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# Realization of Heisenberg models of spin systems with polar molecules in
pendular states

Wenjing Yue  State Key Laboratory of Precision Spectroscopy, East China Normal
University, Shanghai, China  Qi Wei 111 Corresponding email:
<EMAIL_ADDRESS>State Key Laboratory of Precision Spectroscopy, East
China Normal University, Shanghai, China  Sabre Kais  Department of Chemistry,
Purdue University, West Lafayette, Indiana 47907, USA  Bretislav Friedrich
Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195
Berlin, Germany  Dudley Herschbach  Department of Chemistry and Chemical
Biology, Harvard University, Cambridge, Massachusetts, USA

###### Abstract

We show that ultra-cold polar diatomic or linear molecules, oriented in an
external electric field and mutually coupled by dipole-dipole interactions,
can be used to realize the exact Heisenberg XYZ, XXZ and XY models without
invoking any approximation. The two lowest lying excited pendular states
coupled by microwave or radio-frequency fields are used to encode the pseudo-
spin. We map out the general features of the models by evaluating the models’
constants as functions of the molecular dipole moment, rotational constant,
strength and direction of the external field as well as the distance between
molecules. We calculate the phase diagram for a linear chain of polar
molecules based on the Heisenberg models and discuss their drawbacks,
advantages, and potential applications.

##  I Introduction

Introduced by Heisenberg in 1928, the Heisenberg statistical model of spin
systems has been widely used to study phase transitions and critical phenomena
in magnetic systems and strongly correlated electron systems1-Roger
G.Bowers1969 ; 2-J.F.Cooke1970 ; 3-Freeman J.Dyson1976 ; 4-Guang-Shan Tian1997
; 5-Henk W.J2002 ; 6-R.G.Brown2006 . Recently developed powerful tools
developed to unravel the physics of strongly correlated multi-body quantum
systems provide new platforms for understanding quantum
magnetism7-Fernanda2013 . It has been proposed to implement the Heisenberg
model in other systems as well. For example, Pinheiro et al. demonstrated that
in the Mott region, a boson atom in the first excitation band of a two-
dimensional optical lattice can realize the spin-1/2 quantum Heisenberg
model7-Fernanda2013 . Bermudez et al. introduced a theoretical scheme to
simulate the XYZ model using trapped ions8-A. Bermudez2017 .

The molecular axis of polar molecules that are subject to an external electric
field oscillates within a certain angular range about the field direction,
forming pendular states9-B. Friedrich1991 . These pendular states have
specific orientations that give rise to constant projections of the dipole
moment along the external field, resulting in long-range anisotropic
interactions via the electric dipole-dipole coupling. In a field gradient,
pendular molecule can be individually addressed due to its field-dependent
eigenenergy (and orientation). Moreover, the internal structure of polar
molecules is much richer than that of atoms or spins, allowing much richer
physics. Given these unique properties, arrays of polar molecules are
considered to be promising platforms for quantum computing and quantum
information processing10-Book2009 ; 11-D. DeMille2002 ; 12-Philippe
Pellegrini2011 ; 13-Jing Zhu2013 ; 14-Zuo-Yuan Zhang2017 ; 15-Zuo-Yuan
Zhang2020 ; 16-Wei12011 ; 17-Wei22011 ; 32-Micheli ; 33-Charron ; 34-kuz ;
35-ni ; 36-lics ; 37-YelinDeMille ; 38-Wei2010 ; 39-Wei2016 ; 40-kang-Kuen
Ni2018 , which is not unlike spins.

Inspired by the similarity between spins and polar molecules, the simulation
of the spin models with polar molecules has attracted broad interests over the
past decade18-Muller2010 ; 19-Alexey2011 ; 20-Bo Yan2013 ; 21-N.Y.Yao2018 ;
22-Haiyuan Zou2017 ; 22+-Kaden2013 ; M. L. Wall ; A. V. Gorshkov . Müller
described in his thesis the details of how to realize the spin-1/2 XXZ model
as well as t-J model with ultra-cold polar molecules trapped in an optical
lattice18-Muller2010 . Gorshkov et al. demonstrated that the dipole
interactions of ultra-cold alkali metal dimers in optical lattices can be used
to implement the t-J model, providing insights into strong correlation
phenomena in condensed systems19-Alexey2011 . Yan et al. experimentally
observed dipolar spin-exchange interactions with lattice-confined polar
molecules, which laid a foundation for further study of multi-body dynamics in
spin lattices20-Bo Yan2013 . Yao et al. obtained the dipole Heisenberg model
by using polar molecules and found the existence of quantum spin liquids on
the triangular and Kagome lattice21-N.Y.Yao2018 . Zou et al. implemented the
quantum spin model based on the polar molecule KRb in an optical lattice and
discovered the quantum spin liquid on the square lattice22-Haiyuan Zou2017 .

