11institutetext:  Australian National University, Canberra, Australia
11email: {daniyar.itegulov<EMAIL_ADDRESS>22institutetext:
ITMO University, St. Petersburg, Russia

# Handling Verb Phrase Anaphora with Dependent Types and Events

Daniyar Itegulov 11 2 2 Ekaterina Lebedeva We thank Bruno Woltzenlogel Paleo
and Florrie Verity for numerous helpful discussions and insightful remarks.11

###### Abstract

This paper studies how dependent typed events can be used to treat verb phrase
anaphora. We introduce a framework that extends Dependent Type Semantics (DTS)
with a new atomic type for neo-Davidsonian events and an extended @-operator
that can return new events that share properties of events referenced by verb
phrase anaphora.

The proposed framework, along with illustrative examples of its use, are
presented after a brief overview of the necessary background and of the major
challenges posed by verb phrase anaphora.

##  1 Introduction

Davidson [2] observed that some verbs can imply the existence of an “action”.
For example, a sentence “John eats.” represents an action of eating. This
action can be anaphorically referred from a following sentence (e.g.: “The
food is yummy.”). Hence, a framework of natural language semantics should
encompass the notion of action and a mechanism for action reference. Davidson
proposed to equip interpretations of verbs with an additional argument for
events. Then the sentence “John eats.” is interpreted according to Davidson as
$\exists e.eats(e,\textbf{j})$, instead of $eats(\textbf{j})$.

Parsons [8] and Taylor [13] argued that the approach of event semantics
captures the notion of adverbs better than approaches based on higher-order
predicates, such as [14], and is easier to work with. For example, adverbial
modifiers usually affect only the event and not the entity, as the following
example illustrates:

[(1)]  a. John buttered the toast slowly, deliberately, in the bathroom, with
a knife, at midnight. b. $\exists e.butter(e,\textbf{j},\textbf{t})\land
slowly(e)\land deliberately(e)\land in(e,\textbf{b})\land\exists
k.with(e,k)\land at(e,\textbf{m})$

Sentence (1a) contains adverbs that modify the event of buttering the toast.
The corresponding interpretation is shown in (1b).

Additionally to adverbial modifiers, Parsons [8] described two more reasons
for introducing events as a new atomic type: perception verbs and reference to
events.

Parsons [8], furthermore, proposed a framework based on Davidson’s event
theory, called neo-Davidsonian event semantics, that extends it as follows:

  * •

event participants are introduced via thematic roles

  * •

state verbs, in addition to action verbs, are handled with an abstract
variable

  * •

two concepts, event holding and event culmination, are added

  * •

events are decomposed into subevents

The differences between Davidsonian and neo-Davidsonian approaches can be seen
by comparing interpretations (1b) and (1) of Sentence (1a).

[(2)] $\exists e.butter(e)\land agent(e,\textbf{j})\land
patient(e,\textbf{t})\land slowly(e)\land deliberately(e)\land
in(e,\textbf{b})\land\exists k.with(e,k)\land at(e,\textbf{m})$

This paper proposes a framework for solving verb phrase anaphora (also known
as verb phrase ellipsis) based on the neo-Davidsonian event semantics and on
dependent types; and to adapt the existing techniques of handling the
propositional anaphora to Dependent Type Semantics framework. Dependent types
are already used to express pronominal anaphora in [1].

In Section 2, we briefly recall Dependent Type Semantics (DTS), which is a
theoretical foundation for our framework. In Section 3, we discuss major
problems of interpreting verb phrase anaphora. The main contribution of this
paper is presented in Section 4, which describes an extension of the Dependent
Type Semantics, and in Section 5, which discusses an application of subtyping
in the proposed framework.

##  2 Recalling Dependent Type Semantics

Dynamic type semantics (DTS) proposed in [1] is a framework of discourse
semantics based on dependent type theory (Martin-Löf [6]). DTS follows the
constructive, proof-theoretic approach to semantics established by Sundholm
[12] who introduced _Sundholmian_ semantics and by Ranta [11] who introduced
Type Theoretical Grammar.

######  Definition 1 (Dependent function)

For any $(s_{1},s_{2})\in\\{(type,type),(type,\\\
kind),(kind,type),(kind,kind)\\},s\in\\{type,kind\\}$:

$A:s_{1}$ $x:A$ $\vdots$ $B:s_{2}$ $(x:A)\to B:s_{2}$

$A:s$ $x:A$ $\vdots$ $M:B$ $\lambda x.M:(x:A)\to B$

$M:(x:A)\to B$ $N:A$ $MN:B[N/x]$

######  Definition 2 (Dependent pair)

For any $(s_{1},s_{2})\in\\{(type,type),(type,\\\ kind),(kind,kind)\\}$:

$A:s_{1}$ $x:A$ $\vdots$ $B:s_{2}$ $\begin{bmatrix}x:A\\\ B\\\
\end{bmatrix}:s_{2}$

$M:A$ $N:B[M/x]$ $(M,N):\begin{bmatrix}x:A\\\ B\\\ \end{bmatrix}$

$M:\begin{bmatrix}x:A\\\ B\\\ \end{bmatrix}$ $\pi_{1}M:A$

$M:\begin{bmatrix}x:A\\\ B\\\ \end{bmatrix}$ $\pi_{2}M:B[\pi_{1}M/x]$

DTS employs two kinds of dependent types (in addition to simply-typed lambda
calculus): _dependent pair type_ or $\Sigma$-type (notation $(x:A)\to B(x)$)
and _dependent function type_ or $\Pi$-type (notation $(x:A)\times B(x)$). A
dependent pair is a generalization of an ordinary pair. By Curry-Howard
correspondence between types and propositions, the type $(x:A)\times B(x)$
corresponds to an existential quantified formula $\exists x^{A}.B$ and to an
ordinary conjunction $A\land B$ when $x\not\in fv(B)$.111$fv(x)$ denotes all
free variables in $x$. A dependent function is a generalization of an ordinary
function and the type $(x:A)\to B$ corresponds to $\forall x^{A}.B$. Formal
definitions are given through inference rules in definitions 1 and 2.

A comparison between the traditional notation for dependent types and the
notation used in DTS can be seen in fig. 1.

|  $\Pi$-type |  $\Sigma$-type
---|---|---
Initial notation |  $(\Pi x:A)B(x)$ |  $(\Sigma x:A)B(x)$
DTS notation |  $(x:A)\to B(x)$ |  $(x:A)\times B(x),\begin{bmatrix}x:A\\\ B(x)\\\ \end{bmatrix}$
When $x\not\in fv(B)$ |  $A\to B$ |  $\begin{bmatrix}A\\\ B\\\ \end{bmatrix}$
Figure 1: Notation in DTS

The main atomic type in DTS is entity, which represents all entities in a
discourse. With the employment of dependent type constructors, the entity type
can be combined with additional properties. For example, $(\Sigma u:(\Sigma
x:\textbf{entity})\times man(x))\times enter(\pi_{1}(u))$ is a valid DTS
interpretation of “A man entered”. Therefore, in contrast to traditional
approaches to semantical interpretation where entities are not distinguished
by their types, each entity has its own type in DTS.

In the traditional Montague model-theoretic semantics [7] a proposition
denotes a truth value (often defined as an $o$-type). However, DTS does not
follow this convention and instead the meaning of a sentence is represented by
a type. In dependent type theory, types are defined by the inference rules, as
shown in definitions 1 and 2. The rules specify how a dependent type (as a
proposition) can be proved under a given context. Thus, the meaning of a
sentence in proof-theoretic semantics lies in its _verification condition_
similar to the philosophy of language by Dummett [3] [4] and Prawitz [9].

To handle anaphora resolution, DTS distinguishes two kinds of propositions:
static and dynamic. A static proposition $P$ is called true if it is
inhabited, i.e. there exists a term of type $P$. A dynamic proposition is a
function mapping context-proof (a static proposition, representing the
previous discourse) to a static proposition.

@-operator is used to represent anaphora and presupposition triggers. The
operator takes the left context of dynamic propositions it is used in. For
example, Sentence (3a) can be interpreted as (3b) in DTS. The @-operator in
(3b) takes a context as an argument and tries to find a female entity (due to
the interpretation of the word “herself”) in the context passed to it.

Different @-operators can have different types (since context passed to them
can vary in its type). They are distinguished with a numerical subscript.
Since different @-operators can take differently typed contexts, the operators
should have different types as well. That is why contexts’ types of
@-operators are distinguished with a numerical subscript. The full type of
$@_{i}$-operator can look like this: $@_{i}:\gamma_{i}\to\textbf{entity}$.

[(3)]  a. Mary loves herself. b. $\lambda
c.loves(\textbf{m},\pi_{1}(@_{1}c:\begin{bmatrix}x:\textbf{entity}\\\
female(x)\\\ \end{bmatrix}))$

##  3 Verb Phrase Anaphora

Verb phrase anaphora [10] are anaphora with an intentional omission of part of
a full-fledged verb phrase when the ellipsed part can be implicitly derived
from the context. For example, verb phrase anaphora can be observed in (4a)
and (4b):

[(4)]  a. John left before Mary did. b. John left. Mary did too.

In (4a), the word “did” refers to an action John did before Mary. In (4b), the
“did too” clause refers to an action which John and Mary both did. These
sentences can be interpreted in event semantics as the following logical
expressions:

[(5)]  a. $\exists e.agent(e,\textbf{j})\land left(e)\land\exists
e^{\prime}.agent(e^{\prime},\textbf{m})\land left(e^{\prime})\land
before(e,e^{\prime})$ b. $\exists e.agent(e,\textbf{j})\land
left(e)\land\neg\exists e^{\prime}.agent(e^{\prime},\textbf{m})\land
left(e^{\prime})$

Furthermore, an anaphoric verb phrase can “inherit” some properties from its
referent. Consider Example (6a) where “did too” not only refers to the event
of eating performed by John, but also to properties such as “quietly” and
“last night”. Expression (6b) is the interpretation of this sentence in event
semantics.

[(6)]  a. John quietly ate the cake last night. Mary did too. b. $\exists
e.(agent(e,\textbf{j})\land patient(e,\textbf{c})\land ate(e)\land
quietly(e)\land at(e,\textbf{ln}))\land\\\ \exists
e^{\prime}.(agent(e^{\prime},\textbf{m})\land
patient(e^{\prime},\textbf{c})\land ate(e^{\prime})\land
quietly(e^{\prime})\land at(e^{\prime},\textbf{ln}))$

A verb phrase anaphora may have an additional property that can ease the
choice of a correct anaphoric referent-event from the context. This phenomenon
is exemplified in (3), where it is explicit that “too” refers to an action
connected with eating.

John ate pasta and did not feel well. Mary ate too, but nothing happened to
her.

