# Realization of the unidirectional amplification in a cavity magnonic system

Zi-Yuan Wang  Interdisciplinary Center of Quantum Information, State Key
Laboratory of Extreme Photonics and Instrumentation, and Zhejiang Province Key
Laboratory of Quantum Technology and Device, School of Physics, Zhejiang
University, Hangzhou 310027, China  Jie Qian<EMAIL_ADDRESS>Interdisciplinary Center of Quantum Information, State Key Laboratory of
Extreme Photonics and Instrumentation, and Zhejiang Province Key Laboratory of
Quantum Technology and Device, School of Physics, Zhejiang University,
Hangzhou 310027, China  Yi-Pu Wang<EMAIL_ADDRESS>Interdisciplinary
Center of Quantum Information, State Key Laboratory of Extreme Photonics and
Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology
and Device, School of Physics, Zhejiang University, Hangzhou 310027, China
Jie Li  Interdisciplinary Center of Quantum Information, State Key Laboratory
of Extreme Photonics and Instrumentation, and Zhejiang Province Key Laboratory
of Quantum Technology and Device, School of Physics, Zhejiang University,
Hangzhou 310027, China  J. Q. You  Interdisciplinary Center of Quantum
Information, State Key Laboratory of Extreme Photonics and Instrumentation,
and Zhejiang Province Key Laboratory of Quantum Technology and Device, School
of Physics, Zhejiang University, Hangzhou 310027, China

###### Abstract

We experimentally demonstrate the nonreciprocal microwave amplification using
a cavity magnonic system, consisting of a passive cavity (i.e., the split-ring
resonator), an active feedback circuit integrated with an amplifier, and a
ferromagnetic spin ensemble (i.e., a yttrium-iron-garnet sphere). Combining
the amplification provided by the active circuit and the nonreciprocity
supported by the cavity magnonics, we implement a nonreciprocal amplifier with
the functions of both unidirectional amplification and reverse isolation. The
microwave signal is amplified by 11.5 dB in the forward propagating direction
and attenuated in the reverse direction by -34.7 dB, giving an isolation ratio
of 46.2 dB. Such a unidirectional amplifier can be readily employed in quantum
technologies, where the device can simultaneously amplify the weak signal
output by the quantum system and isolate the sensitive quantum system from the
backscattered external noise. Also, it is promising to explore more functions
and applications using a cavity magnonic system with real gain.

††preprint: AIP/123-QED

Nonreciprocal devices, which exhibit the characteristic of unidirectionality,
can play a crucial role in the information technology. To realize the
nonreciprocity, various methods have been proposed, such as the magneto-
optical Faraday rotation Wolfe  _et al._ (1995); Wang and Fan (2005);
Belotelov, Doskolovich, and Zvezdin (2007); Bi  _et al._ (2011); Chin  _et
al._ (2013), nonlinearity Christodoulides and Joseph (1988); Shi, Yu, and Fan
(2015); Fan  _et al._ (2012); Wang  _et al._ (2012); Huang  _et al._ (2018),
spatial-temporal modulation Hwang, Yun, and Kim (1997); Wang  _et al._ (2013);
Doerr, Dupuis, and Zhang (2011); Lira  _et al._ (2012), and reservoir
engineering Metelmann and Clerk (2015); Fang  _et al._ (2017). The main
function of the nonreciprocal devices is to both transmit information in the
desired direction and isolate the backscattered noise. The typical feature is
that the signal is transmitted in one direction with some insertion loss, and
the reverse direction propagation has a large attenuation. On the other hand,
in the process of signal transmission, the signal will inevitably attenuate
continuously when increasing the propagation distance, so the gain medium and
amplification device Xiao  _et al._ (2010); Massel  _et al._ (2011); Fermann
_et al._ (2000); De Leon and Berini (2010); Liu  _et al._ (2007); Khurgin and
Sun (2012); Choksi  _et al._ (2022); Stehlik  _et al._ (2016); Frolov  _et
al._ (1999); Yao  _et al._ (2017, 2023) are also a critical component in the
information technology.

