# Experimental realization of quantum non-reciprocity based on cold atomic
ensembles

Ming-Xin Dong  Key Laboratory of Quantum Information, University of Science
and Technology of China, Hefei, Anhui 230026, China.  Synergetic Innovation
Center of Quantum Information and Quantum Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China.  Yi-Chen Yu  Key Laboratory
of Quantum Information, University of Science and Technology of China, Hefei,
Anhui 230026, China.  Synergetic Innovation Center of Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei, Anhui
230026, China.  Ying-Hao Ye  Key Laboratory of Quantum Information, University
of Science and Technology of China, Hefei, Anhui 230026, China.  Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China.  Wei-Hang Zhang
Key Laboratory of Quantum Information, University of Science and Technology of
China, Hefei, Anhui 230026, China.  Synergetic Innovation Center of Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China.  En-Ze Li  Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei, Anhui
230026, China.  Synergetic Innovation Center of Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei, Anhui
230026, China.  Lei Zeng  Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei, Anhui 230026, China.  Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China.  Guang-Can Guo
Key Laboratory of Quantum Information, University of Science and Technology of
China, Hefei, Anhui 230026, China.  Synergetic Innovation Center of Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China.  Dong-Sheng Ding<EMAIL_ADDRESS>Key
Laboratory of Quantum Information, University of Science and Technology of
China, Hefei, Anhui 230026, China.  Synergetic Innovation Center of Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China.  Bao-Sen Shi<EMAIL_ADDRESS>Key
Laboratory of Quantum Information, University of Science and Technology of
China, Hefei, Anhui 230026, China.  Synergetic Innovation Center of Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China.

(August 27, 2024)

In analog to counterparts widely used in electronic circuits, all optical non-
reciprocal devices are basic building blocks for both classical and quantum
optical information processing. Approaching the fundamental limit of such
devices, where the propagation of a single photon exhibits a good non-
reciprocal characteristic, requires an asymmetric strong coupling between a
single photon and a matter. Unfortunately it has been not realized yet. Here,
we propose and experimentally realize a quantum non-reciprocity device with
low optical losses and a high isolation of larger than 14 dB based on the cold
atoms. Besides, the non-reciprocal transmission of a quantum qubit and non-
reciprocal quantum storage of a true single photon are also realized. All
results achieved would be very promising in building up quantum non-reciprocal
devices for quantum networks.

Non-reciprocal devices, such as circulators and isolators used for
indispensable components in information processing, no matter in electronic
circuits, or in quantum networks, have attracted a lot of attentions. Optical
non-reciprocal devices, as for optical-based functional components controlling
the flow of light, are crucial for advanced optical communication and quantum
information processing Cirac  _et al._ (1997); Lodahl  _et al._ (2017). To
achieve an optical non-reciprocal device, ones have to break Lorentz
reciprocity Fan  _et al._ (2012a); Jalas  _et al._ (2013). Motivated by the
desirable applications in on-chip integration system and quantum networks, a
variety of methods with distinct physical principles have been proposed to
realize optical non-reciprocity. One approach is using the spatio-temporal
modulation Yu and Fan (2009); Kang  _et al._ (2011); Estep  _et al._ (2014);
Sounas  _et al._ (2013); Lira  _et al._ (2012); Tzuang  _et al._ (2014) of the
permittivity of nonlinear materials to break time-reversal symmetry. An
alternative approach is to use nonlinear processes to construct a nonlinear
optical isolation Fan  _et al._ (2012b); Bender  _et al._ (2013); Chang  _et
al._ (2014); Peng  _et al._ (2014). In addition, many other schemes have been
proposed based on various systems, such as opto-mechanical system Hafezi and
Rabl (2012); Xu and Li (2015); Manipatruni  _et al._ (2009); Shen  _et al._
(2016); Ruesink  _et al._ (2016); Fang  _et al._ (2017); Verhagen and Alù
(2017), atomic system Wang  _et al._ (2013); Wu  _et al._ (2014); Horsley  _et
al._ (2013); Megyeri  _et al._ (2018); Xia  _et al._ (2018) and chiral quantum
optics system Lodahl  _et al._ (2017); Scheucher  _et al._ (2016); Sayrin  _et
al._ (2015); Söllner  _et al._ (2015); Xia  _et al._ (2014); Zhang  _et al._
(2018).