An ambiguity between strict and sloppy identity readings of verb phrase
anaphora described by Prust [10] is another intriguing phenomenon. Example
(LABEL:hat-all) illustrates this:

hat-all

John likes his hat. Fred does too.

$\exists x.hat(x)\land owner(x,\textbf{j})\land\exists e.like(e)\land
agent(e,\textbf{j})\land patient(e,x)\land\\\ \exists
e^{\prime}.like(e^{\prime})\land agent(e^{\prime},\textbf{f})\land
patient(e^{\prime},x)$

$\exists x.hat(x)\land owner(x,\textbf{j})\land\exists e.like(e)\land
agent(e,\textbf{j})\land patient(e,x)\land\\\ \exists y.hat(y)\land
owner(y,\textbf{f})\land\exists e^{\prime}.like(e^{\prime})\land
agent(e^{\prime},\textbf{f})\land patient(e^{\prime},y)$

The anaphoric clause in the second sentence of (3) can be interpreted as “Fred
likes John’s hat” (the sloppy identity interpretation (3)) or as “Fred likes
Fred’s hat” (the strict interpretation (3)). A desirable framework should be
able to provide both interpretations.

##  4 Events with Dependent Types

To tackle phenomena discussed in Section 3, we propose to extend DTS with a
new atomic type event for interpreting events. Then, given its left contect
$c$, DTS’s @-operator can be employed for retrieving a variable of type event
analogously to its original use for retrieving a referent of type entity.

As was shown by Parsons [8], event semantics can be employed to represent
propositional anaphora. An example of propositional anaphora is shown in (9a),
where “this” refers to the whole proposition expressed in the first sentence.
Formula (9b) is an interpretation of (9a).

[(9)]  a. John loved Mary. But Mary did not believe this. b. $\exists
e.agent(e,\textbf{j})\land patient(e,\textbf{m})\land loved(e)\land\exists
e^{\prime}.believed(e^{\prime})\land agent(e^{\prime},\textbf{m})\land
patient(e^{\prime},e)$

Dependent typed events allow us to handle more complex types of propositional
anaphora. Similar to entities, events can have various properties provided by
their description. Assume the following three sentences appear in the same
discourse, possibly remotely from each other, but with preservation of the
order:

[(10)]  a. Canberra was hit by a flood on Sunday. b. The fair was held in
London. c. What happened in Canberra is surprising.

Here the anaphoric clause in Sentence (10c) refers to an event discussed
earlier. There are however (at least) two potential events for the reference:
one given by (10a) and another given by (10b). Since the anaphoric clause in
(10c) specifies that it refers to an event happened in Canberra, the anaphora
disambiguates to the event in (10a).

The interpretation of verb phrase anaphora is more challenging, however, than
the interpretation of propositional anaphora: an anaphoric clause in a verb
phrase anaphora usually talks about a new event that inherits properties of
another event. For example, “John left. Bob did too.” conveys two events: one
is about John leaving and the second one is about Bob leaving. In cases of
pronominal and propositional anaphora, however, there is just a reference to
an entity or an event in the context. For example in “John walks. He is
slow.”, pronoun “he” in the second sentence just refers to the entity “John”
from the first sentence.

To handle verb phrase anaphora correctly, it is not enough to just fetch a
referenced variable from the left context; instead a new variable of type
event should be introduced. This new variable _copies_ properties from the
referred event. Furthermore, the agent of the referred event should be changed
to the current agent in the new event. This can be seen in interpretation (6b)
of (6a), where the agent John is replaced with Mary.

Although $@_{i}$-operator has type $\gamma_{i}\to\textbf{entity}$ in DTS for
handling pronominal anaphora, according to DTS syntax for raw terms the
operator can be of any type. We therefore suggest a new type of @-operator
that guarantees that the returned event has a proper agent, necessary for
interpreting the verb phrase anaphora:

[(11)]
$@_{i}:(c:\gamma_{i})\to(x:\textbf{entity})\to\begin{bmatrix}e:\textbf{event}\\\
agent(e,x)\\\ \end{bmatrix}$

Formula (12a) is an interpretation of discourse (4b). The $@_{1}$-operator in
(12b) is applied to its left context $c$ (of type $\gamma_{0}$) and an entity,
and returns a _new_ event of type event with the same properties (apart from
the agent property) of the referenced event. Crucially, the event returned by
$@_{1}$-operator in (12b) is not an event that was in the context previously.
It is a new event with the same properties (e.g. location, time) as a
referenced event from the context, but with a replaced agent.

[(12)]  a. $\lambda c.\begin{bmatrix}e:\textbf{event}\\\
\begin{bmatrix}left(e)\\\ agent(e,\textbf{j}\\\ \end{bmatrix})\\\
\end{bmatrix}$ b. $\lambda
c.(@_{1}c:(x:\textbf{entity})\to\begin{bmatrix}e:\textbf{event}\\\
agent(e,x)\\\ \end{bmatrix})(\textbf{m})$

Note that the entity accepted by the @-operator defined in (4) is the agent in
the new event. For instance, in Example (6a), the interpretation of “did too”
using (4) would have the agent “John” of the referenced event replaced by
“Mary”, but the patient (i.e. the cake) would remain. On the other hand, there
exist cases of verb phrase anaphora where the patient in the referenced event
should be replaced. This usually depends on the voice (active or passive) of
an anaphoric clause, as can be seen from examples in (4).

[(13)]  a. Mary is loved by John. So is Ann.
b. John loves Mary. So does Bob.
c. $\begin{bmatrix}u:\begin{bmatrix}e:\textbf{event}\\\
\begin{bmatrix}agent(e,\textbf{j})\\\ \begin{bmatrix}patient(e,\textbf{m})\\\
loved(e)\\\ \end{bmatrix}\\\ \end{bmatrix}\\\ \end{bmatrix}\\\
\begin{bmatrix}e^{\prime}:\textbf{event}\\\
\begin{bmatrix}agent(e^{\prime},\textbf{j})\\\
\begin{bmatrix}patient(e^{\prime},\textbf{a})\\\ loved(e^{\prime})\\\
\end{bmatrix}\\\ \end{bmatrix}\\\ \end{bmatrix}\\\ \end{bmatrix}$ d.
$\begin{bmatrix}u:\begin{bmatrix}e:\textbf{event}\\\
\begin{bmatrix}agent(e,\textbf{j})\\\ \begin{bmatrix}patient(e,\textbf{m})\\\
loved(e)\\\ \end{bmatrix}\\\ \end{bmatrix}\\\ \end{bmatrix}\\\
\begin{bmatrix}e^{\prime}:\textbf{event}\\\
\begin{bmatrix}agent(e^{\prime},\textbf{b})\\\
\begin{bmatrix}patient(e^{\prime\prime\prime\prime},\textbf{m})\\\
loved(e^{\prime})\\\ \end{bmatrix}\\\ \end{bmatrix}\\\ \end{bmatrix}\\\
\end{bmatrix}$

The first sentences in (13a) and (13b) have the same semantics and hence the
interpretations given to them in (13c) and (13d) coincide. However, despite
both second sentences are written in the same voice as their first sentences,
the second sentences are interpreted differently. Naturally, the second
sentence in (13a) means “Ann is loved by John”, while the second sentence in
(13b) means “Bob loves Mary”. Note that they have replaced different
participants of the first sentences: in (13a) Mary (patient) was replaced by
Ann and in (13b) John (agent) was replaced by Bob.

Furthermore, an interpretation of a sentence may require both the agent and
the patient to be replaced, as for example in the sloppy reading of (3). These
possible cases of anaphora resolution can be tackled with the judgements (4)
assuming they occur in a global context $\mathcal{K}$.

Another important notion in DTS is the felicity condition. The anaphora
resolution for $@_{i}$ operator is launched by type checking of the following
judgement: $\mathcal{K},\gamma_{i}:type\vdash@_{i}:\gamma_{i}\to type$. It
means that the semantical interpretation of a sentence must be of the sort
type assuming that the left context is of type $\gamma_{i}$. A requirement of
a success of the launching the type checker is called _felicity condition_.

In order to preserve the original DTS invariants, we should show how felicity
condition is being fulfilled in the extended DTS. An example of a felicity-
judgement generated by verb phrase anaphora is shown in example (4). It is
different from the felicity condition from original DTS notion since the new
@-operator has a new type as shown in Equation 4.

[(14)]
$\mathcal{K},\gamma_{i}:type\vdash@_{i}:\gamma_{i}\to(x:\textbf{entity})\to\begin{bmatrix}e:\textbf{event}\\\
agent(e,x)\\\ \end{bmatrix}$

Let us assume that the global context $\mathcal{K}$ contains the judgements
from (4). Then one should be able to type check judgements generated by the
verb phrase anaphora.

[(15)]  a. $\begin{aligned} replaceA:\
&(p:\textbf{entity}\to(e:\textbf{event})\to\to\\\
&(original:\textbf{entity})\to(new:\textbf{entity})\to\\\
&(u:\begin{bmatrix}e^{\prime}:\textbf{event}\\\ p\ original\ e^{\prime}\\\
\end{bmatrix})\to(v:\begin{bmatrix}e^{\prime\prime}:\textbf{event}\\\ p\ new\
e^{\prime\prime}\\\ \end{bmatrix})\end{aligned}$ b. $\begin{aligned}
replaceP:\ &(p:\textbf{entity}\to(e:\textbf{event})\to\to\\\
&(original:\textbf{entity})\to(new:\textbf{entity})\to\\\
&(u:\begin{bmatrix}e^{\prime}:\textbf{event}\\\ p\ original\ e^{\prime}\\\
\end{bmatrix})\to(v:\begin{bmatrix}e^{\prime\prime}:\textbf{event}\\\ p\ new\
e^{\prime\prime}\\\ \end{bmatrix})\end{aligned}$ c. $\begin{aligned}
replaceAP:\
&(p:\textbf{entity}\to\textbf{entity}\to(e:\textbf{event})\to\to\\\
&(oagent:\textbf{entity})\to(nagent:\textbf{entity})\to\\\
&(opatient:\textbf{entity})\to(npatient:\textbf{entity})\to\\\
&(u:\begin{bmatrix}e^{\prime}:\textbf{event}\\\ p\ oagent\ opatient\
e^{\prime}\\\
\end{bmatrix})\to(v:\begin{bmatrix}e^{\prime\prime}:\textbf{event}\\\ p\
nagent\ npatient\ e^{\prime\prime}\\\ \end{bmatrix})\end{aligned}$ d.
$j:\textbf{entity}$

Functions $replaceA$, $replaceP$, $replaceAP$ construct a new event $v$ from
an existing event $u$. To express the inheritance of properties and the change
of the agent in $replaceA$ (or patient in $replaceP$), properties are
expressed as a function that accepts two arguments: an agent-entity (or
patient-entity in $replaceP$) and an event; and returns a logical expression
describing the event using the entity. Function $replaceAP$ accounts for cases
where both an agent and a patient are replaced.

