# Quantum Fluctuation Dynamics of Dispersive Superradiant Pulses
in a Hybrid Light-Matter System

Kevin C. Stitely<EMAIL_ADDRESS>Dodd-Walls Centre for Photonic
and Quantum Technologies, New Zealand  Department of Mathematics, University
of Auckland, Auckland 1010, New Zealand  Department of Physics, University of
Auckland, Auckland 1010, New Zealand  Fabian Finger  Institute for Quantum
Electronics, ETH Zürich, 8093 Zürich, Switzerland  Rodrigo Rosa-Medina
Institute for Quantum Electronics, ETH Zürich, 8093 Zürich, Switzerland
Francesco Ferri  Institute for Quantum Electronics, ETH Zürich, 8093 Zürich,
Switzerland  Tobias Donner  Institute for Quantum Electronics, ETH Zürich,
8093 Zürich, Switzerland  Tilman Esslinger  Institute for Quantum Electronics,
ETH Zürich, 8093 Zürich, Switzerland  Scott Parkins  Dodd-Walls Centre for
Photonic and Quantum Technologies, New Zealand  Department of Physics,
University of Auckland, Auckland 1010, New Zealand  Bernd Krauskopf  Dodd-
Walls Centre for Photonic and Quantum Technologies, New Zealand  Department of
Mathematics, University of Auckland, Auckland 1010, New Zealand

###### Abstract

We consider theoretically a driven-dissipative quantum many-body system
consisting of an atomic ensemble in a single-mode optical cavity as described
by the open Tavis-Cummings model. In this hybrid light-matter system the
interplay between coherent and dissipative processes leads to superradiant
pulses with a build-up of strong correlations, even for systems comprising
hundreds to thousands of particles. A central feature of the mean-field
dynamics is a self-reversal of two spin degrees of freedom due to an
underlying time-reversal symmetry, which is broken by quantum fluctuations. We
demonstrate a quench protocol that can maintain highly non-Gaussian states
over long time scales. This general mechanism offers interesting possibilities
for the generation and control of complex fluctuation patterns, as suggested
for the improvement of quantum sensing protocols for dissipative spin-
amplification.

In recent years there has been considerable interest in driven-dissipative
quantum many-body systems, which appear in fields ranging from atomic and
optical physics [1], to condensed matter [2] and quantum information theory
[3]. The in- and outflux of energy, essential for applications, significantly
modifies the dynamics that quantum systems admit. Examples exploiting the
interplay between coherence and dissipation range from limit-cycle and time-
crystalline behavior [4, 5, 6, 7], superradiant oscillations [8, 9], and chaos
[10, 11, 12] to dissipation-induced and topological phases [13, 14, 15, 16,
14]. Studies of many-body nonequilibrium quantum systems typically focus on
mean-field behavior where quantum fluctuations are small. Moreover, treatments
of fluctuations are usually concerned with steady-state effects, such as
critical exponents in the vicinity of quantum phase transitions [17, 18, 19].
Studies of the _dynamics of quantum fluctuations_ in driven-dissipative
quantum systems have so far been limited to spin squeezing [20, 21, 22, 23].
In these works, however, dissipation is deliberately mitigated as it is
detrimental to entanglement.

In this work, we explore the fluctuation dynamics of an atomic ensemble
coupled to a single-mode optical cavity via a driving laser field [Fig.
1(a,b)]. This system, described by the dissipative Tavis-Cummings model [24],
has been thoroughly investigated in its limiting cases where dissipation is
either dominant, leading to superradiant decay [25], or practically
nonexistent, inducing spin-squeezing dynamics [21, 26]. We study the
unexplored regime between these two extremes, and show that the system
supports a build-up of strong non-Gaussian fluctuations, which we term
_dispersive superradiant pulses_ [Fig. 1(c)]. Here, quantum fluctuations
remain significant even as the system size increases towards a regime where
one might naïvely expect the thermodynamic limit to take hold. Moreover, we
show that the appearance of enhanced fluctuations leads to the breaking of an
_inversion-time reversal symmetry_ of the thermodynamic limit. Finally, we
demonstrate that quenching the effective cavity resonance and light-matter
coupling can preserve highly non-Gaussian states over surprisingly long
timescales.

Figure 1:  (a) Schematic of the open Tavis–Cummings model. (b) Atomic energy
level and coupling configuration. (c) Dispersive superradiant pulse
illustrated on the atomic Bloch sphere by six quantum trajectories and the
spin $Q$-function; here, $\kappa=1$, $\lambda=0.5$, $N=200$, with initial
state $\ket{\theta_{0},\phi_{0}}=\ket{\pi/10,\pi/2}$.