To date, most of reported schemes have been performed with a weak coherent
light. However, many practical quantum information processes require a genuine
single photon, not a weak coherent pulse, such as linear quantum
computationKnill  _et al._ (2001). Hence, the ability of realizing single-
photon non-reciprocity is a basic requirement for a quantum network. Nowadays,
many feasible schemes have been proposed to realize single photon non-
reciprocity Zhou  _et al._ (2008, 2013); Tang  _et al._ (2019); Shen and Fan
(2005); Wang  _et al._ (2019), for example, very recently, Liu et at. proposed
a possible experimental scheme to implement phase controlled single-photon
non-reciprocal transmission based on an one-dimensional waveguide Wang  _et
al._ (2019). However, the single photon non-reciprocity has not been realized
yet. The experimental realization of a true single-photon non-reciprocity is
always the focus of people’s attentions for potential applications in quantum
network such as constructing a quantum transistor Chang  _et al._ (2007) and
quantum router Zhou  _et al._ (2013), therefore it is desirable and also a big
challenge to experimentally realize non-reciprocity in full quantum regime.

Here, we propose and experimentally realize a scheme to achieve a true single-
photon non-reciprocity in a cold atomic ensemble. The underlying physics in
our scheme is the asymmetric coupling between a single photon and the specific
atomic transition, which results in an asymmetric permittivity and break the
Lorentz reciprocity. We use a heralded single photon to demonstrate quantum
non-reciprocity with low optical losses and high isolation. Moreover, the
proof-of-principle experimental demonstration of non-reciprocal transmission
of an arbitrary polarized qubit and non-reciprocal quantum storage of a true
single photon are also realized. We believe all results are very promising for
non-reciprocal quantum information processing.

Figure 1: Experimental realization of quantum-regime non-reciprocity. (a)
Schematic diagram of experimental setup. Upper layer of setup is used to
generate a heralded single photon through SFWM process in the cold atomic
cloud 1 and the relevant energy level configuration shown in the dashed-line
rounded rectangle. $\Delta$ represents the detuning of pump 1, which is set to
be $2\pi\times 50$ MHz.
$\left|1\right\rangle=\left|5S_{1/2},F=2\right\rangle,$$\left|2\right\rangle=\left|5S_{1/2},F=3\right\rangle,$$\left|3\right\rangle=\left|5P_{1/2},F^{\prime}=3\right\rangle$
and $\left|4\right\rangle=\left|5P_{3/2},F^{\prime}=3\right\rangle.$ The lower
layer is utilized to demonstrate the quantum non-reciprocity, including a non-
reciprocal apparatus for a qubit as depicted in the dashed-line rectangle. (b)
The exploited multi-Zeeman sub-levels configuration in the forward-propagation
case. Signal field is $\sigma^{+}$ polarized and coupling laser is
$\sigma^{+}$ polarized that both propagate in the forward direction. The
states $\left|g\right\rangle,\left|s\right\rangle,\left|e\right\rangle$
correspond to $\left|1\right\rangle,\left|2\right\rangle,\left|3\right\rangle$
respectively. (c) The multi-Zeeman sub-levels configuration in the backward-
propagation case, and signal field is $\sigma^{-}$ polarized that propagates
in the backward direction.

