# Linear and continuous variable spin-wave processing using a cavity-coupled
atomic ensemble

Kevin C. Cox<EMAIL_ADDRESS>DEVCOM Army Research Laboratory,
Adelphi, MD 20783 USA  Przemyslaw Bienias  Joint Quantum Institute and Joint
Center for Quantum Information and Compute Science, NIST/University of
Maryland, College Park, Maryland 20742, USA  David H. Meyer  DEVCOM Army
Research Laboratory, Adelphi, MD 20783 USA  Donald P. Fahey  DEVCOM Army
Research Laboratory, Adelphi, MD 20783 USA  Paul D. Kunz  DEVCOM Army Research
Laboratory, Adelphi, MD 20783 USA  Alexey V. Gorshkov  Joint Quantum Institute
and Joint Center for Quantum Information and Compute Science, NIST/University
of Maryland, College Park, Maryland 20742, USA

###### Abstract

Spin-wave excitations in ensembles of atoms are gaining attention as a quantum
information resource. However, current techniques with atomic spin waves do
not achieve universal quantum information processing. We conduct a theoretical
analysis of methods to create a high-capacity universal quantum processor and
network node using an ensemble of laser-cooled atoms, trapped in a one-
dimensional periodic potential and coupled to a ring cavity. We describe how
to establish linear quantum processing using a lambda-scheme in a rubidium-
atom system, calculate the expected experimental operational fidelities.
Second, we derive an efficient method to achieve linear controllability with a
single ensemble of atoms, rather than two-ensembles as proposed in [K. C. Cox
et al. “Spin-Wave Quantum Computing with Atoms in a Single-Mode Cavity”,
preprint 2021]. Finally, we propose to use the spin-wave processor for
continuous-variable quantum information processing and present a scheme to
generate large dual-rail cluster states useful for deterministic computing.

††preprint: APS/123-QED

##  I Introduction

Laser-cooled atoms in optical resonators are a building block for many of the
most exquisite demonstrations of quantum electrodynamics. Atom-cavity systems
are the basis for state-of-the-art quantum simulators [1, 2], quantum memories
[3], and entanglement-enhanced atomic clocks [4, 5]. With atom number $N$
commonly between $10^{3}$ and $10^{6}$, an ensemble’s intrinsic capacity to
store quantum information is enormous, with a state space of dimension
$2^{N}$. Designing quantum platforms that are able to access and process this
large amount of quantum information is a grand challenge in atomic science.
Here we analyze a method to store quantum information as collective spin-wave
excitations and realize universal quantum computation in a system where the
spin waves may be efficiently retrieved into a single optical cavity mode.

Recent experiments have introduced a path to use collective spin-wave
excitations, holographically multiplexed, to achieve high-capacity quantum
memories [6, 7, 8, 9, 10], but these experiments have not introduced a method
to achieve full linear controllability of spin-wave excitations, a
prerequisite for a universal quantum processor. Spin-wave quantum systems are
being realized in multiple physical platforms including atomic vapors [7, 8,
9, 10], solid-state crystals [11, 12, 13], and superconducting circuits [14].
Proposals for spin-wave readout of atomic arrays have also been developed [15,
16, 17]. But demonstrating platforms that simultaneously achieve universal
quantum processing and efficient memory readout is still an outstanding
challenge. In a joint Letter publication [18], we present a general scheme for
universal linear-optical quantum processing using spin-wave excitations
coupled to a single optical mode. In this Article, we build upon that work by
describing, in detail, the experimental methods and performance required to
physically realize universal quantum information processing with laser-cooled
atoms inside of an optical ring cavity.

First, we describe a two-ensemble experimental method to realize universal
spin-wave quantum processing in an ensemble of alkali atoms coupled to a bow-
tie ring cavity and discuss the physical operations required. Second, we
present a general proof of linear controllability using only position-space
and momentum-space phase shifts with a single atomic ensemble. We derive the
exact two-mode beamsplitters in this one-ensemble scenario, that can be
accomplished in constant time even when using a large number of spin-wave
modes. Further, we calculate the expected leading operational errors due to
experimental imperfections, and the time required to probabilistically
initialize single photons in the system using heralded quantum memory
initialization. Last, we calculate the amount of multi-mode spin-squeezing
that may be achieved in the system and how this multi-mode squeezing would
perform in a dual-rail cluster state. We calculate that it may be possible to
achieve a continuous-variable cluster state with thousands of modes and
greater than $20$ dB of squeezing per mode, a significant computational
resource for deterministic continuous variable processing [19, 20, 21].

##  II Apparatus

The simplified apparatus and atomic level diagram are displayed in Fig. 1(a)
and (b). We first consider two ensembles, each with $N$ laser-cooled alkali
atoms, approximated as three-level atoms with long-lifetime states $\ket{g}$,
$\ket{e}$ and optically excited state $\ket{i}$ with linewidth $\Gamma$. The
atoms are confined in two one-dimensional periodic potentials, each with $M$
sites, inside of a running-wave optical cavity. The two arrays, labelled $A$
and $B$, have the same cavity couplings and experimental parameters. When
discussing parameters and operators that specifically refer to one array or
the other, we will denote them with a corresponding superscript A or B.

A running-wave cavity is required for this experiment in order to distinguish
between excitations with left or right-travelling photons. The arrays are
optically interrogated using Raman dressing beams (red and violet in Fig. 1)
that stimulate two-photon Raman transitions. Potential gradients (green and
orange in Fig. 1), using laser beams with an intensity variation, are applied
perpendicular to the cavity axis. The off-resonant optical fields are applied
with detuning $\delta_{AC}$ and a spatially varying Rabi frequency
$\Omega_{AC}(x)$.

The optical cavity is described by its finesse $f$, full-width half-maximum
(FWHM) linewidth $\kappa$, Jaynes-Cummings coupling parameter $g$ associated
with the $\ket{g}$-$\ket{i}$ transition and single-atom cooperativity
parameter $C=4g^{2}/\kappa\Gamma$. We consider an apparatus with $C$ near or
less than 1, but large collective cooperativity $NC\gg 1$. Atoms in $\ket{e}$
can be made to interact with the cavity mode by applying Raman dressing lasers
(red and violet in Fig. 1) with Rabi frequency $\Omega_{d}$.

