may-testing and must-testing models — we give the former (slightly simpler)
case first.

Observe that $N\rightarrow T$ denotes a ${\mathbb{N}}$-indexed product of
copies of $[\\![T]\\!]$. Thus we may define the following pairing and
projection operations on terms:

  * •

Given $t_{1}:T$ and $t_{2}:{\mathit{N}}\rightarrow T$, let
$t_{1}\\!\\!::\\!\\!t_{2}:{\mathit{N}}\rightarrow T\triangleq\lambda
u.({\mathtt{Eq}}(u,0)\hskip 2.0ptt_{1})\hskip 2.0pt({\mathtt{pred}}(u)\hskip
2.0ptt_{2})$ and given $t:{\mathit{N}}\rightarrow T$, define
${\mathtt{hd}}(t):T\triangleq t\hskip 2.0pt{\mathtt{0}}$ and
${\mathtt{tl}}(t)\triangleq\lambda v.t\hskip 2.0pt{\mathtt{suc}}(v)$, so that
$[\\![{\mathtt{hd}}(t_{1}\\!\\!::\\!t_{2})]\\!]=t_{1}$ and
$[\\![{\mathtt{tl}}(t_{1}\\!\\!::\\!t_{2})]\\!]=[\\![t_{2}]\\!]$.

  * •

Define ${\mathtt{pair}}:({\mathit{N}}\rightarrow
T)\rightarrow({\mathit{N}}\rightarrow T)\rightarrow{\mathit{N}}\rightarrow
T\triangleq{\mathbf{Y}}\lambda F.\lambda x.\lambda
y.{\mathtt{hd}}(x)\\!\\!::\\!\\!(F\hskip 2.0pty\hskip 2.0pt{\mathtt{tl}}(x))$,
with
${\mathtt{fst}}:({\mathit{N}}\rightarrow{\mathit{o}})\rightarrow({\mathit{N}}\rightarrow{\mathit{o}})\triangleq{\mathbf{Y}}\lambda
F.\lambda x.({\mathtt{hd}}(x)\\!\\!::\\!(F\hskip
2.0pt{\mathtt{tl}}({\mathtt{tl}}(x)))$ and
${\mathtt{snd}}:({\mathit{N}}\rightarrow{\mathit{o}})\rightarrow({\mathit{N}}\rightarrow{\mathit{o}})\triangleq{\mathbf{Y}}\lambda
F.\lambda x.{\mathtt{hd}}({\mathtt{tl}}(x))\\!\\!::\\!\\!(F\hskip
2.0pt{\mathtt{tl}}({\mathtt{tl}}(x))$, so that $[\\![{\mathtt{fst}}\hskip
2.0pt{\mathtt{pair}}(t_{1},t_{2})]\\!]=[\\![t_{1}]\\!]$ and
$[\\![{\mathtt{snd}}\hskip
2.0pt{\mathtt{pair}}(t_{1},t_{2})]\\!]=[\\![t_{2}]\\!]$.

######  Lemma 7.3.

${\mathit{N}}\rightarrow{\mathit{N}}\rightarrow T\unlhd{\mathit{N}}\rightarrow
T$.

###### Proof.

Let ${\mathtt{merge}}:({\mathit{N}}\rightarrow{\mathit{N}}\rightarrow
T)\rightarrow({\mathit{N}}\rightarrow T)\triangleq\lambda
f.({\mathbf{Y}}\lambda F.\lambda u.{\mathtt{pair}}((f\hskip 2.0ptu),(F\hskip
2.0ptf)\hskip 2.0pt{\mathtt{suc}}(u)))\hskip 2.0pt{\mathtt{0}}$,
${\mathtt{unmerge}}:({\mathit{N}}\rightarrow
T)\rightarrow{\mathit{N}}\rightarrow{\mathit{N}}\rightarrow
T\triangleq({\mathbf{Y}}\lambda F.\lambda u.\lambda x.\lambda
v.({\mathtt{Eq}}(u,v)\hskip 2.0pt{\mathtt{fst}}(x))\hskip 2.0pt((F\hskip
2.0pt{\mathtt{suc}}(u))\hskip 2.0pt{\mathtt{snd}}(x)))\hskip
2.0pt{\mathtt{0}}$. ∎

So in particular, ${\mathit{N}}\rightarrow U\unlhd U$. Recall that the
denotation of the type ${\rm
Nature}\triangleq(N\rightarrow{\mathit{o}})\rightarrow{\mathit{o}}$ is
isomorphic to the biorder ${\cal{P}}({\mathbb{N}})_{\uparrow}$.

######  Lemma 7.4.

${\rm Nature}\unlhd U$.

###### Proof.

Observe that we may faithfully represent any subset $X\subseteq{\mathbb{N}}$
as a function $b\in[\\![U]\\!]$ such that $b(i)=I$ if $i\in X$ and
$b(i)=\bot_{E}$ otherwise (where $I\in[\Sigma,\Sigma]$ is the identity
function). This gives the following $\Lambda^{+}(N)\hskip 2.0pt$-definable
retraction:

${\mathtt{inj}}:{\rm Nature}\rightarrow U\triangleq\lambda f.\lambda u.\lambda
x.f\hskip 2.0pt\lambda v.({\mathtt{Eq}}(u,v)\hskip 2.0ptx)\hskip 2.0pt\Omega$
${\mathtt{proj}}:U\rightarrow{\rm Nature}\triangleq\lambda f.\lambda g.{?_{\rm
Nature}}\hskip 2.0pt\lambda w.(f\hskip 2.0ptw)\hskip 2.0pt(g\hskip 2.0ptw)$

For any $g:\widetilde{{\mathbb{N}}}\rightarrow\Sigma$, the set
$\\{\top_{S}[g(n)]_{n}\ |\ n\in{\mathbb{N}}\\}$ is stably bounded above by the
constantly $\top_{S}$ function, and $g=\bigwedge\\{\top_{S}[g(n)]_{n}\ |\
n\in{\mathbb{N}}\\}$. Thus $({\mathtt{proj}}\hskip
2.0pt{\mathtt{inj}}(f))(g)=$
$\bigwedge_{n\in{\mathbb{N}}}f(\top_{S}[g(n)]_{n})=f(\bigwedge_{n\in{\mathbb{N}}}\top_{S}[g(n)]_{n})=f(g)$
by stability. ∎

The key retraction reducing the _order_ of types (depth to which the arrow is
nested on the left) is based on the observation that we may represent a
continuously stable function $f$ from
$[\\![U]\\!]\cong[\Sigma,\Sigma]^{\omega}$ into $\Sigma$ via the following
information:

  * •

Its set of sequentiality indices,

  * •

For each sequentiality index $i$, the function $\lambda x.f(x[I]_{i})$.

######  Lemma 7.5.

For all
$g\in[\\![{\mathit{N}}\rightarrow{\mathit{o}}\rightarrow{\mathit{o}}]\\!]$,
$f(g)=\bigsqcup_{i\in{\mathsf{si}}(f)}(g(i)\hskip 2.0ptf(x[I]_{i}))$.

###### Proof.

If $g(i)=\bot_{S}=\top_{E}$ for some sequentiality index
$i\in{\mathsf{si}}(f)$ then $f(g)=\bot_{S}=g(i)\hskip
2.0ptf(g[I]_{i})=\bigsqcup_{i\in{\mathsf{si}}(f)}(\pi_{i}(x)\hskip
2.0ptf(x[I]_{i}))$. So suppose that $g(i)\in\\{\bot_{E},I\\}$ for all
$i\in{\mathsf{si}}(f)$.
If $g(i)=\bot_{E}$ then $g(i)\hskip 2.0ptf(g[I]_{i})=\bot_{E}$ and if $g(i)=I$
then $g(i)\hskip 2.0ptf(g[I]_{i})=I\hskip 2.0ptf(g)=f(g)$.
So if $g(i)=\bot_{E}$ for every sequentiality index $i$, then
$\bigsqcup_{i\in{\mathsf{si}}(f)}(g(i)\hskip 2.0ptf(x[I]_{i}))=\bot_{E}=f(g)$
by Lemma 3.14.
Otherwise there is some $i\in{\mathsf{si}}(f)$ such that $g(i)=I$ and so
$\bigsqcup_{i\in{\mathsf{si}}(f)}g(i)\hskip 2.0ptf(g[I]_{i})=f(g)$ as
required. ∎

Thus, extracting sequentiality indices for $f,\\{\lambda x.f(x[I]_{i})\ |\
i\in{\mathsf{si}}(f)\\},\\{\lambda x.f(x[I]_{i}[I]_{j})\ |\
i\in{\mathsf{si}}(f),j\in{\mathsf{si}}(\lambda x.f(x[I]_{i})\\},\ldots$
generates a set of finite sequences of indices from which $f$ may be
recovered.

######  Lemma 7.6.

$U\rightarrow{\mathit{o}}\unlhd{\mathit{N}}\rightarrow{\rm Nature}$.

###### Proof.

Let:

${\mathtt{inj}}:(U\rightarrow{\mathit{o}})\rightarrow{\mathit{N}}\rightarrow{\rm
Nature}\triangleq\lambda f.{\mathbf{Y}}_{{\mathit{N}}\rightarrow{\rm
Nature}}\lambda F.(\lambda x.f\hskip 2.0pt\lambda
y.x)\\!\\!::\\!\\!({\mathtt{merge}}(\lambda u.F\hskip 2.0pt\lambda x.f\hskip
2.0ptx[I]_{u}))$. ${\mathtt{p}roj}:({\mathit{N}}\rightarrow{\rm
Nature})\rightarrow U\rightarrow{\mathit{o}}\triangleq{\mathbf{Y}}_{U}\lambda
F.\lambda x.\lambda y.{\mathtt{hd}}(x)\hskip 2.0pt\lambda v.(y\hskip
2.0ptv)\hskip 2.0pt(F\hskip 2.0pt({\mathtt{unmerge}}({\mathtt{tl}}(x))\hskip
2.0ptv))$.

