11institutetext: Princeton University, Princeton, NJ 08544, USA

# Awkward Just-In-Time (JIT) Compilation:

A Developerâs Experience

Ianna Osborne 11<EMAIL_ADDRESS>Jim Pivarski 11 Ioana Ifrim 11 Angus
Hollands 11 Henry Schreiner 11

###### Abstract

Awkward Array is a library for performing NumPy-like computations on nested,
variable-sized data, enabling array-oriented programming on arbitrary data
structures in Python. However, imperative (procedural) solutions can sometimes
be easier to write or faster to run. Performant imperative programming
requires compilation; JIT-compilation makes it convenient to compile in an
interactive Python environment. Various functions in Awkward Arrays JIT-
compile a userâs code into executable machine code. They use several different
techniques, but reuse parts of each othersâ implementations. We discuss the
techniques used to achieve the Awkward Arrays acceleration with JIT-
compilation, focusing on RDataFrame, cppyy, and Numba, particularly Numba on
GPUs: conversions of Awkward Arrays to and from RDataFrame; standalone cppyy;
passing Awkward Arrays to and from Python functions compiled by Numba; passing
Awkward Arrays to Python functions compiled for GPUs by Numba; and header-only
libraries for populating Awkward Arrays from C++ without any Python
dependencies

##  1 Introduction

Awkward Array library Ref-AwkwardArray  offers its users an array-oriented
programming style in a dynamically typed language. The advantage of the array-
oriented calculations is that they fit the data analysis workflow better,
considering the large amounts of data which a physicist typically loads into
an interactive environment. Moreover,the code that performs one operation on
all data is easy enough to write, then observe the intermediate results on
distributions, and reiterate. The imperative code, on the other hand, is
better for the stage after the interactive exploration, but the performant
imperative programming requires compilation. The loops may take longer to
write, especially when fitting into a Just-In-Time (JIT)-compiler and getting
the compile-time types correctly, but are often self-explanatory and faster.
It is not uncommon for a user analysis code to be expressed in a traditional
high-performance, statically typed language - C++, and a statically typed
language requires compilation.

The JIT-compilation makes it convenient to compile in an interactive Python
environment. Several JIT-compilation techniques are used to achieve the
desired acceleration. The Awkward Array functions JIT-compile a userâs code
into executable machine code. They use different techniques, but reuse parts
of each othersâ implementations.

The techniques discussed in this article are focusing on integrating
RDataFrame Ref-RDataFrame , cppyy Ref-cppyy , and Numba Ref-Numba ,
particularly Numba on GPUs. These include Awkward Arrays to and from
RDataFrame conversions, the standalone cppyy integration, passing Awkward
Arrays to and from Python functions compiled by Numba, passing Awkward Arrays
to Python functions compiled for GPUs by Numba, and populating Awkward Arrays
from C++ without any Python dependencies using a header-only library.

##  2 Awkward Arrays and RDataFrame

The Awkward Arrays can be converted to the RDataFrame columns and the
RDataFrame columns can be converted to Awkward Arrays as described in detail
in Ref-Awkward-RDataFrame  and Ref-Awkward-RDataFrame-Tutorial .

The user benefits from a faster execution of both the ROOT C++ functions and
the user-defined pure C++ functions. Here is an example of such conversion
shown beneath.

    import awkward as ak

    df = ak.to_rdataframe({"Events": array})

    rdf = (
        df.Filter("Events.nMuon() == 2")
          .Filter("Events.Muon_charge()[0] != Events.Muon_charge()[1]")
          .Define("dimuon_mass", """
    // this is C++
    return std::sqrt(2 * Events.Muon_pt()[0] * Events.Muon_pt()[1]
        * (std::cosh(Events.Muon_eta()[0] - Events.Muon_eta()[1])
        - std::cos(Events.Muon_phi()[0] - Events.Muon_phi()[1])));
    """))

The ak.to_rdataframe function presents a view of an Awkward Array as an
RDataFrame source. The ak.from_rdataframe function converts the selected
columns as native Awkward Arrays. The implementation leverages on a zero-copy
Awkward Array view and implements all for-loops on data in C++.

The handle to this Array view is a lightweight 40-byte C++ object allocated on
the stack. The generated RDataSource takes pointers into the original array
data via this view. Next, the column readers are generated based on the run-
time type of the views. Finally, the readers are passed to a generated source
derived from ROOT::RDF::RDataSource.

The ak.from_rdataframe function converts the selected columns as native
Awkward Arrays. The templated C++ header-only implementation and the
dynamically generated C++ code are used to extract the columnsâ types and
data.

    import ROOT

    df = ROOT.RDataFrame("Events",
        "root://eospublic.cern.ch//eos/opendata/cms/derived-data/"
        "AOD2NanoAODOutreachTool/Run2012BC_DoubleMuParked_Muons.root")

    array = ak.from_rdataframe(
            df, columns=("Muon_charge", "Muon_eta", "Muon_mass",
                         "Muon_phi", "Muon_pt", "nMuon")
    )

This array has type

    type: 61540413 * {
        Muon_charge: var * int32,
        Muon_eta: var * float32,
        Muon_mass: var * float32,
        Muon_phi: var * float32,
        Muon_pt: var * float32,
        nMuon: uint32
    }

and some of its values look like

    [{Muon_charge: [-1, -1], Muon_eta: [1.07, -0.564], ...},
     {Muon_charge: [1, -1], Muon_eta: [-0.428, 0.349], ...},
     {Muon_charge: [1], Muon_eta: [2.21], Muon_mass: [0.106], ...},
     ...,
     {Muon_charge: [-1, 1], Muon_eta: [-2.15, 0.291], ...}]

On the one hand, the users who do their analysis entirely in Python ecosystem
can benefit from faster execution using ROOT C++ functions, or pure C++, in an
otherwise Awkward analysis at full speed. The performance cost of converting
Awkward Arrays into RDataFrame is negligible, since it is a zero-copy view.

On the other hand, those who prefer C++, ROOT, and RDataFrame, have an ability
to convert their data into Awkward Arrays in order to leverage the tools
available in the wider Scientific Python ecosystem.

##  3 Standalone cppyy

Like ROOT and RDataFrame, cppyy allows a user to write C++ and JIT-compile it
to use it from Python. The C++ code can be included to Python from a C++ file
or written directly as a Python string. But cppyy can be installed without the
entire ROOT package and Awkward Array’s interface to it is more low-level:
users can write C++ functions that operate on whole arrays, multiple arrays,
and return arrays.