We consider $N$ atoms inside a single-mode optical cavity with effective
resonance frequency $\omega$ and decay rate $\kappa$. The atoms are in a
$\Lambda$ configuration with two (nominal) ground states, $\ket{\downarrow}$
and $\ket{\uparrow}$, and an excited state $\ket{e}$ [Fig. 1(a,b)], and are
driven with a cavity-assisted Raman transition: a laser couples to the
transition $\ket{\uparrow}\leftrightarrow\ket{e}$, while
$\ket{\downarrow}\leftrightarrow\ket{e}$ is coupled to the cavity mode. The
atoms are fixed in place at alternate antinodes of the field and couple
identically. If driving laser and cavity mode are far-detuned from atomic
resonance, $\ket{e}$ can be adiabatically eliminated. The coupling scheme can
then be seen as a two-photon transition between $\ket{\downarrow}$ and
$\ket{\uparrow}$, facilitated by a laser and a cavity photon. The system is
captured by the Tavis-Cummings Hamiltonian [24] (with $\hbar=1$)

$\hat{H}_{\text{TC}}=\omega\hat{a}^{\dagger}\hat{a}+\omega_{0}\hat{J}_{z}+\frac{\lambda}{\sqrt{N}}\left(\hat{a}\hat{J}_{+}+\hat{a}^{\dagger}\hat{J}_{-}\right),$
(1)

where $\hat{a}$ is the annihilation operator of the cavity field mode with
effective frequency $\omega$, $\hat{J}_{\pm,z}$ are collective spin-$N/2$
operators, $\omega_{0}$ is the effective energy level splitting, and $\lambda$
is the light-matter coupling strength. The primary dissipative mechanism of
this open quantum system is leakage of cavity photons, which we model with the
Lindblad master equation

$\frac{d\hat{\rho}}{dt}=-i[\hat{H}_{\text{TC}},\hat{\rho}]+\kappa\left(2\hat{a}\hat{\rho}\hat{a}^{\dagger}-\hat{a}^{\dagger}\hat{a}\hat{\rho}-\hat{\rho}\hat{a}^{\dagger}\hat{a}\right),$
(2)

where $\hat{\rho}$ is the reduced density operator and $\kappa$ is the cavity
decay rate. By adiabatically eliminating the cavity field mode (assuming
$\sqrt{\omega^{2}+\kappa^{2}}\gg\lambda,\omega_{0}$) [27, 26] and taking a
rotating frame 111See Supplemental Material at LINK for details on the
semiclassical model, the second-order cumulant expansion, the Husimi spin
$Q$-function, and the quench protocol., we arrive at a one-axis twisting
Hamiltonian

$\hat{H}=-\frac{\xi\lambda^{2}}{N}\left(\frac{N^{2}}{4}-\hat{J}_{z}^{2}\right),$
(3)

with the master equation

$\frac{d\hat{\rho}}{dt}=-i[\hat{H},\hat{\rho}]+\frac{\eta\lambda^{2}}{N}\left(2\hat{J}_{-}\hat{\rho}\hat{J}_{+}-\hat{J}_{+}\hat{J}_{-}\hat{\rho}-\hat{\rho}\hat{J}_{+}\hat{J}_{-}\right).$
(4)

Here $\xi=\omega/(\kappa^{2}+\omega^{2})$,
$\eta=\kappa/(\kappa^{2}+\omega^{2})$, and $\hat{\rho}$ is now the reduced
density operator of the atom-only system. Equation (3) describes the effective
global-range spin-spin interactions within the ensemble, mediated by cavity
photons. In Eq. (4), cavity dissipation gives rise to a collectively enhanced
population inversion towards the spin state
$\bigotimes_{j=1}^{N}\ket{\downarrow_{j}}$, i.e., Dicke superradiance [25,
29]. Since Eqs. (3,4) conserve the collective spin length, the dynamics can be
represented as the evolution of a quasiprobability distribution on the surface
of the Bloch sphere [Fig. 1(c)], which we capture by the Husimi spin
$Q$-function 111. In the semiclassical limit $N\rightarrow\infty$,
fluctuations become negligible (the quasiprobability distribution tends to a
$\delta$ distribution) and the dynamics are described by the mean-field
trajectories $\braket{\hat{J}_{x}(t)}$, $\braket{\hat{J}_{y}(t)}$, and
$\braket{\hat{J}_{z}(t)}$, confined onto the Bloch sphere 111.

The open Tavis-Cummings model has been studied when the dynamics are either
purely dissipative [29, 30, 31, 32, 33, 34] or purely coherent (unitary) [21,
35, 36, 37, 38]. The former case ($\omega/\kappa=0$) reduces to the
superradiance master equation [29, 39]. Here, semiclassical trajectories move
to the south pole along longitudinal lines [Fig. 2(a1)]. Fluctuations increase
uniformly in all spin directions due to the curvature of the Bloch sphere. The
purely coherent case ($\omega/\kappa\rightarrow\infty$), on the contrary, has
no dissipative term and exhibits spin squeezing dynamics [21, 20, 26]. The
semiclassical trajectories form latitudinal circles corresponding to Rabi
oscillations [Fig. 2(a3)], without emission into the cavity, and have an
inversion-dependent Rabi frequency of $2\xi\lambda^{2}\braket{\hat{J}_{z}}/N$,
see Eq. (3) — leading to spin squeezing at the equator [Fig. 2(a3)].