However, in almost all the previous works about implementation of the spin-1/2
Heisenberg model with polar molecules, the ground and first excited pendular
states with $M=0$ were usually chosen as pseudo-spin states, representing spin
up and spin down, respectively18-Muller2010 ; 19-Alexey2011 ; 20-Bo Yan2013 ;
21-N.Y.Yao2018 ; 22-Haiyuan Zou2017 ; 22+-Kaden2013 . In which case the
Hamiltonian is not in the form of the Heisenberg model. Only after applying
the rotating wave approximation can the Heisenberg model be recovered.
Furthermore, it is not a general Heisenberg XYZ model, but its special case,
the XXZ model.

Herein, by choosing the two lowest excited pendular states of a polar molecule
to represent the pseudo-spin states, we show how to achieve spin-1/2
Heisenberg XYZ model as well as XXZ and XY models directly, without any
approximation. We work out the properties of the models by evaluating all
their constants as functions of three dimensionless variables. The first one
is $\mu\varepsilon/B$, the ratio of the Stark energy (magnitude of permanent
dipole moment times electric field strength) to the rotational constant
(proportional to inverse of the molecular moment of inertia); this variable
governs the energy and intrinsic angular shape of the pendular states. The
second one is $\Omega/B$, with $\Omega$ = $\mu^{2}/r^{3}$, the square of the
permanent dipole moment divided by the cube of the separation distance; this
variable governs the magnitude of the dipole-dipole coupling. The third
variable is $\alpha$, the angle between the axis of the molecular array and
the electric field. As a sample application of the Heisenberg model based on
polar molecules, we construct the ground state phase diagram for a linear
array of polar molecules. We also discuss advantages, drawbacks as well as
potential applications of our model.

##  II Pendular states of polar molecules as pseudo-spins

###  II.1 Pendular and pseudo-spin states

Figure 1: Eigenenergies of a polar molecule in an external electric field, as
functions of $\mu\varepsilon/B$, with $\mu$ the permanent dipole moment,
$\varepsilon$ the field strength, B the rotational constant.
$\left|\downarrow\right\rangle$ correlates with the $J=1$, ${M}=1$ and
$\left|\uparrow\right\rangle$ with the $J=1$, ${M}=0$ states. States used as
the pseudo-spin states (red curves) are labeled
$\left|\downarrow\right\rangle$ and $\left|\uparrow\right\rangle$ in the
absence of an external field.

In an electrostatic field, the Hamiltonian of a trapped linear polar molecule
is 16-Wei12011

$H=\frac{{{p^{2}}}}{{2m}}+{V_{trap}}({\bf r})+B{{\bf
J}^{2}}-\bm{\mu}\cdot\bm{\varepsilon},$ (1)

where the molecule, with mass $m$, rotational constant $B$, and body-fixed
dipole moment $\mu$, has translational kinetic energy $p^{2}/2m$, potential
energy $V_{trap}$ within the trapping field and rotational energy
$B\bf{J}^{2}$ as well as interaction energy $\bm{\mu}\cdot\bm{\varepsilon}$
with the external field $\bm{\varepsilon}$. In the trapping well, at ultra-
cold temperatures, the translational motion of the molecule is quite modest
and very nearly harmonic; $p^{2}/2m+V_{trap}(r)$ is thus nearly constant and
can be omitted from the Hamiltonian. There remains the rotational kinetic
energy and Stark interaction,

$H_{s}=B{\bf J}^{2}-\mu\varepsilon\cos\theta,$ (2)

where $\theta$ is the polar angle between the molecular axis (the molecule-
fixed permanent electric dipole moment $\mu$) and the field direction. Under
the action of a strong electrostatic field, the polar molecules are compelled
to undergo pendular oscillations and result in the forming of pendular states,
$|$$\tilde{J}M\rangle$. Here, $\tilde{J}$ wears a tilde to indicate it is no
longer a good quantum number since the Stark interaction mixes the rotational
states, whereas $M$ is still a good quantum number as long as azimuthal
symmetry about $\varepsilon$ is maintained. Figure 1 shows eigenenergies of a
few lowest lying pendular states for a ${}^{1}\Sigma$ diatomic (or linear)
molecule, as functions of $\mu\varepsilon/B$.