Figure 1: Apparatus and level diagram using alkali atoms. (a) Two ensembles
are coupled to a single running-wave cavity, and operations are applied using
potential gradients (green and orange) with operators $\hat{\Delta}^{A}$ and
$\hat{\Delta}^{B}$ and Raman dressing beams (red and violet) with associated
Hamiltonians $\hat{H}^{kA}_{0}$ and $\hat{H}^{kB}_{0}$. (b) Level diagram.
Raman cavity coupling is controlled with the dressing beams (red and violet).
Optical gradients are applied with large detuning $\delta_{AC}$. The $\ket{e}$
states in ensembles $A$ and $B$ are drawn separately to delineate the
ensemble-specific beams.

With large detuning $\delta_{1}$ ($|\delta_{1}|\gg|\delta_{2}|$) between the
cavity and the $\ket{g}$ to $\ket{i}$ transition
($|\delta_{1}|\gg\sqrt{N}g,\Omega_{d}$), the theoretical treatment of Fig.
1(a) may be simplified by adiabatically eliminating $\ket{i}$, creating an
effective two-level system with dressed excited state $\ket{e^{\prime}}$ and
two-photon scattering rate

$\Gamma_{2}(t)=\Gamma\frac{\Omega_{d}^{2}(t)}{4\delta_{1}^{2}},$ (1)

where $\Omega_{d}(t)$ is the Rabi frequency of the Raman dressing laser. This
results in a new two-photon Jaynes-Cummings coupling parameter

$g_{2}(t)=\frac{g\Omega_{d}(t)}{2\delta_{1}}.$ (2)

In addition to the two-photon transition rates, the Raman dressing laser gives
rise to shifts in the cavity resonance frequency and the two-photon transition
frequency, that must be taken into account (see Sec. X).

A diagram that defines the experimental parameters and the relationship
between the three-level system and the two-level model is shown in Fig. 2.
There are several important advantages of this three-level scheme. Critically,
the effective cavity coupling $g_{2}(t)$ and free space scattering rate
$\Gamma_{2}(t)$ are dynamic, and may be turned on and off at high speed via
the dressing laser intensity, proportional to $|\Omega_{d}(t)|^{2}$.

##  III Momentum Basis with Alkali Atoms in Cavity

As described in the joint Letter [18], we can demonstrate linear
controllability by defining a set of orthogonal spin-wave modes in the
momentum basis. In Ref. [18], we focused solely on a two-ensemble apparatus
where qubits are arranged in two banks of one-dimensional arrays. Here, we
first discuss how this proposal may be realized in a three-level Raman system
with laser-cooled and trapped alkali atoms. Then, in Section IV, we show how
to extend the proposal to using only a single one-dimensional array. Using
cold atoms, two arrays may be created via counter-propagating trap beams
within the cavity, or the single sites may be created by projecting an
additional trapping potential transverse to the cavity mode. Such a scheme may
be useful to precisely control the trap dimension and spacing of the array
sites.

Atomic excitations in the array are described by site-specific lowering
operators

$\hat{a}_{x}=\frac{1}{\sqrt{n}}\sum_{l=0}^{n-1}\ket{g_{l}}\\!\bra{e_{l}}.$ (3)

The corresponding collective lowering operators in momentum space are

$\hat{b}_{k}=\frac{1}{\sqrt{M}}\sum_{x=0}^{M-1}e^{i2\pi kx/M}\hat{a}_{x},$ (4)

where $k$ can take integer values from 0 and $M-1$. By initializing and
reading out excitations in the modes defined by the $\hat{b}_{k}$ operators,
we ensure that single-site resolution is unnecessary, and all excitations can
be tuned into full coupling with the single cavity readout mode [18].

The optical cavity and the dressing laser define a unique momentum
$\vec{k}_{0}$ that can interact with the cavity mode at a given time.
$\vec{k}_{0}$ is dictated by the microwave qubit frequency $\omega_{eg}$ and
the angle of incidence of the dressing beam relative to the cavity mode [22].

The set of collective excitations with spin-wave momentum
$\vec{k}\in\vec{k}_{0}+\\{0,1,2...M-1\\}\hat{x}/x_{0}$ is the orthogonal set
of momenta that we consider, where $x_{0}$ is the array spacing. Subsequently,
we leave the momentum offsets as implicit, and simply label the momentum by an
integer $k\in\\{0,1,2...M-1\\}$. The orthogonality condition between spin-wave
modes $\hat{b}_{k}$ is critical to universal computing, since it implies that
collective excitations in mode $\hat{b}_{k}$, for example, are forbidden to
emit into or interact with the cavity mode if $k\neq 0$.

In the limit of large atom number per site $n$ and low excitation number, the
collective spin raising and lowering operators are directly analogous to
harmonic oscillator operators, and the cavity-ensemble system behaves as a
system of coupled harmonic oscillators. This well-established limit is known
as the Holstein-Primakoff approximation. The coupling strength between the
ensemble and cavity is then given by the collective vacuum Rabi splitting,
$\Omega_{2}=2g_{2}\sqrt{N}$.

###  III.1 Operations

The Raman dressing interrogation beams (red and violet) and AC Stark shift
gradient beams (green and orange) shown in Fig. 1 define two Hamiltonians that
may be applied to either ensemble A or B (denoted by superscripts when
necessary). These operations lead to the dynamics described in detail in Ref.
[18]. Here we briefly review the Hamiltonians and the slight modifications
that come from operating in a Raman system. The AC Stark shift gradient
Hamiltonian in the Raman system is

$\hat{H}_{\Delta}=\sum_{x=0}^{M-1}\frac{\hbar\Omega_{AC}^{2}(x)}{4\delta_{AC}}\hat{a}^{\dagger}_{x}\\!\hat{a}_{x}.$
(5)

With a choice of $\Omega^{2}_{AC}(x)\propto x$, this Hamiltonian allows
arbitrary shifts in momentum $k^{A}$ and $k^{B}$, and has a corresponding
unitary that we denote $\hat{\Delta}$:

$\hat{\Delta}=\sum_{x=0}^{M-1}\left[\sum_{l=1}^{n}\left(\ket{g_{l}}\bra{g_{l}}+e^{2\pi
ix/M}\ket{e_{l}}\bra{e_{l}}\right)\right].$ (6)

The behavior of $\hat{\Delta}$ is similar to other gradient quantum memories
using collective ensembles [23, 24]. Equation 5 differs from Eq. 2 in the
joint Letter [18] due to the Raman configuration.