Figure 2:  (a) Semiclassical trajectories (black) and contours of the spin
$Q$-function (blue) during the system’s evolution on the Bloch sphere for
dissipative (a1) and dispersive (a2) pulses initialized at
$\ket{\theta_{0},\phi_{0}}=\ket{\pi/10,\pi/2}$, and for the unitary case (a3)
at $\ket{\pi/2,\pi/2}$. (b) Inversion-time reversal symmetry (blue) is broken
by quantum fluctuations (red). (c) Deformation parameter $\chi^{2}$ as a
function of the inversion $\braket{\hat{J}_{z}(t)}$ for $\omega=0$
(dissipative), and for $\omega=5$ (dispersive); inset: $\chi^{2}$ versus time
for the unitary case with $\omega=5$, $\kappa=0$, and integration time
$T=800$. For all, $\lambda=0.5$, $N=200$.

In the intermediate range $0<\omega/\kappa<\infty$, the interplay between
superradiant-pulsed behavior via dissipation and Rabi oscillations from a
cavity-atom detuning gives rise to rich dynamics of dispersive superradiant
pulses. Here, semiclassical trajectories unwind down the Bloch sphere and
reverse direction at the equator [Fig. 2(a2)], approaching the south pole.
When the mean spin direction is at the equator, the (instantaneous) Rabi
frequency is zero and becomes negative. This results in partial self-reversal
of the trajectories, which stems from the _inversion-time reversal symmetry_
$\mathcal{T}_{1}:(t,\braket{\hat{J}_{z}})\mapsto(-t,-\braket{\hat{J}_{z}})$
111. Hence, any trajectory that intersects the symmetry subspace
${\Sigma_{1}=(\braket{\hat{J}_{x}},\braket{\hat{J}_{y}},\braket{\hat{J}_{z}})\in\mathbb{S}^{2}|\braket{\hat{J}_{z}}=0\\}}$
is invariant under $\mathcal{T}_{1}$ [40, 41]. In other words,
$\braket{\hat{J}_{x}}$ and $\braket{\hat{J}_{y}}$ undergo a reversal of their
dynamics when the equator is crossed.

Since every trajectory tends towards the south pole of the Bloch sphere, any
initially excited trajectory ($\braket{\hat{J}_{z}(0)}>0$) is guaranteed to
intersect $\Sigma_{1}$, and will thus be symmetric under inversion-time
reversal. Unlike existing protocols for engineering time-reversal dynamics,
which require external modification (flipping the sign of the Hamiltonian)
[42, 43, 44], this system is intrinsically _self-reversing_ in the spin
components $\braket{\hat{J}_{x}}$ and $\braket{\hat{J}_{y}}$ [Fig. 2(b)].
Surprisingly, this is dissipation-induced: since the Rabi frequency at the
equator is zero, motion through $\Sigma_{1}$ is purely dissipative. Note that
simultaneous self-reversal of all spin components is impossible, as it would
violate the uniqueness theorem for ordinary differential equations [45, 46].
Rather, only $\braket{\hat{J}_{x}}$ and $\braket{\hat{J}_{y}}$ self-reverse,
while $\braket{\hat{J}_{z}}$ evolves monotonically to
$\braket{\hat{J}_{z}}=-N/2$.

Simulation of Eq. (4) reveals that quantum fluctuations break inversion-time
reversal symmetry [Fig. 2(b)], connected with the appearance of highly non-
Gaussian spin states during dispersive superradiant pulses [Figs. 1(c) and
2(a2)] with a large enhancement of fluctuations in one quadrature. The
physical origin of the non-uniform state expansion can be understood by taking
a quantum trajectories approach [47, 48]. Fig. 1(c) shows six quantum
trajectories initialized in a spin coherent state
$\ket{\theta_{0},\phi_{0}}=\ket{\pi/10,\pi/2}$. The positive divergence of the
semiclassical equations in the northern portion of the Bloch sphere results in
the amplification of the fluctuations of quantum trajectories during the
initial stage of the pulse. This “fanning out” of trajectories causes the
enhancement of fluctuations in certain directions. Once the mean spin
direction crosses the equator, the trajectories converge, so the overall level
of fluctuations is reduced.