We prove by induction on $k\in{\mathbb{N}}$ that for all
$e\in[\\![{\mathit{N}}\rightarrow{\mathit{o}}\rightarrow{\mathit{o}}]\\!]$, if
$|\\{n\in{\rm Nature}\ |\ e(n)=I\\}|\leq k$ then for all
$f\in[\\![U\rightarrow{\mathit{o}}]\\!]$,
${\mathtt{inj}}({\mathtt{p}roj}(f))(e)=f(e)$. For the induction step, unfold
the fixed point, so that:
${\mathtt{inj}}({\mathtt{p}roj}(f))(e)=\bigsqcup_{i\in{\mathsf{si}}(f)}e(i)\hskip
2.0pt{\mathtt{inj}}({\mathtt{p}roj}(\lambda x.f(x[I]_{i})))(e)$. Thus, by
Lemma 7.5 it is sufficient to show that if $i\in{\mathsf{si}}(f)$ then
$e(i)\hskip 2.0pt{\mathtt{inj}}({\mathtt{p}roj}(\lambda
x.f(x[I]_{i})))(e)=e(i)\hskip 2.0ptf(e[I]_{i})$. This is evident if
$e(i)\not=I$ (i.e. $e(i)$ is the constant function $\bot_{S}$ or $\bot_{E}$).
If $e(i)=I$ then
$f(e[I]_{i})={\mathtt{inj}}({\mathtt{p}roj}(\lambda
x.f(x[I]_{i}))(e[\bot_{E}]_{i})\\\
\sqsubseteq_{E}{\mathtt{inj}}({\mathtt{p}roj}(\lambda x.f(x[I]_{i}))(e)\\\
\sqsubseteq_{E}{\mathtt{inj}}({\mathtt{p}roj}(\lambda
x.f(x[I]_{i}))(e[\top_{E}]_{i})=f(e[I]_{i})$ by induction hypothesis applied
to $e[\bot_{E}]_{i}$ and $e[\top_{E}]_{i}$.
So $e(i)\hskip 2.0pt{\mathtt{inj}}({\mathtt{p}roj}(\lambda
x.f(x[I]_{i}))(e)=e(i)\hskip 2.0ptf(e[I]_{i})$, completing the induction.

Since any element $e\in[\\![U]\\!]$ is the $\sqsubseteq_{E}$-supremum of the
approximants $e_{k}$ such that $e_{k}(i)=e(i)$ if $i\leq k$ and
$e_{k}(i)=\bot_{E}$ otherwise (to which the above induction applies), for any
$f\in[\\![U\rightarrow{\mathit{o}}]\\!]$,
${\mathtt{inj}}({\mathtt{p}roj}(f))(e)=f(e)$ by continuity of $f$. ∎

By composing definable retractions we may now show that $U$ is a (definably)
reflexive object (i.e. $U\rightarrow U$ is a definable retract of $U$).

######  Proposition 7.7.

$U\rightarrow U\unlhd U$.

###### Proof.

We have $U\rightarrow U\cong{\mathit{N}}\rightarrow{\mathit{o}}\rightarrow
U\rightarrow{\mathit{o}}\\\
\unlhd{\mathit{N}}\rightarrow{\mathit{o}}\rightarrow{\mathit{N}}\rightarrow{\rm
Nature}\\\
\unlhd{\mathit{N}}\rightarrow{\mathit{o}}\rightarrow{\mathit{N}}\rightarrow
U\\\ \unlhd{\mathit{N}}\rightarrow{\mathit{o}}\rightarrow U\\\
\unlhd{\mathit{N}}\rightarrow U\unlhd U$. ∎

###  7.1. Full Abstraction

Remarking that ${\mathcal{U}}$ also corresponds to a universal object in the
category of countable ocds and continuous functions, we may apply largely the
same argument to establish that $U$ is also a reflexive type in our must-
testing model of bounded $\Lambda^{+}(N)\hskip 2.0pt$: most of the terms used
in the proof of Proposition 7.9 also denote retractions in
${\mathcal{B_{S}}}$, and the same proofs apply. The exception is the
retraction of ${\rm Nature}$ into $U$ (Lemma 7.4), which requires the use of
unbounded nondeterminism. Recall that in our stably continuous model, ${\rm
Nature}$ corresponds to a powerdomain of _finite_ sets of natural numbers: by
using a different representation of this biorder in ${\mathcal{U}}$, we may
construct an alternative retraction.

######  Lemma 7.8.

${\rm Nature}\unlhd U$ in the must-testing model.

###### Proof.

As in Lemma 7.4 we may represent a (finite) set $X$ of integers as an element
$b\in[\\![U]\\!]$ such that $\pi_{i}(b)=I$ if $i\in X$ and
$\pi_{i}(b)=\top_{E}$ otherwise. The problem is that the projection which
sends $b\in[\\![U]\\!]$ to the set of all natural numbers $i$ such that
$b(i)=I$ exhibits unbounded non-determinism (since there may be infinitely
many such $i$) and is therefore not stably continuous. Our solution is to use
a second value $c\in[\\![U]\\!]$ to represent an upper bound for $X$ — i.e.
$c(i)=I$ if there exists $c\in X$ such that $i\leq j$, and $c(i)=\top_{E}$
otherwise. We can then return all natural numbers $i$ such that $b(i)=I$ and
$\forall j\leq i.c(j)=I$ by stepping through values of $i$ in sequence.

Thus we have a definable retraction of ${\rm Nature}$ into $U^{2}$, given by
the terms:

${\mathtt{inj}}_{1}:{\rm Nature}\rightarrow U\triangleq\lambda x:{\rm
Nature}.\lambda u:{\mathit{N}}.\lambda y:{\mathit{o}}.x\hskip 2.0pt\lambda
v.({\mathtt{Eq}}(u,v)\hskip 2.0pty)\hskip 2.0pt\mho$ ${\mathtt{inj}}_{2}:{\rm
Nature}\rightarrow U\triangleq\lambda x:{\rm Nature}.\lambda
u:{\mathit{N}}.\lambda y.{\mathtt{leq}}(u,v)\hskip 2.0pty)\hskip 2.0pt\mho$.
${\mathtt{p}roj}:U\rightarrow U\rightarrow{\rm Nature}\triangleq\lambda
y.\lambda z.\lambda
k.({\mathbf{Y}}_{{\mathit{N}}\rightarrow{\mathit{o}}}\lambda F.\lambda
v:{\mathit{N}}.(z\hskip 2.0ptv)\hskip 2.0pt(((y\hskip 2.0ptv)\hskip
2.0pt(k\hskip 2.0ptv))+(F\hskip 2.0pt{\mathtt{suc}}(v))))\hskip 2.0pt0$

Where
${\mathtt{leq}}:{\mathit{N}}\rightarrow{\mathit{N}}\rightarrow{\mathtt{bool}}\triangleq\lambda
u.\lambda v.\lambda x.\lambda y.({\mathbf{Y}}\lambda f.\lambda
w.({\mathtt{Eq}}(u,w)\hskip 2.0ptx)(({\mathtt{Eq}}(w,v)\hskip 2.0pty)\hskip
2.0pt(f\hskip 2.0pt{\mathtt{suc}}(w))))\hskip 2.0pt{\mathtt{0}}$. ∎

Thus $U\unlhd U\rightarrow U$ in the must-testing model, exactly as in
Proposition 7.7, and we may prove by structural induction:

######  Proposition 7.9.

For every pointed type $P$, $P\unlhd U$ in the may-testing and must-testing
semantics of bounded $\Lambda^{+}(N)\hskip 2.0pt$.

It remains to prove full abstraction for both models. Say that completeness
holds at type $T$ if for all closed $s,t:T$, if
$s{\lesssim^{{\mathit{may}}}}t$ then
$[\\![s]\\!]_{may}\sqsubseteq_{E}[\\![t]\\!]_{may}$ and if
$s{\lesssim^{{\mathit{must}}}}t$ then
$[\\![t]\\!]_{must}\sqsubseteq_{E}[\\![s]\\!]_{must}$.

######  Lemma 7.10.

Completeness holds at the type
$U\triangleq{\mathit{N}}\rightarrow{\mathit{o}}\rightarrow{\mathit{o}}$.

###### Proof.

Suppose e.g. $s{\lesssim^{{\mathit{must}}}}t$. Then by soundness and adequacy,
for any $d\in{\mathbb{N}}$ and $e\in\\{\top_{S},\bot_{S}\\}$ we have
$([\\![s]\\!]_{must}\hskip 2.0ptd)\hskip 2.0pte=\top_{S}=\bot_{E}$ implies
$([\\![t]\\!]_{must}\hskip 2.0ptd)\hskip 2.0pte=\top_{S}=\bot_{E}$, and so
$[\\![t]\\!]_{must}\sqsubseteq_{E}[\\![s]\\!]_{must}$. ∎

Thus completeness holds at all pointed types (it also holds at ${\mathit{N}}$,
since e.g. if $t{\lesssim^{{\mathit{may}}}}s$ then $((({\mathtt{Eq}}\hskip
2.0ptt)\hskip 2.0pt{\mathtt{n}})\hskip 2.0pt\mho)\hskip
2.0pt\Omega\Downarrow^{\mathit{may}}$ implies $((({\mathtt{Eq}}\hskip
2.0pts)\hskip 2.0pt{\mathtt{n}})\hskip 2.0pt\mho)\hskip
2.0pt\Omega\Downarrow^{\mathit{may}}$, and hence
$[\\![t]\\!]_{may}=[\\![s]\\!]_{may}$).

