# Collective modes in imbalanced nodal line semimetal

SK Firoz Islam  Department of Applied Physics, Aalto University, P. O. Box
15100, FI-00076 Aalto, Finland  Alexander A. Zyuzin  Department of Applied
Physics, Aalto University, P. O. Box 15100, FI-00076 AALTO, Finland  Ioffe
Physical–Technical Institute, 194021 St. Petersburg, Russia

###### Abstract

In this work, we investigate collective modes in a nodal line semimetal with
two nodal-lines that have opposite spin polarization in the presence of spin
population imbalance. We find the components of polarization operator taking
into account the electron-electron exchange interaction and obtain the
dispersion relations of collective modes for the bulk and surface states.
There exist four modes in the bulk, among which one is gapless and other three
are gapped. The gapless surface mode is sensitive to the boundary conditions.

Introduction Recently, three dimensional Dirac materials where conduction and
valence bands merge along a line, so called nodal line (or ring) semimetals
(NLSM), have attracted intense research interests due to their unique band
structure Mikitik and Sharlai (1999); Heikkila and Volovik (2011); Burkov  _et
al._ (2011). The nodal line can be associated with the non-trivial Berry phase
and the semimetal supports unusual surface states, for a recent review Yang
_et al._ (2018). The existence of such band structure may be protected by time
reversal, spatial inversion or mirror symmetries Kim  _et al._ (2015); Fang
_et al._ (2015); Chan  _et al._ (2016). A variety of materials have been
reported Bian  _et al._ (2016); Hu  _et al._ (2016); Takane  _et al._ (2016);
Wang  _et al._ (2017); Liu  _et al._ (2018); Takane  _et al._ (2018); Behrends
_et al._ (2017). The presence of almost non-dispersive Landau level was
revealed in Ref. Rhim and Kim (2015). ARPES and quantum oscillation
measurements have revealed the signatures of the nodal ring, see, for example
Hu  _et al._ (2016); Kumar  _et al._ (2017) and review Lv  _et al._ (2021).

The collective excitations are the fundamental optical properties of the
fermionic systems. It was noted that the features of plasmon modes in gapless
Dirac material like grapheneHwang and Das Sarma (2007) drastically differs
from the electronic systems described by quadratic band structure. A similar
investigation was also carried out in siliceneVan Duppen  _et al._ (2014), a
two-dimensional puckered spin-orbit coupled honeycomb lattice of silicon
atoms, exploring the spin and valley dependent plasmon modes. Recently, much
attention has been paid to Dirac-Weyl materials in investigating plasmon modes
with a particular focus on the chiral anomaly Das Sarma and Hwang (2009); Zhou
_et al._ (2015); Panfilov  _et al._ (2014). Very recently, such collective
modes have been studied in NLSMs as well Rhim and Kim (2016); Yan  _et al._
(2016). It was shown that the plasmon modes in three dimensional Weyl and NLSM
are gapped in the long wave-length limit. The gap is determined by the
respective density of states, which in the former case may be tuned by the
parallel electric and magnetic field thanks to the chiral anomaly. Although,
the impact of electron-electron exchange interaction in the presence of
population imbalance (either valley or spin, etc.) on collective excitations
in Dirac materials is far from being understood.

The importance of exchange interaction in the situation with non-equilibrium
distribution of electron spin on spin-wave collective modes was first
emphasized in Ref. Aronov (1977). Later the spin-waves in metals and
semiconductors have been extensively investigated theoretically in Refs.
Aronov, 1977; Deutsch  _et al._ , 2010; Bedell and Dahal, 2006; Zyuzin and
Zyuzin, 2010; Bashkin, 1986. Several experimental works were reported on the
detection of spin-wave in gaseous spin polarized hydrogen, in the mixture of
${}^{3}He$ and ${}^{4}He$, polarized ${}^{3}He$ (see Ref. Bashkin, 1986).
Recently, with analogy to spin, the exchange interaction induced valley wave
in the Dirac materials with valley population imbalance has been discussed in
Ref. Zyuzin and Zyuzin (2019). Apart from graphene Ju  _et al._ (2011); Di
Pietro  _et al._ (2013) and Weyl semimetal Chiarello  _et al._ (2019), the
NLSM plasmon modes have been recently observed experimentallyXue  _et al._
(2021).

In this work, we consider a NLSM hosting two nodal rings with opposite spin
orientation in the presence of the spin imbalance. The model of a semimetal
with concentric loops that come from different spin channels in its band
structure was introduced in Zhang  _et al._ (2020). We consider transverse
spin waves mediated by the electron-electron exchange interaction. In this
model, we find three gapped modes and a gapless mode. These modes are
anisotropic and disperse quadratically in all direction in the long-wave
limit. We also obtain a gapless spin-wave mode corresponding to the surface
states.
Collective modes We start with the effective Hamiltonian describing a model of
semimetal with two nodal-rings that have opposite spin polarization, Zhang
_et al._ (2020)

$H_{\eta}(\mathbf{k})=\epsilon\left(\frac{k_{\perp}^{2}}{Q^{2}}-1\right)\sigma^{x}+v_{z}k_{z}\sigma^{z}-\eta\frac{\lambda
n}{2},$ (1)

where $\epsilon>0$ determines the top and bottom edges of the conductance and
valence bands, respectively, and serves as an energy cut-off for the two-band
model, $Q$ defines the radius of the nodal ring in the plane $k_{z}=0$,
$v_{z}$ is the Fermi velocity along the $z$-direction, and
$\sigma^{i},~{}i=(x,y,z)$ are the Pauli matrices acting on the orbital space.
The last term describes the antisymmetric part of the electron-electron
exchange interaction energy between two nodal rings, where the nodal ring
spin-index is denoted by $\eta=\pm 1$. Note the exchange interaction is taken
to be momentum independent within the simplest approximation for the Fourier
component of screened Coulomb potential by $\lambda>0$.

The low energy band dispersion is given by

$E_{b,\eta}=b\sqrt{\epsilon^{2}(k_{\perp}^{2}/Q^{2}-1)^{2}+v_{z}^{2}k_{z}^{2}}-\eta\frac{\lambda
n}{2}$ (2)

where $b=\pm$ is the band index. The exchange interaction induced term splits
the position of nodal rings in energy by $|\lambda n|$ with respect to each
other. This is similar to the case of Dirac semimetals Zyuzin and Zyuzin
(2019).

