Tunable spin model generation with spin-orbital coupled fermions in optical lattices
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Tunable spin model generation with spin-orbital coupled fermions in optical lattices
Mikhail Mamaev,1, 2, ∗ Itamar Kimchi,1, 2, 3 Rahul M. Nandkishore,2, 3 and Ana Maria Rey1, 2
1
JILA, NIST and Department of Physics, University of Colorado, Boulder, CO 80309, USA
2
Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, USA
3
Department of Physics, University of Colorado, Boulder, CO 80309
(Dated: November 10, 2020)
We study the dynamical behaviour of ultracold fermionic atoms loaded into an optical lattice under the pres-
ence of an effective magnetic flux, induced by spin-orbit coupled laser driving. At half filling, the resulting
system can emulate a variety of iconic spin-1/2 models such as an Ising model, an XY model, a generic XXZ
arXiv:2011.01842v2 [cond-mat.quant-gas] 9 Nov 2020
model with arbitrary anisotropy, or a collective one-axis twisting model. The validity of these different spin
models is examined across the parameter space of flux and driving strength. In addition, there is a parame-
ter regime where the system exhibits chiral, persistent features in the long-time dynamics. We explore these
properties and discuss the role played by the system’s symmetries. We also discuss experimentally-viable im-
plementations.
I. INTRODUCTION these spins. Conventional lattice systems are often captured
by isotropic Heisenberg models, while the presence of mag-
Understanding and quantifying the behaviour of interacting netic flux allows for tunability of the spin interactions. As
quantum particles in lattices is a fundamental goal of modern we show here, the flux leads to more elaborate spin models
quantum science. While there is a plethora of research direc- such as anisotropic XXZ models, collective one-axis twisting
tions, one vital aspect is the response of particles to externally- Hamiltonians [13], or Dzyaloshinskii-Moriya (DM) interac-
imposed magnetic fields. Such fields induce an effective flux tions [14, 15]. The latter is particularly intriguing, as it ex-
that threads through the plaquettes of the lattice [1, 2], cou- hibits chirality, topological features and complex phase dia-
pling the charge and spin degrees of freedom and modifying grams depending on the flux [16, 17]. A system that can re-
the particle dynamics. Interpreting the dynamical response alize several different spin models with easy tunability can be
to an applied flux is important for many applications in con- very useful to the field, as current-generation optical lattice
densed matter, including ferroelectrics [3], spintronics [4, 5] experiments have only recently begun to probe anisotropic in-
and spin-glass physics [6, 7]. teracting spin physics [18, 19].
One of the best ways to study such phenomena is with Here we study the case when the internal atomic states are
ultracold atomic experiments. State-of-the-art ultracold sys- driven by an external laser, which imprints a site-dependent
tems provide exceptional levels of cleanliness, isolation and phase that emulates a magnetic flux. We use the drive strength
tunability. They allow for pristine implementations of iconic and the magnetic flux as tuning parameters, and show the dif-
Fermi- or Bose-Hubbard models that describe interacting par- ferent types of spin interactions that can be realized. We ex-
ticles in a lattice, with additional terms to account for the syn- plore the dynamical properties of these interactions and spec-
thetic magnetic fields [8]. A magnetic flux is easy to impose ify regimes in the parameter space of flux and driving strength
and control with tools such as laser driving, using Raman cou- where the system’s time evolution can be captured by simple
plings or direct optical transitions. models such as Ising, XY, XXZ or collective-spin one-axis
There has been a great deal of theoretical work on Fermi- twisting. We also study a regime where the corresponding
or Bose-Hubbard models with synthetic gauge fields that ul- spin model maps to a Heisenberg model in a twisted frame,
tracold experiments could investigate, exploring ground-state causing the long-time dynamics to develop non-trivial features
phases [9–11] or phenomena such as many-body localiza- such as infinite-time magnetization and chiral spin imbalance.
tion [12]. However, the interplay of magnetic flux together These latter features are not limited to the strongly interact-
with particle interactions can lead to complex dynamical be- ing regime, but also hold in the weakly interacting limit of
haviour that still lacks a good theoretical understanding. In the the Fermi-Hubbard model where atomic motion is relevant.
case of fermions, even non-interacting atoms can exhibit non- We discuss the role that symmetry plays in preventing the
trivial behaviour due to Pauli exclusion, while the addition of system from relaxing. Our predictions can be readily imple-
Hubbard repulsion renders the dynamics even more complex. mented in many ultracold systems, and are especially relevant
In this work, we focus on fermionic atoms loaded into 3D for alkaline-earth or earth-like atoms in 3D lattices or tweezer
optical lattices. In the Mott insulating limit with one atom per arrays, which provide exceptional coherence times [20–22] to
site, each atom acts as an effective spin and the physics can overcome the inherent slow interaction rates while avoiding
be simplified by mapping to an effective spin model. This issues from heating given the very low spontaneous emission
spin model emerges via virtual atomic tunneling processes rates of their low-lying electronic levels.
that lead to second-order superexchange interactions between The outline of the paper is as follows. Section II introduces
the underlying Fermi-Hubbard model and derives the corre-
sponding effective spin models that emerge at half filling. A
dynamical classification of the spin model behaviour in dif-
∗ mikhail.mamaev@colorado.edu ferent parameter regimes is given. Section III focuses on the2
low drive regime, and discusses the persistent magnetization (a) J ( c) |↑⟩= |e⟩
U
or chiral features that can be observed due to the additional e 1 (|↑⟩-|↓⟩)= |←⟩ |→⟩ = 1 (|↑⟩+|↓⟩)
symmetry of a Heisenberg model in a twisted frame. Sec- Ωⅇⅈjϕ 2 2
g
tion IV provides a detailed discussion on possible experimen-
H SE |↓⟩=ⅇⅈjϕ |g⟩
tal implementations. j j+1 ...