Under normal circumstances, the nonreciprocal and gain devices are independent
of each other in the information network, each performing its own function
separately. With the increasingly high requirements for integrating signal
processing devices, whether both nonreciprocity and signal amplification can
be realized in the same device has become an urgent demand. Different systems
were proposed to realize the unidirectional amplification, including the
atomic system Lin  _et al._ (2019), whispering-gallery microcavities Peng  _et
al._ (2014), Josephson circuit Abdo  _et al._ (2013, 2014), microwave
optomechanical device Malz  _et al._ (2018); de Lépinay  _et al._ (2019),
magnonic system Zhao  _et al._ (2022), and others Koutserimpas and Fleury
(2018); Kamal and Metelmann (2017); Jiang  _et al._ (2018); Galiffi, Huidobro,
and Pendry (2019); Song  _et al._ (2019); Taravati and Eleftheriades (2021).

Recently, cavity magnonics has emerged as a new research frontier Rameshti
_et al._ (2022). Based on this platform, various applications in, e.g., memory
Zhang  _et al._ (2015); Shen  _et al._ (2021), cavity optomagnonics for
quantum transduction Osada  _et al._ (2016); Tabuchi  _et al._ (2015);
Hisatomi  _et al._ (2016); Haigh  _et al._ (2015, 2016); Zhang  _et al._
(2016); Kusminskiy, Tang, and Marquardt (2016); Graf  _et al._ (2018); Sharma
_et al._ (2019); Wu  _et al._ (2021); Lauk  _et al._ (2020), and magnon
sensing at the quantum level Lachance-Quirion  _et al._ (2020); Wolski  _et
al._ (2020); Xu  _et al._ (2023a, b) have been reported. In the cavity
magnonic system, a magnon mode (i.e., the Kittel mode) in a ferrite, e.g., the
yttrium-iron-garnet (YIG) sphere, strongly couples to the microwave photons in
the cavity. As a mature commercial material, the YIG owns significant
advantages such as the high spin density Huebl  _et al._ (2013); Zhang  _et
al._ (2014); Tabuchi  _et al._ (2014), flexible adjustment of resonant
frequency Chumak  _et al._ (2015); Wang and Hu (2020), and low damping rate.
Very recently, novel properties of the nonreciprocity and unidirectional
invisibility have also been observed in the cavity magnonic system Wang  _et
al._ (2019), which arise from the interference between the coherent and
dissipative magnon-photon interactions. By combining the easy-to-tune ferrite
and the specially designed cavities, isolators with both nearly infinite
isolation ratios and adjustable working frequencies become implementable Wang
_et al._ (2019); Qian  _et al._ (2020); Zhang  _et al._ (2020); Zhao  _et al._
(2020); Zhang  _et al._ (2021); Kim  _et al._ (2022).

In this work, we construct a nonreciprocal amplifier based on the cavity
magnonic system, consisting of a passive cavity made by the split-ring
resonator (SRR), an active circuit with an embedded amplifier, and a YIG
sphere. These components are all connected to a strip-line waveguide that
supports the traveling photon mode. The active circuit is employed to
compensate the dissipation of the passive cavity until the signal
amplification occurs. The coupling strength between the magnon and cavity
modes can be modified by adjusting the position of the YIG sphere with respect
to the passive cavity. Through the comprehensive and synergistic regulation of
the active circuit and magnon-photon coupling strength, we experimentally
realize the microwave nonreciprocal amplification with a high isolation ratio.
In the optimal conditions, the device exhibits a rightward propagating
amplificition of 11.5 dB and a reverse propagating attenuation of -34.7 dB.
Such devices have potential applications in quantum networks and repeaters
Reiserer and Rempe (2015); Wehner, Elkouss, and Hanson (2018); Zhong  _et al._
(2021); Nguyen  _et al._ (2019); Li  _et al._ (2021). They may also be
promising in protecting the sensitive quantum system from noises associated
with the read-out electronics and amplifying the weak signal leaking out of
the quantum nodes.