Figure 2: Three-level diagram and adiabatic elimination. The experimental
three-level system is treated as a dynamic two-level system with cavity
coupling rate $g_{2}$, excited state lifetime $\Gamma_{2}$, and atom-cavity
detuning $\delta_{2}$.

In this proposal, atom-cavity interactions are governed by the Raman dressing
lasers. We assume that $\delta_{2}\gg\Omega_{2}$, so that the collective atom-
cavity interaction is dispersive, with Hamiltonian

$\hat{H}^{k}_{0}=-\frac{\hbar\Omega_{2}^{2}}{4\delta_{2}}\hat{b}^{\dagger}_{0}\hat{b}_{0},$
(7)

where $\Omega_{2}$ and $\delta_{2}$ are the effective coupling strength and
detuning for the two-photon transition [Eq. (2)].

When $\hat{H}^{k}_{0}$ is applied simultaneously to ensemble $A$ and ensemble
$B$, a spin-wave beamsplitter Hamiltonian results,

$\begin{split}\hat{H}^{BS}&=-\frac{\hbar\Omega_{2}^{2}}{4\delta_{2}}(\hat{b}^{\dagger
A}_{0}+\hat{b}^{\dagger B}_{0})(\hat{b}^{A}_{0}+\hat{b}^{B}_{0})\\\
&=a\begin{pmatrix}1&1\\\ 1&1\end{pmatrix},\end{split}$ (8)

for $a\equiv-\hbar\Omega_{2}^{2}/(4\delta_{2})$, where the second line has
been written in the $(\ket{b^{A}_{0}},\ket{b^{B}_{0}})$ basis. $\hat{H}^{BS}$
can be verified as a beamsplitter Hamiltonian by again calculating the unitary
evolution $\hat{\Pi}=e^{-i\hat{H}^{BS}t/\hbar}$, which can be written as

$\hat{\Pi}=\frac{1}{2}\begin{pmatrix}1+1e^{-2ita}&-1+1e^{-2ita}\\\
-1+1e^{-2ita}&1+1e^{-2ita}\end{pmatrix}.$ (9)

This A-B beamsplitter, the translation operators $\hat{\Delta}$, and mode
phase shifts from Hamiltonian $\hat{H}^{k}_{0}$, together allow one to create
an arbitrary linear unitary in the $2M$ mode system, a capability we refer to
as linear controllability. Linear controllability is the precise requirement
for linear-optical quantum computing, as discussed in the joint Letter [18].
Next, we discuss a general proof of linear controllability that does not
require two separate ensembles, and discuss alternative beamsplitter
constructions.

##  IV Quantum Processing with One Ensemble

###  IV.1 Controllability proof

So far in this work, we have focused on using two atomic ensembles and
performing beamsplitters between spin waves in each. This is a simple
experimental realization, but it is not fundamentally necessary for linear
controllability. We now present a general proof that phase shifts alone are
sufficient for controllability, even with one ensemble. For the general proof,
we allow ourselves to utilize Hamiltonian generators $\ket{a_{x}}\bra{a_{x}}$,
with $x=0,\dots,M-1$, and $\ket{b_{0}}\bra{b_{0}}$. These generators
correspond to the applications of phase shifts in position space and momentum
space, that don’t involve single site addressing.

A necessary and sufficient condition for controllability on the underlying
$M$-dimensional Hilbert space is that our $M+1$ Hamiltonian terms generate the
$(M^{2}-1)$-dimensional Lie algebra $\mathfrak{su}(M)$ [25, 26]. Working in
the $\ket{a_{x}}$ basis, we first construct all the $M-1$ diagonal generators
by taking linear combinations of $\ket{a_{x}}\bra{a_{x}}$. We construct half
(i.e. $M(M-1)/2$) of all the off-diagonal generators by considering, for
$j\neq l$,

$\displaystyle[[\ket{b_{0}}\bra{b_{0}},\ket{a_{j}}\bra{a_{j}}],\ket{a_{l}}\bra{a_{l}}]$
$\displaystyle\propto\frac{1}{\sqrt{M}}[\ket{b_{0}}\bra{a_{l}}-\ket{a_{l}}\bra{b_{0}},\ket{a_{j}}\bra{a_{j}}]$
$\displaystyle\propto-\frac{1}{M}(\ket{a_{l}}\bra{a_{j}}+\ket{a_{j}}\bra{a_{l}}).$
(10)

We construct the remaining $M(M-1)/2$ off-diagonal generators by considering,

$\displaystyle\frac{1}{M}[\ket{a_{l}}\bra{a_{j}}+\ket{a_{j}}\bra{a_{l}},\ket{a_{l}}\bra{a_{l}}]$
$\displaystyle=\frac{1}{M}(i\ket{a_{j}}\bra{a_{l}}-i\ket{a_{l}}\bra{a_{j}}).$
(11)

The generators synthesized in Eqs. (IV.1,11) are precisely the off-diagonal
beamsplitter generators that are not typically accessible in spin-wave quantum
memories. Note that we have only used the $\ket{b_{0}}$ momentum-space phase
shift. However, the off-diagonal elements are reduced by a factor of $1/M$.
For this reason, although the $\hat{H}^{k}_{0}\sim\ket{b_{0}}\\!\bra{b_{0}}$
phase shift generator is sufficient for controllability, the beamsplitter
interactions become weaker as the system size $M$ grows. Accomplishing
arbitrary unitary dynamics would require pulse sequences that grow unfavorably
with $M$. Next, we present a different Hamiltonian generator that allows us to
implement spin-wave beamsplitters in a single ensemble that alleviates this
deleterious scaling, showing that arbitrary two-mode beamsplitters can be
implemented in constant time, even in the large-$M$ limit.

