# Anti-PT-symmetry-enhanced interconversion between microwave and optical
fields

Debsuvra Mukhopadhyay<EMAIL_ADDRESS>Institute for Quantum Science and
Engineering, Texas A$\&$M University, College Station, TX 77843, USA
Department of Physics and Astronomy, Texas A$\&$M University, College Station,
TX 77843, USA  Jayakrishnan M. P. Nair<EMAIL_ADDRESS>Institute
for Quantum Science and Engineering, Texas A$\&$M University, College Station,
TX 77843, USA  Department of Physics and Astronomy, Texas A$\&$M University,
College Station, TX 77843, USA  Girish S. Agarwal  Institute for Quantum
Science and Engineering, Texas A$\&$M University, College Station, TX 77843,
USA  Department of Physics and Astronomy, Texas A$\&$M University, College
Station, TX 77843, USA  Department of Biological and Agricultural Engineering,
Texas A$\&$M University, College Station, TX 77843, USA

###### Abstract

The intrinsic dissipation of systems into a shared reservoir introduces
coherence between two systems, enabling anti-Parity-Time (anti-PT) symmetry.
In this paper, we propose an anti-PT symmetric converter, consisting of a
microwave cavity coupled dissipatively to a ferromagnetic sphere, which
supports significant improvements in the conversion efficiency when compared
to coherently coupled setups. In particular, when only the ferrite sample is
driven, the strong coherence induced by the vacuum of the mediating channel
leads to much stronger enhancements in the intended conversion. The
enhancement is an inalienable artifact of the emergence of a long-lived, dark
mode associated with a quasi-real singularity of the hybrid system. In
addition, we observe considerable asymmetry in the efficiencies of microwave-
to-optical and optical-to-microwave conversions, in spite of the symmetrical
structure of the trilinear optomagnonic coupling stimulating both the
transduction phenomena. The nonreciprocity stems from the intrinsic asymmetry
in the couplings of the microwave and optical fields to the cavity-magnon
network as well as the phase coupling entailed by the spatial separation.

##  I Introduction

Nonlinear three-wave interactions constitute the bedrock of several exotic
nonlinear optical phenomena, including stimulated Raman and Brillouin
scattering Boyd (2020), first observed by Woodbury et al. Woodbury and Ng
(1962) and later explained by Garmier et al Garmire  _et al._ (1963) and
Bloembergen and Shen Bloembergen and Shen (1964); Shen and Bloembergen (1965)
employing a coupled wave formalism. Quantum mechanically, such an interaction
can be understood as the conversion of an incident pump quantum into two
energy-conserving daughter quanta and vice versa. However, such exchanges do
not require all fields to be optical or lie in overlapping frequency domains.
For instance, in the last few decades, such three-mode interactions have been
widely investigated in the context of the interconversion between optical and
microwave fields, both of which are pivotal to an efficient information
processing network. A coherent reversible transduction of signals between
microwave and optical frequencies can leverage the strengths of optical
signals, including low-loss transmission, long-time memory and low thermal
occupancy, while simultaneously facilitating control over the electrical
system with the application of microwave signals. A variety of experimental
systems have been explored, including, for example, optomechanical systems
Agarwal and Jha (2013); Qu and Agarwal (2013); Bagci  _et al._ (2014); Andrews
_et al._ (2014) cold atoms Hafezi  _et al._ (2012); Verdú  _et al._ (2009),
spins Imamoğlu (2009); Marcos  _et al._ (2010), trapped ions Williamson  _et
al._ (2014); Fernandez-Gonzalvo  _et al._ (2015a) and electro-optic Tsang
(2010, 2011). A conversion efficiency close to 10$\%$ between microwave
signals of a few GHz into optical domain was demonstrated in an experiment
Andrews  _et al._ (2014) using optomechanical systems.

Recently, the interconversion between microwave and optical waves was
demonstrated in a hybrid cavity-magnonic setup by coherently coupling a
microwave cavity to the collective spin excitations in a yttrium iron garnet
(YIG) sample through the Purcell effect and by exploiting a trilinear Faraday
interaction between two optical modes and a Kittel mode Hisatomi  _et al._
(2016). Subsequently, Ihn et al. generalized this result to the multimode case
by accounting for both the Kittel mode and a higher-order space-varying
magnetostatic mode Ihn  _et al._ (2020). The primary challenge posed by these
setups is the weakness of the nonlinear parametric interaction between the
optical modes and the magnonic frequencies as they lie in widely disjoint
regimes. While much progress has been accomplished with regard to improving
the conversion efficiency in optomechanical configurations or by coupling an
erbium-doped crystal Fernandez-Gonzalvo  _et al._ (2015b) to both a microwave
and an optical cavity, mitigating the adverse impact of weak optomagnonic
couplings remains a central difficulty. However, all these studies were based
on dispersively coupled systems, while the recent years have seen a
flourishing of interest in anti-PT symmetry achieved via the engineering of
dissipative couplings. While coherent coupling stems from the spatial overlap
between two modes, a dissipative type of coupling Yu  _et al._ (2019) can be
engineered by the inclusion of a shared reservoir coupled independently to the
two modes. Subsidiary to this development is the proliferation of experimental
pursuits in cavity-magnonics Huebl  _et al._ (2013); Tabuchi  _et al._ (2014);
Zhang  _et al._ (2014); Bai  _et al._ (2015); Zhang  _et al._ (2015); Tabuchi
_et al._ (2015); Chumak  _et al._ (2015); Zhang  _et al._ (2016); Osada  _et
al._ (2016); Bai  _et al._ (2017); Zhang  _et al._ (2017); Wang  _et al._
(2018); Lachance-Quirion  _et al._ (2019); Nair  _et al._ (2020, 2021a),
courtesy of the high spin-density and low dissipation rates of YIGs. The
applications of cavity-magnonics include, but are not restricted to, quantum
and classical sensing Tserkovnyak (2020); Lachance-Quirion  _et al._ (2020);
Wolski  _et al._ (2020); Nair  _et al._ (2021b), non-reciprocity Wang  _et
al._ (2019); Zhang  _et al._ (2020), multistability Nair  _et al._ (2020) and
many more. Some experiments have probed dissipative couplings as well Harder
_et al._ (2018); Bhoi  _et al._ (2019).

In an earlier work, a scheme to engineer an augmented response to weak
nonlinear perturbations was expounded in a dissipatively coupled cavity-YIG
apparatus by exploiting the anti-PT symmetry of the configuration Nair  _et
al._ (2021b). In view of the immense potential of anti-PT symmetry, we hereby
extend its application to the problem of interconversion between microwave and
optical fields in the specific context of cavity-magnonics. Our key
observation is that the anti-PT symmetry Peng  _et al._ (2016); Wu  _et al._
(2014, 2015); Wang and Wu (2016); Jiang  _et al._ (2019); Konotop and Zezyulin
(2018); Choi  _et al._ (2018); Li  _et al._ (2019); Yang  _et al._ (2020); Wen
_et al._ (2020) allowed by dissipative frameworks offers a noticeable
improvement in efficiency. In fact, when only the YIG sphere is driven, the
same arrangement will enable improvements by a few orders of magnitude for
strong dissipative couplings in otherwise low-loss environments. This is an
upshot of the emergence of a long-lived eigenmode induced by the coherence.
Furthermore, owing to the dissipative coupling between the two modes, we
observe a strong asymmetry in the efficiencies of microwave-to-optical and
optical-to-microwave conversions. Note that, in general, it is challenging to
engineer non-reciprocity Metelmann and Clerk (2015); Verhagen and Alù (2017);
Nassar  _et al._ (2020); Wang  _et al._ (2021) in physical systems; however,
in waveguide-integrated photonic devices, the reservoir-mediated phase
coupling naturally brings in nonreciprocal attributes.

The structure of the paper is summarized below: Following a concise review of
the theoretical formulation underpinning the conversion model in Sec. II, we
explain the two possible schemes vis-à-vis the microwave-to-optical
transduction mechanism in Sec. III. Finally, in Sec IV, we derive the
efficiency of the reverse conversion, viz. optical to microwave, and
demonstrate the nonreciprocity of the two conversion pathways.

##  II Theoretical Conversion Model

We consider a hybrid cavity-magnonic model in which a rectangular microwave
cavity and a YIG interacts dissipatively via a one-dimensional (1D) microwave
transmission line. Unlike the previously explored scenario where the YIG is
wedged inside the cavity resonator to enhance the coherent coupling between
them through the Purcell effect Hisatomi  _et al._ (2016); Ihn  _et al._
(2020), the interposing waveguide here acts as the mediator of a long-range
coupling between the two. For the purpose of the conversion process, the YIG
sphere is evanescently coupled to an ancillary optical fiber withal. The
transduction mechanism could be initiated by the application of a microwave
drive at a frequency $\omega_{\mu}$, duly complemented by an intense laser
drive at a frequency $\Omega_{0}$ sent along optical fiber targeting the YIG
sample. The converse mechanism of optical-to-microwave transfer proceeds via
the administration of two orthogonally polarized optical inputs at frequencies
$\Omega$ and $\Omega_{0}$ respectively, such that the frequency discord
$\absolutevalue{\Omega-\Omega_{0})}$ is closely resonant with a Kittel mode
frequency, which lies in the microwave regime.

