as function of atom
number. In (a) to (d), the effective number of atoms interacting with the
cavity mode is $N=$ {2000}, {5000}, {21250}, {42500}, respectively, according
to the fitted white-dashed line. Fluorescence is counted for 10 seconds per
pixel of the maps; background from APD dark counts has been subtracted.

The normal mode splitting predicted by Eq. (12) is shown in Fig. 3(a)-(d) as
dashed white lines. The observations indicate that for the range of atom
numbers that we are able to explore, we reach far into the collective strong
coupling regime, where the separation of the coupled resonances of the atom-
cavity system is larger than the characteristic frequencies of all relevant
broadening mechanisms.

###  IV.3 Central feature in the fluorescence

In this section we investigate the fluorescence from the atoms with higher
frequency resolution, in particular the central feature which, conspicuously,
does not appear in the cavity transmission. In Fig. 4(a), the detected
fluorescence is plotted as a function of probe-atom detuning when the cavity-
atom detuning is set to zero. The laser frequency is scanned over the range
$\Delta_{\text{p}}=2\pi\times(\quantity{-6}{MHz}\dots\quantity{6}{MHz})$ in
steps of $2\pi\times\quantity{200}{kHz}$, and for each laser frequency the
fluorescence photons are counted for 5 seconds. The measurement shows two
peaks around $\pm\quantity{5}{MHz}$ and a broad peak centered around atomic
resonance. The side peaks are understood as the atom-cavity coupled states,
and their separation is described by $g_{0}\sqrt{N}$ (Eq. (12) with
$\Delta_{\text{c}}=0$) corresponding to an effective atom number of {22500}.
Our prediction from theory calculated using Eq. (11) is displayed in Fig. 4(b)
and shows reasonable agreement with the experimental observation.

Figure 4:  Central fluorescence maximum. Atomic fluorescence as a function of
laser detuning when the cavity is resonant to the atomic transition,
$\Delta_{\text{c}}=0$. (a) Measured photon counts using an APD vs. probe laser
frequency, with {200}kHz resolution. (b) Calculated fluorescence using Eq.
(11), where an effective atom number of {22500} is used with a cavity drive
strength of $\eta=2\pi\times\quantity{80}{MHz}$ and inhomogeneous broadening
of $\Delta\omega=2\pi\times\quantity{0.62}{MHz}$.

We extend our understanding by spanning the cavity-atom detuning over the same
range as the probe-atom detuning, with the same resolution. Figure 5(a) shows
the measured fluorescence vs. $\Delta_{\text{p}}$ and $\Delta_{\text{c}}$.
Figure 5(b) shows the corresponding excited state population calculated from
Eq. (11). Apart from the coupled states as in Fig. 2, the distinct peak around
the atomic resonance is also visible. We find by exploring our theoretical
model that its width increases with $\Delta\omega$, hence it is a consequence
of the inhomogeneous broadening. It also requires significant driving
strength, which indicates that it is a saturation feature. Remarkably, when
introducing dephasing $\gamma_{d}\sim\Delta\omega$ we find that this central
maximum flattens out and becomes invisible. Detecting the feature in our
experiment therefore indicates that our inhomogeneous broadening exceeds the
homogeneous dephasing rate.

Figure 5:  Higher-resolution study of atomic fluorescence. (a) Measured and
(b) calculated fluorescence are displayed vs. $\Delta_{\text{c}}$ and
$\Delta_{\text{p}}$, exhibiting the atom-cavity coupled states alongside the
broad resonance around $\Delta_{\text{p}}=0$. The parameters used for the
calculation are same as in Fig. 4. A frequency resolution of {200}kHz was used
for both $\Delta_{\text{c}}$ and $\Delta_{\text{p}}$.

##  V Discussion and Conclusion

In this work we have investigated the formation of atom-cavity dressed states
in the collective strong coupling regime on a narrow atomic line, while the
atoms are trapped and cooled in a magneto-optical trap. The coupled states are
observed in the cavity transmission as well as in free-space fluorescence,
which enables us to simultaneously investigate their field character and their
atomic character. A theoretical mean-field description reproduces the main
features of the experimental observations. A broad atomic resonance which only
appears in fluorescence is also found in the theory prediction, confirming our
understanding. Hence it allows us to study how inhomogeneous broadening, by
the much broader transition used for trapping and cooling, effects the atom-
cavity interaction on the narrow line.

Our simplified theory models effects such as frequency broadening arising from
atomic motion, due to magnetic field gradients and fluctuations, and
scattering of blue photons, by two simple mechanisms: inhomogeneous broadening
introduced by $\Delta\omega$ and homogeneous dephasing described by
$\gamma_{\text{d}}$. Despite of these simplifications we find satisfactory
agreement between theory and experiment. In addition, it should be noted that
the parameters used in the model calculations are consistent with other
measurements on the same system [40].