The exchange energy in Eq. (1) can be evaluated self consistently from the
difference of particle densities between two nodal-rings as

$n=\sum_{b=\pm}\int\frac{d^{3}k}{(2\pi)^{3}}\\{f[E_{b,+}]-f[E_{b,-}]\\},$ (3)

where $f[E_{b,\eta}]=[1+e^{(E_{b,\eta}-\mu_{\eta})/T}]^{-1}$ is the Fermi-
Dirac distribution function in the nodal ring $\eta$ at temperature $T$.
Summation is performed over two bands. The spin imbalance is described by the
spin-dependent chemical potential as $\mu_{\pm}=\mu\pm\delta\mu/2$. As noted
above, $\epsilon>\mu$ is the largest energy scale in the problem, so that two-
band approximation in formula (1) may be valid. For our study, we keep zero
temperature $T=0$ throughout the discussion.

At $0<\mu_{\pm}\pm\lambda n/2<\epsilon$, the particle density imbalance is
linearly proportional to $\delta\mu$:

$n=\frac{\nu\delta\mu}{1-\lambda\nu},$ (4)

where $\nu=\mu Q^{2}/4\pi\epsilon v_{z}$ is the density of states at the Fermi
level.

To proceed, let us consider a transverse magnetic field $\propto e^{i(\omega
t-{\bf q\cdot r})}$, which provides the electron transition between the two
nodal rings with spin imbalance. The dispersion of collective modes is given
by the poles of transverse susceptibility. The retarded and advanced Green
functions are given by $G_{\eta}^{R/A}({\bf k},\omega)=[\omega-
H_{\eta}(\mathbf{k})\pm i\delta]^{-1}$ and that can be used to evaluate the
Keldysh Green function as $G_{\eta}^{K}({\bf
k},\omega)=[1-2f_{\eta}(\omega)][G_{\eta}^{R}({\bf
k},\omega)-G^{A}_{\eta}({\bf k},\omega)]$. To obtain the dispersion relation
of the inter-ring collective modes, we follow the standard random phase
approximation and seek the poles of the transverse susceptibility as
$\mathrm{det}(1-\lambda\Pi)=0$, where the inter-ring polarization operator is
a $4\times 4$ matrix, which is given by

$\displaystyle\Pi_{ab,cd}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{d\Omega}{2\pi}\Big{[}G_{-,ab}^{R}(k+q,\omega+\Omega)G_{+,cd}^{K}(k,\Omega)$
(5) $\displaystyle+$ $\displaystyle
G_{-,ab}^{K}(k+q,\omega+\Omega)G_{+,cd}^{A}(k,\Omega)\Big{]}.$

It is convenient to write the polarization operator in new pseudo-spin
representation
$\Pi^{\alpha\beta}=\mathrm{Tr}_{2}\sigma_{bc}^{\alpha}\Pi_{ab,cd}\sigma_{da}^{\beta}/2$
where indices $\alpha,\beta$ take the values $(0,x,y,z)$, i.e, one singlet and
three triplets, and $\mathrm{Tr}_{2}$ denotes trace of Pauli matrices.

The polarization operator in new representation can be written through the sum
of intra-band and inter-band terms as
$\Pi^{\alpha\beta}=\Pi^{\alpha\beta}_{intra}+\Pi^{\alpha\beta}_{inter}$. The
diagonal components which correspond to the inter-band contributions are
obtained as $\Pi^{00}_{inter}=0$,
$\Pi^{xx}_{inter}=2\Pi^{yy}_{inter}=\Pi^{zz}_{inter}=\nu\ln(2\epsilon/\mu)/2$.
On the other hand, the intra-band components are obtained as

$\displaystyle\Pi^{00}_{intra}=-\nu\frac{\delta\mu+\lambda n}{\omega-\lambda
n}-\nu\frac{\delta\mu+\omega}{(\omega-\lambda
n)^{3}}\left(v_{\perp}^{2}q_{\perp}^{2}+\frac{v_{z}^{2}q_{z}^{2}}{2}\right),$
$\displaystyle\Pi^{xx}_{intra}=-\frac{\nu}{2}\frac{\delta\mu+\lambda
n}{\omega-\lambda n}-\nu\frac{\delta\mu+\omega}{(\omega-\lambda
n)^{3}}\left(v_{\perp}^{2}q_{\perp}^{2}+\frac{v_{z}^{2}q_{z}^{2}}{2}\right),$
(6)

$\Pi_{intra}^{yy}=0$, and $\Pi_{intra}^{zz}=\Pi_{intra}^{xx}$, where
$v_{\perp}=|\epsilon|/Q$. The off diagonal intra-band components are obtained
up to linear in $q$ order as $\Pi^{0x}_{intra}=\Pi^{0y}_{intra}=0$,
$\Pi^{0z}_{intra}=\Pi_{intra}^{xz}=\Pi_{intra}^{yz}=\Pi_{intra}^{zz}=\nu
v_{z}q_{z}/(\omega-\lambda n)$, and $\Pi^{xy}_{intra}=0$. The off-diagonal
components describe coupling between different modes.

Figure 1: Dispersions of the collective modes shown for the case $q_{z}=0$.
The particle-hole continuum is depicted by the region confined by two lines
$\omega=\lambda n\pm v_{\perp}q_{\perp}$.