######  Theorem 7.11.

For all terms $s,t$, $s{\lesssim^{{\mathit{may}}}}t$ if and only if
$[\\![s]\\!]_{may}\sqsubseteq_{E}[\\![t]\\!]_{may}$ and
$s{\lesssim^{{\mathit{must}}}}t$ if and only if
$[\\![s]\\!]_{must}\sqsubseteq[\\![t]\\!]_{must}$.

######  Remark 7.12.

Universality of $U$ has several other applications. For example, it allows us
to characterize the $\Lambda^{+}(N)\hskip 2.0pt$-definable infinitary elements
of a (pointed) type-object $[\\![T]\\!]$ as those elements $e\in[\\![T]\\!]$
such that ${\mathtt{inj}}(e):U$ corresponds to a computable (partial) function
from ${\mathbb{N}}$ into $\\{I,\top\\}$. As ${\mathcal{U}}$ is a reflexive
object, endomorphisms on ${\mathcal{U}}$ form a $C$-monoid, providing an
interpretation of the untyped $\lambda$-calculus, similar to other graph
models, but with a weakly sequential, intensional interpretation as sequential
algorithms. Since
$[{\mathcal{U}},{\mathcal{U}}_{\uparrow}]\unlhd{\mathcal{U}}$ and
$[{\mathcal{U}},{\mathcal{U}}]_{\uparrow}\unlhd{\mathcal{U}}$, the $C$-monoid
of endomorphisms on ${\mathcal{U}}$ also furnishes weakly sequential semantics
for the untyped call-by-value and lazy $\lambda$-calculi, to which we can
apply the argument in [Lai03] to prove full abstraction.

###  7.2. Unbounded Nondeterminism

The situation in the non-continuous models is different. Although finitary
elements of these models are still definable, in the absence of continuity
this does not imply full abstraction, which in fact does not hold
(contradicting [Lai06]). We focus on the model of must-testing (since this
requires non-continuity). We have shown by example that some events with
infinite proofs are required to distinguish inequivalent terms in this model,
but hinted that there are many more which may distinguish denotations of terms
which cannot be separated operationally — we now give an example of a stable
and monotone function which can be used to distinguish two must-equivalent
terms, establishing that the model is not fully abstract. Consider the
function $h:[\\![{\rm
Nature}]\\!]\rightarrow[\\![{\mathit{o}}\rightarrow{\mathit{o}}]\\!]$ such
that:

  * •

$h(x)=\top_{E}$ if $x=\top_{E}=\bot_{S}$ (${\mathsf{si}}(x)$ is empty),

  * •

$h(x)=\bot_{E}$, if ${\mathsf{si}}(x)$ is finite but non-empty,

  * •

$h(x)=I$, otherwise (${\mathsf{si}}(x)$ is infinite).

This is monotone and stable, since ${\mathsf{si}}(x)\subseteq{\mathsf{si}}(y)$
and ${\mathsf{si}}(y)$ finite implies ${\mathsf{si}}(x)$ finite, and
$x\leq_{S}y$ in $[\\![{\rm Nature}]\\!]$ implies $x=y$ or
$x=\bot_{S}=\top_{E}$. However, it cannot be computed in $\Lambda^{+}(N)\hskip
2.0pt$, since there is no test which can check for unboundedness and then
return, rather than diverging. Note that $g:[\\![{\rm
Nature}]\\!]\rightarrow[\\![{\mathit{o}}]\\!]$ such that $g(x)=\bot_{E}$ if
${\mathsf{si}}(x)$ is finite, and $g(x)=\top_{E}$ otherwise, is definable as
$\lambda x.({\mathbf{Y}}_{{\mathit{N}}\rightarrow{\mathit{o}}}\lambda
f.\lambda u.x\hskip 2.0pt\lambda v.({\mathtt{leq}}(u,v)\hskip 2.0pt\mho)\hskip
2.0pt(f\hskip 2.0pt{\mathtt{suc}}(v)))\hskip 2.0pt{\mathtt{0}}$.

We may prove that $h$ is not definable in $\Lambda^{+}(N)\hskip 2.0pt$by
showing that all definable functions have the following property:

######  Definition 7.13.

A monotone stable function of extensionally complete biorders $f:D\rightarrow
E$ is _weakly continuous_ if for any $\sqsubseteq_{E}$-directed set $X$,
$f(\bigsqcup X)\leq_{S}\bigsqcup f(X)$.

The function $h$ is not weakly continuous: let $X$ be the
$\sqsubseteq_{E}$-directed set $\\{\bigsqcup_{i\in
J}\pi_{i}:\Sigma^{\omega}\rightarrow\Sigma\ |\
J\subseteq_{\mathit{fin}}{\mathbb{N}}\\}$. Then ${\mathsf{si}}(\bigsqcup
X)={\mathbb{N}}$ and so $h(\bigsqcap X)=I\not\leq_{S}\bot_{E}=\bigsqcup_{x\in
X}h(x)$. However, all $\Lambda^{+}(N)\hskip 2.0pt$terms denote weakly
continuous functions. We prove this using a logical relation — for each type
$T$, we define a set $H(T)\subseteq|[\\![T]\\!]|$ of _hereditarily weakly
continuous_ elements of $[\\![T]\\!]$ as follows:

  * •

$H({\mathit{N}})={\mathbb{N}}$ and $H({\mathit{o}})=\Sigma$

  * •

$f\in H(S\rightarrow T)$ if $f$ is weakly continuous and for any $x\in H(S)$,
$f(x)\in H(T)$.

A function
$f:[\\![S_{1}]\\!]\times\ldots\times[\\![S_{n}]\\!]\rightarrow[\\![T]\\!]$ is
hereditarily weakly continuous if it is weakly continuous and for any $\langle
x_{1},\ldots,x_{n}\rangle\in H(S_{1})\times\ldots\times H(S_{n})$,
$f(x_{1},\ldots,x_{n})\in H(T)$.

######  Proposition 7.14.

Every term of $\Lambda^{+}(N)\hskip 2.0pt$denotes a hereditarily weakly
continuous function.

###### Proof.

A straightforward structural induction. For the fixed point operator, we note
that if $F$ is a $\leq_{S}$-directed set of weakly continuous functions from
$D$ to (bicomplete) $E$, then $\bigvee F$ is weakly continuous, since for any
$\sqsubseteq_{E}$-directed set $X$, $(\bigvee F)(\bigsqcup
X)=\bigvee\\{f(\bigsqcup X)\ |\ f\in F\\}\leq_{S}\bigvee\\{\bigsqcup_{x\in
X}f(x)\ |\ f\in F\\}$. ∎

Hence we have established that:

######  Proposition 7.15.

There is no term $t:{\rm
Nature}\rightarrow{\mathit{o}}\rightarrow{\mathit{o}}$ which denotes the
function $h$.

Moreover, the weak continuity of definable functions may be used to prove that
the model is not fully abstract: we now give a $\Lambda^{+}(N)\hskip
2.0pt$-definable test which distinguishes $h$ from all weakly continuous
functions, and therefore from all definable elements of the model. Define
${\mathtt{?}}:{\mathit{N}}\rightarrow{\rm Nature}\triangleq\lambda u.\lambda
f.?_{N}\hskip 2.0pt\lambda v.({\mathtt{leq}}(u,v)\hskip 2.0pt(f\hskip
2.0ptv))\hskip 2.0pt\mho$ so that ${\mathtt{?}}(n)$ returns a bounded choice
over all values less than $n$. Clearly, $?(n)_{n\in\omega}$ is a
$\sqsubseteq_{E}$-chain with $\bigsqcup_{n\in\omega}?(n)=?_{N}$.

######  Lemma 7.16.

For any weakly continuous $f:[\\![{\rm
Nature}]\\!]\rightarrow[\\![{\mathit{o}}\rightarrow{\mathit{o}}]\\!]$,
$f(?_{N})\bigsqcup_{i\in\omega}(f(?(i))(\bot_{S}))=f(?_{N})(\bot_{S})$.

###### Proof.

If $f(?_{N})$ is a constant function, this is evident. Otherwise $f(?_{N})=I$.
By weak continuity $I=f(?_{N})\leq_{S}\bigsqcup_{n\in\omega}f(?(n))$, and by
extensional monotonicity
$\bigsqcup_{i\in\omega}f(?(i))\sqsubseteq_{E}f(?_{N})=I$, and so
$\bigsqcup_{n\in\omega}f(?(n))=I$. So
$f(?_{N})\bigsqcup_{n\in\omega}(f(?(n))(\bot_{S}))=I(I(\bot_{S})=\bot_{S}=f(?_{N})(\bot_{S})$
as required. ∎

We now give $\Lambda^{+}(N)\hskip 2.0pt$terms of type $({\rm
Nature}\rightarrow{\mathit{o}}\rightarrow{\mathit{o}})\rightarrow{\mathit{o}}$
corresponding to this test. Let $t\triangleq\lambda f.(f\hskip
2.0pt?_{N})\hskip 2.0pt(?_{N}\hskip 2.0pt\lambda u.(f\hskip
2.0pt{\mathtt{?}}(u))\hskip 2.0pt\Omega)$ and $s=\lambda f.(f\hskip
2.0pt?_{N})\hskip 2.0pt\Omega$.

