Quantum Simulation of the First-Quantized Pauli-Fierz Hamiltonian

1]Priyanka Mukhopadhyay<EMAIL_ADDRESS><EMAIL_ADDRESS>2,3]Torin F<EMAIL_ADDRESS>4,5,6]Nathan Wiebe<EMAIL_ADDRESS>
[1]Department of Physical & Environmental Sciences, University of Toronto, ON, Canada
[2]Simons Institute for the Theory of Computing, Berkeley, CA, USA
[3]Berkeley Quantum Information and Computation Center,
University of California, Berkeley, USA
[4]Department of Computer Science, University of Toronto, ON, Canada
[5]Pacific Northwest National Laboratory, Richland, WA, USA
[6]Canadian Institute for Advanced Research, Toronto, ON, Canada

We provide an explicit recursive divide and conquer approach for simulating quantum dynamics and derive a discrete first quantized non-relativistic QED Hamiltonian based on the many-particle Pauli Fierz Hamiltonian.  We apply this recursive divide and conquer algorithm to this Hamiltonian and compare it to a concrete simulation algorithm that uses qubitization.  Our divide and conquer algorithm, using lowest order Trotterization, scales for fixed grid spacing as $\widetilde{O}(\Lambda N^2\eta^2 t^2 /\epsilon)$ for grid size $N$, $\eta$ particles, simulation time $t$, field cutoff $\Lambda$ and error $\epsilon$.  Our qubitization algorithm scales as $\widetilde{O}(N(\eta+N)(\eta +\Lambda^2) t\log(1/\epsilon)) $.  This shows that even a naïve partitioning and low-order splitting formula can yield, through our divide and conquer formalism, superior scaling to qubitization for large $\Lambda$.  We compare the relative costs of these two algorithms on systems that are relevant for applications such as the spontaneous emission of photons, and the photoionization of electrons. We observe that for different parameter regimes, one method can be favored over the other. Finally, we give new algorithmic and circuit level techniques for gate optimization including a new way of implementing a group of multi-controlled-X gates that can be used for better analysis of  circuit cost.

§ ACKNOWLEDGEMENTS

T.F.S. is a Quantum Postdoctoral Fellow at the Simons Institute for the Theory of Computing, supported by the U.S. Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator.  N.W. and P.M. acknowledge funding from
“Embedding QC into Many-body Frameworks for Strongly Correlated Molecular and Materials Systems” project, which is funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, the Division of Chemical Sciences, Geosciences, and Biosciences. The Pacific Northwest National Laboratory is operated by Battelle for the U.S. Department of Energy under Contract DE-AC05-76RL01830.  P.M. and N.W. further acknowledge support from Google inc.

[1]
Matthew Amy, Parsiad Azimzadeh, and Michele Mosca.
On the controlled-NOT complexity of controlled-NOT–phase circuits.
Quantum Science and Technology, 4(1):015002, 2018.

[2]
Israel F Araujo, Daniel K Park, Francesco Petruccione, and Adenilton J
da Silva.
A divide-and-conquer algorithm for quantum state preparation.
Scientific Reports, 11(1):1–12, 2021.

[3]
Dominic W Berry, Graeme Ahokas, Richard Cleve, and Barry C Sanders.
Efficient quantum algorithms for simulating sparse Hamiltonians.
Communications in Mathematical Physics, 270(2):359–371, 2007.

[4]
Bela Bauer, Sergey Bravyi, Mario Motta, and Garnet Kin-Lic Chan.
Quantum algorithms for quantum chemistry and quantum materials science.
Chemical Reviews, 120(22):12685–12717, 2020.

[5]
Ryan Babbush, Dominic W Berry, Jarrod R McClean, and Hartmut Neven.
Quantum simulation of chemistry with sublinear scaling in basis size.
npj Quantum Information, 5(1):1–7, 2019.

[6]
Dominic W Berry, Andrew M Childs, Richard Cleve, Robin Kothari, and Rolando D
Simulating Hamiltonian dynamics with a truncated Taylor series.
Physical Review Letters, 114(9):090502, 2015.

[7]
Oliver Benson.
Assembly of hybrid photonic architectures from nanophotonic constituents.
Nature, 480(7376):193–199, 2011.

[8]
Ryan Babbush, Craig Gidney, Dominic W Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, and Hartmut Neven.
Encoding electronic spectra in quantum circuits with linear T complexity.
Physical Review X, 8(4):041015, 2018.

[9]
Rami Barends, L Lamata, Julian Kelly, L García-Álvarez, Austin G Fowler, A Megrant, Evan Jeffrey, Ted C White, Daniel Sank, Josh Y Mutus, et al.
Digital quantum simulation of fermionic models with a superconducting circuit.
Nature Communications, 6(1):7654, 2015.

[10]
Jan Conrad Baggesen and Lars Bojer Madsen.
Polarization effects in attosecond photoelectron spectroscopy.
Physical Review Letters, 104(4):043602, 2010.

[11]
Alex Bocharov, Martin Roetteler, and Krysta M Svore.
Efficient synthesis of universal repeat-until-success quantum circuits.
Physical Review Letters, 114(8):080502, 2015.

[12]
Steven A Cuccaro, Thomas G Draper, Samuel A Kutin, and David Petrie Moulton.
A new quantum ripple-carry addition circuit.
arXiv preprint quant-ph/0410184, 2004.

[13]
Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein.
Introduction to algorithms.
MIT press, 2022.

[14]
Jorge Casanova, Antonio Mezzacapo, Lucas Lamata, and Enrique Solano.
Quantum simulation of interacting fermion lattice models in trapped ions.
Physical Review Letters, 108(19):190502, 2012.

[15]
Andrew M Childs, Yuan Su, Minh C Tran, Nathan Wiebe, and Shuchen Zhu.
Theory of Trotter error with commutator scaling.
Physical Review X, 11(1):011020, 2021.

[16]
Andrew M Childs and Nathan Wiebe.
Hamiltonian simulation using linear combinations of unitary operations.
Quantum Information & Computation, 12(11-12):901–924, 2012.

[17]
Andrew J Daley, Immanuel Bloch, Christian Kokail, Stuart Flannigan, Natalie Pearson, Matthias Troyer, and Peter Zoller.
Practical quantum advantage in quantum simulation.
Nature, 607(7920):667–676, 2022.

[18]
Roberto Dovesi, Bartolomeo Civalleri, Carla Roetti, Victor R Saunders, and Roberto Orlando.
Ab initio quantum simulation in solid state chemistry.
Reviews in Computational Chemistry, 21:1–125, 2005.

[19]
Thomas G Draper.
Addition on a quantum computer.
arXiv preprint quant-ph/0008033, 2000.

[20]
Richard P Feynman et al.
Simulating physics with computers.
International Journal Theoretical Physics, 21(6/7):467–488, 1982.