We can now construct term $@_{1}$ of type (4), to fulfill the felicity
condition of form (4), as shown in (4):

[(16)] $\mathcal{K},\gamma_{0}:type\vdash\begin{aligned}
&@_{1}:\gamma_{0}\to(x:\textbf{entity})\to\begin{bmatrix}e:\textbf{event}\\\
agent(e,x)\\\ \end{bmatrix}=\\\ &\lambda c.\lambda x.replaceA\ (\lambda
y.\lambda e.\begin{bmatrix}left(e)\\\ agent(e,y)\\\ \end{bmatrix})\
\textbf{j}\ x\ \pi_{1}\pi_{2}(c)\end{aligned}$

A substitution of $@_{1}$ in (12b) with its term defined in (4) leads to the
following semantical interpretation:

[(17)] $\begin{bmatrix}e^{\prime\prime}:\textbf{event}\\\
\begin{bmatrix}left(e^{\prime\prime})\\\ agent(e^{\prime\prime},\textbf{m})\\\
\end{bmatrix}\\\ \end{bmatrix}$

Since anaphora in DTS is resolved using the type checking procedure, the verb
phrase anaphora, just like the pronominal anaphora, can be resolved in various
ways. A type checking algorithm can find different terms which conform to the
specified (by felicity condition) type. For example, in order to handle the
ambiguity between strict and sloppy identity readings, which were discussed in
Example (LABEL:hat-all), our framework can provide both possible
interpretations for Sentence (3). Term (4) shows a generic interpretation of
(3) in the proposed framework.

[(18)] $\lambda
c.\begin{bmatrix}u:\begin{bmatrix}v:\begin{bmatrix}x:\textbf{entity}\\\
\begin{bmatrix}hat(x)\\\ owner(x,\textbf{j})\\\ \end{bmatrix}\\\
\end{bmatrix}\\\ \begin{bmatrix}e:\textbf{event}\\\ \begin{bmatrix}like(e)\\\
\begin{bmatrix}agent(e,\textbf{j})\\\ patient(e,\pi_{1}(v))\\\
\end{bmatrix}\\\ \end{bmatrix}\\\ \end{bmatrix}\\\ \end{bmatrix}\\\
@_{1}(c,u)\textbf{f}:\begin{bmatrix}e^{\prime}:\textbf{event}\\\
agent(e^{\prime},\textbf{f})\\\ \end{bmatrix}\\\ \end{bmatrix}$

In (4), “Fred does too.” is interpreted as the term $@_{1}(c,u)\textbf{f}$,
where $u$ stands for the interpretation of the preceeding sentence “John likes
his hat.”. Recall from Example (LABEL:hat-all) that the latter sentence has an
ambiguous meaning. (4) defines two alternative terms for $@_{0}$, one for each
of the possible meanings. Note that the type of these terms for $@_{0}$
conforms with the felicity condition.

[(19)]  a. $\begin{aligned}
\mathcal{K}\vdash&@_{1}:\gamma_{0}\to(x:\textbf{entity})\to\begin{bmatrix}e^{\prime}:\textbf{event}\\\
agent(e^{\prime},x)\\\ \end{bmatrix}=\\\ &\lambda c.\lambda f.replaceA\
(\lambda y.\lambda e.\begin{bmatrix}like(e)\\\ \begin{bmatrix}agent(e,y)\\\
patient(e,x)\\\ \end{bmatrix}\\\ \end{bmatrix})\ \textbf{j}\ f\
\pi_{1}\pi_{2}\pi_{2}(c)\end{aligned}$ b. $\begin{aligned}
\mathcal{K}\vdash&@_{1}:\gamma_{0}\to(x:\textbf{entity})\to\begin{bmatrix}e^{\prime}:\textbf{event}\\\
agent(e^{\prime},x)\\\ \end{bmatrix}=\\\ &\textbf{let}\ p=\lambda y.\lambda
z.\lambda e.\begin{bmatrix}like(e)\\\ \begin{bmatrix}agent(e,y)\\\
patient(e,z)\\\ \end{bmatrix}\\\ \end{bmatrix}\\\ &\textbf{in}\ \lambda
c.\lambda f.\begin{bmatrix}u:\begin{bmatrix}y:\textbf{entity}\\\ hat(y)\land
owner(y,f)\\\ \end{bmatrix}\\\ replaceAP\ p\ \textbf{j}\ f\
\pi_{1}\pi_{1}\pi_{2}(c)\ \pi_{1}(u)\ \pi_{1}\pi_{2}\pi_{2}(c)\\\
\end{bmatrix}\end{aligned}$

Both terms are valid substitutions for $@_{1}$-operator in (4) and they
represent strict and sloppy anaphora readings respectively. In (19b) let-in
structure is used only as a syntactical sugar for readability and is not
actually a part of DTS term syntax.

##  5 Subtyping

Equation in (4) (i.e. an anaphora resolution solution: a proof of existence of
a term with the required type under the global context $\mathcal{K}$) is not
sound: the type of the right side of the equation is

$\gamma_{0}\to(x:\textbf{entity})\to\begin{bmatrix}e^{\prime\prime}:\textbf{entity}\\\
\begin{bmatrix}left(e^{\prime\prime})\\\ agent(e^{\prime\prime},x)\\\
\end{bmatrix}\\\ \end{bmatrix}$

while the type required by the left side is

$\gamma_{0}\to(x:\textbf{entity})\to\begin{bmatrix}e:\textbf{entity}\\\
agent(e,x)\\\ \end{bmatrix}$

The former type is more specific than the latter type because it has an
additional property “left”.

This is not a problem, as events have a natural subtyping relationship between
them. As described in [5], an event whose agent is $a$ and patient is $p$, is
an event with agent $a$. Despite a different theory underneath, the techniques
described there can be reused for subtyping events in DTS. This leads to the
following subtyping relations in event semantics:

[(20)]  a. $Evt_{AP}(a,p)<:Evt_{A}(a)<:Event\longleftrightarrow\\\
\begin{bmatrix}e:\textbf{event}\\\ \begin{bmatrix}agent(e,a)\\\
patient(e,p)\\\ \end{bmatrix}\\\
\end{bmatrix}<:\begin{bmatrix}e:\textbf{event}\\\ agent(e,a)\\\
\end{bmatrix}<:\begin{bmatrix}e:\textbf{event}\\\ ()\\\ \end{bmatrix}$ b.
$Evt_{AP}(a,p)<:Evt_{P}(p)<:Event\longleftrightarrow\\\
\begin{bmatrix}e:\textbf{event}\\\ \begin{bmatrix}agent(e,a)\\\
patient(e,p)\\\ \end{bmatrix}\\\
\end{bmatrix}<:\begin{bmatrix}e:\textbf{event}\\\ patient(e,p)\\\
\end{bmatrix}<:\begin{bmatrix}e:\textbf{event}\\\ ()\\\ \end{bmatrix}$

Subtyping relations of events can also depend on other properties (e.g. a loud
event performed by John is also an event performed by John). We employ Luo’s
notation to define a new type $Event_{NA}(n,a)$, which is the type of events
with agent $a$ and nature $n$. Nature is a main predicate for each event in
neo-Davidsonian semantics (e.g. “left(e)” for an event of leaving, “ate(e)”
for an event of eating).

The following transformation shows how the dependent event types in DTS
notation from (4) can be converted into dependent event types in Luo notation:

[(21)]  a. $\begin{bmatrix}e^{\prime\prime}:\textbf{event}\\\
\begin{bmatrix}left(e^{\prime\prime})\\\ agent(e^{\prime\prime},x)\\\
\end{bmatrix}\\\ \end{bmatrix}\longleftrightarrow
e^{\prime\prime}:Event_{DA}(left,x)$ b. $\begin{bmatrix}e:\textbf{entity}\\\
agent(e,x)\\\ \end{bmatrix}\longleftrightarrow e:Event_{A}(x)$

A subtyping relationship between these types can be constructed (assuming the
appropriate subtyping rules have been added along with type
$Event_{DA}(d,a)$).

$left:Description$ $x:Agent$ $Event_{DA}(left,x)<:Event_{A}(x)$

The discussed subtyping relationship allows us to obtain (4).

##  6 Conclusion

This paper introduces dependent event types for resolving verb phrase anaphora
with DTS as the underlying framework. To tackle verb phrase anaphora, we
extend DTS’s @-operator, which was originally introduced for handling
pronominal anaphora. The paper also adresses strict and sloppy readings of
verb phrase anaphora and shows that each of them can be achieved solely by
manipulating the interpretation of the @-operator. The previous approaches to
handling the propositional anaphora were also adapted to DTS framework.

Techniques described in this paper could be applied to handle other cases of
anaphora, such as adjectival anaphora, modal and “do so” anaphora. Another
interesting topic would be to study specific behaviours of various thematic
roles, such as experiencer, theme and source.

## References

  * [1] Daisuke Bekki  “Representing Anaphora with Dependent Types”  In _Logical Aspects of Computational Linguistics_ Springer Berlin Heidelberg, 2014, pp. 14–29
  * [2] Donald Davidson  “The Logical Form of Action Sentences”  In _The Logic of Decision and Action_ University of Pittsburgh Press, 1967
  * [3] Michael Dummett  “What is a Theory of Meaning? (II)”  In _Truth and Meaning: Essays in Semantics._ Oxford: Clarendon Press, 1976
  * [4] Michael A. E. Dummett  “What is a Theory of Meaning?”  In _Mind and Language_ Oxford University Press, 1975
  * [5] Zhaohui Luo and Sergei Soloviev  “Dependent Event Types”  In _Logic, Language, Information, and Computation_ Berlin, Heidelberg: Springer Berlin Heidelberg, 2017, pp. 216–228
  * [6] P. Martin-Löf and G. Sambin  “Intuitionistic type theory”, Studies in proof theory  Bibliopolis, 1984
  * [7] Richard Montague  “Formal Philosophy; Selected Papers of Richard Montague”  New Haven: Yale University Press, 1974
  * [8] Terence Parsons  “Events in the Semantics of English: A Study in Subatomic Semantics”  MIT Press, 1990
  * [9] Dag Prawitz  “Intuitionistic Logic: A Philosophical Challenge”  In _Logic and Philosophy / Logique et Philosophie_ Dordrecht: Springer Netherlands, 1980, pp. 1–10
  * [10] Hub Prüst, Remko Scha and Martin Berg  “Discourse grammar and verb phrase anaphora”  In _Linguistics and Philosophy_ 17.3, 1994, pp. 261–327
  * [11] Aarne Ranta  “Type-theoretical Grammar”  Oxford University Press, 1994
  * [12] Göran Sundholm  “Proof Theory and Meaning”  In _Handbook of Philosophical Logic: Volume III: Alternatives in Classical Logic_ Dordrecht: Springer Netherlands, 1986, pp. 471–506
  * [13] Barry Taylor  “Modes of Occurence, Verbs, Adverbs and Events”  In _Revue Philosophique de la France Et de l’Etranger_ 176.3 Presses Universitaires de France, 1986, pp. 406–407
  * [14] Henk J. Verkuyl  “On the Compositional Nature of the Aspects”  Dordrecht, Netherlands: D.Reidel Publishing Company, 1972


# Controlling mode orientations and frequencies in levitated cavity
optomechanics

A. Pontin, H. Fu, J.H. Iacoponi, P.F. Barker and T.S. Monteiro  Department of
Physics and Astronomy, University College London, Gower Street, WC1E 6BT
London, United Kingdom.