The ak.Array, the Python class for all Awkward Arrays, implements a magic
function __cast_cpp__ that is called by cppyy to determine a C++ type of the
array. The Numba implementation Ref-Awkward-Numba  is reused here to generate
a C++ array view on demand. The generated ArrayView C++ class hashed type is
registered as a cpp_type Python string attribute of the ak.Array class. The
cppyy maps the C++ class type as a string to a Python type. The down side is
that the user cannot redefine the function. Each function must have a unique
name. This is similar to the PyROOT interpreter, because an earlier version of
cppyy is used to bind Python and C++ in PyROOT.

The user does not need to know what cpp_type is - the cpp_type is generated on
demand when the array needs to be passed to the C++ function.

    array = ak.Array([
        [{"x": 1, "y": [1.1]}, {"x": 2, "y": [2.2, 0.2]}],
        [],
        [{"x": 3, "y": [3.0, 0.3, 3.3]}],
      ])

    source_code_cpp = """
    template<typename T>
    double go_fast_cpp(T& awkward_array) {
     double out = 0.0;
     for (auto list : awkward_array) {
       for (auto record : list) {
         for (auto item : record.y()) {
           out += item;
         }
       }
     }
     return out;
    }
    """
    cppyy.cppdef(source_code_cpp)

    out = cppyy.gbl.go_fast_cpp[array.cpp_type](array)
    assert out == ak.sum(array["y"])

The cppyy version that is used must be 3.1 or later. The cppyy library can
construct an object of a previously declared type based on an arguments. This
new feature is not available in the earlier versions. cppyy implements an
implicit instantiation from __cast_cpp__ returning a tuple.

##  4 Awkward Arrays and Numba

Numba is a JIT-compiler of functions with Python syntax, and Awkward Arrays
can be passed to and from Numba-compiled functions with a similar interface as
cppyy. Numba can be used in contexts in which acceleration is needed, but C++
is not—it allows users to write accelerated code in the same Python language
as the rest of their code, albeit in a subset of that language (not all Python
code can be compiled).

Numba infers the argument types at call time, and generates optimized code
based on this information. Numba also compiles separate specializations
depending on the input types.

    import numba as nb, numpy as np

    @nb.jit
    def path_length(array):
      result = np.zeros(len(array), dtype=np.float32)
      for i, row in enumerate(array):
        result[i] = 0
        for j, val in enumerate(row):
          result[i] += val
      return result

The implementation to pass Awkward Arrays to and from Python functions
compiled by Numba defines a numba_type property of an ak.Array that is a type
of a generated C++ Array view, as discussed before. Awkward Arrays can be
iterated over in Numba-compiled functions using zero-copy views.

##  5 Python functions compiled for GPUs by Numba

Numba can also compile code for Nvidia GPUs through LLVM’s CUDA backend, and
Awkward Arrays can be passed as arguments into such functions. Currently, the
extension needs to be defined by a user in a function decorator (as
extensions=[ak.numba.cuda] below), but this requirement will be removed in a
future version of Numba.

    import cupy as cp

    # make an Awkward Array of variable-length lists on the GPU with CuPy
    N = 2**20
    counts = cp.random.poisson(1.5, N).astype(np.int32)
    content = cp.random.normal(0, 45, int(counts.sum())).astype(np.float32)
    array = ak.unflatten(content, counts)

    @nb.cuda.jit(extensions=[ak.numba.cuda])
    def path_length(out, array):
        tid = nb.cuda.grid(1)
        if tid < len(array):
            out[tid] = 0
            for i, x in enumerate(array[tid]):
                out[tid] += x

    blocksize = 256
    numblocks = (N + blocksize - 1) // blocksize

    result = cp.empty(len(array), dtype=np.float32)
    path_length[numblocks, blocksize](result, array)

##  6 Header-only libraries

To create Awkward Arrays in C++ without any dependence on Python, we provide a
separate set of header-only libraries that implement an awkward::layoutbuilder
RefH  that builds up an array by appending elements and then shares that array
through basic C types (raw array buffers and a JSON-formatted string the
convey structure).

    #include "awkward/LayoutBuilder.h"
    using awkward::layoutbuilder;

    // instantiating layoutbuilder types (type aliases omitted for brevity)
    enum Field : std::size_t {one, two};
    using UserDefinedMap = std::map<std::size_t, std::string>;
    UserDefinedMap fields_map({
        {Field::one, "one"},
        {Field::two, "two"}
    });

    RecordBuilder<
      RecordField<Field::one, NumpyBuilder<double>>,
      RecordField<Field::two, ListOffsetBuilder<int64_t,
          NumpyBuilder<int32_t>>>
    > builder(fields_map);

    // executable code to create and fill the layoutbuilder
    auto& one_builder = builder.field<Field::one>();
    auto& two_builder = builder.field<Field::two>();

    one_builder.append(1.1);
    auto& two_subbuilder = two_builder.begin_list();
    two_subbuilder.append(1);
    two_builder.end_list();

    one_builder.append(2.2);
    two_builder.begin_list();
    two_subbuilder.append(1);
    two_subbuilder.append(2);
    two_builder.end_list();

##  7 Summary and future plans

The Awkward Array library was designed around array-oriented programming, in
part because imperative programming in Python is impractical at large scales.
However, this library also provides many ways to access the same data—without
copying it—in JIT-compiled contexts, enabling imperative computing at large
scales.

While these bridges may be used directly by users, they are also pathways to
future features. For example, automatic bindings from Kaitai Struct Ref-kaitai
to Awkward Arrays are in development, and these rely heavily on the header-
only awkward::layoutbuilder. JIT-compilation in Julia Ref-julia  is also in
development, borrowing from the patterns set by C++ and Numba integration.

##  8 Acknowledgment

This work is supported by NSF cooperative agreement OAC-1836650 (IRIS-HEP) and
NSF cooperative agreement PHY-2121686 (US-CMS LHC Ops).