###  IV.2 Numerically optimized beamsplitters

Since controllability is possible without two ensembles in principle, it is
worthwhile to describe a construction that achieves efficient linear
controllability in a single ensemble. Unlike in the previous section, we will
work here in the momentum basis $\ket{b_{k}}$. A beamsplitter between spin-
wave modes $\hat{b}_{j}$ and $\hat{b}_{l}$ can be generated by the Hamiltonian

$\displaystyle\hat{H}^{BS}_{jl}$
$\displaystyle\propto(\ket{b_{j}}+\ket{b_{l}})(\bra{b_{j}}+\bra{b_{l}}),$ (12)

with $j\neq l$. Expanding out the state $\ket{b_{j}}+\ket{b_{l}}$ shows that
this Hamiltonian corresponds to both phase modulation and amplitude modulation
across the spin wave:

$\displaystyle\ket{b_{j}}+\ket{b_{l}}$
$\displaystyle=\frac{1}{\sqrt{M}}\sum_{x=0}^{M-1}(e^{2\pi ijx}+e^{2\pi
ilx})\ket{a_{x}}$ (13)

due to the summation of the complex amplitudes at each site. For this reason,
we are not able to apply this Hamiltonian directly using only phase shifts.
This is one reason that previous experiments have not achieved complete
controllability in a spin-wave register.

In order to overcome this challenge, we propose to implement a similar
Hamiltonian, that is generated using only phase shifts but nonetheless yields
efficient unitary controllability. The modified Hamiltonian is

$\displaystyle\hat{H}^{\prime}_{jl}$
$\displaystyle\propto\ket{b^{\prime}}\bra{b^{\prime}},$ (14)
$\displaystyle\ket{b^{\prime}}$
$\displaystyle=\frac{1}{\sqrt{M}}\sum_{x=1}^{M}\operatorname{Exp}\left[i\operatorname{Arg}(e^{2\pi
ijx/M}+e^{2\pi ilx/M})\right]\ket{a_{x}}.$ (15)

$\hat{H}^{\prime}_{jl}$ only applies the phase component of the beamsplitter
Hamiltonian $\hat{H}^{BS}_{jl}$. This Hamiltonian may be constructed using
only phase shifts, by turning on the cavity coupling Hamiltonian
$\hat{H}^{k}_{0}$ [Eq. (7)] to a spin-wave state with the nontrivial phase
$\operatorname{Arg}(e^{2\pi ijx/M}+e^{2\pi ilx/M})$, instead of the $k=0$
mode. Experimentally this would be done in a two-step process, first applying
the phase modulation, and then turning on the cavity coupling. The modified
beamsplitter Hamiltonian $\hat{H}^{\prime}_{jl}$ does not generate an exact
two-mode beamsplitter on its own. However, using $\hat{H}^{\prime}_{jl}$ in
conjunction with the two other available Hamiltonians
$\hat{H}_{j}\propto\ket{b_{j}}\bra{b_{j}}$ and
$\hat{H}_{l}\propto\ket{b_{l}}\bra{b_{l}}$ in a multi-pulse sequence allows us
to do so. Next, we present the procedure to numerically and analytically
generate a precise two-mode beamsplitter using $\hat{H}^{\prime}_{jl}$.

The three operators $\hat{H}^{\prime}_{jl}$, $\hat{H}_{j}$ and $\hat{H}_{l}$
define a three-level system with basis states $\ket{b_{j}}$, $\ket{b_{l}}$,
and $\ket{b^{*}}$. $\ket{b^{*}}$ is defined so that $\ket{b^{*}}$,
$\ket{b_{l}}$, and $\ket{b_{j}}$ form an orthonormal basis of the three-
dimensional space spanned by $\ket{b^{\prime}}$, $\ket{b_{j}}$, and
$\ket{b_{l}}$. To construct $\ket{b^{*}}$, we substract from
$\ket{b^{\prime}}$ its projections on $\ket{b_{j}}$ and $\ket{b_{l}}$ and
normalize the result.

Efficiently generating a beamsplitter requires the generating Hamiltonian
$\hat{H}^{\prime}_{jl}$ to have large off-diagonal element
$\beta=\bra{b_{l}}\hat{H}^{\prime}_{jl}\ket{b_{j}}$. We write
$\hat{H}^{\prime}_{jl}$ in the basis $(\ket{b_{j}},\ket{b_{l}},\ket{b^{*}})$:

$\displaystyle\hat{H}^{\prime}_{lj}$
$\displaystyle=\begin{pmatrix}\alpha&\beta^{*}&\gamma^{*}\\\
\beta&\epsilon&\zeta^{*}\\\ \gamma&\zeta&\theta\end{pmatrix}.$ (16)

The element $\beta$ describes the beamsplitter strength. In Fig. 3 (a), we
plot the magnitude of $\beta$ for the $\hat{H}^{\prime}_{jl}$ Hamiltonian
(purple) as a function of $M$ for$j=1$ and $l=8$. This plot shows that
$\hat{H}^{\prime}_{jl}$ can be used to generate an effective beamsplitter at
large $M$, since $\beta$ remains at a value of nearly 0.4. Figure 3 (b)
displays the magnitude of $\beta$ for $M=115$ as a function of $l$ for $j=1$.
The exact values of the elements of $\hat{H}^{\prime}_{jl}$ depend on $j$,
$l$, and $M$, but critically, they remain large for all values, and approach
the value of $4/\pi^{2}$ for large $M$, shown as a solid dark line in Fig.3(a)
and (b). The resulting value of $4/\pi^{2}$ is derived by calculating
$\braket{b_{l}}{b^{\prime}}$ (or equivalently $\braket{b_{j}}{b^{\prime}}$)
for large $M$. For values where $l-j$ are a multiple of a large divisor of
$M$, departures from the nominal value $\beta=4/\pi^{2}$ are observed. For
example, small deviations at multiples of 5 and 23 can be observed in Fig. 3
for $M=115$ where 23 and 5 are the only nontrivial divisors of 115.

For large $M$, when the value of $l-j$ is not a large integer divisor of $M$,
the Hamiltonian $\hat{H}^{\prime}_{lj}$ becomes,

$\displaystyle\hat{H}^{\prime}_{lj}$
$\displaystyle=\begin{pmatrix}4/\pi^{2}&4/\pi^{2}&\sqrt{\frac{4}{\pi^{2}}(1-\frac{8}{\pi^{2}})}\\\
4/\pi^{2}&4/\pi^{2}&\sqrt{\frac{4}{\pi^{2}}(1-\frac{8}{\pi^{2}})}\\\
\sqrt{\frac{4}{\pi^{2}}(1-\frac{8}{\pi^{2}})}&\sqrt{\frac{4}{\pi^{2}}(1-\frac{8}{\pi^{2}})}&1-\frac{8}{\pi^{2}}\end{pmatrix}.$
(17)

Importantly, in this limit, $\hat{H}^{\prime}_{lj}$ becomes independent of
$l$, $j$, and $M$.