(b)
e
Ω ~J 2 /U
ϕ ϕ ϕ ϕ
H SE :
II. TUNABLE SPIN DYNAMICS g
j-1 j j+1 J J
A. Fermi-Hubbard and spin model FIG. 1. (a) Schematic of Fermi-Hubbard dynamics in an optical lat-
tice, with nearest-neighbour tunneling rate J, onsite repulsion U ,
and on-site driving Ω. When the laser drive has a wavelength that
The system we describe is a three-dimensional (3D) optical
is incommensurate with the underlying optical lattice wavelength,
lattice loaded with ultracold fermionic atoms, each possessing the laser induces spin-orbit coupling (SOC) through a spatially-
two internal states σ ∈ {g, e} corresponding to a spin-1/2 de- dependent phase eijφ in the drive term, with φ controlled by the
gree of freedom. The effective system dimensionality is freely drive implementation (e.g. laser alignment or lattice spacing, see
tuned by changing the lattice confinement strengths. We focus Section IV). (b) The SOC phase creates effective magnetic flux on
on the case where the system is effectively 1D by considering plaquettes of the ladder state structure in 1D, with lattice index j
a strong confinement in two directions that suppresses tun- along the length of the ladder and internal states e, g correspond-
neling along them. Along the direction atoms can tunnel, we ing to the individual rungs. An atom tunneling around one plaquette
assume a lattice of L sites populated by N atoms. The system picks up a total phase of φ. (c) At half filling and strong interac-
is depicted in Fig. 1(a). The Hamiltonian of the system is, tions U/J 1, the atomic spin σ at different lattice site can be
dressed by the laser drive, which modifies the resulting spin dynam-
ics dominated by second-order virtual superexchange processes with
Ĥ = ĤFH + ĤΩ rate ∼ J 2 /U , up to possible normalization from the drive Ω.
X † X
ĤFH = −J ĉj,σ ĉj+1,σ + h.c. + U n̂j,e n̂j,g
j,σ j (1) atoms,
Ω X ijφ †
ĤΩ = (e ĉj,e ĉj,g + h.c.).
2 j σ̂jx = â†j,↑ âj,↓ + h.c. ,
σ̂jy = −i â†j,↑ âj,↓ − h.c. , (3)
Here ĉj,σ annihilates an atom with spin σ on site j. Atoms tun-
nel at rate J and exhibit onsite repulsion of strength U . In ad- σ̂jz = â†j,↑ âj,↑ − â†j,↓ âj,↓ .
dition to the standard Fermi-Hubbard term ĤFH , we include a
This is just the Abrikosov pseudo-fermion representation [27,
laser-driving term ĤΩ that induces the desired flux φ through
28]. Standard second-order perturbation theory then leads
a spatially-dependent phase eijφ , providing model tunability.
to the following general superexchange (SE) spin model (see
The differential phase imprinted by the laser implements a net
Appendix A for derivation),
spin-orbit coupling (SOC) by generating a momentum kick
to the atoms while flipping their spin [1, 2, 23–26]. The re- ĤSE ≈ Jk
X
σ̂jx σ̂j+1
x
+ σ̂jy σ̂j+1
y
+ J⊥
X
σ̂jz σ̂j+1
z
alization of this effective flux in 1D is depicted in Fig. 1(b).
j j
The drive can be implemented with a direct interrogating laser X X (4)
y
σ̂jx σ̂j+1 σ̂jy σ̂j+1
x
σ̂jx .
(if the internal states g, e are split by optical frequency), or + JDM − + JΩ
a Raman coupling (see Section IV for details). We assume j j
φ ∈ [0, π] without loss of generality.
The first two terms correspond to an XXZ model. The third
For sufficiently strong interactions U/J 1 at half fill- term is a DM interaction with a plane axis of ẑ [thus also tak-
ing N/L = 1 with one atom per site, double occupancies of ing the form of ẑ · (~σj × ~σj+1 )]. The interaction coefficients
lattice sites are strongly suppressed. The system dynamics in are,
this regime may be approximated with a spin model, as de-
φ
picted in Fig. 1(c). We define a dressed spin basis at different J 2 U 2 cos(φ) − Ω2 cos2 ( 2 )
lattice sites by making a gauge transformation, defining new Jk = · ,
U U 2 − Ω2
fermionic operators {âj,↑ , âj,↓ } to remove the SOC phase, φ
J 2 U 2 − Ω2 cos2 ( 2 )
J⊥ = · ,
U U 2 − Ω2
ĉ†j,e |0i ≡ â†j,↑ |0i ↔ |↑ij , J 2 (Ω2 − 2U 2 ) sin(φ)
(5)
(2) JDM = · ,
e−ijφ ĉ†j,g |0i ≡ â†j,↓ |0i ↔ |↓ij , U 2(U 2 − Ω2 )
φ
Ω 2J 2 Ω sin2 ( 2 )
and define conventional spin operators for these dressed JΩ = − .
2 U 2 − Ω23
These share the conventional J 2 /U superexchange energy As one limiting regime of the XXZ model, in the strong-
scale, with different normalization factors coming from the drive regime Ω Jk , J⊥ ,JDM if we also have φ ≈ π, the
interplay between the drive and flux. This model is valid for coefficients Jk and J⊥ are approximately equal and opposite
all flux values φ, and all drive frequencies Ω far from the reso- (Jk ≈ −J⊥ ), causing them to cancel each other out and leave
nance point |Ω| = |U |, requiring a spacing of ||U | − |Ω|| J an Ising model,
to prevent higher-order effects. Note that there can also be X X
other resonances such as |Ω| = 6 |U |/2, as considered in ĤIsing = Jk σ̂jx σ̂j+1
x
+ JΩ σ̂jx
e.g. Ref. [29], though the relevant width is far smaller as j j
will be seen in the next section. These spin interactions are [φ ≈ π, Ω Jk , J⊥ ,JDM ].
anisotropic even if the drive is turned off, Ω = 0, because we (7)
are using a dressed basis to absorb the SOC phase (see Ap- As another special regime of the XXZ model, there is a line
pendix A). In the context of the underlying Fermi-Hubbard in parameter space where the Jk coefficient vanishes, requir-
model, one can use a fast pulse of a SOC drive to prepare a p
ing Ω = ±U cos(φ) sec(φ/2), which causes the (σ̂jx σ̂j+1 x
+
desired initial state such as a product state in the dressed ba- y y
sis, after which the drive may be turned off if desired (see σ̂j σ̂j+1 ) terms of Eq. (4) to vanish. Assuming that we also
Section IV for details). There is also a single-particle term still have a strong drive Ω Jk , J⊥ ,JDM , the DM interac-
JΩ corresponding to the drive. This term contains an addi- tion remains averaged out as well, leaving only the σ̂jz σ̂j+1 z
tional single-particle superexchange contribution, but this is term along with the drive (which is now transverse to the in-
typically negligible compared to the bare drive, and so we can teraction). If we again make the rotating-wave approximation
approximate JΩ ≈ Ω/2. σ̂jz σ̂j+1
z
≈ 21 (σ̂jy σ̂j+1
y
+ σ̂jz σ̂j+1
z
), we arrive at an XY model,
If the drive is off, Ω = 0, this spin model commutes with
J⊥ X y y ΩX x
and thus conserves total hŜ z i, where Ŝ γ = 21 j σ̂jγ . When σ̂j σ̂j+1 + σ̂jz σ̂j+1
z
P
ĤXY = + σ̂
2 j 2 j j
the drive is instead very strong, Ω Jk , J⊥ , JDM , total hŜ x i p
is approximately conserved instead because the drive imposes [Ω = U cos(φ) sec(φ/2)].
