# Study of $N(1520)$ and $N(1535)$ structures via $\gamma^{*}p\to N^{*}$
transitions

A. Kaewsnod<EMAIL_ADDRESS>K. Xu<EMAIL_ADDRESS>Z. Zhao  School
of Physics and Center of Excellence in High Energy Physics and Astrophysics,
Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand  X. Y.
Liu  School of Physics and Center of Excellence in High Energy Physics and
Astrophysics, Suranaree University of Technology, Nakhon Ratchasima 30000,
Thailand  School of Physics, Liaoning University, Shenyang 110036, China  S.
Srisuphaphon  Department of Physics, Faculty of Science, Burapha University,
Chonburi 20131, Thailand  A. Limphirat  Y. Yan<EMAIL_ADDRESS>School of
Physics and Center of Excellence in High Energy Physics and Astrophysics,
Suranaree University of Technology, Nakhon Ratchasima 30000, Thailand

###### Abstract

The helicity amplitudes of the $N(1520)$ and $N(1535)$ resonances in the
$\gamma^{*}p\to N^{*}$ electromagnetic transition are studied in the
constituent quark model using the impulse approximation, with the proton and
resonances assumed to be in three-quark configurations. The comparison of
theoretical results and experimental data on the helicity amplitudes
$A_{1/2}$, $A_{3/2}$, and $S_{1/2}$ indicates that the $N(1520)$ and $N(1535)$
resonances are primarily composed of three-quark $L=1$ states but may contain
additional components. However, it is improbable that contributions from meson
clouds will be dominant at low $Q^{2}$.

##  I  Introduction

Low-lying nucleon resonances such as $N(1440)$, $N(1520)$, and $N(1535)$ have
been investigated continuously since their discovery, but their inner
structure remains an open question. In the three-quark picture, for example,
theoretical works always predict that the $N(1440)$ has a greater mass than
the $N(1520)$ and $N(1535)$. Information about the electromagnetic structure
of the nucleon resonances has been accumulated by the experimental facilities
like CLAS (JLab), MAID (Mainz) and MIT-Bates. In recent decades, the
$\gamma^{*}p\to N(1520)/N(1535)$ helicity amplitudes in a wide range of
$Q^{2}$ have been obtained and analyzed at JLab [1, 2, 3, 4, 5, 6, 7, 8, 9]
and by the Mainz group [10, 11, 12, 10] as one parametrization of the inner
structures.

Quark models such as the light-front (LF) relativistic quark model [13, 14,
15, 16], covariant spectator quark model [17, 18], and chiral quark models
[19, 20, 21, 22] have been applied to study the helicity amplitudes of
$N(1520)$ and $N(1535)$. The fact that the results from the quark models are
comparable with the experimental data of the $\gamma^{*}p\to N(1520)$ and
$\gamma^{*}p\to N(1535)$ helicity amplitudes at large square momentum transfer
$Q^{2}$ indicates that these two lowest negative-parity nucleon resonances are
dominated by the three valance quarks. However, at low and medium $Q^{2}$, the
poor theoretical results in the quark models may indicate that other
components, in addition to the three valance quarks, may play an important
role. The meson cloud effects are expected to be significant for both
$N(1520)$ and $N(1535)$ [15, 23, 18, 19, 20, 21]. The inclusion of pentaquark
components $q^{4}\bar{q}$ in Refs. [21, 24] leads to a reasonable description
of the experimental results of $A_{1/2}$ of $N(1535)$. The mass ordering
problem of $N(1440)$, $N(1520)$, and $N(1535)$ is investigated by including
the ground state pentaquark components in the negative-parity nucleon
resonances [25], pointing out that the $N(1520)$ and $N(1535)$ may contain
large pentaquark components.

In this work, we study the helicity transition amplitudes of the $N(1520)$ and
$N(1535)$ resonances in the process $\gamma^{*}p\to N^{*}$ in the three quark
picture. The work is expected to draw the upper limit of the three-quark
contribution and to serve as springboard for a more comprehensive study that
will include meson clouds, pentaquark components, and meson-baryon molecular
components. The proton wave function in this work is adopted from the study of
the proton electric form factor $G_{E}$ in Ref.[26]. This work employs the
impulse approximation, with the electromagnetic current being induced by free
single quark currents. The impulse approximation has been verified as a good
approximation of real processes. For instance, the experimental data on the
proton’s charge form factor and transition $\gamma^{*}p\to N(1440)$ are well
reproduced in the three quark picture using the impulse approximation [26].

The paper is organized as follows. In Sec. II, we briefly introduce the
helicity amplitude formalism before fitting the $A_{1/2}$, $A_{3/2}$, and
$S_{1/2}$ of the $N(1520)$ and $N(1535)$ resonances to the experimental data.
Sec. III contains a concise summary.

##  II Helicity amplitudes of $N(1520)$ and $N(1520)$ resonances

Figure 1: Diagram of photoproduction transition $N\gamma^{*}\rightarrow N^{*}$
where both the initial nucleon ($N$) and final nucleon resonance ($N^{*}$) are
in the three-quark picture.

We extend the constituent quark model in the work [26] to investigate the
helicity amplitudes of electromagnetic transitions $\gamma^{*}N\rightarrow
N(1520)/N(1535)$, where the electromagnetic current is induced by the free
single quark current (the impulse approximation) as illustrated in Fig. 1. We
assumed that both the nucleon $N$ and the resonance $N^{*}$ are composed of
three quarks.

The square of the four-momentum transfer is expressed as
$Q^{2}=-k^{2}=|\bm{k}|^{2}-\omega^{2}$ with the photon energy $\omega$ and the
magnitude of the photon momentum $|\bm{k}|$, defined as

$\displaystyle\omega$ $\displaystyle=\ K-\frac{Q^{2}}{2M_{N^{*}}},$
$\displaystyle|\bm{k}|$ $\displaystyle=\
\left[\omega^{2}+Q^{2}\right]^{\frac{1}{2}}.$ (1)

The $K$ in Eq. (II) is the real-photon momentum in the $N^{*}$ rest frame,
defined as

$\displaystyle K=\frac{M_{N^{*}}^{2}-M_{N}^{2}}{2M_{N^{*}}},$ (2)

where $M_{N^{*}}$ and $M_{N}$ are respectively the $N^{*}$ and $N$ masses.

The transverse helicity amplitudes $A_{1/2}$ and $A_{3/2}$, and the
longitudinal helicity amplitude $S_{1/2}$ are usually defined in the $N^{*}$
rest frame as

$\displaystyle A_{i}$ $\displaystyle=$ $\displaystyle\
\dfrac{1}{\sqrt{2K}}\Braket{N^{*},S^{\prime}_{z}=i}{q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime}}$
$\displaystyle\times T(q_{1}q_{2}q_{3}\rightarrow
q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime})\Braket{q_{1}q_{2}q_{3}}{N,S_{z}=i-1},$
$\displaystyle S_{1/2}$ $\displaystyle=$ $\displaystyle\
\dfrac{1}{\sqrt{2K}}\Braket{N^{*},S^{\prime}_{z}=\frac{1}{2}}{q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime}}$
$\displaystyle\times T(q_{1}q_{2}q_{3}\rightarrow
q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime})\Braket{q_{1}q_{2}q_{3}}{N,S_{z}=\frac{1}{2}},$
(3)

where $i=\frac{1}{2}$ for $A_{1/2}$ and $i=\frac{3}{2}$ for $A_{3/2}$. The $N$
wave function $\Braket{q_{1}q_{2}q_{3}}{N,S_{z}}$ and $N^{*}$ wave function
$\Braket{q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime}}{N^{*},S^{\prime}_{z}}$ in
the three-quark picture have been described in details in Refs.[27, 25], and
we refer the readers to those works for additional information. The
$T(q_{1}q_{2}q_{3}\rightarrow q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime})$ in
Eq. (3) is the transition amplitude of the process $\gamma q\to q^{\prime}$
displayed in Fig. 1, which can be calculated in the standard language of
quantum field theory,

$\displaystyle T(q_{1}q_{2}q_{3}\rightarrow
q_{1}^{\prime}q_{2}^{\prime}q_{3}^{\prime})=\ $ $\displaystyle
e_{3}\bar{u}_{s^{\prime}}(p^{\prime})\gamma^{\mu}u_{s}(p)\epsilon_{\mu}^{\lambda}(k)\Braket{q^{\prime}_{1}q^{\prime}_{2}}{q_{1}q_{2}}$
$\displaystyle=\ $ $\displaystyle
e_{3}T_{s^{\prime}s}^{\lambda}\Braket{q^{\prime}_{1}q^{\prime}_{2}}{q_{1}q_{2}}.$
(4)

Here $e_{3}$ and $u_{s(s^{\prime})}$ denote the electric charge and the Dirac
spinners of the third quark with the single quark spin projections
$s^{\prime},\,s$, and $\lambda$ is the photon helicity. In the work, the mass
of $u$ and $d$ quarks are taken to be $m=5$ MeV. The photon polarization
vectors $\epsilon_{\mu}^{\lambda}(k)$ of the longitudinal ($\lambda=0$) and
transverse ($\lambda=1$) helicity amplitudes in the Lorentz gauge are defined
as $\epsilon^{0}_{\mu}=(1,0,0,0)$ and
$\epsilon^{+}_{\mu}=-\frac{1}{\sqrt{2}}(0,1,i,0)$. The transition amplitude in
the longitudinal helicity amplitude $S_{1/2}$ is defined by the time component
of the electromagnetic current in Refs.[28, 29, 30]. The matrix elements
$T_{s^{\prime}s}^{\lambda}$ are shown in Appendix A. In Eq. (3), one sums over
all possible quantum states of the intermediate three quarks and integrates
over the momenta of all quarks.