We choose the two lowest excited states pendular states, $|$11$\rangle$ and
$|$10$\rangle$, as the pseudo-spin states $\left|\downarrow\right\rangle$ and
$\left|\uparrow\right\rangle$, respectively (see Figure 1). Then use an
external circularly polarized microwave or radio-frequency field to couple the
two states, forming a $\left|\downarrow\right\rangle$ and
$\left|\uparrow\right\rangle$ two-level system. The two pseudo-spin states are
linear superpositions of spherical harmonics $Y_{j}^{1}$and $Y_{j}^{0}$:

$|\downarrow\rangle=\sum_{\begin{subarray}{c}j\end{subarray}}a_{j}Y_{j}^{1}(\theta,\phi)\qquad|\uparrow\rangle=\sum_{\begin{subarray}{c}j\end{subarray}}b_{j}Y_{j}^{0}(\theta,\phi).$
(3)

Figure 2: Coefficients of spherical harmonics for pendular states
$\left|\downarrow\right\rangle$ (left panel) and $\left|\uparrow\right\rangle$
(right panel), see Eq.(3). Dashed curve for $\left|\uparrow\right\rangle$
indicates the coefficients of $Y_{0}^{0}$ is negative.  Figure 3: Wave
functions of the $\left|\downarrow\right\rangle$ and
$\left|\uparrow\right\rangle$ pendular states for values of
$\mu\varepsilon/B=0,4,8,12$ (from left to right), respectively. Panels (a) and
(b) represent the real and imaginary parts of state
$\left|\downarrow\right\rangle$ respectively. Panel (c) represents state
$\left|\uparrow\right\rangle$ (has no imaginary part).

Figure 2 plots the coefficients as functions of $\mu\varepsilon/B$. For
$\mu\varepsilon/B$ = 0, both $\left|\downarrow\right\rangle$ and
$\left|\uparrow\right\rangle$ are purely rotational states with single
component of spherical harmonics of $Y_{1}^{1}$ and $Y_{1}^{0}$, respectively.
As $\mu\varepsilon/B$ increases, more and more components of spherical
harmonics with the same $M$ but different $J$ get involved and the initially
dominant components decrease accordingly. For $\left|\uparrow\right\rangle$,
the dominant component $Y_{1}^{0}$ (shown in brown) decreases so quickly that
it is replaced by $Y_{0}^{0}$ as the leading term for $\mu\varepsilon/B>4.5$.
For $\left|\downarrow\right\rangle$, the initially dominant component
$Y_{1}^{1}$ (show in brown) decreases a little slower but is eventually
replaced by $Y_{1}^{1}$ when $\mu\varepsilon/B$ becomes large enough. Figure 3
displays wave functions of $\left|\downarrow\right\rangle$ and
$\left|\uparrow\right\rangle$ for different magnitudes of the electric field.
For $\left|\uparrow\right\rangle$, since $M=0$, the dipole is rotating with
its ${\mathbf{J}}$-vector perpendicular to the field direction. Without the
external field, the dipole orientation is symmetric in the hemispheres toward
($\theta=0$) or opposite ($\theta=\pi$) to the field direction. With
increasing external field, the pinwheeling dipole favors the opposite
hemisphere because its motion is slowed down there. However, when the external
field becomes large enough, pinwheeling is inhibited and converted into
pendular libration about the field direction, and the dipole orientation
favors the toward hemisphere. For $\left|\downarrow\right\rangle$, since
$M=1$, without the external field, the angular momentum is along the field
direction, and thus the dipole orientation is localized at about
$\theta=\pi/2$ and also symmetric in both hemispheres toward or opposite to
the field direction. As the external field increases, the dipole rotates like
a conical pendulum and its orientation favors more and more the toward
hemisphere.

###  II.2 Hamiltonian of psedo-spins with electric dipole-dipole interaction

Adding a second trapped polar molecule, identical to the first one but a
distance $r_{12}$ apart, introduces the dipole-dipole interaction
term16-Wei12011

${V_{d-d}}=\frac{{{\bm{\mu_{1}}}\cdot{\bm{\mu_{2}}}-3\left({{\bm{\mu_{1}}}\cdot\bm{n}}\right)\left({{\bm{\mu_{2}}}\cdot\bm{n}}\right)}}{{{{\left|{{\bm{r_{1}}}-{\bm{r_{2}}}}\right|}^{3}}}}.$
(4)

Here n is a unit vector along ${\bf r}_{12}$. In the presence of an external
field, $V_{d-d}$ can be expressed in terms of the polar and azimuthal angles:

$\displaystyle V_{d-d}=$
$\displaystyle\Omega\left[\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\cos\varphi_{1}\sin\theta_{2}\cos\varphi_{2}+\sin\theta_{1}\sin\varphi_{1}\sin\theta_{2}\sin\varphi_{2}\right.$
(5)
$\displaystyle\left.-3\left(\sin\theta_{1}\cos\varphi_{1}\sin\alpha+\cos\theta_{1}\cos\alpha\right)\left(\cos\theta_{2}\cos\alpha+\sin\theta_{2}\cos\varphi_{2}\sin\alpha\right)\right],$

where $\Omega={\mu^{2}}/r_{12}^{3}$, $\alpha$ is the angle between the ${\bf
r}_{12}$ vector and the field direction, $\theta_{1}$ and $\theta_{2}$ are the
polar angles between the dipoles (${\bm{\mu}}_{1}$ and ${\bm{\mu}}_{2}$) and
the field direction, $\varphi_{1}$ and $\varphi_{2}$ are the corresponding
azimuths.