In the experimental diagram, a pair of non-classical correlated photons called
Stokes and anti-Stokes photon (labeled as S1 and S2) are emitted through
spontaneously four-wave mixing (SFWM) Balić  _et al._ (2005); Du  _et al._
(2008) process in atomic cloud 1 (the optical depth ($OD$) is 14), as shown in
the top part in Fig.1(a). Two pump beams called pump 1 (780 nm, Rabi frequency
$2\pi\times 0.67$ MHz) and pump 2 (795 nm, Rabi frequency $2\pi\times 4.84$
MHz) with orthogonal circularly polarization counter-propagate through the
atomic cloud 1. The photons S1 and S2 are collected at an angle of
$3^{\text{\textdegree}}$ with respect to the pump lasers. S1 photon is
detected by a single photon counting module, the heralded single photon S2 is
collected and then coupled into a single-mode fiber for the demonstration of
quantum non-reciprocity. Subsequently, we input S2 photon into the atomic
cloud 2 from port 1 and port 2 respectively and study the transmission
properties of the corresponding output ports as shown in the bottom part in
Fig.1(a). Here, the forward-propagation (backward-propagation) case I (II) is
defined as the propagation of S2 photon from port 1 (2) to port 2 (1) while S2
photon co-propagates (counter-propagates) with coupling laser. The marked non-
reciprocal apparatus for a qubit (NRAQ) shown in dotted square frame can be
considered as a whole setup to realize quantum non-reciprocity, it is actually
an optical isolation of a two-port device. The NRAQ mainly consists of the
atomic cloud 2, a coupling laser and a pair of beam displacers (BDs). Here, we
input the coupling laser into the atomic cloud 2 at an angle of
$2.8^{\text{\textdegree}}$ with the respect to the propagation direction of
signal photon S2. In order to minimize the noise scattering from the coupling
laser, we insert two home-made Fabry-Perot (FP) etalons (50% transmittance,
500 MHz bandwidth) for frequency filtering before the single photon detectors.

Fig.1(b), (c) depict the relevant energy-level configurations in forward-
propagation and backward-propagation cases, where the signal (red) and
coupling (blue) fields couple corresponding atomic levels respectively.
$m_{F}$ is the magnetic quantum number of the atomic hyperfine states and the
$\sigma^{+}$/$\sigma^{-}$ polarized signal photon couples the atomic
transition $\Delta{\rm\mathrm{m}}_{{\rm F}}=+1/-1$ respectively. We take into
account all degenerate Zeeman sublevels in theoretical analysis Chen  _et al._
(2000); Wang  _et al._ (2006). In Fig. 1(b), the five Zeeman transitions for
signals (red lines) are all connected with the coupling transitions, which can
be treated as an effectively completed $\Lambda$-type electromagnetically
induced transparency (EIT) system Tiwari  _et al._ (2010), thereby the atoms
initially populated in $\left|g\right\rangle$ are transparent for the forward-
propagated photons due to the quantum interference effect. By comparison, for
backward-propagation case depicted in Fig. 1(c), the transition of
$\left|g,m_{F}=-2\right\rangle\to\left|e,m_{F}=-3\right\rangle$ contributes to
the absorption (see Method) which prevents the propagation of photons in
backward direction. This renders asymmetric interaction between
$\sigma^{+}$/$\sigma^{-}$ polarized photons and atoms, causing the chirality
in our system which can be exploited to realize quantum non-reciprocity.

Figure 2: Experimental results of single-photon non-reciprocity. (a) The
measured single photon wave packets for forward-propagation (red line) and
backward-propagation (blue line) cases. (b) Single-photon EIT transmission
spectra as a function of detuning from atomic resonance $\left|{\rm
g}\right\rangle\to\left|{\rm e}\right\rangle$ in forward-propagation case. The
black line is the theoretical fitting. (c) Single-photon transmission versus
detuning in backward-propagation case and black line is the theoretical
fitting. (d) The transmissions FW (red), BW (blue) and contrast $\eta$
(purple) as a function of $OD$. Here, FW (BW) refers to the forward-
propagation (backward-propagation) case. The solid lines are the theoretical
fittings by using Eq.10. The experiment and theory agreed well.

At first, we study the single-photon non-reciprocity, in this case we block
one optical path L after signal S2 passes through a BD in Fig.1(a). The
measured single-photon wave packets with a resolution of 1.6 ns for forward-
propagation (red line) and backward-propagation cases (blue line) are depicted
in Fig.2(a) respectively. Here, we select a suitable temporal length of S2
photon matching the EIT bandwidth of the atomic cloud 2 to obtain the maximum
transmission rate in forward propagation case. The bandwidth of signal 2 is
set to be $2\pi\times 1.60$ MHz, which can be controlled by the Rabi frequency
of pump 2. Moreover, according to Eq.10 given in Method section, we should
choose a suitable $OD$ of the atomic cloud 2 and Rabi frequency of coupling
laser to obtain high transmission and high isolation simultaneously. The $OD$
of atomic cloud 2 is set to be 19, which is sufficient to prevent signal’s
propagation in the backward direction. Meanwhile, the Rabi frequency of
coupling laser is set to be $2.5\Gamma$ ($\Gamma$ is the decay of level
$\left|3\right\rangle$), satisfying the requirement of high transmission in
forward direction.