Nevertheless, we do not find full quantitative agreement between the theory
and the experiment. The most striking difference is visible in the dressed
state resonances which are far more pronounced in the theory than they are in
the experimental measurement. Our theory predicts stable dressed states even
for driving frequencies close to resonance, $\Delta_{p}\approx 0$. This is not
found in the experiment and therefore requires refined modeling based on
further measurements. One possibility might be a probe-frequency dependent
source of dephasing and broadening which is particularly pronounced close to
atomic resonance. Such a mechanism might be related to opto-mechanical forces
coming from the green laser light. While such forces may be expected to have
even significant effects [3, 1], they have not been considered in the present
paper but will be addressed in the future.

Our experiment and its modeling demonstrate an approach to study the formation
and properties of atom-cavity states on a narrow line, far in the collective
strong coupling regime, and when the atoms are cooled and trapped continuously
on a much ($>$100-x) wider transition. Notably, we simultaneously investigate
the field and the atomic character of the coupled states. Apart from the
coupled states, we highlight a central resonance, visible only in the atomic
signal, which allows us to draw conclusions regarding broadening and dephasing
mechanisms in the atom-cavity system. Apart from fundamentally exploring
collective latter-might phenomena, we expect our work to find significance,
for example, in applications of narrow-line cavity coupling for ultra-stable
lasers or active atomic clocks.

## Acknowledgments

We gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) through the Collaborative Research Center TRR-306
”QuCoLiMa”, sub-project B04. S.B.J. acknowledges stimulating discussions with
S. Schäffer and F. Fama and support from the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) through projects A4 and A5 in TRR-185
“OSCAR”.

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# Rainbow Scars: From Area to Volume Law

Christopher M. Langlett  Department of Physics & Astronomy, Texas A&M
University, College Station, Texas 77843, USA  Zhi-Cheng Yang  Joint Center
for Quantum Information and Computer Science, NIST/University of Maryland,
College Park, Maryland 20742, USA  Joint Quantum Institute, NIST/University of
Maryland, College Park, Maryland 20742, USA  Julia Wildeboer  Department of
Physics & Astronomy, Iowa State University, Ames, Iowa 50011, USA  Alexey V.
Gorshkov  Joint Center for Quantum Information and Computer Science,
NIST/University of Maryland, College Park, Maryland 20742, USA  Joint Quantum
Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
Thomas Iadecola<EMAIL_ADDRESS>Department of Physics & Astronomy, Iowa
State University, Ames, Iowa 50011, USA  Shenglong Xu<EMAIL_ADDRESS>Department of Physics & Astronomy, Texas A&M University, College Station,
Texas 77843, USA

###### Abstract

Quantum many-body scars (QMBS) constitute a new quantum dynamical regime in
which rare “scarred” eigenstates mediate weak ergodicity breaking. One open
question is to understand the most general setting in which these states
arise. In this work, we develop a generic construction that embeds a new class
of QMBS, rainbow scars, into the spectrum of an arbitrary Hamiltonian. Unlike
other examples of QMBS, rainbow scars display extensive bipartite entanglement
entropy while retaining a simple entanglement structure. Specifically, the
entanglement scaling is volume-law for a random bipartition, while scaling for
a fine-tuned bipartition is sub-extensive. When internal symmetries are
present, the construction leads to multiple, and even towers of rainbow scars
revealed through distinctive non-thermal dynamics. To this end, we provide an
experimental road map for realizing rainbow scar states in a Rydberg-atom
quantum simulator, leading to coherent oscillations distinct from the strictly
sub-volume-law QMBS previously realized in the same system.

Statistical mechanics relies on relaxation towards the maximally entropic
state in thermal equilibrium. This process, however, is at odds with the fact
that the entropy of a many-body system prepared in a pure state must remain
identically zero under unitary dynamics. The emergence of statistical
mechanics in such systems, known as quantum thermalization, proceeds by the
relaxation of local sub-regions to a thermal state via the exchange of quantum
correlations with the remainder of the system. This mechanism, whereby a pure
state can become locally indistinguishable from a thermal state, follows from
the eigenstate thermalization hypothesis (ETH) [1, 2, 3, 4]. The ETH
postulates a correspondence between the local reduced density matrix of a
finite-energy-density eigenstate and the Gibbs ensemble.

Many lines of inquiry involve constructing systems where the route to
thermalization detours. For example, quantum integrable systems [5, 6] fail to
thermalize due to extensively many conservation laws; however, these systems
are unstable to perturbations. A more robust violation of the ETH arises in
disordered interacting systems, which may induce many-body localization,
resulting in an extensive number of conservation laws [7, 8, 9, 10].