## References

  * (1) Pivarski, J., Osborne, I., Ifrim, I., Schreiner, H., Hollands, A., Biswas, A., Das, P., Roy Choudhury, S., Smith, N., & Goyal, M. (2018). Awkward Array [Computer software]. https://doi.org/10.5281/zenodo.4341376
  * (2) Harris, C.R., Millman, K.J., van der Walt, S.J. et al. Array programming with NumPy. Nature 585, 357â362 (2020). DOI: 10.1038/s41586-020-2649-2
  * (3) RDataFrame: Easy Parallel ROOT Analysis at 100 Threads Danilo Piparo, Philippe Canal, Enrico Guiraud, Xavier Valls Pla, Gerardo Ganis, Guilherme Amadio, Axel Naumann, Enric Tejedor EPJ Web Conf. 214 06029 (2019) DOI: 10.1051/epjconf/201921406029
  * (4) cppyy [Online] https://cppyy.readthedocs.io/en/latest/
  * (5) Siu Kwan Lam, stuartarchibald, Antoine Pitrou, Mark Florisson, Stan Seibert, Graham Markall, esc, Todd A. Anderson, rjenc29, Guilherme Leobas, luk-f-a, Michael Collison, Jay Bourque, Aaron Meurer, Travis E. Oliphant, Nick Riasanovsky, Kaustubh, Michael Wang, densmirn, … James Bourbeau. (2022). numba/numba: Version 0.56.4 (0.56.4). Zenodo. https://doi.org/10.5281/zenodo.7289231
  * (6) Osborne, I., Pivarski, J. 2022 Awkward to RDataFrame and back [arXiv 2302.09860] `DOI: 10.48550/arXiv.2302.09860`
  * (7) Osborne, I., Pivarski, J., (2022, September 15). Awkward RDataFrame Tutorial. PyHEP 2022 (virtual) Workshop. Zenodo. https://doi.org/10.5281/zenodo.7081586
  * (8) Jim Pivarski, Peter Elmer and David Lange, (16 November 2020). Awkward Arrays in Python, C++, and Numba, EPJ Web of Conferences 245, 05023 (2020). https://doi.org/10.1051/epjconf/202024505023
  * (9) Goyal, M., Osborne, I., Pivarski, J. The Awkward World of Python and C++. ACAT 2022 Workshop. https://doi.org/10.48550/arXiv.2303.02205
  * (10) Kaitai Struct [Online]. https://kaitai.io
  * (11) Julia [Online] https://julialang.org


# Chiral cavity-magnonic system for the unidirectional emission of a tunable
squeezed microwave field

Ji-kun Xie  Sheng-li Ma<EMAIL_ADDRESS>Ya-long Ren  Shao-yan Gao  Fu-li
Li  MOE Key Laboratory for Non-equilibrium Synthesis and Modulation of
Condensed Matter, Shaanxi Province Key Laboratory of Quantum Information and
Quantum Optoelectronic Devices, School of Physics, Xi’an Jiaotong University,
Xi’an 710049, China

###### Abstract

Unidirectional photon emission is crucial for constructing quantum networks
and realizing scalable quantum information processing. In the present work an
efficient scheme is developed for the unidirectional emission of a tunable
squeezed microwave field. Our scheme is based on a chiral cavity magnonic
system, where a magnon mode in a single-crystalline yttrium iron garnet (YIG)
sphere is selectively coupled to one of the two degenerate rotating microwave
modes in a torus-shaped cavity with the same chirality. With the YIG sphere
driven by a two-color Floquet field to induce sidebands in the magnon-photon
coupling, we show that the unidirectional emission of a tunable squeezed
microwave field can be generated via the assistance of the dissipative magnon
mode and a waveguide. Moreover, the direction of the proposed one-way emitter
can be controlled on demand by reversing the biased magnetic field. Our work
opens up an avenue to create and manipulate one-way nonclassical microwave
radiation field and could find potential quantum technological applications.

##  I Introduction

Chiral quantum optics has become a burgeoning field due to its potential
applications in quantum information processing and quantum networks Lodahl
_et al._ (2017); Petersen  _et al._ (2014); Söllner  _et al._ (2015);
Mahmoodian  _et al._ (2016); Tang  _et al._ (2022). It seeks to exploit new
approaches and systems exhibiting chiral light-matter interactions Ramos  _et
al._ (2014); Gong  _et al._ (2018); Sánchez-Burillo  _et al._ (2020); Zhang
_et al._ (2019a); Barik  _et al._ (2020); Vermersch  _et al._ (2016); Mehrabad
_et al._ (2020); Owens  _et al._ (2022); Kim  _et al._ (2015); Dong  _et al._
(2015); Ma  _et al._ (2022). On this subject, nanophotonic systems, such as
nanoscale waveguides and whispering-gallery-mode (WGM) resonators Coles  _et
al._ (2016); Junge  _et al._ (2013); Will  _et al._ (2021), have emerged as
attractive candidates, where the light field is strongly transversely confined
in a subwavelength space and exhibits the optical spin-orbit coupling Bliokh
_et al._ (2015); Mechelen and Jacob (2016). The chiral interaction can be
achieved by coupling the spin-momentum-locked light to quantum emitters with
polarization-dependent dipole transitions Hurst  _et al._ (2018); Mitsch  _et
al._ (2014); Pucher  _et al._ (2022); Kien and Rauschenbeutel (2014); Hallett
_et al._ (2022). And a large number of chiral devices have been proposed
theoretically and demonstrated experimentally, like single-photon diodes Yan
_et al._ (2018a); Scheucher  _et al._ (2016), single-photon routing Shomroni
_et al._ (2014); Yan  _et al._ (2018b); Gonzalez-Ballestero  _et al._ (2016),
circulators Scheucher  _et al._ (2016); Xia  _et al._ (2018), isolators Xia
_et al._ (2018); Sayrin  _et al._ (2015); Xia  _et al._ (2014) and etc. These
elements are key components for building large-scale quantum networks.

In parallel, the chiral light-matter interactions can also be utilized to
control the direction of photon emission Jones  _et al._ (2020); Martin-Cano
_et al._ (2019); Downing  _et al._ (2019); le Feber  _et al._ (2015). By
designing the suitable chiral coupling between quantum emitters and evanescent
fields, the deterministic and the highly directional photon emission can be
achieved along only one of the selected directions Coles  _et al._ (2016);
Hurst  _et al._ (2018); le Feber  _et al._ (2015). Recently, the chiral-
waveguide-based and the chiral-cavity-based systems have been explored to
generate the unidirectional emission of single photons Tang  _et al._ (2019);
Ostrowski  _et al._ (2022); ju Gu  _et al._ (2022); Coles  _et al._ (2017);
Sun  _et al._ (2019). In addition, the unidirectional laser emission has also
been demonstrated through the non-Hermitian scattering induced chirality in
the WGM resonator Peng  _et al._ (2016); Longhi and Feng (2017). The
unidirectional photon emission can facilitate the free-space coupling of
energy, improve the collection efficiency of weak optical signals and on-
demand distribute quantum information Bahari  _et al._ (2017); Jiang  _et al._
(2015); Pichler  _et al._ (2015); Kimble (2008); Wiersig and Hentschel (2008).

