# Broad Spiral-Bandwidth of Orbital Angular Momentum Interface between Photon
and Memory

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.
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.  Wei 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.  Shuai Shi  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.  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.
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)

###### Abstract

The complex interactions between orbital angular momentum (OAM) light and
atoms are particularly intriguing in the areas of quantum optics and quantum
information science. Building a versatile high-dimensional quantum network
needs broad spiral-bandwidth for preparing higher-quanta OAM mode and
resolving the bandwidth mismatch in spatial space and etc. Here, we
experimentally demonstrate a broad spiral-bandwidth quantum interface between
photon and memory. Through twisted fields of the writing and reading, the
correlated OAM distribution between photon and memory is significantly
broadened. This broad spiral-bandwidth quantum interface could be spanned in
multiplexing regime and could work in high-quanta scenario with capability of
$|l|=30$, and we demonstrate the entanglement within 2-D subspace with a
fidelity of $80.5\text{\textpm}4.8\%$ for high $l$. Such state-of-the-art
technology to freely control the spatial distribution of OAM memory is very
helpful to construct high-dimensional quantum networks and provides a
benchmark in the field of actively developing methods to engineer OAM single
photon from matters.

The interaction between orbital angular momentum (OAM) of structured light and
matters has many intriguing applications (Padgett, 2017), including trapping
of particles (He  _et al._ , 1995, 2009) and measuring rotation angular
(Courtial  _et al._ , 1998; Lavery  _et al._ , 2013), OAM-based imaging
(Fürhapter  _et al._ , 2005) and optical communications (Wang  _et al._ ,
2012). In quantum information field, light carried with OAM could
significantly enhance the information capacity, thus advancing the
developments of the high-dimensional (high-D) quantum networks, especially in
OAM entanglement generation (Mair  _et al._ , 2001; Dada  _et al._ , 2011;
Fickler  _et al._ , 2012; Krenn  _et al._ , 2014), OAM-based quantum memory
(Inoue  _et al._ , 2009; Ding  _et al._ , 2013; Nicolas  _et al._ , 2014; Zhou
_et al._ , 2015; Ding  _et al._ , 2015, 2016; Zhang  _et al._ , 2016) and OAM-
based teleportation (Wang  _et al._ , 2015). One of strumbling block of
constructing a high-D quantum network is how to establish a versatile high-D
OAM quantum interface between photon and memory (Krenn  _et al._ , 2017).

Building a high-D OAM quantum interface could be based on the protocol of
Duan-Lukin-Cirac-Zoller (DLCZ) (Duan  _et al._ , 2001) where the
probabilistically generated OAM photon is entangled with memory (Inoue  _et
al._ , 2006, 2009). There are many parameters to characterize the performance
of the interface between photon and memory (Bussières  _et al._ , 2013;
Heshami  _et al._ , 2016), such as lifetime, efficiency and fidelity and etc
(Heshami  _et al._ , 2016; Ma  _et al._ , 2017; Brennen  _et al._ , 2015). The
most unique parameter of high-D OAM quantum interface could be spiral-
bandwidth $\delta l$, which characterizes the mode-matching bandwidth window
(Ding, 2017). The adjacent nodes in high-dimensional quantum networks may be
diverse and different in spatial mode, spiral-bandwidth and etc, for example,
one is encoded in $\pm l$ OAM spaces and the other one is in $\pm(l+m)$ OAM
spaces; or one has the spatial-bandwidth of $\delta l$ and another is
$\delta(l+m)$, thus needing a technology to make the quantum interface be more
flexible and controllable and then people can connect them freely (Kimble,
2008).

In this letter, we experimentally demonstrate a high-D OAM interface between
photon and memory in delayed spontaneously four-wave mixing process. In this
configuration, the write- and read-laser beams are individually encoded, thus
making the joint of correlation against $l$ modes broadened because the
interaction length is increased in transverse azimuthal direction. This offers
the ability to control the spatial distribution, including entangled OAM
eigenmode $\pm l$ and the spiral-bandwidth $\delta l$. Based on that, we
demonstrate the potential applications for OAM multiplexing, and give an
obvious contrast data with inputting $\Delta l=10$. We also have achieved
high-D entanglement with $l$ up to 16 and high-quanta 2-dimensional OAM
entanglement with $l$ up to 30, all of which obey the entanglement properties.
The reported results are useful for realizing broad spiral-bandwidth and
high-D quantum memory and increasing the capacity of quantum communication,
and also is a benchmark of searching ways to explore versatile quantum
interfaces.