Experiments utilizing cold atoms [11, 12, 13, 14, 15, 16, 17], ion traps [18,
19], and superconducting circuits [20, 21] have demonstrated unprecedented
control over the dynamics of many-body systems. Recently, experiments in
Rydberg-atom arrays simulating quantum Ising models in one and two dimensions
[22, 23] observed sustained coherent oscillations of local observables for
special initial states, such as the Néel state. This observation was later
traced to the existence of rare, weakly entangled eigenstates in an otherwise
thermalizing system [24, 25]. This phenomenology was dubbed “quantum many-body
scars” (QMBS) [26], an earlier example of which was found in the Affleck-
Kennedy-Lieb-Tasaki spin chain in Refs. [27, 28]. QMBS have been studied in a
wide range of systems, including the “PXP model” simulated by the Rydberg
experiment [29, 30, 31, 32, 33], the spin-1 XY model [34, 35], Fermi-Hubbard
models [36, 37], Floquet models [38, 39, 40, 41, 42], and other systems [43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53]. Group-theoretic techniques [54, 55,
56, 36, 57, 58], matrix product state methods [59], and projector embeddings
[60, 61] have been employed to systematically generate sub-volume-law QMBS in
the many-body spectrum. It remains an open question to construct QMBS with a
specific entanglement structure in the spectrum of a generic Hamiltonian.

In this work, we develop a general construction for a new class of QMBS,
rainbow scars [62, 63, 64], in the spectrum of an arbitrary Hamiltonian
governing a replicated system. Rainbow scars differ from previous examples of
QMBS in that their entanglement scaling strongly depends on the chosen
bipartition. Specifically, the entanglement is volume-law for a random cut,
but sub-volume-law for a fine-tuned cut. In the presence of symmetries,
multiple and even towers of rainbow scar states emerge, opening the
possibility to probe the scar states with quantum quenches. To make a
connection with current experimental efforts, we propose a realization of
rainbow scars in a system of interacting Rydberg atoms, where these states
lead to coherent oscillatory dynamics whose origin is fundamentally distinct
from the previously studied sub-volume law QMBS.

General Construction.—Imagine two related copies of a quantum many-body system
interacting through a few-body term. We begin with the Hamiltonian:

$\displaystyle H=H_{1}\otimes\mathbbm{1}+\mathbbm{1}\otimes
H_{2}+\lambda_{\text{c}}V_{\text{c}}.$ (1)

Each subsystem $H_{1}$ and $H_{2}$ consists of $N$ sites with a
$d$-dimensional local Hilbert space, spanned by the local computational basis
$\ket{s_{i}}$ at site $i$. The state $\ket{S}=\prod_{i}\ket{s_{i}}$ defines
the global computational basis spanning a Hilbert space of dimension $d^{2N}$.
Moreover, in 1D 111The construction Eq. (1) is valid for arbitrary dimensions,
where the mirror-symmetry operator $\mathcal{M}$ is the map
$\mathcal{M}:i\rightarrow\tilde{i}$. For concreteness we restrict ourselves to
one-dimensional systems., the “copied” Hamiltonian, $H_{2}$, satisfies
$H_{2}=-\mathcal{M}H^{*}_{1}\mathcal{M}$, with the mirror-symmetry operator
$\mathcal{M}$ mapping $i\rightarrow\tilde{i}\equiv 2N-i+1$. Complex
conjugation is defined with respect to the computational basis $\ket{S}$. The
two systems interact through $V_{\text{c}}$, which is expected to thermalize
the combined many-body system, akin to two boxes of gas equilibrating through
a thin connecting wire. Provided the condition
$H_{2}=-\mathcal{M}H_{1}^{*}\mathcal{M}$ is met, the construction is
independent of the microscopic details of $H_{1(2)}$.

Figure 1: Rainbow scars in a random Hamiltonian. (a) Average entanglement for
each bipartition $\ell\in[0,2N]$ with the dotted line indicating maximal
entanglement, here $2N=200$. Inset: Depiction of bipartitions. (b) Second-
order Rényi entropy for a random Hamiltonian drawn from the GUE with a
Heisenberg coupling, $\lambda_{\text{c}}=5.0$. Inset: (i) Chosen entanglement
cuts: standard bipartition (blue) and fine-tuned bipartition (orange). (ii)
Rainbow state for $d=2$ with each bond a Bell state.

We proceed by illustrating how a class of non-thermal states emerges from a
large set of degenerate states through a carefully chosen coupling. First, use
the spectral decomposition to express
$H_{1}=\sum_{n=1}^{d^{N}}E_{n}\ket{\psi_{n}}\bra{\psi_{n}}$, where
$H_{1}\ket{\psi_{n}}=E_{n}\ket{\psi_{n}}$. Similarly, express
$H_{2}=-\sum_{n=1}^{d^{N}}E_{n}\ket{\mathcal{M}\psi_{n}^{*}}\bra{\mathcal{M}\psi_{n}^{*}}$,
where $\ket{\mathcal{M}\psi_{n}^{*}}\equiv(\mathcal{M}\ket{\psi_{n}})^{*}$. At
$\lambda_{\text{c}}=0$, the eigenstates of the overall Hamiltonian $H$, with
eigenvalues $E_{n}-E_{m}$, are
$\\{\ket{\Psi_{nm}}=\ket{\psi_{n}}\otimes\ket{\mathcal{M}\psi_{m}^{*}}:\
\forall\,n,m=1,\ldots,d^{N}\\}$, which have no entanglement between the two
halves. Consequently, $H$ has a $d^{N}$-fold degenerate subspace spanned by
$\ket{\Psi_{nn}}$. Within this degenerate subspace, there exists a special
eigenstate independent of the details of $H_{1}$:

$\displaystyle\ket{I}$
$\displaystyle=\frac{1}{d^{N/2}}\sum_{n=1}^{d^{N}}\ket{\Psi_{nn}}=\frac{1}{d^{N/2}}\sum\limits_{S}\ket{S}\otimes\ket{\mathcal{M}S}$
(2)
$\displaystyle=\frac{1}{d^{N/2}}\bigotimes_{i=1}^{N}\sum_{s=0}^{d-1}\ket{s_{i}}\ket{s_{\tilde{i}}},$

where the second equality follows from inserting a resolution of the identity.
This state is precisely the “rainbow state” [62, 63, 64], named for its
characteristic pattern of entanglement, in which every site $i$ is maximally
entangled with its mirror partner $\tilde{i}$ [see Fig. 1(b) middle inset].
Therefore, the entanglement entropy for the standard bipartition [see Fig.
1(b) top inset] scales linearly with system size, $S=N\log d$, while retaining
a simple structure. More generally, for a random bipartition defining a sub-
region $A$ of size $\ell$, the entanglement scales extensively on average when
$\ell\propto N$: $S_{\text{av}}=(2N-\ell)\ell\log(d)/(2N-1)$ [Fig. 1(a)] (see
SM [66]). The rainbow state is reminiscent of the infinite-temperature
thermofield double state [67, 68], denoted as the state $\ket{I}$
corresponding to the identity operator under the state-channel duality [69,
70]. For $\lambda_{\text{c}}\neq 0$, the rainbow state is selected as an
eigenstate of $H$ from the degenerate subspace provided $\ket{I}$ is an
eigenstate of $V_{\text{c}}$. Specifically, for $d=2$, $\ket{I}$ is a product
of long-range Bell states, $\ket{I}=\bigotimes_{i\leq
N}\left(\ket{\uparrow,\uparrow}+\ket{\downarrow,\downarrow}\right)_{i,\tilde{i}}$.
If the subsystems are coupled through, e.g., a Heisenberg interaction,
$V_{\text{c}}=\vec{S}_{N}\cdot\vec{S}_{N+1}$, then $\ket{I}$ is an eigenstate
of the combined system with energy $E_{I}=\lambda_{\text{c}}/4$.

Figure 2: Second-order Rényi Entropy of the XYZ Model. Blue and orange data
points are defined as in Fig. 1. The system parameters are:
$J_{x}=1.0,J_{y}=1.5,J_{z}=2.0$ and $\tilde{J}=0.5,\lambda_{\text{c}}=1.5$.
(a) The rainbow scar $\ket{I}$ appears at energy $\lambda_{\rm c}/4$ for
$h_{x}=1.0,h_{y}=1.5$ (no symmetries). (b) The rainbow scars
$\\{\ket{I},\ket{Y}\\}$ appear at energies
$\\{\lambda_{\text{c}}/4,-3\lambda_{\text{c}}/4\\}$ for $h_{x}=0$
($\mathbb{Z}_{2}$ symmetry).

To emphasize the construction’s generality, consider a system of $2N$ qubits
for which $H_{1}$ (which fixes $H_{2}$) is randomly drawn from the Gaussian
unitary ensemble (GUE), with a local Heisenberg coupling $V_{\rm c}$ acting
the two middle spins as defined previously. Fig. 1(b) shows the second-order
Rényi entropy, $S^{(2)}\equiv-\log\text{tr}(\rho_{A}^{2})$, for each
eigenstate of $H$, where $\rho_{A}$ is the reduced density matrix of sub-
region $A$ for two different entanglement cuts. Blue points denote the
standard bipartition, while the orange points denote the fine-tuned
bipartition [see Fig. 1(b) top right inset]. The appearance of a
“thermalization band” [71, 72, 73, 74, 75] in both cases indicates that the
coupling brings the combined system to equilibrium, as expected for a random
chaotic model. Additional evidence for this is obtained by calculating the
average level spacing parameter [76, 9, 77, 78], $\langle r\rangle\sim 0.594$,
which is near the GUE random matrix result, $0.60$ [79]. For the standard
bipartition, the rainbow state is found as a non-degenerate eigenstate above
the band with maximal entanglement, markedly distinct from previous examples
of QMBS. By contrast, for the fine-tuned bipartition, the rainbow state is a
product state, thus violating expectations from ETH. A priori, a random
chaotic model is not expected to host QMBS; nevertheless, the local Heisenberg
coupling between the two copies is responsible for selecting the state
$\ket{I}$ from the degenerate subspace and elevating it to a scar.