###### Abstract

Cavity optomechanics offers quantum cooling, control and measurement of small
mechanical oscillators. However the optomechanical backactions disturb the
oscillator motions, shifting their frequencies and generating hybridisation.
This is especially relevant to levitated cavity optomechanics, where optical
trapping also determines the mechanical modes and their frequencies. Here,
using a nanoparticle trapped in a tweezer in a cavity populated only by
coherently scattered (CS) photons, we investigate experimentally the
$S_{xy}(\omega)$ mechanical cross correlation spectra as a function of the
nanoparticle position on the cavity standing wave. We show that the CS field
not only rotates the mechanical modes in the opposite direction to the cavity
dynamics, but also opposes optical spring effects on the mechanical
frequencies. Hence we demonstrate a cancellation point, independent of most
experimental parameters, where it becomes possible to cavity cool and control
unperturbed mechanical modes. The findings have implications for directional
force sensing using the CS set-ups that permit quantum ground state cooling.

There is currently intense interest in optical cooling of levitated
nanoparticles both with cavities as well as active feedback methods
millen2020optomechanics . Cooling with using optical cavities was proposed
well over a decade ago, romero2010toward ; barker2010cavity ; chang2010cavity
; pender2012optomechanical ; monteiro2013dynamics  but only in 2020 was ground
state cooling achieved delic2020cooling : initial experimental efforts were
plagued by technical difficulties of stable trapping in high vacuum
kiesel2013cavity ; asenbaum2013cavity . Several set-ups were investigated for
cavity cooling including tweezer-cavity traps romero2010toward ;
mestres2015cooling , electro-optical traps millen2015cavity ;
fonseca2016nonlinear , and trapping in the near field of a photonic crystal
magrini2018near . Quantum cooling was also previously achieved with feedback
cooling and quantum control Magrini2021quantum ; Novotny2021quantum , but here
we focus on levitated cavity optomechanical systems.

Recently, a 3D coherent scattering (CS) setup was introduced to levitated
cavity optomechanics delic2019cavity ; windey2019cavity  using methods adapted
from atomic physics vuletic2000laser ; vuletic2001three ;
domokos2002collective ; leibrandt2009cavity ; hosseini2017cavity . In contrast
to experiments that consider dispersive coupling, here the cavity is driven
solely by the dipole radiation of the optically trapped silica particle. The
nanoparticle is trapped at the tight focus of the optical tweezer along the
$z$ axis and the tweezer laser polarisation angle and waist set the
orientation and frequencies $\omega^{(0)}_{x},\omega^{(0)}_{y}$ of the
mechanical motion in the $x-y$ plane. The CS set-up yielded unprecedentedly
high optomechanical coupling rates $g_{x},g_{y}$, which subsequently enabled
ground-state cooling of the motion along the cavity axis and thus opened the
door to levitated cavity optomechanics at or near delic2020cooling ; Marin2021
quantum regimes.

Figure 1:  (a) Inset illustrates a nanoparticle, trapped by an optical tweezer
with $x-y$ mechanical modes. However, the presence of a surrounding cavity,
aimed at quantum cooling, hybridises the modes and shifts their unperturbed
frequencies $\omega_{x,y}^{(0)}$. The experimental tweezer polarisation sets
an initial angle $\theta$ between the $x-y$ modes and cavity axis. Then, the
optomechanical cavity hybridisation dynamics adds an effective mode rotation
$\Phi_{dyn}$, where $\Phi_{dyn}\sim-\theta$ leads to formation of dark/bright
modes. However, away from optical nodes, the CS field opposes this effect. (b)
We investigate mode orientation by measuring cross-correlation spectra
$S_{xy}(\omega)$ for $\theta\simeq\pi/4$ as the trapping position is swept
from node (blue, $\phi=\pi/2$) to near the antinode (red, $\phi=0$) of the
cavity standing wave. The $x-y$ motions are always anticorrelated (peaks of
opposite sign), but $S_{xy}$ flips sign at $\phi=\phi_{c}$ (purple line,
$S_{xy}(\omega)\sim 0$) . For this value, the CS field cancels $\Phi_{dyn}$:
the mechanical modes are locked at their unperturbed orientations and
unperturbed frequencies for arbitrary power and $\theta$, but can still be
strongly cooled. The results have implications for directional force sensing
and, in strong coupling regimes, the suppression of dark/bright modes.  Figure
2:  (a) Measured orientation of the mechanical modes in $x-y$ plane as a
function of trapping position. The mode rotation angles, $\Phi$, are obtained
from experimental cross correlation spectra, $S_{xy}(\omega)$: squares (blue)
correspond to the $x$ peak, circles (cyan) to the $y$ peak. $\Phi$ combines a
positive sign contribution, $\Phi_{dyn}$, from the optomechanical dynamics
arising from optical backaction, and $\Phi_{CS}$, a negative sign contribution
from the static CS field. $\Phi_{dyn}$ is dominant at the node ($\phi=\pi/2$)
while $\Phi_{CS}$ dominates at the antinode ($\phi=0$). Where they cancel
(green circle, $\phi=\phi_{c}$ ) the rotation becomes zero. The measured
values of $\Phi$ are compared with $\Phi_{CS}+\Phi_{dyn}$ calculated from
theory (violet lines). (b) Shows that the frequency behavior mirrors the mode
rotation behavior: the optomechanical optical spring effect softens the
mechanical frequencies for red detuning, while in contrast CS field stiffens
the frequencies. At $\phi\simeq\phi_{c}$ the two contributions cancel, leaving
only the unperturbed mechanical frequency set by the tweezer trap. Red lines
denote theoretical values, calculated from linearised optomechanical equations
of motion. Black line shows their combined effect. $\theta\simeq\pi/4$ for all
plots. (c) Experimental set-up, described in the text.

Theoretical analysis MTTM2020 ; MTTM2021  has shown that $x-y$ hybridisation
results from both cavity mediated, back action terms and a direct coupling of
the form $g_{xy}\hat{x}\hat{y}$. There can be three-way hybridisation of the
normal modes in the $x-y$ plane as well as hybridisation with optical modes,
especially in strong-coupling regimes. The interesting case of $g_{x}\simeq
g_{y}$ which is obtained experimentally for tweezer polarisation angle
$\theta=\pi/4$, in the strong-coupling regime produces a mode rotation angle
of $\Phi\sim\pi/4$ and leads to the formation of dark and bright modes
Harris2014 ; MTTM2021 ; Marin2021 .

The set-up is illustrated in Fig.1 : we investigate the behavior of mechanical
modes in the $x-y$ plane experimentally by measuring cross-correlation
$S_{xy}(\omega)$ spectra. We show that rotation of the modes by an angle
$\Phi_{dyn}$ due to a dynamical optical backaction term is, away from the
node, opposed by a rotation $\Phi_{CS}$ due to the CS potential. In addition,
we find here for the first time, that the experimental mechanical frequency
shifts directly mirror this behaviour. At the node, the optical spring effect
due to the cavity dynamics softens the mechanical frequencies; away from the
node, the CS potential stiffens the mechanical frequency. The two effects
cancel at the same point as the cancellation of the mode rotation angle. At
this point the mechanical $x,y$ frequencies both return to their unperturbed
values and the modes are locked to their unperturbed orientation, for
arbitrary input power and polarisation. Yet there is still strong
optomechanical cavity cooling. Trapping away from the node can be less
advantageous: cavity photon occupancies are higher at the antinode relative
the node delic2019cavity . However, as the detuning tends to resonance, the
cancellation point will move closer to the node, mitigating this disadvantage.

Experimental set-up: A schematic overview of the experiment is shown in
Fig.2(c). We use two Nd:YAG lasers at a wavelength $\lambda=1064$ nm. A weak
field from the first is used to implement a Pound-Drever-Hall (PDH) scheme to
lock it to a high finesse Fabry-Perot cavity. The second laser illuminates an
optical tweezer composed of a single aspheric lens of nominal numerical
aperture $NA=0.77$ and a symmetric condenser lens. The tweezer assembly is
monolithic, is mounted on a $3$-axes translational stage and includes an
aspheric collection lens of $NA=0.3$, oriented at $90^{\circ}$ from the
propagation direction, which is exploited for imaging purposes. The tweezer
trapping region is positioned at the center of the optical cavity which has a
length $L_{cav}=12.23\pm 0.02$ mm, a finesse $\mathcal{F}\simeq 31000$
(linewidth $\kappa/2\pi=396\pm 2$ kHz, input rate $\kappa_{in}/2\pi=162\pm 2$
kHz) and a waist of $61\,\mu$m. The two lasers are phase offset locked by
monitoring their beat note and their frequency separation is set to one free
spectral range (FSR=$c/2L_{cav}=12.26\pm 0.02$ GHz). The PDH beam is locked at
resonance and the detuning of the trapping beam can be finely controlled.
Importantly, the PDH beam is orthogonally polarized with respect to the
tweezer beam and interacts with the particle only dispersively. This, in
combination with the low power makes the role of the PDH beam in the dynamics
negligible. The motion of the particle in the tweezer polarization plane is
monitored by distributing the tweezer light collected by the condenser lens to
two balanced detectors. Each detector is balanced using D-shaped mirrors
oriented parallel and perpendicular to the beam polarization thus sensitive to
$x$ and $y$ respectively. The two detectors can measure independently the $x$
and $y$ motion with a rejection ratio potentially exceeding $-30$ dB, however,
any imperfections in the orientations of the D-mirror can result in a small
mixing of $x$ and $y$ in the detected signals.