Figure 1: Overview of experiment. (a) The relevant energy level diagram. Write
and Read represent write-laser pulse and read-laser pulse respectively; S1 and
S2 are Signal 1 photon and Signal 2 photon. States $|1\rangle$, $|2\rangle$,
$|3\rangle$ and $|4\rangle$ correspond to ${}^{85}Rb$ atomic levels of
$5S_{1/2}(F=2)$, $5S_{1/2}(F=3)$, $5P_{1/2}(F=3)$ and $5P_{3/2}(F=3)$
respectively. $\Delta$ represents the detuning of write-laser pulse, which is
set to be $+70\times 2\pi$ MHz. The part in below is the time sequence of
experiment. (b) Experimental setup. MOT is magneto-optical trap, $L1-L8$
represent a series of lenses, $\lambda/4$ is quarter-wave plate, $\lambda/2$
is half-wave plate and PBS is polarization beam splitter. SLM $1\sim 4$ are
spatial light modulators.

The experimental media is an optically thick atomic ensemble of Rubidium 85
(${}^{85}Rb$) that is trapped in two-dimensional magneto-optic trap (MOT). The
involved schematic of the energy levels and the experimental setup are shown
in Fig. 1(a) and Fig. 1(b). We firstly establish the correlation between a
collective spin excited state (spin wave, also called as atomic memory) and a
single photon (Signal 1) through spontaneous Raman scattering (SRS) in atomic
ensemble. In this process, the write-laser is set to blue-detuned with atomic
transition $|2\rangle\rightarrow|3\rangle$. After reflecting from SLM 1 as
depicted in Fig. 1 (b), the write-laser has carried on the OAM phase message
loaded by a computer. Then, a 4-f image system with unequal arms, which is
consisted of two lenses L1 and L2 with focal length of 300 mm and 500 mm
respectively, is utilized to map the OAM phase of the write-laser to the
center of atomic ensemble accurately. The Signal 1 photon emitted from atomic
ensemble is mapped onto another SLM 2 for detecting the OAM modes. Due to the
angular momentum is conserved in SRS process, hence the spatial modes of the
spin wave and Signal 1 are entangled in OAM degree of freedom. This OAM
correlation can be flexibly demonstrated by mapping and checking the OAM modes
on SLM 1 and SLM 2 respectively.

The OAM-based DLCZ quantum memory is built when the entanglement between the
spin wave and Signal 1 photon is created. After a storage time of $\Delta t$,
we use another SLM 3 to load OAM structured light to read the spin wave out to
Signal 2, the Signal 2 is also mapped onto another SLM 4. Ultimately, in order
to check the quantum correlation between Signal 1 and atomic spin wave, we
measure the coincidence counts between Signal 1 and Signal 2 by projecting
them onto SLM 2 and SLM 4 respectively, in which the different phase
structures on both of SLM 2 and 4 are loaded for measurement. Here, two
couples of 4-f systems with unequal arms are used to map the OAM phase of
signal photons to SLM accurately, see supplementary information. The reflected
photons from SLMs are collected into two single-mode fibers, which are
detected by two detectors (avalanche diode, PerkinElmer SPCM-AQR-16-FC, 60%
efficiency, maximum dark count rate of 25/s) respectively.

Figure 2: The measured correlated OAM distribution under different OAM modes
of write- and read-beams. The correlated OAM distribution with
$l_{w}=-l_{r}=0$ (black line) and $l_{w}=-l_{r}=10$ (red line). These data are
fitted by the function $y={y_{0}}+2aw/[\pi(4{x^{2}}+{w^{2}})]$ with parameters
($y_{0}=15.8$, $w=11.4\pm 2.3$, $a=5501$) and ($y_{0}=0,w=27.0\pm
1.8,a=11534$).

In previous work (Ding  _et al._ , 2016) for demonstrating high-D OAM quantum
interface with Gaussian mode input, it is hard to generate higher-D
entanglement because the correlated coincidences of photons decay very quickly
against $l$. Here, we turn the OAM quanta of write- and read-beams to modulate
the light-atom interaction length. We input the write-laser with OAM quanta of
$l_{W}$. Due to the fact that SRS process conserves angular momentum, we have
created OAM entanglement between Signal 1 and atomic memory, which can be
specified by the formula