an energy penalty to flipping spins along the ±x Bloch sphere (8)
direction. Note that while the superexchange coefficients can We emphasize that this XY model with a strong (commuting)
be modified by the drive, as a first rough estimate the high- field, and the underlying Ising model with a strong transverse
drive condition Ω Jk , J⊥ , JDM may be interpreted as a field are only equivalent under unitary time-evolution [30].
drive faster than the bare superexchange rate, Ω J 2 /U . For the strong drive regime Ω Jk , J⊥ ,JDM with small
flux, the coefficients Jk , J⊥ are almost equal to J 2 /U . We
can thus collect the XXZ model of Eq. (6) into an isotropic
B. Spin model regimes Heisenberg term and a perturbative σ̂jx σ̂j+1x
component. In
this regime the system can be approximated with a collective-
While the general spin model of Eq. (4) is complex, there spin one-axis twisting (OAT) model, whose dynamical prop-
are parameter regimes where its form simplifies, allowing the erties can be explored using restrictions to the fully-symmetric
dynamical emulation of other more conventional spin models. Dicke manifold [31–34]. The model is written as,
For a strong drive Ω JDM , the DM interaction is averaged
out and can be neglected. Furthermore, assuming the drive ĤOAT = P̂Dicke ĤXXZ P̂Dicke
also satisfies Ω Jk , J⊥ , the σ̂jy σ̂j+1y
and σ̂jz σ̂j+1
z
terms are 2(Jk + J⊥ ) ~ ~ 2(Jk − J⊥ ) x x
equivalent in unitary evolution under a rotating-wave approx- = S·S+ Ŝ Ŝ + 2JΩ Ŝ x ,
L−1 L−1
imation, allowing us to interchange and collect them together
[φL 1, Ω Jk , J⊥ , JDM ].
via σ̂jy σ̂j+1
y
≈ σ̂jz σ̂j+1
z
≈ 21 (σ̂jy σ̂j+1
y
+ σ̂jz σ̂j+1
z
). This leaves
(9)
an XXZ-type model,
The operator P̂Dicke projects to the Dicke manifold, spanned
X Jk + J⊥ X y y by the collective-spin states |S = L/2, M i which are eigen-
σ̂jx σ̂j+1
x
σ̂j σ̂j+1 + σ̂jz σ̂j+1
z
ĤXXZ = Jk + states of Ŝ x |S, M i = M |S, M i and S ~ ·S
~ |S, M i = S(S +
2
j j
1) |S, M i, with S ~ = (Ŝ , Ŝ , Ŝ ). Here M takes values
x y z
X
+ JΩ σ̂jx [Ω Jk , J⊥ , JDM ]. −S, −S + 1, . . . S − 1, S. The flux must be small compared
j to 1/L rather than to 1, because the collective regime validity
(6) depends on the Dicke manifold gap, which shrinks with sys-
Note that this model is not the same as simply taking the XXZ- tem size for nearest-neighbour interactions. Note that while
like piece from the first line of Eq. (4), as here the single- the above model requires a strong drive, like the XXZ model
particle drive term commutes with the XXZ term, making it we can also write a one-axis twisting model in the Ω = 0
easy to account for in unitary evolution. One could also write regime as well (see Appendix B). Similar low-drive collective
an XXZ model for the no-drive limit Ω = 0 (see Appendix B), physics were explored in Ref. [35] for the weakly-interacting
for which we would just keep the XXZ portion of Eq. (4); this regime; in contrast, here we have a strongly-interacting model
can be valid in the no-drive limit if the JDM term vanishes that nonetheless allows us to map the nearest-neighbour su-
parametrically due to its sin(φ) factor for φ ≈ 0, π. perexchange interactions to a collective-spin model through4
gap protection. Moreover, as discussed in Ref. [34], the col- drive. The second evolution is,
lective behaviour can be more robust when mapping from an
XXZ model with anisotropy slightly below unity (on the easy- (X)
O 2 x
|ψ0 i= |→ij measuring C (X) = hŜ i, (12)
plane side), which this model can realize as discussed in the j
L
next section.
√
Finally, for a small drive Ω . Jk , J⊥ ,JDM , we must use where |→ij = (|↑ij + |↓ij )/ 2. Here we label our con-
the full model of Eq. (4). If the drive is turned off completely trast C (X) as the magnetization. For a large drive Ω
(Ω = 0), the resulting interaction can be written as, Jk , J⊥ , JDM only the first C (Z) evolution will see non-trivial
dynamics, as C (X) is approximately conserved by the drive.
J2 X On the other hand, for a weak drive Ω . Jk , J⊥ , JDM we can
ĤHeisen+T = cos(φ) σ̂jx σ̂j+1
x y
+ σ̂jy σ̂j+1 + σ̂jz σ̂j+1
z
U j have non-trivial dynamics for both evolutions depending on
y the model in question. In principle a full rigorous compari-
− sin(φ) σ̂jx σ̂j+1 − σ̂jy σ̂j+1
x
, [Ω = 0]. son should consider all possible high-energy state properties
(10) rather than the two selected above. Since we mainly seek a
We label the spin interaction in this regime as Heisen- qualitative understanding of model regimes, we have chosen
berg+twist (Heisen+T). While a DM interaction is present, its a pair of experimentally-simple evolutions for which at least
effect for the above parameters can be simplified to Heisen- one will have non-trivial dynamics at every point in parameter
berg model physics in a twisted frame of reference, as will be space.
discussed in Section III. Having a small non-zero Ω . JDM To determine the validity of a particular spin model,
will not significantly change this picturePaside from adding the we compute the time-dependence of C (α) for both evo-
corresponding single-particle term JΩ j σ̂jx , since the coef- (α)
lutions α ∈ {Z, X} using a spin model CSpin (where
ficients of the model are only weakly dependent on Ω in this
regime. For a larger Ω JDM , the DM term will be rotated Spin ∈ {Ising, XY, OAT, XXZ, Heisen + T}) and the
(α)
out. Fermi-Hubbard model CFermi [re-written to reflect the basis
In addition to all of the above, we can also add an extra field rotation of Eq. (2)], out to a time of,
through the use of theP laser drive’s detuning δ, which would tf = 4/max(|Jk |, |J⊥ |), (13)
take the form of 2δ j (n̂j,e − n̂j,g ) in the basis of the bare
Fermi-Hubbard model, thus adding a spin term of the form which is four times the timescale of the fastest superexchange
∼ 2δ j σ̂jz . While such a term does not commute with the
P
interaction strength at any given point in parameter space.