Symmetries.—When symmetries are present, multiple rainbow scars emerge. Let
$\mathcal{O^{\alpha}}$ be symmetry generators satisfying
$[H_{1},\mathcal{O^{\alpha}}]=0$. Then the state
$\ket{\mathcal{O^{\alpha}}}=\left(\mathcal{O^{\alpha}}\otimes\mathbb{1}\right)\ket{I}$
also belongs to the $d^{N}$-fold degenerate subspace at $\lambda_{\rm c}=0$
and is independent of the details of $H_{1}$. Provided the
$\ket{\mathcal{O}^{\alpha}}$ are eigenstates of $V_{\rm c}$, they will emerge
as scars in the spectrum. For example, consider the case where $H_{1}$ has a
$\mathbb{Z}_{2}$ symmetry generated by $\mathcal{O}^{x}=\prod_{i\leq
N}\sigma^{x}_{i}$, where $\sigma^{x}$ is a Pauli operator. The result is an
additional rainbow state, $\ket{X}=\bigotimes_{i\leq
N}\left(\ket{\downarrow,\uparrow}+\ket{\uparrow,\downarrow}\right)_{i,\tilde{i}}$.
If the commutator $[H_{1},\mathcal{O}^{\alpha}]=0$ for each
$\mathcal{O}^{\alpha}=\prod_{i\leq N}\sigma^{\alpha}_{i}$
($\alpha=\\{x,y,z\\}$), then a set of orthogonal rainbow scars,
$\\{\ket{I},\ket{X},\ket{Y},\ket{Z}\\}$ arises in the spectrum. Moreover, if
$H$ possesses a global symmetry or kinetic constraints leading to disconnected
sub-sectors, then the projection of the rainbow state into each sub-sector
yields an eigenstate of $H$.

We examine the consequence of symmetries by studying two coupled XYZ chains of
$N$ spins in a magnetic field:

$\displaystyle H_{1}$
$\displaystyle=\sum_{i=1}^{N-1}\left(J_{x}S_{i}^{x}S_{i+1}^{x}+J_{y}S_{i}^{y}S_{i+1}^{y}+J_{z}S_{i}^{z}S_{i+1}^{z}\right)$
(3)
$\displaystyle+\sum_{i=1}^{N}\left(h_{x}S_{i}^{x}+h_{y}S_{i}^{y}\right)+\tilde{J}\sum\limits_{i=1}^{N-2}S^{z}_{i}S^{z}_{i+2},$
$\displaystyle H_{2}$ $\displaystyle=-\mathcal{M}H_{1}^{*}\mathcal{M}.$

Here, $S_{i}^{\alpha}$ are spin-$1/2$ operators on site $i$. The next-nearest
neighbor interaction $\tilde{J}$ is included to prevent integrability. The
chains are coupled by $V_{\text{c}}=\vec{S}_{N}\cdot\vec{S}_{N+1}$. Depending
on $J_{\alpha}$ and $h_{x(y)}$, the Hamiltonians $H_{1(2)}$ realize different
internal symmetry groups including $\mathbb{Z}_{2}$ and $U(1)$, leading to a
variety of rainbow scars, as we discuss below.

First, when $J_{x}\neq J_{y}\neq J_{z}$ and both $h_{x(y)}$ are non-zero,
$H_{1}$ has no internal symmetries, and the only scar is $\ket{I}$, as
illustrated in Fig. 2(a). Next, setting $h_{x}=0$, $H_{1}$ acquires a
$\mathbb{Z}_{2}$ symmetry generated by $\mathcal{O}^{y}$. Consequently, the
rainbow scars $\ket{I}$ and $\ket{Y}$ are eigenstates with energies
$\lambda_{\text{c}}/4$ and $-3\lambda_{\text{c}}/4$, respectively [see Fig.
2(b)]. With $h_{x}=h_{y}=0$, $H_{1}$ gains a symmetry generated by
$\mathcal{O}^{\alpha}$ for $\alpha\in\\{x,y\\}$, resulting in four orthogonal
rainbow scars, $\\{\ket{I},\ket{X},\ket{Z},\ket{Y}\\}$. Here the first three
states correspond to the triplet states degenerate at $\lambda_{\text{c}}/4$,
while the final state is the singlet state at energy $-3\lambda_{\text{c}}/4$.

Figure 3: $U(1)$ rainbow tower of many-body scars. (a) Overlap between the
rainbow state $\ket{I}$ and each energy eigenstate of Eq. (3), both with
($J_{x}=J_{y}=1.5$, orange) and without ($J_{x}=1.0$, $J_{y}=1.5$, blue)
$U(1)$ symmetry. (b) Second-order Rényi entropy with the standard bipartition
in the $U(1)$-symmetric case ($J_{x}=J_{y}=1.5$). The orange curve highlights
the projected rainbow state in each allowed magnetization sector. Each dot
along the orange curve is doubly degenerate, corresponding to projections of
$\ket{I}$ and $\ket{Z}$, respectively. (c) Krylov time evolution of
$\langle(S^{x}(t))^{2}\rangle/N$ in a system of $2N=20$ spins prepared in
$\ket{I}$, with time step $dt=0.1$. Lighter colors correspond to increasing
$J_{y}=J_{x}+D$, i.e $D=[0.0,0.25,0.50]$ with $J_{x}=1.0$ The remaining
parameters used in (a), (b) and (c) are
$J_{z}=2.0,\mu=0.5,\tilde{J}=0.5,\lambda_{\text{c}}=1.5$.