$\left|\psi\right\rangle_{{}_{photon-
atom}}^{l_{W}}{\rm=}\sum\nolimits_{l=-\infty}^{l=\infty}c_{l}\left|l\right\rangle_{{\rm
S1}}\otimes\left|l_{W}-l\right\rangle_{a}$ (1)

here, $\left|c_{l}\right|^{2}$ represents excitation probability,
$\left|l\right\rangle_{{\rm S1}}$ is the OAM eigenmode of Signal 1 with quanta
of $l$. $\left|l_{W}-l\right\rangle_{a}$ is the OAM eigenmode of atomic memory
with quanta of $l_{W}-l$. Through this method, the atomic memory could carry
the arbitrary OAM topological charge with the term of $l_{W}-l$, thus
resulting in the redistributed quantum interface. We also check the
conservation of OAM modes at two situations $l_{W}=2$, $l_{R}=0$ and $l_{W}=1$
and $l_{R}=2$, which can be found in the supplement. Most importantly, the
spiral-bandwidth of generated photons is broadened when we increase the OAM
quanta of write- and read-beams.

Figure 3: Multiplexing OAM modes. (a) The multiplexing with different OAM
modes of write- and read-lasers. The cross and check marks in up/down equation
shows the weak/strong correlation under the different ($|l_{W}|\neq|l_{R}|$)
/same OAM orders ($|l_{W}|=|l_{R}|$). (b) The coincidence counts for the
different situations of $\lvert l_{W}\rangle=\lvert 0\rangle/\lvert
10\rangle$, $\lvert l_{S1}\rangle=\lvert 0\rangle/\lvert 10\rangle$, $\lvert
l_{R}\rangle=\lvert 0\rangle/\lvert-10\rangle$, $\lvert l_{S2}\rangle=\lvert
0\rangle/\lvert-10\rangle$. (c) The memory decay function with different OAM
modes (black for $|l_{W}|=0$, red for $|l_{W}|=10$). The memory decays
exponentially with coherence time of $\tau=1655$ ns. (d) The multiplexing in
radial direction with different OAM modes in inner and outer rings. When the
OAM orders in the inner and outer rings are different (up equation), the
correlation is weak; On the contrary, the correlation is strong when the OAM
orders are same (down equation). (e) The normalized coincidence counts with
$|l_{W}\rangle=|0\rangle+|10\rangle$, $|l_{R}\rangle=|-10\rangle+|0\rangle$
and $|l_{W}\rangle=|0\rangle+|10\rangle$,
$|l_{R}\rangle=|0\rangle+|-10\rangle$ in the left two bars. The right two bars
are the corresponding OAM mode of $|l_{W}\rangle=|0\rangle+|9\rangle$,
$|l_{R}\rangle=|-9\rangle+|0\rangle$ and $|l_{W}\rangle=|0\rangle+|9\rangle$,
$|l_{R}\rangle=|0\rangle+|-9\rangle$. (f) The detected contrast of coincidence
counts against different $\Delta l$. The storage time is set to be 500 ns. The
different signs of OAM quanta set in (a) and (d) are required to maintain OAM
conservation.

Due to the broadening effect of spiral-bandwidth with larger $l$ laser beam
input, the distribution of generated OAM signal 1 and memory would be
redistributed in more flat range. This is because the generated OAM modes are
dependent on the interaction length and the waist of the write- and read-beams
(Du  _et al._ , 2008). The vector mismatching $\Delta k$ from transverse
azimuthal phase would increase the value of $\Delta k\cdot L$, where $L$ is
the interaction length. This effect is very promising because it is regarded
as a concentration operation. In order to achieve high-D OAM quantum memory,
we utilize the above method to extend the quanta of write-laser, we set
$l_{W}=10$. In addition, we set $l_{R}=-10$ for reading process. The writing
and reading process of DLCZ quantum memory is essentially a delayed
spontaneously four-wave mixing process. Based on the unique advantages of
individually modulating write- and read-beams of four-wave mixing process (not
like a single pump field used in spontaneous parametric down conversion), the
write- and read-laser beams can be individually loaded OAM modes with opposite
signs whilst the input total angular momentum can be zero, thus making the
joint spectrum of correlation broadeded. We map different OAM phases onto SLM
2 and SLM 4, and record the coincidence between Signal 1 and Signal 2 photons.
The spiral-bandwidth of OAM entanglement is measured in the red line in Fig.
2. The spiral-bandwidth of generated single photons becomes much flat than the
scheme with inputting Gaussian mode. The spiral-bandwidth of Gaussian mode is
about $\delta l=11.4\pm 2.3$, whilst for $l_{W}=-l_{R}=10$ the spiral-
bandwidth is $\delta l=27.0\pm 1.8$ obviously enhanced by a factor of $\sim
2.4$. We also check the high-D OAM entanglement with the broadened spiral-
bandwidth, and give a high-D entanglement properties with OAM quanta up to
$\Delta l=16$, see supplement.