Physical model: In a coherent scattering (CS) approach, the optical cavity is
not externally driven but it is populated exclusively by light scattered by
the nanoparticle. The corresponding Hamiltonian results from the coherent
interference between the electric fields of the tweezer and cavity
$\hat{H}=-\frac{\alpha}{2}|\mathbf{\hat{E}}_{\text{cav}}+\mathbf{\hat{E}}{}_{\text{tw}}|^{2},$
where $\alpha$ is the polarizability of the nanosphere. The interference term
$\propto(\mathbf{\hat{E}}_{\text{cav}}^{\dagger}\mathbf{\mathbf{\hat{E}}}_{\text{tw}}+\mathbf{\hat{E}}_{\text{cav}}\mathbf{\mathbf{\hat{E}}}_{\text{tw}}^{\dagger})$
gives rise to an effective CS potential:

$\hat{V}_{\text{CS}}/\hbar=-E_{d}\text{cos}(\phi+k\hat{Y}_{c})e^{-({\hat{x}^{2}/w_{x}^{2}+\hat{y}^{2}/w_{y}^{2}})}\left[\hat{a}+\hat{a}^{\dagger}\right].$
(1)

In the above, we omitted the $\hat{z}$ dependence. The CS dynamics decouples
into a 2+1 dynamics with $x-y$ motion close in frequency and prone to
hybridise; and a (typically) off-resonant $z$ motion that is largely
decoupled. We note that the $z$ motion can be strongly cooled using feedback
cooling Magrini2021quantum ; Novotny2021quantum  but, in most experimental
implementations, $z$ is anyway moderately cooled due to a small unavoidable
tilt relative to the cavity $z$ axis. This effect and the full $z$ dynamics is
fully taken into account in numerical simulations, however, for brevity, it is
not included in the present discussion.

In Eq. 1, $Y_{c}$ represents the cavity axis. Relating the cavity coordinates
to the tweezer frame is a simple rotation of the coordinate frames $[X_{c}\
Y_{c}]^{T}=R_{z}(\theta)[x\ y]$, where $R_{z}(\theta)$ is the 2D rotation
matrix. $\phi$ is the displacement of the trap from an antinode, $w_{y}\simeq
1.068\mu$m and $w_{x}\simeq 0.928\mu$m are the tweezer waists.

Linearised equations of motion are obtained using Eq.1; operators are expanded
about equilibrium values, thus the optical field operators
$\hat{a}\to\overline{\alpha}+\hat{a}(t)$ are expanded about the mean field,
where $n_{p}=|\overline{\alpha}|^{2}$ is the mean photon occupancy of the
cavity. This linearisation yields also optomechanical coupling terms
$g_{x}\hat{x}(\hat{a}+\hat{a}^{\dagger})$ and
$g_{y}\hat{y}(\hat{a}+\hat{a}^{\dagger})$. In MTTM2020 ; MTTM2021  it was
shown further that the CS potential also yields direct coupling terms
$g_{xy}\hat{x}\hat{y}$ that are of similar order to the usual linearised
terms, thus do not vanish as the quantum occupancies are approached.

In MTTM2020 ; MTTM2021  it was shown that $x-y$ hybridisation requires a
correction to the 1D unhybridised solutions, hence
$\hat{x}(\omega)\simeq\hat{x}^{1D}(\omega)+\mathcal{R}_{xy}(\omega)\hat{y}(\omega)$
and
$\hat{y}(\omega)\simeq\hat{y}^{1D}(\omega)+\mathcal{R}_{yx}(\omega)\hat{x}(\omega)$,
where $\mathcal{R}_{xy},\mathcal{R}_{yx}$ are hybridisation functions that are
small in the weak coupling regimes of the present experiments. In that case if
$\mathcal{R}_{xy}\simeq-\mathcal{R}_{yx}$ the above linear $x-y$ hybridisation
relation is already suggestive of a simple frame rotation.

One can show (see Appendix for details) that the resulting mechanical
correlation spectra:

$S_{xy}(\omega)\approx\text{Re}({\mathcal{R}_{yx}}(\omega))S_{xx}(\omega)+\text{Re}({\mathcal{R}_{xy}}(\omega))S_{yy}(\omega)$
(2)

depend on the real parts of the hybridisation functions. For
$\omega\sim\omega_{x,y}$, one can show
$\mathcal{R}_{xy}\simeq-\mathcal{R}_{yx}\equiv
G(\omega)/(\omega_{x}-\omega_{y})$ where
$G(\omega)=\left[i\eta_{c}(\omega)g_{x}g_{y}+g_{xy}\right]$ and
$\eta_{c}=\chi(\omega)-\chi^{*}(-\omega)$,
$\chi(\omega)=1/(-i(\omega+\Delta)+\kappa/2)$ is the cavity susceptibility
function. Hence we can write:

$S_{xy}(\omega)\simeq\Phi\ [S_{yy}(\omega)-S_{xx}(\omega)].$ (3)

The above expression is generic to a 2D optomechanical system with non-zero
$g_{x},g_{y},g_{xy}$ in many typical regimes. However, the CS version has
additional and unexpected features. The angle $\Phi=\Phi_{dyn}+\Phi_{CS}$ can
be decomposed into two separate contributions: (i) a cavity mediated term
$\Phi_{dyn}\equiv\text{Re}({i\eta_{c}g_{x}g_{y}})/(\omega_{x}-\omega_{y})$ and
(ii) a direct contribution $\Phi_{CS}\equiv g_{xy}/(\omega_{x}-\omega_{y})$
arising from the static CS potential. It was found in MTTM2020  that the
optomechanical couplings are $g_{x}\simeq-E_{d}\ k\sin\theta\sin\phi X_{zpf}$,
$g_{y}\simeq-E_{d}\ k\cos\theta\sin\phi Y_{zpf}$ and

$g_{xy}\simeq g_{x}g_{y}2\Delta\cot^{2}{\phi}/(\kappa^{2}/4+\Delta^{2})$ (4)

thus the combined dynamical and CS rotation becomes:

$\Phi=\Phi_{dyn}+\Phi_{CS}=\frac{g_{x}g_{y}}{(\omega_{x}-\omega_{y})}\left[\text{Re}({i\eta_{c}})+\frac{2\Delta\cot^{2}{\phi}}{(\kappa^{2}/4+\Delta^{2})}\right].$
(5)

The two terms are of oppposite sign, hence their effect is to rotate the modes
in opposing directions. $X_{zpf}=\sqrt{\hbar/(2m\omega_{x})}$,
$Y_{zpf}=\sqrt{\hbar/(2m\omega_{y})}$.

The experimental PSDs $S_{xx},S_{yy}$ and correlation spectra at different
trap positions are acquired at a constant pressure of $3\times 10^{-3}$ mbar
with each time trace covering an observation time of $10$ s. From these we
obtain $\Phi$ as a function of the trapping positions $\phi$.

The measured rotation in the mode orientations are shown in Fig.2(a) for two
separate detunings which represent the two interesting limit cases. To model
the experiments, for all results, we employed nanosphere radius $R=60.1$ nm,
input power to the tweezer $P_{tw}=0.485$ W and $\theta=49^{\circ}$.

Behavior of the mechanical frequencies: An interesting and unexpected
observation is that the behavior of the frequencies mirrors the mode
rotations; at the $\phi\simeq\phi_{c}$ position, they return to their
unperturbed values. This is shown in Fig.2(b), for two values of the optical
detuning. Here, the experimental values are obtained by fitting the PSDs.

The unperturbed mechanical frequencies of this levitated optomechanical system
are set by the tweezer trap:

$(\omega_{(x,y)}^{(0)})^{2}=\frac{\alpha\epsilon^{2}_{tw}}{mw_{x,y}^{2}}$ (6)

where $\epsilon^{2}_{tw}=4P_{tw}/(w_{x}w_{y}\pi c\epsilon_{0})$ is related to
the input power from the tweezer. In the presence of the cavity the coupling
to the optical mode dynamics introduces an optical spring ‘softening’ (for
red-detuned light) that is generic to all cavity optomechanical set-ups. In
strong coupling regimes this can be a very large shift. Neglecting a
correction for 2D $x-y$ coupling MTTM2020 ; MTTM2021 , the optical spring
shift
$(\delta\omega^{(j)}_{OS})^{2}=\text{Re}({-i2g^{2}_{j}\omega_{j}^{(0)}\eta_{c}})$
with $j=x,y$.

However, for the coherent scattering set-up, there is a countervailing
potential, obtained by linearising Eq.1, that ‘stiffens’ the mechanical
frequencies. It is a static contribution, dependent on the mean photon
occupancy of the cavity, thus can be considered an effect of co-trapping by
the CS field. It takes the form
$(\delta\omega^{(j)}_{CS})^{2}\simeq-(E_{d}k^{2}\sin^{2}\theta/m)2\Delta\cos^{2}\phi[(\kappa/2)^{2}+\Delta^{2}]^{-1}$.

Hence the corrected frequencies combine the two contributions:
$(\omega_{j})^{2}=(\omega_{j}^{(0)})^{2}+(\delta\omega^{(j)}_{CS})^{2}+(\delta\omega^{(j)}_{OS})^{2}$
and may be written:

$(\omega_{j})^{2}\simeq(\omega_{j}^{(0)})^{2}-2g_{j}^{2}\omega_{j}^{(0)}\left[\text{Re}({i\eta_{c}})+\frac{2\Delta\cot^{2}{\phi}}{(\kappa^{2}/4+\Delta^{2})}\right]$
(7)

Figure 3:  (a) To measure mode rotation, we compare cross correlation spectra
$S_{xy}(\omega)$ with the rescaled difference between PSDs
$S_{xy}(\omega)\simeq\Phi(\phi)\ [S_{yy}(\omega)-S_{xx}(\omega)]$ to extract
$\Phi(\phi)$. Plots show experimental $S_{xy}$ at a node, $\Delta/2\pi=-360$
kHz and rescaled, measured, $S_{yy}-S_{xx}$. The rescaling gives excellent
agreement, but there is a bias $\Phi_{\beta}\approx 0.03$ arising from an
imperfection in the orientation of the D-mirror ($\sim 2^{\circ}$) for the $y$
detector. This yields a systematic shift between the scaling of all points
obtained from the $y$ peak (cyan) and the $x$ peak (dark blue). In Fig.2 all
$y$ (cyan) data points are shifted by a constant,
$\Phi(\phi)\to\Phi(\phi)-\Phi_{\beta}$. Upper red curve shows $x-y$
hybridisation mediated by the cavity mode. (b) Illustrates the ‘locking’ of
the mode orientation at $\phi=\phi_{c}$. (i) (upper panel): for
$\Delta/2\pi=-176$ kHz, nearing resonance at $\omega_{x,y}/(2\pi)\sim 150$
kHz, $\phi_{c}/2\pi=0.145$. At this point, for arbitrary tweezer polarisation
$\theta$ or input power, the modes remain at the unperturbed orientations.
(ii) (lower panel) for large detuning $-\Delta/2\pi>350$ kHz the locking point
is $\phi_{c}/2\pi=0.125$. For lower detunings, $\phi_{c}$ moves towards the
node and at $\Delta/2\pi=-150$ kHz, $\phi_{c}/2\pi\simeq 0.15$.

It is clear from a comparison with Eq.5 that the multiplicative terms in
square brackets are equivalent; thus if they cancel for the mode orientations,
they cancel for the frequency contributions. A more refined analysis, with
consideration of weaker corrections and higher order terms will show that the
$\phi=\phi_{c}$ point is not identical for the frequencies and orientations,
but it is quite close.