(a)

a
[6] [10] [12] [8] [31] [32]

(b)

b
[4] [6] [10] [12] [32]

(c)  Figure 2: Helicity amplitudes of the $\gamma^{*}p\rightarrow N(1520)$
transition compared to the measurements. The solid and dash-dotted curves are
the results in Fitting A and Fitting B, respectively. The experimental data
are taken from [4, 6, 8, 31, 10, 12, 32].

c
[10] [12] [4] [6]

We fit the experimental data on the helicity amplitudes $A_{1/2}$, $A_{3/2}$,
and $S_{1/2}$ of the $N(1520)$ and $N(1535)$ resonances in the $\gamma^{*}p\to
N^{*}$ electromagnetic transition in the constituent quark model in the
impulse approximation with assuming that the proton and the resonances are in
the three-quark configurations. The proton spatial wave function is imported
from the theoretical results of the proton electric form factor [26]. The
spin, flavor, and color wave functions of the proton and the resonances are
characterized by the $SU(2)$, $SU(3)$, and $SU(3)$ symmetries, respectively.
The following calculations are performed:

Fitting A: The helicity amplitudes $A_{1/2}$, $A_{3/2}$, and $S_{1/2}$ of the
$N(1520)$ and $N(1535)$ resonances are fitted together, where the $N(1520)$
and $N(1535)$ take the same $L=1$ three quark state.

Fitting B: The helicity amplitudes $A_{1/2}$, $A_{3/2}$, and $S_{1/2}$ of the
$N(1520)$ resonance are fitted, where the $N(1520)$ is a $L=1$ three quark
state.

Fitting C: The helicity amplitudes $A_{1/2}$, and $S_{1/2}$ of the $N(1535)$
resonances are fitted, where the $N(1535)$ is a $L=1$ three quark state.

The fitting results of the helicity amplitudes of the $N(1520)$ and $N(1535)$
resonances are shown in Figs. 2c and 3 with the following notations: solid
lines indicate Fitting A, dash-dotted lines indicate Fitting B, and dashed
lines indicate Fitting C. The experimental data of the $A_{1/2}$ of the
$N(1520)$ resonance clearly split into two clusters, as shown in Fig. 2c: the
upper one from Ref. [6] and the lower one from Ref. [10, 12]. We fit in
Fitting A and Fitting B all the $A_{1/2}$ experimental data together.

The experimental data of all helicity amplitudes are reasonably fitted in the
three quark picture at large square momentum transfers, $Q^{2}>2.0$ GeV2 as
shown in Figs. 2c and 3, suggesting that the two resonances are primarily
three quark $L=1$ states. The results are consistent with other quark model
calculations.

The fitting results for the amplitudes $A_{3/2}$ and $S_{1/2}$ of the
$N(1520)$ resonance, as well as the amplitude $S_{1/2}$ of the $N(1535)$
resonance, are largely discrepant with the experimental data in the low and
medium $Q^{2}$ ranges, which may indicate that the calculations lack the
degrees of freedom necessary to fit the experimental data. The $N(1520)$ and
$N(1535)$ may contain other components rather than the $L=1$ three-quark
state, such as meson clouds, meson-baryon molecules and pentaquarks.

Note that the helicity amplitudes $A_{1/2}$ for both the $N(1520)$ and
$N(1535)$ resonances are well fitted over the entire $Q^{2}$ range. The
results contradict the argument that meson-cloud contributions are dominant at
low $Q^{2}$.

It is noted that Fitting B can not significantly improve the results of
Fitting A. It may conclude that the spatial wave functions of the $N(1520)$
and $N(1535)$ resonances are not significantly different in the three-quark
picture, which is consistent with mass spectrum calculations, where the tiny
$\vec{L}\cdot\vec{S}$ coupling results in a very small difference between the
two resonances’ wave functions.

##  III  Summary

We have studied the helicity amplitudes of the $N(1520)$ and $N(1535)$
resonances in the process $\gamma^{*}p\to N^{*}$. The impulse approximation is
applied in the work, where the electromagnetic current is induced by the free-
single-quark currents.

The very similar fitting results in Fitting A, B and C indicate that the quark
distributions of the $N(1520)$ and $N(1535)$ resonances in the three quark
picture are not significantly different, which is consistent with mass
spectrum calculations.

The experimental data of all the helicity amplitudes are reasonably fitted in
the three quark picture at large square momentum transfers, $Q^{2}>2.0$ GeV2,
implying that the $N(1520)$ and $N(1535)$ resonances are dominated by the
three valance quarks.

The amplitudes $A_{3/2}$ and $S_{1/2}$ of the $N(1520)$ and the amplitude
$S_{1/2}$ of the $N(1535)$ can not be fit in the three quark picture. It might
expect that the $N(1520)$ and $N(1535)$ may consist of other components
besides the $L=1$ three quark states, such as meson-clouds, meson-baryon
molecules, and pentaquarks.

It is arguable that meson-cloud contributions are dominant at low $Q^{2}$, as
the helicity amplitudes $A_{1/2}$ for both the $N(1520)$ and $N(1535)$
resonances are reasonably fit in the three quark picture over the entire
$Q^{2}$ range.

Calculations of higher-order contributions from the meson cloud and gluon
exchange diagrams, as well as pentaquark components, are currently underway.

a
[6] [2] [10] [12] [32]

(a)

b
[4] [6] [10] [12]

(b)

Figure 3: Helicity amplitudes of the $\gamma^{*}p\rightarrow N(1535)$
transition compared to the measurements. The solid and dashed curves are the
results in Fitting A and Fitting C, respectively. The experimental data are
taken from [4, 6, 2, 10, 12, 32].

## ACKNOWLEDGMENTS

This research has received funding support from the NSRF via the Program
Management Unit for Human Resources & Institutional Development, Research and
Innovation [grant number B05F640055]. A.K. and Y.Y. gratefully acknowledge
funding from TSRI-Royal Golden Jubilee Ph.D. (RGJ-PHD) Program (Grant No.
PHD/0242/2558) and SUT-OROG scholarship (contract no. 46/2558).

##  Appendix A Matrix elements of $\gamma q\to q^{\prime}$ for helicity
amplitudes

The matrix elements of the single quark transition $\gamma q\to q^{\prime}$
for the helicity $\lambda=0,1$ are derived in detail,

$\displaystyle T^{0}_{\uparrow\uparrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[1+\dfrac{p_{z}^{\prime}p_{z}+2p_{-}^{\prime}p_{+}}{(E^{\prime}+m)(E+m)}\right],$
$\displaystyle T^{0}_{\uparrow\downarrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[\dfrac{\sqrt{2}(p_{z}^{\prime}p_{-}-p_{-}^{\prime}p_{z})}{(E^{\prime}+m)(E+m)}\right],$
$\displaystyle T^{0}_{\downarrow\uparrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[\dfrac{\sqrt{2}(-p_{z}^{\prime}p_{+}+p_{+}^{\prime}p_{z})}{(E^{\prime}+m)(E+m)}\right],$
$\displaystyle T^{0}_{\downarrow\downarrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[1+\dfrac{p_{z}^{\prime}p_{z}+2p_{+}^{\prime}p_{-}}{(E^{\prime}+m)(E+m)}\right],$
$\displaystyle T^{+}_{\uparrow\uparrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[\dfrac{2p_{+}}{E+m}\right],$
$\displaystyle T^{+}_{\uparrow\downarrow}$ $\displaystyle=$ $\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[-\dfrac{\sqrt{2}p_{z}}{E+m}+\dfrac{\sqrt{2}p_{z}^{\prime}}{E^{\prime}+m}\right],$
$\displaystyle T^{+}_{\downarrow\uparrow}$ $\displaystyle=$ $\displaystyle\
0,$ $\displaystyle T^{+}_{\downarrow\downarrow}$ $\displaystyle=$
$\displaystyle\
\left[\dfrac{(E^{\prime}+m)(E+m)}{4E^{\prime}E}\right]^{\frac{1}{2}}\left[\dfrac{2p_{+}^{\prime}}{E^{\prime}+m}\right],$
(5)

where $E$ and $E^{\prime}$ are respectively the energies of the initial and
final interacting quarks with the dynamical quark mass of $u$ and $d$ quarks
as $m$, and $p_{\pm}=\frac{1}{\sqrt{2}}(p_{x}\pm ip_{y})$.

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# Quantum Dynamics with Stochastic Non-Hermitian Hamiltonians

Pablo Martinez-Azcona<EMAIL_ADDRESS>Department of Physics and
Materials Science, University of Luxembourg, L-1511 Luxembourg  Aritra Kundu
Department of Physics and Materials Science, University of Luxembourg, L-1511
Luxembourg
Avadh Saxena Theoretical Division, Los Alamos National Laboratory, Los Alamos,
New Mexico 87545, USA  Adolfo del Campo Department of Physics and Materials
Science, University of Luxembourg, L-1511 Luxembourg  Donostia International
Physics Center, E-20018 San Sebastián, Spain  Aurélia Chenu
<EMAIL_ADDRESS>Department of Physics and Materials Science, University
of Luxembourg, L-1511 Luxembourg

###### Abstract

We study the quantum dynamics generated by a non-Hermitian Hamiltonian subject
to stochastic perturbations in its anti-Hermitian part, describing fluctuating
gains and losses. The master equation governing the noise-average dynamics
describes a new form of dephasing. We characterize the resulting state
evolution and analyze its purity. The novel properties of such dynamics are
illustrated in a stochastic dissipative qubit. Our analytical results show
that adding noise allows for a rich control of the dynamics, with a greater
diversity of steady states and the possibility of state purification.