rest of the Hamiltonian, if |δ| |U 2 − Ω2 |, the superex- This timescale is used for every model except the OAT, for
change model will remain the same to good approximation. which we instead use tf = 8/|Jk − J⊥ | (16 time units of
This permits the addition of an extra single-particle term with-
the twisting term (Ŝ x )2 without the L-dependence, which is
out changing the spin interactions. We do not explicitly do so
sufficient to observe entanglement properties such as spin-
in this work, but such a detuning nonetheless provides yet an-
squeezing). A dynamical error metric for a given spin model
other tuning parameter that can be implemented without the
is defined as,
need for additional experimental ingredients.
s ˆ
(α) 1 tf h (α) (α)
i2
∆Spin = dt CFermi (t) − CSpin (t) . (14)
tf 0
C. Dynamical model comparisons
This metric is a root-mean-square error giving the average dif-
ference between a contrast measurement of the Fermi and spin
It is useful to know where the various simplified models
model being considered over the time interval [0, tf ]. For ev-
discussed in the previous section are applicable. The most rig-
ery choice of φ, Ω in the parameter space there will be two
orous metric of dynamical model agreement is state fidelity,
error metrics for the two evolutions α ∈ {Z, X}, and we take
but such a comparison tends to be unnecessarily harsh be-
the worse of the two,
cause the fidelity can drop with increasing system size while
experimentally-relevant observables remain in agreement. We h
(Z) (X)
i
instead evaluate the validity of the spin models through com- ∆Spin = max ∆Spin , ∆Spin . (15)
parison of simple collective observables.
We examine two typical time evolutions of the system, Of course, the disagreement between the models will tend to
starting from product initial states. The first evolution is, grow at longer times, and to an extent this analysis is quali-
tative. We choose four times the superexhange rate because
this should be sufficient to see non-trivial contrast decay, and
2
q
(Z)
O
(Z) be useful for applications such as spin-squeezing or entangled
|ψ0 i = |↑ij measuring C = hŜ y i2 + hŜ z i2 .
j
L state generation.
(11) Fig. 2(a) shows a color plot of the resulting error metrics for
The observable C (Z) is the spin contrast, chosen such that the different spin models. The color scheme is chosen such
there are no fast single-particle oscillations coming from the that a given model’s color completely vanishes when its error5
80 (a) (b)C (Z ) (c)C (Z )
Ising
ϕ=π ϕ=1.25
XY
1. Ω/J=15 1. Ω/J=62
0.8 0.8
OAT 0.6 0.6
XXZ 0.4 0.4
Heisen+T 0.2 0.2
60 tJ tJ
50. 100.150.200. 50. 100.150.200.
(d)C (Z ) ϕ=0.5 (e)C (X) ϕ=π/2
Ω=U 1. Ω/J=15 1. Ω/J=0
0.8
Ω 40 0.95 0.6
J 0.4
0.9 0.2
tJ tJ
0. 50. 100.150.200. 50. 100.150.200.
20 Ω=U/2 (f )Δ 80
2
td ( JU )
4. (g)
J∥ =0 60 102
2.
Ω 40 101
J
ϕ
0 1. 2. 20 100
-2. Ω/U
0 π/4 π/2 3π/4 π 0.3
0 10-1
0 π/4 π/2 3π/4 π
ϕ -4. 1.3 ϕ
FIG. 2. (a) Dynamical regimes of the driven Fermi-Hubbard model compared to various spin models. The system (size L = 8) is evolved to a
fixed time tf (four times the fastest superexchange rate), and the contrast observable of the Fermi model is compared to each listed spin model
through an error metric [Eq. (15)]. The color scheme depicts regions where the error metric of the corresponding spin model is small; the color
of a given model fully vanishes when ∆Spin ≥ 0.25, corresponding to an average error of 0.25 in a contrast measurement (see Appendix B
for more details on the color scaling). Regions in black indicate regimes where the spin model description breaks down altogether due to
higher-order processes from resonances such as Ω = U or Ω = U/2, with the RGB color coordinates scaled down by the error of the full
spin model ∆SE . (b-e) Snapshots of contrast evolutions for the Ising, XY, OAT, XXZ and Heisen+T (Heisenberg+twist) models respectively,
for specific points in the parameter regime as indicated by filled circles in panel (a). The former three plot the evolution of the high-drive
contrast C (Z) , while the Heisen+T plot shows evolution of the low-drive contrast C (X) . (f) Anisotropy of the XXZ model as a function of flux
for different fixed values of drive strength relative to Hubbard repulsion. (g) Characteristic timescale td needed for the contrast to decay down
to 1/e, using C (Z) for high drive Ω/J J 2 /U and C (X) for no drive Ω = 0. Black regions are points where Ω = U , and the spin model
description is invalid due to resonance. Purple points indicate parameters for which the contrast does not decay below 1/e at any time. This
occurs trivially for φ = 0 (where the spin model is a pure Heisenberg model and no dynamics occur for product states), and for the special
regime of φ 1, Ω = 0 (which is discussed in Section III).
metric reaches ∆Spin ≥ 0.25, corresponding to an average er- Finally, for low drive Ω = 0 we have the Heisenberg+twist
ror of 0.25 in measurements of C (X) or C (Z) . The Hubbard model (blue). Note that the system in this regime can also be
interaction strength is chosen to be U/J = 50, well into the described by a low-drive version of the XXZ or OAT mod-
Mott insulating regime to ensure that it does not contribute els (see Appendix B), which is why the corresponding colors
additional error. We see the parameter regimes as described in are also present for Ω = 0. The regions in blue are points in
Section II B. Large flux φ ≈ π and strong drive corresponds to parameter space where the chiral DM interaction exclusive to
the Ising model (red). The line in parameter space satisfying the Heisenberg+twist is necessary to capture the correct time-
Jk = 0 corresponds to the XY model (green). There is a nar- evolution.
row red line adjacent to this regime where the Ising model Figs. 2(b-e) show snapshots of the relevant models’ time-
also looks to be valid, but this is a spurious effect caused evolution from different points of the diagram. The dark re-
by the Ising evolution happening to align with the Fermi- gion near Ω = U is the resonance point where the superex-
Hubbard out to the specific timescale tf we use. For small change denominators vanish and second-order perturbation
flux φL 1 and large drive, we have the OAT model (yel- theory breaks down, causing no spin model to be valid. There
low). Underlying the other models is the XXZ model in white, is an additional resonance point near Ω = U/2 corresponding
which is valid throughout most of the parameter regime. The to a second-order resonant process not captured by the spin
XXZ anisotropy parameter ∆ = 2Jk /(Jk + J⊥ ) is plotted in model [29], although the width of this resonance is smaller.