To describe the non-uniform expansion, we introduce the _deformation
parameter_

$\chi^{2}=\frac{\displaystyle\max_{\Phi}\braket{\Delta\hat{J}_{\perp}(\Phi)^{2}}}{\displaystyle\min_{\Phi}\braket{\Delta\hat{J}_{\perp}(\Phi)^{2}}}\geq
1,$ (5)

where $\braket{\Delta\hat{J}_{\perp}(\Phi)^{2}}$ is the spin variance in a
direction orthogonal to the mean spin direction, parameterized by the angle
$\Phi$, relative to the longitudinal direction. Figure 2(c) shows $\chi^{2}$
during a purely dissipative and a dispersive pulse. In the dissipative case,
the state expands uniformly [Fig. 2(a1)] and $\chi^{2}\approx 1$ throughout.
In contrast, the dispersive case has a clear initial rise in $\chi^{2}$, then
peaks and quickly decreases as the system approaches its steady state
($\braket{\hat{J}_{z}}\rightarrow-N/2$) with uniform fluctuations. The
(nonequilibrium) steady state is the spin coherent state at the south pole
$\ket{\theta,\phi}=\ket{\pi,\phi}$ which is also a dark state of the jump
operator $\hat{J}_{-}$, i.e., $\hat{J}_{-}\ket{\pi,\phi}=0$. Figure 2(c) also
demonstrates that quantum fluctuations break inversion-time symmetry, since
$\chi^{2}$ is not symmetric about $\braket{\hat{J}_{z}}=0$.

Figure 3:  Generation of non-Gaussian correlations in a dispersive
superradiant pulse. (a) Semiclassical (green), second-order cumulant (blue),
and master equation (red) computations of $\braket{\hat{J}_{x}(t)}$. (b) Total
degree of correlation at third-order $\mathcal{C}_{3}(t)$. (b1)–(b3) Snapshots
of the spin $Q$-function at different times of the pulse, with
$\ket{\theta_{0},\phi_{0}}=\ket{\pi/10,\pi/2}$ and $\kappa=1$, $\omega=5$,
$\lambda=0.5$, $N=200$.

We apply a cumulant expansion approach to further investigate these emerging
non-Gaussian quantum states, as shown in Figs. 1(c) and 2(a2). Also known as
connected correlation functions [49] or Ursell functions [50], cumulants
quantify the effects of higher-order moments by subtracting out redundant
information determined by moments of lower order. They have been used recently
as the basis for a truncation method in a variety of contexts [51, 52, 53, 54,
55, 56, 57].

In typical light-matter interaction models, the equations of motion for
moments at order $n$ depend on the moments at order $n+1$, thus, creating an
infinite hierarchy of differential equations. In the $n$th cumulant expansion,
one assumes that the cumulants at order $n+1$ vanish, thus closing the system
of equations. The first-order expansion describes the semiclassical (mean-
field) approximation, which is valid when the strength of fluctuations is
small. The second-order cumulant expansion, with cumulants
$\braket{\hat{J}_{j}\hat{J}_{k}}_{c}=\braket{\hat{J}_{j}\hat{J}_{k}}-\braket{\hat{J}_{j}}\braket{\hat{J}_{k}}$,
$j,k\in\\{x,y,z\\}$, is valid for approximately Gaussian states and enables
investigations of fluctuations, e.g., for spin squeezing. The build-up of
third-order cumulants

$\displaystyle\braket{\hat{J}_{j}\hat{J}_{k}\hat{J}_{\ell}}_{c}=$
$\displaystyle\braket{\hat{J}_{j}\hat{J}_{k}\hat{J}_{\ell}}-\braket{\hat{J}_{j}}\braket{\hat{J}_{k}\hat{J}_{\ell}}-\braket{\hat{J}_{k}}\braket{\hat{J}_{j}\hat{J}_{\ell}}$
$\displaystyle-\braket{\hat{J}_{\ell}}\braket{\hat{J}_{j}\hat{J}_{k}}+2\braket{\hat{J}_{j}}\braket{\hat{J}_{k}}\braket{\hat{J}_{\ell}},$
(6)

with $j,k,\ell\in\\{x,y,z\\}$, thus signals the appearance of non-Gaussianity.
Since correlations shift between cumulants of different operator permutations
dynamically, we define the _total degree of correlation at third-order_

$\mathcal{C}_{3}(t)=\sum_{j,k,\ell}|\braket{\hat{J}_{j}\hat{J}_{k}\hat{J}_{\ell}}_{c}|$
(7)

by summing over all possible cumulant permutations.

The time evolution of $\braket{J_{x}}$ and $\mathcal{C}_{3}$ during a
dispersive superradiant pulse is shown in Figs. 3(a,b). When the state is
approximately coherent [Fig. 3(b1)], $\mathcal{C}_{3}$ is small and the
semiclassical and Gaussian approximations hold. Near the midpoint of the
pulse, correlations build up and the state becomes highly non-Gaussian [Fig.
3(b2)], leading to failure of the second-order cumulant expansion (Gaussian
approximation) [Fig. 3(a)]. The final state is again a coherent spin state,
for which $\mathcal{C}_{3}\to 0$. In the second-order cumulant expansion,
inversion-time reversal symmetry is reflected by invariance under the
transformation
${\mathcal{T}_{2}:(t,\braket{\hat{J}_{z}},\braket{\hat{J}_{z}\hat{J}_{\ell}})\mapsto(-t,-\braket{\hat{J}_{z}},-\braket{\hat{J}_{z}\hat{J}_{\ell}})},$
where $\ell\in\\{x,y\\}.$ The corresponding symmetry subspace features the
additional conditions $\braket{\hat{J}_{z}\hat{J}_{\ell}}=0$. Contrary to the
semiclassical case, where expectation factorization ensures
$\braket{\hat{J}_{z}\hat{J}_{\ell}}=0$ if $\braket{\hat{J}_{z}}=0$, these are
generally not satisfied. As a result, trajectories will generally not feature
inversion-time reversal symmetry with quantum fluctuations.