Now the total Hamiltonian is ${H_{total}}$ = ${H_{s1}}$ \+ ${H_{s2}}$ \+
${V_{d-d}}$. When set up in the basis set of the direct product of pseudo-spin
states
$\\{|\downarrow\downarrow\rangle,|\downarrow\uparrow\rangle,|\uparrow\downarrow\rangle,|\uparrow\uparrow\rangle\\}$,
it takes the form

${H_{s1}}+{H_{s2}}=\left({\begin{array}[]{*{20}{c}}{2{E_{0}}}&0&0&0\\\
0&{{E_{0}}+{E_{1}}}&0&0\\\ 0&0&{{E_{1}}+{E_{0}}}&0\\\
0&0&0&{2{E_{1}}}\end{array}}\right),$ (6)
${V_{d-d}}=\Omega\left({\begin{array}[]{*{20}{c}}{{P_{\alpha}}C_{0}^{2}}&0&0&{{Q_{\alpha}}C_{X}^{2}}\\\
0&{{P_{\alpha}}{C_{0}}{C_{1}}}&{-{P_{\alpha}}C_{X}^{2}}&0\\\
0&{-{P_{\alpha}}C_{X}^{2}}&{{P_{\alpha}}{C_{1}}{C_{0}}}&0\\\
{{Q_{\alpha}}C_{X}^{2}}&0&0&{{P_{\alpha}}C_{1}^{2}}\end{array}}\right),$ (7)

where $E_{0}$ and $E_{1}$ are eigeneneries of the pendular pseudo spin states
$|$$\downarrow$$\rangle$ and $|$$\uparrow$$\rangle$, respectively (see Figure
1). $P_{\alpha}$ and $Q_{\alpha}$ are simple functions of $\alpha$:
${P_{\alpha}}=1-3{\cos^{2}}\alpha$, ${Q_{\alpha}}=-3{\sin^{2}}\alpha$. In
$V_{d-d}$, the basis states are linked by matrix elements containing $C_{0}$
and $C_{1}$, the field-induced dipole moments orientation cosines, and
$C_{X}$, the transition dipole moments between the pseudo-spin states
$|\downarrow\rangle$ and $|\uparrow\rangle$. These are given by

$C_{0}=\langle\downarrow|\cos\theta|\downarrow\rangle\qquad
C_{1}=\langle\uparrow|\cos\theta|\uparrow\rangle\qquad
C_{X}=\langle\downarrow|\sin\theta\cos\varphi|\uparrow\rangle.$ (8)

In contrast to a real spin state which has a constant dipole moment, here, the
values of $C_{0}$, $C_{1}$ and $C_{X}$ are functions of external electric
fields, which are displayed in Figure 4. When $\mu\varepsilon/B$ increases,
$C_{0}$ becomes increasingly positive, whereas $C_{1}$ is increasingly
negative upto about $\mu\varepsilon/B=2$, then climbs to zero at about
$\mu\varepsilon/B=4.9$ and thereafter is increasingly positive. The fact that
$C_{X}=0$ at $\mu\varepsilon/B$ = 0 means that without the external electric
field, the transition between $\left|\downarrow\right\rangle$ and
$\left|\uparrow\right\rangle$ is not allowed as a one-photon electric dipole
transition. Fortunately, increasing the external field introduces sufficient
mixing of other spherical harmonics, particularly admixing of $Y_{0}^{0}$ and
$Y_{2}^{0}$ into $\left|\uparrow\right\rangle$ and admixing of $Y_{2}^{1}$
into $\left|\downarrow\right\rangle$ (see Figure 2), such that $C_{X}$
increases sharply from zero to a considerable value, enabling the
$\left|\downarrow\right\rangle\leftrightarrow\left|\uparrow\right\rangle$
transition to occur as a one-photon transition.

Figure 4: Matrix elements of $C_{0}$, $C_{1}$ and $C_{X}$ as functions of
$\mu\varepsilon/B$. The dotted green line is ${C_{0}}-{C_{1}}$.