[21]
Thomas Faulkner, Thomas Hartman, Matthew Headrick, Mukund Rangamani, and Brian Swingle.
Snowmass white paper: Quantum information in quantum field theory and quantum gravity.
arXiv preprint arXiv:2203.07117, 2022.

[22]
Johannes Flick, Michael Ruggenthaler, Heiko Appel, and Angel Rubio.
Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (qed) chemistry.
Proceedings of the National Academy of Sciences,
114(12):3026–3034, 2017.

[23]
Johannes Feist, Oleg Zatsarinny, Stefan Nagele, Renate Pazourek, Joachim Burgdörfer, Xiaoxu Guan, Klaus Bartschat, and Barry I Schneider.
Time delays for attosecond streaking in photoionization of neon.
Physical Review A, 89(3):033417, 2014.

[24]
L García-Álvarez, IL Egusquiza, Lucas Lamata, Adolfo Del Campo, Julian Sonner, and Enrique Solano.
Digital quantum simulation of minimal AdS/CFT.
Physical Review Letters, 119(4):040501, 2017.

[25]
Iulia M Georgescu, Sahel Ashhab, and Franco Nori.
Quantum simulation.
Reviews of Modern Physics, 86(1):153, 2014.

[26]
Vlad Gheorghiu, Jiaxin Huang, Sarah Meng Li, Michele Mosca, and Priyanka Mukhopadhyay.
Reducing the CNOT count for Clifford+T circuits on NISQ
IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2022.

[27]
Craig Gidney.
Halving the cost of quantum addition.
Quantum, 2:74, 2018.

[28]
Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone.
Architectures for a quantum random access memory.
Physical Review A, 78(5):052310, 2008.

[29]
Vlad Gheorghiu, Michele Mosca, and Priyanka Mukhopadhyay.
T-count and T-depth of any multi-qubit unitary.
npj Quantum Information, 8(1):1–10, 2022.

[30]
András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe.
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics.
In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 193–204, 2019.

[31]
Jacky Huyghebaert and Hans De Raedt.
Product formula methods for time-dependent Schrodinger problems.
Journal of Physics A: Mathematical and General, 23(24):5777, 1990.

[32]
Jeongwan Haah, Matthew B Hastings, Robin Kothari, and Guang Hao Low.
Quantum algorithm for simulating real time evolution of lattice Hamiltonians.
SIAM Journal on Computing, (0):FOCS18–250, 2021.

[33]
Yong He, Ming-Xing Luo, E Zhang, Hong-Ke Wang, and Xiao-Feng Wang.
Decompositions of n-qubit Toffoli gates with linear circuit complexity.
International Journal of Theoretical Physics, 56(7):2350–2361, 2017.

[34]
Kade Head-Marsden, Johannes Flick, Christopher J Ciccarino, and Prineha Narang.
Quantum information and algorithms for correlated quantum matter.
Chemical Reviews, 121(5):3061–3120, 2020.

[35]
Stuart Hadfield and Anargyros Papageorgiou.
Divide and conquer approach to quantum Hamiltonian simulation.
New Journal of Physics, 20(4):043003, 2018.

[36]
Walter Hofstetter and Tao Qin.
Quantum simulation of strongly correlated condensed matter systems.
Journal of Physics B: Atomic, Molecular and Optical Physics, 51(8):082001, 2018.

[37]
Cody Jones.
Low-overhead constructions for the fault-tolerant Toffoli gate.
Physical Review A, 87(2):022328, 2013.

[38]
AS Kheifets and IA Ivanov.
Delay in atomic photoionization.
Physical Review Letters, 105(23):233002, 2010.

[39]
Vadym Kliuchnikov, Dmitri Maslov, and Michele Mosca.
Practical approximation of single-qubit unitaries by single-qubit quantum Clifford and T circuits.
IEEE Transactions on Computers, 65(1):161–172, 2015.

[40]
Christian Kokail, Christine Maier, Rick van Bijnen, Tiff Brydges, Manoj K Joshi, Petar Jurcevic, Christine A Muschik, Pietro Silvi, Rainer Blatt, Christian F Roos, et al.
Self-verifying variational quantum simulation of lattice models.
Nature, 569(7756):355–360, 2019.

[41]
Natalie Klco, Alessandro Roggero, and Martin J Savage.
Standard model physics and the digital quantum revolution: thoughts about the interface.
Reports on Progress in Physics, 2022.

[42]
Ian D Kivlichan, Nathan Wiebe, Ryan Babbush, and Alán Aspuru-Guzik.
Bounding the costs of quantum simulation of many-body physics in real space.
Journal of Physics A: Mathematical and Theoretical,
50(30):305301, 2017.

[43]
Alan J Laub.
Matrix analysis for scientists and engineers, volume 91.
Siam, 2005.

[44]
Guang Hao Low and Isaac L Chuang.
Optimal Hamiltonian simulation by quantum signal processing.
Physical Review Letters, 118(1):010501, 2017.

[45]
Guang Hao Low and Isaac L Chuang.
Hamiltonian simulation by qubitization.
Quantum, 3:163, 2019.

[46]
Jianping Li.
General explicit difference formulas for numerical differentiation.
Journal of Computational and Applied Mathematics,
183(1):29–52, 2005.

[47]
Guang Hao Low, Yuan Su, Yu Tong, and Minh C Tran.
Complexity of implementing Trotter steps.
PRX Quantum, 4(2):020323, 2023.

[48]
Junyu Liu and Yuan Xin.
Quantum simulation of quantum field theories as quantum chemistry.
Journal of High Energy Physics, 2020(12):1–48, 2020.

[49]
Wayne C Myrvold, Joy Christian, and Lucien Hardy.
Quantum gravity computers: On the theory of computation with indefinite causal structure.
Quantum reality, relativistic causality, and closing the epistemic circle: essays in honour of Abner Shimony, pages 379–401, 2009.

[50]
H Mabuchi and AC Doherty.
Cavity quantum electrodynamics: coherence in context.
Science, 298(5597):1372–1377, 2002.

[51]
LR Moore, MA Lysaght, JS Parker, HW Van Der Hart, and KT Taylor.
Time delay between photoemission from the 2$p$ and 2$s$ subshells of neon.
Physical Review A, 84(6):061404, 2011.

[52]
Michele Mosca and Priyanka Mukhopadhyay.
A polynomial time and space heuristic algorithm for T-count.
Quantum Science and Technology, 7(1):015003, 2021.

[53]
David Marcos, Peter Rabl, Enrique Rico, and Peter Zoller.
Superconducting circuits for quantum simulation of dynamical gauge fields.
Physical Review Letters, 111(11):110504, 2013.

[54]
Mikko Möttönen, Juha J Vartiainen, Ville Bergholm, and Martti M Salomaa.
Transformation of quantum states using uniformly controlled rotations.
Quantum Information & Computation, 5(6):467–473, 2005.