With increasing the OAM quanta in DLCZ writing and reading processes, we
exhibit a potential application for multiplexing with different OAM modes. If
we select $l_{W}=10$ for writing and $l_{R}=-10$ for reading out, we can
detect the correlated coincidence; while for $l_{W}=10$, $l_{R}=0$ given in
Fig. 3(a), there would be almost no coincidence counts exhibiting
orthogonality-like property shown by Fig. 3(b). The nonlinearity of the
interleaved OAM modes is strongly dependent on the overlap of beam profiles of
write- and read-beams, it would become small if the mismatch between write-
and read-beams is large, then the coincidence counts would jump down, see
supplement. The detected contrast
$(C_{\text{same}}-C_{\text{diff}})/(C_{\text{same}}+C_{\text{diff}})$ is
$0.85\pm 0.03$ (where $C_{\text{same}}$ and $C_{\text{diff}}$ are defined as
coincidence counts with same ($|l_{W}|=|l_{R}|$) and different
($|l_{W}|\neq|l_{R}|$) write- and read-beam OAM modes), near to the
theoretical estimation of $0.91$. In this process, the storage coherence time
is almost same for $l_{W}=0$ and $l_{W}=10$, see Fig. 3(c). Furthermore, we
map the different OAM modes in the inner ($l_{in}$) and outer rings
($l_{out}$) to detect the multiplexing property along radial direction, see
Fig. 3(d). Since the nonlinearity of interleaved OAM modes (for example
$l_{W}=0$ and $l_{W}=10$) in the center of ensemble can be distinguished by
inputting distinct OAM modes (Fig. 3(e)), this may result in multiplexing
along radial direction. We map the different OAM modes with $\Delta
l=1,2\ldots 10$ in inner and outer rings ($\Delta l=l_{out}-l_{in}$) and
detect the correlation given in Fig. 3(e). The crosstalk between different OAM
modes is detected by setting same phase structure or the different phase
structure. The contrast of coincidence counts is increased against with
$\Delta l$ because the large difference $\Delta l$ means that the mismatch
between write- and read-beams decrease the correlation (Fig. 3(f)), agreeing
with the above analysis. This whole process could be regarded as the
multiplexing of two OAM spectra, which are created by inputting two distinct
OAM writing and reading ($l_{out}$, $l_{in}$), thus satisfying some quantum
information protocols with one OAM spectra docking to another OAM spectra.

At last, broad spiral-bandwidth offers an ability for demonstrating high-
quanta OAM quantum interface in 2-D subspace. For this, we set $l_{W}=30$,
$l_{R}=-30$, the storage time is set to be twice of the width of write pulse,
the decoherence from the transverse azimuthal momentum mismatch between write-
laser and the Signal 1 photons is ignored. The nonlinearity of the DLCZ
process at large quanta $l$ is relatively small, we then only consider the two
OAM modes for verifying entanglement. We choose the OAM modes of $l=28,32$ to
verify the high-quanta OAM entanglement. The photonic entangled state is
expressed as:

$\left|\psi\right\rangle_{photon-photon}^{{\rm{30}},{\rm{-30}}}{\rm{=}}{\raise
2.15277pt\hbox{${\scriptstyle{\rm{1}}}$}\kern-1.00006pt/\kern-1.49994pt\lower
1.07639pt\hbox{${\scriptstyle{\sqrt{2}}}$}}\left({{{\left|{-28}\right\rangle}_{{\rm{S1}}}}{{\left|{28}\right\rangle}_{{\rm{S2}}}}+{{\left|{-32}\right\rangle}_{{\rm{S1}}}}{{\left|{32}\right\rangle}_{{\rm{S2}}}}}\right)$
(2)

Through quantum state tomography, we obtain the reconstructed density matrix
as shown in Fig. 4(a) and (b). The fidelity of reconstructed density matrix is
calculated as $80.5\pm 4.8\%$ by comparing with the ideal density matrix. We
also check the violation of Clauser-Horne-Shimony-Holt (CHSH) inequality
(Clauser  _et al._ , 1969; Freedman and Clauser, 1972; Clauser and Horne,
1974) to demonstrate the nonlocality of the entangled state. The CHSH
parameter S (Leach  _et al._ , 2009) is represented as following:
$S=\left|E\left(\theta_{S2},\theta_{S1}\right)-E\left(\theta_{S2},\theta^{\prime}_{S1}\right){\rm+}E\left(\theta^{\prime}_{S2},\theta_{S1}\right){\rm+}E\left(\theta^{\prime}_{S2},\theta^{\prime}_{S1}\right)\right|$.
Here, the correlation function $E\left({{\theta_{S2}},{\theta_{S1}}}\right)$
can be calculated from the rates of coincidence at several particular
orientations, $\theta/\theta^{\prime}$ represents the angle of phase
distribution on the surface of SLM 2 and SLM 4. The calculated S is $2.22\pm
0.07$ which is larger than 2 violating the CHSH inequality, thus it
demonstrates the real entanglement between Signal 1 and Signal 2 photons.