##  III Realization of the Heisenberg model of spin systems with polar
molecules.

###  III.1 General Heisenberg XYZ model based on pseudo-spins.

The total Hamiltonian of the two-dipoles molecular system can be mapped onto a
two-qubit spin-1/2 general Heisenberg XYZ model:

${H_{XYZ}}={J_{x}}\sigma_{1}^{x}\sigma_{2}^{x}+{J_{y}}\sigma_{1}^{y}\sigma_{2}^{y}+{J_{z}}\sigma_{1}^{z}\sigma_{2}^{z}-\gamma\left({\sigma_{1}^{z}+\sigma_{2}^{z}}\right),$
(9)

here ${\sigma_{x}}$, ${\sigma_{y}}$ and ${\sigma_{z}}$ are Pauli operators;
$J_{x},J_{y},J_{z}$ and $\gamma$ are coupling constants given by

$\begin{split}{J_{x}}&=\Omega\left({3{{\cos}^{2}}\alpha-2}\right)C_{X}^{2},\\\
{J_{y}}&=\Omega C_{X}^{2},\\\
{J_{z}}&=\frac{{\Omega\left({1-3{{\cos}^{2}}\alpha}\right){{\left({{C_{0}}-{C_{1}}}\right)}^{2}}}}{4},\\\
\gamma&=\frac{{2\left({{E_{1}}-{E_{0}}}\right)+\Omega\left({3{{\cos}^{2}}\alpha-1}\right)\left({C_{0}^{2}-C_{1}^{2}}\right)}}{4}.\end{split}$
(10)

Figure 5: Contour plots of $J_{x}$$/\Omega$ (panel a), $J_{z}$$/\Omega$ (panel
b) and second part of $\gamma$$/\Omega$ (panel c) as functions of reduced
variable $\mu\varepsilon/B$ and angle $\alpha$. $J_{y}$ is
$\alpha$-independent and the same as $J_{x}$ when $\alpha=0$.

Equations (9-10) demonstrate how to realize the spin-1/2 anisotropic
Heisenberg model with polar molecules in pendular states. The model constants
($J_{x},J_{y},J_{z}$ and $\gamma$) are functions of $\mu\varepsilon/B$,
$\Omega$ and $\alpha$, which means the model can be adjusted by modifying
these parameters. For all the constants, the relations to $\Omega$ are simply
linear, whereas the relations to $C$’s are quadratic. $J_{y}$ is $\alpha$
independent and equal to $J_{x}$ with $\alpha=0$. The term $\gamma$ consists
of two parts. One part is related to the energy gap $\Delta E=E_{1}-E_{0}$
which is shown in Figure 1. The other part is proportional to $\Omega$ which
is similar to $J$’s. The contour plots in Figure 5 illustrate how
$J_{x}/\Omega$, $J_{z}/\Omega$ and the second part of $\gamma/\Omega$ change
with $\mu\varepsilon/B$ and $\alpha$. When $\mu\varepsilon/B$ increases from 0
to 12, the magnitude of the coupling coefficients ($J_{x}/\Omega$,
$J_{y}/\Omega$ and $J_{z}/\Omega$) changes in the order of 0 to $10^{-1}$.
Similar results are obtained for the second part of $\gamma$$/\Omega$ .
Maximum or minimum values of $J_{x}$ and $J_{y}$ appear at large
$\mu\varepsilon/B$, whereas for $J_{z}$ they appear around
$\mu\varepsilon/B=3$.

For given $\Omega$ and $\alpha$, the coupling constants ${J_{x}}$, ${J_{y}}$,
${J_{z}}$ and $\gamma$ depend only on $x=\mu\varepsilon/B$, which enters those
constants through ${C_{0}}$, ${C_{1}}$, ${C_{X}}$ and $\Delta E$. To provide a
convenient means to evaluate Equation (10), we fitted our numerical results to
obtain accurate approximation formulas,

$\displaystyle{{\left({{E_{1}}-{E_{0}}}\right)}\mathord{\left/{\vphantom{{\left({{E_{1}}-{E_{0}}}\right)}B}}\right.\kern-1.2pt}B}={A_{1}}x+{A_{2}}{x^{2}}+{A_{3}}{x^{3}}+{A_{4}}{x^{4}}+{A_{5}}{x^{5}},$
(11) $\displaystyle
C(x)={A_{0}}+\frac{{{A_{1}}}}{{1+\exp[(x-{x_{1}})/{k_{1}}]}}+\frac{{{A_{2}}}}{{1+\exp[-(x-{x_{2}})/{k_{2}}]}}.$
(12)

These functions are plotted in Figure 6. The fitted parameters are given in
Tables (1-2).

Figure 6: Comparison of exact results (blue curves) with fitted approximation
functions (dashed red curves) cf. Eqs. (11) and (12) : energy difference,
${{\left({{E_{1}}-{E_{0}}}\right)}\mathord{\left/{\vphantom{{\left({{E_{1}}-{E_{0}}}\right)}B}}\right.\kern-1.2pt}B}$,
the field-induced dipole moments, ${C_{0}}$ and ${C_{1}}$ and the transition
dipole moments, ${C_{X}}$. Table 1: Values of the parameters for Eq.11.

