# Topology and its detection in a dissipative Aharonov-Bohm chain

Haowei Li  CAS Key Laboratory of Quantum Information, University of Science
and Technology of China, Hefei 230026, China  Wei Yi<EMAIL_ADDRESS>CAS Key
Laboratory of Quantum Information, University of Science and Technology of
China, Hefei 230026, China  CAS Center For Excellence in Quantum Information
and Quantum Physics, Hefei 230026, China

###### Abstract

In a recent experiment, a dissipative Aharaonov-Bohm (AB) chain was
implemented in the momentum space of a Bose-Einstein condensate. Formed by a
series of dissipative AB rings threaded by synthetic magnetic flux, the chain
exhibits the non-Hermitian skin effect, necessitating the non-Bloch band
theory to account for its topology. In this work, we systematically
characterize topological features of the dissipative AB chain, particularly
beyond the experimentally realized parameter regime. Further, we show that an
atom-injection spectroscopy is not only capable of revealing topological edge
states, as has been demonstrated in the experiment, but also the general band
structure of the system. We then discuss alternative dynamic detection schemes
for the topological edge states. Given the generality of the model and the
detection schemes, our work is helpful to future study of topological models
with non-Hermitian skin effects across a variety of quantum simulators.

##  I Introduction

The state-of-the-art quantum control in systems such as photonics [1, 2, 3],
cold atoms [4, 5, 6, 7], or trapped ions [8, 9, 10] offer unprecedented access
to the rich dynamics and exotic phenomena in open quantum systems that undergo
particle or energy exchange with their environment. A non-Hermitian
description applies therein, for instance, by imposing post selection [11, 12,
13, 14, 15], or by mapping the density-matrix dynamics to an enlarged Hilbert
space [16, 17, 18, 19]. The resulting non-Hermitian physics provides an
unconventional perspective of open systems, and has attracted extensive
interest in recent years. Dictated by a non-Hermitian effective Hamiltonian,
exotic spectral or dynamic properties, such as the parity-time symmetry [20,
21], enhanced sensing [22, 23, 24] and topological transfer [3, 7], non-
Hermitian topology [25, 26, 27, 28, 29, 30] and so on, have been
systematically studied and experimentally confirmed in a wide range of
physical systems.

The recent discovery of the non-Hermitian skin effect has stimulated further
research activities [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42]. Under
the non-Hermitian skin effect, eigenstates of a system become exponentially
localized at boundaries, leading to dramatic changes in the system’s band and
spectral topology [36, 37], dynamics [16, 18, 19, 40, 41, 42], and spectral
symmetry [43, 44, 45]. Experimentally, the non-Hermitian skin effect and its
consequences have been observed in classical and photonic systems [46, 47, 48,
49], as well as in a Bose-Einstein condensate of ultracold atoms [50]. In the
last case, a dissipative Aharonov-Bohm (AB) chain was implemented in the
momentum and hyperfine-spin space of the condensate atoms. As illustrated in
Fig. 1, the AB chain consists of a series of triangular AB rings [6], each
threaded by a synthetic magnetic flux, realized by engineering the phases of
the nearest-neighbor hopping rates. Dissipation is introduced through on-site
particle loss for each ring, such that dynamics of atoms that remain in the
chain is driven by a non-Hermitian Hamiltonian that features non-trivial band
topology. Importantly, the interplay of synthetic flux and dissipation gives
rise to a non-reciprocal flow in the bulk that lies at the origin of the non-
Hermitian skin effect. In the experiment, the non-Hermitian skin effect was
observed through a directional propagation of atoms along the chain, while the
topological edge states were probed through an inverse spectroscopy, where
atoms are injected into an empty dissipative AB chain from a bystander state.
Despite its experimental implementation, a systematic study of the topological
properties of the dissipative AB chain is missing in the literature. Further,
given the intrinsic difficulty of detecting topological edge states in the
presence of non-Hermitian skin effects (as both are localized at the
boundary), more variety of detection schemes is desirable.

Figure 1: Schematic illustration of a dissipative Aharonov-Bohm chain. Each
unit cell consists of three sublattice sites $a$, $b$, and $c$, forming a
triangular loop threaded by a flux $\phi$. The green, blue and red bounds
denote hopping between adjacent sites. See main text for definition of
variables.