To generate exact beamsplitters, we numerically optimize amplitudes in an
interleaved pulse sequence. The desired 50-50 beamsplitter unitary is

$\displaystyle\hat{U}^{BS}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&i&0\\\ i&1&0\\\
0&0&\sqrt{2}\end{pmatrix}.$ (18)

For large $M$, we find that $\hat{U}^{BS}$ can be achieved in a seven-pulse
sequence of the form

$\begin{split}\hat{U}^{BS}&=\hat{U}^{\prime}_{lj}(\theta_{7})\cdot\hat{U}_{l}(\theta_{6})\cdot\hat{U}_{j}(\theta_{5})\\\
&\quad\cdot\hat{U}^{\prime}_{lj}(\theta_{4})\cdot\hat{U}_{l}(\theta_{3})\cdot\hat{U}_{j}(\theta_{2})\cdot\hat{U}^{\prime}_{lj}(\theta_{1}),\end{split}$
(19)

where each unitary is derived from it’s respective Hamiltonian:
$\hat{U}_{l}(\theta)=e^{-i\hat{H}_{l}\theta}$,
$\hat{U}_{j}(\theta)=e^{-i\hat{H}_{j}\theta}$ and
$\hat{U}^{\prime}_{lj}(\theta)=e^{-i\hat{H}^{\prime}_{lj}\theta}$. The
solutions for the rotation angles $\\{\theta_{1},...\theta_{7}\\}$ are given
analytically in a supplemental file 111A pdf printout of a Mathematica
notebook that calculates these values is included as Supplementary Material,
available online.. The approximate numerical values are shown below. This
solution is valid for arbitrary values of $j$ and $l$.

$i$ | $\theta_{i}/2\pi$
---|---
$1$ | 0.347136
$2$ | 0.222136
$3$ | 0.222136
$4$ | 0.125
$5$ | 0.652864
$6$ | 0.652864
$7$ | 0.777864
Table 1: Approximate numeric rotation angles for each unitary of the optimized
50-50 beamsplitter in Eq. 19.

The key achievement of this construction is that exact two-mode beamsplitters
may be achieved between any two spin-wave modes in constant (independent of
$M$) time, even for large $M$. This type of connectivity is unique relative to
most optical setups where only the two-mode beamsplitters that operate between
adjacent modes are usually easy to implement. In the future, other useful
beamsplitter constructions may be obtained by considering phase-modulation
theory, using phase modulation and single-mode phase shifts to create
arbitrary unitary operations. Investigations into other experimentally
convenient tools for linear control will remain an area for further research.

Figure 3: Beamsplitters with a single ensemble. (a) The off-diagonal matrix
element $\beta$ is plotted for the optimized momentum-space beamsplitter
$\hat{H}^{\prime}_{jl}$. $\beta$ remains large for all values of $M$
indicating an effective beamsplitter, and approaches a constant value
$4/\pi^{2}$ for large $M$ (solid line). (b) $\beta$ is plotted for $M=115$ and
$j=0$ versus $l$, indicating that beamsplitters are possible for all values of
$l-j$. Slight variation in $\beta$ is observed for values of $l-j$ that are a
large divisor of $M$, evident in the plot for values of $l$ that are multiples
of 5 and 23.

##  V Operational Fidelity

The fundamental sources of error present in the atomic spin-wave processor are
discussed in the joint Letter [18]. These errors, present for any
implementation using optical qubits in a cavity, arise from atomic saturation
and atomic emission into the cavity and into free space. Here we discuss
additional technical sources of error that will likely arise in the cold-atom
implementation.

###  V.1 Effect of momentum displacement errors

First we consider the effect of small amplitude errors in the momentum
displacement operator $\hat{\Delta}$. This operator works correctly when the
amount of phase shift leads to an integer change in the momentum index. We
consider the effect of small imperfections $\epsilon$ in the amplitude of this
operation, that results in a non-integer momentum $k\rightarrow k+1+\epsilon$.
The error $\eta_{\Delta}$ is calculated to be,

$\displaystyle\eta_{\Delta}$ $\displaystyle=1-|\\!\langle
b_{k+\epsilon}|b_{k}\rangle|^{2}$
$\displaystyle=1-\bigg{|}\frac{1}{M}\sum_{j=1}^{M}e^{2\pi i\epsilon
j/M}\bigg{|}^{2}$ $\displaystyle\approx\pi^{2}\epsilon^{2},$ (20)

in the limit of small $\epsilon$ and large $M$. The loss of quantum fidelity
is second order in the error $\epsilon$. But nonetheless, the $\hat{\Delta}$
operation will require good amplitude control. More complex pulse sequences
that are amplitude independent to higher order—similar to those used in
Nonlinear Magneto-Optical Rotation (NMOR), pulsed spectroscopy, dynamic
decoupling, and optimal control [28, 29, 30]—may be useful to eliminate this
error in experimental settings.

###  V.2 Errors from variation in atom number

The goal of this apparatus will be to achieve approximately constant atom
number per site. However, some variation will likely remain. We estimate the
errors from this variation. In the case of non-uniform atom number per site,
the cavity dressing interaction is re-written using a non-uniform projector
$\ket{b^{\prime}_{0}}$

$\displaystyle H^{k*}_{0}$
$\displaystyle=\frac{\hbar\Omega_{2}^{2}}{4\delta_{2}}\ket{b_{0}^{\prime}}\bra{b_{0}^{\prime}},$
(21) $\displaystyle\ket{b_{0}^{\prime}}$
$\displaystyle=\sum_{j=0}^{M-1}\sqrt{\frac{n^{\prime}_{j}}{N}}\ket{a_{j}},$
(22) $\displaystyle\ket{a_{j}}$
$\displaystyle=\frac{1}{\sqrt{n_{j}^{\prime}}}\sum_{l=0}^{n^{\prime}_{j}-1}\ket{g_{l}}\\!\bra{e_{l}},$
(23)

where $n^{\prime}_{j}=n(1+\epsilon_{j})$ is the erroneous factor describing
the non-uniform atom number at site $j$ and the errors $\epsilon_{j}$ are
assumed to sum to zero. With atom number variation, this cavity coupling
Hamiltonian is not equivalent to $H^{k}_{0}\propto\ket{b_{0}}\\!\bra{b_{0}}$.
The cavity interaction leads to a phase shift in a new mode
$\ket{b_{0}^{\prime}}$, a mode that is not trivially decomposable into the
orthogonal basis. Assuming the orthogonal basis must be maintained for the
desired operations, this leads to an error of

$\displaystyle\eta_{N}$
$\displaystyle=1-|\\!\braket{b_{0}}{b_{0}^{\prime}}\\!|^{2}$
$\displaystyle\approx\frac{1}{M}\sum_{j=0}^{M-1}\frac{\epsilon_{j}^{2}}{4}$
(24)

for small errors $\epsilon_{j}$. $\eta_{N}$ will likely be dominated by static
inhomogeneities in atom number, and dealing with non-uniform ensembles may
require additional work in the future. These static errors may be correctable
using compensation techniques in the pulses or perhaps appropriate re-
definition of the basis.