In Fig.3(a), the procedure for measuring the mode angles $\Phi$ as a function
of $\phi$ is illustrated. A small imperfection in the orientation of the
D-mirror, of the order of $2^{\circ}$, for the detection of $y$ introduces a
nonvanishing cross correlation with $x$ even in absence of the cavity
interaction. This translates into a constant bias in the evaluation of $\Phi$,
which can be easily removed.

Behavior of $\phi_{c}$: the position of the cancellation point is shown in
Fig.3(b). Clearly, from Eq.5 and Eq.7, there is no dependence of $\phi_{c}$ on
the experimental polarisation or input power. The upper panel shows this
behavior for $\Delta/2\pi=-176$ kHz. The rate of rotation of the mode
orientation varies considerably, but the cancellation point remains fixed at
$\phi_{c}/2\pi=0.145$. The lower panel shows that for large detuning,
$\phi_{c}$ is independent of $\Delta$. As shown in MTTM2020 , at large
detuning,
$\text{Re}({i\eta_{c}})\to\frac{-2\Delta}{(\kappa^{2}/4+\Delta^{2})}$ is real,
we obtain $1-2\cot^{2}\phi_{c}=0$, implying $\phi_{c}/2\pi=0.125$ for all
$\Delta$. However, at resonance ($-\Delta\sim\omega_{x,y}$), $\phi_{c}$ moves
towards the node. For this particular cavity, $\phi_{c}/2\pi\simeq 0.15$ near
resonance. For lower $\kappa$ and below resonance, one can approach
$\phi_{c}/2\pi=0.2$, where the cavity photon occupancy is only $\sim 4-5$
times larger than at the node. But Eq.5 assumes weak coupling; a more detailed
analysis is required to obtain the $\Phi\simeq 0$ point in regimes with
significant mechanical-optical hybridisation.

Conclusions CS set-ups approaches have had a disruptive effect on levitated
cavity optomechanics for two key reasons: (i) they offer optomechanical
coupling strengths an order of magnitude larger relative to earlier dispersive
approaches and (ii) the possibility of trapping at a cavity node, not possible
in dispersive set-ups, offers an elegant solution to the detrimental impact of
technical noise, including classical laser frequency noise, because of the
much lower cavity photon occupancy. Hence most CS experiments trap at a cavity
node. However, a theoretical study MTTM2020  indicated trapping away from the
node (for $-\Delta\gg\omega_{x,y}$, might offer a regime where the CS
potential cancels the dynamics-induced hybridisation of the mechanical modes.

Here this effect is demonstrated and investigated experimentally. In addition,
it was also found, for the first time, that the frequency shifts precisely
mirror this effect and that the cancellation point, for lower detunings, moves
closer to the node.

Future studies will combine technical improvements to the frequency noise with
operation closer to (but not exactly at) the node and will thus be able to
maximise the potential of this regime. This has important potential
consequences for sensing applications. For example, the directional
sensitivity of levitated nanoparticles aids the search for dark matter
candidates since rejection of background events can be enhanced by the
knowledge of the direction of the incoming dark matter candidate dark_matter_1
; dark_matter_2 ; dark_matter_3 . The same process can impede the formation of
bright/dark modes in the strong coupling regime allowing efficient 2D quantum
cooling.

Acknowledgements The authors would like to acknowledge helpful discussions
with J. Gosling and M. Toroš. The authors acknowledge funding from the
Engineering and Physical Sciences Research Council (EPSRC) Grant No.
EP/N031105/1. H.F. and J.H.I. acknowledge EPSRC studentship funding via grant
number EP/L015242/1 (H.F.) and EP/R513143/1 (J.H.I.).

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APPENDIX

##  I Mechanical frequencies

The combined tweezer-cavity Hamiltonian takes the form:

$\hat{H}=-\frac{\alpha}{2}|\mathbf{\hat{E}}_{\text{cav}}+\mathbf{\hat{E}}{}_{\text{tw}}|^{2},$
(8)

where $\mathbf{\hat{E}}_{\text{cav}}$ ($\mathbf{\hat{E}}_{\text{tw}}$) denotes
the cavity (tweezer) field,
$\alpha=3\epsilon_{0}V_{s}\frac{\epsilon_{R}-1}{\epsilon_{R}+2}$ is the
polarizability of the nanosphere, $V_{s}$ is the volume of the nanosphere,
$\epsilon_{0}$ is the permittivity of free space, and $\epsilon_{R}$ is the
relative dielectric permittivity.

We assume a coherent Gaussian tweezer field and replace the modes with
c-numbers to find:

$\mathbf{\mathbf{\hat{E}}}_{\text{tw}}=\frac{\epsilon_{tw}}{2}\frac{1}{\sqrt{1+(\frac{z}{z_{R}})^{2}}}e^{-\frac{\hat{x}^{2}}{w_{x}^{2}}}e^{-\frac{\hat{y}^{2}}{w_{y}^{2}}}e^{ik\hat{z}+i\Phi(\hat{z})}e^{-i\omega_{\text{tw}}t}\mathbf{e}_{y}+\text{cc}$
(9)

where $\Phi(z)=-\arctan\frac{z}{z_{R}}$ is the Gouy phase, $z_{R}=\frac{\pi
w_{x}w_{y}}{\lambda}$ is the Rayleigh range, $w_{x}$ ($w_{y}$) are the beam
waist along the $x$ ($y$) axis,
$\epsilon_{tw}=\sqrt{\frac{4P_{\text{tw}}}{w_{x}w_{y}\pi\epsilon_{0}c}}$ is
the amplitude of the electric field, $c$ is the speed of light,
$P_{\text{tw}}$ is the laser power, $\omega_{\text{tw}}$ is the tweezer
angular frequency, $t$ is the time, and
$\hat{\boldsymbol{r}}=(\hat{x},\hat{y},\hat{z})$ is the position of the
nanoparticle. $\mathbf{e}_{j}$ are the unit vectors: $\mathbf{e}_{z}$ is
aligned with the symmetry axis of the tweezer field and $\mathbf{e}_{y}$ is
aligned with the polarization of the tweezer field.

The cavity field is given by:

$\mathbf{\hat{E}}_{\text{cav}}=\epsilon_{c}\text{cos}(k(Y_{0}^{\text{(c)}}+\hat{Y}^{\text{(\text{c)}}}))\mathbf{e}_{x}^{c}\left[\hat{a}+\hat{a}^{\dagger}\right],$
(10)

where $\epsilon_{c}=\sqrt{\frac{\hbar\omega_{c}}{2\epsilon_{0}V_{c}}}$ is the
amplitude at the center of the cavity, $V_{c}$ is the cavity volume,
$\omega_{c}$ is the cavity frequency, $\hat{a}$ ($\hat{a}^{\dagger}$ )is the
annihilation (creation) operator, $Y_{0}^{\text{(c)}}$ is an offset of the
cavity coordinate system (centered at a cavity antinode) with respect to the
tweezer coordinate system.

The cavity $X_{c}$-$Y_{c}$ plane is rotated by an angle $\theta$ with respect
to the tweezer $x$-$y$ plane. For $\theta=0$, the tweezer polarization
($y$-axis) becomes aligned with the cavity symmetry axis ($Y_{c}$-axis). In
particular, we have
$\hat{Y}^{\text{(\text{c)}}}=\text{sin}(\theta)\hat{x}+\text{cos}(\theta)\hat{y}$
while
$\hat{X}^{\text{(\text{c)}}}=\text{cos}(\theta)\hat{x}-\text{sin}(\theta)\hat{y}$.

We expand the Hamiltonian in Eq. (8), exploiting Eqs. (9) and (10) to obtain
three terms:

$\hat{H}=-\frac{\alpha}{2}|\mathbf{\mathbf{\hat{E}}}_{\text{tw}}|^{2}-\frac{\alpha}{2}|\mathbf{\hat{E}}_{\text{cav}}|^{2}-\frac{\alpha\text{sin}(\theta)}{2}(\mathbf{\hat{E}}_{\text{cav}}^{\dagger}\mathbf{\mathbf{\hat{E}}}_{\text{tw}}+\mathbf{\hat{E}}_{\text{cav}}\mathbf{\mathbf{\hat{E}}}_{\text{tw}}^{\dagger}),$
(11)

where the terms on the right hand-side represent the tweezer trapping
potential, the cavity intensity field, and the tweezer-cavity interaction term
(from left to right). The first (tweezer field) term dominates the trapping
and primarily sets the three mechanical frequencies $\omega_{x}$,
$\omega_{y}$, and $\omega_{z}$. In the discussion below we focus only on the
$x,y$ modes.

The second term represents trapping by the cavity intensity potential. In
earlier experiments in levitated cavity optomechanics with no tweezer trap,
this field determined the mechanical frequencies. However, achieving
reasonable frequencies required very high photon occupancies $n_{p}\sim
10^{10}$. In CS setups this term provides a negligible correction to the
frequencies but is included in the numerics for precision.

The third term, which we will denote as $\hat{V}_{\text{CS}}$, is the most
interesting, as this is the coherent scattering potential that has had a
transformative effect in the levitated cavity optomechanics field:

$\displaystyle\frac{\hat{V}_{\text{CS}}}{\hbar}$ $\displaystyle=$
$\displaystyle-
E_{d}\text{cos}(\phi+k(\hat{x}\sin\theta+\hat{y}\cos\theta))e^{-\frac{\hat{x}^{2}}{w_{x}^{2}}}e^{-\frac{\hat{y}^{2}}{w_{y}^{2}}}\hat{A};$
(12) where $\displaystyle\
\hat{A}=\left[\hat{a}e^{-i(k\hat{z}+\Phi(\hat{z}))}+\hat{a}^{\dagger}e^{+i(k\hat{z}+\Phi(\hat{z}))}\right]$

$\textrm{and}\
E_{d}=\frac{\alpha\epsilon_{c}\epsilon_{tw}\sin\theta}{2\hbar},$ (13)

while $\phi=kY_{0}^{\text{(c)}}$ represents the effect of the shift between
the origin of the tweezer field and the cavity standing wave. Linearising
$\hat{V}_{\text{CS}}$ about equilibrium displacements yields the
optomechanical coupling strengths:

$\displaystyle g_{x}$ $\displaystyle\simeq$ $\displaystyle-E_{d}\
k\sin\theta\sin\phi X_{zpf}$ $\displaystyle g_{y}$ $\displaystyle\simeq$
$\displaystyle-E_{d}\ k\cos\theta\sin\phi Y_{zpf},$ (14)

$X_{zpf}=\sqrt{\hbar/(2m\omega_{x})}$, $Y_{zpf}=\sqrt{\hbar/(2m\omega_{y})}$.