In the forward propagation case, the transmission rate is estimated to be
92.9%. By contrast, the transmission rate is near zero exhibiting a high
isolation in backward propagation case. In addition, we define a contrast
$\eta$ between two propagation cases described by the following formula

$\eta=\frac{CC_{{\rm fw}}-CC_{{\rm bw}}}{CC_{{\rm fw}}+CC_{{\rm bw}}}$ (1)

where $CC_{{\rm fw}}$/$CC_{{\rm bw}}$ represents the total integral
coincidence counts in forward propagation/backward propagation case. The
obtained $\eta$ is 0.96, implying a sharp contrast between two propagation
cases, this clearly characterizes a strong non-reciprocity in our scheme.

Subsequently, we explore the relation between the transmission and the
frequency of signal 2 in two propagation cases. In the forward-propagation
case, we obtain a single-photon EIT spectrum via changing the detuning of S2
photon from $-2{\rm\pi}\times 18\sim+2{\rm\pi}\times 22$ MHz as depicted in
Fig.2(b). Here, the frequency detuning of S2 photon is achieved by varying the
frequency of P2 laser controlled by an acousto-optic modulator (AOM).
Accordingly, we observe a single photon transmission spectra as shown in
Fig.2(c) in the backward-propagation case. Obviously, a high contrast is
available as long as the single photons is resonant with the atomic transition
of $\left|{\rm g}\right\rangle\to\left|{\rm e}\right\rangle$. In Fig.2(d), we
plot the transmissions for forward propagation (red), backward propagation
(blue) and the contrast $\eta$ (purple) as a function of $OD$. When the $OD$
is close to zero, the transmission becomes symmetric due to the nearly
identical transmissions in two propagation cases resulting in the vanish of
non-reciprocal behavior. We could increase the isolation rate by increasing OD
to obtain a better non-reciprocity.

Figure 3: The non-reciprocity for polarization qubits. (a-d) The reconstructed
density matrices for single photons encoded in $\left|H\right\rangle$ ,
$\left|V\right\rangle,$ $\left|R\right\rangle$ and $\left|D\right\rangle$
polarization respectively in the forward-propagation case. The height of bars
denotes the magnitude ${\rm Abs}[\rho]$ and color represents phase. (e-h) The
contrast of forward-propagation and backward-propagation cases for four
polarization qubits.

Next, we characterize the transmission of qubits by inputting four
polarization states $\left|H\right\rangle$, $\left|V\right\rangle,$
$\left|R\right\rangle=\left(\left|H\right\rangle-i\left|V\right\rangle\right)/\sqrt{2}$
and
$\left|D\right\rangle=\left(\left|H\right\rangle+\left|V\right\rangle\right)/\sqrt{2}$
into our non-reciprocal apparatus. The preparation of an arbitrary
polarization qubit state of $\left|\psi\right\rangle={\rm
cos}\frac{\theta}{2}\left|H\right\rangle+e^{i\phi}{\rm
sin}\frac{\theta}{2}\left|V\right\rangle$ could be achieved by adjusting a
quarter-wave plate (QWP) and a half-wave plate (HWP). Here, $\theta$ and
$\phi$ represent the angle of HWP and QWP respectively. As sketched in
Fig.1(a), the NRAQ is a spatially multiplexed dual-rail geometry constructed
by a pair of BDs. We exploit BDs to spatially separate two orthogonal
polarization components of qubits which transmit simultaneously in NRAQ. In
addition, before signals enter into the atomic cloud 2, we make two components
have identical circular polarization $\sigma^{+}$ for satisfying the non-
reciprocity conditions described in the above context. Hence, we insert a HWP
on optical path R to convert the polarization of the photon from vertical to
horizontal and a common QWP on both optical paths (L and R) to convert the
polarization of photon from horizontal to circular. Subsequently, the signals
are focused into the centre of atoms by a lens with a focal length of 300 mm.
Symmetrically, we recombine two paths into a single spatial mode with the
assistance of a QWP, lens, HWP, and BD. Here, a pair of BDs form a long-time
and passively-stabilized Mach-Zehnder (M-Z) interferometer and the relative
phase between L and R is set to be zero by adjusting the tilt of BDs. In our
experiment, we utilize two non-reciprocal processes simultaneously in a single
NRAQ to realize non-reciprocal transmission of qubits. Ultimately, a
polarization analyzer consisting of a QWP, HWP and PBS is exploited to measure
the output state. For forward-propagation case, we characterize the output
polarized states by using the quantum state tomography (see Method sections)
and the reconstructed density matrices are shown in Fig.3(a-d). By comparison,
we measure the transmission in backward (blue) and forward directions (red)
for four polarization qubit states as shown in Fig.3(e-h). It is obvious that
we can observe high contrasts of 0.96, 0.94, 0.93, 0.95 respectively for four
polarized states according to Eq.1.