Figure 4:  Scaling of maximum correlations with $N$ at second order (a) and
third order (b). Initial state is
$\ket{\theta_{0},\phi_{0}}=\ket{\pi/10,\pi/2}$ with $\kappa=1$ and
$\lambda=0.5$.

So far, we studied the dissipative generation of non-Gaussian states for
$N=200$. However, many cold atom experiments operate in the regime $N>10^{3}$
[58]. Surprisingly, strong correlations created in the dispersive regime decay
slowly as the atom number $N$ is increased. Figure 4(a) shows how the maximum
total degree of normalized correlations at order two,
$\max(\mathcal{C}_{2})/N^{2}$, scales with $N$ as $\omega$ is varied. Linear
fits over a logarithmic scale demonstrate an approximate power law. The purely
dissipative case ($\omega=0$) has the largest gradient (magnitude), with
correlations falling off rapidly (approximately $\propto 1/N$), which is
compatible with the behavior of coherent states. For larger $\omega$, both
second- and third-order correlations, $\max(\mathcal{C}_{2})/N^{2}$ and
$\max(\mathcal{C}_{3})/N^{3}$ [Fig. 4] are maintained for significantly larger
atom numbers, as the pulses become increasingly more dispersive. In this
regime, we observe the formation of non-Gaussian states for systems with up to
at least $N=1000$ particles. The onset of classical behavior, in the sense
that higher-order cumulants vanish as $N$ increases, occurs for dissipative
superradiant pulses well before dispersive pulses. Interestingly, this
indicates that for typical atom numbers in cold atom experiments, a regime
exists where even for large $N\gtrsim 10^{3}$ the full distribution of quantum
fluctuations must be taken into account.

Figure 5:  Preserving non-Gaussian correlations with a quench protocol. (a)
$\mathcal{C}_{3}(t)$ for no ramp (red), an instantaneous ramp (dark green),
and for ramp times $T_{\text{ramp}}=500/\kappa,1000/\kappa$ (lighter greens).
The quench protocol is shown in the inset. (b1)–(b4) Snapshots of spin
$Q$-function for $T_{\text{ramp}}=500/\kappa$. Here, $\kappa=1$,
$\omega_{\text{max}}=5$, $\omega_{\text{min}}=0$, $\lambda_{\text{max}}=0.2$,
$\lambda_{\text{min}}=0.02$, $N=200$, and the initial state is
$\ket{\theta_{0},\phi_{0}}=\ket{\pi/10,\pi/2}$.

In the dispersive pulses discussed so far, non-Gaussian states appear and
disappear rapidly. As we now show, it is possible to form and subsequently
maintain non-Gaussian states with a quenching protocol for the effective
cavity resonance $\omega$ and coupling $\lambda$. When the mean spin direction
is roughly at the equator of the Bloch sphere, so that non-Gaussian
correlations are maximal, $\omega$ and $\lambda$ are ramped down. In this way,
a dispersive pulse becomes dissipative and the evolution is slowed, thereby
preserving non-Gaussian correlations are preserved [Fig. 5]. To account for
experimentally feasible, non-instantaneous quenches, we model the quenching
protocol with a piecewise linear ramp in $\omega$ and $\lambda$ 111.
Initially, the pulse has a dispersive stage with $\omega=\omega_{\text{max}}$
and $\lambda=\lambda_{\text{max}}$. The initially excited atoms evolve as in
Fig. 3 and begin to exhibit strong correlations, signalling the appearance of
non-Gaussianity. When the correlations reach their peak near the midpoint of
the pulse, $\omega$ and $\lambda$ are ramped down linearly over a time
$T_{\text{ramp}}$ to a dissipative stage with $\omega=\omega_{\text{min}}$ and
$\lambda=\lambda_{\text{min}}$. In a purely dissipative regime with
$\omega_{\text{min}}=0$, the non-Gaussian state formed over the initial
dispersive stage of the pulse is “frozen” and transported down the Bloch
sphere as in Fig. 2(a1). Since the speed is determined by $\lambda$, a quench
can maintain the non-Gaussian states formed in the dispersive stage of the
pulse over long periods of time. The preservation of correlations is monitored
with $\mathcal{C}_{3}(t)$ in Fig. 5(a), with the quenching protocol shown in
the inset panel. With no ramping, the superradiant pulse remains dispersive,
and there is a rapid build-up of correlations which then disappear quickly.
For maintaining non-Gaussian correlations, the best-case scenario is an
instantaneous quench with $T_{\text{ramp}}=0$. In this case, we observe a
rapid onset of correlations in the dispersive stage and then, in the
dissipative stage, a decay that preserves third-order correlations over a long
timescale (ensured by setting $\lambda_{\text{min}}/\lambda_{\text{max}}\ll
1$). For a more realistic quench with finite $T_{\text{ramp}}$ (the time taken
for the driving laser to change frequency and power), we observe similar
behavior, with the maximal degree of correlation only slightly below the
instantaneous quench. The evolution of quantum states on the Bloch sphere for
$T_{\text{ramp}}=500/\kappa$ is shown in Fig. 5(b1)–(b4), showcasing the
dispersive stage, followed by a very slow decay of the state.