In this work, we carry out a systematic study of the dissipative AB chain,
focusing on its topology and detection. We identify an additional topological
phase transition beyond the experimentally demonstrated parameter regime, and
derive analytical expressions for the topological transition points using the
non-Bloch band theory. Invoking the theoretical framework of Feshbach
projection [11, 52], we derive the transfer rates of the experimentally
implemented atom-injection spectroscopy, which are in good agreement with
numerical simulations. In the experiment, atoms were injected to an open edge
of the AB chain, to detect the topological edge states. Here we show that by
injecting atoms into a bulk site far away from the boundary, spectral
information under the periodic boundary condition can be obtained from the
transfer rate. We then propose a dynamic detection scheme for the topological
edge states.

The paper is organized as follows. In Sec. II, we review the model Hamiltonian
for the dissipative AB chain, and show that it has the non-Hermitian skin
effect. In Sec. III, we characterize its topological properties using the non-
Bloch band theory. We then provide a theoretical characterization of the
injection spectroscopy in Sec. IV. In Sec. V, we discuss the dynamic detection
of topological edge states. We summarize in Sec. VI.

##  II Model

The non-Hermitian Hamiltonian of the dissipative AB chain illustrated in Fig.
1 is given by [6, 50]

$\displaystyle H=$
$\displaystyle\sum_{n=1}^{N}[J_{p}b^{\dagger}_{n}a_{n}+J_{s}c^{\dagger}_{n}b_{n}+J_{s}e^{i\phi}a^{\dagger}_{n}c_{n}+{H.c.}]$
(1)
$\displaystyle+\sum_{n=1}^{N-1}[J_{t}a^{\dagger}_{n+1}b_{n}+{H.c.}]+\sum_{n=1}^{N}(\Delta-i\gamma)c_{n}^{\dagger}c_{n}.$

Here $a_{n}(a_{n}^{\dagger})$, $b_{n}(b_{n}^{\dagger})$ and
$c_{n}(c_{n}^{\dagger})$ are respectively the annihilation (creation)
operators for the $a$, $b$ and $c$ sublattice sites of the $n$th unit cell;
$J_{p}$, $J_{s}$ and $J_{t}$ are the nearest-neighbour hopping rates; $\Delta$
and $\gamma$ are respectively the on-site potential and the loss rate on site
$c$; the phase $\phi\in[0,2\pi)$ corresponds to a synthetic magnetic flux.

Figure 2: Eigenspectra of a dissipative AB chain under the open boundary
condition. We take $N=100$ unit cells, $J_{s}/J_{p}=2$, $\phi=\pi/2$,
$\Delta/J_{p}=-2$, and $\gamma/J_{p}=1$ for numerical calculations. (a)(b) The
real ($\text{Re}(E)$) and imaginary ($\text{Im}(E)$) components of
eigenenergies as functions of $J_{t}$. The red and blue lines in (a)(b) denote
topological edge states, each two-fold degenerate. The two topological
transition points are $J_{t,c1}/J_{p}=1.56$ (associated with edge states in
red) and $J_{t,c2}/J_{p}=3.41$ (associated with those in blue), respectively.

Hamiltonian (1) hosts topological edge states under the open boundary
condition. For finite $\gamma$ and $\phi\notin\\{0,\pi\\}$, all eigenstates
accumulate to the boundaries under the non-Hermitian skin effect. As
demonstrated in Ref. [6], this can be understood in the limit
$\Delta,\gamma\gg J_{s},J_{p},J_{t}$, when Hamiltonian (1) can be
perturbatively reduced to a non-Hermitian Su-Schrieffer-Heeger model with
asymmetric hopping. Physically, this is because the interplay of the synthetic
flux and on-site loss gives rise to a non-reciprocal flow along the chain.
Beyond such a limit, the dissipative AB chain is qualitatively different from
a non-Hermitian Su-Schrieffer-Heeger model, particularly for the lack of
chiral symmetry. Nevertheless, both the non-Hermitian skin effect and band
topology persist as salient features of the dissipative AB chain.

In Fig. 2, we show typical eigenspectra of the model under the open boundary
condition. Two gap-closing points can be identified, particularly visible in
$\text{Re}(E)$, where topological edge states emerge. The topological
transitions are robust under variations of $\phi$ and $\gamma$, their
locations however, sensitively depend on these parameters. In the following,
we denote the location of the topological phase transitions as $J_{t,c1}$ and
$J_{t,c2}$, which are associated with the edge states in red and blue,
respectively, in Fig. 2. We further denote the corresponding eigenenergies of
the topological edge states as $E_{c1}$ and $E_{c2}$, respectively. Note that
the transition at $J_{t,c1}$ was experimentally probed in Ref. [50], but not
the one at $J_{t,c2}$.