Finally, when $h_{x}=h_{y}=0$ and $J_{x}=J_{y}$, the total magnetization
$S^{z}=\sum^{2N}_{i=1}S^{z}_{i}$ of the combined system is conserved. In this
case, the four scarred states from the previous example are still present, and
their projections into each magnetization sector (if nonzero) are eigenstates.
For instance, the rainbow states $\ket{X}$ and $\ket{Y}$ lie within the
$S^{z}=0$ magnetization sector. While $\ket{I}$ and $\ket{Z}$, instead, have
finite projections onto all magnetization sectors with
$\sum_{i=1}^{N}S_{i}^{z}=\sum_{i=N+1}^{2N}S_{i}^{z}$; these projections
coincide up to a global phase, leading to $(N+1)$ degenerate eigenstates.
Adding a magnetic field $\mu\,S^{z}$ breaks this degeneracy, resulting in an
exact tower of rainbow eigenstates with equal energy spacing $2\mu$. Climbing
the ladder is accomplished by $N$ applications of the operator
$\hat{J}^{+}=\frac{1}{2}\sum_{i=1}^{N}S_{i}^{+}S_{\tilde{i}}^{+}$ to the fully
polarized state, $\ket{\Omega}=\bigotimes_{i}\ket{\downarrow}$. In Fig. 3(b),
we plot $S^{(2)}$ for each eigenstate, with the non-thermal states spanning
the tower highlighted in orange 222Here the tower of rainbow states are hidden
within the bulk of the spectrum rather than above because these projected
rainbow states only explore a subset of configurations within a specific
$S^{z}$ sector. In [66], we demonstrate that the tower has volume-law
entanglement scaling for the standard bipartition, while the find-tuned cut
scales logarithmically. The other scar states $\\{\ket{X},\ket{Y}\\}$ in the
$S^{z}=0$ sector are non-zero because they are exact eigenstates of the
magnetic field term.

Performing a quantum quench from any state with a finite overlap with each
eigenstate of the tower leads to perfect coherent oscillatory dynamics [54,
56, 57, 58]. In particular, preparing the Hamiltonian Eq. (3) in either
$\ket{I}$ or $\ket{Z}$ results in perfect oscillations, quantified through the
non-local correlator, $\langle(S^{x}(t))^{2}\rangle/N$ for $2N=20$ spins,
where $S^{x}=\sum S^{x}_{i}$. These oscillations are found to be remarkably
robust to perturbations. We perturb Eq. (3) with $J_{y}=J_{x}+D$ [81], where,
at $D=0$, the $U(1)$ symmetry is exact and the correlator has the analytical
form, $\langle(S^{x}(t))^{2}\rangle/N=\langle S^{x}(0)^{2}\rangle\cos^{2}(\mu
t)/N$. When $D$ is nonzero, the $U(1)$ is explicitly broken, yet the
oscillations remain strong for deviations up to $D\sim 0.50$ upon which
thermalization occurs after a transient [see Fig. 3(c)].

Spectral Symmetry and Experimental Realization.—A drawback of the construction
Eq. (1) is the minus sign on $H_{2}$, which is challenging to implement
experimentally. This dilemma is solved in models with a spectral-reflection
symmetry [82], where each non-zero eigenstate $\ket{E}$ has a chiral partner
$\ket{-E}$. This results from considering the operator $\mathcal{O}$
satisfying $\\{\mathcal{O},H_{1}\\}=0$; we can then define
$H_{2}=+\mathcal{M}H_{1}^{*}\mathcal{M}$ and the state
$\ket{\mathcal{O}}=\left(\mathcal{O}\otimes\mathbb{1}\right)\ket{I}$ as an
eigenstate of $H_{1}+H_{2}$.