To further quantify the performance of the NRQA, we have defined several
parameters such as transmission rate, transmission fidelity for forward
propagation case and isolation for backward propagation case. Here, the
transmission fidelity $\mathcal{F}$ is defined in Method sections, and the
transmission rate $T_{{\rm fw}}$ and isolation $I_{{\rm bw}}$ are defined as

$T_{{\rm fw}}=\frac{CC_{{\rm fw}}}{CC_{{\rm in}}},I_{{\rm bw}}=10{\rm
log}\frac{CC_{{\rm in}}}{CC_{{\rm bw}}}$ (2)

here, $CC_{{\rm in}}$ is the coincidence counts of initial input photons. The
calculated values of $T_{{\rm fw}}$, $\mathcal{F}$ and $I_{{\rm bw}}$ for four
polarization qubit states without any noise correction are presented in Table
Experimental realization of quantum non-reciprocity based on cold atomic
ensembles. It is very clear that we realize the high transmission and fidelity
of polarization qubits in forward direction and the low transmission in
backward direction.

Ultimately, we study the non-reciprocity of single photon storage process in
our system. The non-reciprocity light storage has attracted many interests due
to potential applications in asymmetric quantum networks, and it was
demonstrated in a whispering-gallery-mode optical microresonantor Dong  _et
al._ (2015) based on Brillouin-scattering-induced transparency recently by
using a coherent light as input. Here, we demonstrate a quantum non-reciprocal
storage of a true single photon based on our non-reciprocity device.

We input single photons in forward and backward directions respectively with a
fixed coupling laser. We block one path L of M-Z interferometer as shown in
Fig.1(a). Note that the input signals (blue lines in Fig.4) in two propagation
cases are the same when we take the overall loss of photons into account. In
the forward propagation case, the dark state is formed leading to a high
transparency of signal, which is the first step to realize single photon
storage based on EIT storage protocol (see Method sections for more details).
After we shut off the coupling adaibatically, the single photon is stored in
the atomic ensemble. We utilize sequential coupling laser pulses to
dynamically write and read the single photon schematically illustrated in the
green line of Fig.4(a). The underlying mechanism in storage process can be
considered as the conversion between atomic collective spin excitation and
optical mode. After a programmed storage time, we read out the signal and the
retrieved component of signal (red part in read out process) is depicted in
Fig.4(a). Here, the $OD$ of the atomic cloud 2 in storage process is set to be
54, and the storage efficiency is close to 9% which is mainly limited by the
low $OD$ of atoms in our system. The storage efficiency can be further
improved by enlarging the $OD$ of the atomic cloud 2 and by optimizing the
input pulse wave packet to match the EIT bandwidth of the atomic cloud 2. For
comparison, we try the single photon storage in backward case as shown in
Fig.4(b). It is obvious that we cannot observe a clear retrieved signal,
implying a high isolation for backward-propagation signal. The forbidden
storage process arises from the reason that optical modes cannot satisfy the
condition of EIT process in backward-propagation.

Figure 4: The non-reciprocal quantum storage. (a) The input signal (blue line)
and stored signal (red line) before and after storage process in forward-
propagation case. (b) The input signal (blue line) and output signal (red
line) in backward-propagation case.