Figure 1: (a) Schematic diagram of the experimental setup, consisting of a YIG
sphere glued at the end of a displacement cantilever, a passive cavity made by
the split-ring resonator (SRR), and an active circuit with an embedded
amplifier. The DC voltage (V) applied on the amplifier is used to sustain the
gain in our device. (b) Schematic diagram of the cooperative dissipation as
well as the complex coupling between the cavity mode and the magnon mode.

The nonreciprocal amplification device is depicted in Fig. 1(a), where the
microwave circuit consists of a strip-line waveguide, a passive cavity made by
the split-ring resonator (SRR), and an active circuit with an embedded
amplifier. For the passive cavity, a SRR wih parameters $g=18$ mm, $b=5.2$ mm,
$a=25$ mm, and $w=1.5$ mm is side-coupled to the strip-line waveguide with a
distance of $d=0.2$ mm, and the resonant frequency of the SRR is designed to
be $\omega_{\rm{c}}/2\pi=3.03$ GHz to match the optimal working frequency of
the active circuit Zarifi, Thundat, and Daneshmand (2015). The active circuit
is capacitively coupled to the SRR via a gap of $t=0.8$ mm. By introducing a
gain through the amplifier, the active circuit can compensate the loss of the
passive cavity. In the experiment, we apply a DC voltage on the amplifier to
continuously adjust the gain provided by the active circuit. The strip-line
waveguide has a width of $v=2.53$ mm for 50 $\Omega$ impedance matching. Both
the cavity and waveguide are fabricated on a F4B substrate. A 1 mm-diameter
YIG sphere is glued at the end of a displacement cantilever, and the relative
position between the YIG sphere and the planar microwave circuit can be finely
adjusted by a three-dimensional (3D) motor-controlled robot arm. An external
magnetic field is applied perpendicularly to the planar device, which is used
to both saturate the magnetization of the YIG sphere and tune the frequency of
the magnon mode. To characterize this device, a vector network analyzer (VNA)
with a signal power of -20 dBm is used to measure the transmission spectrum
$|S_{12(21)}|$, where the subscript 21 (12) indicates that the microwave field
is loaded to the port 1 (2) and propagates to the port 2 (1) (see
Supplementary Material Sec. I).

Our device is schematically shown in Fig. 1(b), where the cavity mode and the
magnon mode dissipate cooperatively to the traveling wave-type dissipative
reservoir (waveguide). Here, $\alpha$ and $\beta$ are the intrinsic damping
rates of the magnon mode and the passive cavity mode, respectively, and
$\beta$ can be reduced by the active circuit, or even turned to be negative.
The external damping rates of the magnon mode and the cavity mode, i.e.,
$\gamma$ and $\kappa$, reflect their interactions with the waveguide traveling
photon modes. The cooperative dissipation gives rise to the dissipative
coupling between the cavity mode and the magnon mode Metelmann and Clerk
(2015); Fang  _et al._ (2017); Wang  _et al._ (2019). Meanwhile, the cavity
mode and the magnon mode can directly interact with each other via the spatial
mode overlapping, which is attributed to the coherent magnon-photon coupling.

When the coupling mechanism of our system is dominated by both coherent and
dissipative couplings, the interference between them has a significant
modulation on the transmission spectrum of the system. Due to the asymmetric
location of the YIG sphere with respect to the central line of the device, the
phase difference between the coherent and dissipative couplings will be
different when the microwave is loaded to the ports 1 and 2. Under the
circumstances, the overall coupling between the cavity mode and the magnon
mode can become direction-dependent. In other words, the nonreciprocity stems
from the inconsistent interference effect between the coherent and dissipative
couplings when the microwave transmission direction is reversed Wang  _et al._
(2019).