[55]
Priyanka Mukhopadhyay, Nathan Wiebe, and Hong Tao Zhang.
Synthesizing efficient circuits for Hamiltonian simulation.
npj Quantum Information, 9(1):31, 2023.

[56]
Michael A Nielsen and Isaac L Chuang.
Quantum Computation and Quantum Information.
Cambridge University Press, 2010.

[57]
Philipp Niemann, Rhitam Datta, and Robert Wille.
Logic synthesis for quantum state generation.
In 2016 IEEE 46th International Symposium on Multiple-Valued Logic (ISMVL), pages 247–252. IEEE, 2016.

[58]
Benjamin Nachman, Davide Provasoli, Wibe A De Jong, and Christian W Bauer.
Quantum algorithm for high energy physics simulations.
Physical Review Letters, 126(6):062001, 2021.

[59]
Stefan Nagele, Renate Pazourek, Johannes Feist, Katharina Doblhoff-Dier, Christoph Lemell, Karoly Tőkési, and Joachim Burgdörfer.
Time-resolved photoemission by attosecond streaking: extraction of time information.
Journal of Physics B: Atomic, Molecular and Optical Physics, 44(8):081001, 2011.

[60]
Yunseong Nam, Yuan Su, and Dmitri Maslov.
Approximate quantum fourier transform with $O(n\log (n))$ T gates.
npj Quantum Information, 6(1):1–6, 2020.

[61]
Juan J Omiste and Lars Bojer Madsen.
Attosecond photoionization dynamics in neon.
Physical Review A, 97(1):013422, 2018.

[62]
Juan J Omiste and Lars Bojer Madsen.
Photoionization of aligned excited states in neon by attosecond laser  pulses.
Journal of Physics B: Atomic, Molecular and Optical Physics, 54(5):054001, 2021.

[63]
Martin Plesch and Časlav Brukner.
Quantum-state preparation with universal gate decompositions.
Physical Review A, 83(3):032302, 2011.

[64]
Renate Pazourek, Johannes Feist, Stefan Nagele, and Joachim Burgdörfer.
Attosecond streaking of correlated two-electron transitions in  helium.
Physical Review Letters, 108(16):163001, 2012.

[65]
Ketan N Patel, Igor L Markov, and John P Hayes.
Optimal synthesis of linear reversible circuits.
Quantum Information & Computation, 8(3):282–294, 2008.

[66]
Renate Pazourek, Stefan Nagele, and Joachim Burgdörfer.
Attosecond chronoscopy of photoemission.
Reviews of Modern Physics, 87(3):765, 2015.

[67]
Helmut Ritsch, Peter Domokos, Ferdinand Brennecke, and Tilman Esslinger.
Cold atoms in cavity-generated dynamical optical potentials.
Reviews of Modern Physics, 85(2):553, 2013.

[68]
Krupa Ramasesha, Stephen R Leone, and Daniel M Neumark.
Real-time probing of electron dynamics using attosecond time-resolved spectroscopy.
Annual Review of Physical Chemistry, 67:41–63, 2016.

[69]
Lidia Ruiz-Perez and Juan Carlos Garcia-Escartin.
Quantum arithmetic with the Quantum Fourier Transform.
Quantum Information Processing, 16(6):1–14, 2017.

[70]
Abhishek Rajput, Alessandro Roggero, and Nathan Wiebe.
Hybridized methods for quantum simulation in the interaction picture.
Quantum, 6:780, 2022.

[71]
Neil J Ross and Peter Selinger.
Optimal ancilla-free Clifford+T approximation of Z-rotations.
Quantum Information & Computation, 16(11&12):901–953, 2016.

[72]
Yuan Su, Dominic W Berry, Nathan Wiebe, Nicholas Rubin, and Ryan Babbush.
Fault-tolerant quantum simulations of chemistry in first
PRX Quantum, 2(4):040332, 2021.

[73]
Martin Schultze, Markus Fieß, Nicholas Karpowicz, Justin Gagnon, Michael Korbman, Michael Hofstetter, S Neppl, Adrian L Cavalieri, Yannis Komninos, Th Mercouris, et al.
Delay in photoemission.
Science, 328(5986):1658–1662, 2010.

[74]
Alexander F Shaw, Pavel Lougovski, Jesse R Stryker, and Nathan Wiebe.
Quantum algorithms for simulating the lattice Schwinger model.
Quantum, 4:306, 2020.

[75]
Maria Schuld and Francesco Petruccione.
Machine learning with quantum computers.
Springer, 2021.

[76]
Herbert Spohn.
Dynamics of charged particles and their radiation field.
Cambridge University Press, 2004.

[77]
Illya Shapoval, Vincent Paul Su, Wibe de Jong, Miro Urbanek, and Brian Swingle.
Towards quantum gravity in the lab on quantum processors.
arXiv preprint arXiv:2205.14081, 2022.

[78]
Masuo Suzuki.
General theory of fractal path integrals with applications to many-body theories and statistical physics.
Journal of Mathematical Physics, 32(2):400–407, 1991.

[79]
Yu Tong, Victor V Albert, Jarrod R McClean, John Preskill, and Yuan Su.
Provably accurate simulation of gauge theories and bosonic systems.
Quantum, 6:816, 2022.

[80]
Teague Tomesh, Nicholas Allen, and Zain Saleem.
Quantum-classical tradeoffs and multi-controlled quantum gate decompositions in variational algorithms.
arXiv preprint arXiv:2210.04378, 2022.

[81]
Jimmy Vinbladh, Jan Marcus Dahlström, and Eva Lindroth.
Many-body calculations of two-photon, two-color matrix elements for attosecond delays.
Physical Review A, 100(4):043424, 2019.

[82]
Reinier van der Meer, Zichang Huang, Malaquias Correa Anguita, Dongxue Qu, Peter Hooijschuur, Hongguang Liu, Muxin Han, Jelmer J Renema, and Lior Cohen.
Experimental simulation of loop quantum gravity on a photonic chip.
npj Quantum Information, 9(1):32, 2023.

[83]
Nathan Wiebe, Dominic Berry, Peter Høyer, and Barry C Sanders.
Higher order decompositions of ordered operator exponentials.
Journal of Physics A: Mathematical and Theoretical,
43(6):065203, 2010.

[84]
Dave Wecker, Matthew B Hastings, Nathan Wiebe, Bryan K Clark, Chetan Nayak, and Matthias Troyer.
Solving strongly correlated electron models on a quantum computer.
Physical Review A, 92(6):062318, 2015.

[85]
Xiaoling Wu, Xinhui Liang, Yaoqi Tian, Fan Yang, Cheng Chen, Yong-Chun Liu, Meng Khoon Tey, and Li You.
A concise review of Rydberg atom based quantum computation and quantum simulation.
Chinese Physics B, 30(2):020305, 2021.