Using the solution of the self-consistency equation (4) and assuming $\lambda
n>v_{\perp}q_{\perp},v_{z}q_{z}$ and $\lambda n>|\omega|$, the dispersion of
collective modes can be obtained as

$\omega_{0}=-\frac{1-\lambda\nu}{\lambda
n}\left(v_{\perp}^{2}q_{\perp}^{2}+\frac{1+\gamma}{2}v_{z}^{2}q_{z}^{2}\right),$
(7)

where $\gamma=\lambda^{2}\nu^{2}[(1-\lambda\nu)(1-C)]^{-1}\ll 1$ with
$C=2\lambda\nu\ln(2\epsilon/\mu)$. The other modes are given by

$\displaystyle\omega_{x}=\frac{\lambda
n}{2}\left(1-C\right)-\frac{1-\lambda\nu}{\lambda
n}\left(v_{\perp}^{2}q_{\perp}^{2}+\frac{1+\gamma}{2}v_{z}^{2}q_{z}^{2}\right),$
$\displaystyle\omega_{y}=\lambda
n\left(1-\frac{C}{2}\right)-2\frac{(\lambda\nu)^{2}}{\lambda
n}\frac{v_{z}^{2}q_{z}^{2}}{1-C}$ (8)

and $\omega_{z}=\omega_{x}$. We note that three modes
($\omega_{x},\omega_{y},\omega_{z}$), corresponding to the triplet components
of the polarization operator are gapped whereas the singlet component
($\omega_{0}$) is gapless. The degree of gap is determined by the strength of
electron-electron exchange interaction. The dispersions of collective modes
are schematically shown in the Fig. (1). All these modes become damped while
crossing the boundary of particle-hole continuum region, which, for example
for $q_{z}=0$, is defined by $\lambda
n-v_{\perp}q_{\perp}\leq\omega\leq\lambda n+v_{\perp}q_{\perp}$.

Let us quickly comment on the zero pumping case when $\delta\mu=0$. First at
$n=0$, the components of polarization operator reduce to

$\displaystyle\Pi^{00}_{intra}$ $\displaystyle=$
$\displaystyle-\frac{\nu}{\omega^{2}}\left(v_{\perp}^{2}q_{\perp}^{2}+\frac{v_{z}^{2}q_{z}^{2}}{2}\right)$
(9)

and $\Pi^{xx}_{intra}=\Pi^{yy}_{intra}=\Pi^{00}_{intra}$. The collective mode
with dispersion

$\omega=\sqrt{-\lambda\nu}\left(v_{\perp}^{2}q_{\perp}^{2}+\frac{v_{z}^{2}q_{z}^{2}}{2}\right)^{1/2}$
(10)

exists provided $\lambda<0$. Indeed, without the particle imbalance, the
plasmon modes are expected. As it was shown in Ref. Yan  _et al._ (2016). In
this work, contrary to our study, the interaction term was taken to be
$\lambda\sim 1/q^{2}$ which leads to the gapped spectrum of the plasmon modes.

Surface states One of the unique aspects of nodal-ring semimetals is the
existence of the so-called drum-head surface states. Let us consider NLSM
occupying a region $z>0$ with a boundary at $z=0$. We seek for a wave function
at the boundary as
$\Psi=[A,B]^{T}e^{-z\alpha+i\mathbf{k}_{\perp}\cdot\mathbf{r}}$, where
$\mathrm{Re}\alpha>0$ and $A,B$ are some coefficients. The eigenvalue equation
reads

$[\epsilon(k_{\perp}^{2}/Q^{2}-1)\sigma^{x}+i\alpha
v_{z}\sigma^{z}]\Psi=(E_{\eta}+\eta\frac{\lambda_{s}n_{s}}{2})\Psi.$ (11)

Here the exchange energy at the surface is denoted by $\lambda_{s}$ and
$E_{\eta}$ is the energy eigenvalue for nodal ring $\eta=\pm 1$. Using the
hermiticity of Hamiltonian one determines the boundary condition and obtains a
wave function up to an arbitrary phase $\psi$ in the form

$\Psi\propto[1,e^{i\psi}]^{T}e^{-z\alpha+i\mathbf{k}_{\perp}\cdot\mathbf{r}}.$
(12)

Generally $\psi$ may be a function of position at the boundary. Here we
consider it to be coordinate independent. Separating the real and imaginary
part from equation
$[iv_{z}\alpha-(E_{\eta}+\eta\lambda_{s}n/2)]+\epsilon(k_{\perp}^{2}/Q^{2}-1)e^{i\psi}=0$,
one gets the dispersion of surface state in the form

$E_{\eta}=\epsilon(k_{\perp}^{2}/Q^{2}-1)\cos\psi-\eta\frac{\lambda_{s}n_{s}}{2}$
(13)

provided condition

$\alpha v_{z}\equiv-\epsilon(k_{\perp}^{2}/Q^{2}-1)\sin\psi>0$ (14)

holds. At $\psi=\pm\pi/2$, the flat band surface state is strictly localized
at the surface where as the state is delocalized for $\psi=0$. Without loosing
the generality, we consider $\psi\in[0,\pi/2)$.

Let us now discuss the collective modes for surface states in the presence of
population imbalance. We neglect the interference between bulk and localized
surface states, as we are looking at two limiting cases only. Following the
same approach as in the bulk we shall obtain the density imbalance at the
surface. The chemical potential for the surface states with spin imbalance is
taken to be $\mu_{\eta}=\mu+\eta\delta\mu/2$. One obtains
$n_{s}=\nu_{s}\delta\mu/(1-\lambda_{s}\nu_{s})$ provided $\mu<0$ and
$|\mu|<\epsilon\cos\psi$. Here $\nu_{s}=Q^{2}/(4\pi\epsilon\cos\psi)$ is the
2D surface density of states. The polarization operator is obtained as

$\displaystyle\Pi_{s}$ $\displaystyle=$
$\displaystyle-\nu_{s}\frac{\delta\mu+\lambda_{s}n_{s}}{\omega-\lambda_{s}n_{s}}$
$\displaystyle-$
$\displaystyle\nu_{s}\Big{(}1-\frac{|\mu|}{\epsilon\cos\psi}\Big{)}\frac{\omega+\delta\mu}{(\omega-\lambda_{s}n_{s})^{3}}\left(v_{\perp}q_{\perp}\cos\psi\right)^{2}.$

Hence, the surface mode dispersion at $\lambda_{s}\nu_{s}>v_{\perp}q_{\perp}$
and $\lambda_{s}\nu_{s}>|\omega|$ is given by