In recent years, the hybrid quantum system of yttrium iron garnet (YIG) and
superconducting circuit has been considered as a powerful platform for quantum
information applications Lachance-Quirion  _et al._ (2019); Rameshti  _et al._
(2022); Clerk  _et al._ (2020); Yuan  _et al._ (2022). Due to the high spin
density and low damping rate of the ferrimagnetic insulator YIG, the strong
and even ultrastrong coupling between a magnetostatic mode in a YIG sphere and
a microwave cavity mode have been experimentally observed Tabuchi  _et al._
(2014); Zhang  _et al._ (2014); Li  _et al._ (2022); Goryachev  _et al._
(2014); Wang  _et al._ (2018); Maier-Flaig  _et al._ (2016). Many intriguing
phenomena have been studied in this hybrid system, such as the generation of
various quantum states Li  _et al._ (2018); Zhang  _et al._ (2019b); Li  _et
al._ (2019); Nair and Agarwal (2020); Yuan  _et al._ (2020); Kong  _et al._
(2021); kun Xie  _et al._ (2020a); Liu  _et al._ (2019); Zhao  _et al._
(2020); Wu  _et al._ (2021); long Ren  _et al._ (2022a); Tan and Li (2021),
nonreciprocity Wang  _et al._ (2019); Kong  _et al._ (2019); long Ren  _et
al._ (2022b), non-Hermitian physics Zhang  _et al._ (2017); Lu  _et al._
(2021); Harder  _et al._ (2018), and Floquet engineering Xu  _et al._ (2020);
Zhang  _et al._ (2022); fan Qi and Jing (2022); Xie  _et al._ (2023).
Remarkably, magnons also carry angular momentum or “spin”. For instance, the
Kittel mode magnetization precesses counterclockwise around the applied
magnetic field Yu  _et al._ (2023). Analogous with the chiral coupling of
spin-polarized atoms or quantum dots with spin-momentum-locked light Coles
_et al._ (2016); Junge  _et al._ (2013); Will  _et al._ (2021), the Kittel
mode can only couple to photons with the same polarization in ring-shaped
cavity or waveguide Yu  _et al._ (2020a, b); Zhang  _et al._ (2020); Bourhill
_et al._ (2023); Yu  _et al._ (2020c). Therefore, this hybrid magnonic system
can act as a promising architecture for studying chiral quantum optics Zhu
_et al._ (2020); Wang  _et al._ (2021); long Ren  _et al._ (2022c); Xu  _et
al._ (2021); Zhan  _et al._ (2022).

Figure 1: (Color online) Schematic diagram of the hybrid quantum setup. A
torus-shaped cavity supports both CCW ($a$) and CW ($b$) rotating microwave
modes, which are evanescently coupled to a nearby microwave waveguide. A small
ferromagnetic YIG sphere is placed inside the cavity and biased
perpendicularly by a static magnetic field $B_{0}$ to establish the magnon-
photon coupling. Additionally, the magnon mode is driven by a two-color
Floquet driving field (not shown) along the bias field direction.

In this paper, we develop a method for the unidirectional emission of a
tunable squeezed microwave field based on a chiral cavity magnonic system. In
our scheme, a torus-shaped microwave cavity that supports a pair of degenerate
counter-rotating microwave modes is exploited to chirally coupled to a YIG
sphere. The chiral interaction can be achieved by locating the YIG sphere on a
special chiral line in the cavity, where the circular polarization of the
cavity mode is locked to its propagation direction Zhang  _et al._ (2020);
Bourhill  _et al._ (2023); Yu  _et al._ (2020c); Zhu  _et al._ (2020); Wang
_et al._ (2021). As a result, the Kittel mode will only couple to one of the
microwave modes with the same polarization. By applying a two-color Floquet
driving field to the magnetic sphere to induce sideband transitions, we can
engineer a squeezing-type interaction between the magnon mode and the selected
microwave mode, which can be steered into a squeezed state with the aid of
large decay rate of the magnon mode. By further considering the cavity
evanescently coupled to a microwave waveguide, we can generate a
unidirectionally squeezed microwave source. Notably, the emission direction
can be conveniently adjusted by reversing the bias magnetic field, and the
amount of squeezing is tunable by changing the external parameters. Moreover,
the numerical simulations indicate that the unidirectional emission is robust
against the imperfect chiral coupling. Our work could find a wide range of
practical applications in quantum networks and quantum sensing.

##  II Model

As schematically shown in Fig. 1, the hybrid cavity magnonic system under
consideration consists of a torus-shaped microwave cavity with a small highly
polished YIG sphere placed inside Bourhill  _et al._ (2023); Yu  _et al._
(2020c). Due to the geometrical rotational symmetry, the cavity supports a
pair of degenerate counterclockwise (CCW) and clockwise (CW) microwave modes,
which are evanescently coupled to a nearby microwave waveguide. Under a static
out-of-plane bias magnetic field at a strength $B_{0}$, many magnetostatic
modes will be excited in the YIG sphere Tabuchi  _et al._ (2014); Zhang  _et
al._ (2014); Li  _et al._ (2022). In our scheme, we focus only on the Kittel
mode, which has the uniform spin precessions in the whole volume of the
magnetic sphere. The two microwave modes are coupled to the Kittel mode
through the collective magnetic-dipole interaction, respectively. Now, we can
give the free Hamiltonian of the Kittel mode and the cavity modes (we set
$\hbar=1$ hereafter)

$H_{0}=\omega_{m}m^{{\dagger}}m+\omega_{0}\sum_{\alpha=a,b}\alpha^{{\dagger}}\alpha.$
(1)

Here, $m$ $(m^{{\dagger}})$ is the boson operator of the magnon mode with the
resonance frequency $\omega_{m}=\gamma B_{0}$, where $\gamma=2\pi\times 28$
GHz/T is the gyromagnetic ratio. $a$ $(a^{{\dagger}})$ and $b$
$(b^{{\dagger}})$ represent the annihilation (creation) operators for the CCW
and CW microwave modes with the resonance frequency $\omega_{0}$.

We then consider the chiral photon-magnon interaction, which takes the form
Zhang  _et al._ (2020); Bourhill  _et al._ (2023); Yu  _et al._ (2020c); Zhu
_et al._ (2020); Wang  _et al._ (2021)

$H_{int}=\sum_{\alpha=a,b}g_{\alpha}(\alpha^{{\dagger}}+\alpha)(m^{{\dagger}}+m)$
(2)

with $g_{\alpha}(\alpha=a,b)$ describing the coupling strength between the
cavity mode $\alpha$ and the magnon mode $m$. To be specific, the Kittel mode
magnetization precesses around the effective magnetic field in an
anticlockwise manner and couples preferentially to photons with the same
polarization. Here, the magnetic field of each TE mode is transversally
confined in the cavity, so that it has a strong longitudinal polarization
component along the propagation direction Lodahl  _et al._ (2017); Shomroni
_et al._ (2014); Aiello  _et al._ (2015). Moreover, the longitudinal and
transverse polarization components oscillate $\pm 90^{\circ}$ out of phase
with each other, where $+$ ($-$) sign depends on the propagation direction of
the cavity modes. At the special radial locations, the magnetic field can be
circularly polarized when the longitudinal and transverse polarization
components have the same amplitudes. Also, since the polarization direction is
locked to the sign of their linear momentum, the counterpropagated CW and CCW
modes can thereby possess mutually orthogonal polarizations and opposite
chiralities Zhang  _et al._ (2020). As a result, the magnon mode can only
couple to one of them with the same chirality, i.e., $g_{a}\neq 0$ and
$g_{b}=0$, or $g_{b}\neq 0$ and $g_{a}=0$.