[86]
Xiao Yuan, Suguru Endo, Qi Zhao, Ying Li, and Simon C Benjamin.
Theory of variational quantum simulation.
Quantum, 3:191, 2019.

[87]
Xiao Yuan, Jinzhao Sun, Junyu Liu, Qi Zhao, and You Zhou.
Quantum simulation with hybrid tensor networks.
Physical Review Letters, 127(4):040501, 2021.

[88]
Erez Zohar, J Ignacio Cirac, and Benni Reznik.
Quantum simulations of lattice gauge theories using ultracold atoms in optical lattices.
Reports on Progress in Physics, 79(1):014401, 2015.


# A long-lived solid-state optical quantum memory for high-rate quantum
repeaters

Mohsen Falamarzi Askarani  These authors contributed equally to this work.
QuTech and Kavli Institute of Nanoscience, Delft University of Technology,
2628 CJ Delft, The Netherlands  Antariksha Das∗ QuTech and Kavli Institute of
Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands
Jacob H. Davidson  QuTech and Kavli Institute of Nanoscience, Delft University
of Technology, 2628 CJ Delft, The Netherlands  Gustavo C. Amaral  QuTech and
Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft,
The Netherlands  Neil Sinclair  John A. Paulson School of Engineering and
Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
Division of Physics, Mathematics and Astronomy, and Alliance for Quantum
Technologies (AQT), California Institute of Technology, Pasadena, California
91125, USA  Joshua A. Slater  QuTech and Kavli Institute of Nanoscience, Delft
University of Technology, 2628 CJ Delft, The Netherlands  Sara Marzban  QuTech
and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ
Delft, The Netherlands  Charles W. Thiel  Department of Physics, Montana State
University, Bozeman, MT, USA  Rufus L. Cone  Department of Physics, Montana
State University, Bozeman, MT, USA  Daniel Oblak  Institute for Quantum
Science and Technology, and Department of Physics & Astronomy, University of
Calgary, Calgary, Alberta, T2N 1N4, Canada  Wolfgang Tittel  QuTech and Kavli
Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The
Netherlands

###### Abstract

We argue that long optical storage times are required to establish
entanglement at high rates over large distances using memory-based quantum
repeaters. Triggered by this conclusion, we investigate the 3H6
$\leftrightarrow$ 3H4 transition at 795.325 nm of Tm:Y3Ga5O12 (Tm:YGG). Most
importantly, we show that the optical coherence time can reach 1.1 ms, and,
using laser pulses, we demonstrate optical storage based on the atomic
frequency comb protocol up to 100 $\mu$s as well as a memory decay time TM of
13.1 $\mu$s. Possibilities of how to narrow the gap between the measured value
of Tm and its maximum of 275 $\mu$s are discussed. In addition, we demonstrate
quantum state storage using members of non-classical photon pairs. Our results
show the potential of Tm:YGG for creating quantum memories with long optical
storage times, and open the path to building extended quantum networks.

_Introduction –_ The future quantum internet Kimble (2008); Wehner  _et al._
(2018) will enable one to share entanglement and hence quantum information
over large distances – ultimately between any two points on earth. To overcome
attenuation in optical fibers and enable quantum communication in a scalable
manner, quantum repeaters are needed Briegel  _et al._ (1998); Duan  _et al._
(2001); Sangouard  _et al._ (2011); Sinclair  _et al._ (2014); Jiang  _et al._
(2009).

Optical quantum memories Lvovsky  _et al._ (2009) are vital building blocks in
many quantum repeater architectures Jiang  _et al._ (2009). They allow storing
qubits, encoded into photons that have travelled over long distances, until
feed-forward information becomes available. This information specifies which
optical mode—including spectral and temporal modes—a stored qubit should
occupy once it has been re-emitted from the memory. Note that the required
mode assignment (also referred-to as mode mapping operation) can happen in a
memory-internal manner, e.g. by controlling the moment at which a photon is
re-emitted (aka read-out on demand) Fleischhauer and Lukin (2000); Duan  _et
al._ (2001); Kraus  _et al._ (2006); Afzelius  _et al._ (2010), or externally,
e.g. by shifting the spectrum of the emitted photon Sinclair  _et al._ (2014);
Grimau Puigibert  _et al._ (2017).

To increase the rate at which a quantum repeater distributes entanglement, it
is important that qubits can be added continuously to the memory – not only
once a previously stored qubit has been re-emitted but also while it is still
being stored. Such multiplexed storage implies (a) the use of ensembles of
absorbers and (b) that any memory-specific control operation, triggered by the
absorption of a newly arriving qubit, must neither affect re-emission nor the
possibility for mode mapping of a previously absorbed qubit. Stated
differently, any control operation required after absorption of a qubit or a
train of qubits must not introduce deadtime that prevents the memory from
storing additional qubits.

Unfortunately, the latter requirement of qubit independence (condition b) can
be at odds with a high repetition rate. As we show below, one example is that
of temporal multiplexing and read-out on demand in the so-called atomic
frequency comb (AFC) quantum memory protocol, which requires one to
temporarily map qubit states from optical coherence to spin coherence Afzelius
_et al._ (2009). This leads us to conclude that it is important to optimize
the _optical storage time_ , i.e. the time during which qubits are stored as
optical coherence, which can be excited using visible or near infrared light.
Triggered by this conclusion, we investigate purely optical storage using
thulium-doped yttrium gallium garnet (Tm3+:Y3Ga5O12 or Tm:YGG) – a rare-earth-
ion doped crystal (REIC) whose promising spectroscopic properties have been
demonstrated previously Thiel  _et al._ (2014a, b). However, the potential of
Tm:YGG for storing photonic qubits has not yet been established. Here we show
that its optical coherence time T2 can reach 1.1 ms, which is one of the
longest times reported for any REIC Böttger  _et al._ (2009); Equall  _et al._
(1994). Motivated by this promising result, we investigate the prospects of
Tm:YGG for optical quantum storage using the two-level AFC protocol, and
demonstrate storage of laser pulses for up to 100 $\mu$s. This time is
comparable with recent results obtained using Yb:Y2SiO5 Businger  _et al._
(2020), and exceeds all other reported results of storage of light in optical
coherence with any REIC by at least a factor of 10. However, we also find that
the memory decay time Tm of around 13 $\mu$s is 20 times smaller than the
T2-imposed maximum of 275 $\mu$s, showing the presence of decoherence that
does not affect the T2 measurements. Before addressing possible reasons for
this large gap, we confirm the possibility for feed-forward-based spectral
mode mapping—an alternative to read-out on demand Sinclair  _et al._
(2014)—and we also show that quantum correlations between members of photon
pairs persist throughout storage, i.e. that our memory can operate in the
quantum regime. We conclude by briefly discussing how to improve memory
efficiencies by means of cavity-enhanced light-matter interaction.