In Fig. 3, we show the spatial probability distribution of eigen wavefunctions
under different boundary conditions. We choose the parameter $J_{t}/J_{p}=5$,
such that two pairs of topological edge states (indicated by red and blue)
exist under the open boundary condition. For finite $\gamma$ and
$\phi\notin\\{0,\pi\\}$, all eigenstates are localized toward the boundaries,
indicating the presence of non-Hermitian skin effect. It follows that
topological edge states in Fig. 2 can only be accounted for by a non-Bloch
topological invariants under the non-Bloch band theory.

Figure 3: Spacial distribution of the eigenstates for an AB chain with
$N=100$. We take $J_{t}/J_{p}=5$, while other parameters are the same as those
in Fig. 2. Gray: periodic boundary condition. Black: bulk eigenstates under
the open boundary condition. Red and blue: degenerate topological edge states
with eigenenergies $E_{c1}$ and $E_{c2}$, respectively.

##  III Topology under the non-Bloch band theory

Topological edge states of the dissipative AB chain are characterized by the
non-Bloch band theory [31, 33]. The idea is to take into account the
deformation of the bulk eigenstates under the non-Hermitian skin effect,
replacing the phase factor $e^{ik}$ of the Bloch waves (under the periodic
boundary condition) with a spatial mode factor $\beta(k)=|\beta(k)|e^{ik}$.
Here the quasimomentum $k\in[0,2\pi)$, and the trajectory of $\beta(k)$ on the
complex plane is known as the generalized Brillouin zone (GBZ), which can be
calculated from the Schrödinger’s equation as shown below.

In the spirit of the non-Bloch band theory, we write the non-Bloch Hamiltonian
as

$H(\beta)=\left(\begin{array}[]{ccc}0&J_{p}+J_{t}\beta^{-1}&J_{s}e^{i\phi}\\\
J_{p}+J_{t}\beta&0&J_{s}\\\
J_{s}e^{-i\phi}&J_{s}&\Delta-i\gamma\end{array}\right).$ (2)

The Schrödinger’s equation in the GBZ is then
$[H(\beta)-E]|\varphi^{R}_{j}(\beta)\rangle=0$, where $E$ is the eigenenergy,
$|\varphi^{R}_{j}(\beta)\rangle$ is the right eigenstate, and $j$ is the band
index. Sending the determinant of the eigen equation to zero, we have

$\displaystyle\left[J_{t}\beta+J_{p}+\frac{e^{-i\phi}J_{s}^{2}}{E-(\Delta-i\gamma)}\right]\left[J_{t}\beta^{-1}+J_{p}+\frac{e^{i\phi}J_{s}^{2}}{E-(\Delta-i\gamma)}\right]$
(3) $\displaystyle-\left[E-\frac{J_{s}^{2}}{E-(\Delta-i\gamma)}\right]^{2}=0.$

The spatial mode functions $\beta(k)$ can be solved by requiring the two roots
of Eq. (3) to have the same magnitude, with $|\beta_{1}|=|\beta_{2}|$. We then
have [50]

$|\beta(k)|=\sqrt{\left|\frac{J_{p}+\frac{e^{-i\phi}J_{s}^{2}}{E-(\Delta-i\gamma)}}{J_{p}+\frac{e^{i\phi}J_{s}^{2}}{E-(\Delta-i\gamma)}}\right|}.$
(4)

It is then straightforward to solve for $E$ and $|\beta(k)|$ from Eqs. (3) and
(4) for each $k$. The resulting eigenenergy $E$ gives the eigenspectrum under
an open boundary condition.

In Fig. 4(a)(b)(c), we show the eigenspectra for different parameters, under
both the periodic (dots) and the open boundary conditions (solid curves).
Under the periodic boundary condition, the eigenenergies of each band form a
closed spectra loop, consistent with the well-known spectral topology of the
non-Hermitian skin effect. By contrast, under the open boundary condition, the
eigenenergies collapse to open arcs within the closed loops. In Fig. 4(b)(c),
the discrete red and blue dots outside the spectra loops correspond to the
topological edge states in Fig. 2.