###  V.3 Readout

Another important ingredient for the spin-wave quantum processor is readout.
Readout is required for almost all photonic processes, and is necessary for
linear optical quantum computing. The readout process is not a focus of this
manuscript, because spin-wave readout has been studied in-depth by many
previous quantum memory experiments [31, 6, 3]. In particular, atom-cavity
systems demonstrate the most efficient readout of any type of quantum memory
with intrinsic readout probabilities of well over 90% possible [3]. Many
effective quantum networking protocols are stable to inefficiencies at this
level [32].

##  VI Initialization time

Initialization of single-photon excitations in the spin-wave memory may be
achieved by several different methods. Here, we consider probabilistically
creating, in rapid succession, single excitations in a large array of momentum
eigenstates. The level diagram for this write process is shown in the inset of
Fig. 4(b). The level scheme is the inverse of the diagram in Fig. 1 and
requires one additional longitudinal cavity mode, that can easily be selected
with the frequency of the initialization laser (also called the write laser).
The initialization laser must counter-propagate relative to the dressing laser
to maintain phase matching of both the read and write photons into the cavity
mode [22].

Memory initialization is accomplished using the standard atomic memory
heralded write process into the $\ket{b_{0}}$ mode [33], followed by a unit
displacement $\hat{\Delta}$, repeated until a large fraction of the spin-wave
modes are initialized. It is important to keep the probability of double
excitation low, since such errors are not detected by heralding. In atomic
memories, the double excitation error $\eta_{2}$ is proportional to the write
probability $p_{1}$, $\eta_{2}\propto p_{1}$, meaning that the write
probability must be kept small [32]. However, when excitations are initialized
within an $M$-mode register, the standard double-excitation error $\eta_{2}$
due to atomic emission into free space is amplified. Normally, the full error
from the write process $\eta_{w}$ is of scale $\eta_{2}$ [32]. However, in an
M-mode spin-wave register, each mode gains an independent error of scale
$\eta_{2}$ for every write process, so that the total error in each mode
compounds to a larger value $\eta_{w}\sim M\eta_{2}$.

In order to overcome this unfortunate scaling, we propose a modified heralded
initialization scheme that works in the two-ensemble configuration. The pulse
sequence is displayed in Fig. 4(a). By writing excitations into a single
ensemble (chosen to be mode $\hat{b}_{0}^{A}$ here), the excitation can be
initialized and transferred with a low-error beamsplitter operation into
ensemble $B$. Ensemble $A$ can be cleared with a standard optical pumping
pulse (labeled “clear”), before subsequent excitations are written. The
optical pumping prevents errors in the initial write procedure from
compounding in later steps.

In Fig. 4(b), we plot the approximate initialization time required to
initialize 1000 modes. The speed limits for the memory write process are
dictated by the excited state linewidth $\Gamma$ and the cavity linewidth
$\kappa$. Using these rates, and maintaining a single write error
$\eta_{w}\sim\eta_{2}\sim p_{1}$ of less than 0.001, we plot the estimated
time $T_{1000}$ required to create 1000 single excitations in Fig. 4 versus
cavity finesse $f$:

$T_{1000}\sim\frac{1000}{\eta_{w}}(\frac{1}{\Gamma}+\frac{1}{\kappa}).$ (25)

The cavity linewidth $\kappa$ is related to finesse by $\kappa=2\pi c/(lf)$
where $c$ is the speed of light and $l$ is the round-trip cavity length. We
see that the cavity lifetime becomes the limiting factor at a finesse of
around 2000, for a cavity length of 2 cm, assuming ideal detection efficiency.
Although the initialization time for 1000 excitations and $f=10^{4}$ is still
well below the maximum atomic lifetimes observed in spin-wave memories [34],
this speed limit may be a significant concern for future high capacity
memories in high finesse cavities. Deterministic initialization methods
involving single photon sources or Rydberg excitations may be necessary to
consider in the future [35, 36].

Figure 4: (a) Initialization sequence. Excitations are written into ensemble
$A$ and transfered to ensemble $B$ to prevent compounding errors. (b)
Approximate initialization time for 1000 single-photon excitations in the
ensemble for a 2 cm length optical cavity. The initialization time is limited
by both the cavity lifetime (green dot-dashed, given by
$1000/(\eta_{w}\kappa$)) and atomic excited state lifetime (blue dashed, given
by $1000/(\eta_{w}\Gamma$)) with the total time shown in orange (solid). The
level diagram for memory initialization is shown in the inset.

##  VII Squeezing and continuous variable processing

Atom-cavity experiments have recently generated record amounts of squeezing,
entanglement useful for enhancing quantum sensors [37, 38]. Squeezing of 20
dB, or a factor 100 in variance, is now achievable in systems similar to the
one proposed here. Additionally, recent optical experiments have shown how to
use squeezing as a powerful computational resource to create dual-rail cluster
states [39]. Cluster states are particularly appealing for future quantum
processors because they are deterministic and are amenable to powerful
continuous variable quantum error correction schemes [40, 21]. By creating an
$M$-mode processor, we now open the possibility to combine the cluster state
concept with large amounts of atomic spin squeezing to build a continuous
variable atomic processor. Here, we describe how to implement the optical
dual-rail cluster state scheme in the atom-cavity system. The cold atom system
leads to numerous advantages relative to the photonic implementation since the
qubits are stationary and amenable to feedback and high-fidelity processing
with no losses.