Quantum systems are always in contact with an environment. For this reason,
any realistic description of quantum dynamics should include the effects of
such coupling. The theory of open quantum systems (OQS) offers a wide range of
approaches to this end Breuer and Petruccione (2007); Rivas and Huelga (2012).
Two prominent approaches involve non-Hermitian (NH) Ashida  _et al._ (2020);
Plenio and Knight (1998); Moiseyev (2011) and stochastic Budini (2001); Chenu
_et al._ (2017); Kiely (2021) Hamiltonians. NH Hamiltonians are generally
associated with effective descriptions that account only for a subset of the
total number of degrees of freedom of a quantum mechanical system. They can be
obtained unraveling the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation
Gorini  _et al._ (1976); Lindblad (1976) and post-selecting the trajectories
with no quantum jumps Ashida  _et al._ (2020); Roccati  _et al._ (2022), or
using projector techniques, as often done in quantum optics, nuclear physics,
and molecular quantum chemistry Plenio and Knight (1998); Moiseyev (2011);
Feshbach (1958, 1962); Cohen‐Tannoudji  _et al._ (1998). Recently, non-
Hermitian Hamiltonians have regained interest because they display novel
physical phenomena and topological phases Shen  _et al._ (2018); Gong  _et
al._ (2018); Kawabata  _et al._ (2018); Yao and Wang (2018); Takata and Notomi
(2018); Yang  _et al._ (2018); Guo  _et al._ (2023) and provide understanding
of the quantum Zeno effect Dubey  _et al._ (2023); Snizhko  _et al._ (2020).

NH Hamiltonians generally exhibit complex energy eigenvalues in which the
imaginary part describes the lifetime of a state Gamow (1928); Cohen‐Tannoudji
_et al._ (1998). Time evolution ceases then to be trace-preserving due to the
leakage of probability density between the subspace under consideration and
that associated with the remaining degrees of freedom. The evolution can be
made trace-preserving (TP) by renormalizing the density matrix at all times,
which renders the evolution nonlinear in the state Carmichael (2008); Brody
and Graefe (2012); Alipour  _et al._ (2020); Geller (2023); Rembieliński and
Caban (2021). TP nonlinear dynamics has been realized experimentally by post-
selection of any dissipative evolution, i.e., discarding the trajectories with
quantum jumps Naghiloo  _et al._ (2019); Chen  _et al._ (2021). In this
setting, the renormalization of the trace comes naturally as a restriction to
the subset of trajectories with no quantum jumps. Furthermore, it allows us to
study a plethora of dissipation-induced phenomena Abbasi  _et al._ (2022);
Erdamar  _et al._ (2024); Harrington  _et al._ (2022); Chen  _et al._ (2022);
Wang  _et al._ (2023).

Stochastic Hamiltonians allow for a treatment of OQS in which the information
of the environment is included through the noise statistics and the coupling
to the noise Van Kampen (2011); Budini (2001); Kiely (2021). These appear
naturally in the stochastic Schrödinger equations obtained by unraveling the
GKSL equation in quantum trajectories Wiseman (1996); Wiseman and Milburn
(2009); Carmichael (2008). Stochastic Hamiltonians can be used to engineer
many-body and long-range interactions Chenu  _et al._ (2017), and their
behavior beyond the noise average relates to quantum information scrambling
Martinez-Azcona  _et al._ (2023).

Non-Hermitian Hamiltonians are typically studied in a deterministic setting.
In this Letter, we go beyond this paradigm by studying the dynamics of NH
Hamiltonians subject to classical noise, focusing on the effect of noise in
the anti-Hermitian part of the Hamiltonian. Stochastic non-Hermitian
Hamiltonians appear, e.g., in homodyne detection of the degenerate parametric
oscillator Carmichael (1991). The stability of exceptional points under noise
has also been investigated Wiersig (2020). Here, we consider the stochastic
dynamics generated by a fluctuating non-Hermitian operator and derive an exact
nonlinear master equation for the noise-averaged state dynamics, which
recovers noise-induced quantum jumps. In general, this master equation is not
of GKSL form and exhibits novel features. We derive an exact equation of
motion for the purity and characterize the dynamics. Specifically, we identify
the stable steady states in terms of the eigenvalues and eigenstates of the
Liouvillian and show that they live in the subspace spanned by the
eigenvectors whose eigenvalues’ real part is the largest. We illustrate these
regimes by studying the dynamics in a stochastic dissipative qubit Naghiloo
_et al._ (2019). Our analytical results provide the possible steady states,
the spectral properties governing the dynamics, and the evolution of the
purity.

_Dynamics under anti-Hermitian fluctuations._ We consider a system with
Hermitian Hamiltonian $\hat{H}_{0}$ that is subject to classical noise coupled
to the anti-Hermitian operators $i\hat{L}$, i.e.,

$\hat{H}_{t}=\hat{H}_{0}-i(1+\sqrt{2\gamma}\xi_{t})\hat{L}.$ (1)

This describes fluctuations around the mean in the Hermitian operator
$\hat{L}=\hat{L}^{\dagger}$, with standard deviation $\sqrt{2\gamma}$. The
classical noise is taken as Gaussian real white noise for simplicity,
characterized by $\mathbb{E}(\xi_{t})=0$ and
$\mathbb{E}(\xi_{t}\xi_{t^{\prime}})=\delta(t-t^{\prime})$, where
$\mathbb{E}(\bullet)$ denotes the classical stochastic average. The
(unnormalized) system density matrix for a single trajectory
$\tilde{\varrho}_{t}$ evolves, over a small time increment, as
$\tilde{\varrho}_{t+\mathrm{d}t}=\hat{U}_{\mathrm{d}t}\tilde{\varrho}_{t}\hat{U}^{\dagger}_{\mathrm{d}t}$,
where the propagator
$\hat{U}_{\mathrm{d}t}=\exp(-i\hat{H}_{0}\mathrm{d}t-\hat{L}\mathrm{d}t-\sqrt{2\gamma}\hat{L}\mathrm{d}W_{t})$
depends on the Wiener process $\mathrm{d}W_{t}=\xi_{t}\mathrm{d}t$. For
Gaussian white noise, the latter obeys Itō’s rules Gardiner (1985):
$\mathrm{d}W_{t}^{2}=\mathrm{d}t$ and
$(\mathrm{d}W_{t})^{n}=0,\;\forall\;n>2$. Expanding the exponential dictating
the evolution of the density matrix with these rules, we find a stochastic
differential equation (SDE) for its evolution,

$\displaystyle\mathrm{d}\tilde{\varrho}_{t}=$
$\displaystyle\left(-i[\hat{H}_{0},\tilde{\varrho}_{t}]-\\{\hat{L},\tilde{\varrho}_{t}\\}+\gamma\\{\hat{L},\\{\hat{L},\tilde{\varrho}_{t}\\}\\}\right)\mathrm{d}t$
$\displaystyle-\sqrt{2\gamma}\\{\hat{L},\tilde{\varrho}_{t}\\}\mathrm{d}W_{t}.$
(2)

We denote $\tilde{\rho}\equiv\mathbb{E}(\tilde{\varrho})$ the noise-averaged
density matrix and use the tilde $\tilde{\bullet}$ to highlight that it is not
normalized. Trace preservation can be imposed either by (i) renormalizing all
single-trajectories and then average over the noise Carmichael (2008); Wiseman
and Milburn (2009); Dubey  _et al._ (2023); Benoist  _et al._ (2021), or (ii)
first averaging over the noise, and only then imposing trace preservation.
These two approaches will generally yield different dynamics. We focus on the
latter to capture the evolution of ensembles of trajectories and defer the
former for future work.

The average over the noise of the unnormalized evolution reads

$\displaystyle\frac{\mathrm{d}\tilde{\rho}_{t}}{\mathrm{d}t}=-i[\hat{H}_{0},\tilde{\rho}_{t}]-\\{\hat{L},\tilde{\rho}_{t}\\}+\gamma\\{\hat{L},\\{\hat{L},\tilde{\rho}_{t}\\}\\}\equiv\tilde{\mathcal{L}}[\tilde{\rho}_{t}]$
(3)

for the noise-averaged density matrix, where $\tilde{\mathcal{L}}[\bullet]$
denotes the non-trace-preserving Liouvillian superoperator. This master
equation can be formally solved as
$\tilde{\rho}_{t}=e^{\tilde{\mathcal{L}}t}[\hat{\rho}_{0}]$. Imposing trace
preservation at the average level yields the nonlinear master equation

$\displaystyle\frac{\mathrm{d}\hat{\rho}_{t}}{\mathrm{d}t}=$
$\displaystyle\;\tilde{\mathcal{L}}[\hat{\rho}_{t}]-\mathrm{Tr}(\tilde{\mathcal{L}}[\hat{\rho}_{t}])\hat{\rho}_{t},$
(4) $\displaystyle=$
$\displaystyle-i[\hat{H}_{0},\hat{\rho}_{t}]-\\{\hat{L},\hat{\rho}_{t}\\}+\gamma\\{\hat{L},\\{\hat{L},\hat{\rho}_{t}\\}\\}$
$\displaystyle+2\mathrm{Tr}(\hat{L}\hat{\rho}_{t})\hat{\rho}_{t}-4\gamma\mathrm{Tr}(\hat{L}^{2}\hat{\rho}_{t})\hat{\rho}_{t},$

the solution of which is simply given by normalizing the non-trace-preserving
(nTP) evolution, i.e.,
$\hat{\rho}_{t}\equiv\frac{\tilde{\rho}_{t}}{\mathrm{Tr}(\tilde{\rho}_{t})}=e^{\tilde{\mathcal{L}}t}[\hat{\rho}_{0}]/\mathrm{Tr}(e^{\tilde{\mathcal{L}}t}[\hat{\rho}_{0}]).$
Since the NH Hamiltonian is stochastic, the evolution leads to quantum
jumps—terms of the form $\gamma\hat{L}\hat{\rho}_{t}\hat{L}$. Note that this
master equation is not of the GKSL form; rather, it describes a new form of
dephasing in which the jump operators act on the density matrix through a
double anticommutator and do not conserve the norm of the state. A more
general form is detailed in App. F. Similar equations, with a double
anticommutator but with a minus sign, appear when studying the adjoint
Liouvillian with fermionic jump operators and evolving operator Liu  _et al._
(2023); Clark  _et al._ (2010).