In addition, the chiral coupling can be well controlled by reversing the
direction of the biased magnetic field, because it is accompanied with the
reversion of the precession of the Kittel mode. Alternatively, each of the two
degenerate counter-propagating microwave modes can have opposite chirality at
different radial locations Bourhill  _et al._ (2023); Yu  _et al._ (2020c), so
the chiral coupling can also be controlled by shifting the magnet position.

Furthermore, a two-color Floquet driving field is applied to the YIG sphere
along the bias magnetic field direction, under which the oscillating frequency
of the magnon mode is modulated accordingly. So the coupling between the
Floquet driving field and the Kittel mode is described by Xu  _et al._ (2020);
Zhang  _et al._ (2022); fan Qi and Jing (2022); Xie  _et al._ (2023)

$H_{d}(t)=[\Omega_{1}\cos(\omega_{d1}t)+\Omega_{2}\cos(\omega_{d2}t+\theta)]m^{{\dagger}}m,$
(3)

where $\Omega_{j}$ $(j=1,2)$ denotes the driving amplitudes with the driving
frequencies $\omega_{dj}$, $\theta$ is the relative phase. Experimentally,
this manipulation can be implemented through a small coil being looped tightly
around the magnetic sphere to modulate the bias magnetic field Xu  _et al._
(2020). Under this kind of periodical modulation, the desired parametric
magnon-photon interaction can be sculpted to realize the one-way squeezed
microwave source.

Figure 2: (Color online) Schematic diagram of sideband transitions of the
coupled magnon-photon modes, where $\omega_{d1}=\omega_{m}-\omega_{0}$ and
$\omega_{d2}=\omega_{m}+\omega_{0}$ correspond to the red and blue sidebands,
respectively.

##  III Unidirectional emission of a tunable squeezed microwave field

In this section, we will detail the procedure for unidirectionally creating a
tunable squeezed microwave field based on above introduced chiral cavity
magnonic system. The chirality stems from the selective coupling of the magnon
mode to one of the two degenerate rotating microwave modes with the same
polarization. Combining with the Floquet driving to engineer a desired
squeezing-type interaction, we can generate a chiral squeezed microwave
radiation under the assistance of a dissipative magnon mode and a waveguide.

###  III.1 The effective Hamiltonian for the generation of a chiral squeezed
state

We start to derive the effective Hamiltonian to selectively generate a chiral
squeezed state; that is, only one of the two cavity modes will be squeezed. To
this end, the Floquet driving to the magnon mode plays a key role. We define
the following rotating transformation:

$\displaystyle U(t)$ $\displaystyle=$
$\displaystyle\mathcal{T}\exp\\{-i\int_{0}^{t}[H_{0}+H_{d}(\tau)]d\tau\\}$ (4)
$\displaystyle=$
$\displaystyle\exp\\{-i(\omega_{m}m^{{\dagger}}m+\omega_{0}\sum_{\alpha=a,b}\alpha^{{\dagger}}\alpha)t$
$\displaystyle-i[\xi_{1}\sin(\omega_{d1}t)+\xi_{2}\sin(\omega_{d2}t+\theta)]m^{{\dagger}}m\\},$

where $\mathcal{T}$ is the time order operator and
$\xi_{j}=\Omega_{j}/\omega_{dj}(j=1,2)$. In the rotating frame with respect to
$U(t)$, the total Hamiltonian $H_{t}=H_{0}+H_{int}+H_{d}(t)$ of the system
will become

$\displaystyle H_{I}$ $\displaystyle=$ $\displaystyle
U^{{\dagger}}(t)H_{t}U(t)+i\frac{dU^{{\dagger}}(t)}{dt}U(t)$ (5)
$\displaystyle=$
$\displaystyle\sum_{\alpha=a,b}g_{\alpha}m^{{\dagger}}e^{i[\omega_{m}t+\xi_{1}\sin(\omega_{d1}t)+\xi_{2}\sin(\omega_{d2}t+\theta)]}$
$\displaystyle\times(\alpha
e^{-i\omega_{0}t}+\alpha^{{\dagger}}e^{i\omega_{0}t})+H.c.$

By using the Jacobi-Anger identity

$e^{ix\sin\phi}=\sum_{n=-\infty}^{n=\infty}J_{n}(x)e^{in\phi}$ (6)

with the $n$th Bessel functions of the first kind $J_{n}(x)$, the Hamiltonian
can be rewritten as

$\displaystyle H_{I}$ $\displaystyle=$
$\displaystyle\sum_{\alpha=a,b}g_{\alpha}m^{{\dagger}}e^{i\omega_{m}t}\sum_{n=-\infty}^{\infty}J_{n}(\xi_{1})e^{in\omega_{d1}t}\sum_{k=-\infty}^{\infty}J_{k}(\xi_{2})$
(7) $\displaystyle\times e^{ik(\omega_{d2}t+\theta)}(\alpha
e^{-i\omega_{0}t}+\alpha^{{\dagger}}e^{i\omega_{0}t})+H.c.$

To produce the desired magnon-photon coupling, as displayed in Fig. 2, we
choose the Floquet driving frequencies satisfying
$\omega_{d1}=\omega_{m}-\omega_{0}$ and $\omega_{d2}=\omega_{m}+\omega_{0}$,
which correspond to the red and blue sideband transitions, respectively.
Furthermore, under the condition
$g_{\alpha}\ll\\{\omega_{dj},\omega_{m},\omega_{0}\\}$, we can only keep the
resonant terms in Eq. (7), but safely discard those fast oscillating terms by
means of the rotating-wave approximation. Then, we can obtain the effective
Hamiltonian of the system

$H_{eff}=\sum_{\alpha=a,b}G_{\alpha}m^{{\dagger}}(\alpha+\varepsilon
e^{-i\theta}\alpha^{{\dagger}})+H.c.,$ (8)

where we have $G_{\alpha}=g_{\alpha}J_{-1}(\xi_{1})J_{0}(\xi_{2})$, and
$\varepsilon
e^{-i\theta}=J_{0}(\xi_{1})J_{-1}(\xi_{2})e^{-i\theta}/J_{-1}(\xi_{1})J_{0}(\xi_{2})$
is the ratio of the parametric-amplifier interaction to the Jaynes-Cummings-
type coupling. In what follows, we exclusively consider the situation of
$\left|\varepsilon\right|<1$, which ensures that Eq. (8) is stable. It is now
clear that $H_{eff}$ describes a squeezing-type interaction, and can be used
to produce the squeezed state. However, due to the chiral coupling, i.e.,
$G_{a}\neq 0$ and $G_{b}=0$, or $G_{b}\neq 0$ and $G_{a}=0$, we can only
generate a squeezed state in one of the two cavity modes.