_The need for long optical storage_ – To support our claim that qubit
independence can be at odds with a high repetition rate, let us discuss the
example of REICs and temporal mode mapping using the AFC spin-wave memory
Afzelius  _et al._ (2010). As depicted in Fig. 1a, a pair of optical control
pulses ($\pi$-pulses that resonantly couple the excited state $|e\rangle$ with
a ground state $|s\rangle$) allows one to reversibly map optical
coherence—excited through the absorption of photonic qubits—onto a spin
transition. In this case, the timing of the second control pulse determines
the moment at which the photons will be re-emitted from the memory, allowing
for readout on demand.

Let us now assume that a number of qubits—a first train—has already been
absorbed by the memory, that the first control pulse has been applied, and
that a second train of qubits has just been added to the memory (Figs. 1b-d
depict this sequence). At this point, the first train (R1) is stored in terms
of spin coherence, and the second (R2) in terms of optical coherence.
Unfortunately, the subsequent control pulse, applied to the memory with the
goal to map the second train onto spin coherence, simultaneously maps the
first train back onto optical coherence. This causes re-emission of these
qubits at a time that is determined by the need to transfer the second train,
rather than by feed-forward information that specifies what to do with the
first.

Fortunately, this problem can be avoided by storing only one train at a time.
But in order to maximize the repetition rate of the repeater, this block, and
hence the time during which qubits are stored in optical coherence, should be
as long as possible – ideally as long as the total storage time. Obviously,
long optical storage times are also needed in all quantum memories that do not
employ mapping between optical and spin coherence Sinclair  _et al._ (2014).

Figure 1: Recall on demand using the AFC spin-wave protocol. a. Pulse sequence
for AFC spin-wave storage. A three-level lambda system is formed by spin
states $|g\rangle$ and $|s\rangle$, and by excited state $|e\rangle$. b - e.
Memory input and output, as well as atomic coherence (indicated by light blue
and green figures of eight) for different moments during storage (see text for
details). Trains of qubits are labelled R1-R4. Figure 2: Schematic of the
experimental setup. AOM: acousto-optic modulator; PM: phase modulator; OS:
optical switch; EOM: electro-optic modulator; PD: photo detector; CCD: charge-
coupled device camera; B: Magnetic field. Inset: Simplified energy level
diagram of Tm:YGG showing the 3H${}_{6}\leftrightarrow^{3}$H4 zero-phonon
line.

_Tm:YGG and experimental setup –_ Due to their unique spectroscopic properties
Liu and Jacquier (2006), REICs have been broadly explored over the last two
decades for quantum technology Thiel  _et al._ (2011); Zhong and Goldner
(2019), including as ensemble-based quantum memory for light Lvovsky  _et al._
(2009); Bussieres  _et al._ (2013); Heshami  _et al._ (2016) or for quantum
processing Saglamyurek  _et al._ (2016); Kinos  _et al._ (2021). But while
significant effort has been spent on achieving long storage times of qubits in
spin coherence (values up to one hour have recently been reported Ma  _et al._
(2021); Jobez  _et al._ (2014); Seri  _et al._ (2017, 2019)), much less work
has been devoted to pushing and better understanding the limitations of
storage in optical coherence.

To address this shortcoming, we use a 25-mm long, 1% Tm:YGG single crystal,
mounted inside an adiabatic demagnetization refrigerator and cooled to around
500 mK. YGG forms a cubic lattice in which Tm3+ replaces yttrium ions without
charge compensation in six crystallographically equivalent sites of local D2
point group symmetry Dillon and Walker (1961). A superconducting solenoid
allows applying a magnetic field of up to 2 T along the crystal $<111>$ axis,
resulting in the splitting of all electronic levels into two hyperfine sub-
levels. Fig. 2 (Inset) depicts a simplified level structure.

The 3H${}_{6}\leftrightarrow^{3}$H4 zero-phonon line at 795.325 nm is
addressed by a tunable continuous-wave diode laser that propagates along the
$<110>$ crystal direction. The polarization state of the input light is
unknown but constant during the measurements, due to the use of a non-
polarization maintaining fiber at the input of our crystal. The laser is
frequency-locked to a high finesse cavity using the Pound-Drever-Hall (PDH)
method, resulting in a frequency instability of less than 20 kHz over around
100 $\mu$s. To intensity- and frequency-modulate the light, we steer the beam
into a single-pass acousto-optic modulator (AOM) (driven by a synthesized
signal generator) and a phase modulator (PM) (driven by an arbitrary waveform
generator). After passing through the crystal, the light is directed to a
photo-detector. This setup is used for AFC creation (see Puigibert  _et al._
(2020) for a description), initial memory characterization, and storage of
optical pulses in a single spectral mode. To allow for frequency-multiplexed
storage and feed-forward recall, additional phase modulators are used to add
frequency side-bands to the laser light, each of which creates a memory in a
different spectral segment, and to frequency shift the light after re-emission
so that only the desired spectral mode passes through an optical filter cavity
Sinclair  _et al._ (2014). See Fig. 2 for a schematic.

Figure 3: Storage of data in a single spectral mode. a. Exponential decay of
the two-pulse photon echo signal at 257 G b. Optical coherence time T2 as a
function of magnetic field. The dashed line is a guide for the eye. c. AFC of
1 MHz bandwidth tailored for 5 $\mu$s storage time. d. Memory efficiency as a
function of storage time using AFCs with finesse 2. The error bars in panels a
and d are smaller than the data points.

_Measurements and results –_ First, as a key property that determines the
maximum optical storage time, we characterized the optical coherence time
$T_{2}$ as a function of magnetic field using 2-pulse photon echoes Sun  _et
al._ (2012). As an important difference compared to our previous studies Thiel
_et al._ (2014a, b), the temperature was lowered from 1.2 K to 500 mK. As
shown in Figs. 3a and 3b and predicted by our earlier studies Thiel  _et al._
(2014a), this resulted in a very significant improvement of the coherence time
from 490 $\mu$s to around 1.1 ms, which makes the transition almost
radiatively limited (the radiative lifetime T1 of the 3H4 level is 1.3 ms
Thiel  _et al._ (2014b)) and corresponds to one of the longest reported
optical coherence times for any rare-earth crystal.