Parameters | Values
---|---
${A_{1}}$ | 0.00794
${A_{2}}$ | 0.16531
${A_{3}}$ | -0.02838
${A_{4}}$ | 0.00206
${A_{5}}$ | $-5.55762\times{10^{-5}}$

  * •

${R^{2}}=0.9999$.

Table 2: Values of the parameters for Eq.12.

Parameters | Values for ${C_{0}}$ | Values for ${C_{X}}$ | Values for ${C_{1}}$
---|---|---|---
${A_{0}}$ | -0.24612 | 0.21844 | -0.91801
${A_{1}}$ | -0.56893 | -0.53637 | 0.9
${A_{2}}$ | 0.95967 | 0.02855 | 1.36773
${x_{1}}$ | -0.09066 | -04403 | 0.09317
${x_{2}}$ | -1.25815 | 4.28747 | 2.52364
${k_{1}}$ | 2.17868 | 1.18595 | 0.80729
${k_{2}}$ | 6.7313 | 0.94214 | 3.38213

  * •

${R^{2}}=1$ for ${C_{0}}$ , ${R^{2}}=0.9999$ for ${C_{X}}$ and ${C_{1}}$.

The Heisenberg model given by Equations (9-10) is a general XYZ model. But we
can get two different special cases of the Heisenberg model by changing the
direction of the external field. One is the XXZ model which obtains by taking
$\alpha={0^{\circ}}$. In that case, we have ${J_{x}}={J_{y}}\neq{J_{z}}$ and
${J_{x}}={J_{y}}\neq 0$, ${J_{z}}\neq 0$. The other one is the XY model which
obtains for $\alpha={54.7^{\circ}}$, known as the magic angle. In that case,
we have ${J_{x}}\neq 0$, ${J_{y}}\neq 0$ and ${J_{z}}=0$.

###  III.2 The Heisenberg XXZ model and quantum phase diagram of polar
molecules.

Figure 7: (a) Quantum phase diagram of the XXZ model associated with $J_{z}/J$
and $\gamma/J$ for a linear spin chain. (b) The ratio of the coupling
constants of the XXZ model, $J_{z}/J$, as a function of $\mu\varepsilon/B$ in
dipole system of polar molecules. (c) The ratio of the coupling constants of
the XXZ model, $\gamma/J$, as a function of reduced variables,
$\mu\varepsilon/B$ and $\Omega/B$, in dipole system of polar molecules.

In order to demonstrate the application of the Heisenberg model based on
pendular polar molecules, we take the XXZ model ($\alpha={0^{\circ}}$) as an
example. If only pairwise interaction is considered, for a system with $N$
polar molecules trapped in a linear array with external electric field along
the array, the Hamiltonian has the form of the XXZ model,

${H_{XXZ}}=\sum\limits_{i=1}^{N-1}{\left[{J\left({\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}}\right)}+{J_{z}}\sigma_{i}^{z}\sigma_{i+1}^{z}\right]}-\gamma\sum\limits_{i=1}^{N}{\sigma_{i}^{z}}$
(13)

with couplings given by

$\begin{split}J&=\Omega C_{X}^{2},\\\
{J_{z}}&=-\frac{{\Omega{{\left({{C_{0}}-{C_{1}}}\right)}^{2}}}}{2},\\\
\gamma&=\frac{{\left({{E_{1}}-{E_{0}}}\right)+\Omega\left({C_{0}^{2}-C_{1}^{2}}\right)}}{2}.\end{split}$
(14)

Figure 7(a) displays the ground state phase diagram of a spin-1/2 XXZ chain
with nearest-neighbor interaction23-Christian2010 ; 24-Mykhailo2019 ; 25-D. C.
Cabra1998 . The abscissa is the scaled anisotropy parameter $J_{z}/J$, and the
ordinate is the scaled magnetic field $\gamma/J$. There are two gapped phases:
one is the ferromagnetic phase for $J_{z}/J<-1$; the other is the
antiferromagnetic phase for $J_{z}/J>1$. In between is the Luttinger liquid
phase26-F.D.M. Haldane1980 . According to Equation 14, for polar molecules,
$J_{z}/J$ only depends on $\mu\varepsilon/B$, so in Figure 7(b) we show how
$J_{z}/J$ changes when $\mu\varepsilon/B$ increases from 0 to 12. The critical
value of $J_{z}/J=-1$ appears at $\mu\varepsilon/B=6.1$ ($\varepsilon$ =13.5
kV/cm for the SrO molecule). In order to obtain phase information about the
polar molecule system, we still need to know the range of values of
$\gamma/J$. According to Equation 14, $\gamma/J$ depends on both
$\mu\varepsilon/B$ and $\Omega$. So in Figure 7(c) we plot of $\gamma/J$ as a
function of $\mu\varepsilon/B$ for different $\Omega/B$. Finally we obtain a
ground state phase diagram associated with $\mu\varepsilon/B$ and $\Omega/B$
for a linear array of polar molecules, which is shown in Figure 8.