By introducing a hybridized mode
$A=(G_{a}a+G_{b}b)/\sqrt{G_{a}^{2}+G_{b}^{2}}$, the Hamiltonian in Eq. (8) can
be rewritten as

$H_{eff}=\sqrt{G_{a}^{2}+G_{b}^{2}}[m^{{\dagger}}(A+\varepsilon
e^{-i\theta}A^{{\dagger}})+H.c.].$ (9)

To clearly reveal the mechanism for generating the chiral squeezed state, we
perform the unitary squeezing transformation
$H_{S}=S_{A}^{{\dagger}}(\zeta)H_{eff}S_{A}(\zeta)$ kun Xie  _et al._ (2020b),
where $S_{A}(\zeta)=\exp[(\zeta A^{2}-\zeta^{\ast}A^{{\dagger}2})/2]$
represents the single-mode squeezing operator with $\zeta=re^{i\theta}$. Here
$r$ is squeezing parameter defined by $\tanh r=\varepsilon$ and $\theta$
denotes the squeezing angle, both of which can be controlled on-demand by
adjusting the two-color Floquet driving field. In the squeezed frame, the
effective Hamiltonian (9) is transformed to the form

$H_{S}=\sqrt{\left(G_{a}^{2}+G_{b}^{2}\right)\left(1-\varepsilon^{2}\right)}(m^{{\dagger}}A+mA^{{\dagger}}).$
(10)

Obviously, $H_{S}$ describes a beam-splitter interaction. If we set $G_{a}\neq
0$ and $G_{b}=0$ ($G_{b}\neq 0$ and $G_{a}=0$) via the selective coupling
rule, the mode $A=a$ ($A=b$) can be cooled down to the vacuum state in the
squeezed picture via the dissipation of the magnon mode. Reversing the
squeezing transformation, the cavity mode $a$ ($b$) is actually steered into a
single-mode squeezed state. In this way, a chiral squeezed state is created.
By further considering the microwave cavity coupled to a waveguide, the
squeezed output field can be emitted along the right (left) direction of the
waveguide. This is the basic idea for the unidirectional emission of a
squeezed source.

###  III.2 Unidirectional squeezing emission

We now study the unidirectional squeezing emission by considering the coupling
of the microwave cavity coupled to a transmission line that acts the role of a
waveguide. According to the Hamiltonian (8) and the standard input–output
theory, the general quantum Langevin equations of the system is given by

$\displaystyle\dot{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{2}\alpha-iG_{\alpha}(m+\varepsilon
e^{-i\theta}m^{{\dagger}})+\sqrt{\kappa_{ex}}\alpha_{in,ex}$ (11a)
$\displaystyle+\sqrt{\kappa_{0}}\alpha_{in,0},$
$\dot{m}=-\frac{\gamma_{m}}{2}m-i\sum_{\alpha=a,b}G_{\alpha}(\alpha+\varepsilon
e^{-i\theta}\alpha^{{\dagger}})+\sqrt{\gamma_{m}}m_{in}.$ (11b) Without loss
of generality, the cavity modes $a$ and $b$ are assumed to have the same
damping rate $\kappa=\kappa_{0}+\kappa_{ex}$, where $\kappa_{0}$ is the
intrinsic decay rate, and $\kappa_{ex}$ denotes the external coupling between
the cavity mode and the waveguide. In addition, $\alpha_{in,0}$ and
$\alpha_{in,ex}$ $(\alpha=a,b)$ represent the noise operators. $m_{in}$ is the
noise operator of the magnon mode, and $\gamma_{m}$ is the associated damping
rate. These noise operators obey the following correlation relations
$\left\langle\alpha_{in,ex}(t)\alpha_{in,ex}^{{\dagger}}(t^{\prime})\right\rangle=\delta(t-t^{\prime}),$
(12a)
$\left\langle\alpha_{in,0}(t)\alpha_{in,0}^{{\dagger}}(t^{\prime})\right\rangle=\delta(t-t^{\prime}),$
(12b) $\left\langle
m_{in}(t)m_{in}^{{\dagger}}(t^{\prime})\right\rangle=\delta(t-t^{\prime}),$
(12c) where we have neglected the thermal effects. Because, at the
experimental working temperature $T=20$ mK, the average thermal excitation
number of a boson mode with resonance frequency $6.5$ GHz is about $10^{-7}$,
which is thus negligible.

By introducing the Fourier transformation
$o(t)=\int_{-\infty}^{+\infty}o(\omega)e^{-i\omega t}d\omega/2\pi$ for an
arbitrary operator $o$, we can rewrite the Eqs. (11a) and (11b) in the
frequency domain as

$\displaystyle\alpha(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{\frac{\kappa}{2}-i\omega}\\{-iG_{\alpha}[m(\omega)+\varepsilon
e^{-i\theta}m^{{\dagger}}(-\omega)]$ (13a)
$\displaystyle+\sqrt{\kappa_{ex}}\alpha_{in,ex}(\omega)+\sqrt{\kappa_{0}}\alpha_{in,0}(\omega)\\},$
$\displaystyle m(\omega)$ $\displaystyle=$
$\displaystyle\frac{1}{\frac{\gamma_{m}}{2}-i\omega}\\{-i\sum_{\alpha=a,b}G_{\alpha}[\alpha(\omega)$
(13b) $\displaystyle+\varepsilon
e^{-i\theta}\alpha^{{\dagger}}(-\omega)]+\sqrt{\gamma_{m}}m_{in}(\omega)\\}.$
Correspondingly, the correlation functions of Eqs. (12a)-(12c) in the
frequency domain yield
$\left\langle\alpha_{in,ex}(\omega)\alpha_{in,ex}^{{\dagger}}(-\omega^{\prime})\right\rangle=2\pi\delta(\omega+\omega^{\prime}),$
(14a)
$\left\langle\alpha_{in,0}(\omega)\alpha_{in,0}^{{\dagger}}(-\omega^{\prime})\right\rangle=2\pi\delta(\omega+\omega^{\prime}),$
(14b) $\left\langle
m_{in}(\omega)m_{in}^{{\dagger}}(-\omega^{\prime})\right\rangle=2\pi\delta(\omega+\omega^{\prime}).$
(14c) Our interest is the output fields of the cavity. Hence, in terms of the
standard input-output relation
$\alpha_{out,ex}=\sqrt{\kappa_{ex}}\alpha-\alpha_{in,ex},$ (15)