Fig. 2(f); we can attain any ferromagnetic or easy-plane value We also note that the associated timescales speed up as the
for Ω/U < 1, and any antiferromagnetic value for Ω/U > 1. resonance point Ω = U is approached. Fig. 2(g) plots a6
characteristic timescale td defined as the time needed for the of the DM term. In our system the coefficients are (again,
contrast to decay down to 1/e, using the C (Z) contrast for maintaining Ω = 0),
Ω/J J 2 /U (the high drive limit, all data points except
the Ω = 0 line) and C (X) for Ω = 0. This allows for an evalu- ∆Ω = sec(φ), D = − tan(φ). (17)
ation of experimental tradeoff, where one can move closer to
the resonance for faster timescales at the cost of weaker model The non-zero flux causes this model to deviate away from a
agreement. There is also a region of φ 1, Ω = 0 where the conventional Heisenberg model by a set of position-dependent
contrast does not fully decay at any time despite no obvious local rotations of the spin variables [37] because we are in a
conservation law protecting it. This persistent magnetization dressed basis, see Eq. (2). However, we can map back to a
effect will be discussed in Section III. Heisenberg model at the price of changing the initial state.
While these simulations are for relatively small system size More concretely, the dynamics of
L = 8, in general the regimes of validity do not undergo sig- (X)
O
nificant change as the size increases because the interactions ĤHeisen+T evolving |ψ0 i = |→ij , (18)
are nearest-neighbour. The only exception is the OAT model, j
for which the regime will shrink with increasing L because it
requires φL 1 (the yellow region looks relatively large here which correspond to the low-drive evolution from the prior
because we use a small L). We also note that open boundary section, are equivalent to:
conditions are used for the above simulations with all mod-
J2 X
els except the OAT; this is to ensure that the chiral proper- ĤHeisen = ~σj · ~σj+1
ties of the DM interaction are captured, as will be explored in U j
(19)
Section III. Comparisons with the OAT use a periodic Fermi- i
P z O
Hubbard model because open boundaries can lead to a minor evolving |ψ0Spiral i = e 2 j jφσ̂j |→ij ,
j
but non-zero offset to the contrast even in the thermodynamic
limit. with the initial state becoming a spiral in the plane of the DM
For the simpler regimes such as the Ising or one-axis twist- interaction, rotating by an angle φ per lattice site under a uni-
ing model, the dynamics are well understood and can have i
P z
tary transformation Û = e 2 j jφσ̂j . The mapping is exact
analytic solutions [36]. More general XXZ-type dynamics
for open boundary conditions and Ω = 0. Periodic boundaries
can be complex to treat, as even exact 1D Bethe ansatz tech-
can lead to incommensurate mismatch if the flux is not a mul-
niques are difficult for full dynamical evolution. However,
tiple of 2π/L (see discussion in Appendix D). Hereafter, we
the parameter regime of low drive Ω = 0 where the Heisen-
berg+twist model is valid offers a special case. The dynamics refer to the dynamics under ĤHeisen+T as the gauged frame
there are non-trivial, but exhibit special long-time features that (working in a dressed basis), and the equivalent dynamics un-
can be understood from even the non-interacting limit, offer- der ĤHeisen as the un-gauged frame (for which the quantiza-
ing analytic tractability while still simulating the dynamical tion axis of the Bloch sphere is set by the bare atomic states g,
behaviour of a strongly interacting model. In the next section, e with no site-dependent phases).
we will focus on this regime in more detail. The spiral structure from non-zero flux causes the system’s
dynamics to exhibit non-trivial features due to the additional
underlying symmetry of the Heisenberg model. As the sim-
III. PERSISTENT LONG-TIME BEHAVIOUR plest example of such features, in Fig. 3(a-d) we plot the time-
evolution of the collective magnetization hŜ x i in the gauged
(X)
A. Long-time magnetization profiles frame (starting from |ψ0 i) for different values of flux. We
consider both the Fermi-Hubbard model in the appropriate ba-
sis for different values of Hubbard repulsion U/J [panels (a-
Having shown the different regimes of spin models that can
be realized with laser-driven SOC optical lattice systems, we c)] and the spin model ĤHeisen+T [panel (d)]. We find that
focus on the regime of Ω = 0 where the DM interaction plays in all cases, after an initial decay there is a non-zero mean
a role. Conventionally, the ground-state properties of systems magnetization that persists to infinite time,
including DM interactions can already be quite complex [17]. ˆ T
1
In our case, we can work directly with the Heisenberg+twist hŜ x it→∞ = lim dthŜ x i(t), (20)
model of Eq. (10). For clarity, we write it again, T →∞ T 0
J2 X while the other in-plane component hŜ y i averages to zero.
ĤHeisen+T = cos(φ) σ̂jx σ̂j+1
x
+ σ̂jy σ̂j+1
y
+ ∆σ̂jz σ̂j+1
z
This dependence of this mean magnetization on the flux is
U j shown in Fig. 3(e). If we had started with a product state
y
+ D σ̂jx σ̂j+1 − σ̂jy σ̂j+1
x
. along the ŷ-direction of the Bloch sphere we would have a
(16) non-zero mean hŜ y i and zero mean hŜ x i instead. Interest-
Here ∆Ω=0 is an XXZ anisotropy (note that it differs from ingly enough, we find that while the amplitude of fluctuations
the anisotropy of the model in Eq. (6) because we are in the about the mean depends strongly on the Hubbard parameter
low-drive limit here), and D describes the relative strength U/J (recall that the spin model is only valid for U/J 1),7
(a)2⟨S x ⟩/L ϕ=0 (b)2⟨S x ⟩/L (c)2⟨S x ⟩/L (d)2⟨S x ⟩/L
U/J = 0 U/J = 0.5 U/J = 1 U/J=∞ (Spin model)
1.0 ϕ=0.2 1.0 1.0 1.0
0.8 ϕ=0.4 0.8 0.8 0.8
0.6 ϕ=0.6 0.6 0.6 0.6
0.4 0.4 0.4 0.4
ϕ=0.8
0.2 0.2 0.2 0.2 2 tJ
tJ tJ tJ
-0.2 200 400 600 800 1000 -0.2 200 400 600 800 1000 -0.2 200 400 600 800 1000 -0.2 200 400 600 8001000U
-0.4 -0.4 -0.4 -0.4
(e)2⟨S x ⟩t→∞ /L (f )2⟨S x ⟩t→∞ /L (g)⟨σyj ⟩ Δy
0.8
1.0 0.6
1.0 1.0 0.4
0.2
0.5 0. tJ2 /U
U/J=0 (free) 400. 800.