Note that, in general, the unnormalized evolution (3) can be trace-decreasing
(TD), trace-preserving (TP), or trace-increasing (TI). While TD dynamics can
easily be interpreted through post-selection, TI evolutions are conceptually
challenging, even if sometimes deemed non-physical Nielsen and Chuang (2010)
they can describe a biased ensemble of trajectories Garrahan and Lesanovsky
(2010). Note that the present TI dynamics can always be made TD or TP by a
gauge transformation in the master equation, namely, using an imaginary offset
$-ia$ in (1) (cf. App. A).

We now characterize the dynamics generated by (4) in terms of the purity
evolution and the stable steady states in the general setup before
illustrating them in a stochastic dissipative qubit.

_Purity dynamics._ The purity $P_{t}=\mathrm{Tr}(\hat{\rho}_{t}^{2})$ evolves
according to the differential equation

$\displaystyle\partial_{t}P_{t}=$
$\displaystyle-4\mathrm{Tr}(\hat{L}\hat{\rho}_{t}^{2})+4\gamma\left(\mathrm{Tr}(\hat{L}^{2}\hat{\rho}_{t}^{2})+\mathrm{Tr}(\hat{L}\hat{\rho}_{t}\hat{L}\hat{\rho}_{t})\right)$
$\displaystyle+4\mathrm{Tr}(\hat{L}\hat{\rho}_{t})P_{t}-8\gamma\mathrm{Tr}(\hat{L}^{2}\hat{\rho}_{t})P_{t}.$
(5)

Evaluating this expression at $t=0$ yields the decoherence time
$1/\tau_{P}\equiv\partial_{t}P_{t}|_{t=0}$. Whenever the state is pure,
$\hat{\rho}_{t}=\ket{\Psi_{t}}\bra{\Psi_{t}}$, the terms coming from the
deterministic anti-Hermitian part cancel out and the purity evolution
$\partial_{t}P_{t}=-4\gamma{\rm Var}_{\Psi}(\hat{L})$ is dictated by the
variance ${\rm
Var}_{\Psi}(\hat{L})=\braket{\Psi_{t}}{\hat{L}^{2}}{\Psi_{t}}-\braket{\Psi_{t}}{\hat{L}}{\Psi_{t}}^{2}$.
Thus, initial pure states exhibit a purity decay unless they are eigenstates
of $\hat{L}$, in which case the purity remains constant at first order in $t$
($\tau_{P}\rightarrow\infty$). This result is identical to the decoherence
time of pure states in a dephasing channel and quantum Brownian motion Zurek
(2003); Breuer and Petruccione (2007); Beau  _et al._ (2017). Interestingly,
this expression contains out-of-time-order terms reminiscent of OTOC’s
Martinez-Azcona  _et al._ (2023) and generalizes the known evolution for NH
Hamiltonians Brody and Graefe (2012).

_Long time dynamics: Stable steady states._ To characterize the long-time
dynamics, we use the right and left eigendecompositions of the non-trace-
preserving Liouvillian superoperator,
$\tilde{\mathcal{L}}[\tilde{\rho}_{\nu}]=\lambda_{\nu}\tilde{\rho}_{\nu}$ and
$\tilde{\mathcal{L}}^{\dagger}[\tilde{\rho}_{\nu}^{\textsc{(l)}}]=\lambda_{\nu}^{*}\tilde{\rho}_{\nu}^{\textsc{(l)}}$,
respectively. We define the eigenoperator basis as
$\hat{\sigma}_{\nu}=\hat{\rho}_{\nu}/\mathrm{Tr}(\hat{\rho}_{\nu})$ if the
operator has non-zero trace and as $\hat{\sigma}_{\nu}=\hat{\rho}_{\nu}$ if
the operator is traceless. Furthermore, the eigenstate is physical when it is
trace one, $\mathrm{Tr}(\sigma_{\nu})=1$, and positive semidefinite
$\hat{\sigma}_{\nu}\geq 0$. The basis is ordered such that the physical states
appear first $\nu<N_{\textsc{p}}$, the traceful states appear second
$\nu<N_{\textsc{t}}$, and the traceless states come last. In each of the
subsets, the states are further ordered by decreasing
$\mathrm{Re}(\lambda_{\nu})$. Assuming the Liouvillian $\tilde{\mathcal{L}}$
to be diagonalizable, the evolution of the density matrix in this operator
basis follows as

$\displaystyle\\!\\!\\!\hat{\rho}_{t}$
$\displaystyle=\frac{\sum_{\nu=0}^{N^{2}-1}c_{\nu}e^{\lambda_{\nu}t}\hat{\sigma}_{\nu}}{\sum_{\mu=0}^{N_{\textsc{t}}-1}c_{\mu}e^{\lambda_{\mu}t}}\stackrel{{\scriptstyle
t\rightarrow\infty}}{{\sim}}\hat{\sigma}_{0}{+}e^{-\Delta
t}\sum_{\nu\in\mathbb{M}_{2}}\frac{c_{\nu}e^{i\omega_{\nu}t}}{c_{0}}\hat{\sigma}_{\nu},$
(6)

where the coefficients are
$c_{\nu}=\mathrm{Tr}(\hat{\sigma}_{\nu}^{\textsc{(l)}}\hat{\rho}_{0})/{\rm
Tr}(\hat{\sigma}_{\nu}^{\textsc{(l)}}\hat{\sigma}_{\nu})$. All the physical
states ($\hat{\sigma}_{\nu}$ with $\nu<N_{\textsc{p}}$) are steady states.
However, only those whose eigenvalues have the largest real part
$\hat{\sigma}_{0}$ are stable under all perturbations 111The stability and
restrictions imposed by complete positivity will be studied in a follow-up
article (in preparation)..

At long times, the stable steady state $\hat{\sigma}_{0}$ is the eigenvector
associated with the eigenvalue with the largest real part over which there is
an initial support, i.e.,
$\mathrm{Re}(\lambda_{0})=\max_{\nu}\mathrm{Re}(\lambda_{\nu})$ such that
$c_{0}\neq 0$. The corrections to this state are suppressed at a rate dictated
by the dissipative gap $\Delta=\mathrm{Re}(\lambda_{0}-\lambda_{1})$. They can
oscillate with a frequency $\omega_{\nu}=\mathrm{Im}(\lambda_{\nu})$,
depending on the presence or absence of complex eigenvalues in the set of
states with the second largest real part, denoted by $\mathbb{M}_{2}$.

The long-time limit above assumes that the largest eigenvalue is real and non-
degenerate. In a more general case, denoting $\mathbb{M}_{1}$ the set of
states with the largest real part among those on which the initial state has
nonzero support, the steady state follows as
$\hat{\rho}^{\textsc{s}}=\frac{\sum_{\nu\in\mathbb{M}_{1}}^{N^{2}-1}c_{\nu}e^{i\omega_{\nu}t}\hat{\sigma}_{\nu}}{\sum_{\mu\in\mathbb{M}_{1}}^{N_{\textsc{t}}-1}c_{\mu}e^{i\omega_{\mu}t}}$.

_Illustration: The stochastic dissipative qubit._ Many of the exciting
phenomena displayed by non-Hermitian Hamiltonians can be experimentally
observed in a dissipative qubit Naghiloo  _et al._ (2019); Chen  _et al._
(2021); Abbasi  _et al._ (2022); Erdamar  _et al._ (2024); Harrington  _et
al._ (2022); Chen  _et al._ (2022); Wang  _et al._ (2023). The setting
corresponds to the Lindblad evolution of a 3-level system
$\\{\ket{g},\ket{e},\ket{f}\\}$, where all quantum jumps to $\ket{g}$ are
discarded. This post-selection process leads to the effectively non-Hermitian
Hamiltonian $\hat{H}=J\hat{\sigma}_{x}-i\Gamma\hat{\Pi}$, where $J$ is the
hopping between $\ket{e}$ and $\ket{f}$, $\Gamma$ is the loss rate from
$\ket{e}$ to $\ket{g}$ and $\hat{\Pi}=\ket{e}\bra{e}$ denotes the projector
over state $\ket{e}$. The (shifted) spectrum
$\varepsilon_{\pm}+i\frac{\Gamma}{2}=\pm\sqrt{J^{2}-\frac{\Gamma^{2}}{4}}$ is
real for $J\geq\frac{\Gamma}{2}$, and imaginary for $J<\frac{\Gamma}{2}$.
These two regimes respectively correspond to (passive) $\mathcal{PT}$ unbroken
and broken symmetry phases Bender and Boettcher (1998).

Figure 1: Purity evolution for different strengths of the noise as given by
(9) in a cross-section of the Bloch sphere. Figure 2: Spectral and steady-
state phase diagrams: (a) dissipative gap $\Delta$, (b) maximum imaginary part
of the eigenvalues $\omega_{\nu}$, (c) $z$, and (d) $y$ components of the
Bloch vector for the stable steady state. The relevant parameters are the
strength of the noise $\gamma$ and the decay rate $\Gamma$. The transition to
the noise-induced phase is at $\gamma^{*}=1/(2\Gamma)$ (black). The transition
from $\mathcal{PT}$ broken to the $\mathcal{PT}$ unbroken phase happens at
$\Gamma/J=2$ (gray dashed). In (b) the power laws $\gamma=1/(3\Gamma)$ and
$\gamma=1/\Gamma$ are also shown (orange dotted).