There are several viable paths to create spin squeezing in the spin-wave
memory including one- and two-axis twisting and quantum non-demolition (QND)
measurements. For example, the squeezing operation may be described by an
operator
$\hat{S}(\alpha)=\exp(\frac{1}{2}(\alpha\hat{b}_{0}^{2}-\alpha\hat{b}_{0}^{\dagger
2}))$ that squeezes only the $k=0$ mode. In this case, the mode operators are
transformed as

$\displaystyle\hat{b}_{0}$
$\displaystyle\rightarrow\hat{S}^{\dagger}(\alpha)\hat{b}_{0}\hat{S}(\alpha)=\hat{b}_{0}\cosh(\alpha)-\hat{b}_{0}^{\dagger}\sinh(\alpha),$
(26) $\displaystyle\hat{b}_{k}$
$\displaystyle\rightarrow\hat{S}^{\dagger}(\alpha)\hat{b}_{k}\hat{S}(\alpha)=\hat{b}_{k}\,\,\,(k\neq
0).$ (27)

Critically, the operator $\hat{S}$ does not affect modes $\hat{b}_{k}$ with
$k\neq 0$, that are orthogonal to $\hat{b}_{0}$. This operation allows us to
create independently squeezed spin-wave modes. In the next section, we present
a description of how to create an $M$-mode squeezed state where each $k$-mode
is spin squeezed. This state, when passed through $2M-1$ beamsplitters,
transforms into a dual-rail cluster state that may be used for universal
quantum computation. This method is a direct adaptation of seminal results in
the optical regime, creating continuous variable cluster states of light [41].

##  VIII Cluster state generation

Spin-wave continuous-variable quantum computing can be achieved using the same
basis as linear optical spin-wave computing. We consider a similar protocol to
experiments in the optical regime that have generated dual-rail cluster states
with over 10000 nodes [41].

A continuous-variable cluster state is a large entangled state defined by
nullifiers, analagous to the stabilizers of a discrete cluster state [19]. The
nullifiers are joint operators that describe noise projection of nearest-
neighbor spin-waves. The nullifiers in our spin wave case are,

$\displaystyle\epsilon^{x}_{k}$
$\displaystyle=\hat{\mathcal{X}}^{A}_{k}+\hat{\mathcal{X}}^{B}_{k}+\hat{\mathcal{X}}^{A}_{k+1}-\hat{\mathcal{X}}^{B}_{k+1},$
(28) $\displaystyle\epsilon^{p}_{k}$
$\displaystyle=\hat{\mathcal{P}}^{A}_{k}+\hat{\mathcal{P}}^{B}_{k}-\hat{\mathcal{P}}^{A}_{k+1}+\hat{\mathcal{P}}^{B}_{k+1},$
(29)

for any mode $k$ (modulo $M$) where $\hat{\mathcal{X}}^{A}_{k}$ and
$\hat{\mathcal{P}}^{A}_{k}$ are the quadrature operators for the
$\hat{b}^{A}_{k}$ spin wave. Optical experiments have achieved entanglement,
as detected by a reduction in $|\\!\braket{\epsilon^{x}}\\!|^{2}$ and
$|\\!\braket{\epsilon^{p}}\\!|^{2}$ below a value of 1/2. Current
demonstrations have reached values of around $-6$ dB [41]. State-of-the-art
spin squeezing may be able to reduce these quadrature values to $-20$ dB or
smaller, potentially reaching the fault tolerant threshold for GKP-type
quantum error correction [42, 21].

An experimental diagram is shown in Fig. 5. Two ensembles (labeled A and B)
are loaded into the bowtie cavity, one on each side. The ensembles each
provide a basis of momentum states $\ket{b^{A}_{k}}$ and $\ket{b^{B}_{k}}$ for
$0\leq k<M$. The goal is to use collective cavity quantum non-demolition (QND)
measurement or other technique [37, 38] to generate spin-squeezing in each
momentum mode, and then use collective cavity interactions to emulate the
beamsplitters required to transform the squeezed modes into a cluster state.

Figure 5: Cluster state generation scheme. (a) Optical diagram for generating
a spin-wave cluster state. QND measurements are sequentially applied to
squeeze each momentum state. Using two beamsplitters and a phase shift
$\hat{\Delta}^{A}$, a dual rail cluster state is created. Additional unshown
momentum shift operators $\hat{\Delta}^{A}$ and $\hat{\Delta}^{B}$ are
necessary to transfer spin waves in and out of the $k=0$ interaction mode. (b)
Example pulse sequence for generating a dual-rail cluster state with $M=2$.
Pulse colors are set to match Fig. 1 and the squeezing pulses are labeled with
the affected initial value of $k$.

First, cavity QND measurements or other squeezing operations $\hat{S}$ are
performed on each individual spin wave to create a stream of squeezed states.
Spin waves are transferred into and out of cavity coupling with the
$\hat{\Delta}^{A}$ and $\hat{\Delta}^{B}$ operators. Using the well-known
construction for creating dual-rail cluster states [41], two-mode
beamsplitters $\hat{H}^{BS}$ and phase shifts $\hat{\Delta}^{A}$ and
$\hat{\Delta}^{B}$ are sufficient for creation. Unlike optical cluster states,
the atomic dual rail cluster state is stationary with long coherence time, and
hence amenable to real-time computation.

In Fig. 6(a) and (b), we estimate the maximum capacity $M$ of the spin-wave
processor for storing a large cluster state. The amount of squeezing and the
capacity will be limited by several factors, including the beamsplitter errors
[18] and the ability for the atom-cavity system to generate squeezing in the
first place. But one fundamental limitation is the capacity of the atomic
system to store spin-squeezed states without a subsequent reduction in
coherence due to nonlinearity (i.e., curvature of the Bloch sphere leading to
nonlinear projections of the Bloch vector onto a 2-D plane).

State-of-the-art atom-cavity experiments can generate spin squeezing with
spectroscopic enhancement near a factor of one hundred [37, 38]. Spectroscopic
enhancement, or amount of squeezing, is defined as the entanglement-generated
improvement in the sensor’s ability to resolve a quantum phase. For a single
spin wave, we write the spectroscopic enhancement [43]

$S_{s}=R_{s}\mathcal{C}^{2},$ (30)

where $R_{s}=2\textrm{Var}(\hat{J}_{z})/N$ is the reduction in variance of the
expectation value of the collective spin operator $\hat{J}$ along a particular
axis (chosen as $z$ here). $\mathcal{C}$ is the spin coherence of the ensemble
defined as $\mathcal{C}=J/(N/2)$, where $J\equiv\langle\hat{J}\rangle$ is the
expectation value of the total projection of $\hat{J}$. $J$ can have values
between $N/2$ (full spin coherence) and $0$ (no spin coherence). $S_{s}>1$ is
both a witness for atomic entanglement as well as a measure of the
entanglement-generated improvement in the quantum sensor.