At the outset, we overview a first-principle description of the exchange
dynamics enabling the mode conversions. In the most general case, the full
Hamiltonian
$\mathcal{H}=\mathcal{H}_{s}+\mathcal{H}_{\text{micro}}+\mathcal{H}_{\text{optical}}$
of the driven cavity-YIG system would comprise of the following contributions:

$\displaystyle\mathcal{H}_{s}=$
$\displaystyle\hbar\omega_{a}a^{\dagger}a-\hbar\gamma_{e}B_{0}S_{z},$
$\displaystyle\mathcal{H}_{\text{micro}}=$
$\displaystyle\mathcal{H}_{\text{micro}}^{(a)}+\mathcal{H}_{\text{micro}}^{(m)},$
$\displaystyle\mathcal{H}_{\text{optical}}=$
$\displaystyle\int_{0}^{T}dt\>\hbar gM_{x}(t)s_{x}(t)c.$ (1)

Here, $\mathcal{H}_{s}$ represents the free Hamiltonian of the cavity-YIG
composite, where $a$ ($a^{\dagger}$) denotes the annihilation (creation)
operator of the cavity, $B_{0}$ the applied bias magnetic field and $S_{z}$
the collective spin operator of the YIG along the z direction, and
$\gamma_{e}$ is the gyromagnetic ratio. The Hamiltonian
$\mathcal{H}_{\text{micro}}$ captures the typical interplay between incident
microwave photons and the cavity-YIG network, with the first term
$\mathcal{H}_{\text{micro}}^{(a)}$ encapsulating the coupling of the microwave
drive to the cavity. The second term
$\mathcal{H}_{\text{micro}}^{(m)}=-\hbar\gamma_{e}\vec{S}.\vec{B}_{\text{micro}}$
characterizes the exchange interaction between a large number of spins in the
YIG and the local microwave field $\vec{B}_{\text{micro}}$. The specific
expressions for these two contributions would be introduced in due course
based on contextual considerations about the incident fields. In addition, two
linearly polarized optical field modes selectively couple to the ferromagnetic
material which is magnetized along the z direction. The parameter $T$
corresponds to the Faraday interaction time, $g$ is the coupling strength
between the optical field modes and the ferromagnetic sample, $c$ is the
velocity of light, $M_{x}$ and $s_{x}$ denote the x-components of the
magnetization and the relevant stokes operator respectively. Owing to the
circular birefringence of the medium, the linearly polarized light rotates
with components along the perpendicular directions, producing the trilinear
interaction $\mathcal{H}_{\text{optical}}$. This phenomenon is known as the
Faraday effect, generating optical sidebands displaced by
$\omega_{m}=\gamma_{e}B_{0}$ on either flank of the $\Omega_{0}$-band, and
thereby, converting those incident microwave photons into optical photons. By
the same token, the application of two orthogonally polarized optical beams
stimulates the production of a microwave field oscillating at a frequency
equaling the difference between the two optical frequencies, a phenomenon
commonly known as the inverse Faraday effect. These two phenomena constitute
the physical foundation of the conversion scheme.

Figure 1: First scheme for microwave-to-optical conversion. The bottom
microwave waveguide couples to both the cavity and the YIG sphere, whereas the
itinerant vacuum mode $b_{y}$ in the top fiber, polarized along $y$-axis,
addresses the collective magnonic excitation. A microwave signal $a_{\mu}$ is
launched through the bottom channel, in conjunction with an intense optical
laser drive $b_{z}$, polarized along the $z$-axis and shone on the YIG. This
induces both the Stokes and the anti-Stokes optical sidebands at the output
port of the optical fiber.

YIG is a ferrite compound having a high ferric iron (Fe3+) density which is
approximately $4.22\times 10^{27}$m-3, diameter $d=1$ mm and total number of
spins $N\approx 10^{18}$. Employing the Holstein-Primakoff transformation in
the high-spin-density limit, we can recast the raising and lowering spin
operators into $S^{+}=\sqrt{5N}m,\ S^{-}=\sqrt{5N}m^{\dagger}$, where
$\\{m,m^{\dagger}\\}$ follow the Bosonic algebra. This reduces the Hamiltonian
of the magnons effectively to $\hbar\omega_{m}m^{\dagger}m$, with
$\omega_{m}=\gamma_{e}B_{0}$. The magnetostatic Kittel mode being a spatially
uniform mode, the magnetization has a simple form in terms of the collective
spin operator, $M=\gamma_{e}S/V$, where $V$ is the volume of the YIG sample.
Further, the Stokes operator $s_{x}$ of the input optical field can be
expressed as $s_{x}=-i[b_{z}^{\dagger}b_{y}-b_{z}b_{y}^{\dagger}]$, with
$b_{y}(t)$ and $b_{z}(t)$ representing two optical field operators along the y
and z directions respectively. Now, since $b_{z}(t)$ is a strongly driven
optical mode, we switch to a classical treatment of this mode whereby
$b_{z}(t)$ is superseded by the c-number
$\sqrt{\frac{P_{0}}{\hbar\Omega_{0}}}e^{-i(\Omega_{0}t-\theta)}$, where
$P_{0}$ is the input laser power and $\theta$ is the phase acquired by the
wave in its transit across the fiber from the cavity to the YIG. Finally,
assuming a clear distinction between the timescales of the interaction and the
magnonic oscillation, i.e. $T\ll 1/{\omega_{m}}$, we can tailor the
optomagnonic interaction in (1) into

$\displaystyle\mathcal{H}_{\text{optical}}=-i\hbar\sqrt{\kappa_{o}}(m+m^{\dagger})[b_{y}e^{i\Omega_{0}t}-b_{y}^{\dagger}e^{-i\Omega_{0}t}],$
(2)

where $\kappa_{o}=\frac{g^{2}c^{2}T^{2}NP_{0}}{8V^{2}\hbar\Omega_{0}}$ is the
opto-magnonic coupling strength Hisatomi  _et al._ (2016), and a redefinition
$b_{y}\rightarrow b_{y}e^{i\theta}$ has been used to eliminate the
$\theta$-dependence. The dissipative non-Hermitian coupling between the cavity
and the YIG has to be formalized at the level of a master equation for the
system by treating the common waveguide as a thermal bath. We can obtain the
master equation for the density operator by adiabatically eliminating the
slowly evolving waveguide degrees of freedom. A rigorous derivation of the
same appears in the Appendix.

##  III Transduction from Microwave to Optical Field

Apropos of the microwave-to-optical conversion process, the magnons in mode
$m$ are driven by a classical microwave drive frequency $\omega_{\mu}$, while
concurrently interfacing with an optical waveguide that delivers and channels
the optical component $b_{y}$. In 1(a), the input microwave drive is launched
through the bottom fiber, while in 1(b), it directly impinges on the YIG
sample. The Faraday interaction $\mathcal{H}_{\text{optical}}$, prompted by
the pump laser $b_{z}$ at the frequency $\Omega_{0}$ sent along the optical
channel, entails the Stokes and anti-Stokes sidebands relative to $\Omega_{0}$
in the optical spectrum. The indirect coupling $\Gamma$ engineered between the
cavity and the magnons through the interceding transmission line consolidates
the coupling of the YIG to the microwave photons via the Purcell effect. The
sideband outputs can be empirically reconstructed from the amplified, beat-
down, heterodyne signal between the carrier and the side modes. In principle,
there could be two possible schemes of effecting the conversion: (i) by
guiding the microwave drive field along the microwave fiber, or (ii) by
directly shining a maser on the YIG. In what follows, we develop explicit
results for the two schemes separately.

###  III.1 Scheme 1: Microwave field launched through the waveguide

When the microwave field is launched through the transmission line, the
magnons encounter a phase-translated input relative to the intracavity field,
which promptly simplifies the components in $\mathcal{H}_{\text{micro}}$ to

$\displaystyle\mathcal{H}_{\text{micro}}^{(a)}$
$\displaystyle=-i\hbar\sqrt{\kappa_{a}}[a^{\dagger}{a}_{\mu}e^{-i\omega_{\mu}t}-aa_{\mu}^{\dagger}e^{i\omega_{\mu}t}],$
$\displaystyle\mathcal{H}_{\text{micro}}^{(m)}$
$\displaystyle=-i\hbar\sqrt{\kappa_{m}}[m^{\dagger}{a}_{\mu}e^{-i(\omega_{\mu}t-\phi)}-ma_{\mu}^{\dagger}e^{i(\omega_{\mu}t-\phi)}],$
(3)

where $\kappa_{a}$ and $\kappa_{m}$ symbolize the leakage rates of the cavity
and the magnons respectively into the interfacing microwave line, and
$a_{\mu}$ is the microwave drive. Fig. 1 portrays the overarching ladder
network. It is convenient to redefine the system variables as
$\tilde{a}=ae^{i\omega_{\mu}t}$, $\tilde{m}=me^{i\omega_{\mu}t}$, and,
depending on whether we consider the Stokes or anti-Stokes process,
$\tilde{b}_{y}=b_{y}e^{i(\Omega_{0}-\omega_{\mu})t}$ or
$\tilde{b}_{y}=b_{y}e^{i(\Omega_{0}+\omega_{\mu})t}$. The operation dispels
the explicit time-dependence in $\mathcal{H}$, when the fast-oscillating terms
are expunged. The first process pertains to a parametric-amplification type
process, whereas the second is associated with a beam-splitter type
interaction. We now develop the Langevin equations for the two respective
processes, to the lowest order in the optomagnonic coupling rate
$\sqrt{\kappa_{0}}$. For notational simplicity, we drop the tildes over the
mode variables.