$\omega=-\left(1-\frac{|\mu|}{\epsilon\cos\psi}\right)(1-\lambda_{s}\nu_{s})\cos^{2}(\psi)\frac{v_{\perp}^{2}q_{\perp}^{2}}{\lambda_{s}n_{s}},$
(16)

which is quadratic in wave-vector and the slope is strongly determined by the
properties of the surface. Note that similar to $\omega_{0}$ in Eq. (7), the
surface mode varies inversely with $\lambda_{s}n_{s}$. However, the mode
strongly depends on the properties of the boundary, vanishing in the limit
$\cos\psi\rightarrow|\mu|/\epsilon$. Now we comment on the dispersion in
absence of imbalance i.e., $\delta\mu=0$. In this case, the polarization
operator reduces as

$\displaystyle\Pi_{s}$ $\displaystyle=$
$\displaystyle-\nu_{s}\Big{(}1-\frac{|\mu|}{\epsilon\cos\psi}\Big{)}\left(\frac{v_{\perp}q_{\perp}\cos\psi}{\omega}\right)^{2}.$
(17)

which leads the spectrum as

$\omega=\sqrt{-\lambda_{s}\nu_{s}}\Big{(}1-\frac{|\mu|}{\epsilon\cos\psi}\Big{)}^{1/2}v_{\perp}q_{\perp}\cos\psi$
(18)

indicating that surface mode exists only for $\lambda_{s}<0$. It is also
interesting to note that the mode is linearly dispersive, whereas in presence
of imbalance it disperses quadratically.
Effects of tilt Now we comment on the possible effects of time-reversal
symmetry breaking perturbation on the spectrum of the mode. The Hamiltonian
can be written for tilted nodal ring as

$H_{\eta}=\epsilon\left(\frac{k_{\perp}^{2}}{Q^{2}}-1\right)(\sigma^{x}+\eta
t_{\perp})+v_{z}k_{z}(\sigma^{z}+\eta t_{z})-\eta\frac{\lambda n}{2},$ (19)

The spin index ($\eta$) indicates that the two nodal rings are tilted in
opposite direction. We note that for the weak tilting $|t_{\perp}|\ll
1,|t_{z}|\ll 1$, the intra-band component of the polarization operator is
slightly affected in terms of the coefficients at $q_{\perp},q_{z}$. Hence,
the tilting modifies velocities $v_{\perp}$ and $v_{z}$ of valley-wave modes
but not the wave-vector power law dependence. It is also noteworthy to mention
that the weak tilting does not affect the mode gap.

Finally we draw a comparison with the valley-wave in gapless Dirac-Weyl
systems, studied recently Zyuzin and Zyuzin (2019). We note that the gapless
collective mode in the presence of the population imbalance in NLSM, graphene,
or Weyl semimetals exhibit almost similar behaviour except the degree of
curvature, which depends on system parameters. However, the degree of gap in
the spectrum is strongly determined by the dimension and exchange interaction.
For example, we note that the gapless mode exists in the NLSM case for any
sign of the relative Berry flux of the two nodal-rings. This is not the case
for three dimensional Dirac-Weyl materials as discussed in Zyuzin and Zyuzin
(2019). Another important difference with Dirac-Weyl systems is the feature of
collective modes which arise due to surface states.

Although, collective charge dynamics of Fermi arc surface states in Weyl
semimetals has been studied Gorbar  _et al._ (2019); Adinehvand  _et al._
(2019), to the best of our knowledge, the exchange interaction mediated
collective modes in Fermi arcs surface states have not been discussed yet. To
support the population imbalance scenario and resulting collective modes, the
system has to posses at least two orthogonal surface states. This criteria may
be satisfied, for example, in antiferromagnetic Dirac semimetals, where
counter-propagating Fermi arc surface states are expected Tang  _et al._
(2016).

Summary To conclude, we investigate the electron-electron exchange interaction
generated collective modes in nodal-line semimetal, which arise due to the
population imbalance between two nodal rings. We evaluate the transverse
polarization function, which enables to obtain the dispersion relations of
inter-node collective modes. We find a gapless mode with quadratic dispersion
relation for both the bulk and surface states. Very recently, plasmon modes
have been observed in NLSMXue  _et al._ (2021) in equilibrium situation, which
can be extended further to spin-non-equilibrium cases with double/multiple
nodal rings to realize our findings. We also reveal that the plasmon
dispersion for surface states switches to linearly dispersive from quadratic
one while turning off the imbalance between two rings.

Acknowledgements: This work is supported by the Academy of Finland Grant No.
308339.

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# Cavity Sub- and Superradiance Enhanced Ramsey Spectroscopy

Christoph Hotter  Institut für Theoretische Physik, Universität Innsbruck,
Technikerstr. 21a, A-6020 Innsbruck, Austria  Laurin Ostermann  Institut für
Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020
Innsbruck, Austria  Helmut Ritsch  Institut für Theoretische Physik,
Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria

(August 29, 2024)

###### Abstract

Ramsey spectroscopy in large, dense ensembles of ultra-cold atoms trapped in
optical lattices suffers from dipole-dipole interaction induced shifts and
collective superradiance limiting its precision and accuracy. We propose a
novel geometry implementing fast signal readout with minimal heating for large
atom numbers at lower densities via an optical cavity operated in the weak
single atom but strong collective coupling regime. The key idea is controlled
collective transverse $\pi/2$-excitation of the atoms to prepare a macroscopic
collective spin protected from cavity superradiance. This requires that the
two halves of the atomic ensemble are coupled to the cavity mode with opposite
phase, which is naturally realized for a homogeneously filled volume covering
odd and even sites of the cavity mode along the cavity axis. The origin of the
superior precision can be traced back to destructive interference among sub-
ensembles in the complex nonlinear collective atom field dynamics. In the same
configuration we find surprising regular self-pulsing of the cavity output for
suitable continuous illumination. Our simulations for large atom numbers
employing a cumulant expansion are qualitatively confirmed by a full quantum
treatment of smaller ensembles.

##  I Introduction

Precise measurement of the energy difference of atomic transitions and their
corresponding frequencies has been at the heart of atomic physics for more
than a century. Nowadays, laser spectroscopy features incredible accuracy and
precision of up to $20$ digits [1]. These advances allow for precise tests of
fundamental physics, from general relativity and astrophysics to effects of
super-symmetry in strong interactions with nuclei or even the spectral
properties of antimatter [2, 3]. Yet, there is still a strong desire to push
these limits even further, while, in parallel, one also aims at
technologically simpler, faster and more robust methods to reach a desired
precision.