######  Lemma 7.17.

$[\\![t]\\!]_{\mathit{must}}\not=[\\![s]\\!]_{\mathit{must}}$.

###### Proof.

Observe that $h(?_{N})=I$ but $h(?(n))$ is the constantly $\bot_{E}$ function
for each $n$, and hence
$[\\![t]\\!](h)=h(?_{N})(\bigsqcup_{n\in\omega}h(?(n)(\bot_{S})))=\bot_{E}$
but $[\\![s]\\!](h)=h(?_{N})(\bot_{S})=\bot_{S}$. ∎

######  Lemma 7.18.

$t$ and $s$ are must-equivalent in $\Lambda^{+}(N)\hskip 2.0pt$.

###### Proof.

By extensionality of $\Lambda^{+}(N)\hskip 2.0pt$, it suffices to show that if
$r:{\rm Nature}\rightarrow{\mathit{o}}\rightarrow{\mathit{o}}$ is any closed
term of $\Lambda^{+}(N)\hskip 2.0pt$then $t\hskip 2.0ptr$ is equivalent to
$s\hskip 2.0ptr$. By Proposition 7.14 $r$ denotes a weakly continuous function
and so $[\\![t\hskip
2.0ptr]\\!]=[\\![r]\\!](?_{N})(\bigsqcup_{i\in\omega}[\\![r]\\!](?(i))(\bot_{S}))=[\\![r]\\!](?_{N})(\bot_{S})=[\\![s\hskip
2.0ptr]\\!]$, and so by computational adequacy, $t\hskip
2.0ptr\Downarrow_{\mathit{must}}\mho$ if and only if $s\hskip
2.0ptr\Downarrow_{\mathit{must}}\mho$ as required. ∎

Thus we have shown that:

######  Theorem 7.19.

The must testing semantics of $\Lambda^{+}(N)\hskip 2.0pt$in ${\mathcal{B}}$
is not fully abstract.

###  7.3. Further Directions

We have established new extensional and intensional representations of higher-
order functional computation with bounded and unbounded nondeterminism, and
shown how they can be used to interpret a simple but expressive programming
language. Further study of this model may shed light on the ongoing debate
over the relevance of the concept of _fairness_ [Dji76], or bear relevance to
abstract interpretation of functional languages.

In this paper, we have considered only simply typed languages, but a more
general domain theoretic approach would extend our results to recursive types.
As well as fixed points for morphisms, we may also construct fixed points
(minimal invariants) for functors on our categories of (complete) biorders, as
coinciding limit/colimits of ordinal chains of embedding projection pairs (in
the continuous cases, these will be $\omega$-chains) obtained by standard
constructions. We refer to [Lai03] for one particular application — the
construction of fully abstract models of call-by-value and lazy call-by-name
$\lambda$-calculi by solving the domain equations $D\cong[D,D_{\uparrow}]$ and
$D\cong[D,D]_{\uparrow}$ (respectively) in the category of stably complete
biorders and stably continuous functions (the models described [Lai03] are in
the category of bicpos [Ber78], but all go through in ${\mathcal{B_{S}}}$).

######  Theorem 7.20 ([Lai03]).

The semantics of $\lambda_{v}$ in ${\mathcal{B_{S}}}$ is fully abstract.

By our results relating stably continuous functions to sequential algorithms
we may therefore construct fully abstract intensional models of $\lambda_{v}$
and $\lambda_{l}$ which (unlike e.g. the games model of the lazy
$\lambda$-calculus in [AM95]) is fully abstract without any “intrinsic
preorder quotient”. We leave as further work the extension of our semantics of
unbounded non-determinism to recursive types.

Further questions suggested by our semantics include:

  * •

Can the notion of weak continuity be completed to give a characterization of
the functions and/or sequential algorithms which are definable in unbounded
$\Lambda^{+}(N)\hskip 2.0pt$? We have also left open some important questions
regarding the algebraicity properties of biorders arising from ocds, which
might suggest further constraints on them.

  * •

Can we interpret nondeterministic languages with richer type systems (for
example, explictly controlling the use of effects such as nondeterminism). In
particular, can we relate powerdomain constructions on biorders to ordered
concrete data structures.

  * •

Can we describe models of unbounded non-determinism which make the “games-
like” structure of ordered concrete data structures more explicit — for
example, using graph games [HS02] or concurrent games [RW11] ? Is there a
corresponding linear decomposition of our model?

  * •

Jones et al. [JB08] have described powerful (but necessarily incomplete)
methods for deciding properties of terms in weakly sequential languages such
as the untyped call-by-value $\lambda$-calculus — can we relate these to our
intensional and extensional models?

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# Enhanced Persistent Orientation of Asymmetric-Top Molecules Induced by
Cross-Polarized Terahertz Pulses

Long Xu<EMAIL_ADDRESS>Department of Physics, Xiamen University, Xiamen
361005, China  AMOS and Department of Chemical and Biological Physics, The
Weizmann Institute of Science, Rehovot 7610001, Israel  Ilia Tutunnikov
<EMAIL_ADDRESS>AMOS and Department of Chemical and Biological Physics, The
Weizmann Institute of Science, Rehovot 7610001, Israel  Department of
Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA  Yehiam Prior<EMAIL_ADDRESS>AMOS and Department of
Chemical and Biological Physics, The Weizmann Institute of Science, Rehovot
7610001, Israel  Ilya Sh. Averbukh<EMAIL_ADDRESS>AMOS and
Department of Chemical and Biological Physics, The Weizmann Institute of
Science, Rehovot 7610001, Israel  Department of Chemistry, University of
British Columbia, Vancouver, British Columbia V6T 1Z1, Canada

###### Abstract

We investigate the persistent orientation of asymmetric-top molecules induced
by time-delayed THz pulses that are either collinearly or cross polarized. Our
theoretical and numerical results demonstrate that the orthogonal
configuration outperforms the collinear one, and a significant degree of
persistent orientation - approximately 10% at 5 K and nearly 3% at room
temperature - may be achieved through parameter optimization. The dependence
of the persistent orientation factor on temperature and field parameters is
studied in detail. The proposed application of two orthogonally polarized THz
pulses is both practical and efficient. Its applicability under standard
laboratory conditions lays a solid foundation for future experimental
realization of THz-induced persistent molecular orientation.

##  I Introduction

Molecular alignment and orientation induced by short laser pulses have
attracted widespread interest across various disciplines within ultrafast
science. This is mainly due to their pivotal role in numerous physical
processes, such as laser-induced ionization, high-order harmonic generation,
molecular imaging, etc. Existing methods for molecular rotational control and
their applications have been reviewed extensively over the past two decades
[1, 2, 3, 4, 5, 6]. Besides molecular alignment, various strategies have been
proposed and implemented to achieve molecular orientation. One such tool
employs optical-frequency two-color laser pulses consisting of a fundamental
wave and its second harmonic. Such pulses were shown to be effective in
inducing molecular orientation via two distinct intensity-dependent mechanisms
[7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27]. At low (nonionizing) intensities, the dominant mechanism depends on the
interaction of the two-color pulse with the molecular hyperpolarizability,
inducing asymmetric torques that orient the molecules [7, 8, 9, 10, 11, 12,
13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. The other mechanism,
which is dominant at high (ionizing) intensities, is caused by the different
probability of ionization along or against the polarization direction of the
asymmetric electric field generated by the collinearly polarized two-color
pulse [15, 16, 17, 27]. Another method combines weak electrostatic and intense
non-resonant laser fields [28, 29, 30, 31, 32, 33, 34, 35, 36]. Advances in
Terahertz (THz) technology paved the way for achieving molecular orientation
using pulsed THz fields, either standalone [37, 38, 39, 40, 41, 42, 43, 44] or
in conjunction with optical-frequency pulses [45, 46, 47, 48]. Recently,
attention has also shifted towards the study of enantioselective orientation,
which employs laser or THz pulses with twisted polarization [49, 50, 51, 52,
53, 54, 55, 56], as well as two-color cross-polarized laser pulses [10, 25].

In addition to the well-known transient orientation that appears immediately
following excitation by the laser pulses, a new phenomenon of persistent
orientation has recently been studied. The effect was observed for the first
time in chiral molecules excited by laser pulses with twisted-polarization
[53, 54]. Furthermore, it has been predicted in non-linear molecules
stimulated by THz pulses [44, 55] or two-color laser pulses [24, 25, 27]. In
this context, “persistent” refers to a nonzero average orientation that
persists on a time scale several orders of magnitude longer than the duration
of the applied external field.

While persistent orientation in chiral molecules has been experimentally
achieved using polarization-shaped laser pulses [54], similar effects are yet
to be realized using THz or two-color laser pulses. One reason is that the
value of the persistent orientation is generally small, especially at high
temperatures. Thus, an efficient approach is required to enhance the effect
and make it experimentally feasible.

Here, we investigate the persistent orientation of asymmetric-top molecules
excited by THz pulses. We show that the persistent orientation may be
significantly improved using a pair of time-delayed cross-polarized THz
pulses, compared to excitation by collinearly polarized pulses. We further
investigate the dependencies on the field amplitude, the polarization angle,
the time delay between pulses, and the temperature. The paper is organized as
follows. Section II introduces the theoretical methods. The persistent
orientation induced by a single THz pulse and a pair of time-delayed THz
pulses are presented in Sec. III and Sec. IV, respectively. Finally, we
summarize the work in Sec. V.