Figure 2: Stokes conversion efficiencies for the microwave-to-optical
conversion pertaining to scheme 1, are plotted under anti-PT symmetric
conditions and comparable couplings $\Gamma=g$ in both dissipative and
coherent setups. In (a), we impose $\gamma=2\Gamma$ and in (b), we consider
$\gamma=1.1\Gamma$. The absolute efficiencies have been scaled up by a factor
of $S=1.4\crossproduct 10^{5}$ in both (a) and (b), for
$\Gamma\approx\pi\crossproduct 25$ MHz and $\Gamma_{0}\approx\pi\crossproduct
0.3$ mHz.

Stokes output: Pursuant to the master equation derived in Appendix A, the
mean-value dynamics of the cavity and the magnon modes reduces, in this case,
to

$\displaystyle\dot{a}$ $\displaystyle=-(i\Delta_{a}+\gamma_{a})a-\Gamma
e^{i\phi}m-\sqrt{\kappa_{a}}a_{\mu},$ $\displaystyle\dot{m}$
$\displaystyle=-(i\Delta_{m}+\gamma_{m})m-\Gamma
e^{i\phi}a-\sqrt{\kappa_{m}}a_{\mu}e^{i\phi}+\sqrt{\kappa_{o}}b_{y}^{\dagger}.$
(4)

where $\gamma_{a}=\kappa_{a}+\tilde{\kappa}_{a}$ and
$\gamma_{m}=\kappa_{m}+\tilde{\kappa}_{m}+\kappa_{o}\approx\kappa_{m}+\tilde{\kappa}_{m}$
encompass the dissipative effects. The decay rate $\tilde{\kappa}_{a}$
($\tilde{\kappa}_{m}$) stands for the intrinsic damping of the cavity (magnon)
mode into the non-waveguide modes. Based on the input-output formulation, the
output Stokes wave would be accorded by the relation

$\displaystyle b_{y,\text{out}}=-\sqrt{\kappa_{o}}m^{\dagger},$ (5)

since the input mode $b_{y}$, in this case, can be taken to be the thermal
vacuum. Eq. (4) can be condensed into the form

$\displaystyle\dot{X}=-i\mathcal{M}X-\mathcal{F}-\sqrt{\kappa_{o}}\mathcal{G},$
(6)

where $X=(a\hskip 5.69054ptm)^{T}$,
$\mathcal{F}=a_{\mu}(\sqrt{\kappa_{a}}\hskip
5.69054pt\sqrt{\kappa_{m}}e^{i\phi})^{T}$, $\mathcal{G}=(0\hskip
5.69054ptb_{y}^{\dagger})^{T}$, and
$\mathcal{M}=\begin{pmatrix}\Delta_{a}-i\gamma_{a}&-i\Gamma e^{i\phi}\\\
-i\Gamma e^{i\phi}&\Delta_{m}-i\gamma_{m}\\\ \end{pmatrix}$. If
$b_{y,\text{out}}$ has to be solved to the lowest (here, linear) order in
$\sqrt{\kappa_{0}}$, it follows that the long-time solution to $X$ could be
obtained by dropping the final term in (6), leading to

$\displaystyle X=i\mathcal{M}^{-1}\mathcal{F}.$ (7)

Hence, the solutions to the intracavity field and the magnonic oscillation
stand as

$\displaystyle a$
$\displaystyle=\frac{ia_{\mu}[\sqrt{\kappa_{a}}(\Delta_{m}-i\gamma_{m})+i\sqrt{\kappa_{m}}e^{2i\phi}\Gamma]}{(\Delta_{a}-i\gamma_{a})(\Delta_{m}-i\gamma_{m})+\Gamma^{2}e^{2i\phi}},$
$\displaystyle m$
$\displaystyle=\frac{ia_{\mu}e^{i\phi}[i\sqrt{\kappa_{a}}\Gamma+\sqrt{\kappa_{m}}(\Delta_{a}-i\gamma_{a})]}{(\Delta_{a}-i\gamma_{a})(\Delta_{m}-i\gamma_{m})+\Gamma^{2}e^{2i\phi}}.$
(8)

By virtue of Eq. (5), we educe the Stokes conversion efficiency to be

$\displaystyle\eta_{s}^{(d.c)}=\absolutevalue{\frac{b_{y,\text{out}}}{a_{\mu}}}=\absolutevalue{\frac{i\Gamma\sqrt{\kappa_{o}\kappa_{a}}+(\Delta_{a}-i\gamma_{a})\sqrt{\kappa_{o}\kappa_{m}}}{(\Delta_{a}-i\gamma_{a})(\Delta_{m}-i\gamma_{m})+\Gamma^{2}e^{2i\phi}}},$
(9)

For comparison, we recall the efficiency factor in a coherent environment,
where the cavity and the magnons address each other by means of a Hermitian
coupling $J(a^{\dagger}m+am^{\dagger})$:

$\displaystyle\eta_{s}^{(c.c)}=\absolutevalue{\frac{b_{y,\text{out}}}{a_{\mu}}}=\absolutevalue{\frac{J\sqrt{\kappa_{o}\kappa_{a}}}{(\Delta_{a}-i\gamma_{a})(\Delta_{m}-i\gamma_{m})-J^{2}}}.$
(10)

Since the solution to $X$ determines the Stokes sideband of the optical
spectra, its efficiency of conversion from the microwave input pivots on the
symmetry properties of $\mathcal{M}$. In this context, it is useful to note
that anti-PT symmetry could be engineered in this bimodal framework by
devising the constraints $\Delta_{a}=-\Delta_{m}=\Delta$,
$\gamma_{1}=\gamma_{2}=\gamma$ and $\phi=n\pi$, with $n$ being a natural
number. While, from the above expressions, it is clear that the order of
magnitude does not change dramatically by switching to a dissipative
configuration, it does turn out that dissipative couplings can predominantly
outstrip the efficiency of coherent setups under anti-PT conditions. From Eqs.
(9) and (10), the figure of merit quantified as an advantage gained through
anti-PT symmetry in dissipative environments can be simplified as

$\displaystyle\dfrac{\eta_{s}^{(d.c)}}{\eta_{s}^{(c.c)}}=\bigg{[}\frac{\\{\Delta^{2}+(\gamma-\Gamma)^{2}\\}\\{\Delta^{2}+\gamma^{2}+\Gamma^{2}\\}}{\Gamma^{2}\\{\Delta^{2}+\gamma^{2}-\Gamma^{2}\\}}\bigg{]}^{1/2},$
(11)

where $\kappa_{a}\approx\kappa_{m}$ and $\Gamma\approx J$ has been assumed.
Figures 2(a) and 2(b) plot the conversion factors as well as the relative
efficiencies w.r.t coherent couplings for comparable system parameters and
similar coupling strengths, under the constraint of anti-PT symmetry. It is
interesting to note the broadband nature of the profiles for anti-PT symmetry.
For large dampings or frequency detunings in the range $\Delta\geq\Gamma$, the
expression above exceeds 1, demonstrating the superiority of these couplings.
In the complementary regime, the situation is not so cut and dry, and
numerical means become the imperative.

Anti-Stokes output: An identical calculation for the anti-Stokes process
yields the same efficiency factor as above, i.e.,

$\displaystyle\eta_{as}^{(d.c)}=\absolutevalue{\frac{b_{y,\text{out}}}{a_{\mu}}}=\absolutevalue{\frac{i\Gamma\sqrt{\kappa_{o}\kappa_{a}}+(\Delta_{a}-i\gamma_{a})\sqrt{\kappa_{o}\kappa_{m}}}{(\Delta_{a}-i\gamma_{a})(\Delta_{m}-i\gamma_{m})+\Gamma^{2}e^{2i\phi}}},$
(12)

where the relevant input-output relation,
$b_{y,\text{out}}=-\sqrt{\kappa_{o}}m$, has been employed. Thus, the input
microwave photons of frequency $\omega_{\mu}$, upon interacting with the
optical wave of frequency $\Omega_{0}$, are converted into output traveling
light modes with frequencies $\Omega_{0}-\omega_{\mu}$ and
$\Omega_{0}+\omega_{\mu}$ pertaining to the Stokes and anti-Stokes bands
respectively.

Figure 3: Second scheme for microwave-to-optical conversion. The microwave
input as well as the laser drive are targeted on the YIG sphere, generating
the optical sidebands.

###  III.2 Scheme 2: Direct excitation of the magnons by a microwave pump

Figure 4: Stokes conversion efficiencies for the scheme 2 with the chosen
parameters $\gamma=1.1\Gamma$ (i.e., $\varepsilon=0.1$) and
$\gamma=1.01\kappa$ (i.e., $\varepsilon=0.01$), under anti-PT symmetry. The
absolute efficiencies have been scaled up by
$S(\varepsilon=0.1)=1.4\crossproduct 10^{5}$ and
$S(\varepsilon=0.01)=1.4\crossproduct 10^{4}$. Since the peaks in the two
graphs are in the vicinity of each other, the proportionality in Eq. (15) is
vindicated.