As an example, we consider a chain of interacting Rydberg atoms with a non-
uniform spacing [see Fig. 4(a)] governed by the Hamiltonian

$\displaystyle H=$
$\displaystyle\frac{\Omega}{2}\sum_{i=1}^{2N}\sigma^{x}_{i}+\sum_{i<j}V_{i,j}n_{i}n_{j}-\sum_{i=1}^{2N}\Delta_{i}n_{i}\,.$
(4)

Here, we set the interatomic spacing $a=1$ except between sites $N$ and $N+1$,
where the spacing is $\tilde{a}$. The operator $\sigma^{x}_{i}$ connects the
internal ground state $\ket{g}_{i}$ to the Rydberg state $\ket{r}_{i}$ of the
$i$-th atom, with parameters $\Omega$ (Rabi frequency) and $\Delta_{i}$
(detuning) characterizing the drive laser. Rydberg states interact through
$V_{i,j}=V_{0}/r_{i,j}^{6}$, with operators
$n_{i}=\left(1+\sigma^{z}_{i}\right)/2$. In the limit $V_{i,i+1}\gg\Omega\gg
V_{i,i+2}$, we take $V_{N,N+1}=V_{0}/\tilde{a}^{6}$ to be comparable to
$\Omega$; equivalently, we take $\tilde{a}>1.0$. In addition, we take
$\Delta_{i}=0$ except for the two central sites, where
$\Delta_{N}=\Delta_{N+1}=\Delta_{\text{opt}}=V_{N,N+1}/2$. The coupling
between the central sites then becomes
$V_{0}\sigma_{N}^{z}\sigma_{N+1}^{z}/4\tilde{a}^{6}$.

In the limit $V_{i,i+1}\gg\Omega\gg V_{i,i+2}$, a pair of $U(1)$ conservation
laws emerge, with generators
$n^{rr}_{1(2)}=\sum^{N}_{i(\tilde{i})=1}n_{i(\tilde{i})}n_{i(\tilde{i})+1}$
that count the number of nearest-neighbor pairs of Rydberg excitations in each
half of the chain. The projection of $H$ onto a sector with fixed
$n^{rr}_{1,2}$ reads

$\displaystyle H$
$\displaystyle=H_{1}+H_{2}+\frac{V_{0}}{4\tilde{a}^{6}}\sigma^{z}_{N}\sigma^{z}_{N+1}+V_{0}\left(n^{rr}_{1}+n^{rr}_{2}\right),$
(5) $\displaystyle H_{1}$
$\displaystyle=\mathcal{P}_{1}\left(\frac{\Omega}{2}\sum_{i=1}^{N}\sigma^{x}_{i}\right)\mathcal{P}_{1},\quad
H_{2}=\mathcal{P}_{2}\left(\frac{\Omega}{2}\sum_{i=1}^{N}\sigma^{x}_{i}\right)\mathcal{P}_{2},$

where $\mathcal{P}_{1(2)}$ projects the left (right) half of the chain into a
sector with fixed $n^{rr}_{1(2)}$. The Hamiltonians $H_{1}$ and $H_{2}$
individually have a spectral-reflection symmetry, since
$\\{\mathcal{O}^{z},H_{1(2)}\\}=0$ where the product is over the first (last)
$N$ sites 333Here the projection operators, $\mathcal{P}_{1(2)}$ commute with
the spectral-symmetry generators, i.e.,
$[\mathcal{P}_{1(2)},\mathcal{O}^{z}]=0$.. When
$\mathcal{P}_{1}=\mathcal{M}\mathcal{P}_{2}\mathcal{M}$ (note
$\mathcal{P}_{1}^{*}=\mathcal{P}_{1}$), then $n^{rr}_{1}=n^{rr}_{2}$ and
$H_{2}=+\mathcal{M}H_{1}^{*}\mathcal{M}$. Together with the spectral-
reflection symmetry, this implies that the rainbow state
$\left(\mathcal{P}_{1}\otimes\mathcal{P}_{2}\right)\ket{Z}$ is an eigenstate
of $H_{1}+H_{2}$. This state is also an eigenstate of the coupling, and
therefore of the overall $H$ in Eq. (5).

Figure 4: Dynamical signature in a chain of interacting Rydberg atoms. (a)
Depiction of a non-uniformly spaced Rydberg chain. (b) Maximum overlap of
$\ket{Z}$ projected into the sub-sector absent of neighboring Rydberg states
for different interaction strengths. Nearest-neighbor (all-to-all)
interactions is denoted by blue(red). (c) Dynamics of the average expectation
value between inversion pairs,
$\overline{\langle\sigma^{x}_{i}(t)\sigma^{x}_{\tilde{i}}(t)\rangle}$ prepared
in $\ket{Z}$. Inset: Short time dynamics for $t\Omega\sim 1.5$. (d) Néel state
dynamics for the correlator,
$\overline{\langle\sigma^{z}_{i}(t)\sigma^{z}_{\tilde{i}}(t)\rangle}$.
Parameters used in (b), (c) and (d): $\Omega/2\pi=2$MHz, $V_{0}=12\Omega$,
$\Delta_{\text{opt}}=V_{0}/2\tilde{a}^{6}$ with $\tilde{a}\sim 1.51$ and
$2N=16$.