##  II Theoretical methods

Asymmetric-top molecules are modeled as rigid bodies with three orthogonal
inertia axes denoted by $a$, $b$, and $c$, with the moments of inertia
$I_{a}<I_{b}<I_{c}$. We perform quantum mechanical and classical simulations
of the molecular rotations following the THz excitation. The model Hamiltonian
is $H(t)=H_{r}+H_{\mathrm{int}}(t)$ [57, 5], where $H_{r}$ is the rotational
kinetic energy Hamiltonian and
$H_{\mathrm{int}}(t)=-\bm{\mu}\cdot\mathbf{E}(t)$ describes the field-molecule
interaction. Here $\bm{\mu}$ is the molecular dipole moment vector, and
$\mathbf{E}(t)$ is the electric field vector. The symmetric-top eigenfunctions
$|JKM\rangle$ are used as a basis set [58]. The quantum numbers $J$, $K$, and
$M$ correspond to the total angular momentum and its projections on the
molecular $a$ and the laboratory $X$ axes, respectively. The nonzero matrix
elements of $H_{r}$ are given by [58]

$\displaystyle\langle
JKM|H_{r}|JKM\rangle=\frac{B+C}{2}\left[J(J+1)-K^{2}\right]+AK^{2},$
$\displaystyle\langle JKM|H_{r}|J,K\pm 2,M\rangle=\frac{B-C}{4}f(J,K\pm 1),$
(1)

where $A=\hbar^{2}/2I_{a}$, $B=\hbar^{2}/2I_{b}$, $C=\hbar^{2}/2I_{c}$, and
$f(J,K)=\sqrt{(J^{2}-K^{2})[(J+1)^{2}-K^{2}]}$. The eigenfunctions of the
asymmetric-top molecule, $|J\kappa M\rangle=\sum_{K}c_{K}^{(J\kappa
M)}|JKM\rangle$, with $-J\leq\kappa\leq J$, can be obtained by numerical
diagonalization of $H_{r}$.

The finite initial rotational temperature is accounted for using the random
phase wave functions method [59]. In this method, the initial state is defined
as

$\displaystyle|\Psi_{n}(t_{i})\rangle=\sum_{J\kappa
M}\sqrt{\frac{e^{-\varepsilon_{J\kappa
M}/(k_{B}T)}}{\mathcal{Z}}}e^{i\varphi_{n,J\kappa M}}|J\kappa M\rangle,$ (2)

where $t_{i}$ is the initial time, $\varepsilon_{J\kappa M}$ is the rotational
kinetic energy of $|J\kappa M\rangle$, $k_{B}$ is the Boltzmann constant, $T$
is the temperature, and $\mathcal{Z}$ is the partition sum. Here
$\varphi_{n,J\kappa M}\in[0,2\pi)$ is a random number and the sum runs over
all the thermally populated eigenstates $|J\kappa M\rangle$. The time-
dependent Schrödinger equation
$i\hbar\partial_{t}|\Psi_{n}(t)\rangle=H(t)|\Psi_{n}(t)\rangle$ is solved
using numerical exponentiation of the Hamiltonian matrix (see Expokit [60]).
The expectation value gives the degree of orientation of the molecular $a$
axis,

$\displaystyle\braket{\cos(\theta_{aj})}(t)=\frac{1}{N}\sum\limits_{n=1}^{N}\langle\Psi_{n}(t)|\bm{a}\cdot\mathbf{e}_{j}|\Psi_{n}(t)\rangle,$
(3)

where $N$ is the number of initial states, $\mathbf{e}_{j}$ is a unit vector
along the laboratory $j$ ($j=X,Y,Z$) axis, $\bm{a}$ is a unit vector along the
molecular $a$ axis, and $\theta_{aj}$ is the angle between the molecular $a$
axis and the laboratory $j$ axis. Given the temperatures considered ($T\geq
5\,$K), we found that $N=100$ is sufficient for convergent results, where the
absolute error in the degree of orientation is less than 0.001.

Here we use formaldehyde ($\mathrm{CH_{2}O}$) as an example asymmetric-top
molecule. In this molecule, the molecular dipole moment points along the
molecular $a$ axis. Table 1 summarizes the molecular properties as taken from
NIST [density functional theory (DFT), Coulomb-attenuating method with Becke
three-parameter Lee-Yang-Parr functional and augmented correlation-consistent
polarized valence triple zeta Gaussian basis set (CAM-B3LYP/aug-cc-pVTZ)]
[61]. Since formaldehyde is a planar molecule belonging to the $C_{2v}$ point
symmetry group, the statistical factor due to nuclear spin [62] should be
included in the sum in Eq. (2). According to Eq. (1), the rotational states
with odd and even $K$ are decoupled. As a result, the initial rotational
$\ket{J\kappa M}$ states formed by $\ket{JKM}$ states with odd $K$ have triple
the statistical weight of $\ket{J\kappa M}$ states formed by $\ket{JKM}$ with
even $K$ [62].

Table 1: Molecular properties of $\mathrm{CH_{2}O}$: moments of inertia (in atomic units) and nonzero elements of the dipole moment (in Debye) in the frame of principal axes of inertia.  Molecule | Moments of inertia | Dipole components
---|---|---
Formaldehyde $\mathrm{CH_{2}O}$ | $I_{a}=11560$ | $\mu_{a}=2.389$
$I_{b}=84122$ |
$I_{c}=95682$ |

In the classical limit, we model the ensemble behavior using the Monte Carlo
approach. Initially, sample molecules are isotropically distributed in space,
their angular velocities ($\Omega_{i}$) are random and follow the Boltzmann
distribution $P(\Omega_{i})\propto\exp[-I_{i}\Omega_{i}^{2}/(2k_{B}T)]$, where
$i=a,b,c$. The rotational dynamics of each sample molecule is determined by
Euler’s equations in the rotating molecular frame [63]

$\mathbf{I}\bm{\dot{\Omega}}=(\mathbf{I}\bm{\Omega})\times\bm{\Omega}+\mathbf{T},$
(4)

where $\mathbf{I}=\mathrm{diag}(I_{a},I_{b},I_{c})$ is the moment of inertia
tensor, $\bm{\Omega}=(\Omega_{a},\Omega_{b},\Omega_{c})$ is the angular
velocity vector, and $\mathbf{T}=(T_{a},T_{b},T_{c})$ is the torque due to the
interactions with the THz field, given by
$\mathbf{T}=\bm{\mu}\times\mathbf{E}_{\mathrm{mol}}(t)$. Here,
$\mathbf{E}_{\mathrm{mol}}(t)$ is the electric field vector expressed in the
frame of principal axes of inertia. A quaternion,
$q=(q_{0},q_{1},q_{2},q_{3})$ parametrizes the relationship between the
molecular and laboratory frames [64, 65]. The equation of motion for a
quaternion is given by $\dot{q}=q\Omega/2$, where the multiplication rule of
quaternions is adopted [64, 65], and
$\Omega=(0,\Omega_{a},\Omega_{b},\Omega_{c})$. The ensemble average gives the
degree of orientation,

$\displaystyle\braket{\cos(\theta_{aj})}(t)=\frac{1}{N^{\prime}}\sum\limits_{n^{\prime}=1}^{N^{\prime}}\cos[\theta_{aj}(n^{\prime},t)],$
(5)

where $N^{\prime}=10^{6}$ is the number of sample molecules used in the
classical ensemble and $\theta_{aj}(n^{\prime},t)$ is the time-dependent angle
of the $n^{\prime}$-th molecule. A more detailed description of the classical
simulations can be found in [44].

##  III Persistent orientation induced by a linearly polarized THz pulse

We begin with the case of excitation by a single pulse. Consider asymmetric-
top $\mathrm{CH_{2}O}$ molecules excited by a single THz pulse propagating
along the laboratory $Z$ axis and polarized along the $X$ axis. The electric
field of the THz pulse is given by

$\displaystyle\mathbf{E}(t)=E_{0}f(t)\mathbf{e}_{X},$ (6)

where the time dependence is described by [66]

$\displaystyle f(t)=(1-2\sigma t^{2})e^{-\sigma t^{2}},$ (7)

$E_{0}$ is the peak amplitude, $\sigma$ determines the temporal width of the
THz pulse (in our simulations we use $\sigma=3.06\,\mathrm{ps}^{-2}$), and
$\mathbf{e}_{X}$ is a unit vector along the laboratory $X$ axis.

Figure 1:  (a) Classically and quantum mechanically calculated time-dependent
degree of orientation of $\mathrm{CH_{2}O}$ molecules excited by a linearly
polarized THz pulse. Here, $T=300\,\mathrm{K}$ and the field amplitude is
$E_{0}=6.5\,\mathrm{MV/cm}$ [see Eq. (6)]. (b) Magnified view of the
rectangular region marked in (a). The value obtained using the classical
formula [$\braket{L_{a}L_{X}/L^{2}}$, see Eq. (13)] is shown by the dotted
green line. (see text)

Figure 1 shows the time-dependent degree of orientation,
$\braket{\cos(\theta_{aX})}$. The rotational temperature is set to
$T=300\,\mathrm{K}$. As expected, a strong transient orientation appears in
response to the THz pulse. The results of quantum and classical simulations
are in excellent agreement on the short time scale [see Fig. 1(a)]. On a
longer time scale, the classical orientation factor tends to a nonzero
constant value [see Fig. 1(b)], which we denote by
$\braket{\cos(\theta_{aX})}_{p}$, while the quantum expectation value
oscillates around the same nonzero baseline. This effect induced by a single
THz pulse has been previously studied [44]. The long-lasting orientation
metric, $\braket{\cos(\theta_{aX})}_{p}$ was termed the _persistent
orientation_ factor.