This protocol of selectively driving the YIG sphere, as demonstrated in Fig.
3, can exploit a spectral singularity of dissipatively interacting systems in
drawing out an enhanced steady-state response in the hybrid system. The vacuum
of the quantized field in a reservoir induces coherence between any two
systems coupled to it. When optimally strong, the coherence can push one of
the poles in the linear response to the real axis, under anti-PT symmetric
conditions. With the corresponding linewidth suffering a stark suppression,
the resonant response shoots up, only to be regularized by intrinsic
anharmonicites present in any of the modes. This feature in a two-mode
dissipatively coupled system has been tailored into a convenient mechanism for
sensing weak nonlinear perturbations Nair  _et al._ (2021b). Although, for our
current analysis, we do not meander into the nonlinear domain, we throw light
on the feasibility of ramping up the efficiency of our conversion model by
operating in the neighborhood of this singularity. With this constraint, it is
easy to see that the poles in Eq. (9) approach zero as
$\Gamma\rightarrow\gamma$. This is possible under the circumstance when
extraneous decoherence is strongly overshadowed by the coherence produced by
the shared reservoir. However, around this point, the numerator also becomes
small, precluding sizable enhancements in the output fields procreated via
scheme 1. To mitigate the pernicious role of the numerator, we can merely
resort to shining a maser beam on the YIG sample. While the microwave-YIG
interaction becomes
$\mathcal{H}_{\text{micro}}^{(m)}=-i\hbar\sqrt{2\kappa_{m}}[m^{\dagger}{a}_{\mu}e^{-i(\omega_{\mu}t-\phi)}-ma_{\mu}^{\dagger}e^{i(\omega_{\mu}t-\phi)}]$,
the cavity gets decoupled from any external driving fields. Therefore, the
term $\mathcal{H}_{\text{micro}}^{(a)}$ drops out. The Langevin equations for
the mode amplitudes corresponding to the Stokes process would, then, be
remodeled as

$\displaystyle\dot{a}$ $\displaystyle=-(i\Delta_{a}+\gamma_{a})a-\Gamma
e^{i\phi}m,$ $\displaystyle\dot{m}$
$\displaystyle=-(i\Delta_{m}+\gamma_{m})m-\Gamma
e^{i\phi}a-\sqrt{2\kappa_{m}}a_{\mu}e^{i\phi}+\sqrt{\kappa_{o}}b_{y}^{\dagger}.$
(13)

The input-output relation pertaining to the optical mode $b_{y}$ remains
intact, which yields the Stokes efficiency factor to be

$\displaystyle\eta_{s}^{(d.c)}=\absolutevalue{\frac{b_{y,\text{out}}}{a_{\mu}}}=\frac{\sqrt{2\kappa_{o}\kappa_{m}(\Delta_{a}^{2}+\gamma_{a}^{2})}}{(\Delta_{a}-i\gamma_{a})(\Delta_{m}-i\gamma_{m})+\Gamma^{2}e^{2i\phi}}$
(14)

To bring out the significance of the anti-PT symmetry, we look at the behavior
of the above expression under this constraint, i.e.,
$\Delta_{a}=-\Delta_{m}=\Delta$, $\gamma_{1}=\gamma_{2}=\gamma$ and
$\phi=n\pi$ . The efficiency is plotted in Fig. 4, which unveils a spike
around the origin $\Delta/\kappa=0$. When $\gamma$ is only slightly larger
than $\Gamma$, such that $\gamma=\Gamma(1+\varepsilon)$ for $\varepsilon\ll
1$, the above expression reduces, in the limit $\Delta/\Gamma\rightarrow 0$,
to

$\displaystyle\eta_{s}^{(d.c)}\approx(\varepsilon/2)^{-1}(\kappa_{o}/\kappa_{a})^{1/2}.$
(15)

Consequently, the smallness of $\varepsilon$ has a direct bearing on the
scaling up of the figure of merit characterizing the conversion scheme. For
instance, when $\varepsilon$ becomes one-tenth, there is about a tenfold
amplification observed in the optical output. This is evident from the
comparative plot in Fig. 4, since the deviations of $\gamma$ from $\Gamma$ are
respectively one and two orders of magnitude smaller than $\Gamma$. For
$\varepsilon\approx 0.1$, the value of $\eta_{s}^{(d.c)}$ approximates to
$0.025\%$, which signifies a remarkable improvement over the coherent setting.

This property in a dissipatively coupled system can also be justified from the
perspective of bright and dark states pertaining to radiating and non-
radiating modes of the system. On defining two linearly independent modes
$c_{\pm}=1/{\sqrt{2}}(a\pm m)$, we find that the mode $c_{-}$ evolves as
$\dot{c}_{-}=-\varepsilon\Gamma c_{-}-i\Delta
c_{+}-\sqrt{\kappa_{m}}a_{\mu}+\sqrt{\kappa_{0}}b_{y}^{\dagger}$. Thus, the
mode $c_{-}$ acts as a long-lived mode, akin to a dark state, as $\varepsilon$
becomes small. The other mode $c_{+}$ decays significantly faster at the rate
of $2\Gamma$. The emergence of a dark state explains the coherent buildup in
the output signal around the singularity $\varepsilon=0$.

##  IV Optical-to-microwave conversion: Nonreciprocity due to dissipative
coupling

Figure 5: Nonreciprocal Stokes’ conversion efficiencies (M: Microwave, O:
Optical) pertaining to the scheme 1, for the system parameters
$\Delta_{a}=-\Delta_{m}=\delta$ (for M$\rightarrow$O) and
$\delta_{a}^{(+)}=-\delta_{m}^{(+)}=\delta$ (for O$\rightarrow$M), plotted
against $\delta$for two sets of the phase separation $\phi$. The conversion
is, however, reciprocal when $\phi$ is an integer multiple of $\pi$. The
M$\rightarrow$O graphs refer to Eq. (9), and the O$\rightarrow$M graphs refer
to Eq. (19).

The reverse procedure of light getting transformed into microwave photons is
made feasible by the inverse Faraday effect. The initial conditions are now
translated into injecting two copropagating phase-coherent laser inputs
($b_{y}$ and $b_{z}$) along the optical transmission line, with the relevant
frequencies in the optical domain and separated by a microwave frequency. If
the difference frequency coincides with a Kittel mode frequency, the resonance
imparts an oscillatory magnetization to the magnons in the YIG, which, in
turn, elicit traveling microwave photons at this difference frequency. With
the plane-wave ansatz $b_{y}=\sqrt{\frac{P}{\hbar\Omega}}e^{-i\Omega t}$,
where $P$ is the drive power, we can derive the induced microwave output at
frequency $\omega_{+}=\Omega_{0}-\Omega$ (Stokes scattering) or
$\omega_{-}=\Omega-\Omega_{0}$ (anti-Stokes scattering), depending on
whichever is positive, by solving the dynamical equations in time domain. On
rotating the variables as $m\rightarrow me^{i\omega_{\mu}t}$ and $a\rightarrow
ae^{i\omega_{\mu}t}$, and defining the input amplitude
$\beta=\sqrt{\frac{P}{\hbar\Omega}}$, we reduce the evolution equations as

$\displaystyle\dot{a}$
$\displaystyle=-(i\delta_{a}^{(\pm)}+\gamma_{1})a-\Gamma e^{i\phi}m,$
$\displaystyle\dot{m}$
$\displaystyle=-(i\delta_{m}^{(\pm)}+\gamma_{2})m-\Gamma
e^{i\phi}a\pm\sqrt{\kappa_{o}}\beta,$ (16)

where the detunings $\delta_{a}^{(\pm)}=\omega_{a}-\omega_{\pm}$ and
$\delta_{m}^{(\pm)}=\omega_{m}-\omega_{\pm}$ have been introduced, and the
rapid oscillations neglected. Since no classical microwave drive needs to be
applied, the corresponding interaction term has been dropped. It is now a
simple exercise to evaluate the scattered microwave field in the long-time
limit. Clearly, the long-time solutions $X_{\pm}$ would correspond to fresh
microwave oscillations at $\omega_{+}$ or $\omega_{-}$, as the case may be, as
an artifact of the nonlinear interaction between the two optical beams and the
driven magnons. Eq. (16) yields the solutions

$\displaystyle X_{\pm}=\mp
i\sqrt{\kappa_{0}}\beta\mathcal{H}^{-1}\begin{pmatrix}0\\\ 1\\\
\end{pmatrix}.$ (17)

Invoking the input-output relation for the field transmitted across the
microwave waveguide, we find that

$\displaystyle a_{\text{out}}^{\pm}$
$\displaystyle=-(\sqrt{\kappa_{a}}e^{i\phi}a_{\pm}+\sqrt{\kappa_{m}}m_{\pm})$
$\displaystyle=\mp i\beta\bigg{[}\frac{i\Gamma
e^{2i\phi}\sqrt{\kappa_{0}\kappa_{a}}+\sqrt{\kappa_{o}\kappa_{m}}(\delta_{a}^{\pm}-i\gamma_{1})}{(\delta_{a}^{\pm}-i\gamma_{1})(\delta_{m}^{\pm}-i\gamma_{2})+\Gamma^{2}e^{i\phi}}\bigg{]}.$
(18)

This allows us to infer the conversion efficiencies for the two possible
microwave bands,

$\displaystyle\absolutevalue{\frac{a_{\text{out}}^{\pm}}{\beta}}=\absolutevalue{\frac{i\Gamma
e^{2i\phi}\sqrt{\kappa_{o}\kappa_{a}}+(\delta_{a}^{\pm}-i\gamma_{1})\sqrt{\kappa_{o}\kappa_{m}}}{(\delta_{a}^{\pm}-i\gamma_{1})(\delta_{m}^{\pm}-i\gamma_{2})+\Gamma^{2}e^{2i\phi}}}.$
(19)

where the upper sign refers to the Stokes band and the lower to the anti-
Stokes generation.