we can derive out the frequency components of the output field operators

$\displaystyle a_{out,ex}(\omega)$ $\displaystyle=$
$\displaystyle[\frac{N_{b}(\omega)\kappa_{ex}}{D(\omega)}-1]a_{in,ex}(\omega)+\frac{1}{D(\omega)}\\{N_{b}(\omega)\sqrt{\kappa_{ex}\kappa_{0}}a_{in,0}(\omega)+8G_{a}G_{b}(1-\varepsilon^{2})$
(16a)
$\displaystyle\times[\kappa_{ex}b_{in,ex}(\omega)+\sqrt{\kappa_{ex}\kappa_{0}}b_{in,0}(\omega)]+4iG_{a}\kappa_{-}\sqrt{\kappa_{ex}\gamma_{m}}[m_{in}(\omega)+\varepsilon
e^{-i\theta}m_{in}^{{\dagger}}(-\omega)]\\},$ $\displaystyle
b_{out,ex}(\omega)$ $\displaystyle=$
$\displaystyle[\frac{N_{a}(\omega)\kappa_{ex}}{D(\omega)}-1]b_{in,ex}(\omega)+\frac{1}{D(\omega)}\\{N_{a}(\omega)\sqrt{\kappa_{ex}\kappa_{0}}b_{in,0}(\omega)+8G_{a}G_{b}(1-\varepsilon^{2})$
(16b)
$\displaystyle\times[\kappa_{ex}a_{in,ex}(\omega)+\sqrt{\kappa_{ex}\kappa_{0}}a_{in,0}(\omega)]+4iG_{b}\kappa_{-}\sqrt{\kappa_{ex}\gamma_{m}}[m_{in}(\omega)+\varepsilon
e^{-i\theta}m_{in}^{{\dagger}}(-\omega)]\\}$

with $\kappa_{-}=\kappa-2i\omega$, $\gamma_{-}=\gamma_{m}-2i\omega$,
$N_{\alpha}(\omega)=8G_{\alpha}^{2}(\varepsilon^{2}-1)-2\kappa_{-}\gamma_{-}$
and
$D(\omega)=4\kappa_{-}(\varepsilon^{2}-1)(G_{a}^{2}+G_{b}^{2})-\kappa_{-}^{2}\gamma_{-}$.

Now, the spectrum of the output microwave field for the $\alpha$ mode is given
by

$S_{\alpha,out}(\omega)=\frac{1}{4\pi}\int_{-\infty}^{+\infty}d\omega^{\prime}e^{-i(\omega+\omega^{\prime})t}[\left\langle
X_{\alpha,out}^{\theta}(\omega)X_{\alpha,out}^{\theta}(\omega^{\prime})\right\rangle+\left\langle
X_{\alpha,out}^{\theta}(\omega^{\prime})X_{\alpha,out}^{\theta}(\omega)\right\rangle],$
(17)

where
$X_{\alpha,out}^{\theta}(\omega)=\alpha_{out,ex}(\omega)e^{\frac{i\theta}{2}}+\alpha_{out,ex}^{{\dagger}}(-\omega)e^{-\frac{i\theta}{2}}$
is the associated quadrature operator of the output fields. Substituting Eqs.
(16a) and (16b) into Eq. (17), we can obtain the squeezing spectra of the
output fields

$\displaystyle S_{a,out}(\omega)$ $\displaystyle=$
$\displaystyle\left|\frac{N_{b}(\omega)\kappa_{ex}}{D(\omega)}-1\right|^{2}+\left|\frac{N_{b}(\omega)\sqrt{\kappa_{ex}\kappa_{0}}}{D(\omega)}\right|^{2}+\left|\frac{8G_{a}G_{b}(1-\varepsilon^{2})\kappa_{ex}}{D(\omega)}\right|^{2}$
(18a)
$\displaystyle+\left|\frac{8G_{a}G_{b}(1-\varepsilon^{2})\sqrt{\kappa_{ex}\kappa_{0}}}{D(\omega)}\right|^{2}+\left|\frac{4G_{a}\kappa_{-}(1-\varepsilon)\sqrt{\kappa_{ex}\gamma_{m}}}{D(\omega)}\right|^{2},$
$\displaystyle S_{b,out}(\omega)$ $\displaystyle=$
$\displaystyle\left|\frac{N_{a}(\omega)\kappa_{ex}}{D(\omega)}-1\right|^{2}+\left|\frac{N_{a}(\omega)\sqrt{\kappa_{ex}\kappa_{0}}}{D(\omega)}\right|^{2}+\left|\frac{8G_{a}G_{b}(1-\varepsilon^{2})\kappa_{ex}}{D(\omega)}\right|^{2}$
(18b)
$\displaystyle+\left|\frac{8G_{a}G_{b}(1-\varepsilon^{2})\sqrt{\kappa_{ex}\kappa_{0}}}{D(\omega)}\right|^{2}+\left|\frac{4G_{b}\kappa_{-}(1-\varepsilon)\sqrt{\kappa_{ex}\gamma_{m}}}{D(\omega)}\right|^{2}.$

To quantify the amount of squeezing of the output fields, we now define the
quantum noise reduction
$F_{\alpha,out}(\omega)=-10\log_{10}[S_{\alpha,out}(\omega)/S_{\alpha,out}^{vac}(\omega)]$
in dB units with $S_{\alpha,out}^{vac}(\omega)=1$ being the squeezing spectrum
of the vacuum state. Note that if the output field is squeezed, the quantum
noise reduction $F_{\alpha,out}(\omega)$ is larger than zero. Clearly, to
achieve a perfect one-way squeezing emitter, the condition
$F_{a,out}(\omega)>0$ and $F_{b,out}(\omega)=0$ ($F_{b,out}(\omega)>0$ and
$F_{a,out}(\omega)=0$) should be satisfied, which indicates that the squeezed
field is only emitted along the right (left) direction of the waveguide.