Figure 8: Ground state phase diagram of the XXZ model associated with
$\Omega/B$ and $\mu\varepsilon/B$ for polar molecules in a linear array.

##  IV Discussion and Prospects

In this paper, our chief aim was to demonstrate that the Heisenberg model of
spin systems can be realized with ultra-cold diatomic or linear ${}^{1}\sum$
molecules, oriented in an external electrostatic field and coupled by the
electric dipole-dipole interaction. This requires use of pendular states
comprised of superpositions of spherical harmonics. Here the two lowest lying
excited states coupled by microwave or radio-frequency fields are used to
mimic the two-level spin system. This provides a new physical platform for the
study of the Heisenberg model. Since the dipole is encoded in the rotational
states of the molecules, the field-induced electric dipole-dipole interactions
between the molecules reproduce magnetic dipole-dipole interactions between
spins. In order to map out the general features of the model, we have
considered a wide range of parameters defined by sets of unitless reduced
variables, involving the dipole moments, rotational constant, dipole-dipole
coupling, electric field strength and direction.

The external field plays an essential role. In order to induce extensive
hybridization of rotational states, the field strength needs to be
sufficiently high. This has a dual purpose. Firstly, to make the molecules
undergo pendular oscillations about the field direction; otherwise rotational
tumbling would average out the molecule’s dipole moment in the laboratory
frame. Secondly, to make the transition dipole moment $C_{X}$ deviate from
zero such that a one-photon transition
$\left|\downarrow\right\rangle\leftrightarrow\left|\uparrow\right\rangle$
would be fully allowed.

Using optical lattices to trap the molecules limits the distance between
adjacent molecules to a few hundreds nanometers, so that the dipole-dipole
coupling is weak ($\Omega/B$ typically of order $10^{-6}$ to $10^{-4}$)
compared with the energy gap $\Delta E$ and thus the model parameter
$\gamma/J$ becomes very large ($>10^{4}$). For that weak coupling realm, the
ground state of the Heisenberg model obtained with polar molecules is always
in the ferromagnetic phase (see Figure 8). In order to enhance the dipole-
dipole coupling, the molecular distance $r$ has to be shortened. For the SrO
molecule as an example, the molecular distance of less than 10 nm is required
for $\gamma/J\sim 1$. Such distance is much shorter than what can be achieved
in typical optical lattices, but might be obtained with arrays of nanoscale
plasmon-enhanced electro-optical traps27-Brian Murphy2009 ; 28-D.E.Chang2009
or molecular Wigner crystals29-P. Rabl2007 ; 30-H. P. Buchler2007 . That will
be a promising approach to extend the experimental scope of the model.

Polar molecules also offer significant advantages for achieving the Heisenberg
spin model, due to their high controllability and the presence of strong and
long range interactions. Stark energy is quite large, so for instance the
energy gaps between pseudo-spin states $\left|\downarrow\right\rangle$ and
$\left|\uparrow\right\rangle$ are typically in the range of microwave
frequencies, as opposed to the radiofrequencies separating of real spin
states. This enables a faster optically controlled transition between the two
energy levels of polar molecules. Electric dipole-dipole interaction is also
much stronger than that of spins, resulting in a larger frequency shift, which
is essential for building quantum logic gates.

The influence of the $\left|{1-1}\right\rangle$ pendular state which is
degenerate with $\left|{11}\right\rangle$ was also numerically analyzed. If we
use $\left|{1-1}\right\rangle$ instead of $\left|{11}\right\rangle$ as the
pseudo spin states $\left|\downarrow\right\rangle$, we will obtain the same
field-induced dipole moments ${C_{0}}$, ${C_{1}}$ and the transition dipole
moment ${C_{X}}$. That indicates that the pseudo-spin state
$\left|\downarrow\right\rangle$ can also be $\left|{1-1}\right\rangle$, since
the Hamiltonian matrix is the same as that for $\left|{11}\right\rangle$. But
if the pseudo spin state $\left|\downarrow\right\rangle$ is a superposition of
$\left|{1-1}\right\rangle$ and $\left|{11}\right\rangle$, ${C_{0}}$ and
${C_{1}}$ will remain unchanged, whereas ${C_{X}}$ will be different. In this
case, the Hamiltonian is still in the form of Heisenberg model, but the
Hamiltonian matrix elements related to ${C_{X}}$ take on different values.
Including the $\left|{1-1}\right\rangle$ state would increase the flexibility
and complexity of the model, and will not be elaborated upon here. However,
this problem can be avoided by introducing a tilt angle $\beta$ between the
polarization vector of the optical trapping field that confines the molecules
and the electrostatic field such that $\beta$ $\neq$ 0, $\pi$. In that case,
the degeneracy of the $\pm$M states is lifted Bretislav1 ; Bretislav2 .
Alternatively, for molecules with a nuclear electric quadrupole moment, a
superimposed magnetic field would lift the $\pm$M degeneracy via the
interaction between this moment and the magnetic moment generated by molecular
rotation 20-Bo Yan2013 ; 31-S.Ospelkaus2010 .