In summary, we propose and experimentally demonstrate quantum non-reciprocity
based on cold atomic ensembles. We exploit an asymmetric interaction between
single photons and cold atoms to realize the no-reciprocity in our experiment.
Besides, we realize the non-reciprocal transmission of polarization qubits and
non-reciprocal quantum storage of a true single photon, which shows potential
applications in quantum information processing. It is an outlook that this
non-reciprocity mechanism would be realized by using single resonator enhanced
atom Sayrin  _et al._ (2015) initially prepared in a specific Zeeman state
like $\left|{\rm g_{-2}}\right\rangle$ in this work or integrated vapor cell
Overstolz  _et al._ (2014), these could give prospective for future integrated
quantum information processing.

## Method Sections

Preparation of a true single photon. We generate heralded single photons
through SFWM process in cold ${}^{85}{\rm Rb}$ atoms trapped in a 2D magneto-
optical trap (MOT). The repetition rate of our experiment is 100 Hz and
experimental window is 1.3 ms during which the MOT magnetic field is switched
off completely. The pumps 1 and 2 are controlled by two AOMs modulated by
arbitrary function generators (Tektronix, AFG3252). Two lenses each with a
focal length of 500 mm are used to couple the signal fields S1 and S2 emitted
from atomic cloud 1. The pumps 1 and 2 are collinear, and hence their
respective signal fields are collinear to due to the phase matching condition
$k_{p1}-k_{S1}=k_{p2}-k_{S2}$ in the SFWM process, as in our previous work
Ding  _et al._ (2013). The two signal photons are ultimately collected into
their respective single-mode fibers and are detected by two single photon
detectors (avalanche diode, PerkinElmer SPCM-AQR-16-FC, $60$ efficiency,
maximum dark count rate of 25/s). The signals from the two detectors are then
sent to a time-correlated single photon counting system (TimeHarp 260) to
measure their time-correlated function.

Asymmetric interaction between single photons and atoms

The optical properties of atomic system are fundamentally determined by their
intrinsic energy-level configuration. In our work, the photons couple with
specific atomic transitions, contributing to the direction-dependent
transparency and absorption, inducing this non-reciprocity. The photons
interactions with atoms in forward and backward directions become
asymmetrical, the interaction Hamiltonian can be written as follow:

In forward-propagation case, the interaction Hamiltonian $H_{\mathop{{\rm
int}}}^{{\rm fw}}$ can be described as

$\displaystyle H_{\mathop{{\rm int}}}^{{\rm
fw}}=-\frac{\hbar}{2}(\sum\limits_{i=-2}^{2}\Omega_{{\rm
p}i}^{+}\left|e_{i+1}\right\rangle\langle g_{i}|+\Omega_{{\rm
c}i}^{+}\left|e_{i+1}\right\rangle\langle s_{i}|+{\rm c}.{\rm c})$ (3)
$\displaystyle-\hbar\sum\limits_{i=-2}^{2}(\Delta\omega_{p}\left|e_{i+1}\right\rangle\langle
e_{i+1}|+\delta\left|s_{i}\right\rangle\langle s_{i}|)$

here, $\left|g_{i}\right\rangle$ refer to the Zeeman states of level $g$,
where $-F\leq i\leq F$. The similar definitions are adapted to levels $s$ and
$e$. $\Omega_{{\rm p}i}^{+}=-\mu_{e_{i+1},g_{i}}E_{p}^{+}/\hbar$ and
$\Omega_{ci}^{+}=-\mu_{e_{i+1},s_{i}}E_{c}^{+}/\hbar$ denote the Rabi
frequencies of $\sigma^{+}$ photons and coupling laser respectively.
$\mu_{e_{i+1},g_{i}}$ represents the dipole moment between Zeeman
states$\left|e_{i+1}\right\rangle$ and$\left|g_{i}\right\rangle$.
$\Delta\omega_{p}=\omega_{p}-\omega_{ge}$ is the single-photon detuning,
$\delta=\Delta\omega_{p}-\Delta\omega_{c}$ represents the two-photon detuning.