Figure 4: The reconstructed density matrix for photonic entangled state
$\left|\psi\right\rangle_{photon-photon}^{{\rm{30}},{\rm{-30}}}$. (a) and (b)
are the real and imaginary parts of density matrix respectively. Each data for
reconstructing density matrices are recorded in 3000 s

In this work, we demonstrate a broad spiral-bandwidth OAM DLCZ memory, the OAM
distribution and the quanta of OAM quantum interface are freely manipulated.
In this state-of-the-art quantum interface, we have achieved high-D OAM
entanglement with OAM modes difference $\Delta l$ up to 16, the quanta of OAM
2-D subspace can be accessible to $l=\pm 30$. The experiment reported here
would be very promising to demonstrate high-quanta OAM quantum interface and
study the fundamental physics in OAM-based light and matter interaction.

### ACKNOWLEDGMENTS

Dong-Sheng Ding and Ming-Xin Dong contributed this paper equally. We thank
Guo-Yong Xiang professor for loaning a SLM. 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 Chinese Academy of
Sciences.

## Supplementary

### Experimental time sequence.

The repetition rate of our experiment is $100\,\mathrm{Hz}$, and the MOT
trapping time is 8.7 ms. Besides, the operation window of 1.3 ms consists of
2600 cycles with a cycle time of 500 ns. Write-laser and read-laser are pulsed
by acousto-optic modulator with pulse width of 50 ns and 200 ns respectively
in each cycle. The optical depth in MOT is about 40. The storage time is
controlled by changing the delay time between write- and read-laser through an
arbitrary function generator. The magnetic field for trapping is shut down in
the experiment window.

### 4-F image system for four SLMs.

The SLM 1 acts as a mask plane, and the center of atomic ensemble in MOT is
the image plane. Two lenses L1 and L2 with focal length of 300 mm and 500 mm
are utilized to map the phase message of SLM 1 to the atomic ensemble. Due to
the phase matching condition $k_{W}-k_{S1}=k_{R}-k_{S2}$, the imaging system
can be easily optically aligned. The Signal 1 and Signal 2 fields are
collinear, the Signal 1 beam is completely overlapped by the write beam
through demonstrating electromagnetically induced transparency effect. Here,
the write-laser carrying high OAM quanta diffracts very strongly and results
in the waist of laser beam too large in the center of atomic ensemble, which
results in weak interaction between write-laser and atomic ensemble. Through
the 4-f image system with unequal arms, we can not only map the OAM phase
message to the center of atomic ensemble accurately but also decrease the
waist of write-laser with high OAM quanta. Similarly, the single photon
carried with OAM phase message from the center of atomic ensemble is retrieved
to project on SLM 1 via the other 4-f image system, and ultimately we collect
photons by single-mode fibers.

### Theoretical analysis.

In the interaction picture, despite the decay of spin wave, the effective
Hamiltonian for the delayed four-wave mixing process is written as (Wen and
Rubin, 2006)

$\displaystyle{\hat{H}_{I}}$
$\displaystyle=\frac{{\varepsilon_{0}}}{4}\int_{-L/2}^{L/2}{dz{\chi^{(3)}}{{\vec{E}}_{W}}{{\vec{E}}_{R}}{{\vec{E}}^{*}}_{S1}{{\vec{E}}^{*}}_{S2}}+H.c$
(3)

where $H.c.$ means the Hermitian conjugate. ${\chi^{(3)}}$ is the third-order
nonlinear susceptibility for resonant signal 2 photon, which is given (Braje
_et al._ , 2004):

$\displaystyle{\chi^{(3)}}$
$\displaystyle=\frac{{N{\mu_{13}}{\mu_{32}}{\mu_{24}}{\mu_{41}}/({\varepsilon_{0}}{\hbar^{3}})}}{{({\Delta_{W}}+i{\gamma_{23}})[{{\left|{\Omega_{R}}\right|}^{2}}-4(\omega+i{\gamma_{24}})(\omega+i{\gamma_{21}})]}}$
(4)

here, ${\mu_{ij}}$ are the electric dipole matrix elements. ${\gamma_{ij}}$
are the dephasing rates. ${\Omega_{R}}$ is the Rabi frequency of read laser.
The probability to generate the Signal 1 and Signal 2 in modes
$|l_{S1}\rangle$, $|l_{S2}\rangle$ is given by the overlap with write- and
read-laser beam profiles:

$\begin{split}c_{l_{W}l_{R}l_{S1}l_{S2}}&\sim\int_{-L/2}^{L/2}\int_{0}^{r}\int_{0}^{2\pi}\varepsilon_{0}\chi^{(3)}rLG_{0}^{l_{W}}(r,\phi)LG_{0}^{l_{R}}(r,\phi)\\\
&LG_{0}^{l_{W}}(r,\phi)LG_{0}^{l_{R}}(r,\phi)[LG_{0}^{l_{S1}}(r,\phi)]^{*}[LG_{0}^{l_{S2}}(r,\phi)]^{*}d\phi
drdz\end{split}$ (5)

The integral over the azimuthal coordinate is

$\displaystyle\int_{0}^{2\pi}{d\phi}exp[i({l_{W}}+{l_{R}}-{l_{S1}}-{l_{S2}})\phi]$
$\displaystyle=2\pi{\delta_{{l_{W}}+{l_{R}},{l_{S1}}+{l_{S2}}}}$ (6)

From which, we can obtain the topological charge conservation law in OAM space
is ${l_{W}}+{l_{R}}={l_{S1}}+{l_{S2}}$. According to Eq. (4), we can find the
probability of ${l_{S1}}$-Signal 1 and ${l_{S2}}$-Signal 2 photons with
${l_{W}}$-write and ${l_{R}}$-read lasers, which strongly depends on the
profiles match between the four fields.

In order to illustrate the topological charge conservation law in our OAM
quantum interface in DLCZ memory, we input the write-laser with OAM quanta of
$l_{W}$. Due to the fact that SRS process conserves angular momentum, we have
created OAM entanglement between Signal 1 and atomic spin wave, which can be
specified by the formula $\left|\psi\right\rangle_{{}_{photon-
atom}}^{{l_{W}}}{\rm{=}}\sum\nolimits_{l=-\infty}^{l=\infty}{c_{l}}{\left|l\right\rangle_{{\rm{S1}}}}\otimes{\left|{{l_{W}}-l}\right\rangle_{a}}$,
here, ${\left|{c_{l}}\right|^{2}}$ represents excitation probability,
${\left|l\right\rangle_{{\rm{S1}}}}$ is the OAM eigenmode of Signal 1 with
quanta of $l$. ${\left|{{l_{W}}-l}\right\rangle_{a}}$ is the OAM eigenmode of
atomic spin wave with quanta of ${l_{W}}-l$. Through this method, the atomic
spin wave could carry the arbitrary OAM topological charge with the term of
${l_{W}}-l$, thus resulting in the redistributed quantum interface.

Figure 5: Reconstructed density matrices for Modulated OAM entanglement. The
real (a,c) and imaginary (b, d) parts of density matrices for photonic OAM
entangled state $\left|\psi\right\rangle_{photon-photon}^{2,0}$ and
$\left|\psi\right\rangle_{photon-photon}^{1,2}$. Each data for reconstructing
density matrices are recorded in 1000 s.