Such a rainbow state exists for each sub-sector satisfying
$\mathcal{P}_{1}=\mathcal{M}\mathcal{P}_{2}\mathcal{M}$, leading to an
equally-spaced tower of scar states with energies
$V_{0}/4\tilde{a}^{6}+2V_{0}\,n_{1}^{rr}$. We emphasize this tower is distinct
from the strictly sub-volume-law scars of the PXP model, which reside in the
sector with $n^{rr}_{1}=n^{rr}_{2}=0$ [scartheo1, 24]. This tower of states
becomes exact in the limit $V_{i,i+1}\gg\Omega\gg V_{i,i+2}$; remarkably, it
is also robust away from this limit.

In Fig. 4(b), we determine the maximum overlap between each eigenstate and the
projection of $\ket{Z}$ into the $n^{rr}_{1}=n^{rr}_{2}=0$ sector. For
strictly nearest-neighbor interactions (blue), the maximum overlap asymptotes
to unity as $V_{0}\to\infty$. However, this is not the case when the full van
der Waals interaction is accounted for (red); here, the overlap grows slowly,
never exceeding $\sim 0.5$. This is a result of the next-nearest-neighbor
interactions breaking the spectral-reflection symmetry of $H_{1,2}$ in Eq.
(5).

Fig. 4(c) shows the quench dynamics of the $\ket{Z}$ rainbow state under the
Hamiltonian Eq. (4). We consider both nearest-neighbor (blue) and full van der
Waals interactions (red) with parameters $V_{0}=12\,\Omega$ and interchain
spacing $\tilde{a}\sim 1.51$. Remarkably, for nearest-neighbor interactions,
the oscillations are robust, persisting well beyond the local thermalization
timescale $1/\Omega$. In the limit $V_{0}\to\infty$, the coherent dynamics
become exactly periodic with a period $\tau=\pi/V_{0}$ as a consequence of the
rainbow tower. Including long-range interactions leads to faster relaxation
dominated by next-nearest-neighbor terms on a timescale $1/V_{i,i+2}$. This
dynamical behavior is confirmed by measuring the average expectation value
between inversion partners,
$\overline{\langle\sigma^{x}_{i}(t)\sigma^{x}_{\tilde{i}}(t)\rangle}=\sum_{i}\langle\sigma^{x}_{i}(t)\sigma^{x}_{\tilde{i}}(t)\rangle/2N$.
Interestingly, the sub-volume-law scars of the PXP model [24, 25] coexist with
the rainbow scars, still displaying a strong dynamical signature, illustrated
in Fig. 4(d) by preparing the system in the Néel state. We emphasize that the
dynamical signature of the rainbow tower is more robust than that of the PXP
scars for nearest-neighbor interactions. This results from the fact that
$\ket{Z}$ has unit overlap with the rainbow tower in the limit
$V_{0}\to\infty$, whereas the PXP tower remains approximate in this limit.

Experimental preparation.—Within our construction, rainbow state preparation
requires non-local gates to entangle inversion partners at sites $i$ and
$\tilde{i}$, posing an experimental challenge. One solution is quantum state
reversal, which takes a system of nearest-neighbor Bell pairs and reverses the
state of every other site through the center of the chain [84, 85, 86].
Another possibility in the Rydberg platform is to utilize the power of optical
tweezers to control the geometry. Specifically, if the geometry is altered to
a ladder configuration, rainbow state preparation becomes a local operation;
returning to the chain platform is accomplished by “unfolding” the traps.
However, as we show in [66], the non-ergodic dynamics discussed here persist
even in the ladder geometry, making it a promising alternative to probe the
rainbow scars experimentally.

Outlook and conclusion.—This work gives a general recipe to realize a new
class of QMBS, dubbed rainbow scars, that are related to the infinite-
temperature thermofield double states. These scar states emerge in any system
of the form (1), so long as $(i)$ $H_{2}=-\mathcal{M}H^{*}_{1}\mathcal{M}$ and
$(ii)$ the rainbow state (2) is an eigenstate of the term $V_{\text{c}}$
coupling the sub-systems. Symmetries enrich the construction, leading to
orthogonal sets or even towers of rainbow scars. These non-thermal states
display volume-law entanglement for random bipartitions and sub-volume law
scaling for a fine-tuned bipartition, as well as perfect coherent dynamics in
the presence of towers. Our work serves as an experimental blueprint for
Rydberg simulators, where we find a robust dynamical signature distinct from
previous studies. A possible future research direction is generalizing this
method to the finite-temperature thermofield double state, where non-
equilibrium dynamics is not well understood. Moreover, this construction is
readily generalized to bilayer systems in higher dimensions, opening a route
to new QMBS manifestations.

Acknowledgments.—Z.-C.Y. and A.V.G. acknowledge funding by AFOSR, AFOSR MURI,
NSF PFCQC program, DoE ASCR Quantum Testbed Pathfinder program (award No. DE-
SC0019040), U.S. Department of Energy Award No. DE-SC0019449, DoE ASCR
Accelerated Research in Quantum Computing program (award No. DE-SC0020312),
ARO MURI, and DARPA SAVaNT ADVENT. Portions of this research were conducted
with the advanced computing resources provided by Texas A&M High Performance
Research Computing.

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