Figure 6: Nonreciprocity in the Stokes’ conversion efficiencies (M: Microwave,
O: Optical) pertaining to scheme 2, for the system parameters
$\Delta_{a}=-\Delta_{m}=\delta$ (for M$\rightarrow$O) and
$\delta_{a}^{(+)}=-\delta_{m}^{(+)}=\delta$ (for O$\rightarrow$M), and
$\gamma_{a}=\gamma_{m}=1.1\kappa$.The M$\rightarrow$O graphs refer to Eq.
(14), and the O$\rightarrow$M graphs to Eq. (19).

Strikingly, there is a structural disparity between the expression in Eqs. (9)
or (14), and that in Eq. (19), indicating nonreciprocal transductions. In the
case of scheme 1, for $\phi=n\pi$, the microwave-to-optical conversion
efficiency and vice versa resemble each other up to a reinterpretation of the
detunings. However, there is a palpable quantitative asymmetry in the two
mechanisms pertaining to an arbitrary phase for an otherwise commensurate set
of system parameters. The discrepancy between the two efficiency factors is
even more evident in scheme 2, whereby the two conversion processes ensue with
starkly disparate efficiencies, regardless of the choice of phase. Fundamental
to this nonreciprocity in either scheme is the factor that microwave fields
are directly or indirectly linked up with both the cavity and the YIG, whereas
the optical fields interact purely with the YIG. In addition, the phase-
sensitive asymmetry pertinent to scheme 1 can be traced down to the existence
of the reservoir-mediated phase-coupling between the cavity and the YIG, that
is a characteristic of dissipatively coupled systems. This is veritably
distinct from the symmetric nature of conversion observed in coherently
coupled setups, where both the conversion mechanisms unfold with equal
efficiencies. Note that this nonreciprocity in the conversion mechanisms
exists in spite of the symmetrical interaction between the two optical modes
and the magnons, as embodied in $\mathcal{H}_{\text{optical}}$.

##  V Conclusion

In conclusion, we have demonstrated the efficient interconversion between
optical and microwave fields in the context of cavity-magnonicsa dissipatively
coupled optomagnonic setup. As reported in the manuscript, dissipatively
coupled systems with anti-PT symmetry perform significantly better than
coherent settings for comparable system parameters. We have explicated two
disparate schemes for the microwave-to-optical conversion, one of which
involves injecting the microwave field through the shared waveguide and the
other where the ferromagnetic sample is directly subject to an external pump.
While the two schemes showcase improved conversion efficiencies compared to
the coherently coupled systems, the second scheme, which directly drives the
magnetic sample, manifests remarkable improvements when the dissipative
coupling dominates the extraneous dissipations. This superior conversion
efficiency stems from the emergence of a long-lived dark mode, and
consequently, the linear response suffers a tremendous boost. The transduction
protocols achieved via strong phase-sensitive dissipative couplings also
demonstrate strong nonreciprocity, with apparent discrepancies in the
efficiencies of microwave to optical conversion and vice versa.

##  VI Acknowledgements

The authors acknowledge the support of The Air Force Office of Scientific
Research [AFOSR award no FA9550-20-1-0366], The Robert A. Welch Foundation
[grant no A-1943] and the Herman F. Heep and Minnie Belle Heep Texas A&M
University endowed fund.

##  Appendix A Derivation of the master equation for a chain of emitters
coupled to a waveguide

Figure 7: An array of $N$ single-mode quantum emitters coupled to the
evanescent field of a one-dimensional waveguide, with
$\gamma_{\alpha}=\kappa_{\alpha\alpha}$ denoting the individual coupling
rates. The waveguide can be adiabatically eliminated to yield the master
equation of the emitter chain.

As depicted in Fig. 7, we consider an $N$-mode system (S) interacting
dissipatively through a shared one-dimensional bath (B) aligned along the
$x$-axis. Keeping the model very basic, we split the net Hamiltonian
$\mathcal{H}$ into three contributions given by

$\displaystyle\mathcal{H}_{S}$
$\displaystyle=\hbar\sum_{\lambda=1}^{N}\omega_{\lambda}c_{\lambda}^{\dagger}c_{\lambda},$
$\displaystyle\mathcal{H}_{B}$
$\displaystyle=\hbar\sum_{k}\omega_{k}a_{k}^{\dagger}a_{k},$
$\displaystyle\mathcal{H}_{SB}$
$\displaystyle=i\hbar\sum_{k}\sum_{\lambda=1}^{2}g_{k\lambda}(a_{k}e^{ikx_{\lambda}}-a_{k}^{\dagger}e^{-ikx_{\lambda}})(c_{\lambda}+c_{\lambda}^{\dagger}),$
(20)

where we assume that the mode $c_{\lambda}$ is coupled to the waveguide at the
location $x=x_{\lambda}$. The coupling coefficients $g_{k\lambda}$’s are
assumed to be real. Here, $\mathcal{H}_{SB}$ exemplifies a typical two-body
interaction among spatially separated modes with electromagnetic field
quantized in a one-dimensional geometry. We can adiabatically eliminate the
reservoir degrees of freedom to obtain the master equation under the Markov
approximation as

$\displaystyle\dot{\rho}_{S}(t)=$
$\displaystyle-\frac{i}{\hbar}[\mathcal{H}_{S},\rho_{S}(t)]$
$\displaystyle-\frac{1}{\hbar^{2}}\int_{0}^{\infty}\differential{\tau}\Tr_{B}[\mathcal{H}_{SB},[\mathcal{H}_{SB}(-\tau),\rho_{S}(t)\rho_{B}]]$
(21)

where
$\mathcal{O}(-\tau)=\exp[-\frac{i}{\hbar}(\mathcal{H}_{S}+\mathcal{H}_{B})\tau]\mathcal{O}\exp[\frac{i}{\hbar}(\mathcal{H}_{S}+\mathcal{H}_{B})\tau]$,
and $\rho_{B}$ is the initial state of the bath Agarwal (2012). Idealizing the
bath to be a thermalized vacuum at zero temperature, the reservoir signatures
would encoded as
$\expectationvalue{a_{k}a_{k^{\prime}}}=\expectationvalue{a_{k}^{\dagger}a_{k^{\prime}}}=0$
and $\expectationvalue{a_{k}a_{k^{\prime}}^{\dagger}}=\delta_{k,k^{\prime}}$.
Substituting these expressions into (21), we obtain $\tau-$integrals which,
for the moment, can be expressed in terms of

$\displaystyle\mathcal{T}_{k\mu}^{(\pm)}=\int_{0}^{\infty}\differential\tau
e^{i(\omega_{\mu}\pm\omega_{k})\tau}=\pi\delta(\omega_{\mu}\pm\omega_{k})+i\text{P}\frac{1}{(\omega_{\mu}\pm\omega_{k})},$
(22)

where $\text{P}(1/x)$ denotes the Cauchy Principal Value of its argument.
Since $\omega_{k}$’s are all positive, terms of the form
$\delta(\omega_{\mu}+\omega_{k})$ can be stamped out. In light of these
simplifications, we can compactify (21) into the form

$\displaystyle\dot{\rho}_{S}=-\frac{i}{\hbar}[\mathcal{H}_{S},\rho_{S}]-\frac{1}{\hbar^{2}}\sum_{\alpha,\beta=1}^{N}(\mathcal{W}_{\alpha\beta}+\mathcal{W}_{\alpha\beta}^{\dagger}),$
(23)

where a typical contribution would appear as

$\displaystyle\mathcal{W}_{\alpha\beta}=\sum_{k}g_{k\alpha}g_{k\beta}e^{ik(x_{\alpha}-x_{\beta})}\bigg{[}\mathcal{T}_{k\beta}^{(-)}(c_{\alpha}^{\dagger}c_{\beta}\rho_{S}-c_{\beta}\rho_{S}c_{\alpha}^{\dagger})+$
$\displaystyle\mathcal{T}_{k\beta}^{(+)}(c_{\alpha}c_{\beta}^{\dagger}\rho_{S}-c_{\beta}^{\dagger}\rho_{S}c_{\alpha})\bigg{]}.$
(24)