Ramsey spectroscopy, employing a carefully engineered sequence of time delayed
short phase-coherent pulses, can be used to resonantly excite a particle to a
long-lived and thus energetically very narrow state. It is one of the most
commonly used and well proven methods in precision spectroscopy. Applied to
large ensembles of ultra-cold atoms trapped in a magic wavelength optical
lattice it constitutes the current state of the art [4]. In order to realize
precise measurements fast, a large number of particles needs to be
interrogated. They should be confined to a small volume in order to ensure a
homogeneous environment. Here, the remaining weak inter-particle interactions
are a source of inaccuracy and the emerging fast superradiant collective decay
limits the maximal measurement time and precision [5].

The measurement readout requires a precise determination of the fraction of
excited atoms for a given probing frequency. In principle, one has to measure
the induced fluorescence after the Ramsey sequence. However, for very long
lived clock states this is too slow, inefficient and imprecise in order to be
practical and more elaborate procedures are necessary. Hence, besides methods
relying on additional fast transitions [6], using cavity enhanced detection of
dispersive shifts or collective decay has been suggested [7, 2]. However,
while cavity superradiance is fast and precise for counting excitations [8,
9], the presence of the cavity perturbs the free evolution between the Ramsey
pulses, which again limits measurement precision.

Figure 1: _Cavity Enhanced Spectroscopy Setup._ We consider a homogeneous,
dilute ensemble of narrow line two-level atoms at random, but fixed positions
in a standing wave optical resonator coherently driven by a transverse plane
wave laser. We assume a weak single atom but strong collective coupling
regime, i.e. $\kappa,\,\sum_{j}\,g_{j}^{2}/\kappa\gg\Gamma\gg
g_{j}^{2}/\kappa$. The Cavity Ramsey fringes and the photon number self-
pulsing are indicated as cavity output signals for the respective drive laser
operations.

In this work we propose an improved implementation of cavity enhanced Ramsey
spectroscopy combining fast collective readout with high precision. The
central idea is to address special collective excited states of the atomic
ensemble, which decouple from the cavity during the free evolution phase [10,
11]. The final readout is still performed via the superradiant manifold
allowing for fast detection of the ensemble. Surprisingly, despite sounding
challenging and complicated, there is a very natural and simple possibility to
implement this scheme for a large but dilute ensemble within a standing wave
cavity.

_Model._ – We consider $N$ two-level atoms with a narrow transition at
frequency $\omega_{\mathrm{a}}$ coupled to a single mode cavity. The atoms are
coherently driven with a detuning between the laser and the atomic transition
frequency of $\delta_{\mathrm{a}}=\omega_{\mathrm{l}}-\omega_{\mathrm{a}}$,
the corresponding Rabi frequency is denoted by $\Omega$. The cavity is detuned
by $\delta_{\mathrm{c}}=\omega_{\mathrm{l}}-\omega_{\mathrm{c}}$ from the
laser and we have an atom-cavity coupling of $g_{j}$ for the $j$-th atom. The
system is depicted in Fig. 1. Its Hamiltonian in the rotating frame of the
pump laser reads

$\displaystyle H$
$\displaystyle=-\delta_{\mathrm{c}}a^{\dagger}a+\sum_{j=1}^{N}\Big{[}-\delta_{\mathrm{a}}\sigma_{j}^{22}$
(1)
$\displaystyle+g_{j}(a^{\dagger}\sigma_{j}^{12}+a\sigma_{j}^{21})+\frac{\Omega}{2}(\sigma_{j}^{21}+\sigma_{j}^{12})\Big{]},$

with the cavity photon creation (annihilation) operator $a^{\dagger}$ ($a$)
and the atomic transition operator $\sigma_{j}^{kl}=\ket{k}_{j}\bra{l}_{j}$
for the $j$-th atom. The coherent interaction is accompanied by dissipative
processes, which are accounted for by the Liouvillian
$\mathcal{L}\left[\rho\right]$ in the master equation,

$\dot{\rho}=i\left[\rho,H\right]+\mathcal{L}\left[\rho\right].$ (2)

In the Born-Markov approximation [12] we can write the Liouvillian in Lindblad
form as

$\mathcal{L}\left[\rho\right]=\sum_{i}R_{i}\left(2J_{i}\rho
J_{i}^{\dagger}-J_{i}^{\dagger}J_{i}\rho-\rho J_{i}^{\dagger}J_{i}\right),$
(3)

with the jump operator $J_{i}$ and its corresponding rate $R_{i}$ for the
$i$-th dissipative process. We list these processes in Tbl. 1, including
cavity photon losses as well as decay and dephasing of the atoms with rate
$\Gamma$ and $\nu$, respectively.

Table 1: _Dissipative Processes._ The system features a damped cavity mode as well as atomic decay and dephasing. $i$ | $J_{i}$ | $R_{i}$ | Description
---|---|---|---
1 | $a$ | $\kappa$ | cavity photon losses
2 | $\sigma_{j}^{12}$ | $\Gamma$ | decay from $\ket{2}_{j}$ to $\ket{1}_{j}$
3 | $\sigma_{j}^{22}$ | $\nu$ | dephasing of the $j$-th atom

As we are targeting narrow clock transitions for our atoms the system is
operated in the bad and large volume cavity regime $\kappa\gg\Gamma$ with only
a small single atom cooperativity $C_{j}=g_{j}^{2}/(\kappa\Gamma)\ll 1$ but a
sufficiently large ensemble to enter the strong collective coupling regime
$NC=\sum_{j}C_{j}\gg 1$. Typically, this parameter regime implies a very large
atom number $N\gg 1$, which does not allow for a full quantum simulation, but
we can very well treat this problem in a second order cumulant expansion [13,
14]. A comparison with a full quantum simulation for a small atomic ensemble
is shown in the appendix [15]. For a qualitative illustration of our
deliberations even first order cumulant equations, i.e. mean field, (see
appendix) suffice for most calculations. Additionally, we neglect dipole-
dipole interaction [16], as our atomic ensemble is sufficiently dilute such
that neither dispersive, nor dissipative collective processes among the
individual atoms will play a considerable role.