0.8 U/J=0.5 0.8 0.0 j
2 4 6 8 10 12 14
U / J= 1 L=14 - 0.5
0.6 0.6 tJ 2 /U
U/J=∞ (spin) L=12 - 1.0 0 50 100 250 500
L=10
0.4 0.4 L=8 (h) Δy L=8
0.4 L=10
L=12
0.2 0.2 0.3 L=14
0.2
0.1
ϕ ϕL ϕL
0.5 1.0 1.5 2.0 2.5 3.0 10 20 30 40 - 0.1 2 4 6 8
FIG. 3. (a-d) Time-evolution of the magnetization hŜ x i for the Fermi-Hubbard model with U/J = 0, 0.5, 1 and Ω = 0 [panels (a-c)] and the
Heisenberg+twist spin model (U/J → ∞ limit) [panel (d)], for different values of flux φ. System size is L = 8. After a short initial decay, the
magnetization stabilizes to a nonzero value depending on the flux. (e) Long-time mean magnetization for the different models in panels (a-d).
Up to small deviations, the mean is the same for all interaction strengths. (f) Long-time mean magnetization for the spin model as a scaling
function of flux times system size φL for different system sizes. For smaller flux, the curves fall on top of each other. Larger flux causes
them to deviate, as the system is periodic under φ → φ + 2π (with no L scaling). Error bars denote 1 standard deviation of the long-time
fluctuations about the mean. (g) Site-resolved transverse magnetization hσ̂jy i for different time snapshots using the spin model (L = 14). An
infinite-time chiral lattice-wide imbalance ∆y is established, as plotted in the inset. (h) Plot of infinite-time imbalance, also showing scaling
behaviour. Dots are the spin model, while the dark blue line is the U/J=0 exact analytic result in the thermodynamic limit L → ∞.
the mean value remains largely the same independently of pression for the mean by solving the U/J = 0 interactionless
U/J. This permits us to write an approximate analytic ex- case (with open boundaries),
L 2 φL
2 x 4 X
πjk
0
πj k
1 sin 2
hŜ it→∞ ≈ sin2 sin2 cos [φ(j − j 0 )] = as L → ∞. (21)
L L(L + 1)2 L+1 L+1 2L2 sin2 φ
j,j 0 ,k=1 2
As an even more intriguing feature, in Fig. 3(f) we find that relaxation to zero magnetization, as the system has rotational
at small flux φL . 1 there is a scaling behaviour as a func- symmetry in the x̂-ŷ plane of the Bloch sphere, and there is no
tion of φL, or flux times total system size, which eventually obvious conservation law that discriminates hŜ x i from hŜ y i.
peels off once φ gets large enough. The first minimum of this One could expect this to be a consequence of 1D integrabil-
scaling long-time magnetization occurs at φL = 2π, which ity, since the model maps to a Heisenberg model in the un-
corresponds to a full-period twisting of a spiral state in the un- gauged frame, but we also find similar effects in equivalent
gauged frame. We see a non-zero magnetization even at a full 2D systems (see Sec. III B). One may also consider this to be
period twist because we use open boundary conditions. Pe- a fine-tuned regime, but in addition to its persistence for all
riodic boundaries see a similar profile, except with hŜ x it→∞ U/J Hubbard repulsion strengths, we find that this non-zero
falling to zero at φ = 2πn/L for any n ∈ N (see Appendix D magnetization is robust to perturbative effects such as har-
for further discussion on boundary effects). monic trapping or imperfect filling fraction N/L < 1, which
An infinite-time non-zero magnetization independent of only slightly change the outcome (See Appendix C). Further-
U/J for this model is surprising. Naively, one would expect a more, the scaling behaviour maintaining a non-zero average at8
φL = 2π implies that boundary conditions play a non-trivial Heisenberg and will have dynamics that are chaotic even in
role even in the thermodynamic limit (this is unique to the 1D [38], causing the magnetization to decay to zero for all
interacting spin model, see Appendix D for details). φ 6= 0.
Aside from simple observables like magnetization, the sys- Recall that in general, the Heisenberg model conserves
tem can also develop long-time lattice-wide chiral spin imbal- both total angular momentum S ~ 2 [with eigenvalues S(S + 1)]
ances. Fig. 3(g) shows the other site-resolved in-plane mag- and angular momentum projection Ŝ x (with eigenvalues M ),
netization hσ̂jy i across the lattice for different snapshots of the ~ 2 shells and Ŝ x sectors within
splitting the Hilbert space into S
time-evolution (with open boundaries). While the mean hŜ y i each shell:
is zero, we find that the system establishes a tilt in the spin-
projection along the ŷ direction, indicating that the DM inter- ~ 2 ] = 0, L L
[ĤHeisen , S S= , − 1, . . .
action maintains a non-zero spin current at all times, opposed 2 2 (24)
by the relaxation dynamics of the XXZ model. This tilt can [ĤHeisen , Ŝ x ] = 0, M = S, S − 1 · · · − S.
be quantified by,
With φ 6= 0, the spiral initial state |ψ0Spiral i in the un-
L/2 L gauged frame will be distributed among these symmetry sec-
1 X y X y
∆y = hσ̂ i − hσ̂j i , (22) tors. For sufficiently small flux, only the highest angular mo-
L j=1 j
j=L/2+1 mentum shells are populated. These include the Dicke mani-
fold S = L2 and the spin-wave manifold S = L2 − 1, followed
which is plotted in Fig. 3(h) showing the same characteristic by S = L2 − 2, etc. The higher angular momentum shells have
scaling behaviour as the long-time magnetization. The sys- few states per symmetry sector [1 in Dicke, L−1 in spin-wave,
tem generates an extensive spin imbalance depending on the then O(L2 ), O(L3 ) and so on]. When enough of the initial
scaled flux φL. We can again approximate it using the non- state population sits in these highest shells, there are insuffi-
interacting limit U/J = 0, cient states for the system to relax and an infinite-time magne-
tization is generated. The equivalence between the Fermi- and
φL 2 φL
2 sin 2 sin 4 spin models can also be understood from this argument; the
∆y ≈ − 2 as L → ∞. (23) undriven Fermi-Hubbard model in the un-gauged frame also
L sin2 φ
2 has SU(2) symmetry regardless of the value of U/J, mean-
ing that the lack of relaxation should persist. Spin-insensitive
This imbalance is again surprising, especially because it man- perturbations such as external harmonic trapping or imperfect
ifests as an extensive chiral feature resulting from an initial filling fraction likewise maintain SU(2) symmetry and pre-
excitation with non-extensive energy in the thermodynamic serve the magnetization or imbalance. Non-negligible bound-
limit. Furthermore, for φL ≤ 2π the spin is imbalanced in ary effects in the thermodynamic limit are also sensible, as
one direction, whereas for φL slightly higher than that the di- the structure of the highest angular momentum shells depends
rection is reversed, even though the first 2π only makes a full- strongly on the boundaries (sinusoidal vs plane-wave), caus-
period revolution of the spins and should not set a preferential ing the relevant populations as a scaling function of φL to
spin pumping direction. be different. Note that the non-negligible boundary effects are
only maintained in the thermodynamic limit for the interacting
spin model, however, as discussed in Appendix D. In a sense,
B. Symmetry-restricted features the system thermalizes within a restricted set of Hilbert space
manifolds that prevent full relaxation in the conventional man-
The reason that non-trivial long-time behaviour occurs ner.