Next, we derive an approximate analytic expression for the persistent
orientation factor in the case of asymmetric-top molecules. Starting from the
wave function of an asymmetric-top molecule after the THz excitation, at
$t\,(t>t_{f})$

$\displaystyle\ket{\Psi(t)}$
$\displaystyle=e^{-\frac{i}{\hbar}\int_{t_{f}}^{t}H_{r}dt^{\prime}}\ket{\Psi(t_{f})}$
$\displaystyle=\sum\limits_{J\kappa M}C_{J\kappa M}\ket{J\kappa
M}e^{-\frac{i}{\hbar}\varepsilon_{J\kappa M}(t-t_{f})},$ (8)

where $t_{f}$ is a moment at the end of the pulse, $C_{J\kappa
M}=\braket{J\kappa M}{\Psi(t_{f})}$, and

$\displaystyle\ket{\Psi(t_{f})}=\mathcal{T}e^{-\frac{i}{\hbar}\int_{t_{i}}^{t_{f}}H(t^{\prime})\,dt^{\prime}}\ket{\Psi(t_{i})}$
(9)

is the wave function immediately at the end of the pulse. $\mathcal{T}$ is the
time-ordering operator. Recall that the thermal effects are accounted for in
the initial wave function with the random phases, as shown in Eq. (2). Here,
we use a single $|\Psi_{n}(t_{i})\rangle$ which suffices to describe the
dynamics at high enough temperatures. Accordingly, the time-dependent degree
of orientation is given by

$\displaystyle\braket{\cos(\theta_{aX})}(t)=\braket{\Psi(t)}{\cos(\theta_{aX})}{\Psi(t)}$
$\displaystyle=$
$\displaystyle\sum\limits_{JKM}\sum\limits_{J^{\prime},\kappa,\kappa^{\prime}}A_{JKM,\kappa}^{*}A_{J^{\prime}KM,\kappa^{\prime}}\braket{JKM}{\cos(\theta_{aX})}{J^{\prime}KM}$
$\displaystyle\times
e^{-\frac{i}{\hbar}(\varepsilon_{J^{\prime}\kappa^{\prime}M}-\varepsilon_{J\kappa
M})(t-t_{f})},$ (10)

where $A_{JKM,\kappa}=\braket{JKM}{J\kappa M}\braket{J\kappa M}{\Psi(t_{f})}$.
The mean orientation factor approximates the persistent orientation factor,

$\displaystyle\braket{\cos(\theta_{aX})}_{p}$
$\displaystyle\approx\frac{1}{T_{\mathrm{rev}}}\int_{t_{f}}^{t_{f}+T_{\mathrm{rev}}}\braket{\cos(\theta_{aX})}(t)dt$
$\displaystyle\approx\sum\limits_{JKM}\sum\limits_{\kappa,\kappa^{\prime}}A_{JKM,\kappa}^{*}A_{JKM,\kappa^{\prime}}\frac{KM}{J(J+1)},\\!\\!\\!$
(11)

where $T_{\mathrm{rev}}$ is a “revival period”. It should be noted that
several revival times can be defined for asymmetric-top molecules. In the
context of our derivation, $T_{\mathrm{rev}}$ refers to a typical rotational
time scale on the order of tens of picoseconds, and variations in the exact
value are insignificant. Here, we neglect the oscillatory behavior caused by
the nonzero energy difference between $|J\kappa M\rangle$ and
$|J^{\prime}\kappa^{\prime}M\rangle$,
$\Delta\varepsilon=\varepsilon_{J^{\prime}\kappa^{\prime}M}-\varepsilon_{J\kappa
M}$. Nonzero $\Delta\varepsilon$ produces oscillations, sign changes, and even
the vanishing of the persistent orientation factor on the nanosecond time
scale. Proper description of the molecular rotations on the nanosecond time
scale may require considering intermolecular collisions, complicating the
model considerably. Thus, we leave the study of persistent orientation on such
a long time scale for future studies.

In the special case of a symmetric-top molecule, for which $K$ is a good
quantum number, the formula in Eq. (11) reduces to (see Appendix A for
details)

$\displaystyle\braket{\cos(\theta_{aX})}_{p}$
$\displaystyle\approx\sum\limits_{JKM}|A_{JKM,K}|^{2}\frac{KM}{J(J+1)}$
$\displaystyle=\sum\limits_{JKM}|\braket{JKM}{\Psi(t_{f})}|^{2}\frac{KM}{J(J+1)},$
(12)

which is the same as in [67]. The corresponding classical formula,
$\braket{L_{a}L_{X}/L^{2}}$, can be derived from a single classical symmetric
top persistent orientation, $L_{a}L_{X}/L^{2}$ [44, 24]. Here, angle brackets
denote the ensemble average, $L$ is the magnitude of the angular momentum,
$L_{a}$ and $L_{X}$ are the angular momentum projections on the molecular $a$
axis, and the laboratory $X$ axis (polarization direction of the THz pulse),
respectively. The direct relation between the quantum and classical formulas
is evident due to the correspondence between the quantum and classical
quantities, $J\leftrightarrow L$, $K\leftrightarrow L_{a}$, and
$M\leftrightarrow L_{X}$. After the pulse, $L$, $L_{a}$, and $L_{X}$ are
conserved in the case of a symmetric-top molecule, while $L_{a}$ and $L_{X}$
are conserved even during the pulse.

The case of asymmetric-top molecules is more complicated because $L_{a}$ of
the individual molecules is not a constant of the motion [63]. Nevertheless,
we found that after the pulse, the distribution of $L_{a}L_{X}/L^{2}$ reaches
a dynamic equilibrium, such that the ensemble average
$\braket{L_{a}L_{X}/L^{2}}$ is effectively time-independent (like in the
symmetric-top case). Thus, the classical formula, corresponding to the
quantum-mechanical one in Eq. (11) is

$\displaystyle\braket{\cos(\theta_{aX})}_{p}\approx\left\langle\frac{L_{a}L_{X}}{L^{2}}\right\rangle=\frac{1}{N^{\prime}}\sum\limits_{n^{\prime}=1}^{N^{\prime}}\frac{L_{a}(n^{\prime},t)L_{X}(n^{\prime})}{L^{2}(n^{\prime})},$
(13)

where $L(n^{\prime})$, $L_{a}(n^{\prime},t)$, and $L_{X}(n^{\prime})$ are the
angular momentum magnitude and components of the $n^{\prime}$-th molecule.
After the pulse, $L_{a}(n^{\prime},t)$ is time-dependent, while
$L_{X}(n^{\prime})$ and $L(n^{\prime})$ are conserved. Figure 1(b) compares
the prediction of the formula in Eq. (13) with the results of fully time-
dependent simulation. Equation (13) shows an excellent approximation to the
persistent orientation factor and reflects the following qualitative symmetry
argument. If the molecules are initially isotropically distributed in space,
the distribution of $L_{a}L_{X}/L^{2}$ is fully symmetric, which implies that
$\braket{L_{a}L_{X}/L^{2}}=0$. The THz pulse(s) breaks the symmetry of the
molecular ensemble, resulting in nonzero average
$\braket{L_{a}L_{X}/L^{2}}\neq 0$ during and after the pulse [44].

Figure 2:  Classically calculated persistent orientation factor as a function
of field amplitude at low ($5\,\mathrm{K}$) and at room ($300\,\mathrm{K}$)
rotational temperatures.

It is instructive to consider the dependence of the persistent orientation
factor, $\braket{\cos(\theta_{aX})}_{p}$ on the peak amplitude of the THz
pulse, $E_{0}$ [see Eq. (6)]. Figure 2 shows the classically calculated
persistent orientation factor as a function of the field amplitude,
$\braket{\cos(\theta_{aX})}_{p}(E_{0})$. The degree of orientation is taken at
$t=50\,\mathrm{ps}$, after the steady-state value has been reached, see Fig.
1(b). For linearly polarized THz fields, with increasing field amplitudes, the
value of the persistent orientation factor fluctuates between positive and
negative values, and when $E_{0}\leq 30\,\mathrm{MV/cm}$, its absolute value
is limited: $|\braket{\cos(\theta_{aX})}_{p}|\leq 0.013$ at
$T=300\,\mathrm{K}$ and $|\braket{\cos(\theta_{aX})}_{p}|\leq 0.016$ at
$T=5\,\mathrm{K}$. The persistent orientation factor,
$\braket{\cos(\theta_{aX})}_{p}(E_{0})$, is a non-monotonic function at both
low and high temperatures, which is consistent with our previous study [44].
Thus, increasing the field amplitude won’t necessarily lead to a higher value
of the persistent orientation.

For field polarization fixed along the $X$ axis, $L_{X}$ remains constant
during and after the pulse. Thus, considering the formula in Eq. (13), when
$E_{0}\rightarrow 0$, the symmetry of the distribution of $L_{a}L_{X}/L^{2}$
remains mostly unchanged, resulting in vanishing persistent orientation. When
$E_{0}\rightarrow\infty$, the ensemble symmetry may be completely broken, but
the post-pulse angular momentum $L$ also tends to infinity, resulting in
$L_{a}L_{X}/L^{2}\rightarrow 0$. Thus, between these limits, there will be a
value of the field amplitude that will give rise to the maximal persistent
orientation.

One way to potentially enhance the persistent orientation factor is by shaping
the _field polarization_. The qualitative arguments above do not apply in this
case because the angular momentum projection $L_{X}$ is no longer conserved
when the field polarization is not fixed (e.g., a pair of time-delayed cross-
polarized THz pulses). The following section explores this potential route to
enhanced persistent orientation.