Figure 4: (a)(b)(c) Eigenspectra of Hamiltonian Eq. (1) with $N=100$ unit
cells on the complex plane. Black: eigenspectra under PBC. Orange, green, and
purple: eigenspectra for three different bands under the open boundary
condition. Red and blue triangle denote the topological edge states with
eigenenergies $E_{c1}$ and $E_{c2}$, respectively. (d)(e)(f) GBZs on the
complex plane. Orange, green, and purple: GBZs for the three different bands
in (a)(b)(c). Black dashed line is the unit circle, which corresponds to the
conventional Brillouin zone. For (a)(d), $J_{t}/J_{p}=0.5$; for (b)(e),
$J_{t}/J_{p}=2.5$; for (c)(f), $J_{t}/J_{p}=5$. Other parameters are the same
as those in Fig. 2. Figure 5: Bloch (gray) and non-Bloch (black) winding
numbers. The vertical dashed lines in red and blue denote $J_{t,c1}$ and
$J_{t,c2}$, respectively. All parameters are the same as those in Fig. 2.

In Fig. 4(d)(e)(f), we plot the GBZs of the three bands under the parameters
of Fig. 4(a)(b)(c), respectively. For all cases, the calculated
$|\beta(k)|<1$, and the GBZs are within the unit circle. This indicates that
under the open boundary condition, all eigenstates accumulate to the left
boundary (toward small unit-cell index $n$).

Figure 6: Transfer rate for the inverse spectroscopy, for $|f\rangle$ located
at an open boundary. (a)(b)(c) The probe Hamiltonian couples the bystander
state to the $b$ sublattice site of the $N$th unit cell (the right-most unit
cell on the edge), with (a) $J_{t}/J_{p}=0.5$, (b) $J_{t}/J_{p}=2.5$, and (c)
$\delta_{pb}/J_{p}=\mathrm{Re}(E_{c1})$. Here $E_{c1}$ is the energy of the
topological edge state. The dashed vertical lines in (a)(b) correspond to
$\mathrm{Re}(E_{c1})$, and the dashed line in (c) corresponds to $J_{t,c1}$.
(d)(e)(f) The probe Hamiltonian couples the bystander state to the $c$
sublattice site of the $N$th unit cell (the right-most unit cell on the edge),
with (d) $J_{t}/J_{p}=2.5$, (e) $J_{t}/J_{p}=5$, and (f)
$\delta_{pb}/J_{p}=\mathrm{Re}(E_{c2})$. The dashed vertical lines in (d)(e)
correspond to $\mathrm{Re}(E_{c2})$, and the dashed line in (f) corresponds to
$J_{t,c2}$. For all subplots, the black solid lines and the magenta dashed
lines are respectively the theoretically predicted transfer rate using Eq.
(14), and the numerically simulated transfer rates from Eq. (16). For all
figures, $J_{pb}/J_{p}=0.01$, $\tau J_{p}=40\pi$. Other parameters are the
same as those in Fig. 2. Figure 7: Transfer rate for the inverse spectroscopy,
as the probe Hamiltonian is coupled to the $b$ sublattice site of the $51$st
unit cell, which is deep in the bulk. (a)(b) Hermitian case with $\gamma=0$.
(c)(d) Non-Hermitian case with $\gamma/J_{p}=1$. We also take
$J_{t}/J_{p}=0.5$ in (a)(c), and $J_{t}/J_{p}=2.5$ in (b)(d), while other
parameters are the same as those in Fig. 6. For all subplots, $\tau
J_{p}=40\pi$, the black solid lines and the magenta dashed lines are
respectively the theoretically predicted transfer rate using Eq. (14), and the
numerically simulated transfer rates from Eq. (16). The shaded regions in gray
indicate the real components of the spectra under the periodic boundary
condition.  Figure 8: Boundary dynamics for the detection of topological edge
states. (a)(b) Color contour for the normalized occupation distribution as a
function of time. (b)(d) The normalized occupation distribution at the time
$\tau J_{p}=6$. We take $J_{t}/J_{p}=1$ in (a)(c), and $J_{t}/J_{p}=5$ in
(b)(d). Other parameters are the same as those in Fig. 2.