We have dropped the fast-oscillating terms that go as $c_{\alpha}c_{\beta}$ or
$c_{\alpha}^{\dagger}c_{\beta}^{\dagger}$ under the purview of the rotating
wave approximation. Using a linearized approximation to the reservoir
frequencies by letting $\omega_{k}\approx v_{g}\absolutevalue{k}$, we take the
continuum limit $\sum_{k}\rightarrow\frac{L}{2\pi}\int\differential k$ in
computing $\mathcal{W}_{\alpha\beta}$ and
$\mathcal{W}_{\alpha\beta}^{\dagger}$. With the identification
$\Lambda_{\pm}=\frac{1}{\pi}\text{P}\int_{-\infty}^{\infty}\differential
k\frac{e^{ik(x_{\alpha}-x_{\beta})}}{\omega_{\beta}\pm
v_{g}\absolutevalue{k}}$ and the assumption that the coupling to the modes is
independent of the field’s propagation direction, the cardinal intermediate
relations could be codified as

$\displaystyle\int_{-\infty}^{\infty}\differential ke^{\pm
ikx_{\alpha\beta}}\delta(\omega_{\beta}-v_{g}\absolutevalue{k})$
$\displaystyle=\frac{2}{v_{g}}\cos(k_{\beta}x_{\alpha\beta}),$
$\displaystyle\Lambda_{+}+\Lambda_{-}$
$\displaystyle=\frac{2}{v_{g}}\sin(k_{\beta}x_{\alpha\beta}),$ (25)

where $x_{\alpha\beta}=\absolutevalue{x_{\alpha}-x_{\beta}}$ and
$k_{\beta}=\omega_{\beta}/v_{g}$. Terms like
$\text{P}\int_{-\infty}^{\infty}\differential k\frac{1}{\omega_{\beta}\pm
v_{g}\absolutevalue{k}}$ get reflected as small frequency shifts in
$\omega_{1}$ and $\omega_{2}$, which can be neglected. Then, collecting the
like terms together in (23) and exploiting the preceding relations in (24) and
(25), we obtain the full master equation for the dissipative dynamics of S:

$\displaystyle\dot{\rho}_{S}=-\frac{i}{\hbar}[\mathcal{H}_{S},\rho_{S}]-\sum_{\alpha,\beta=1}^{N}\kappa_{\alpha\beta}(c_{\alpha}^{\dagger}c_{\beta}\rho_{S}-2c_{\beta}\rho_{S}c_{\alpha}^{\dagger}-\rho_{S}c_{\alpha}^{\dagger}c_{\beta})$
$\displaystyle-i\sum_{\alpha\neq\beta}\Omega_{\alpha\beta}[c_{\alpha}^{\dagger}c_{\beta},\rho_{S}],$
(26)

where the off-diagonal coefficients $\kappa_{\alpha\beta}$, for
$\alpha\neq\beta$, signify dissipative couplings, while $\Omega_{\alpha\beta}$
simulate dispersive interactions. The relevant coefficients are given by

$\displaystyle\kappa_{\alpha\beta}$
$\displaystyle=\frac{g_{\alpha}^{2}L}{v_{g}}\delta_{\alpha\beta}+\Gamma_{\alpha\beta}\cos\phi_{\alpha\beta}(1-\delta_{\alpha\beta}),$
$\displaystyle\Gamma_{\alpha\beta}$
$\displaystyle=(\kappa_{\alpha\alpha}\kappa_{\beta\beta})^{1/2}=\frac{g_{\alpha}g_{\beta}L}{v_{g}},$
$\displaystyle\Omega_{\alpha\beta}$
$\displaystyle=\Gamma_{\alpha\beta}\sin\phi_{\alpha\beta},$ (27)

where, in view of the proximity between the transition frequencies, it is
assumed that $k_{\lambda}\approx k_{0}=\omega_{0}/v_{g}$
$\forall\lambda\in\\{1,N\\}$. $g_{k\alpha}\approx g_{\alpha}$, and
$\\{k_{0},\omega_{0}\\}$ is the central waveguide mode in the vicinity of
which the linear dispersion holds valid. The phases $\phi_{\alpha\beta}$ are
defined as $\phi_{\alpha\beta}=k_{0}x_{\alpha\beta}$, and $g_{k\lambda}$’s are
taken to be $k$-independent. When $\phi_{\alpha\beta}$’s are integral
multiples of $\pi$, the couplings are purely dissipative. Note that the decay
parameter $\kappa_{\alpha\alpha}=\frac{g_{\alpha}^{2}L}{v_{g}}$ deduced here
accounts only for the waveguide’s contribution to the dynamics. When other
decohering channels are considered in parallel, additional dissipative effects
are tacked onto these terms.

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# Geometric phase and wave–particle duality of the photon

Elvis Pillinen<EMAIL_ADDRESS>Atri Halder  Ari T. Friberg  Tero Setälä
Andreas Norrman  Center for Photonics Sciences, University of Eastern Finland,
P.O. Box 111, FI-80101 Joensuu, Finland

(August 28, 2024)

###### Abstract

The concepts of geometric phase and wave–particle duality are interlinked to
several fundamental phenomena in quantum physics, but their mutual
relationship still forms an uncharted open problem. Here we address this
question by studying the geometric phase of a photon in double-slit
interference. We especially discover a general complementarity relation for
the photon that connects the geometric phase it exhibits in the observation
plane and the which-path information it encases at the two slits. The relation
can be seen as quantifying wave–particle duality of the photon via the
geometric phase, thus corroborating a foundational link between two ubiquitous
notions in quantum physics research.

_Introduction_.—The geometric phase [1], which is the phase that a physical
system acquires as it evolves along a curved trajectory in the underlying
parameter space, is a universal concept within the physical sciences [2]. It
is encountered in particle physics, condensed-matter physics, fluid dynamics,
and astrophysics, among other branches [3], and offers unique opportunities
for emerging quantum technologies [4, 5]. Wave–particle duality is another
central notion in modern physics that restricts the coexistence of
interferometric “which-path information” (particle behavior) and fringe
visibility (wave behavior) of quantum objects [6]. It is perhaps the most
recognized manifestation of quantum complementarity [7, 8, 9] and has been
observed in a wide variety of quantum physical systems, such as elementary
particles [10, 11, 12], atoms [13, 14, 15], molecules [16, 17], and even
antimatter [18].

In optical physics, the geometric phase arises from the change in the light’s
polarization state [19, 20, 21, 22, 23] and it has found numerous applications
in advanced light manipulation [24]. Even a single photon can carry the
geometric phase [25], which can be seen as a specific wave facet of the
photon. For classical light fields, the geometric phase was very recently
observed in the continuous, periodic polarization-state pattern in two-slit
interference [26, 27]. The recognition that light has the ability to exhibit
such interferometric polarization-state modulation has further revealed
fundamental aspects of wave–particle duality of the photon [28, 29]. These
facts hint that the geometric phase is profoundly linked to the dual
wave–particle nature of light at the single-photon level, but exactly how has
remained an unresolved physical problem.

In this Letter, we investigate the geometric phase of the photon in double-
slit interference and show that it is directly connected, in a deeply
complementary manner, to the photon’s particle characteristics at the two
slits. In particular, we formulate a fundamental wave–particle duality
relation for the photon in terms of the geometric phase that it displays in
the interference plane and the which-path information that it carries in the
slit plane. Our work thus establishes a link between two elementary notions in
physics and provides foundational insights into the nature of the photon.

Figure 1: System under study. A photon in a four-mode state
$\ket{\Psi}=c_{1x}\\!\ket{1,0,0,0}+c_{1y}\\!\ket{0,1,0,0}+c_{2x}\\!\ket{0,0,1,0}+c_{2y}\\!\ket{0,0,0,1}$
strikes screen $\mathcal{A}$ with two small slits at positions
$\mathbf{r}_{1}$ and $\mathbf{r}_{2}$. The interfering light is observed at
position $\mathbf{r}$ in plane $\mathcal{B}$ where periodic intensity and
polarization-state fringes appear after repeated experimental runs.

_System under study_.—Let us consider monochromatic quantum light impinging on
two identical slits (pinholes) located at $\mathbf{r}_{1}$ and
$\mathbf{r}_{2}$ in an opaque screen $\mathcal{A}$ (see Fig. 1). The light
emerging from the slits is observed on another screen $\mathcal{B}$ far from
$\mathcal{A}$ at position $\mathbf{r}$ in the paraxial domain. Under these
circumstances, the electric field operator at $\mathcal{B}$ is [28, 30]

$\hat{\mathbf{E}}(\mathbf{r})=K\Big{[}\hat{\mathbf{E}}(\mathbf{r}_{1})\frac{e^{ikr_{1}}}{r_{1}}+\hat{\mathbf{E}}(\mathbf{r}_{2})\frac{e^{ikr_{2}}}{r_{2}}\Big{]},$
(1)

where $K$ is a constant, $k$ is the wave number in free space, and
$r_{m}=|\mathbf{r}-\mathbf{r}_{m}|$ with $m\in\\{1,2\\}$. Moreover, each of
the electric field operators $\hat{\mathbf{E}}(\mathbf{r}_{1})$ and
$\hat{\mathbf{E}}(\mathbf{r}_{2})$ in the slit plane $\mathcal{A}$ contains
two orthogonal polarization modes ($x$ and $y$), characterized by the
annihilation operators $\hat{a}_{1x},\hat{a}_{1y}$ and
$\hat{a}_{2x},\hat{a}_{2y}$. We are especially interested in the case where
the light field is in an arbitrary, pure single-photon state

$\displaystyle\ket{\Psi}=$
$\displaystyle\>c_{1x}\ket{1,0,0,0}+c_{1y}\ket{0,1,0,0}$ (2)
$\displaystyle+c_{2x}\ket{0,0,1,0}+c_{2y}\ket{0,0,0,1}.$

Here $\ket{n_{1x},n_{1y},n_{2x},n_{2y}}$ is a four-mode Fock state, with
$n_{m\mu}$ denoting the photon number in the mode $\mu\in\\{x,y\\}$ at slit
$m\in\\{1,2\\}$, and $|c_{1x}|^{2}+|c_{1y}|^{2}+|c_{2x}|^{2}+|c_{2y}|^{2}=1$,
with $|c_{m\mu}|^{2}$ giving the probability to find the photon in the
corresponding mode.