_Collective Cavity mediated Super- and subradiance._ – In the following, for
simplicity, we assume the atoms located close to cavity mode anti-nodes with
half of the atoms at the maxima and half at the minima of the mode function
along the cavity axis. Hence their respective effective coupling is well
approximated by $+g$ and $-g$. As confirmed by more involved simulations,
investigating various different distributions for the atom-field coupling,
this simplification already captures the essential physics discussed below.

Figure 2: _Cavity Mediated Collective Decay._ Time evolution of the intra-
cavity photon number (a) and the single atom excited state population (b)
after a short pulse excitation preparing each atom in the same coherent
superposition. The black line represents an inverted ensemble with
$\left\langle\sigma^{22}\right\rangle=80\%$ excited state fraction ($\approx
3\pi/4$-pulse) and the orange line depicts an ensemble after a $\pi/2$-pulse
to create $\left\langle\sigma^{22}\right\rangle=50\%$. For uniform cavity
coupling (dashed line) in both cases we see an immediate superradiant
population decay to the ground state creating a photon pulse in the cavity.
Cavity coupling with alternating signs (solid line) leads to a weaker and time
delayed pulse for an inverted ensemble, while the atomic excitation is almost
perfectly protected from cavity decay without inversion (solid line in (b)).
We have assumed $N=2\cdot 10^{5}$ atoms with
$g=10\Gamma,\,\kappa=10^{4}\Gamma$ and
$\delta_{\mathrm{a}}=\delta_{\mathrm{c}}=\nu=0$.

It is well established that inverting all atoms with a short $\pi$-pulse
induces the emission of a delayed intense light pulse due to cavity enhanced
superradiant decay [17, 18, 19, 20, 21, 22]. Synchronized stimulated emission
in a cavity occurs even for a dilute ensemble, which does not exhibit free
space superradiance. Figure 2a shows typical trajectories for the
corresponding time evolution of the intra-cavity photon number $\left\langle
a^{\dagger}a\right\rangle$, when all atoms are initially coherently prepared
at $\left\langle\sigma^{22}\right\rangle=80\%$ (black line). Figure 2b depicts
the corresponding time evolution of the excited state population. For
comparison we show the behavior for all atoms equally coupled to the cavity
(dashed line, $g_{j}=g$) as well.

However, for the system we consider ($g_{j}=(-1)^{j}g$), we will observe such
pulsed emission for an ensemble of inverted atoms only. If the excited state
population is below $50\%$ the atoms are not able to synchronize and thus will
not emit a significant amount of photons into the cavity, see solid orange
line in Fig. 2. Figure 3a shows the total number of emitted photons $\langle
a^{\dagger}a\rangle_{\mathrm{out}}$ for different values of the coherently
prepared initial excited state population. We can clearly observe that almost
no photons leak through the cavity mirrors until the atoms are inverted. This
is due to subradiant suppression as the photons emitted by the atoms coupled
to the cavity with opposite $g$ will interfere destructively [23, 24]. For a
fully inverted ensemble we obtain approximately one photon for each atom.
However, for an initial inversion with
$\left\langle\sigma^{22}\right\rangle<1$ we only get
$N\cdot(2\langle\sigma^{22}\rangle-1)$ photons. The solid lines in Fig. 2b
also indicate that the atoms retain a population of
$1-\left\langle\sigma^{22}\right\rangle$ after the photon emission into the
cavity and subsequently experience free-space decay with the rate $\Gamma$
only. Again, the dashed line represents the case of all atoms coupling equally
($g_{j}=g$), where we see that even not fully inverted atoms will
superradiantly emit photons into the cavity. Additionally we plot the peak
intra-cavity photon number (blue) illustrating the same behavior. Figure 3b
shows the delay time of the peak photon number as a function of the atomic
excitation. For uniform coupled atoms (dashed line) a higher excitation leads
to a later pulse. Whereas for alternating coupling (solid line) the delay time
of the peak decreases for higher inversion, with a maximum at
$\langle\sigma^{22}\rangle_{t=0}=50\%$.

Note, that a crucial requirement for cavity subradiance is that the cumulative
dipole moment of the atoms projected on the cavity mode
$\sum_{j}g_{j}\langle\sigma^{12}_{j}\rangle$ vanishes [10, 11]. Here, for
simplicity, we chose the same laser excitation phase for all atoms, but a
variable excitation phase works equally well as long as the overall relative
phase disappears. Fortunately, this case is typically automatically realized
for a random distribution of a sufficiently large dilute intra-cavity atomic
ensemble. Note that this is also true for a ring-cavity featuring a continuous
atom-cavity coupling phase along the cavity axis [25].

Figure 3: _Cavity Sub- and Superradiance._ (a) Comparison of the integrated
cavity-output photon number $\left\langle
a^{\dagger}a\right\rangle_{\mathrm{out}}=\kappa\int\left\langle
a^{\dagger}a\right\rangle\mathrm{d}t$ (black) and peak intra-cavity photon
number (blue) as a function of the single atom excitation probability for
alternating coupling (solid line) and uniform coupling (dashed line). Note the
strong suppression of superradiant emission in the alternating coupling case
as long as no net-inversion is created initially. (b) Delay time of the peak
photon number. For alternating coupling the pulse appears earlier for higher
inversion, whereas for equally coupled atoms the pulse delay time increases
with growing excited state population. The circles indicate the operating
parameters chosen for Fig. 2a.

_Cavity Enhanced Ramsey Probing._ – As argued above, transverse pumping with
an overall vanishing phase of the atom-cavity coupling thus allows for a
$\pi/2$-pulse excitation of the atomic ensemble without an immediate rapid
superradiant decay through the cavity. This intriguing feature will be an
essential ingredient for a new implementation of cavity assisted Ramsey
spectroscopy as discussed below, when we combine it with fast direct
measurements of the number of excited atoms via the emitted cavity photons
after the second Ramsey pulse. The crucial advantage of this scheme is that it
can be very fast with no additional manipulation of the atoms needed for the
read out, hence the signal is less perturbed. Furthermore the atoms are not
significantly heated by this measurement and can therefore be reused, thus,
the dead time between measurements can be significantly reduced. Another
advantage in comparison with other non-demolition measurements for atomic
clocks [7, 26, 27] is that the signal, i.e. the number of photons, scales
linearly with the number of atoms. So, in principle, an arbitrarily large
number of atoms can be employed, which drastically increases the signal to
noise ratio.