Furthermore, in backward-propagation case, the interaction Hamiltonian
$H_{{\rm int}}^{{\rm bw}}$ can be written as

$\displaystyle H_{{\rm int}}^{{\rm
bw}}=-\frac{\hbar}{2}[\sum\limits_{i=-1}^{2}\left(\Omega_{{\rm
p}i}^{-}\left|e_{i-1}\right\rangle\langle g_{i}|+\Omega_{{\rm
c}i}^{+}\left|e_{i-1}\right\rangle\langle s_{i-2}|\right)$
$\displaystyle+\Omega_{{\rm p-2}}^{-}\left|e_{-3}\right\rangle\langle
g_{-2}|+{\rm c}.{\rm
c}.]-\hbar\sum\limits_{i=-1}^{2}(\Delta\omega_{p}\left|e_{i-1}\right\rangle\langle
e_{i-1}|$ (4) $\displaystyle+\delta\left|s_{i-2}\right\rangle\langle
s_{i-2}|)-\hbar\Delta\omega_{p}\left|e_{-3}\right\rangle\langle e_{-3}|$

$\Omega_{{\rm p}i}^{-}=-\mu_{e_{i-1},g_{i}}E_{p}^{-}/\hbar$ represents the
Rabi frequency of $\sigma^{-}$ photons.

The respective susceptibility $\chi_{fw(bw)}$ can be formula as Wang  _et al._
(2006)

$\chi_{fw}=-\frac{2N}{5\hbar\varepsilon_{0}}\sum\limits_{i=-2}^{2}\frac{\left|\mu_{e_{i+1},g_{i}}\right|^{2}}{\Omega_{p_{i}}^{+}}\rho_{g_{i},e_{i+1}}$
(5)

$\chi_{bw}=-\frac{2N}{5\hbar\varepsilon_{0}}(\sum\limits_{i=-1}^{2}\frac{\left|\mu_{e_{i-1},g_{i}}\right|^{2}}{\Omega_{p_{i}}^{-}}\rho_{g_{i},e_{i-1}}+\frac{\left|\mu_{e_{-3},g_{-2}}\right|^{2}}{\Omega_{p_{-2}}^{-}}\rho_{g_{-2},e_{-3}})$
(6)

here, $N$ is atomic density. The density matrix $\rho_{g_{i},e_{i+1}}$can be
calculated from master equation

$\frac{\partial\rho}{\partial
t}=-\frac{i}{\hbar}[\mathop{H_{int}},\rho]-\frac{1}{2}\left\\{\Gamma,\rho\right\\}$
(7)

Ultimately, susceptibility $\chi_{fw(bw)}$ can be written as

$\chi_{fw}=\frac{N}{5\hbar\varepsilon_{0}}\sum\limits_{i=-2}^{2}\frac{4\left|\mu_{e_{i+1},g_{i}}\right|^{2}(\delta+i\gamma_{gs})}{\left|\Omega_{ci}^{+}\right|^{2}-4(\delta+i\gamma_{gs})(\Delta\omega_{p}+i\gamma_{ge})}$
(8)

$\displaystyle\chi_{bw}=$
$\displaystyle\frac{N}{5\hbar\varepsilon_{0}}(\sum\limits_{i=-1}^{2}\frac{4\left|\mu_{e_{i-1},g_{i}}\right|^{2}(\delta+i\gamma_{gs})}{\left|\Omega_{ci}^{+}\right|^{2}-4(\delta+i\gamma_{gs})(\Delta\omega_{p}+i\gamma_{ge})}$
$\displaystyle-\frac{\left|\mu_{e_{-3},g_{-2}}\right|^{2}}{\Delta\omega_{p}+i\gamma_{ge}})$
(9)

We consider an effective
$OD=NLk_{0}\left|\mu_{e,g}\right|^{2}/\left(\varepsilon_{0}\gamma_{ge}\hbar\right)$
, $\mu_{e,g}$ is the effective dipole moments. Moreover, we take into account
different Clebsch-Gordan coefficients of various transitions between different
Zeeman sublevels and the relations between $\mu_{e,g}$ and
$\mu_{e_{i+1},g_{i}}$($\mu_{e_{i-1},g_{i}}$) referred to Steck (2001). $k_{0}$
is the wave vector, $L$ is the length of atomic cloud,
$\gamma_{gs},\gamma_{ge}$ are the mean dephasing rates of atomic transition
$\left|g\right\rangle\to\left|s\right\rangle$ and
$\left|g\right\rangle\to\left|e\right\rangle$ respectively.