In this work, we are interested in the effect of noise in the anti-Hermitian
part of the Hamiltonian. We thus consider a Stochastic Dissipative Qubit (SDQ)
with time-dependent Hamiltonian

$\displaystyle\hat{H}^{\textsc{sdq}}_{t}=J\hat{\sigma}_{x}-i\Gamma(1+\sqrt{2\gamma}\xi_{t})\hat{\Pi},$
(7)

which physically corresponds to considering Gaussian fluctuations in the decay
parameter $\Gamma$, with strength $\gamma$. The master equation (4) for this
system reads

$\displaystyle\partial_{t}\hat{\rho}_{t}$
$\displaystyle=-iJ[\hat{\sigma}_{x},\hat{\rho}_{t}]-(\Gamma-\gamma\Gamma^{2})\\{\hat{\Pi},\hat{\rho}_{t}\\}+2\gamma\Gamma^{2}\hat{\Pi}\hat{\rho}_{t}\hat{\Pi}$
$\displaystyle\quad+(2\Gamma-4\gamma\Gamma^{2})\mathrm{Tr}(\hat{\Pi}\hat{\rho}_{t})\hat{\rho}_{t}.$
(8)

Note that the trace of the unnormalized Liouvillian acting on any state reads
$\mathrm{Tr}(\tilde{\mathcal{L}}[\hat{\rho}])=(4\gamma\Gamma^{2}-2\Gamma)\mathrm{Tr}(\hat{\Pi}\hat{\rho})$.
As $\mathrm{Tr}(\hat{\Pi}\hat{\rho})\geq 0$, the dynamics is trace-decreasing
when $4\gamma\Gamma^{2}-2\Gamma<0$ and trace-increasing when
$4\gamma\Gamma^{2}-2\Gamma>0$. Therefore, the dynamics exhibits a transition
from TD to TI at $\gamma^{*}=\frac{1}{2\Gamma}$. At this critical value, the
nonlinear term vanishes: the dynamics is Completely Positive and Trace
Preserving (CPTP), and its generator admits the standard GKSL form, with jump
operator $\hat{\Pi}$ and rate $\Gamma$. When the Hamiltonian is Hermitian
($\Gamma=0$), the dynamics is trivially unitary. The noise can be used to tune
the success rate, i.e. the number of trajectories with no jumps. For TD
dynamics, this success rate corresponds to the trace
$\mathrm{Tr}(\tilde{\rho}_{t})\propto\exp((4\gamma\Gamma^{2}-2\Gamma)t)$. If
the noise increases this decay is slower, thus, a stronger noise $\gamma$ can
be used to mitigate the decay of the success rate, especially near the
critical value $\gamma^{*}$.

Every state of a qubit can be expressed in the Bloch sphere as
$\hat{\rho}_{t}=\frac{1}{2}(\mathbb{1}+\mathbf{r}_{t}\cdot\hat{\boldsymbol{\sigma}})$
Nielsen and Chuang (2010). Expressing the Bloch vector
$\mathbf{r}_{t}=(x_{t},y_{t},z_{t})$ in spherical coordinates
$(r_{t},\theta_{t},\phi_{t})$, the purity (5) evolves for any state
$\hat{\rho}_{t}$ as

$\\!\\!\\!\partial_{t}P_{t}=\Gamma
r_{t}\left[(2\gamma\Gamma-1)(r_{t}^{2}-1)\cos\theta_{t}-\gamma\Gamma
r_{t}\sin^{2}\theta_{t}\right].$ (9)

Note that this expression shows a change in behavior at the TD-TI transition
$\Gamma\gamma=\frac{1}{2}$. When $\Gamma\gamma<\frac{1}{2}$, the purifying
states are in the north hemisphere of the Bloch ball, while for
$\Gamma\gamma>\frac{1}{2}$, these states are shifted to the southern
hemisphere, see Fig. 1.

We now investigate the spectral properties of the SDQ Liouvillian (8). The
phase diagrams in Fig. 2 have three distinct phases: at weak noise $\gamma$,
the notions of $\mathcal{PT}$ broken $\Gamma/J>2$ and unbroken $\Gamma/J<2$
symmetry of the NH Hamiltonian govern the dynamics, while at large noise
strength, we see a transition to the TI dynamics at $\gamma^{*}$. We refer to
these phases as $\mathcal{PT}$ broken ($\mathcal{PT}$b), $\mathcal{PT}$
unbroken ($\mathcal{PT}$u) and Noise Induced (NI). Figure 2(a) shows the
dissipative gap $\Delta$ as a function of the noise strength $\gamma$ and the
decay rate $\Gamma$. In the $\mathcal{PT}$u phase, the dissipative gap is very
small, and thus the convergence to the stable steady state is very slow. In
the $\mathcal{PT}$b phase the gap is larger, therefore the steady state is
reached faster than in the $\mathcal{PT}$u phase. In the NI phase, the gap is
very large, ensuring fast convergence to the stable steady state [cf. Eq.
(6)]. Note that the gap is always smaller around the transition to the NI
phase, which implies that the GKSL dynamics have slower convergence to the
stable steady state. Fig. 2(b) shows the maximum imaginary part of the
eigenvalues of the Lindbladian, which dictate the oscillatory behavior of the
dynamics. Deep in the $\mathcal{PT}$b and NI phases, the imaginary part
vanishes, implying that the dynamics is not oscillatory. However, it is
nonzero in the $\mathcal{PT}$u phase. This fact, along with the very small
dissipative gap, shows what features of the $\mathcal{PT}$u phase survive the
application of classical noise.

We now characterize the non-degenerate stable steady state
$\hat{\rho}^{\textsc{s}}$. Figure 2(c) shows the $z$ component of the Bloch
vector of $\hat{\rho}^{\textsc{s}}$, which reads
$z(\hat{\rho}^{\textsc{s}})=-\frac{\lambda_{0}(\lambda_{0}-A)}{4+\lambda_{0}(\lambda_{0}-A)}$
(cf. App. B), where $\lambda_{0}$ denotes the eigenvalue with the largest real
part and the constant $A=\frac{\Gamma}{J}(\gamma\Gamma-1)$. In the
$\mathcal{PT}$u phase, the $z$ component is small, and the steady state is
close to the completely mixed state. Thus, when a very small noise is added to
the $\mathcal{PT}$ symmetric NH Hamiltonian, the time evolution leads to a
highly mixed state at very long times. In the $\mathcal{PT}$b phase, $z$ is
close to unity—so the stable steady state is close to $\ket{f}$, as losses
induced by $\Gamma$ remove all population in the $\ket{e}$ state. In the NI
phase, the steady state is $\ket{e}$—the state that originally leaked to the
ground state. This can be interpreted as a noise-induced transition to
stability Horsthemke and Lafever (2006) of the originally unstable state, a
feature shared by other noisy dynamics Martinez-Azcona  _et al._ (2023). The
$y$ Bloch coordinate phase diagram in Fig. 2(d) shows that the transition from
the mixed state to the $\ket{f}$ ($\ket{e}$) state happens by acquiring a
negative (positive) value of the $y$ coordinate in the Bloch sphere. The
analytic expression for this quantity is
$y(\hat{\rho}^{\textsc{s}})=\frac{2\lambda_{0}}{4+\lambda_{0}(\lambda_{0}-A)}$
(cf. App. B).

Figure 3: Fidelity between the time-evolved state and the stable steady state
as a function of the loss parameter $\Gamma$ and time for noise strength
$\gamma J=0.05$. $\mathcal{PT}$u to $\mathcal{PT}$b phase transition (gray
dashed) and $\mathcal{PT}$b to NI phase transition (black). Time-scales of the
dissipative gap $\Delta^{-1}$ (white dotted) and oscillatory dynamics
$2\pi/\omega$ (red dotted).

Let us now turn to the dynamics generated by the master equation (8). The
dynamics can be solved numerically in two complementary ways (cf. App. C, D).
We characterize the distinguishability between the evolving and stable steady
states using the Uhlmann fidelity
$F^{\textsc{s}}_{t}=F(\hat{\rho}_{t},\hat{\rho}^{\textsc{s}})$ Uhlmann (1976);
Bures (1969), which admits a compact form in a qubit system Jozsa (1994);
Hübner (1992) (cf. App. E). Figure 3 shows the evolution of the fidelity as a
function of time and the decay parameter $\Gamma$ for a small noise, $\gamma
J=0.05$. This value ensures that the three different phases are manifested:
(i) for $\Gamma/J<2$, the system is in the $\mathcal{PT}$u phase, and the
fidelity to the steady state exhibits the anticipated oscillatory behavior
until it vanishes at a time close to the inverse dissipative gap
$t\sim\Delta^{-1}$ (white dotted line). The period of these oscillations is
perfectly characterized by $2\pi/\omega$ where
$\omega=\max_{\nu}\mathrm{Im}(\lambda_{\nu})$, as highlighted by the red
dotted line. Note that this oscillatory frequency is affected by the increase
of $\Gamma$ as we approach the $\mathcal{PT}$ phase transition, as also
observed experimentally in the dissipative qubit Naghiloo  _et al._ (2019).
(ii) For $2<\Gamma/J<\frac{1}{2J\gamma}$, the system is in the $\mathcal{PT}$b
phase, with a larger dissipative gap [cf. Fig. 2(a)], and thus a faster
convergence to the steady state. Interestingly, the oscillatory timescale can
be finite outside of the $\mathcal{PT}$u phase as $\omega$ does not vanish
identically. In particular, it shows minima at $\Gamma=\frac{1}{3\gamma}$ and
$\Gamma=\frac{1}{\gamma}$, as observed in Fig. 2(b). Exactly at the transition
$\Gamma=\frac{1}{2\gamma}$, the dynamics is CPTP, and its convergence is
slower, as explained by the smaller gap at the transition line. (iii) For
$\Gamma>\frac{1}{2\gamma}$, the system is in the NI phase and exhibits the
fastest convergence to the steady state. Experimentally, the decay rate is
$\Gamma=6.7\;\mu\mathrm{s}^{-1}$ Naghiloo  _et al._ (2019), which upper bounds
the noise strength $\gamma<\gamma^{*}\approx 0.074\;\mu\mathrm{s}$ given that
no NI phase was observed.