##  IV Persistent orientation induced by a pair of time-delayed cross-
polarized THz pulses

Figure 3:  Classical persistent orientation factor,
$\braket{\cos(\theta_{aX})}_{p}$ as a function of field amplitude, $E_{0}$ and
time delay, $\tau$ for several relative angles, $\alpha$ [see Eqs. (14) and
(15)] at $T=300\,\mathrm{K}$ (left panels) and $T=5\,\mathrm{K}$ (right
panels). As expected, the (absolute) values are lower at room temperature
(note the difference in color bar scales). The persistent orientation factor
along $\mathbf{e}_{2}$ (the polarization direction of the second THz pulse)
satisfies
$\braket{\cos(\theta_{a\mathbf{e}_{2}})}_{p}(E_{0},-\tau)=\braket{\cos(\theta_{aX})}_{p}(E_{0},\tau)$.
Figure 4:  Classically calculated $\phi_{\mathrm{optimal}}$ (along which the
maximum persistent orientation factor is reached) as a function of field
amplitude and time delay for different angles $\alpha$ [see Eqs. (14) and
(15)] at $T=300\,\mathrm{K}$ (left panels) and $T=5\,\mathrm{K}$ (right
panels). Only persistent orientation factor values larger than 0.01 are
considered. The field parameters used are the same as in Fig. 3.  Figure 5:
Classically calculated maximum persistent orientation factor in the $XY$ plane
as a function of field amplitude and time delay (in absolute value) for
various angles $\alpha$ [see Eqs. (14) and (15)] at $T=300\,\mathrm{K}$ (left
panels) and $T=5\,\mathrm{K}$ (right panels). The field parameters used are
the same as in Fig. 3. Equal maximum persistent orientation factors are
achievable for positive or negative time delays but along different $\phi$.

Consider two pulses propagating along the laboratory $Z$ axis; the first THz
pulse is polarized along the $X$ direction, while the second one is polarized
in the $XY$ plane at angle $\alpha$ to the $X$ direction. The resulting
electric field is described by

$\mathbf{E}(t)=E_{0}\Big{[}f(t)\mathbf{e}_{X}+f(t-\tau)\mathbf{e}_{2}\Big{]},$
(14)

where the polarization direction of the second THz pulse is given by

$\mathbf{e}_{2}=\cos(\alpha)\mathbf{e}_{X}+\sin(\alpha)\mathbf{e}_{Y},$ (15)

$\mathbf{e}_{Y}$ is a unit vector along the laboratory $Y$ axis, $f(t)$ is
defined in Eq. (7), and $\tau$ is the time delay between the peaks of two THz
pulses.

We wish to investigate the dependence of the persistent orientation on the
various parameters: field amplitude $E_{0}$, the delay between the two pulses
$\tau$, and the angle between their individual polarizations $\alpha$. Figure
3 depicts this dependence for two rotational temperatures: $T=5\,\mathrm{K}$
(right side) and room ($T=300\,\mathrm{K}$) (left side). For each temperature
we show five panels, each for a different angle $\alpha$ between the two
polarizations:
$\alpha=0^{\circ},45^{\circ},90^{\circ},135^{\circ},180^{\circ}$. Each panel
shows the molecular persistent orientation factor along the laboratory $X$
axis as a function of the field amplitude (horizontal axis) and the time delay
(vertical axis), $\braket{\cos(\theta_{aX})}_{p}(E_{0},\tau)$. The value of
the persistent orientation is given by the color code.

A sizable persistent orientation factor appears at short delays,
$0.2\,\mathrm{ps}<|\tau|<0.6\,\mathrm{ps}$, when the two time-delayed THz
pulses partially overlap. Note, however, for $\tau\approx 0$, the two THz
pulses coalesce into a single linearly polarized pulse, resulting in a small
persistent orientation factor.

By observation, under some conditions, the cross-polarized pulses induce a
higher persistent orientation factor (colors blue or red in the figure). At
$\alpha=45^{\circ},135^{\circ}$ and especially for $\alpha=90^{\circ}$, the
values are much higher than for the collinearly polarized pulses
($\alpha=0^{\circ},180^{\circ}$). The most efficient configuration is for
$\alpha=90^{\circ}$. Indeed, the optimal polarization angle, $\alpha$, depends
on the temperature, the field amplitude, and the time delay. Figure 3(c) shows
that at $T=300\,\mathrm{K}$, the highest (in absolute value) persistent
orientation factor, $\braket{\cos(\theta_{aX})}_{p}\approx 0.029$, is achieved
when $E_{0}=17\,\mathrm{MV/cm}$ and $\tau=0.32\,\mathrm{ps}$. Note that a
slightly higher value ($\braket{\cos(\theta_{aX})}_{p}\approx 0.032$) is
achieved in the case of $\alpha=135^{\circ}$ [see Fig. 3(d)], but it requires
a much higher field amplitude ($E_{0}=30\,\mathrm{MV/cm}$).

The use of cross-polarized pulses at room temperature almost doubles the
persistent orientation induced by a pair of collinearly polarized pulses
[$|\braket{\cos(\theta_{aX})}_{p}|<0.019$ in Fig. 3(a) and
$|\braket{\cos(\theta_{aX})}_{p}|<0.008$ in Fig. 3(e)]. The enhancement is
more pronounced at low temperatures. As shown in Fig. 3(h) for
$T=5\,\mathrm{K}$, a value of $\braket{\cos(\theta_{aX})}_{p}\approx-0.101$ is
reached when $E_{0}=21\,\mathrm{MV/cm}$ and $\tau=-0.58\,\mathrm{ps}$. This is
more than five times higher (in absolute value) compared to collinearly
polarized THz pulses [$|\braket{\cos(\theta_{aX})}_{p}|<0.019$ in Figs. 3(f)
and 3(j)]. These findings confirm that polarization shaping (e.g., combining
two time-delayed orthogonally polarized pulses) may be efficient for
increasing the persistent orientation.

A single THz pulse induces persistent orientation along its polarization
direction. However, cross-polarized THz pulses induce orientation in the $XY$
plane, and it is not apriori clear that the polarization along the $X$
direction is the maximal one. For fixed field parameters (polarization angle,
field amplitude, and time delay), we consider the persistent orientation along
a direction in the $XY$ plane defined by
$\mathbf{e}_{XY}=\cos(\phi)\mathbf{e}_{X}+\sin(\phi)\mathbf{e}_{Y}$, where the
angle $\phi$ runs from $0^{\circ}$ to $360^{\circ}$. The direction along which
the maximum persistent orientation factor appears is defined as
$\phi_{\mathrm{optimal}}$. We plot $\phi_{\mathrm{optimal}}$ and maximum
persistent orientation factor (measured along the optimal direction) as
functions of the field amplitude and the time delay, for each set of field
parameters in Figs. 4 and 5, respectively. For better visibility, Fig. 4 shows
only the points where the maximum persistent orientation factor is larger than
0.01. As can be seen, $\phi_{\mathrm{optimal}}$ has a non-monotonic dependence
on the field amplitude and the time delay.

Figure 5 shows that sizable persistent orientation factor appears when
$0.2\,\mathrm{ps}<|\tau|<0.6\,\mathrm{ps}$, similar to
$\braket{\cos(\theta_{aX})}_{p}$ in Fig. 3. For the collinearly polarized THz
pulses ($\alpha=0^{\circ},180^{\circ}$), the maximum persistent orientation
factor is the same as in Fig. 3. However, the maximum persistent orientation
factor is slightly higher in the case of cross-polarized THz pulses. Figure
5(c) shows that at $T=300\,\mathrm{K}$, the global maximum at
$E_{0}=17\,\mathrm{MV/cm}$ and $\tau=0.32\,\mathrm{ps}$ is about 0.0293, which
is a slightly higher than $\braket{\cos(\theta_{aX})}_{p}$ in Fig. 3. At
$T=5\,\mathrm{K}$, the global maximum is about 0.116 at
$E_{0}=30\,\mathrm{MV/cm}$ and $\tau=0.38\,\mathrm{ps}$, as shown in Fig.
5(h).

Figure 6: Classically calculated persistent orientation factor as a function
of temperature for different time delays. Here the field amplitude is fixed at
$E_{0}=21\,\mathrm{MV/cm}$ and the polarization angle is fixed at
$\alpha=90^{\circ}$, see Eqs. (14) and (15).

Temperature is a crucial parameter affecting the persistent orientation [53,
44, 55, 24, 27]. Figure 6 shows the temperature dependence of the persistent
orientation factor, $\braket{\cos(\theta_{aX})}_{p}$ for different time
delays. Here, we consider orthogonally polarized THz pulses
($\alpha=90^{\circ}$) and the field amplitude is fixed at
$E_{0}=21\,\mathrm{MV/cm}$. Recall that the persistent orientation factor
along the laboratory $X$ axis is approximately given by
$\braket{L_{a}L_{X}/L^{2}}$. In the case of a single $X$-polarized THz pulse,
there is no persistent orientation at $T=0\,\mathrm{K}$ because the conserved
$L_{X}$ is zero [44]. In contrast, the orthogonally polarized pulses change
$L_{X}$ and can induce nonzero persistent orientation factor in the $XY$ plane
even at $T=0\,\mathrm{K}$.

For long time delays, the maximum (in absolute value)
$\braket{\cos(\theta_{aX})}_{p}\approx-0.131$ appears at $T=0\,\mathrm{K}$,
and $\braket{\cos(\theta_{aX})}_{p}$ decays with the temperature (see the case
of $\tau=-0.6\,\mathrm{ps}$ in Fig. 6). The persistent orientation factor
slowly decreases with temperature for shorter time delays (see the cases of
$\tau=-0.5,-0.4\,\mathrm{ps}$). The optimal time delay at higher temperatures
is shorter than at low temperatures. For instance, at $T=0\,\mathrm{K}$, the
optimal time delay is $\tau=-0.6\,\mathrm{ps}$, while at $T=300\,\mathrm{K}$,
the optimal time delay is $\tau=-0.4\,\mathrm{ps}$. The effective rotational
time scale is shorter at high temperatures, requiring shorter time delays. As
noted, for vanishingly short time delays, the two pulses coalesce to a
linearly polarized pulse, leading to a smaller persistent orientation factor
that may even change its sign with temperature (see the case of
$\tau=-0.3\,\mathrm{ps}$).