In addition, as a feature of the CS set up, the linearisation yields also
direct $x-y$ couplings $g_{xy}\simeq-
g_{x}g_{y}\frac{2\text{Re}(\bar{\alpha})\cos{\phi}}{E_{d}\sin^{2}{\phi}}$
related to the mean photon occupancy $n_{p}=|\bar{\alpha}|^{2}$ in the cavity.
They are negligible at the nodes ($\phi=\pi/2$), but become stronger as
$\phi\to 0$.

###  I.1 tweezer trap frequencies

The mechanical $x,y$ frequencies are set mainly by the tweezer trap
$-\frac{\alpha}{2}|\mathbf{\mathbf{\hat{E}}}_{\text{tw}}|^{2}$. Linearising
leads to

$(\omega^{(0)}_{x,y})^{2}=\frac{\alpha\epsilon^{2}_{tw}}{mw_{x,y}^{2}}$ (15)

###  I.2 co-trapping

The linearisation of ${\hat{V}_{\text{CS}}}$ also yields corrections to the
tweezer trap frequencies: the zero-th order frequency must be corrected by co-
trapping by the coherent scattering potential in Eq.(12). These stiffen the
tweezer frequencies by corrections of the form:

$\displaystyle(\delta\omega_{x})^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{E_{d}\hbar}{m}2\alpha_{R}\cos\phi[k^{2}\sin^{2}\theta+\frac{2}{w_{x}^{2}}]$
$\displaystyle(\delta\omega_{y})^{2}$ $\displaystyle\simeq$
$\displaystyle\frac{E_{d}\hbar}{m}2\alpha_{R}\cos\phi[k^{2}\cos^{2}\theta+\frac{2}{w_{y}^{2}}]$
(16)

For the $x,y$ frequencies, the stiffening depends on the mean cavity field:

$\overline{\alpha}=\alpha_{R}+i\alpha_{I}=\frac{-iE_{d}\cos\phi}{i\Delta-\kappa/2}$
(17)

so $\alpha_{R}=\frac{-\Delta E_{d}\cos\phi}{\kappa^{2}/4+\Delta^{2}}$.

We note that the second term in the square brackets in Eq.16 is small, so
$(\delta\omega_{x})^{2}\simeq\frac{E_{d}\hbar}{m}2\alpha_{R}\cos\phi
k^{2}\sin^{2}\theta$ represents already a good approximation to the frequency
correction.

###  I.3 optical spring

A well-studied effect in optomechanics is the so called ‘optical spring’ shift
of the mechanical frequencies that arises from the dynamical interplay between
the fluctuations in the optical mode and the mechanical motion. If we write
the self energy as
$\Sigma(\omega\simeq\omega_{j})=-i2g^{2}_{j}\omega_{j}\eta_{c}(\omega\simeq\omega_{j})$,
for $j\equiv x,y$, its imaginary part related to the optomechanical damping,
while its real part is related to the optical spring shift squared:

$\displaystyle(\delta\omega_{(OS,x)})^{2}$ $\displaystyle\simeq$
$\displaystyle\omega_{x}\text{Re}{\left\\{-i2g^{2}_{x}\eta_{c}(\omega=\omega_{x})\right\\}}$
$\displaystyle(\delta\omega_{(OS,y)})^{2}$ $\displaystyle\simeq$
$\displaystyle\omega_{y}\text{Re}{\left\\{-i2g^{2}_{y}\eta_{c}(\omega=\omega_{y})\right\\}}$
(18)

For red detuning, it softens the mechanical frequencies. It is strongest at
the node and tends to zero as the antinode is approached. In [24], an
additional correction was found, resulting from the intrinsic 2D dynamics,
that here is negligible.

##  II The $S_{xy}$ spectrum

It was shown in [24] that one can correct the 1D mechanical displacement
spectra to allow for hybridisation. For the $x$ displacement, for example, one
can write:

$\hat{x}^{\text{3D}}(\omega)=\hat{x}^{\text{1D}}(\omega)+\mathcal{R}_{xy}(\omega)\hat{y}^{\text{3D}}(\omega)+\mathcal{R}_{xz}(\omega)\hat{z}^{\text{3D}}(\omega)$
(19)

using the appropriate hybridisation functions, $\mathcal{R}_{jk}(\omega)$,
introduced in [24], to correct the unhybridised spectra,
$\hat{x}^{\text{1D}}(\omega)$. Analogous expressions are obtained for $y,z$.
Since the unhybridised spectra are well-known and given in terms of optical
noises, the $x,y,z$ expressions can be rearranged and solved in closed form to
obtain PSDs and arbitrary correlation spectra. It is assumed that
linearisation of the equations of motion is valid and we can use quantum
linear theory (QLT).

We obtained ‘exact’ QLT PSDs $S_{xx}(\omega),S_{yy}(\omega)$ and correlation
spectra $S_{xy}(\omega)$ to compare with experiment. However, for physical
insight, we also obtain below a simplified analysis that gives excellent
agreement with the full 3D QLT. As the motion is approximately 2D, we neglect
the $z$ motion and we further approximate:

$\displaystyle\hat{x}^{\text{3D}}(\omega)$
$\displaystyle\simeq\hat{x}^{\text{1D}}(\omega)+\mathcal{R}_{xy}(\omega)\hat{y}^{\text{1D}}(\omega)$
(20) $\displaystyle\hat{y}^{\text{3D}}(\omega)$
$\displaystyle\simeq\hat{y}^{\text{1D}}(\omega)+\mathcal{R}_{yx}(\omega)\hat{x}^{\text{1D}}(\omega).$
(21)

where for a modest hybridisation correction, we substitute the 1D expressions
in the last term. Below, we drop the 3D superscript and assume that
$\hat{x},\hat{y}$ inlude hybridisation. To compare with experiment, we
consider the symmetrised mechanical correlations:

$S_{xy}(\omega)=\frac{1}{2}\left\langle[\hat{x}]^{\dagger}\hat{y}\rangle+\langle[\hat{y}]^{\dagger}\hat{x}\rangle\right)$
(22)

The current experiments are in weak coupling regimes and there is little
mechanical-optical hybridisation. This means that
$\langle[\hat{x}^{1D}]^{\dagger}\hat{y}^{1D}\rangle\simeq 0$. But in regimes
of strong optical back-actions even the unhybrised modes can be correlated
hence $\langle[\hat{x}^{1D}]^{\dagger}\hat{y}^{1D}\rangle\neq 0$. We have
verified numerically that the 1D components have negligible cross
correlations, so $S_{xy}^{\text{1D}}\simeq 0$ and hence:

$S_{xy}(\omega)\approx\left(\text{Re}({\mathcal{R}_{yx}}(\omega))S_{xx}(\omega)+\text{Re}({\mathcal{R}_{xy}}(\omega))S_{yy}(\omega)\right)$
(23)

We can see that the cross-corrrelations are closely related to the real part
of the hybridisation functions. The latter can be given explicitly:

$\displaystyle\mathcal{R}_{xy}(\omega)$ $\displaystyle=$
$\displaystyle\frac{i\mu_{x}(\omega)}{M_{x}(\omega)}G(\omega)\ \textrm{and}$
$\displaystyle\mathcal{R}_{yx}(\omega)$ $\displaystyle=$
$\displaystyle\frac{i\mu_{y}(\omega)}{M_{y}(\omega)}G(\omega)$ (24)

where $G(\omega)=\left[i\eta_{c}(\omega)g_{x}g_{y}+g_{xy}\right]$ is a term
that represents the interference between the ‘direct’ static coupling between
$x$ and $y$ (proportional to $g_{xy}$); and an indirect, cavity mediated,
coupling term ( proportional to $g_{x}g_{y}$). The prefactors
$M_{j}(\omega)=1+g_{j}^{2}\mu_{j}(\omega)\eta_{c}(\omega)$ include a small
optical backaction correction to each displacement. For our simplified
analysis, we take $M_{j}\simeq 1$. Numerical tests showed this is an excellent
approximation. The reason for this is that the small backaction correction is
peaked around each of the mechanical frequencies, i.e. at
$M_{x}(\omega\approx\omega_{x})$, and $M_{y}(\omega\approx\omega_{y})$ while
for the cross-correlation, we show below the values around
$M_{x}(\omega\approx\omega_{y})\approx 1$, and
$M_{y}(\omega\approx\omega_{x})\approx 1$ are most important.

We note that, in the present discussion, we refer to both the cavity mediated
couplings $\eta_{c}(\omega)g_{x}g_{y}$ as well as the usual optomechanical
back-action terms $g_{j}^{2}\eta_{0}(\omega)$ as ‘optical backaction’ terms,
but clearly, in the former case, the optical backaction acts on different
mechanical modes.

Optical and mechanical susceptibilities: The $\mu_{j}(\omega)$ are mechanical
susceptibilities, while $\eta_{c}$ is the optical susceptibility. We have the
usual mechanical susceptibility
$\mu_{j}(\omega)=\chi(\omega,\omega_{j})-\chi^{*}(-\omega,\omega_{j})$ and
optical susceptibility
$\eta_{c}(\omega)=\chi(\omega,-\Delta)-\chi^{*}(-\omega,-\Delta)$, where eg
$\chi(\omega,\omega_{x})=[-i(\omega-\omega_{x})+\frac{\Gamma}{2}]^{-1}$ and
$\chi(\omega,\Delta)=[-i(\omega-\Delta)+\frac{\kappa}{2}]^{-1}$

###  II.1 Anticorrelation of $S_{xy}$

We can readily show that the $x-y$ modes are in general anti-correlated (and
this was observed in the experimental data) by showing that the hybridisation
functions $\text{Re}{\mathcal{R}_{xy}}\approx-\text{Re}{\mathcal{R}_{yx}}$.

We have shown that $\mathcal{R}_{xy}(\omega)\simeq G(\omega)i\mu_{x}(\omega)$
and $\mathcal{R}_{yx}(\omega)\simeq G(\omega)i\mu_{y}(\omega)$ and:

$S_{xy}(\omega)\approx\left(\text{Re}\left[i\mu_{y}(\omega)G(\omega)\right]S_{xx}+\text{Re}\left[i\mu_{x}(\omega)G(\omega)\right]S_{yy}\right).$
(25)

However, the PSDs are sharply peaked about the mechanical frequencies
$S_{xx}(\omega\approx\omega_{x})$ and $S_{yy}(\omega\approx\omega_{y})$, hence
we are interested in the value prefactors at those frequencies, namely
$\mu_{y}(\omega\sim\omega_{x})G(\omega\approx\omega_{x})$ and
$\mu_{x}(\omega\sim\omega_{y})G(\omega\approx\omega_{y})$.

Since $\kappa\gg|\omega_{x}-\omega_{y}|$, the cavity susceptibility function
$\eta_{c}(\omega)$, and hence $G(\omega)$ is insensitive to frequency: i.e.
$G(\omega_{x})\sim G(\omega_{y})$. Hence the anticorrelation behavior must
originate in the mechanical susceptibilities.