The average intensity and polarization-state distributions of the one-photon
light at $\mathcal{B}$ are completely specified by the four Stokes parameters
[31]

$S_{j}(\mathbf{r})=\bra{\Psi}\\!\hat{\mathbf{E}}^{\dagger}(\mathbf{r})\bm{\sigma}_{j}\hat{\mathbf{E}}(\mathbf{r})\\!\ket{\Psi},\
\ j\in\\{0,1,2,3\\},$ (3)

where the dagger stands for the adjoint, $\bm{\sigma}_{0}$ is the $2\times 2$
identity matrix, and $\\{\bm{\sigma}_{1},\bm{\sigma}_{2},\bm{\sigma}_{3}\\}$
are the Pauli matrices. From Eqs. (1)–(3) we obtain

$\displaystyle\\!\\!\\!S_{j}(\mathbf{r})=$
$\displaystyle\;S_{j}^{\prime}(\mathbf{r})+S_{j}^{\prime\prime}(\mathbf{r})+2[S_{0}^{\prime}(\mathbf{r})S_{0}^{\prime\prime}(\mathbf{r})]^{1/2}$
(4)
$\displaystyle\times\\!|s_{j}(\mathbf{r}_{1},\mathbf{r}_{2})|\cos(\alpha_{j}-k\Delta
r),\ \ j\in\\{0,1,2,3\\},$

with $S_{j}^{\prime}(\mathbf{r})$ and $S_{j}^{\prime\prime}(\mathbf{r})$ being
the Stokes parameters on $\mathcal{B}$ when only the slit at $\mathbf{r}_{1}$
or $\mathbf{r}_{2}$ is open, respectively, and $\Delta r=r_{1}-r_{2}$.
Furthermore, $|s_{j}(\mathbf{r}_{1},\mathbf{r}_{1})|$ and $\alpha_{j}$ are the
magnitudes and phases of

$s_{j}(\mathbf{r}_{1},\mathbf{r}_{2})=\frac{S_{j}(\mathbf{r}_{1},\mathbf{r}_{2})}{[S_{0}(\mathbf{r}_{1})S_{0}(\mathbf{r}_{2})]^{1/2}},\
\ j\in\\{0,1,2,3\\},$ (5)

which in turn are the intensity-normalized versions of

$S_{j}(\mathbf{r}_{1},\mathbf{r}_{2})=\bra{\Psi}\\!\hat{\mathbf{E}}^{\dagger}(\mathbf{r}_{1})\bm{\sigma}_{j}\hat{\mathbf{E}}(\mathbf{r}_{2})\\!\ket{\Psi},\
\ j\in\\{0,1,2,3\\}.$ (6)

The quantities in Eq. (6) are the coherence (two-point) Stokes parameters that
contain all the information on the first-order vector-field correlations
between $\mathbf{r}_{1}$ and $\mathbf{r}_{2}$ in the slit plane $\mathcal{A}$
[29, 32, 33, 34]. The conventional (one-point) Stokes parameters at the slits
are obtained from Eq. (6) as
$S_{j}(\mathbf{r}_{1},\mathbf{r}_{1})=S_{j}(\mathbf{r}_{1})$ and
$S_{j}(\mathbf{r}_{2},\mathbf{r}_{2})=S_{j}(\mathbf{r}_{2})$.

Equation (4) particularly shows that the polarization state (and intensity) on
$\mathcal{B}$ varies periodically when moving the observation point
$\mathbf{r}$ transversally along the screen. In addition, since any such one-
photon light is completely first-order coherent at $\mathcal{A}$ [28] and
thereby fully polarized on $\mathcal{B}$ [35], so that
$S_{1}^{2}(\mathbf{r})+S_{2}^{2}(\mathbf{r})+S_{3}^{2}(\mathbf{r})=S_{0}^{2}(\mathbf{r})$,
the polarization-state evolution in the observation plane occurs on the
surface of the associated Poincaré sphere (cf. Fig. 2) [31, 34]. These
observations will be of central importance when we next assess the geometric
phase on screen $\mathcal{B}$.

_Geometric phase_.—To ascertain the geometric phase exhibited by the photon,
we employ the quantum kinematic approach introduced by Mukunda and Simon [36].
Let us consider a generic pure quantum state that evolves along some smooth
curve $\gamma$ within the Hilbert space, i.e.,
$\ket{\psi(\gamma_{1})}\rightarrow\ket{\psi(\gamma_{2})}$, where
$\ket{\psi(\gamma_{1})}$ is the initial state and $\ket{\psi(\gamma_{2})}$ is
the final state. In this case, the geometric phase $\Phi_{\mathrm{G}}$ can be
expressed as the difference

$\Phi_{\mathrm{G}}=\Phi_{\mathrm{T}}-\Phi_{\mathrm{D}},$ (7)

where the first term

$\Phi_{\mathrm{T}}=\arg\braket{\psi(\gamma_{1})}{\psi(\gamma_{2})}$ (8)

is the total phase between the initial and final states, and the second term

$\Phi_{\mathrm{D}}=\mathrm{Im}\left[\int_{\gamma_{1}}^{\gamma_{2}}\bra{\psi(\gamma)}\\!\frac{d}{d\gamma}\\!\ket{\psi(\gamma)}d\gamma\right]$
(9)

is the dynamic phase along the path.

Figure 2: Polarization-state evolution of the photon on the Poincaré sphere
spanned by the intensity-normalized Stokes parameters
$\\{s_{1},s_{2},s_{3}\\}$. The polarization states are specified by the
spherical polar angles $0\leq\theta\leq\pi$ and $0\leq\phi\leq 2\pi$, with
$\theta=0$ and $\theta=\pi$ representing right-handed and left-handed
polarization states $\ket{R}$ and $\ket{L}$, respectively. The blue curve
represents the dynamical phase $\Phi_{\mathrm{D}}$ between the initial state
$\ket{\psi_{1}}$ and final state $\ket{\psi_{2}}$, whereas the red curve is
the geodesic between these states. The geometric phase $\Phi_{\mathrm{G}}$
equals half the green surface area enclosed by the curves.

We recall from our above discussion that light in the pure single-photon state
given by Eq. (2) is fully polarized in the observation plane, with its
polarization state evolving cyclically on the surface of the Poincaré sphere.
It has further been shown that for any such polarized light field the
evolution path has the form of a circle [37]. We can therfore describe the
polarization-state evolution of the photon on screen $\mathcal{B}$ in terms of
the following qubit:

$\ket{\psi(\gamma)}=\cos(\theta/2)\ket{R}+e^{i\gamma\phi}\sin(\theta/2)\ket{L}.$
(10)

Here $\theta\in[0,\pi]$ and $\phi\in[0,2\pi]$ are the polar and azimuthal
angles on the Poincaré sphere as depicted in Fig. 2, $\ket{R}$ and $\ket{L}$
are right-handed and left-handed circular polarization bases, respectively,
and $\gamma\in[0,1]$ is a continuous real parameter. Our choice of a constant
angle $\theta$ simplifies but does not reduce the generality of the analysis,
since circular paths of any other orientation can always be transformed into
the form of Eq. (10) by a suitable rotation. The chosen value range for
$\phi$, corresponding to one circular loop (cycle) on the Poincaré sphere,
spans over a single spatial period in the interference pattern. We note,
however, that the relation between $\phi$ and $\Delta r$ is in general not
linear.

We can now connect the polarization-state evolution on the Poincaré sphere
with the quantum kinematic approach. On substituting Eq. (10) into Eqs.
(7)–(9), with $\gamma_{1}=0$ and $\gamma_{2}=1$, we find that the geometric
phase displayed by the photon in the observation plane $\mathcal{B}$ is

$\displaystyle\Phi_{\mathrm{G}}=$
$\displaystyle\arctan\left[\frac{\sin^{2}(\theta/2)\sin\phi}{\cos^{2}(\theta/2)+\sin^{2}(\theta/2)\cos\phi}\right]$
(11) $\displaystyle-\frac{\phi}{2}\big{(}1-\cos\theta\big{)}.$

Equation (11) is fully general in the sense that it covers both cyclic
($\phi=2\pi$) and noncyclic ($\phi<2\pi$) evolution. It is formally similar to
the classical geometric phase [27] and consistent with the so-called geodesic
rule [38, 39, 40, 41]: the start and end points in any noncyclic evolution
should be connected with a geodesic, and the geometric phase is specified by
half the enclosed solid angle (surface area). Some important features can be
concluded from Eq. (11). The phase magnitude is bounded as
$0\leq|\Phi_{\mathrm{G}}|\leq\pi$, and in any cyclic evolution it reduces to
$|\Phi_{\mathrm{G}}|=\pi(1-\cos{\theta})$. For the special cases $\theta=0$
and $\theta=\pi$, corresponding to the north and south poles on the Poincaré
sphere, respectively, we have $|\Phi_{\mathrm{G}}|=0$. Likewise, when the
polarization path is on the equator, $\theta=\pi/2$, the geometric phase is
zero for any trajectory within the range $0\leq\phi<\pi$, while it suddenly
changes to $|\Phi_{\mathrm{G}}|=\pi$ for $\pi<\phi\leq 2\pi$.