We are now in a position to calculate the non-Bloch winding number, which can
restore the bulk-boundary correspondence and predict the existence and number
of topological edge states. Unlike the non-Hermitian Su-Schrieffer-Heeger
model, the dissipative AB chain features three bands. The non-Bloch winding
number $\nu$ is defined through the global Berry phase, which is the sum of
the Berry phases of all three bands, with

$\nu=\frac{1}{2\pi}\sum_{j}\Theta_{j}.$ (5)

Here the Berry phase of the $j$th band is given by

$\Theta_{j}=i\oint_{\mathrm{GBZ}_{j}}\mathrm{d}\beta\left\langle\varphi^{L}_{j}(\beta)\left|\partial_{\beta}\right|\varphi^{R}_{j}(\beta)\right\rangle,$
(6)

where the right and left eigenstates of $H(\beta)$ are defined as
$H(\beta)|\varphi_{j}^{R}(\beta)\rangle=E_{j}|\varphi_{j}^{R}(\beta)\rangle$
and
$H^{\dagger}(\beta)|\varphi_{j}^{L}(\beta)\rangle=E_{j}^{*}|\varphi_{j}^{L}(\beta)\rangle$,
respectively. The integration in Eq. (6) is over the GBZ of the $j$th band.
When $\beta$ is replaced by $e^{ik}$ in Eq. (6), the non-Bloch winding number
is reduced to the Bloch winding number which characterizes the band topology
of the system under the periodic boundary condition.

In Fig. 5, we show the Bloch (gray) and non-Bloch (black) winding numbers. The
non-Bloch winding number is quantized to integers, and changes its value at
topological transitions that are consistent with the gap-closing points in
Fig. 2. By contrast, the Bloch winding number can take half-integer values,
and it does not indicate the topological transitions of the system under the
open boundary condition. Note that half-integer winding numbers have
previously been reported in the non-Hermitian, asymmetric Su-Schrieffer-Heeger
model [38, 51], where explicit geometric interpretations can be found based on
its chiral symmetry. While chiral symmetry is absent in our model, the origin
of these half-integer winding numbers, and their general relation to the non-
Hermitian skin effect, are interesting open questions.

The topological transition points can be analytically determined from the gap-
closing condition. At the gap-closing pint, GBZs of two different bands
intersect on the complex plane at the same eigenenergies. It follows that Eq.
(3) features a double root at the topological transition. This is satisfied
for

$\displaystyle J_{t,cj}=$
$\displaystyle\sqrt{\left|(J_{p}+\frac{e^{i\phi}J_{s}^{2}}{E_{cj}-(\Delta-i\gamma)})\cdot(J_{p}+\frac{e^{-i\phi}J_{s}^{2}}{E_{cj}-(\Delta-i\gamma)})\right|},$
(7)

with $j=1,2$. And the roots are given by

$\displaystyle
E_{c1}=\frac{(\Delta-i\gamma)+\sqrt{(\Delta-i\gamma)^{2}+4J_{s}^{2}}}{2},$ (8)
$\displaystyle
E_{c2}=\frac{(\Delta-i\gamma)-\sqrt{(\Delta-i\gamma)^{2}+4J_{s}^{2}}}{2},$ (9)

which correspond to the energies of the topological edge states emerging at
the two transition points in Fig. 1. Both $J_{t,cj}$ and $E_{cj}$ calculated
from Eqs. (7)(8)(9) are in excellent agreement with the numerically calculated
eigenspectra in Fig. 2. Note that we take the positive branch for the square
roots in Eqs. (8)(9).

To close this section, we discuss the symmetry of Hamiltonian (2). While it
does not have chiral symmetry, Hamiltonian (2) is symmetric under the
following transformation: $\Gamma H^{T}(\beta)\Gamma^{-1}=H(\beta)$, where
$\Gamma=\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\
0&0&e^{-i\phi}\end{array}\right)$. In the Hermitian limit with $\gamma=0$, the
symmetry is reduced to $\Gamma H^{T}(k)\Gamma^{-1}=H(k)$, where $k$ is then
the quasi-momentum in the conventional Brillouin zone. We have checked that
such a symmetry protects the two-fold degeneracy of the topological edge
states emerging from either phase transitions. Note that, while the Berry
phases $\Theta_{j}$ are quantized to multiples of $\pi$ in the presence of
such a symmetry, they are no longer so when the symmetry is broken. By
contrast, the non-Bloch winding number is always quantized, since the global
Berry phase, when integrated over the GBZ, is always integer multiples of
$2\pi$.