One potential application of polar molecules is in quantum computing, as
originally proposed by DeMille two decades ago 11-D. DeMille2002 . Since then,
many aspects and variants have been extensively studied, including for both
diatomic linear molecules and symmetric top molecules 12-Philippe
Pellegrini2011 ; 13-Jing Zhu2013 ; 14-Zuo-Yuan Zhang2017 ; 15-Zuo-Yuan
Zhang2020 ; 16-Wei12011 ; 17-Wei22011 ; 32-Micheli ; 33-Charron ; 34-kuz ;
35-ni ; 36-lics ; 37-YelinDeMille ; 38-Wei2010 ; 39-Wei2016 ; 40-kang-Kuen
Ni2018 . For linear molecules, the $\left|{00}\right\rangle$ and
$\left|{10}\right\rangle$ pendular states are the most commonly used qubit
states12-Philippe Pellegrini2011 ; 13-Jing Zhu2013 ; 15-Zuo-Yuan Zhang2020 ;
16-Wei12011 ; 32-Micheli ; 33-Charron ; 34-kuz ; 35-ni ; 36-lics ;
37-YelinDeMille ; 38-Wei2010 ; 39-Wei2016 ; 40-kang-Kuen Ni2018 . For
symmetric top molecules, different choices of qubit states have been
explored17-Wei22011 ; 14-Zuo-Yuan Zhang2017 . In most cases, the Hamiltonian
matrices are complex and no existing model can be used directly. In the
meantime, spin systems are also considered to be a promising platform to
implement a quantum computer. In fact, most of the work on quantum computers
is based on spin systems (Note: superconducting loops are actually artificial
spins)41-L.M.K.2000 ; 42-J.Zhang2005 ; 43-L.M.K2004 ; 44-Kavita2000 ;
45-Chiu2004 ; 46-V.W.Scarola2005 ; 47-F.B.M2009 ; 48-M.Asoudeh2004 ;
49-Meng2015 ; 50-V.V.Aristov2004  and Heisenberg model is the most popular
model used in treating such systems45-Chiu2004 ; 46-V.W.Scarola2005 ;
47-F.B.M2009 ; 48-M.Asoudeh2004 ; 49-Meng2015 . If we take our pseudo-spin
states $\left|\downarrow\right\rangle$ and $\left|\uparrow\right\rangle$ as
qubits states, then the two methods coincide. This opens up the prospect of
directly transplanting methods and techniques developed for spins to polar
molecules.

So far, most proposals for implementing quantum computing with polar molecules
have been based on the gate model. Our new choices of qubits states also
invite the possibility of adiabatic quantum computing 51-BRIW+14 ; 52-SQVL14 ;
53-VMBR+14 ; 54-VMKO15 ; 55-HJAR+15 ; 56-KHZO+15 ; 57-Ta ; 58-Gre ; 59-Ari ;
60-Sha ; 61-Ke . This follows primarily from the fact that the energy gap
between the two qubit states $\Delta E=E_{1}-E_{0}$ can be arbitrarily tuned
from 0 to 3.7B (see Figure 1) by changing electric fields. For adiabatic
quantum computing, the tunable energy gap between $|0\rangle$ and $|1\rangle$
need to be large compared with the interaction energy $V_{d-d}$. In this case,
it is around $10^{4}$ times larger than the coupling energy $V_{d-d}$, which
is far beyond the current limit that spin systems can achieve 57-KYNH+15 .
Moreover, one requirement for adiabatic quantum computing is that the energy
gap between the ground and the first excited state be maintained during the
adiabatic evolution such that no phase transitions could occur. This
requirement is also satisfied, since for a practical coupling constant
($\Omega/B<10^{-2}$) during the adiabatic process of reducing the electric
field, the entire polar molecular system remains in the ferromagnetic phase,
without undergoing any phase transition (see Figure 8).

## ACKNOWLEDGEMENTS

We are grateful for support from National Natural Science Foundation of China
(Grant Nos. 11974113 and 11674098). SK would like to acknowledge the support
of the National Science Foundation under award number 1955907. BF gratefully
acknowledges the hospitality of John Doyle and Hossein Sadeghpour during his
stay at Harvard Physics and at the Harvard-Smithsonian Institute for
Theoretical Atomic, Molecular, and Optical Physics (ITAMP).

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