In conclusion, we investigated the open Tavis-Cummings model in the dispersive
regime between the well-explored limits of dominant dissipative and unitary
processes. The interplay of both processes generates and sustains superradiant
pulses with highly non-Gaussian quantum correlations. Importantly, enhanced
fluctuations are largely preserved with increasing atom number, showing that
the quantum-to-classical crossover can depend strongly on the dynamical
behavior of fluctuations. Our results open up exciting possibilities for
generation and control of specified non-Gaussian correlations in driven-
dissipative many-body quantum systems. The non-Gaussian quantum states studied
could be used in optimal state preparation in quantum metrology [59]. Owing to
the strong state deformation, our protocol for creating dispersive
superradiant pulses might be exploited to improve magnetic field sensing using
superradiant spin-amplification [32]. Its implementation with ultracold atoms
in the dispersive regime of cavity QED [8, 16] has several advantages. Since
cavity detuning and coupling strength are determined by the properties of the
drive laser, the parameters we considered can indeed be adjusted dynamically
during the experiment. Moreover, it is possible to non-destructively measure
the cavity output field, enabling real-time readout of the spin inversion,
which is crucial for optimal superradiant spin amplification.

###### Acknowledgements.

We acknowledge funding from the Swiss National Science Foundation: project
numbers IZBRZ2, 186312, 182650, and 212168 and NCCR QSIT, from EU Horizon2020:
ERC advanced grant TransQ (project number 742579).

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# Sensitivity analysis of random two-body interactions

Calvin W. Johnson  Department of Physics, San Diego State University, 5500
Campanile Drive, San Diego, CA 92182-1233  Plamen G. Krastev  Department of
Physics, San Diego State University, 5500 Campanile Drive, San Diego, CA
92182-1233  Lawrence Livermore National Laboratory, P.O. Box 808, L-414,
Livermore, CA 94551

###### Abstract

The input to the configuration-interaction shell model includes many dozens or
hundreds of independent two-body matrix elements. Previous studies have shown
that when fitting to experimental low-lying spectra, the greatest sensitivity
is to only a few linear combinations of matrix elements. Here we consider
interactions drawn from the two-body random ensemble, or TBRE, and find that
the low-lying spectra are also most sensitive to only a few linear
combinations of two-body matrix elements, in a fashion nearly
indistinguishable from an interaction empirically fit to data. We find in
particular the spectra for both the random and empirical interactions are
sensitive to similar matrix elements, which we analyze using monopole and
contact interactions.

††preprint: INT-PUB-10-008††preprint: LLNL-JRNL-423216

##  I Introduction and Motivation

The configuration-interaction shell model is a useful framework for a detailed
understanding of low-energy nuclear structure BG77 ; br88 ; ca05 . The many-
body basis is a large dimension ($10^{3-10}$) set of Slater determinants,
which are antisymmeterized products of single-particle states. One must
truncate the single-particle states, corresponding to one or a few shells
(typically using the harmonic oscillator as an approximation to the mean-
field); the many-body basis may be further truncated. For phenomenological
calculations one writes the Hamiltonian in terms of single-particle energies
and two-body matrix elements, while for ab initio calculations one may extend
this to three-body interactions NO03 .

The two-body matrix elements are the matrix elements of the residual
interaction in the lab frame,

$V_{JT}(ij,kl)=\langle ij;JT|\hat{V}|kl;JT\rangle$ (1)

where $|ij;JT\rangle$ is the normalized, antisymmeterized product of particles
in orbits labeled by $i$ and $j$ and coupled to good angular momentum $J$ and
isospin $T$. If one starts from a translationally invariant interaction
between particles, one can either compute the integral in the lab frame or
start in the relative frame and then transform to the lab frame; in either
case there are correlations between the matrix elements, although they are not
obvious to the casual observer.

Often for semi-phenomenological calculations, one starts from a “realistic”
interaction, and then adjusts the two-body matrix elements until the rms error
on a set of experimentally known energy levels is minimized BG77 . In the
$1s$-$0d$ shell, such a semi-phenomenological interaction has been recently
derived br06 , improving on an earlier interactionWildenthal .