In the real lattice space, the symmetry operation can be further decomposed
into $\Gamma=PC_{+}$, where $P:a_{n}\rightarrow a_{N-n},\,\,b_{n}\rightarrow
b_{N-n},\,\,c_{n}\rightarrow c_{N-n}$; $C_{+}:a_{n}\rightarrow
b_{N-n},\,\,b_{n}\rightarrow a_{N-n},\,\,c_{n}\rightarrow e^{-i\phi}c_{N-n}$.

We identify $P$ and $C_{+}$ as the inversion and the non-Hermitian variant of
the time-reversal operators, respectively. In particular, $C_{+}$ can be
identified with the $\text{TRS}^{\dagger}$ symmetry in Ref. [26]. Physically,
the combined inversion and time-reversal symmetry is understood from the
observation that the dissipative AB chain remains invariant by simultaneously
reversing the flux and the lattice, but not either alone.

##  IV Detecting topological edge states and band structure

In the experiment [50], a momentum-resolved Bragg spectroscopy was applied to
detect the topological edge states, where atoms are injected into an edge site
of the AB chain. In this section, we provide a theoretical description for the
atom-injection spectroscopy, and show that a similar detection scheme can be
applied to probe the band structure. We then propose an alternative dynamic
detection scheme for the topological edge states.

###  IV.1 Injection spectroscopy

We consider coupling atoms in a bystander state $|d\rangle$ to a local site
$|f\rangle$ of the dissipative AB chain. Site $|f\rangle$ can be any one of
the sublattice sites $|a\rangle$, $|b\rangle$ or $|c\rangle$. It can be on the
edge, as is the case in the experiment, or in the bulk, far away from any
boundaries. The chain is originally empty, such that the scheme is similar in
spirit to the inverse radio-frequency spectroscopy. The probe Hamiltonian
reads

$H_{pb}=J_{pb}d^{\dagger}f+J_{pb}f^{\dagger}d+\delta_{pb}d^{\dagger}d.$ (10)

Here, $d$ ($d^{\dagger}$), $f$ ($f^{\dagger}$) respectively denote the
annihilation (creation) operators for state $|d\rangle$ and $|f\rangle$.
$J_{pb}$ is the coupling rate between $|d\rangle$ and $|f\rangle$,
$\delta_{pb}$ is the detuning of the coupling frequency with respect to the
transition $|d\rangle\rightarrow|f\rangle$. Here the overall dynamics is
governed by the Hamiltonian $H^{\prime}=H+H_{pb}$.

Following the practice of Feshbach projection [52, 53, 54], we define the
projection operators $P=|d\rangle\langle d|$ and $Q=\mathbf{I}-P$. The
effective Hamiltonian in the subspace of the bystander state $|d\rangle$ is
then

$H_{\mathrm{eff}}(E)=H^{\prime}_{PP}+H^{\prime}_{PQ}\frac{1}{E-H^{\prime}_{QQ}}H^{\prime}_{QP},$
(11)

where

$\displaystyle H^{\prime}_{PP}=\delta_{pb}d^{\dagger}d,$ $\displaystyle\quad
H^{\prime}_{PQ}=J_{pb}d^{\dagger}f,$ (12) $\displaystyle
H^{\prime}_{QP}=J_{pb}f^{\dagger}d,$ $\displaystyle\quad H^{\prime}_{QQ}=H.$

It can be shown straightforwardly that

$H_{\mathrm{eff}}(E)=\left(\delta_{pb}+\sum_{j}\frac{J_{pb}^{2}\psi_{j,f}^{R}\psi_{j,f}^{L*}}{E-E_{j}}\right)d^{\dagger}d,$
(13)

where $\psi_{j,f}^{R}=\langle f|\psi_{j}^{R}\rangle$ and
$\psi_{j,f}^{L*}=\langle\psi_{j}^{L}|f\rangle$, and the right and left
eigenstates of $H$ are defined as
$H|\psi_{j}^{R}\rangle=E_{j}|\psi_{j}^{R}\rangle,H^{\dagger}|\psi_{j}^{L}\rangle=E_{j}^{*}|\psi_{j}^{L}\rangle$.
The effctive Hamiltonian Eq. (13) is dissipative, where the dissipation is due
to the atom transfer from state $|d\rangle$ to the AB chain. We define the
transfer rate

$T(\tau)=1-\exp[-2R(\delta_{pb})\tau],$ (14)

which describes the probability for an atom to be transferred from $|d\rangle$
to the chain within the evolution time $\tau$. Here