We now consider simultaneous equal squeezing in all $M$ modes of an ensemble.
In this case, we reduce the quantum noise in all spin-wave modes by an equal
amount $R_{s}$. $S_{s}$ is defined to be the squeezing that would be observed
in a spin-wave, if no other modes were squeezed. However, when all modes are
squeezed at the same time, we observe a lower amount of squeezing in each
mode, that we denote $S_{M}$. To calculate $S_{M}$, we must include the
additional reduction in total $\mathcal{C}$ due to Bloch sphere curvature,
that leads to a compounding reduction in squeezing in every mode. Then, the
observed squeezing of a single mode in the presence of squeezing in all other
modes is

$\displaystyle S_{M}$ $\displaystyle\approx R_{s}\prod_{j=0}^{M-1}C_{j}^{2},$
(31)

where $C_{j}$ is the spin coherence in each mode that additionally limits the
total squeezing. The coherence of each mode is limited by quantum back action
and the curvature of the Bloch sphere [44]. Each spin-wave has an rms (root
mean squared) back action around the Bloch sphere of angle
$\theta_{rms}=\theta_{SQL}\sqrt{R_{s}}$ where $\theta_{SQL}=1/\sqrt{N}$ is the
standard quantum limit in radians (see, for example, Ref. [38]). This leads to
a fundamental loss in coherence in each spin wave, due to Bloch sphere
curvature:

$\displaystyle C_{j}$ $\displaystyle\approx e^{-\theta_{rms}^{2}/2}\approx
e^{-\theta_{SQL}^{2}R_{s}/2}.$ (32)

By combining Eq. (32) and Eq. (31), the maximum $M$-mode squeezing is found to
be

$\displaystyle S_{M}$ $\displaystyle\approx R_{s}e^{-M\theta^{2}_{SQL}R_{s}}.$
(33)

Quantum inefficiency and other decoherence sources that would cause further
loss of coherence are not considered.

In Fig. 6 we plot the value of $M$ that is achieved for $S_{M}=100$, that is,
20 dB of squeezing versus $N$ and $R_{s}$. The maximum capacity $M_{opt}$ is
found at $R_{s}=eS_{M}$. The maximized value is

$M_{opt}=\frac{N}{2eS_{M}}.$ (34)

A processor reaching the levels of performance in Fig. 6 would be state-of-
the-art and likely useful for deterministic quantum networking and
entanglement distribution. Further, 20 dB of squeezing in each mode would
reach or nearly reach the fault-tolerant threshold [21] for continuous
variable quantum computing.

Figure 6: Capacity $M$ of a spin-wave cluster that can achieve 20 dB of spin
squeezing simultaneously in each mode ($S_{M}=100$) as a function of the atom
number $N$ and squeezing $R_{s}$, the quantum noise reduction in each mode.
$M$ is optimized at $R_{s}=eS_{M}$ (pink line)

##  IX Near-term Applications

Overall, deterministic continuous-variable quantum computing has significantly
better prospects for scalability than linear optics with single-quanta
excitations, because the linear optics scheme requires a significant resource
overhead to achieve deterministic processing [21]. However, in the near term,
small-scale linear optical processing inside of a multiplexed quantum memory
may be a significant boon toward realizing a quantum repeater with medium to
high speed of entanglement generation over long distances (over 100 km) [8,
45]. Meanwhile, in the continuous-variable scheme, the creation of spin-wave
entangled states and the study of their uses for quantum sensing and
networking applications should be one of the first experimental goals.

The multimode quantum processor presented here is also ideal for certain
classes of quantum sensing problems that involve data distributed between
multiple modes. The spin wave processor utilizes $2M$ independent modes within
the two $N$-atom ensembles, yielding enhanced capability for certain classes
of measurements. Recent work has shown that distributed quantum sensors enable
new sets of applications involving measurements of extended systems [46, 47,
48, 49], and the spin wave processor may extend these protocols to sensing and
receiving data distributed into multiple spatial or temporal modes.

##  X Technical Challenges

Additional technical challenges will likely arise when building an experiment
to accomplish this proposal. The application of the Raman dressing laser
results in a shift in the cavity resonance frequency by amount
$\Delta\omega_{c}\approx N_{g}g^{2}/\delta_{1}$ and a shift in the two-photon
resonance frequency $\omega_{eg}$ by amount
$\Delta\omega_{eg}\approx\Omega_{d}^{2}/\delta_{1}$. These shifts will need to
be taken into account to achieve accurate dynamics in the two-photon system.
To achieve effective operations, the laser detunings may need to be actively
stabilized to account for these cavity and state shifts, as has been done in
recent entanglement-generation experiments [37, 38].

Rapidly addressing a large capacity of quantum bits or modes is a ubiquitous
challenge in nearly every experimental quantum platform. Here, we achieve that
end simply, using a single optical cavity mode and an off-resonant light
source. One advantage of the proposal is that the $\hat{\Delta}$ operator can
be implemented with light that is at a wavelength far from atomic resonance,
with no stringent wavelength or power requirements. We envision a fast
electro-optic system operating in the near-infrared, which can operate with
switching ranges of well over 1 GHz.

##  XI Conclusion

Overall, we are optimistic that holographic spin-wave excitations in a cavity-
coupled ensemble may become a useful platform for quantum information
processing. The system combines several attractive characteristics including
efficient readout into a single optical cavity mode, large capacity, and
universal processing capabilities able to achieve high fidelity. In the long
term, many quantum information processing devices will likely require
networked operation, and atom-cavity systems will be a preeminent platform to
achieve this.

##  XII Acknowledgements

P.B. and A.V.G. acknowledge funding by ARO MURI, DARPA SAVaNT ADVENT, AFOSR
MURI, AFOSR, NSF PFCQC program, DoE ASCR Accelerated Research in Quantum
Computing program (award No. DE-SC0020312), the DoE ASCR Quantum Testbed
Pathfinder program (award No. DE-SC0019040), and U.S. Department of Energy
Award No. DE-SC0019449.

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