It has been found that the fits are empirically dominated by a few linear
combinations of matrix elements. The physical meaning of those dominant
combinations is not immediately obvious. One might naively guess the linear
correlations are due to an underlying translationally invariant interaction
(although a density dependence would destroy this). Somewhat more
phenomenologically, it has been argued by appealing to mean-field properties
that one can improve fits primarily through adjusting the monopole-monopole
part of the interaction, that is, interaction terms that look like
$n_{a}(n_{b}-\delta_{ab})$, where $n_{a}$ is the number of particles in the
$a$th orbit. This protocol for shifting monopole strengths has been
successfully applied to several semi-empirical interactions po81 ; ma97 ; ut99
; ho04 ; su06

A related study BJ09 , investigating the origin of many-body forces from
truncation of the model space, also found an empirical fit dominated by a few
linear combinations of matrix elements. While much of the fit was dominated by
the monopole interactions, even better agreement was brought about using a
contact interaction motivated by its usage in mean-field calculations sk56 ;
be03  and effective field theoryvk99 .

In investigating the character and origin of the dominant matrix elements, it
is useful to ask if there is anything special about the nuclear interaction.
One way to ask this question is to compare with interactions drawn from the
two-body random ensemble (TBRE), which despite their arbitrary nature are
known to echo some features of real nuclear spectra JBD98 ; ZV04 ; ZAY04 ;
PW07 . In this paper we conduct a sensitivity analysis of the low-lying
spectra of random interactions and compare against a standard empirical
interaction, USDB. We find that for more measures all the interactions are
nearly indistinguishable, at least on a statistical level.

##  II Methodology and results

Our methodology follows previous work BG77 ; br06 ; BJ09 ; we work in the
$1s$-$0d$ valence space with an inert 16O core. Given an input set of two-body
matrix elements (we leave aside single-particle energies and any $A$-dependent
scaling), which we write as a vector $\vec{v}$, we can calculate the
eigenvalues $E_{\alpha}(\vec{v})$ of the many-body Hamiltonian. For this work
the label $\alpha$ ranged over all nuclides with $0\leq
Z_{\mathrm{valence}}\leq N_{\mathrm{valence}}\leq 10$ and took the ground
state binding energy and the first five excitation energies.

If one has a target spectrum $E_{\alpha}^{0}$, say from experiment, then the
goal of the fit is to minimize

$\sum_{\alpha}\left(E_{\alpha}(\vec{v})-E_{\alpha}^{0}\right)^{2}$ (2)

(for simplicity we leave off the experimental uncertainty in each state).
Expanding to first order

$E_{\alpha}(\vec{v}+\delta\vec{v})\approx E_{\alpha}(\vec{v})+\sum_{i}\delta
v_{i}\frac{\partial E_{\alpha}}{\partial v_{i}}$ (3)

then minimizing (2) yields

$\sum_{\alpha}\left(E_{\alpha}(\vec{v})-E_{\alpha}^{0}\right)\frac{\partial
E_{\alpha}}{\partial v_{i}}=\sum_{j}\sum_{\alpha}\frac{\partial
E_{\alpha}}{\partial v_{i}}\frac{\partial E_{\alpha}}{\partial v_{j}}\delta
v_{j}.$ (4)

This equation is in the form $\vec{b}=\mathbf{A}\vec{x}$ where

$A_{ij}=\sum_{\alpha}\frac{\partial E_{\alpha}}{\partial v_{i}}\frac{\partial
E_{\alpha}}{\partial v_{j}}.$ (5)

The derivatives come via the Hellmann-Feynman theorem F39

$\frac{\partial E_{\alpha}}{\partial
v_{i}}=\left\langle\Psi_{\alpha}\left|\hat{H}_{i}\right|\Psi_{\alpha}\right\rangle$
(6)

where $\hat{H}_{i}$ is the Hamitonian operator whose strength is $v_{i}$.

We then find the eigenvalues of $\mathbf{A}$ (which is equivalent to finding
the squares of the eigenvalues in the singular-value decomposition of
$\partial E_{\alpha}/\partial v_{i}$). We do this for both the USDB
interaction and for an ensemble of 100 sets of random two-body interactions,
also called the two-body random ensemble (TBRE). The results are shown for the
TBRE in Fig. 1, where we have separated out the sensitivity just for the
binding energies (ground state energies) and the excitation energies. Although
not shown, the equivalent SVD eigenvalues for USDB are completely within the
TBRE results.

Figure 1:  (Color online)The spectra of eigenvalues from a singular-value
decomposition of the sensitivity matrix $\mathbf{A}$ (Eq. 5) for the two-body
random ensemble. The lower curve is for ground state energies, while the upper
curve is for excitation energies.