##  V Conclusions

We set out to investigate the persistent orientation of asymmetric-top
molecules excited by a pair of cross-polarized THz pulses, aiming to overcome
the limitations of the existing scheme that yields relatively small persistent
orientation. Our simulations revealed that using time-delayed, orthogonally
polarized THz pulses significantly enhances the persistent orientation,
especially at lower temperatures. These findings not only outperform previous
methods but also suggest that measurements at room temperature are now within
experimental feasibility.

Furthermore, we derived an approximate expression describing the field-free
persistent orientation of asymmetric-top molecules. One key distinction of our
cross-polarized scheme is that it disrupts the conservation of angular
momentum along the field polarization direction, contributing to the observed
enhancement. While this qualitative insight provides an initial explanation, a
more detailed and rigorous understanding of the underlying mechanism still
warrants further research. Moreover, simultaneous optimization of all
controllable parameters (including, for example, pulse duration) may yield
even greater improvements.

The persistent orientation may be measured by probing the free-induction-decay
[40, 43], by measuring the second (or higher order) harmonic generation in the
gas phase [15, 68, 69], or via Coulomb explosion [17, 47, 54]. Naturally,
intermolecular collisions will eventually destroy the persistent orientation.
However, this may be useful, as time-resolved measurement of persistent
orientation decay may provide new insights into the molecular gas relaxation
dynamics.

###### Acknowledgements.

This work is supported by the startup funding from Xiamen University. I.A.
gratefully acknowledges the hospitality extended to him during his stay at the
Department of Chemistry of the University of British Columbia. This research
was made possible in part by the historic generosity of the Harold Perlman
Family.

##  Appendix A Persistent orientation factor of asymmetric-top molecules

The time-dependent degree of orientation along the laboratory $X$ axis,
according to Eq. (III), is given by

$\displaystyle\braket{\cos(\theta_{aX})}(t)=\braket{\Psi(t)}{\cos(\theta_{aX})}{\Psi(t)}$
$\displaystyle=$ $\displaystyle\sum\limits_{J\kappa
M}\sum\limits_{J^{\prime}\kappa^{\prime}M^{\prime}}C_{J\kappa
M}^{*}C_{J^{\prime}\kappa^{\prime}M^{\prime}}\braket{J\kappa
M}{\cos(\theta_{aX})}{J^{\prime}\kappa^{\prime}M^{\prime}}e^{-\frac{i}{\hbar}(\varepsilon_{J^{\prime}\kappa^{\prime}M^{\prime}}-\varepsilon_{J\kappa
M})(t-t_{f})}$ $\displaystyle=$ $\displaystyle\sum\limits_{J\kappa
M}\sum\limits_{J^{\prime}\kappa^{\prime}M^{\prime}}\sum\limits_{K,K^{\prime}}C_{J\kappa
M}^{*}C_{J^{\prime}\kappa^{\prime}M^{\prime}}\braket{J\kappa
M}{JKM}\braket{JKM}{\cos(\theta_{aX})}{J^{\prime}K^{\prime}M^{\prime}}\braket{J^{\prime}K^{\prime}M^{\prime}}{J^{\prime}\kappa^{\prime}M^{\prime}}e^{-\frac{i}{\hbar}(\varepsilon_{J^{\prime}\kappa^{\prime}M^{\prime}}-\varepsilon_{J\kappa
M})(t-t_{f})}$ $\displaystyle=$
$\displaystyle\sum\limits_{JKM}\sum\limits_{J^{\prime},\kappa,\kappa^{\prime}}C_{J\kappa
M}^{*}C_{J^{\prime}\kappa^{\prime}M}\braket{J\kappa
M}{JKM}\braket{JKM}{\cos(\theta_{aX})}{J^{\prime}KM}\braket{J^{\prime}KM}{J^{\prime}\kappa^{\prime}M}e^{-\frac{i}{\hbar}(\varepsilon_{J^{\prime}\kappa^{\prime}M}-\varepsilon_{J\kappa
M})(t-t_{f})}$ $\displaystyle=$
$\displaystyle\sum\limits_{JKM}\sum\limits_{J^{\prime},\kappa,\kappa^{\prime}}A_{JKM,\kappa}^{*}A_{J^{\prime}KM,\kappa^{\prime}}\braket{JKM}{\cos(\theta_{aX})}{J^{\prime}KM}e^{-\frac{i}{\hbar}(\varepsilon_{J^{\prime}\kappa^{\prime}M}-\varepsilon_{J\kappa
M})(t-t_{f})},$ (16)

where $A_{JKM,\kappa}=C_{J\kappa M}\braket{JKM}{J\kappa
M}=\braket{JKM}{J\kappa M}\braket{J\kappa M}{\Psi(t_{f})}$.

The persistent orientation factor is approximated as the mean orientation
factor averaging over a revival period,

$\displaystyle\braket{\cos(\theta_{aX})}_{p}$
$\displaystyle\approx\frac{1}{T_{\mathrm{rev}}}\int_{t_{f}}^{t_{f}+T_{\mathrm{rev}}}\braket{\cos(\theta_{aX})}(t)dt$
$\displaystyle\approx\sum\limits_{JKM}\sum\limits_{\kappa,\kappa^{\prime}}A_{JKM,\kappa}^{*}A_{JKM,\kappa^{\prime}}\frac{KM}{J(J+1)},\\!\\!\\!$
(17)

where $T_{\mathrm{rev}}$ is a “revival period”. It should be noted that
several revival times can be defined for asymmetric-top molecules. In the
context of our derivation, $T_{\mathrm{rev}}$ refers to a typical rotational
time scale on the order of tens of picoseconds, and variations in the exact
value are insignificant. Here, we neglect the oscillatory behavior caused by
the coherence between $|J\kappa M\rangle$ and
$|J^{\prime}\kappa^{\prime}M\rangle$, which has a nonzero energy difference,
$\Delta\varepsilon=\varepsilon_{J^{\prime}\kappa^{\prime}M}-\varepsilon_{J\kappa
M}$. On the long-time scale, such nonzero energy difference may result in
oscillation, sign change, and even the vanishing of the persistent orientation
factor. This is in sharp contrast to the case of symmetric-top molecules,
where the persistent orientation is permanent.

The states $|JKM\rangle$ are the eigenfunctions of the symmetric-top
molecules, thus the time-dependent degree of orientation [see Eq. (16)] can be
expressed as

$\displaystyle\braket{\cos(\theta_{aX})}(t)$ $\displaystyle=$
$\displaystyle\sum\limits_{J^{\prime},JKM}A_{JKM,K}^{*}A_{J^{\prime}KM,K}\braket{JKM}{\cos(\theta_{aX})}{J^{\prime}KM}$
$\displaystyle\times
e^{-\frac{i}{\hbar}(\varepsilon_{J^{\prime}KM}-\varepsilon_{JKM})(t-t_{f})}.$
(18)

For symmetric-top molecules [$B=C$, see Eq. (1)], the rotational kinetic
energy takes the form [58]

$\displaystyle\begin{split}\varepsilon_{JKM}=BJ(J+1)+(A-B)K^{2}.\end{split}$
(19)

As a result, the energy difference between $|JKM\rangle$ and
$|J^{\prime}KM\rangle$ is
$\varepsilon_{J^{\prime}KM}-\varepsilon_{JKM}=B(J^{\prime}+J+1)(J^{\prime}-J)=2mB$,
where $m$ is an integer. Averaging over a revival period,
$T_{\mathrm{rev}}=\pi\hbar/B$ [70], the accumulated phase is
$-(\varepsilon_{J^{\prime}KM}-\varepsilon_{JKM})T_{\mathrm{rev}}/\hbar=-2m\pi$,
resulting in the persistent orientation factor

$\displaystyle\braket{\cos(\theta_{aX})}_{p}$
$\displaystyle\approx\sum\limits_{JKM}A_{JKM,K}^{*}A_{JKM,K}\frac{KM}{J(J+1)}$
$\displaystyle=\sum\limits_{JKM}|\braket{JKM}{\Psi(t_{f})}|^{2}\frac{KM}{J(J+1)},$
(20)

which is consistent with the results in the quantum [67] and classical cases
[44, 24].

The persistent orientation factors in the cases of symmetric- [Eq. (20)] and
asymmetric-top [Eq. (17)] molecules are similar, and both depend on the three
quantum numbers: $J$, $K$, $M$. However, there are two main differences: (i)
The persistent orientation factor is permanent for symmetric-top molecules,
while it oscillates and eventually vanishes for asymmetric-top molecules due
to the nonzero energy difference,
$\Delta\varepsilon=\varepsilon_{J^{\prime}\kappa^{\prime}M}-\varepsilon_{J\kappa
M}$. (ii) According to Eqs. (17) and (20), the persistent orientation factor
depends on the quantum number $K$. As a result, each symmetric-top molecule
has a specific contribution to the persistent orientation factor. On the other
hand, in the case of an asymmetric-top molecule, $K$ is not a good quantum
number; the contribution of each molecule is time-dependent and only the
ensemble has a well-defined, approximately time-independent persistent
orientation factor.

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