_Which-path information_.—To quantify the which-path information (WPI) carried
by the photon at the two slits, we utilize the following two measures
(applicable to any mixed state represented by a density operator $\hat{\rho}$)
[28]:

$D_{0}=\frac{|S_{0}(\mathbf{r}_{1})-S_{0}(\mathbf{r}_{2})|}{S_{0}(\mathbf{r}_{1})+S_{0}(\mathbf{r}_{2})},\
\
D_{S}=\frac{|\mathbf{S}(\mathbf{r}_{1})-\mathbf{S}(\mathbf{r}_{2})|}{S_{0}(\mathbf{r}_{1})+S_{0}(\mathbf{r}_{2})}.$
(12)

The quantity $D_{0}$ is called the intensity distinguishability and it
describes the intensity difference between the slits. The quantity $D_{S}$,
with
$\mathbf{S}(\mathbf{r}_{m})=[S_{1}(\mathbf{r}_{m}),S_{2}(\mathbf{r}_{m}),S_{3}(\mathbf{r}_{m})]$
in slit $m\in\\{1,2\\}$, is the Stokes (or polarization) distinguishability
that characterizes the polarization-state difference in the slit plane. The
denominators in Eq. (12) ensure that $0\leq D_{0}\leq 1$ and $0\leq D_{S}\leq
1$.

For the single-photon state in Eq. (2), the two general measures in Eq. (12)
turn into

$D_{0}=|p_{1}-p_{2}|,\ \
D_{S}=\sqrt{1-4|c_{1x}^{\ast}c_{2x}+c_{1y}^{\ast}c_{2y}|^{2}},$ (13)

where $p_{m}=|c_{mx}|^{2}+|c_{my}|^{2}$ is the path probability of the photon
to pass via the slit $m\in\\{1,2\\}$. At this one-photon level $D_{0}$
represents the path predictability [6, 28], i.e., the possibility to correctly
guess which of the slits the photon traverses based on its initial state
preparation. For example, when $p_{1}\gg p_{2}$ the probability of detecting
the photon in slit 1 is much larger than finding it in slit 2, yielding a high
path predictability ($D_{0}\approx 1$). In contrast, for $p_{1}\approx p_{2}$
the path predictability is negligible ($D_{0}\approx 0$). Likewise, in this
single-photon case $D_{S}$ translates into the path distinguishability [6,
28], which describes one’s ability to discriminate the photon’s path with
respect to its polarization state at the two slits. Especially, maximum path
distinguishability ($D_{S}=1$) is reached whenever the photon is orthogonally
polarized in the slit plane. For instance, if we were to measure $x$-polarized
($y$-polarized) light when $c_{1y}=c_{2x}=0$, then a count signal in the
detection plane would directly reveal that the photon has passed the first
(second) slit. On the other hand, minimum path distinguishability ($D_{S}=0$)
occurs if $\mathbf{S}(\mathbf{r}_{1})=\mathbf{S}(\mathbf{r}_{2})$. Generally,
however, the WPI of the photon is characterized by partial path predictability
($0<D_{0}<1$) and distinguishability ($0<D_{S}<1$).

We further introduce the quantity

$d=\frac{D_{0}}{D_{S}},$ (14)

which is the ratio between the two different WPI species. For any polarized
light, including the one-photon state of Eq. (2), the two measures in Eq. (12)
obey $D_{0}\leq D_{S}$ [28]. The WPI ratio in Eq. (14) therefore satisfies
$0\leq d\leq 1$, physically meaning that the photon carries more WPI in terms
of its polarization state than in terms of its path probabilities. The lower
limit $d=0$ is reached only if the path predictability is zero ($D_{0}=0$),
whereas the upper bound $d=1$ is saturated only for scalar light
($D_{0}=D_{S}$). In particular, for any polarized light field under cyclic
evolution the geometric phase is related to the intensity and polarization-
state differences at the slits according to
$|\Phi_{\mathrm{G}}|=\pi[1-|S_{0}(\mathbf{r}_{1})-S_{0}(\mathbf{r}_{2})|/|\mathbf{S}(\mathbf{r}_{1})-\mathbf{S}(\mathbf{r}_{2})|]$
[26, 37]. On connecting this expression with Eqs. (11) and (14), we then find
that the polar angle $\theta$ in the Poincaré sphere representation is
directly linked to the WPI ratio $d$ as

$d=\cos{\theta}.$ (15)

Equation (15) covers also the noncyclic case as $\theta$ remains unaffected
when varying the azimuthal angle $\phi$.

_Wave–particle duality_.—We are now in a position to present the main result
of this Letter. By using Eq. (15) and standard trigonometric identities, we
can first write the geometric phase in Eq. (11) as

$\Phi_{\mathrm{G}}=\arctan\left(\frac{\sin{\phi}}{\eta+\cos{\phi}}\right)-\frac{\phi}{2}(1-d),$
(16)

where $\eta=(1+d)/(1-d)$. We observe that the geometric phase in Eq. (16) is
now expressed solely in terms of the WPI ratio $d$ and the azimuthal angle
$\phi$ (corresponding to the distance along the observation screen). Equation
(16) thus provides an exact link between the WPI ratio at the slits and the
geometric phase in the detection plane.

By eventually introducing the $\pi$-normalized geometric phase
$\Phi_{\mathrm{G}}^{\prime}=\Phi_{\mathrm{G}}/\pi$, which is bounded as
$0\leq|\Phi_{\mathrm{G}}^{\prime}|\leq 1$, we discover from Eq. (16) the
complementarity relation

$|\Phi_{\mathrm{G}}^{\prime}|+d\leq 1.$ (17)

This result is obtained by first considering the maximum possible value of
$|\Phi_{\mathrm{G}}^{\prime}|$ when $0<d<1$, and then separately analyzing the
special scenarios of $d=0$ and $d=1$. When $0<d<1$, we find the maximum by
taking the derivative of Eq. (16) with respect to $\phi$. The corresponding
zero yields $\phi=2\pi$, which stands for cyclic evolution. Substituting this
value into Eq. (16) results in a strict complementarity identity for any
cyclic evolution, i.e.,

$|\Phi_{\mathrm{G}}^{\prime}|+d=1,\ \ \mathrm{if}\ \phi=2\pi.$ (18)

For noncyclic evolution we then necessarily have

$|\Phi_{\mathrm{G}}^{\prime}|+d<1,\ \ \mathrm{if}\ \phi<2\pi.$ (19)

The special case of $d=0$ is encountered solely if the path predictability is
zero ($D_{0}=0$). In this situation Eq. (16) leads to
$\Phi_{\mathrm{G}}^{\prime}=-\lfloor(\phi+\pi)/2\pi\rfloor$, where
$\lfloor\cdots\rfloor$ denotes the floor function. We thus find
$|\Phi_{\mathrm{G}}^{\prime}|=0$ if $0\leq\phi<\pi$ and
$|\Phi_{\mathrm{G}}^{\prime}|=1$ if $\pi<\phi\leq 2\pi$, both of which satisfy
Eq. (17). The last case with $d=1$ is the trivial scalar-light scenario
($D_{0}=D_{S}$) for which Eq. (16) directly gives
$|\Phi_{\mathrm{G}}^{\prime}|=0$, which is also encompassed by Eq. (17).

Equation (17) constitutes the main result of our work. It can be viewed as a
fundamental quantificator of wave–particle duality of the photon in terms of
the geometric phase in the observation plane $\mathcal{B}$ (wave aspect) and
the WPI ratio in the slit plane $\mathcal{A}$ (particle aspect). As underlined
by Eq. (18), for cyclic evolution these two attributes are strictly mutually
exclusive, i.e., reducing (increasing) $d$ increases (reduces)
$|\Phi_{\mathrm{G}}|$ such that their combined sum equals exactly one.

_Conclusions_.—In summary, we have explored the geometric phase of a single
photon in the double-slit setup. In particular, we discovered a fundamental
wave–particle duality relation for the photon that interconnects the geometric
phase it exhibits in the observation plane and the which-path information it
encompasses in the slit plane. This complementarity relation sets no
restrictions on the polarization state at the slits and covers both cyclic and
noncyclic polarization evolution in the interference plane. In the cyclic
case, the general inequality turns into a tight identity which states that the
geometric phase and which-path information are of strictly complementary
nature. Due to their universal physical characters, we expect that similar
features between the geometric phase and wave–particle duality exist in other
quantum systems as well. Our work thereby unifies fundamenal notions in
physics, reveals uncharted facets of the dual wave–particle nature of light,
and identifies future directions towards research on the geometric phase.

_Acknowledgments_.—The authors would like to thank Robert Fickler, Rafael
Barros, and Jaime Moreno for fruitful discussions. This research was supported
by the Research Council of Finland (Grant Nos. 354918, 349396, and 346518).

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