_Conclusions._ We have considered the time evolution governed by a fluctuating
non-Hermitian Hamiltonian describing a quantum mechanical system subject to
stochastic gain and loss. The resulting noise-averaged dynamics is described
by a novel kind of master equation beyond the GKSL form. We characterized the
purity dynamics and found that the stable steady states live in the
Liouvillian eigenspace whose eigenvalues have the largest real part. The
salient features of such dynamics are manifested in a stochastic dissipative
qubit. The addition of noise allows for control over the steady states and the
rate of convergence. We find three main dynamical phases: the $\mathcal{PT}$
unbroken and broken regimes, complemented with a noise-induced phase. Our
results are amenable to current experimental platforms realizing non-Hermitian
evolution. Our findings suggest that engineering fluctuating operators
associated with gain and loss may open new avenues for quantum state
preparation Verstraete  _et al._ (2009). Our results may also be applied to
understand the stability of imaginary time evolution Motta  _et al._ (2020)
against classical noise.

Acknowledgements— We thank Oskar Prosniak, Kohei Kawabata, and Martin Plenio
for their insightful discussions and Shiue-Yuan Shiau for comments on the
manuscript. This research was funded by the Luxembourg National Research Fund
(FNR, Attract grant 17132054 and grant 17132054). The work at Los Alamos was
supported by the U.S. Department of Energy.

##  Appendix A Gauge transformations of the nonlinear master equation

We discuss below how trace-increasing dynamics can be changed, such as to
conserve or reduce the state trace. For this, we explicitly distinguish the
stochastic $\hat{L}_{\textsc{s}}$ and deterministic $\hat{L}_{\textsc{d}}$
parts of the anti-Hermitian part of the Hamiltonian, and add an imaginary
deterministic shift $a\in\mathbb{R}$. Namely, we consider

$\hat{H}_{t}^{\prime}=\hat{H}_{0}-i(\hat{L}_{\textsc{d}}+a\,\hat{\mathbb{1}})-i\sqrt{2\gamma}\xi_{t}\hat{L}_{\textsc{s}},$
(10)

which generates the nonlinear master equation

$\displaystyle\partial_{t}\hat{\rho}_{t}=$
$\displaystyle-i[\hat{H}_{0},\hat{\rho}_{t}]-\\{\hat{L}_{\textsc{d}}+a\,\hat{\mathbb{1}},\hat{\rho}_{t}\\}+\gamma\\{\hat{L}_{\textsc{s}},\\{\hat{L}_{\textsc{s}},\hat{\rho}_{t}\\}\\}$
$\displaystyle+2\mathrm{Tr}\big{(}(\hat{L}_{\textsc{d}}+a\,\hat{\mathbb{1}})\hat{\rho}_{t}\big{)}\hat{\rho}_{t}-4\gamma\mathrm{Tr}(\hat{L}_{\textsc{s}}^{2}\hat{\rho}_{t})\hat{\rho}_{t}.$
(11)

Systems with any shift $a$ have the same dynamics, since the terms
$-\\{a\,\hat{\mathbb{1}},\hat{\rho}_{t}\\}$ and
$+2\mathrm{Tr}(a\hat{\rho}_{t})\hat{\rho}_{t}$ exactly cancel each other. This
provides a non-Hermitian generalization of the well-known fact that shifting
the Hamiltonian by a constant $\hat{H}+a\mathbb{1}$ has no consequence on the
dynamics. Thus, a complex zero of energy also has no dynamical effect,
provided that the trace of the density matrix is renormalized. It follows that
the dynamics can be made TD or even TP with a suitable choice of $a$, namely

$4\gamma\mathrm{Tr}(\hat{L}^{2}_{\textsc{s}}\hat{\rho}_{t})-2\mathrm{Tr}(\hat{L}_{\textsc{d}}\hat{\rho}_{t})-a\leq
0,$ (12)

where ‘$<$’ corresponds to a TD map and ‘$=$’ to TP dynamics.

For the SDQ considered in the main text,
$\hat{L}_{\textsc{d}}=\hat{L}_{\textsc{s}}=\Gamma\hat{\Pi}$, so the shift has
to obey the condition

$a\geq(4\gamma\Gamma^{2}-2\Gamma)\mathrm{Tr}(\hat{\Pi}\hat{\rho}),$ (13)

for any $\hat{\rho}$. Since the expectation value has the property
$\mathrm{Tr}(\hat{\Pi}\hat{\rho})=\rho_{ee}\leq 1$, a looser but state-
independent condition on the shift reads

$a\geq 4\gamma\Gamma^{2}-2\Gamma=BJ.$ (14)

Alternatively, consider a gauge transformation that shifts the stochastic
component by a constant $b\in\mathbb{R}$

$\hat{H}_{t}^{\prime}=\hat{H}_{0}-i\hat{L}^{\prime}_{\textsc{d}}-i\sqrt{2\gamma}\xi_{t}(\hat{L}_{\textsc{s}}+b\hat{\mathbb{1}}),$
(15)

which generates the nonlinear master equation

$\displaystyle\partial_{t}\hat{\rho}_{t}=$
$\displaystyle-i[\hat{H}_{0},\hat{\rho}_{t}]-\\{\hat{L}^{\prime}_{\textsc{d}}-4\gamma
b\hat{L}_{\textsc{s}},\hat{\rho}_{t}\\}+\gamma\\{\hat{L}_{\textsc{s}},\\{\hat{L}_{\textsc{s}},\hat{\rho}_{t}\\}\\}$
$\displaystyle+2\mathrm{Tr}((\hat{L}^{\prime}_{\textsc{d}}-4\gamma
b\hat{L}_{\textsc{s}})\hat{\rho}_{t})\hat{\rho}_{t}-4\gamma\mathrm{Tr}(\hat{L}^{2}_{\textsc{s}}\hat{\rho}_{t})\hat{\rho}_{t}.$
(16)

It follows that the transformation

$\begin{cases}\hat{L}_{\textsc{s}}\rightarrow\hat{L}_{\textsc{s}}+b\,\hat{\mathbb{1}},\\\
\hat{L}_{\textsc{d}}\rightarrow\hat{L}_{\textsc{d}}+4\gamma
b\,\hat{L}_{\textsc{s}},\end{cases}$ (17)

also leaves the nonlinear master equation invariant. This transformation,
similar to the GKSL case Breuer and Petruccione (2007), allows choosing jump
operators $\hat{L}_{\textsc{s}}$ that are traceless.

##  Appendix B Liouvillian spectrum of the Stochastic Dissipative Qubit

The master equations given in the main text can be formulated in Liouville
superspace. To this end, the density operator is written as a vector
$|\tilde{\rho})$ of the form

$\tilde{\rho}=\sum_{n,m}\tilde{\rho}_{nm}\ket{n}\bra{m}\;\mapsto\;|\tilde{\rho})=\sum_{n,m}\tilde{\rho}_{nm}\ket{n}\otimes\ket{m}^{*};$
(18)

we refer the reader to Gyamfi (2020) for a detailed description of this
procedure. Superoperators are mapped to operators on the superspace following
the Choi-Jamiołkowski isomorphism Choi (1975); Jamiołkowski (1972)

${\hat{X}}\tilde{\rho}{\hat{Y}}\;\mapsto\;({\hat{X}}\otimes{\hat{Y}}^{\intercal})|\tilde{\rho}),$
(19)

where $\otimes$ denotes the Kronecker product and $\bullet^{\intercal}$ the
transpose.

Vectorization requires fixing the inner product to be taken between operators;
in the above procedure, the inner product is chosen as the standard Hilbert-
Schmidt inner product $(X|Y)=\mathrm{Tr}(\hat{X}^{\dagger}\hat{Y})$, which for
vectorized operators conveniently reduces to the standard euclidean inner
product for vectors, $(X|Y)=\sum_{m,n}X_{mn}^{*}Y_{mn}$.