Next, we investigated the possibility for optical data storage, both using
laser pulses as well as quantum states of light. Towards this end, we employed
the two-level atomic frequency comb protocol Afzelius  _et al._ (2009). An AFC
is characterized by an absorption profile composed of evenly-spaced peaks in
the frequency domain, which can be created using frequency-selective optical
pumping of population from the troughs of the AFC to other atomic levels. Note
that Tm:YGG is well suited for this task due to long-lived hyperfine levels
within the electronic ground state manifold Thiel  _et al._ (2014b).
Absorption of a photon by an AFC results in the creation of a collective
atomic excitation described by
$|\psi\rangle_{A}=N^{-1/2}\sum_{j=1}^{N}c_{j}e^{i2\pi\delta_{j}t}e^{-kz_{j}}$,
where $N$ is the number of atoms in the ensemble, $\delta_{j}$ the frequency
detuning of the jth atom’s transition with respect to the input photon’s
carrier frequency, and $z_{j}$ and $c_{j}$ denote the position and the
excitation probability amplitude of the jth atom, respectively. After initial
dephasing, the coherence rephases, resulting in re-emission of the photon
after a time $\tau$ that is determined by the inverse peak spacing $\Delta$,
where $\tau=1/\Delta$. The AFC protocol is described in great detail in
Afzelius  _et al._ (2009); Jobez  _et al._ (2016), and an example of an AFC in
Tm:YGG is depicted in Fig. 3c.

_a) Long-lived storage of laser pulses –_ Given the promisingly long optical
coherence time, it is interesting to assess how the memory efficiency evolves
as a function of storage time. To this end, we used laser pulses of 1 $\mu$s
duration, and created AFCs with finesse F—the ratio between AFC peak spacing
$\Delta$ and peak width $\delta$—of 2 (see Sinclair  _et al._ (2014);
Puigibert  _et al._ (2020) for more information about AFC creation). Peak
spacings varied between 100 kHz and 10 kHz, corresponding to storage times
between 10 and 100 $\mu$s, respectively. The choice of F=2 maximizes the
storage efficiency of the memory, which was limited due to the crystal’s small
optical depth of around 1. The AFC bandwidth for all storage times was 0.5 MHz
except for 100 $\mu$s, where it was reduced to 0.2 MHz. The preparation of the
AFCs took 1 s; it was followed by a waiting time of 20 ms to avoid detecting
spurious photons caused by spontaneous decay of ions excited during the AFC
preparation. The magnetic field in all measurements was around 100 G, which
maximizes the optical coherence time.

As shown in Fig. 3d, we detected re-emitted pulses after up to 100 $\mu$s and
found that the memory storage efficiency $\eta$ decreases exponentially as a
function of storage time with a 1/e decay time of $T_{m}=$13.1$\pm$0.8 $\mu$s.
This value is much smaller than the ultimate limit imposed by T2 of around 275
$\mu$s Jobez  _et al._ (2016). This discrepancy shows that additional
decoherence mechanisms are at play. Possibilities, some of which will be
discussed in more detail in a forthcoming paper, include vibrations of the
crystal inside the pulse-tube cooler, remaining laser frequency jitter,
magnetic field instabilities, and spectral diffusion due to the interaction
between thulium and neighboring ions in the crystal.

_b) Frequency-multiplexed storage with feed-forward mode mapping –_ To
demonstrate spectral multiplexing, we prepared 11 different AFCs (featuring
F=2), each of 1 MHz bandwidth and spaced by 10 MHz, over a total bandwidth of
100 MHz. Laser pulses of 1$\mu$s duration were created in each spectral mode.
They were stored and recalled after 5$\mu$s (see Fig. 4a). Assuming that five
subsequent pulses fit into this time, this results in a multimode capacity
over spectral and temporal degrees of freedom of 55. Note that the storage
time—significantly shorter than our maximum of 100 $\mu$s in this and all
subsequent measurements—was limited by a trade-off between more complex AFC
tailoring procedures and memory efficiency. Otherwise, all parameters used to
create the AFCs remained the same as before.

For these measurements, the internal storage efficiency was close to 1.3%.
(This value was calculated by comparing the energies of input and re-emitted
pulses and by taking coupling loss into account). Note that both the total
bandwidth and the number of spectral channels can easily be increased with
more laser power. Given the inhomogeneous broadening of the
3H6$\leftrightarrow$3H4 transition of Tm:YGG of 56 GHz, this would in
principle allow creating thousands of spectral modes. At the same time, our
optimization of the comb-preparation parameters shows that more laser power
would also improve the AFC quality per spectral mode and thereby allow
increasing the storage time by an order of magnitude and hence the multi-mode
capacity by another factor of ten.

To meet the requirement of feed-forward-based mode mapping imposed by the
targeted use of the memory in a quantum repeater, we furthermore demonstrated
shifting of the recalled laser pulses such that only the desired spectral mode
is subsequently transmitted through a filter cavity. For this proof-of-
principle demonstration, which followed the approach introduced in Sinclair
_et al._ (2014), we tailored with the help of PM2 three AFCs, again spaced by
10 MHz, in which spectrally matched optical pulses were stored. Pulses in
different spectral modes were separated by 20 $\mu$s in order to make them
distinguishable in time – a necessary feature for analysis since the filter
cavity after the memory erased all spectral distinguishability.

Figure 4: Storage of data in multiple spectral modes. a. Spectrally-
multiplexed AFC quantum memory used to simultaneously store optical pulses in
11 spectral modes during 5 $\mu$s. The individual modes were resolved by
changing the resonance frequency of the Fabry-Perot filtering cavity. b. Feed-
forward mapping between different spectral modes. The resonance frequency of
the filtering cavity only allowed transmission of the spectral mode at zero
frequency detuning (depicted in red). The detuned frequency modes at 10 MHz
and 20 MHz (shown in blue and pink, respectively) were resolved by shifting
the frequency using a phase modulator (PM3). See description in the main text.
The stored and re-emitted pulses are magnified by a factor of 2, and crosstalk
is indicated using dotted circles.

First, we set the resonance frequency of the filtering cavity such that it
allowed transmission of the spectral mode at zero frequency detuning– both for
the light that was directly transmitted through the AFC as well as the stored
and re-emitted pulse. The measured signal is depicted in Fig. 4b in red. At
the same time, the cavity largely suppressed the pulses at 10 and 20 MHz
detuning. However, because the cavity linewidth of 7.5 MHz is comparable with
the spectral mode spacing of 10 MHz, small fractions of the neighboring modes
leaked through. This crosstalk is visible in the peaks in the red-colored
signal trace centered at 20 and 40 $\mu$s.

Next, we drove PM3, positioned in-between the memory and the filter cavity,
using a 10 MHz serrodyne signal that shifted all pulses emitted by the memory
by -10 MHz. Consequently, the signal encoded originally at +10 MHz detuning,
depicted in Fig. 4b in blue, became resonant with the cavity, resulting in its
transmission through the cavity and the detection of a large peak at 20 $\mu$s
followed by a smaller peak—the recalled pulse—5 $\mu$s later. As before, some
leakage of signals in neighboring modes lead to peaks at 0 and 40 $\mu$s.

Finally, the same experiment was repeated with a serrodyne shift of -20 MHz.
The result is depicted by the magenta-colored signal trace in Fig. 4b.