Figure 4a shows the output signal of the cavity Ramsey method, the total
number of photons leaking through the cavity mirrors as a function of the
laser-atom detuning $\delta_{\mathrm{a}}$. Similar to the conventional Ramsey
method, fringes appear [28, 29, 30]. One striking difference, however, is that
a non-inverted ensemble does not produce a signal, corresponding to the flat
zero-photon regions. This narrows the FWHM of the cavity-Ramsey fringes
slightly compared to the conventional Ramsey fringes (see Fig. 4b). Including
an atomic dephasing with $\nu=10\Gamma$ (dashed line) merely weakens the
signal, yet, the shape of the curve is essentially the same. Note, that by
choosing $\delta_{\mathrm{c}}=\delta_{\mathrm{a}}$ we have implicitly assumed
that the cavity is perfectly on resonance with the atomic transition.
Therefore, one might wonder if a detuned cavity impairs the signal. But, since
we operate deeply in the bad cavity regime, only shifts of the cavity
resonance frequency on the order of $\kappa$ are important.

Overall, this means that enhancing the Ramsey spectroscopy by adding a cavity
achieves the same (or even slightly improved) accuracy, but has the advantage
of a convenient, fast, non-destructive measurement scaling linearly in the
atom number, which can substantially reduce the measurement dead time. At this
point we want to mention that the atoms need to be initially in the ground
state for each subsequent measurement. As we saw in Fig. 2b a not fully
inverted ensemble will retain some population in the excited state. This means
one needs to bring these atoms back to the ground state. Unfortunately, this
is not possible in a straight-forward way via a coherent drive on the clock
transition only. However, there are other ways to achieve this, for example
with an induced decay via another transition or by depleting the ground state
population temporarily to some other level to enable the depopulation of the
excited clock state.

Note that this cavity assisted Ramsey procedure will not work for
$\pi/2$-pulse excitations through the cavity mirrors. The reason for this is
that all atoms exhibit the same relative atom-cavity phase and will therefore
already decay superradiantly after the first $\pi/2$-pulse.

Figure 4: _Cavity Ramsey Method._ (a) Fringes in the photon number obtained
via the cavity Ramsey method with characteristic flat zero-photon regions. (b)
Zoom-in on the central fringe. The FWHM ($\pi/T$) of an optimal independent
atom Ramsey sequence with waiting time $T$ and the cavity Ramsey fringe ($\sim
2\pi/3T$) are highlighted. The parameters are $N=2\cdot
10^{5},\,g=10\Gamma,\,\kappa=10^{4}\Gamma,\,\delta_{\mathrm{c}}=\delta_{\mathrm{a}},\,\Omega=1000\Gamma$
and $\nu=0$ (solid) or $\nu=10\Gamma$ (dashed). The free evolution time
between the two $\pi/2$-pulses is $T=\pi/100\Gamma$.

_Self-Pulsing under Continuous Illumination._ – Continuously driving the
atomic ensemble with a suitable Rabi frequency leads to striking self-pulsing
of the system as shown in Fig. 5a. Yet, the explanation for this initially
surprising behavior is rather simple: as we have seen, the photon emission
into the cavity for a non-inverted atomic ensemble with vanishing relative
phase is strongly suppressed. Therefore, with atoms initially in the ground
state, there is no significant cavity photon number at least until
$t=\pi/2\Omega$ (see Fig. 5b). But, as soon as a significant population
inversion is achieved the ensemble emits a superradiant photon pulse into the
cavity. Subsequently the excited state population is depleted below $50\%$ and
the photon number quickly reduces to almost zero due to the very fast cavity
photon decay. Since the laser is still on, the procedure starts over again and
we obtain another pulse. As we can see in Fig. 5a the peak photon number
reduces from pulse to pulse, which can be traced back to decay and decoherence
damping the Rabi oscillations. Additionally the timeevolution for all equally
coupled atoms (dashed blue line, $g_{j}=g$) is plotted in (a) and (b). We see
that in this case the self-pulsing does not occur and the cavity photon number
reaches a steady state at $\langle a^{\dagger}a\rangle=\Omega^{2}/4g^{2}$ very
quickly. The steady state value for the excited state population
($\langle\sigma^{22}\rangle\approx 1.7\cdot 10^{-3}$) is reached much later.
Note, that the system with vanishing relative phase also reaches a steady
state with, surprisingly, the same photon number $\langle
a^{\dagger}a\rangle=\Omega^{2}/4g^{2}$ but a much higher excited state
population ($\langle\sigma^{22}\rangle\approx 0.25$).

As we need an excited state population of at least $50\%$ to observe the
photon peaks, and the atoms basically do Rabi oscillations to reach this,
there should be a lower bound for the Rabi frequency $\Omega$, depending on
the laser-atom detuning $\delta_{\mathrm{a}}$, the atomic decay $\Gamma$ and
the dephasing rate $\nu$. This is exactly what we see in Fig. 5c and Fig. 5d.
The condition $\Omega>|\delta_{\mathrm{a}}|,\,(\Gamma+\nu)/2$ needs to be
satisfied in order for the peak intra-cavity photon number of the first pulse
to appear. Furthermore, the collective photon emission from the atoms into the
cavity is determined by the frequency $NC\Gamma$. Thus, for the superradiant
photon pulse to dominate over the coherent photon scattering from the drive,
we need to ensure that $NC\Gamma>\Omega/2$. This threshold is also shown in
Fig. 5c and Fig. 5d. Additionally we find that self-pulsing can also occur
when only half of the atoms in the cavity are driven, in this case the atom-
cavity coupling phase can be identical for all atoms.

Figure 5: _Self-Pulsing._ Time evolution of the cavity photon number (a) and
excited state population (b) for a continuous drive, resulting in photon
number self-pulsing. (c) and (d) Scans over $\Omega$, $NC$ and
$\delta_{\mathrm{a}}$ for the peak photon number of the first pulse (red
circle in (a)). The dashed white lines represent the threshold
$2NC\Gamma>\Omega>|\delta_{\mathrm{a}}|,(\Gamma+\nu)/2$. The parameters when
kept constant are $N=2\cdot
10^{5},\,g=10\Gamma,\,\kappa=10^{4}\Gamma,\,\delta_{\mathrm{c}}=\delta_{\mathrm{a}}=0,\,\Omega=100\Gamma$
and $\nu=10$.