The non-trace-preserving state dynamics is then dictated by
$\frac{\mathrm{d}}{\mathrm{d}t}|\tilde{\rho})=\tilde{\mathcal{L}}|\tilde{\rho})$
with the vectorized Liouvillian superoperator in (3) given by

$\begin{split}\tilde{\mathcal{L}}=&-i(\hat{H}_{0}\otimes\hat{\mathbb{1}}-\hat{\mathbb{1}}\otimes\hat{H}_{0}^{\intercal})-(\hat{L}\otimes\hat{\mathbb{1}}+\hat{\mathbb{1}}\otimes\hat{L}^{\intercal})\\\
&+\gamma(\hat{L}^{2}\otimes\hat{\mathbb{1}}+\hat{\mathbb{1}}\otimes(\hat{L}^{2})^{\intercal}+2\hat{L}\otimes\hat{L}^{\intercal}).\end{split}$
(20)

Similarly, the Liouvillian for the SDQ (8) is given in matrix form by

$\tilde{\mathcal{L}}=J\left(\begin{array}[]{cccc}0&i&-i&0\\\ i&A&0&-i\\\
-i&0&A&i\\\ 0&-i&i&B\end{array}\right),$ (21)

where we have defined the constants $A=\frac{\Gamma}{J}(\gamma\Gamma-1)$ and
$B=2\frac{\Gamma}{J}(2\gamma\Gamma-1)$. The characteristic polynomial of the
Liouvillian thus reads

$(A-\Lambda)f(\Lambda)=0,$ (22)

with the cubic polynomial
$f(\Lambda)=\Lambda^{3}-\Lambda^{2}(A+B)+\Lambda(4+AB)-2B$. To find its roots,
we first shift the variable, using $z=\Lambda-(A+B)/3$ and get a depressed
cubic, i.e. a cubic equation without quadratic term

$f(z)=z^{3}+3Cz+D=0,$ (23)

with the constants $C=-\big{(}\frac{A+B}{3}\big{)}^{2}+\frac{4+AB}{3}$ and
$D=\frac{A+B}{3}(4+AB)-2B-2\big{(}\frac{A+B}{3}\big{)}^{3}$. We then use
Cardan’s trick Nickalls (1993), also known as Vieta’s substitution, that is,
set $z=U+\frac{a}{U}$, choosing $a$ to remove the terms in $U$ and $1/U$,
i.e., $a=-C$. This leads to a quadratic equation in $U^{3}$,

$(U^{3})^{2}+DU^{3}-C^{3}=0,$ (24)

with solutions
$U_{m,\pm}=e^{mi\frac{2\pi}{3}}\big{[}-\frac{D}{2}\pm\sqrt{\big{(}\frac{D}{2}\big{)}^{2}+C^{3}}\big{]}^{1/3}$,
with $m=(0,\pm 1)$. The Liouvillian eigenvalues follow as

$\Lambda_{m,\pm}=J\left(U_{m,\pm}-\frac{C}{U_{m,\pm}}+\frac{A+B}{3}\right).$
(25)

Note that we seem to have obtained 6 solutions from a cubic equation. However,
three pairs of solutions are the same. To show this, let
$V=-\frac{D}{2}\pm\sqrt{\frac{D^{2}}{4}+C^{3}}\equiv U_{m,\pm}^{3}$ denote any
nonzero root of the quadratic equation (24). If $V$ is a root then
$\frac{-C^{3}}{V}$ is also a root; this implies that if we change
$+\rightarrow-$ in $\Lambda_{m,\pm}$ we exchange the terms $U_{m,\pm}$ with
the term $-\frac{C}{U_{m,\pm}}$. So it is enough to consider the eigenvalues
$\Lambda_{m,+}$.

We denote the eigenvalues
$\\{\lambda_{0}\equiv\Lambda_{0,+},\lambda_{1}\equiv\Lambda_{1,+},\lambda_{2}\equiv\Lambda_{-1,+},\lambda_{3}=AJ\\}$
that diagonalize the Liouvillian (21),

$\tilde{\mathcal{L}}=\sum_{\nu=0}^{3}\lambda_{\nu}|B_{\nu})(B_{\nu}|.$ (26)

In the domain of interest, with $\gamma$ and $\Gamma$ positive, $\lambda_{0}$
is the eigenvalue with the maximum real part. This feature is verified for a
wide range of parameters in Fig. 4, where the maximum difference of the real
part of eigenvalues $\lambda_{\nu}$ for $\nu\geq 1$ and $\lambda_{0}$ is
always shown to be negative. The full spectrum can be checked for many
different parameters in Fig. 5. This plot allows us to see the large
dissipative gap and no oscillatory behavior in the NI phase (a3, a4, b4), as
well as the non-zero imaginary components and a small gap in the
$\mathcal{PT}$u phase (c1-3), and the intermediate gap and no oscillations in
the $\mathcal{PT}$b phase (a1), as well as the transitions between them

Figure 4: Maximum difference of the real part of $\lambda_{\nu},\nu\geq 1$
with $\lambda_{0}$. The difference is always negative for the displayed range
of parameters, therefore $\lambda_{0}$ is always the eigenvalue with the
largest real part. The transitions between the $\mathcal{PT}$u,
$\mathcal{PT}$b, and NI phases are shown in the dashed and solid line as in
Fig. 2 of the main text.  Figure 5: Spectrum of the SDQ Liouvillian for
different locations in the phase diagram of the system. The parameters span
the different phases: (a1) $\mathcal{PT}$ broken, (c1-3) $\mathcal{PT}$
unbroken, and (a3, a4, b4) Noise Induced; as well as the transitions between
them: (b1,2) $\mathcal{PT}$ breaking transition, (a2,b3) TD-TI transition and
(c4) transition from mixed state to $\ket{e}$ state. Note that the $x$-axis
has been rescaled by $\Gamma/J$.

The eigenvector
$|B_{0})=(b_{0}^{(0)},b_{1}^{(0)},b_{2}^{(0)},b_{3}^{(0)})^{\intercal}$
associated to the eigenvalue $\lambda_{0}$ is given by the solution of the
system of equations

$\begin{cases}+i(b^{(0)}_{1}-b^{(0)}_{2})=\lambda_{0}b^{(0)}_{0},\\\
+i(b^{(0)}_{0}-b^{(0)}_{3})=(\lambda_{0}-A)b^{(0)}_{1},\\\
-i(b^{(0)}_{0}-b^{(0)}_{3})=(\lambda_{0}-A)b^{(0)}_{2},\\\
-i(b^{(0)}_{1}-b^{(0)}_{2})=(\lambda_{0}-B)b^{(0)}_{3}.\\\ \end{cases}$ (27)

By substitutions, given that $\lambda_{0}$ is the largest eigenvalue, in
particular $\lambda_{0}>A$, we find

$b^{(0)}_{3}=b^{(0)}_{0}\left(1+\lambda_{0}\frac{\lambda_{0}-A}{2}\right)$
(28)

and $b^{(0)}_{1}=-b^{(0)}_{2}=-ib^{(0)}_{0}\lambda_{0}/2$. We choose
$b^{(0)}_{0}$ such as to normalize the eigenvector for real $\lambda_{0}$,
yielding the eigenvector associated to the stable steady state

$|B_{0})=\frac{1}{2+\lambda_{0}\frac{\lambda_{0}-A}{2}}\left(\begin{array}[]{c}1\\\
-\frac{i}{2}\lambda_{0}\\\ \frac{i}{2}\lambda_{0}\\\
1+\lambda_{0}\frac{\lambda_{0}-A}{2}\end{array}\right).$ (29)

##  Appendix C Bloch sphere dynamics for the Stochastic Dissipative Qubit

Any density matrix of a qubit $\hat{\rho}$ is completely characterized by its
Bloch coordinates $\mathbf{r}=(x,y,z)$ from the decomposition

$\hat{\rho}=\frac{1}{2}(\hat{\mathbb{1}}+\mathbf{r}\cdot\hat{\boldsymbol{\sigma}}),$
(30)

where $\hat{\boldsymbol{\sigma}}$ is a vector containing the Pauli matrices
obeying the standard commutation
$[\hat{\sigma}_{n},\hat{\sigma}_{m}]=2i\epsilon_{nml}\hat{\sigma}_{l}$, and
anticommutation
$\\{\hat{\sigma}_{n},\hat{\sigma}_{m}\\}=2\delta_{nm}\hat{\mathbb{1}}$
relations.

For the SDQ, the master equation (4) in the main text dictates the evolution
of these coordinates according to the coupled differential equations

$\begin{cases}\dot{x}=-(\gamma\Gamma^{2}+z\Gamma(1-2\gamma\Gamma))x,\\\
\dot{y}=-2Jz-(\gamma\Gamma^{2}+z\Gamma(1-2\gamma\Gamma))y,\\\
\dot{z}=2Jy-\Gamma(1-2\gamma\Gamma)(z^{2}-1).\end{cases}$ (31)

The streamlines of this vector field, as well as the analytical steady state
$|B_{0})$ (29) are compared in Fig. 6. We see perfect agreement between the
analytical steady state (red diamond) and the evolution of the streamlines
showing convergence to it.

The behavior discussed in the main text is also apparent from these
coordinates: (i) the $\mathcal{PT}$b phase has a steady state in the north
pole, very close to the $\ket{f}$ state [cf. Fig. 6(a1)]; (ii) the
$\mathcal{PT}$u phase [cf. Fig. 6(c1-3)] has a steady state close to the
maximally mixed state in the center of the sphere, with a small $y$ component,
as observed in Fig. 2(d) of the main text and (iii) the NI phase [cf. Fig.
6(a3-4, b4)] has a steady state very close to the $\ket{e}$ state in the south
pole of the sphere.

The transition between the different phases also exhibits an interesting
behavior: In the $\mathcal{PT}$u to $\mathcal{PT}$b phase transition (cf. Fig.
6 (b1-2)), we see a convergence to a state close to
$\ket{-y}=\frac{\ket{f}-i\ket{e}}{\sqrt{2}}$, interestingly, the color scale
shows that the speed of this convergence is larger the dissipative gap or the
oscillatory timescale; the transition from TD to TI dynamics is also
interesting (cf. Fig. 6 (a2, b3)), in this transition the dynamics is CPTP,
and the steady state is the maximally mixed state in the center of the Bloch
ball, to which the convergence shows almost no oscillations in (a2) and an
oscillatory behavior in (b3), see Fig. 5 to understand this. Lastly, the
transition from $\mathcal{PT}$u to the NI phases (cf. Fig. 6 (c4)) shows an
oscillatory convergence to the steady state, which has a positive component of
$y$, as already known from Fig. 2 (d).

Figure 6: Streamlines of the vector fields in the cross-section of the Bloch
sphere with $x=0$. The parameters span the different phases: (a1)
$\mathcal{PT}$ broken, (c1-3)$\mathcal{PT}$ unbroken, and (a3, a4, b4) Noise
Induced; as well as the transitions between them: (b1,2) $\mathcal{PT}$
breaking transition, (a2,b3) TD-TI transition and (c4) transition from mixed
state to $\ket{e}$ state. The analytical steady state (29) is shown as the red
diamond. The color and linewidth represent the Euclidean speed
$v_{E}=\sqrt{(\dot{y})^{2}+(\dot{z})^{2}}$, divided by the maximum of the two
main frequency units: the dissipative gap $\Delta$ and the maximum imaginary
part of the eigenvalues $\omega=\max_{\nu}(\mathrm{Im}(\lambda_{\nu}))$.