After a period of storage, we check photon-atom entanglement by inputting
read-laser with OAM quanta of $l_{R}$, and checking the entanglement between
Signal 1 and Signal 2. The entanglement between Signal 1 and Signal 2 can be
written as $\left|\psi\right\rangle_{photon-
photon}^{{l_{W}},{l_{R}}}{\rm{=}}\sum\nolimits_{l=-\infty}^{l=\infty}{c_{l}}{\left|l\right\rangle_{{\rm{S1}}}}\otimes{\left|{{l_{W}}+{l_{R}}-l}\right\rangle_{{\rm{S2}}}}$.
At first, we set $l_{W}=2$ and $l_{R}=0$, it means using OAM quanta of 2 and 0
to write and read respectively. Thus, the photonic entangled state is a sum of
${\left|l\right\rangle_{{\rm{S1}}}}\otimes{\left|{2-l}\right\rangle_{{\rm{S2}}}}$
with different $l$, this is a modulated asymmetric OAM entangled state. Here,
we only post-select the OAM mode of entangled state into two-dimensional
subspace ${\left|0\right\rangle_{{\rm{S1}}}}{\left|2\right\rangle_{S2}}$ and
${\left|2\right\rangle_{{\rm{S1}}}}{\left|0\right\rangle_{S2}}$, that is
$\left|\psi\right\rangle_{photon-photon}^{2,0}{\rm{=}}{\raise
2.15277pt\hbox{${\scriptstyle 1}$}\kern-1.00006pt/\kern-1.49994pt\lower
1.07639pt\hbox{${\scriptstyle{\sqrt{2}}}$}}\left({{{\left|0\right\rangle}_{{\rm{S1}}}}{{\left|2\right\rangle}_{{\rm{S2}}}}+{{\left|2\right\rangle}_{{\rm{S1}}}}{{\left|0\right\rangle}_{{\rm{S2}}}}}\right)$.
To characterize the OAM entanglement between Signal 1 and Signal 2, we
reconstruct the density matrices by projecting Signal 1 and Signal 2 onto OAM
bases of $\left|0\right\rangle$, $\left|2\right\rangle$,
${{\left({\left|0\right\rangle-i\left|2\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|0\right\rangle-i\left|2\right\rangle}\right)}{2^{1/2}}}}\right.\kern-1.2pt}{2^{1/2}}}$,
${{\left({\left|0\right\rangle+\left|2\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|0\right\rangle+\left|2\right\rangle}\right)}{2^{1/2}}}}\right.\kern-1.2pt}{2^{1/2}}}$
for demonstrating quantum tomography. Then we use the obtained 16 coincidence
rates to reconstruct the density matrix of state as shown in Fig. 5 (a) and
(b). According to the formula
$F{\rm{=Tr(}}\sqrt{\sqrt{\rho}{\rho_{{\rm{ideal}}}}\sqrt{\rho}}{{\rm{)}}^{2}}$,
which compares the constructed density matrix $\rho$ with the ideal density
matrix ${\rho_{{\rm{ideal}}}}$, we obtain the fidelity of $83.3\pm 3.5$%. We
also try another data set of $m_{1}=1$ and $m_{2}=2$, and obtain the photonic
entangled state $\left|\psi\right\rangle_{photon-photon}^{1,2}{\rm{=}}{\raise
2.15277pt\hbox{${\scriptstyle 1}$}\kern-1.00006pt/\kern-1.49994pt\lower
1.07639pt\hbox{${\scriptstyle{\sqrt{2}}}$}}\left({{{\left|0\right\rangle}_{{\rm{S1}}}}{{\left|3\right\rangle}_{{\rm{S2}}}}+{{\left|3\right\rangle}_{{\rm{S1}}}}{{\left|0\right\rangle}_{{\rm{S2}}}}}\right)$.
Similarly, we reconstruct the density matrix of this state, the real and
imaginary parts of reconstructed density matrix are shown in Fig. 5 (c) and
(d), with fidelity of $81.1\pm 4.2$%. In this process, although the fidelity
is not very high, but it reveals that in DLCZ quantum memory, the OAM modes
are conserved in the whole writing and reading process.

### The entanglement dimensionality witness.

In order to demonstrate the high-D entanglement between Signal 1 and atomic
memory, we avoid the crosstalk between neighboring OAM modes. We select the
modes of $l=0,4,8,12,16$ in which three modes between adjacent terms are
removed for better isolation. We read the photon-atom entanglement out to
photon-photon entanglement for verification. So, the entangled state is
$\left|\psi\right\rangle_{photon-
photon}^{{\rm{10}},-{\rm{1}}0}{\rm{=}}{c_{1}}{\left|0\right\rangle_{{\rm{S1}}}}{\left|0\right\rangle_{{\rm{S2}}}}+{c_{2}}{\left|{-4}\right\rangle_{{\rm{S1}}}}{\left|4\right\rangle_{{\rm{S2}}}}+{c_{3}}{\left|{-8}\right\rangle_{{\rm{S1}}}}{\left|8\right\rangle_{{\rm{S2}}}}+{c_{4}}{\left|{-12}\right\rangle_{{\rm{S1}}}}{\left|{12}\right\rangle_{{\rm{S2}}}}+{c_{5}}{\left|{-16}\right\rangle_{{\rm{S1}}}}{\left|{16}\right\rangle_{{\rm{S2}}}}$here,
${c_{5}}\sim{c_{5}}$ are the corresponding amplitudes of different terms
${\left|0\right\rangle_{{\rm{S1}}}}{\left|0\right\rangle_{{\rm{S2}}}}\sim{\left|{\rm{-16}}\right\rangle_{{\rm{S1}}}}{\left|{\rm{16}}\right\rangle_{{\rm{S2}}}}$.
For verifying the high-D state, it is very promising to use high-D
entanglement dimensionality witness Agnew  _et al._ (2012); Krenn  _et al._
(2014) to characterize the entanglement existed in our system. The
entanglement dimensionality witness is expressed as
${W_{d}}=3\frac{{D(D-1)}}{2}-D(D-d)$, here, $D$ is the number of measured OAM
modes, and $d$ is associated with dimensions of entanglement. If $W>{W_{d}}$,
the two photons entangled in at least $d+1$ dimensions, where $W$ is obtained
from calculating the sum of visibilities $N=V_{x}+V_{y}+V_{z}$ in total two
dimensional subspace. The $V_{x}$, $V_{y}$ and $V_{z}$ represent the
visibilities of two-photon interference in the diagonal/anti-diagonal, left-
circular/right-circular and horizontal/vertical bases respectively in each OAM
mode of $a$ and $b$, here $a$ and $b$ are selected from $l=0,4,8,12,16$. A
disadvantage of quantum tomography for high-D entanglement is that the needed
measurement data is the order of $d^{4}$, which is a large challenge in its
realization and is impractical while $d$ is set to 5 in our experiment.
Therefore, we adopt the method of dimensionality witness to certificate the
existence of high-D entanglement and characterize the dimensionality. We
calculate the value $W$ is 21.93±0.55, which violates the bound $W_{d}$ of 20
(the number of measured OAM mode $D$ is 5 and $d$ is 3) by 3 s.d’s, thus there
is at least a 4-D OAM entanglement between Signal 1 and Signal 2 photons. In
these measurements, the atom-photon entangled states are both detected in
photonic regime, we assume the fidelity of reading out from ensembles is near
unit. Although there are definitely some noise or inefficient elements from
reading process, making the degree of the measured entanglement lower than
that existed in ensembles.