_Conclusions._ – We have proposed and studied a new variant of cavity enhanced
Ramsey spectroscopy which simplifies and accelerates the measurement procedure
via tailored atom-field coupling. As it is particularly advantageous for long
lived atoms it should be beneficial for the performance of optical atomic
clocks. The underlying phenomena are, on the one hand, the well-known cavity
superradiance in an inverted atomic ensemble in order to achieve a fast
readout, and, on the other hand, the subradiant behavior of specific atomic
ensembles with an overall vanishing collective cavity coupling. Furthermore,
we found that the chosen operating conditions with weak single atom coupling
but strong collective coupling also induce a striking self-pulsing instability
for continuous drive at suitable Rabi frequencies. Interestingly the necessary
operating conditions are within the reach of current experimental setups with
only minimal adjustments required.

Some preliminary studies on the influence of imperfections in the setup as
variable coupling strengths, slow atomic motion or fluctuations in the
excitation procedure qualitatively yield very similar results for
experimentally realistic assumptions. However, a more detailed study of these
and other aspects as heating and loss is required for a quantitative
prediction of the practical system performance.

###### Acknowledgements.

We thank Stefan Schäffer for helpful discussions. We acknowledge funding from
the European Union’s Horizon 2020 research and innovation program under Grant
Agreement No. 820404 iqClock. The numerical simulations were performed with
the open-source framework QuantumCumulants.jl [14] and QuantumOptics.jl [31].

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##  Appendix A Mean-field equations

Throughout the paper we calculate the dynamics with a second order cumulant
expansion [14]. Nevertheless, the mean-field equations also contain the key
physics, therefore we show these much simpler equations for a qualitative
description of the system:

$\displaystyle\frac{d}{dt}\langle a\rangle=$
$\displaystyle-\left(i\delta_{\mathrm{c}}+\frac{\kappa}{2}\right)\langle
a\rangle-i\sum_{j=1}^{N}g_{j}\langle{\sigma_{j}}^{{12}}\rangle$ (4)
$\displaystyle\frac{d}{dt}\langle{\sigma_{j}}^{{22}}\rangle=$
$\displaystyle-\Gamma\langle{\sigma_{j}}^{{22}}\rangle+i\frac{\Omega}{2}\left[\langle{\sigma_{j}}^{{12}}\rangle-\langle{\sigma_{j}}^{{21}}\rangle\right]$
(5) $\displaystyle+$ $\displaystyle ig_{j}\left[\langle
a^{\dagger}\rangle\langle{\sigma_{j}}^{{12}}\rangle-\langle
a\rangle\langle{\sigma_{j}}^{{21}}\rangle\right]$
$\displaystyle\frac{d}{dt}\langle{\sigma_{j}}^{{12}}\rangle=$
$\displaystyle\left(i\delta_{\mathrm{a}}-\frac{\Gamma+\nu}{2}\right)\langle{\sigma_{j}}^{{12}}\rangle$
(6) $\displaystyle+$ $\displaystyle i\left(\frac{\Omega}{2}+g_{j}\langle
a\rangle\right)\left[2\langle{\sigma_{j}}^{{22}}\rangle-1\right]$

In equation (4) we see that for a vanishing coupling phase and equal coherence
for all atoms, the cavity field does not grow. We also observe this in Fig. 6.
On some point, however, the atoms will gain a sufficient inversion to emit a
superradiant pulse into the cavity. Note that there is an initial inaccuracy
(e.g. in the cavity field) needed in the mean-field description, otherwise the
cavity photon number will never be unequal zero. Due to this initial
inaccuracy the timeevolution for the two ensembles is striking different. For
example we see that for the second pulse the excited state population is
drastically decreased for one ensemble, but increased for the other. This is a
feature of the mean-field description, which does not occur in that way in a
higher order cumulant expansion.

Adiabatically eliminating the cavity field shows that the two frequencies
$\Omega$ and $Ng^{2}/\kappa$ compete with each other. Furthermore we see that
for a small cavity field the atoms basically do independent Rabi-oscillations
with frequency $\Omega$.

Figure 6: _Mean-field self-pulsing._ The timeevolution of the photon number
(a) excited state population (b) and coherence (c) for a continuous drive. Due
to the initial seed the behavior of the atoms in the two ensembles ($g_{j}=+g$
and $g_{j}=-g$) is striking different. The parameters are the same as in Fig.
5 (a).

##  Appendix B Comparison with full quantum model

To ensure the qualitative validity of our second order cumulant expansion we
compare the results with a full quantum model. Of course this is only possible
for a relative small number of atoms. To push the number of atoms as far as
possible we use the Monte Carlo wave-function method [32, 33], and describe
the atoms in the Dicke basis which means that only collective atomic effects
can be investigated. Thus we neglect individual atomic decay and dephasing.
Fig. 7 shows the comparison between the second order cumulant expansion and
the full quantum model for the cavity subradiance (a)-(c), cavity Ramsey
method (d) and self-pulsing (e)-(f). Overall we find a good qualitative
agreement. A perfect quantitative correspondence is not to be expected for
such small atom numbers, due to the needed strong single atom cooperativity.

Figure 7: _Full quantum model._ The $2^{\mathrm{nd}}$ order cumulant expansion
(solid line) is compared with a full quantum model (dashed line) for the
cavity subradiance in (a)-(c), the cavity Ramsey method in (d) and the self-
pulsing in (e) and (f). In all plots we used $\kappa=200,\Gamma=\nu=\eta=0$
and $\delta_{\mathrm{c}}=\delta_{\mathrm{a}}$. For (a)-(c) the remaining
parameters are $N=20,\,g=10$ and $\delta_{\mathrm{a}}=0$, for (d)
$N=20,\,g=10,\,\Omega=100$ and $T=\pi/10$ and for (e)-(f) $N=2\cdot
50,\,g=4,\,\delta_{\mathrm{a}}=0$ and $\Omega=4$.