However, since $\omega_{x}+\omega_{y}\gg|\omega_{x}-\omega_{y}|$ and the
mechanical damping $\Gamma\ll|\omega_{x}-\omega_{y}|$ is negligible at
ultrahigh vacuum, we can write:

$\mu_{x}(\omega\sim\omega_{y})\simeq[-i(\omega_{y}-\omega_{x})+\frac{\Gamma}{2}]^{-1}\simeq
i/(\omega_{x}-\omega_{y})\\\ $ (26)

and

$\mu_{y}(\omega\sim\omega_{x})\simeq[-i(\omega_{x}-\omega_{y})+\frac{\Gamma}{2}]^{-1}\simeq
i/(\omega_{y}-\omega_{x})\\\ $ (27)

Figure 4:  (left panel) PSDs for $x$ and $y$ for $\Delta/2\pi=-176$ kHz, near
the node, for experiment and QLT numerics. (right panel) for the same
parameters, plots the cross correlation spectra, showing that the approximate
expression Eq.28 gives excellent agreement with the full theory as well as the
experimental $x$ peak; the experimental $y$ peak requires a systematic
correction for all data points as discussed in the main text.

Finally, we obtain an approximate expression for the mechanical correlation
spectra:

$S_{xy}(\omega)\approx\frac{G(\omega)}{\omega_{x}-\omega_{y}}[S_{yy}(\omega)-S_{xx}(\omega)]$
(28)

showing clearly that the $x$ and $y$ peaks have opposite signs. So the
anticorrelation arises because the susceptibility for $y$ involves
upconverting in frequency whereas $x$ represents a downconversion. The overall
sign flips when $G(\omega)$ changes sign. For $\theta=3\pi/4$ then
$g_{x}=-g_{y}$ and $g_{xy}\to-g_{xy}$ so there is a global sign flip of
$S_{xy}$ relative to $\theta=3\pi/4$.

In the figure we test this simple expression against the exact QLT
anticorrelation spectrum. We can see that there is remarkable agreement
everywhere: the expression is remarkably accurate. Since
$G(\omega\sim\omega_{x})\simeq G(\omega\sim\omega_{y})\equiv G$, the prefactor
is a constant and we can equate it to the rotation angle
$\Phi\sim\frac{G}{\omega_{x}-\omega_{y}}$. In practice, comparisons with
experimental spectra employ the average of the full QLT expressions
$\Phi=\frac{1}{2}\left(\text{Re}[{R}_{xy}]-\text{Re}[{R}_{yx}]\right)$, but
the simplified expressions are generally quite accurate.

###  II.2 Suppression of hybridisation

If the term $G(\omega)\left[i\eta_{c}(\omega)g_{x}g_{y}+g_{xy}\right]\simeq
0$, the destructive interference between $x-y$ coupling and indirect, cavity-
mode mediated coupling suppresses hybridisation and hence $S_{xy}\simeq 0$.

Since the direct coupling $g_{xy}\simeq-
g_{x}g_{y}\frac{2\text{Re}(\bar{\alpha})\cos{\phi}}{E_{d}\sin^{2}{\phi}}$, and
$\bar{\alpha}\simeq-iE_{d}\text{cos}(\phi)[i\Delta-\kappa/2]^{-1}$:

$g_{xy}\simeq
g_{x}g_{y}\left[\frac{2\Delta\cot^{2}{\phi}}{\Delta^{2}+\frac{\kappa^{2}}{4}}\right].$
(29)

Thus depending on the positioning, $\Delta$ or $\kappa$, the direct couplings
contribution can be similar or exceed the cavity mediated coupling. Direct and
indirect contributions, in general,interfere destructively. We can show that
$i\eta_{c}(\omega)\to\frac{-2\Delta}{(\kappa/2)^{2}+\Delta^{2}}$ if
$\text{-}\Delta\gg\omega$ (and we are interested primarily in the region
$\omega\sim\omega_{j}$). Thus for large $-\Delta$:

$G(\omega)\approx
g_{x}g_{y}\left[\frac{-2\Delta}{\Delta^{2}+(\kappa/2)^{2}}\right]\left[1-\cot^{2}\phi\right],$
(30)

and we see that the $G$ is real and frequency independent. Furthermore, at
$\phi=\pi/4$ the $x$-$y$ hybridisation almost fully vanishes so we have a
cancellation point where the $S_{xy}$ correlation spectra are near zero.

Figure 5:  Density plot of the cancellation point
$\phi_{c}=\tan^{-1}(\sqrt{C_{\phi}})$ as a function of the cavity linewidth
$\kappa$ and optical detuning $\Delta$, with black contours at notable values.
$C_{\phi}$ is shown in Eq.33. White-dashed lines show the experimental
parameters, with white dots representing the two sets of data taken where
$\phi_{c}$ was investigated. As highlighted by Fig.2 in the main text, the
$\phi_{c}$ predictions agree well with experimental observations. The blank
ellipse in the top left is the case where $C_{\phi}<0$, leading to complex-
valued $\phi_{c}$ that is not plotted. This condition leads to
$\Delta^{2}+\kappa^{2}/4<\omega_{y}^{2}$, showing the ellipse’s radii to be
$2\omega_{y}$ and $\omega_{y}$ in the $\kappa$ and $\Delta$ axes,
respectively. The large detuning limit of $\phi_{c}/2\pi\to 0.125$ can be seen
in the lower half. The plot suggests that experiments with this cavity
($\kappa=396\,\text{kHz}\cdot 2\pi\simeq 2.9~{}\omega_{y}$) are limited to
$\phi_{c}/2\pi\sim 0.17$, even in a very small detuning limit
$-\Delta\ll\omega_{y}$. Nevertheless, the higher-valued (brighter) area around
the $\phi_{c}/2\pi=0.2$ contour suggests that a better cavity with a
realistically lower $\kappa\gtrsim 2\omega_{y}$, may have cancellation closest
to the node with $\phi_{c}/2\pi\sim 0.2$, so long as the detuning is below
resonance. However, if the detuning is above resonance, the
$\phi_{c}/2\pi=0.1$ contour highlights a region where an increasingly perfect
cavity $\kappa\ll\omega_{y}$ may even move $\phi_{c}$ toward the antinode.

##  III Behavior of $\phi=\phi_{c}$

In the present experimental study we are not necessarily in the large detuning
limit so we consider other regimes including $-\Delta\sim\omega_{x,y}$ and
find the cancellation $\phi=\phi_{c}$ moves towards the node.

Although the present experimental data is not conclusive, it is certainly
consistent with a $\phi_{c}/2\pi\simeq 0.15$ for $-\Delta/2\pi=176$ kHz,
closer to the node than for $-\Delta/2\pi=360$ kHz, for which
$\phi_{c}/2\pi\simeq 0.125$. Operating even closer to the node would be even
more advantageous for quantum optomechanics as it reduces cavity photon
occupancies and thus the deleterious effects of optical noise.

In the present study, we consider only mode rotation in the $x-y$ plane. For
experiments in the good cavity limit and nearer resonance, a model of mode
rotation must consider the tripartite interaction between $x$, $y$ and the
optical mode. Below we neglect optical mixing and present a simplified
analysis where the mode rotation angle does not consider the neglected
$\langle[\hat{x}^{1D}]^{\dagger}\hat{y}^{1D}\rangle\simeq 0$ terms.

We analyse simply the behavior of of the simplified model
$\Phi(\phi)\sim\frac{G}{\omega_{x}-\omega_{y}}$. We write:

$\Phi(\phi)=\frac{A}{\omega_{y}-\omega_{x}}\left(\text{Re}[i\eta_{\text{c}}(\omega_{y})]\sin^{2}\phi+B\cos^{2}\phi\right)$
(31)

where
$A=\frac{g_{x}g_{y}}{\sin^{2}\phi}=E_{\text{d}}^{2}k^{2}X_{\text{zpf}}Y_{\text{zpf}}\sin\theta\cos\theta$,
and $B=\frac{g_{xy}}{g_{x}g_{y}}=\frac{2\Delta}{\Delta^{2}+\kappa^{2}/4}$.

Solving for $\Phi(\phi_{c})=0$ leads to:

$\phi_{c}=\tan^{-1}\left(\sqrt{C_{\phi}}\right)$ (32)

where $C_{\phi}=-B/\text{Re}[i\eta_{\text{c}}(\omega_{y})]$.

###  III.1 Moving $\phi_{c}$ closer to the node

We plot Eq.32 in Fig.5, showing how the cancellation point $\phi_{c}$ varies
with $\kappa$ and $\Delta$. First, we highlight that the predictions for the
two experimentally explored sets of parameters (white dots) match the
observation of the cancellation point moving closer to node as the detuning
approaches resonance
$-\Delta/2\pi=176\,\text{kHz}\sim\omega_{y}/2\pi=136\,\text{kHz}$.

Next, we note that Fig.5 (and hence Eq.32) verifies that, as $C_{\phi}$ gets
larger ($\kappa,\Delta\gg\omega_{y}$), $\phi_{c}/2\pi\to 0.125$, consistent
with the analysis from the previous section.

Eq.32 also shows that $\phi_{c}$ is real-valued as long as $C_{\phi}\geq 0$.
Using the full form of the cavity susceptibility function
$\eta_{\text{c}}(\omega)$, we obtain:

$C_{\phi}=\frac{(\Delta^{2}+\kappa^{2}/4-\omega_{y}^{2})^{2}+(\kappa\omega_{y})^{2}}{(\Delta^{2}+\kappa^{2}/4-\omega_{y}^{2})(\Delta^{2}+\kappa^{2}/4)}$
(33)

By inspection, it is seen from the first parentheses in the denominator (other
terms being always positive) that $C_{\phi}\geq 0$ holds so long as
$\Delta^{2}+\kappa^{2}/4>\omega_{y}^{2}$. This condition is seen clearly in
Fig.5 to be the form of the blank ellipse in the top left, which has a radius
in the $\kappa~{}(\Delta)$ axis of $2\omega_{y}~{}(\omega_{y})$.

Lastly, the higher-valued area between the $\phi_{c}/2\pi=0.2$ and $0.17$
contours suggests that – although this particular
$\kappa/2\pi=396\,\text{kHz}$ might be limited to $\phi_{c}/2\pi\sim
0.17-0.18$ (even in an extreme $-\Delta\ll\omega_{y}$ limit) – for operation
in the good cavity regime, with $\kappa\gtrsim 2\omega_{y}\simeq
270\,\text{kHz}\cdot 2\pi$, then $\phi_{c}/2\pi\sim 0.2$ may be possible with
a detuning below resonance.

A future analysis will investigate how best to extract the mode orientation in
the $x-y$ plane in the good cavity regime and with strong light-matter
coupling.