$R(\delta_{pb})=-\operatorname{Im}\left(\langle
d|H_{\mathrm{eff}}(E)|d\rangle\right)=-\operatorname{Im}\sum_{j}\frac{J_{pb}^{2}\psi_{j,f}^{R}\psi_{j,f}^{L*}}{\delta_{pb}-E_{j}+i0^{+}}.$
(15)

The term $i0^{+}$ in the denominator ensures that Eq. (15) recovers the
familiar form of the Fermi’s golden rule in the Hermitian case. Note that the
same Eq. (15) can be derived using the linear response theory (see Appendix).

The transfer rate can also be evaluated through numerical simulation of the
system dynamics. Initialized in the state $|d\rangle$ at the initial time
$t=0$, the time evolved state at time $\tau$ is then
$|\psi(\tau)\rangle=e^{-iH^{\prime}\tau}|d\rangle$. Note that the non-
normalized nature of $|\psi(\tau)\rangle$ corresponds to loss of atoms from
the dissipative AB chain. The transfer rate can be expressed as

$T_{evo}(\tau)=1-|\langle d|e^{-iH^{\prime}\tau}|d\rangle|^{2}.$ (16)

Due to the perturbative nature of Eq. (14), we expect that the transfer rates
calculated from Eq. (14) and Eq. (16) are very close to each other, provided
that $J_{pb}/J_{p}\ll 1$ and $J_{pb}^{2}\tau\ll 1\ll J_{p}\tau$. This is
indeed the case as we show below.

For the detection of the topological edge states, we follow the practice of
Ref. [50], and consider $|f\rangle$ to be on the edge of the AB chain. The
calculated transfer rates are plotted in Fig. 6. While results from Eq. (14)
and Eq. (16) agree well with one another, signals of the edge states are
visible near the appropriate detuning $\delta_{pb}$. Specifically, in Fig.
6(a)(b), we aim to detect the edge state with energy $E_{c1}$. The state has a
large support on sublattice site $b$, we therefore set $|f\rangle$ on site $b$
of the right-most unit cell on the edge. When $J_{t}<J_{t,c1}$, $T(\tau)$
exhibits a valley at $\delta_{pb}=\mathrm{Re}(E_{c1})$ [Fig. 6(a)], indicating
the presence of a band gap. When $J_{t}>J_{t,c1}$, a peak emerges at
$\delta_{pb}=\mathrm{Re}(E_{c1})$ [Fig. 6(b)], suggesting the presence of an
in-gap edge state. In Fig. 6(c), we show the transfer rate at
$\delta_{pb}=\mathrm{Re}(E_{c1})$ as a function of $J_{t}$, which clearly
indicates a phase transition near $J_{t,c1}$.

Similarly, in Fig. 6(d)(e), we aim to probe the edge state with energy
$E_{c2}$. Since the edge states now have a large support on sublattice site
$c$, we set $|f\rangle$ on site $c$ of the right-most unit cell. We find that
a peak appears near $\delta_{pb}=\mathrm{Re}(E_{c2})$ in the transfer rate
when $J_{t}>J_{t,c2}$ [Fig. 6(d)(e)], consistent with the emergence of the
topological edge states. Likewise, the topological phase transition is clearly
visible near $J_{t}=J_{t,c2}$ in Fig. 6(f).

Compared to the Bragg spectroscopy implemented in Ref. [50], for the numerical
simulations here, we consider a weaker probe ($J_{\text{pb}}\sim h\times 12$
Hz, $h$ being the planck constant), and a longer probe time ($\tau\sim 16$
ms). Such optimization leads to a more faithful detection with a better
resolution.

Alternatively, we can couple the bystander state to a sublattice site in the
bulk to reveal the global spectral features under the periodic boundary
condition. The results are plotted in Fig. 7. In the Hermitian case with
$\gamma=0$ [Fig. 7(a)(b)], the transfer rate shows sharp edges at the band
edge, revealing both the band continuum and the band gaps. For the dissipative
AB chain with finite $\gamma$ [Fig. 7(c)(d)], the transfer-rate profiles are
broadened due to the imaginary components of the eigenspectra. Nevertheless,
the band gaps are still visible as valleys in the profile. In relation to the
experiment in Ref. [50], injecting atoms into the bulk offers a complementary
detection scheme for the topological phase transition, by observing the
closing of the band gaps.