##  Appendix D Comparison of different numerical approaches

We now compare two different numerical resolutions of the SDQ dynamics. The
first is to numerically integrate the nonlinear system of differential
equations (31)—we do so by a standard 4th-order Runge-Kutta (RK4) method. The
second approach is to get the time evolution of the density matrix from the
formal solution of the master equation, i.e.
$|\tilde{\rho}_{t})=e^{\tilde{\mathcal{L}}t}|\rho_{0})$, where the equation is
vectorized to compute the map $e^{\tilde{\mathcal{L}}t}[\bullet]$ simply
through matrix exponentiation. Additionally, this method requires
normalization of the state
$|\rho_{t})=|\tilde{\rho}_{t})/\mathrm{Tr}(\tilde{\rho}_{t})$, where
$\mathrm{Tr}(\tilde{\rho}_{t})=(\mathbb{1}|\tilde{\rho}_{t})=\tilde{\rho}_{ee}+\tilde{\rho}_{ff}$
and $\tilde{\rho}_{ee}=\bra{e}\tilde{\rho}_{t}\ket{e}$. To avoid computing a
matrix exponential for each time $t$, we Trotterize the evolution as
$e^{\tilde{\mathcal{L}}t}=\prod_{n=1}^{N_{\textsc{t}}}e^{\tilde{\mathcal{L}}\Delta_{\textsc{t}}}$,
subdividing the evolution in $N_{\textsc{t}}$ steps of length
$\Delta_{\textsc{t}}=t_{\rm end}/N_{\textsc{t}}$.

These two approaches are compared in Fig. 7 to compute the purity, showing a
perfect agreement. Additionally, the first-order evolution of the purity
obtained from $\partial_{t}P$ in (9) also agrees with the simulated evolution.

Figure 7: Purity computed from 4th-order Runge Kutta (black line) and using
the vectorized Liouvillian (red dotted line). The first-order Taylor
approximation, determined from the $\partial_{t}P$ formula (blue dashed line),
matches the two approaches. The parameters for the simulation are
$\gamma=0.5,\;\Gamma=0.5$ and the initial state has Bloch components
$\mathbf{r}=(0.2,\;0.8,\;0.4)$. The absolute value of the difference between
the two solutions is of order $10^{-11}$ with a timestep
$J\Delta_{\textsc{t}}=0.004$.

##  Appendix E Uhlmann fidelity for a qubit

The fidelity between two pure states $\ket{\psi}$ and $\ket{\phi}$ measures
how distinguishable the two states are, and is given by
$F(\psi,\phi)=|\braket{\psi}{\phi}|^{2}=\mathrm{Tr}(\ket{\psi}\braket{\psi}{\phi}\bra{\phi})$
Nielsen and Chuang (2010). The generalization to any two mixed states
$\hat{\rho},\;\hat{\sigma}$ requires the introduction of the Uhlmann fidelity
Uhlmann (1976); Bures (1969) as

$F(\hat{\rho},\hat{\sigma})=\Big{(}\mathrm{Tr}\sqrt{\sqrt{\hat{\rho}}\hat{\sigma}\sqrt{\hat{\rho}}}\Big{)}^{2}.$
(32)

This expression is cumbersome to compute due to the matrix square roots, which
in particular require dealing with a non-vectorized density matrix. For a two-
dimensional Hilbert space, there is a simpler expression for the Uhlmann
fidelity given by Jozsa (1994); Hübner (1992)

$F(\hat{\rho},\hat{\sigma})=\mathrm{Tr}(\hat{\rho}\hat{\sigma})+2\sqrt{\det{\hat{\rho}}\det{\hat{\sigma}}}.$
(33)

This expression was used in Fig. 3 to compute the distinguishability between
the instantaneous state and the stable steady state.

##  Appendix F Standard form of the master equation

We here look for the general structure of master equations describing open
systems with balanced gain and loss, specifically, for the generator of the
dynamics of an unnormalized density matrix $\tilde{\rho}$

$\displaystyle\frac{d}{dt}\tilde{\rho}=\tilde{\mathcal{L}}[\tilde{\rho}],$
(34)

where $\tilde{\mathcal{L}}$ need not be trace-preserving. Balanced gain and
loss is satisfied provided that the normalized density matrix $\hat{\rho}$
evolves according to

$\displaystyle\frac{d}{dt}\hat{\rho}=\tilde{\mathcal{L}}[\hat{\rho}]-{\rm
Tr}\big{(}\tilde{\mathcal{L}}[\hat{\rho}]\big{)}\hat{\rho},$ (35)

making the equation of motion manifestly nonlinear. To determine the structure
of $\tilde{\mathcal{L}}$, one can introduce an orthonormal basis of
$N$-dimensional operators $\hat{F}_{i}=1,\dots,N^{2}$ such that ${\rm
Tr}(\hat{F}_{i}^{\dagger}\hat{F}_{j})=\delta_{ij}$, with $N$ the dimension of
the Hilbert space. It is convenient to choose
$\hat{F}_{N^{2}}=\hat{\mathbb{1}}/\sqrt{N}$ so that the $\hat{F}_{i}$ are
traceless for $i=1,\dots,N^{2}-1$. The Liouvillian $\tilde{\mathcal{L}}$ can
be determined analogously to the procedure used to establish the structure of
Markovian semigroups and the Lindblad master equation Gorini  _et al._ (1976);
Breuer and Petruccione (2007), in our case leading to

$\displaystyle\tilde{\mathcal{L}}[\tilde{\rho}]=-i[\hat{H},\tilde{\rho}]+\\{\hat{G},\tilde{\rho}\\}+\sum_{ij=1}^{N^{2}}a_{ij}\hat{F}_{i}\tilde{\rho}\hat{F}_{j}^{\dagger},$
(36)

where the Hermitian operator $\hat{H}=i(\hat{F}-\hat{F}^{\dagger})/2$ and the
operator

$\displaystyle\hat{G}=\frac{1}{2N}a_{N^{2}N^{2}}\hat{\mathbb{1}}+\frac{1}{2}(\hat{F}^{\dagger}+\hat{F}),$
(37)

are defined in terms of

$\displaystyle\hat{F}=\frac{1}{\sqrt{N}}\sum_{i=1}^{N^{2}}a_{iN^{2}}\hat{F}_{i},$
(38)

for some positive expansion coefficients $a_{ij}$ ($i,j=1,\dots,N^{2}$). The
standard Lindblad equation follows from imposing trace preservation in (36)
Gorini  _et al._ (1976); Breuer and Petruccione (2007), which leads to

$\displaystyle\hat{G}=-\frac{1}{2}\sum_{ij=1}^{N^{2}}a_{ij}\hat{F}_{j}^{\dagger}\hat{F}_{i}.$
(39)

However, by relaxing this condition, the general structure of
$\tilde{\mathcal{L}}[\hat{\rho}]$ is considered. The rate of norm change in
this case is

$\displaystyle\frac{d}{dt}{\rm Tr}(\tilde{\rho})=2{\rm
Tr}(\hat{G}\tilde{\rho})+\sum_{ij=1}^{N^{2}}a_{ij}{\rm
Tr}(\hat{F}_{j}^{\dagger}\hat{F}_{i}\tilde{\rho}).$ (40)

It follows that the general structure of the nonlinear master equation for
open systems with balanced gain and loss reads

$\displaystyle\frac{d}{dt}\hat{\rho}$ $\displaystyle=$
$\displaystyle-i[\hat{H},\hat{\rho}]+\\{\hat{G},\hat{\rho}\\}+\sum_{ij=1}^{N^{2}}a_{ij}\hat{F}_{i}\hat{\rho}\hat{F}_{j}^{\dagger}$
$\displaystyle-\Big{(}2{\rm
Tr}(\hat{G}\hat{\rho})+\sum_{ij=1}^{N^{2}}a_{ij}{\rm
Tr}(\hat{F}_{j}^{\dagger}\hat{F}_{i}\hat{\rho})\Big{)}\hat{\rho}.$

This master equation can be brought to a diagonal form by considering the
unitary transformation
$[\hat{u}\hat{a}\hat{u}^{\dagger}]_{k\ell}=\gamma_{k}\delta_{k\ell}$ and
writing $\hat{F}_{i}=\sum_{k=1}^{N^{2}}u_{ki}\hat{A}_{k}$ in terms of the new
set of operators $\\{\hat{A}_{k}\\}$, to find

$\displaystyle\frac{d}{dt}\hat{\rho}$ $\displaystyle=$
$\displaystyle-i[\hat{H},\hat{\rho}]+\\{\hat{G},\hat{\rho}\\}+\sum_{ij=1}^{N^{2}}\gamma_{k}\hat{A}_{k}\hat{\rho}\hat{A}_{k}^{\dagger}$
$\displaystyle-\Big{(}2{\rm
Tr}[\hat{G}\hat{\rho}]+\sum_{k=1}^{N^{2}}\gamma_{k}{\rm
Tr}[\hat{A}_{k}^{\dagger}\hat{A}_{k}\hat{\rho}]\Big{)}\hat{\rho},$

with

$\displaystyle\hat{G}=\frac{1}{2N}a_{N^{2}N^{2}}\hat{\mathbb{1}}+\frac{1}{2\sqrt{N}}\gamma_{N^{2}}(\hat{A}_{N^{2}}+\hat{A}_{N^{2}}^{\dagger}).$
(43)

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