To well grasp the main result of this work, we start our analysis with the
ideal chiral coupling, i.e., $g_{a}\neq 0$ and $g_{b}=0$. Then, the output
squeezing spectra in Eqs. (18a) and (18b) yield

$\displaystyle S_{a,out}(\omega)$ $\displaystyle=$
$\displaystyle\left|\frac{\kappa_{-}\gamma_{-}(2\kappa_{ex}-\kappa_{-})-4\kappa_{-}(1-\varepsilon^{2})G_{a}^{2}}{\kappa_{-}^{2}\gamma_{-}+4\kappa_{-}(1-\varepsilon^{2})G_{a}^{2}}\right|^{2}$
(19a)
$\displaystyle+\left|\frac{2\kappa_{-}\gamma_{-}\sqrt{\kappa_{ex}\kappa_{0}}}{\kappa_{-}^{2}\gamma_{-}+4\kappa_{-}(1-\varepsilon^{2})G_{a}^{2}}\right|^{2}$
$\displaystyle+\left|\frac{4G_{a}\kappa_{-}(\varepsilon-1)\sqrt{\kappa_{ex}\gamma_{m}}}{\kappa_{-}^{2}\gamma_{-}+4\kappa_{-}(1-\varepsilon^{2})G_{a}^{2}}\right|^{2},$
$\displaystyle S_{b,out}(\omega)$ $\displaystyle=$ $\displaystyle 1,$ (19b)

Figure 3: (Color online) The squeezing spectra $F_{a,out}(\omega)$ [(a)] and
$F_{b,out}(\omega)$ [(b)] versus the dimensionless frequency
$\omega/\gamma_{m}$ under the different coupling ratios $g_{b}/g_{a}=0$,
$0.05$, $0.1$. The related parameters are chosen as $G_{a}/\gamma_{m}=2.5$,
$\varepsilon=0.95$, $\kappa_{0}/\gamma_{m}=0.05$, $\kappa_{ex}/\gamma_{m}=2.5$
and $\gamma_{m}/2\pi=2$ MHz.

It is now clear that the spectra $S_{a,out}(\omega)<1$ and
$S_{b,out}(\omega)=1$ can be satisfied under the situation of perfect
chirality, which is equivalent to $F_{a,out}(\omega)>0$ and
$F_{b,out}(\omega)=0$. So, a desired one-way squeezed source can be created;
that is, the squeezed microwave field can only be emitted in one direction. To
exhibit the unidirectional squeezing emission, we choose $\varepsilon=0.95$,
and show $F_{a,out}(\omega)$ and $F_{b,out}(\omega)$ versus the dimensionless
frequency $\omega/\gamma_{m}$ in Fig. 3. It is observed that
$F_{a,out}(\omega)$ is larger than zero within a large frequency range, where
the largest value of $F_{a,out}(\omega)$ is about $13.45$ dB at the central
resonance frequency. On the contrary, $F_{b,out}(\omega)$ is zero over the
whole frequency range. This means that we can obtain a squeezed output field
in the right side of the waveguide, but there is no squeezing of the output
field in the left side of the waveguide. Thus, the unidirectional emission of
a squeezed source can be achieved. Notably, we can readily switch the emission
direction of the squeezed field by reversing the direction of the external
biased magnetic field.

In practice, our system will inevitably suffer from the non-ideal chiral
coupling, i.e., $g_{a}\neq 0$ and $g_{b}\neq 0$, such that that the completely
one-way squeezing emission will be spoiled. In the presence of non-ideal
chiral coupling, we know from Eq. (10) that the hybridized mode $A$ will be
steered into a squeezed state via the dissipation of the magnon mode $m$,
i.e., both the modes $a$ and $b$ are thereby squeezed. As a result, the
squeezed fields will be emitted in both directions of the waveguide. With the
increase of $g_{b}$, the associated output squeezing $F_{a,out}(\omega)$ will
be reduced, while the output squeezing $F_{b,out}(\omega)$ is gradually
increased. Nonetheless, for $g_{b}/g_{a}=0.1$, the value of
$F_{a,out}(\omega)$ only has a negligible reduction, and the maximal value of
$F_{b,out}(\omega)$ is about $0.04$ dB [See Fig. 3]. So, our scheme is robust
against the imperfect chiral coupling, indicating that we can still implement
a high-performance unidirectional squeezing emitter.

Figure 4: (Color online) The squeezing degree $F_{a,out}(0)$ [(a)] and
$F_{b,out}(0)$ [(b)] versus the parameter $\varepsilon$ under the different
coupling ratios $g_{b}/g_{a}=0$, $0.05$, $0.1$. The other parameters are the
same as in Fig. 3.

Finally, we emphasize here that the proposed one-way emitter with a tunable
amount of squeezing can also be generated in our scheme. Fig. 4 displays the
$F_{a,out}(0)$ and $F_{b,out}(0)$ versus the parameter $\varepsilon$. We can
see that $F_{a,out}(0)$ behaves as a nonmonotonic function of $\varepsilon$,
implying that a tunable squeezed output field can be created by tuning the
parameter $\varepsilon$. This can be implemented by adjusting the Floquet
driving parameters $\Omega_{j}(j=1,2)$.

As presented in Fig. 4(a), there exists an optimum value of $\varepsilon$,
where a maximal amount of output squeezing is produced. To illustrate this
point, we recall that the intra-cavity squeezing depends on the parameter
$\varepsilon$. Meanwhile, according to the standard input-output relation, the
intra-cavity squeezing determines the output one. So, the output squeezing is
also determined by the parameter $\varepsilon$. Based on Eqs. (9) and (10), we
know that the dissipative magnon mode can be used to cool the cavity mode down
to a squeezed state. So, it seems like that the intra-cavity squeezing will be
continuously increased with the increase of $\varepsilon$. However, as the
parameter $\varepsilon$ increasing, the effective magnon-photon coupling,
which plays the role of cooling, is gradually decreased in the squeezed
representation [See Eq. (10)]. Besides, the cavity vacuum noise is transformed
to the effective thermal noise, which is also be amplified. As a result, the
cavity can not approach a squeezed vacuum state, which is actually a thermal
squeezed state. Therefore, the competition of these two processes leads to an
optimum value of $\varepsilon$ that can generate a peak of output squeezing.

##  IV Conclusion

In summary, we have presented an approach for the unidirectional emission of a
tunable squeezed microwave field in a chiral cavity magnonic system of a YIG
sphere and a torus-shaped cavity. The chirality comes from the selective
coupling between the Kittel mode to one of the two counter-propagating
microwave modes with the same polarization. Under a bichromatic Floquet
driving field to induce sidebands, the unidirectional emission of a tunable
squeezed microwave source can be generated with the help of a dissipative
magnon mode and a waveguide. Moreover, the emission direction can be
conveniently changed by varying the direction of the external applied magnetic
field. With the rapid development of hybrid cavity magnonic system, our scheme
is expected to be implemented in a realistic experiment, which can stimulate a
wide range of quantum technological applications.

## Acknowledgement

This work was supported by the National Natural Science Foundation of China
(Grant Nos. 11704306 and 12074307).

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