###  IV.2 Dynamic detection of edge states

Topological edge states can also be detected through dynamics close to the
boundary. Under the non-Hermitian skin effect, eigenstates of the AB chain
accumulate to one of the edges. The idea is to initialize the state near the
opposite edge, and observe the time-dependent population along the chain.
While the non-Hermitian skin effect would drive the population toward the
other edge, topological edge states should remain near the initial site. To
quantitatively characterize the phenomena, we define the normalized occupation

$|\psi^{\prime}(\tau,n)|^{2}=\sum_{j=a,b,c}\frac{|\langle
n,j|\psi(\tau)\rangle|^{2}}{\langle\psi(\tau)|\psi(\tau)\rangle},$ (17)

which indicates the spatial distribution of the state at the time $\tau$.

In Fig. 8, we show the time evolution of the probability in the topological
trivial [Fig. 8(a)(c)] and non-trivial regions [Fig. 8(b)(d)]. In the
topological trivial region, the time-evolved state diffuses into the bulk
without much occupation at the boundary. By contrast, in the topological non-
trivial region, the time-evolved state still exhibits a peak at the boundary,
together with the diffusive dynamics into the bulk. Note that the normalized
occupation at the boundary decreases with time because there are bulk
eigenstates that decay slower than the edge states. The detection scheme
therefore should only work at intermediate times. Nevertheless, such a dynamic
detection is readily accessible in experiments, and provides a direct signal
of the non-Hermitian skin effect.

##  V Conclusion

In conclusion, we have characterized the topological features of a dissipative
AB chain in detail, and provided a theoretical description for the recently
implemented atom-injection spectroscopy. We show that the injection
spectroscopy can be applied to the bulk sites and resolve the band structure
of the system under a periodical boundary condition. We further propose an
alternative detection scheme for the topological edge states. Our studies are
helpful for future experimental study of the dissipative AB chain in relevant
quantum simulation systems.

###### Acknowledgements.

This work has been supported by the Natural Science Foundation of China (Grant
Nos. 11974331) and the National Key R&D Program (Grant Nos. 2016YFA0301700,
2017YFA0304100).

## Appendix

In this Appendix, we show that the expression in Eq. (15) can also be derived
from the linear response theory [55, 56]. Given the probe Hamiltonian Eq.
(10), we consider the correlation function

$D(t,t^{\prime})=-i\theta(t-t^{\prime})\left\langle\left[f^{\dagger}(t)d(t),d^{\dagger}(t^{\prime})f(t^{\prime})\right]\right\rangle.\
$ (A1)

In the frequency space, the correlation function becomes

$D(i\delta_{pb})=\frac{1}{\beta}\sum_{j}\sum_{n}G_{d}(i\omega_{n}-i\delta_{pb})G_{f}(E_{j},i\omega_{n}),$
(A2)

where $G_{d}(i\omega_{n})=1/i\omega_{n}$ is the Green’s function for state
$|d\rangle$, and $G_{f}$ is the Green’s functions for state $|f\rangle$, with

$G_{f}(E_{j},i\omega_{n})=\frac{\langle
0|f|\psi_{j}\rangle\langle\psi_{j}|f^{\dagger}|0\rangle}{i\omega_{n}-E_{j}}=\frac{\psi_{j,f}^{R}\psi_{j,f}^{L*}}{i\omega_{n}-E_{j}}.$
(A3)

Here $\omega_{n}$ are the bosonic Matsubara frequencies, $E_{j}$ and
$|\psi_{j}\rangle$ are respectively the eigenenergy and eigenstate of the bulk
Hamiltonian Eq. (1). The initial state $|0\rangle$ corresponds to an empty
lattice, particularly with no occupation on site $f$. After analytic
continuation, we derive the response function $R(\delta_{pb})$ as

$\displaystyle R(\delta_{pb})$
$\displaystyle=-J_{pb}^{2}\operatorname{Im}D\left(i\delta_{pb}\rightarrow\delta_{pb}+i0^{+}\right)$
$\displaystyle=-\operatorname{Im}\sum_{j}\frac{J_{pb}^{2}\psi_{j,f}^{R}\psi_{j,f}^{L*}}{\delta_{pb}-E_{j}+i0^{+}}.$
(A4)

We thus reproduces Eq. (15) by applying the linear response theory to our non-
Hermitian system.

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