_c) Storage of heralded single photons –_ Finally, we verified that Tm:YGG, in
conjunction with the two-level AFC protocol, is suitable for quantum state
storage. As described in detail in Grimau Puigibert  _et al._ (2017), we
created pairs of quantum-correlated photons at 795 and 1532 nm wavelength by
means of spontaneous parametric down-conversion of strong laser pulses in a
periodically-poled LiNbO3 crystal. The detection of a 1532 nm photon using a
superconducting nanowire single-photon detector (SNSPD) heralded the presence
of a 795 nm photon, which was directed into, stored in, and released after 43
ns from the Tm:YGG memory. Note that the memory creation procedure remained
unchanged except that the AFC bandwidth was increased to 4 GHz to better match
the photon bandwidth, and that the magnetic field was increased to 12 kG to
match the Tm level splitting with the spacing between a trough and the
neighboring tooth in the AFC Davidson  _et al._ (2020); Saglamyurek  _et al._
(2011). The photons were then detected using a single-photon detector based on
a silicon avalanche photodiode. The storage efficiency, assessed by comparing
photon detection rates with and without memory and after taking coupling loss
into account, was around 0.35%.

To verify that the non-classical correlations with the 1532 nm photons persist
throughout the storage process, we measured the 2nd order cross-correlation
coefficient $g^{(2)}_{12}(t)$ between the two photons belonging to the same
pair using time-resolved coincidence detectionPuigibert  _et al._ (2020).
Before storage, we found $g^{(2)}_{12}(0)_{no\>mem}$= 18$\pm$0.02 and,
importantly, after storage $g^{(2)}_{12}(43)_{mem}$= 4.58$\pm$0.46. Both
values surpass the classical upper bound of 2, confirming the quantum nature
of both the photon source as well as of our memory.

_Outlook and Conclusion –_ Our investigations have resulted in an optical
coherence time $T_{2}$ up to 1.1 ms and optical storage times of up to 100
$\mu$s. However, they also revealed a memory decay time of 13.1 $\mu$s – a
factor of 20 less than the limit imposed by T2. To further increase the memory
performance, the most important task is to identify and remove the source of
this excessive decoherence. Even though our laser is frequency stabilized and
the 500 mK plate inside the cryostat on which the crystal is mounted features
vibration isolation, it is very likely that remaining instabilities affect our
measurements Louchet-Chauvet  _et al._ (2019). Another possible source of
spectral diffusion is that of ion-ion interaction inside the Tm:YGG crystal.
Reducing the Tm doping concentration (which affects Tm-Tm interactions), e.g.
by using a longer crystal with the same optical depth, or choosing a magnetic
field direction for which the 795 nm transition becomes insensitive to first
order to magnetic field fluctuations Tongning  _et al._ (2015); McAuslan  _et
al._ (2012); Ruskuc  _et al._ (2020); Ortu  _et al._ (2018) may help improving
memory performance.

In addition to the small memory decay time, our demonstration currently
suffers from small recall efficiencies for all storage times. Selecting a
specific polarization state at the input of the crystal to optimize the
photon-ion interaction is expected to improve the observed memory efficiency.
In addition, we note that embedding the crystal inside an impedance-matched
cavity allows circumventing the effect of small optical depth provided AFCs
with large finesse and small background absorption d0 can be created Afzelius
and Simon (2010); Sabooni  _et al._ (2013); Davidson  _et al._ (2020). For
instance, tailoring and analyzing an AFC with F = 4 allowed us to predict an
efficiency of 15% for 30 $\mu$s storage time, assuming the quality of the AFCs
does not change when embedding the crystal inside the cavity. This would
already allow using the memory for a proof-of-principle demonstration of an
elementary quantum repeater link. Naturally, the creation of AFCs with further
reduced background will result in a further increase of the efficiency. We
anticipate that these improvements will make Tm:YGG suitable for efficient
establishment of entanglement across extended quantum networks as a part of a
quantum repeater.

## Acknowledgments

The authors would like to thank M. Grimau Puigibert, T. Chakraborty and O. P.
Casas for help during various stages of the experiment, and M. Afzelius for
discussions. We acknowledge funding through the Netherlands Organization for
Scientific Research (NWO), the European Union’s Horizon 2020 Research and
Innovation Program under Grant Agreement No. 820445 and Project Name Quantum
Internet Alliance, as well as by Alberta Innovates Technology Futures (AITF),
the National Sciences and Engineering Research Council of Canada (NSERC) and
the Alberta Ministry of Jobs, Economy and Innovation’s Major Innovation Fund
Project on Quantum Technologies. Furthermore, W.T. acknowledges funding as a
Senior Fellow of the Canadian Institute for Advanced Research (CIFAR), and
N.S. thanks the National Science Foundation Science and Technology Center for
Integrated Quantum Materials under Cooperative Agreement No. DMR-1231319 as
well as the AQT Intelligent Quantum Networks and Technologies (INQNET)
research program for support. This material is based in part on research at
Montana State University sponsored by Air Force Research Laboratory under
agreement number FA8750-20-1-1004.