is because of the underlying exact SU(2) symmetry of the To help quantify the above arguments, in Fig 4(a) we plot
Heisenberg+twist model, together with its associated reduc- the overlap of the initial state wavefunction |ψ0Spiral i in the
tion of available phase space at small values of the flux. While un-gauged frame with the different symmetry sectors of the
the model we study is in a twisted (gauged) frame, the local Heisenberg model ĤHeisen ,
basis rotations still preserve the associated conserved quanti- XX
ties, just in a twisted form. With no flux φ = 0 we have a PS = |hψ0Spiral |φS,M,n i|2 , (25)
Heisenberg model, which is a critical point of the XXZ model M n
(∆Ω=0 = 1, D = 0). The addition of the DMp term causes
this critical point to extend into a line D = ± ∆2Ω=0 − 1 where |φS,M,n i is the n-th eigenstate of ĤHeisen within the
(with a corresponding branch for the ∆Ω=0 = −1 critical symmetry sector of angular momentum S and projection M .
point). For the parameters in Eq. (17), the system remains At φ =
N0, all population sits in the maximally-polarized Dicke
on the critical line at some position determined by the flux. state j |→ij . Increasing φ causes the deeper shells to be-
The additional symmetries of the Heisenberg model, while come populated, increasing the number of states that the sys-
twisted, still cause the Hilbert space to break into symmetry tem can explore. We give a metric of this property by defining,
sectors and restrict the number of states the system can re-
lax into. By comparison, a model sitting off the critical line 1 X
ζ= PS N S , (26)
of ∆Ω=0 = sec(φ), D = − tan(φ) cannot be mapped to the Nmax
S9
(a)PS (b) ζ
1.0 S
L
Dicke L
-4 0.8
2 2
0.8 L
- 1 Spin-wave L
-5 L=8
2 2 Entropy
L
- 2 L
-6 0.6 ϕL=7 L=9
2 2
0.6 L
-3 1.5
ϕL=6
ϕL=5
L=10
2
0.4 ϕL=4 L=11
0.4 1. ϕL=3
ϕL=2 L=12
0.5
0.2 0.2 ϕL=1
2
L=13
t JU
200. 400.
ϕL 0. ϕL
5 10 15 5. 10. 15.
FIG. 4. (a) Distribution of initial spiral state |ψ0Spiral i wavefunction population in different total angular momentum shells PS of the Heisenberg
model ĤHeisen as a function of scaled flux φL, for a system of size L = 12. (c) Weighted average ζ of shell population times the number
of states per symmetry sector of each shell. Note that each shell has a number of symmetry sectors corresponding to the further conserved
quantity S x . The Dicke manifold has NS = 1, spin-waves NS = L − 1, then O(L2 ), O(L3 ) and so on. The dashed line corresponds to
φL = 2π. The inset shows the bipartite entanglement entropy dynamics for an L = 10 system for specific snapshots of flux.
which is a weighted average of the population in each shell 2D, using a similar spiral initial state with a 2D structure [i.e.
iφ(i+j) z
times the number of states per symmetry sector in that shell a unitary transformation of the form e 2 σ̂i,j for lattice co-
NS , normalized by the number of states in the largest sec- ordinates (i,j)] . While the qualitative profile is changed, we
tor Nmax = NS=1 (for even L). The metric ζ can be un- still find non-zero persistent averages, which actually remain
derstood as a measure of how big a Hilbert space the system higher out to longer values of φL (with L = Lx × Ly for
can explore. When ζ 1, the wavefunction has most of its lattice length Lx and width Ly ). This occurs because 2D sys-
weight in symmetry sectors much smaller in dimension NS tems retain more population in the highest shells for the same
than the largest-size ones (with size Nmax ), and the system flux (since the energy gaps between shells scale with coor-
will have trouble relaxing. For ζ approaching of order one, dination number), and those shells have the same symmetry
most the wavefunction weight is in the biggest possible sec- sector sizes NS independent of dimension. Fig. 5(b) confirms
tors and we can expect a more conventional decay of magne- this prediction by plotting the same metric ζ, showing that it
tization/imbalance. Note that NS is not simply the number saturates at ζ ≈ 1 at the same flux that we see the infinite-
of states in the S-shell, but the number of states per sector of time magnetization drop to zero. There have also been studies
fixed M as well. For example, the Dicke manifold has L + 1 of similar physics in 3D using approximate numerical meth-
states in total, but each symmetry sector of M = − L2 , . . . , L2 ods [39], although there persistent infinite-time magnetization
within it only has 1 state, thus NS=L/2 = 1. In Fig. 4(b) was found for φ = 0, π.
we plot this metric as a function of φL, finding a character-
istic scaling crossover in behaviour. The regime where mean
infinite time magnetization falls near zero corresponds to the
regime where ζ saturates to a value near one.
(a)2⟨S x ⟩t→∞ /L (b) ζ
To connect with more conventional metrics, in the inset of
1. 0.8
Fig 4(b) we also plot the dynamics of the bipartite entangle-
0.8
ment entropy (partitioning the lattice into left/right halves) for 0.6
different values of φL. As ζ ≈ 1 is approached, the entangle- 0.6 1D 1D
2D 0.4 2D
ment entropy saturates at its maximum permitted value based 0.4
on the system size, while for small flux φL . 1 it never 0.2
0.2
reaches that value. ϕL 0. ϕL
5. 10. 15. 20. 5. 10. 15. 20.
The unusual dynamics described above appear reminiscent
to other kinds of unusual dynamics associated with integra-
bility, and indeed the 1D nearest neighbor Heisenberg model FIG. 5. (a) Long-time magnetization for the model of Eq. (16) and
is integrable. However here we find that integrability is not its 2D equivalent as a function of φL (in 2D, L = Lx × Ly ). Sys-
a necessary (and is in general not a sufficient) ingredient for tem size is L = 12 in 1D and (Lx , Ly ) = (4, 3) in 2D, using open
boundaries. Error bars indicate one standard deviation of the fluc-
the observed long-time dynamics. Beyond the mechanism we
tuations about the mean. Dashed lines estimate the point where the
propose above, which is unrelated to integrability, additional magnetization first reaches a minimum. (b) Hilbert space fragmenta-
evidence for the unimportance of integrability here is our sur- tion metric from Eq. (26) for the same 1D and 2D systems.
prising observation of analogous long-time behaviour in an
equivalent 2D Heisenberg model, which is not thought to be
integrable. Fig. 5(a) plots the infinite-time magnetization in10
IV. EXPERIMENTAL IMPLEMENTATIONS The same techniques may be used for measuring collective
observables. Total hŜ z i is simply measured from the bare
atomic excitation fraction, as hŜ z i = 21 j (hn̂j,e i − hn̂j,g i).