Figure 6: (a) The post-selected correlated OAM matrix between Signal 1 and
Signal 2 photons with OAM modes difference $|\Delta l|$ up to 16. (b) The each
sum of visibilities for 2-D subspaces for detecting the high-D entanglement
dimensionality witness.

### 2-D high-$l$ Entanglement and state tomography

If we considered the OAM modes of $a$ and $b$ with $l$=32 and 28, the Signal 1
and Signal 2 are entangled in OAM space and entangled state is expressed as

$\left|{\Psi_{3}}\right\rangle{\rm{=}}{\raise
2.15277pt\hbox{${\scriptstyle{\rm{1}}}$}\kern-1.00006pt/\kern-1.49994pt\lower
1.07639pt\hbox{${\scriptstyle{\sqrt{{\rm{2}}}}}$}}\left({{{\left|{-28}\right\rangle}_{{\rm{S1}}}}{{\left|{28}\right\rangle}_{{\rm{S2}}}}+{{\left|{-32}\right\rangle}_{{\rm{S1}}}}{{\left|{32}\right\rangle}_{{\rm{S2}}}}}\right)$
(7)

Here, ${\left|{-28}\right\rangle_{{\rm{S1}}}}$ represents the Signal 1
carrying with OAM quanta of -28. By using two computers, we project two
photons onto two SLMs respectively and four state of $\left|{\phi_{1\sim
4}}\right\rangle$ ($\left|{-28}\right\rangle$, $\left|{-32}\right\rangle$,
${{\left({\left|{-28}\right\rangle-i\left|{-32}\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|{-28}\right\rangle-i\left|{-32}\right\rangle}\right)}{2^{1/2}}}}\right.\kern-1.2pt}{2^{1/2}}}$,
${{\left({\left|{-28}\right\rangle+\left|{-32}\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|{-28}\right\rangle+\left|{-32}\right\rangle}\right)}{2^{1/2}}}}\right.\kern-1.2pt}{2^{1/2}}}$)
are programmed onto SLM 2 and $\left|{\varphi_{1\sim 4}}\right\rangle$
($\left|{28}\right\rangle$, $\left|{32}\right\rangle$,
${{\left({\left|{28}\right\rangle-i\left|{32}\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|{28}\right\rangle-i\left|{32}\right\rangle}\right)}{2^{1/2}}}}\right.\kern-1.2pt}{2^{1/2}}}$,
${{\left({\left|{28}\right\rangle+\left|{32}\right\rangle}\right)}\mathord{\left/{\vphantom{{\left({\left|{28}\right\rangle+\left|{32}\right\rangle}\right)}{2^{1/2}}}}\right.\kern-1.2pt}{2^{1/2}}}$)
are programmed onto SLM 4. Then, we obtain a set of 16 data for reconstructing
the density matrix given in the main text. The error bars in our experiment
are estimated by Poisson statistics and using Monte Carlo simulations with the
aid of Mathematica software.

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