## References

  * Kimble (2008) H. J. Kimble, Nature 453, 1023 (2008).
  * Wehner  _et al._ (2018) S. Wehner, D. Elkouss, and R. Hanson, Science 362 (2018), 10.1126/science.aam9288.
  * Briegel  _et al._ (1998) H.-J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998).
  * Duan  _et al._ (2001) L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, Nature 414, 413 (2001).
  * Sangouard  _et al._ (2011) N. Sangouard, C. Simon, H. De Riedmatten, and N. Gisin, Reviews of Modern Physics 83, 33 (2011).
  * Sinclair  _et al._ (2014) N. Sinclair, E. Saglamyurek, H. Mallahzadeh, J. A. Slater, M. George, R. Ricken, M. P. Hedges, D. Oblak, C. Simon, W. Sohler, and W. Tittel, Phys. Rev. Lett. 113, 053603 (2014).
  * Jiang  _et al._ (2009) L. Jiang, J. M. Taylor, K. Nemoto, W. J. Munro, R. Van Meter, and M. D. Lukin, Phys. Rev. A 79, 032325 (2009).
  * Lvovsky  _et al._ (2009) A. I. Lvovsky, B. C. Sanders, and W. Tittel, Nature photonics 3, 706 (2009).
  * Fleischhauer and Lukin (2000) M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. 84, 5094 (2000).
  * Kraus  _et al._ (2006) B. Kraus, W. Tittel, N. Gisin, M. Nilsson, S. Kröll, and J. I. Cirac, Phys. Rev. A 73, 020302 (2006).
  * Afzelius  _et al._ (2010) M. Afzelius, I. Usmani, A. Amari, B. Lauritzen, A. Walther, C. Simon, N. Sangouard, J. Minář, H. de Riedmatten, N. Gisin, and S. Kröll, Phys. Rev. Lett. 104, 040503 (2010).
  * Grimau Puigibert  _et al._ (2017) M. Grimau Puigibert, G. H. Aguilar, Q. Zhou, F. Marsili, M. D. Shaw, V. B. Verma, S. W. Nam, D. Oblak, and W. Tittel, Phys. Rev. Lett. 119, 083601 (2017).
  * Afzelius  _et al._ (2009) M. Afzelius, C. Simon, H. de Riedmatten, and N. Gisin, Phys. Rev. A 79, 052329 (2009).
  * Thiel  _et al._ (2014a) C. W. Thiel, N. Sinclair, W. Tittel, and R. L. Cone, Phys. Rev. B 90, 214301 (2014a).
  * Thiel  _et al._ (2014b) C. W. Thiel, N. Sinclair, W. Tittel, and R. L. Cone, Phys. Rev. Lett. 113, 160501 (2014b).
  * Böttger  _et al._ (2009) T. Böttger, C. W. Thiel, R. L. Cone, and Y. Sun, Phys. Rev. B 79, 115104 (2009).
  * Equall  _et al._ (1994) R. W. Equall, Y. Sun, R. L. Cone, and R. M. Macfarlane, Phys. Rev. Lett. 72, 2179 (1994).
  * Businger  _et al._ (2020) M. Businger, A. Tiranov, K. T. Kaczmarek, S. Welinski, Z. Zhang, A. Ferrier, P. Goldner, and M. Afzelius, Phys. Rev. Lett. 124, 053606 (2020).
  * Liu and Jacquier (2006) G. Liu and B. Jacquier,  _Spectroscopic properties of rare earths in optical materials_ , Vol. 83 (Springer Science & Business Media, 2006).
  * Thiel  _et al._ (2011) C. Thiel, T. Böttger, and R. Cone, Journal of luminescence 131, 353 (2011).
  * Zhong and Goldner (2019) T. Zhong and P. Goldner, Nanophotonics 8, 2003 (2019).
  * Bussieres  _et al._ (2013) F. Bussieres, N. Sangouard, M. Afzelius, H. de Riedmatten, C. Simon, and W. Tittel, Journal of Modern Optics 60, 1519 (2013).
  * Heshami  _et al._ (2016) K. Heshami, D. G. England, P. C. Humphreys, P. J. Bustard, V. M. Acosta, J. Nunn, and B. J. Sussman, Journal of Modern Optics 63, 2005 (2016), pMID: 27695198, https://doi.org/10.1080/09500340.2016.1148212 .
  * Saglamyurek  _et al._ (2016) E. Saglamyurek, M. G. Puigibert, Q. Zhou, L. Giner, F. Marsili, V. B. Verma, S. W. Nam, L. Oesterling, D. Nippa, D. Oblak, _et al._ , Nature communications 7, 11202 (2016).
  * Kinos  _et al._ (2021) A. Kinos, D. Hunger, R. Kolesov, K. Mølmer, H. de Riedmatten, P. Goldner, A. Tallaire, L. Morvan, P. Berger, S. Welinski, K. Karrai, L. Rippe, S. Kröll, and A. Walther, “Roadmap for rare-earth quantum computing,” (2021), arXiv:2103.15743 [quant-ph] .
  * Ma  _et al._ (2021) Y. Ma, Y.-Z. Ma, Z.-Q. Zhou, C.-F. Li, and G.-C. Guo, Nature Communications 12, 2381 (2021).
  * Jobez  _et al._ (2014) P. Jobez, I. Usmani, N. Timoney, C. Laplane, N. Gisin, and M. Afzelius, New Journal of Physics 16, 083005 (2014).
  * Seri  _et al._ (2017) A. Seri, A. Lenhard, D. Rieländer, M. Gündoğan, P. M. Ledingham, M. Mazzera, and H. de Riedmatten, Phys. Rev. X 7, 021028 (2017).
  * Seri  _et al._ (2019) A. Seri, D. Lago-Rivera, A. Lenhard, G. Corrielli, R. Osellame, M. Mazzera, and H. de Riedmatten, Phys. Rev. Lett. 123, 080502 (2019).
  * Dillon and Walker (1961) J. F. Dillon and L. R. Walker, Phys. Rev. 124, 1401 (1961).
  * Puigibert  _et al._ (2020) M. l. G. Puigibert, M. F. Askarani, J. H. Davidson, V. B. Verma, M. D. Shaw, S. W. Nam, T. Lutz, G. C. Amaral, D. Oblak, and W. Tittel, Phys. Rev. Research 2, 013039 (2020).
  * Sun  _et al._ (2012) Y. Sun, C. W. Thiel, and R. L. Cone, Phys. Rev. B 85, 165106 (2012).
  * Jobez  _et al._ (2016) P. Jobez, N. Timoney, C. Laplane, J. Etesse, A. Ferrier, P. Goldner, N. Gisin, and M. Afzelius, Phys. Rev. A 93, 032327 (2016).
  * Davidson  _et al._ (2020) J. H. Davidson, P. Lefebvre, J. Zhang, D. Oblak, and W. Tittel, Phys. Rev. A 101, 042333 (2020).
  * Saglamyurek  _et al._ (2011) E. Saglamyurek, N. Sinclair, J. Jin, J. A. Slater, D. Oblak, F. Bussieres, M. George, R. Ricken, W. Sohler, and W. Tittel, Nature 469, 512 (2011).
  * Louchet-Chauvet  _et al._ (2019) A. Louchet-Chauvet, R. Ahlefeldt, and T. Chanelière, Review of Scientific Instruments 90, 034901 (2019), https://doi.org/10.1063/1.5080086 .
  * Tongning  _et al._ (2015) R.-C. Tongning, T. Chanelière, J.-L. Le Gouët, and M. F. Pascual-Winter, in  _Journal of Physics: Conference Series_ , Vol. 605 (IOP Publishing, 2015) p. 012037.
  * McAuslan  _et al._ (2012) D. L. McAuslan, J. G. Bartholomew, M. J. Sellars, and J. J. Longdell, Phys. Rev. A 85, 032339 (2012).
  * Ruskuc  _et al._ (2020) A. Ruskuc, J. M. Kindem, J. G. Bartholomew, J. Rochman, Y. Q. Huan, and A. Faraon, in  _OSA Quantum 2.0 Conference_ (Optical Society of America, 2020) p. QTh6A.7.
  * Ortu  _et al._ (2018) A. Ortu, A. Tiranov, S. Welinski, F. Fröwis, N. Gisin, A. Ferrier, P. Goldner, and M. Afzelius, Nature Materials 17, 671 (2018).
  * Afzelius and Simon (2010) M. Afzelius and C. Simon, Phys. Rev. A 82, 022310 (2010).
  * Sabooni  _et al._ (2013) M. Sabooni, S. T. Kometa, A. Thuresson, S. Kröll, and L. Rippe, New Journal of Physics 15, 035025 (2013).