We observe a asymmetric permittivity in forward and backward propagation
cases, which induces the non-reciprocity in our scheme. The transmission for
forward $T_{fw}$ and backward $T_{bw}$ propagation cases can be described by

$T_{fw(bw)}=e^{-2{\rm Im}[(\omega_{p}/c)(1+\chi_{fw(bw)}/2)]L}$ (10)

here, $\omega_{p}$ is the frequency of photon, $c$ is the speed of light in
vacuum.

Quantum state tomography of qubits. To characterize the output polarization
qubit state, we project single photon states onto four bases
$\left|H\right\rangle$, $\left|V\right\rangle,$
$\left(\left|H\right\rangle+i\left|V\right\rangle\right)/\sqrt{2}$ and
$\left(\left|H\right\rangle+\left|V\right\rangle\right)/\sqrt{2}$ by a
polarization analyzer consisting of a QWP, HWP and PBS. Then, we obtain a set
of data for reconstructing the density matrix. The fidelity is calculated by
the formula $\mathcal{F}={\rm Tr(}\sqrt{\sqrt{\rho}\rho_{{\rm
ideal}}\sqrt{\rho}}{\rm)}^{2}$, which compares the reconstructed density
matrix $\rho$ with ideal density matrix $\rho_{{\rm ideal}}$. Ultimately, we
evaluate the standard deviation in our experiment by the Poisson statistics
using Monte Carlo simulation with assistance of Mathematica software.

EIT quantum memory. In the forward-propagation case, after propagating through
a 200-m optical fiber (corresponding to a time delay of $\sim$1 $\mu s$), S2
photon is focus onto the atomic cloud 2 by using a lens with focus length of
300 mm. We adiabatically switch off the coupling laser, and a stored atomic
collective excitation is obtained given by $1/\sqrt{m}\sum
e^{i\mathbf{\mathit{k}}_{S}\mathbf{\cdot\mathit{r}}_{i}}\left|g\right\rangle_{1}\cdot\cdot\cdot\left|s\right\rangle_{i}\cdot\cdot\cdot\left|g\right\rangle_{m}$
Fleischhauer and Lukin (2000), also called as a spin wave.
$\mathbf{\mathrm{\mathit{k}}}_{S}=\mathbf{\mathrm{\mathit{k}}}_{c}-\mathbf{\mathrm{\mathit{k}}}_{s2}$
is the wave vector of atomic spin wave, $\mathbf{\mathrm{\mathit{k}}}_{c}$ and
$\mathbf{\mathit{k}}_{s2}$ are the vectors of coupling and S2 fields,
$\mathbf{\mathit{r}}_{i}$ denotes the position of the $i$-th atom in atomic
cloud 2. After a programmed storage time, the spin wave is converted back into
photonic excitation by switching on the coupling laser again.

Author contributions M.X.D conceived the idea with discussions with D.S.D. and
B.S.S.. M.X.D carried out the experiments with assistance from D.S.D., W.H.Z.
and Y.C.Y.. All authors contributed to the discussions and analysis of
results. M.X.D. and D.S.D. wrote the manuscript with contributions from
B.S.S.. D.S.D., B.S.S. and G.C.G. supervised the project.

Competing financial interests The authors declare no competing financial
interests.

Acknowledgments The authors thank Prof. Chun-Hua Dong, Dr. Wei Zhang, and Dr.
Meng-Jun Hu for fruitful discussions. This work was supported by National Key
R&D Program of China (2017YFA0304800), the National Natural Science Foundation
of China (Grant Nos. 61525504, 61722510, 61435011, 11174271, 61275115,
11604322), and the Innovation Fund from CAS, Anhui Initiative in Quantum
Information Technologies (AHY020200), the Fundamental Research Funds for the
Central Universities, and the Youth Innovation Pro motion Association of CAS
under Grant No. 2018490.

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