P
The general system described in Eq. (1) can be realized
in several ways. The most straightforward implementation Total hŜ x i can be measured by reversing the above state
is with a 3D optical lattice. Dynamics can be restricted to preparation protocol; one skips the phase by −π/2, makes
1D as explored in this work by increasing the transverse lat- a π/2 pulse, then measures excitation fraction.
tice depths, e.g. a deep lattice along directions ŷ, ẑ and a
shallower depth along a tunneling axis x̂ (though still deep
enough to maintain U/J 1). Spin-orbit coupled driving Initialize π/2 pulse Skip phase
y
may be realized through a direct optical transition between x Ωⅇⅈjϕ ⅇ-ⅈπ/2 Ωⅇⅈjϕ
long-lived internal states (g, e), such as clock states used in z
conventional atomic clock protocols, which will ensure the
j j +1 j +2
coherence times needed for observing spin dynamics. Flux is 2ϕ
generated whenever the drive laser wavevector ~kL has some = ⊗ j |g⟩j
ϕ = ⊗j|→⟩j
projection along the tunneling axis x̂, and will take the value
φ = cos(θ)(2πa)/λL , with a the lattice spacing, λL the driv- FIG. 6. Schematic for preparing product states in the dressed eigen-
ing laser wavelength and θ the angle between ~kL and the tun- basis Nof the drive. The system is initialized in a bare atomic product
neling axis x̂. state j |gij . The spin-orbit coupled drive is turned on, and a π/2
The most promising platform candidates are alkaline earth pulse is made, preparing a spiral state on the equator of the Bloch
sphere (the red lines indicate the drive axes of rotation on each lat-
or earth-like atom experiments, as they provide long-lived op-
tice site). The phase of the drive is then skipped by π/2, shifting the
tically separated internal clock states and magnetic field insen- axes to match the currentN spin direction on each site (blue lines). This
sitivity [24–26]. Alternatively, one may emulate these types results in a product state j |→ij . Measuring hŜ x i can also be done
of spin physics with nuclear-spin states [40] such as hyperfine P
by reversing this protocol, then measuring j (hn̂j,e i − hn̂j,g i).
states within a given manifold where the spin-orbit coupling is
implemented via Raman transitions. The relevant flux in this
case will come from the difference in the overall projection of In addition to optical lattices, much of the physics can also
the two Raman beams, e.g. φ = [cos(θ1 ) − cos(θ2 )](2πa)/λL be done with optical tweezer arrays placed close enough to
with θ1 , θ2 the angles of the beams to the tunneling axis x̂. allow tunneling. These have seen significant recent develop-
Depending on the duration of spin dynamics one wishes to ment due to their tunability and control [42–47], and offer
emulate, other atomic platforms such as alkali atoms may also an interesting alternative platform for spin dynamics experi-
prove useful, especially in regimes near resonance |U | ≈ |Ω| ments. If using atoms with long-lived internal clock states, a
where the spin model still holds, but the timescales are faster. single interrogating laser can be applied in exactly the same
There have also been discussions on the use of Lanthanide way as for the lattice; one simply needs to control either
atoms [41], which can avoid some of the heating issues typi- the angle or tweezer spacing to realize the desired φ. Spin-
cally found in alkali atoms. orbit coupled Raman schemes are likewise straightforward to
Preparing the desired product initial states is straightfor- adapt. As further possibility, higher-dimensional systems are
(Z) N also simple to generate; one makes the lattice shallower along
ward. A product state |ψ0 i = j |↑ij in the dressed basis
more than one tunneling direction, or uses a 2D tweezer array.
is trivial to prepare, as it is equal to a product state of all atoms
Each direction’s corresponding flux will be controlled by the
in the bare atomic basis up to an overall phase,
projection of the driving laser(s) onto the relevant axis. This
(Z)
O permits studies of the same types of spin models in 2D, as well
|ψ0 i = |eij , (27) as interplay between interactions of various types along differ-
j ent dimensions (such as for example an isotropic interaction
along one dimension and an anisotropic one along another),
and can thus be initialized with standard optical pumping tech-
(X) N which can be relevant to emulating real condensed matter ma-
niques. Creating a product state |ψ0 i = j |→j i requires terials.
a little more effort, because it is an eigenstate of the drive
and cannot be generated with the same drive alone. How-
ever, such a state can be prepared by skipping the laser phase. V. CONCLUSIONS
Fig. 6 shows such a protocol.
N One initializes all atoms in the
bare atomic ground-state j |gij , implements a π/2 pulse We have shown that using a single laser drive to induce
with the same laser used for driving, then skips its phase
magnetic flux in a fermionic optical lattice system can realize
ahead by π/2. The pulse will create a site-dependent rota-
a wide variety of different spin models across the parameter
tion that
N transforms the state into the desired dressed product regime of flux and driving strength. This system is readily im-
state j |→ij ,
plementable in modern optical lattice or tweezer experiments
P iπ using highly coherent atomic states. It opens a path to greatly
− iπ e− eijφ ĉ†j,e ĉj,g +h.c.
2
O O
e 4 j
|gij = |→ij . (28) improve quantum simulation capabilities using tools already
j j in reach in current experiments. In addition to studying well-11
understood models such as the Ising or one-axis twisting mod- and free models (see Appendix D), which could become more
els, more exotic physics such as lack of relaxation imposed by complex in higher dimensions. A more in-depth study of the
symmetry constraints can also be explored with this setup. persistent magnetization behaviour’s breakdown from exter-
nal perturbations can also be done. Finally, one can study the
There are many possible future directions to explore on steady state spin imbalances in the context of spin transport.
both experimental and theoretical fronts. One may inquire Acknowledgements. This material is based in part upon
further as to what happens to similar long-time magnetization work supported by the AFOSR grant FA9550-19-1-0275 9
behaviour in higher-dimensional systems; we find that 2D also (AMR) and FA9550-20-1-0222 (RN), by the DARPA and
exhibits non-zero steady state averages, but there are quali- ARO grant W911NF-16-1-0576, the ARO single investiga-
tative differences in the resulting profiles. Boundary effects tor award W911NF-19-1-0210, the NSF PHY1820885, NSF
can be probed, as we find discrepancies between interacting JILA-PFC PHY-1734006 grants, and by NIST.
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