The lower curve is for ground states only, while the upper curve is for
excitations energies relative to the ground state. Clearly, and perhaps
unsurprisingly, the ground state energies are predominantly sensitive to just
a few linear combinations of matrix elements–significantly fewer than
excitations energies.

Fig. 1 characterizes eigenvalues of $\mathbf{A}$. The next step is to
characterize the eigenvectors associated with the dominant eigenvalues,
specifically by comparing with monopole and contact interactions. To do so we
first discuss a method for quantifying the overlap between two vector
subspaces MC ; SK08 . Consider two vectors subspaces, $S_{1}$ and $S_{2}$. Let
$\mathbf{V}_{1}$ be a matrix whose column vectors are the (orthonormal) basis
vectors of $S_{1}$, and similarly with $\mathbf{V}_{2}$. From these one
constructs the overlap matrix
$\mathbf{\Omega}=\mathbf{V}_{1}^{\dagger}\mathbf{V}_{2}$. Note that if the
subspaces are not of equal dimension then $\mathbf{\Omega}$ is not a square
matrix. In any case we do a singular value decomposition of $\mathbf{\Omega}$;
the SVD eigenvalue spectrum then is a measure of the overlap of the two
spaces. If the two spaces perfectly overlap then all eigenvalues are 1, if
just $N$ of the dimensions perfectly overlap than $N$ eigenvalues will be 1
and the rest zero. Note that this method is invariant under arbitrary choice
of orthonormal bases.

We begin with the monopole-monopole interaction of the form
$n_{a}(n_{b}-\delta_{ab})$, which has six unique terms, and thus six vectors
or linear combinations of matrix elements, in the $sd$-shell. These we combine
with the $k$ most dominant linear combinations that arise from the previous
analysis; somewhat arbitrarily we chose $k=6$ (our results do not change
qualitatively for other small values of $k$). The results, the SVD eigenvalues
of $\Omega$, are shown in Fig. 2. It is important to note we compute the
spectrum separately for each randomly generated interaction and then compute
the distribution.

Figure 2:  (Color online) The SVD spectrum that measures the overlap of the
subspace defined by the six largest eigenvectors from Fig. 1, with the
subspace defined by the monopole-monopole interaction. (Black) squares are for
the TBRE, while (red) diamonds are for USDB.

The results for ground states and for excited states are similar, so we
combine all states into a single calculation. The eigenvalues for USDB are
roughly $50\%$ higher than for the TBRE, but otherwise qualitatively very
similar.

We also compared for contact interactions; we took only two terms, the
$S=0,T=1$ channel and $S=1,T=0$ channel (there being only the $s$-wave channel
in relative coordinates). These results we summarize in Table 1.

Table 1: Leading eigenvalues from SVD of subspace overlaps of two-term contact interaction  Interaction | ground states | excited states | all states
---|---|---|---
USDB | 0.60 | 0.58 | 0.62
TBRE | $0.55\pm 0.04$ | $0.41\pm 0.06$ | $0.55\pm 0.04$

For comparison, the leading eigenvalue for the overlaps of USDB versus the
six-term monopole is 0.94, while that of the TBRE versus monopole is $0.66\pm
0.03$. There is somewhat more sensitivity to the monopole interaction than the
contact interaction; however, the reader should keep in mind that is not the
whole story. Recall that when fitting an interaction, the linearized equations
are cast in the form $\mathbf{A}\vec{x}=\vec{B}$. Our analysis in this paper
is entirely with the eigenvalues of $\mathbf{A}$, but in any fit one must also
look to $\vec{b}$ (which in practice is the deviation of the theoretical
spectra from experiment). For example, in BJ09  it was found that using a
contact interaction brought better agreement than a monopole interaction. One
can understand this in terms of conjugate gradient methods NR : the direction
of local steepest gradient may not in fact point towards the global minimum.

Figure 3:  SVD spectrum from the overlap of the dominant eigenvectors from
USDB and the TBRE.

By our measures so far, both the TBRE and the empirically-fit USDB look
qualitatively similar. Therefore we take a final analysis by comparing the
dominant linear combinations of the USDB with those from the TBRE. This is
show in Fig. 3, using the same analysis as for Fig. 2. For comparison with the
previous results, the leading eigenvalue is $0.72\pm 0.06$.

##  III Conclusion

We have analyzed the sensitivity of the low-lying spectra of the random two-
body ensemble of interactions to variations of the Hamiltonian matrix
elements; by using singular value decomposition, we find the dominant linear
combinations, which would be important in any fit to experimental data. We
found the SVD eigenvalues follow a pattern remarkably similar to that shown by
semi-realistic/semi-phenomenological interactions such as USDB. We also
analyzed the most dominant linear combinations of matrix elements by computing
the overlap with monopole and contact interactions. Overall, both the TBRE and
the empirical USDB had qualitatively similar results.

The U.S. Department of Energy supported this investigation through contracts
DE-FG02-96ER40985 and DE-FC02-09ER41587, and through subcontract B576152 by
Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344.

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