We define $\mathbb{C}_{R(L)}=M+iN$ to characterize the direction-dependent
complex coupling, where $M$ denotes the coherent coupling, $iN$ corresponds to
the dissipative coupling, with $i$ indicating the non-Hermiticity of the
dissipative coupling, and the subscript $R(L)$ represents the case when the
signal is loaded to the port 1 (2). When the traveling wave propagates in the
opposite direction, $\mathbb{C}_{R}\neq\mathbb{C}_{L}$, so the nonreciprocity
emerges. Under the rotating-wave approximation, the Hamiltonian of the non-
Hermitian system can be written as

$H/\hbar=\tilde{\omega}_{\rm{c}}a^{{\dagger}}a+\tilde{\omega}_{\rm{m}}b^{{\dagger}}b+\mathbb{C}_{R(L)}(a^{{\dagger}}b+b^{{\dagger}}a),$
(1)

where $\tilde{\omega}_{\rm{c}}=\omega_{\rm{c}}-i(\beta+\kappa)$ and
$\tilde{\omega}_{\rm{m}}=\omega_{\rm{m}}-i(\alpha+\gamma)$ are the complex
frequencies of the cavity mode and the magnon mode, respectively, while
$a^{{\dagger}}$ ($a$) and $b^{{\dagger}}$ ($b$) are the creation
(annihilation) operators of the cavity mode and the magnon mode, respectively.

Figure 2: (a) The measured transmission spectra $|S_{\rm{T}}|$ versus the
amplifier bias voltage $V$. The arrow labels the critical point, where the
intrinsic dissipation is exactly compensated by the active circuit. (b) and
(c) The effective intrinsic damping rate $\beta_{\rm{e}}$ and the effective
external damping rate $\kappa_{\rm{e}}$ of the cavity mode versus the
amplifier bias voltage $V$. In the blue region of (b), the cavity is lossy,
while in the red region, the intrinsic loss of the cavity mode is compensated.
Figure 3: (a) Schematic diagram of three different positions (i.e., positions
1, 2, and 3), at which the YIG sphere is placed, respectively, in the
experiment. (b) Complex coupling strength obtained when the YIG sphere is
placed at these three positions, respectively. (c)-(e) Measured rightward
($|S_{21}|$) and leftward ($|S_{12}|$) transmission mappings versus the field
detuning $\Delta_{\rm{m}}=\omega-\omega_{\rm{m}}$ and the probe frequency
detuning $\Delta_{\rm{c}}=\omega-\omega_{\rm{c}}$, when the YIG sphere is
placed at positions 1, 2, and 3, respectively. The measured transmission
spectra $|S_{21}|$ and $|S_{12}|$ at the arrow-marked fields in (d) are
plotted as the inset panel in (b), which shows a clear feature of
nonreciprocal transmission.

Below we first characterize the loss compensation of the active circuit to the
passive cavity, and find out the relationship between the bias voltage of the
amplifier and the compensated dissipation of the passive cavity. Figure 2(a)
shows the response of the cavity mode to the bias voltage applied on the
amplifier, where the measured transmission spectra $|S_{\rm{T}}|$ at a series
of bias voltages are plotted. When the voltage is zero, $|S_{\rm{T}}|<0$, and
the spectrum shows a dip at the resonant frequency of the cavity mode. With
the increase of the amplifier voltage, the amplitude of $|S_{\rm{T}}|$
decreases until an ultra-sharp dip ($\sim$55 dB) appears. Then, it gradually
increases and the spectrum turns into a resonance peak with $|S_{\rm{T}}|>0$.
This process is accompanied with the intrinsic damping rate of the cavity mode
compensated to be zero and then negative. The transmission coefficient of the
side-coupled cavity mode can be described by the following equation:

$S_{\rm{T}}=1+\frac{\kappa_{\rm{e}}}{i(\omega-\omega_{\rm{c}})-(\kappa_{\rm{e}}+\beta_{\rm{e}})},$
(2)

where $\kappa_{\rm{e}}=\kappa\cdot\eta$ and $\beta_{\rm{e}}=\beta-g$ are the
effective external and intrinsic damping rates of the cavity mode,
respectively. The parameter $\eta$ indicates that the external damping rate of
the cavity mode is affected by the active circuit when the bias voltage is
applied. The parameter $g$ is introduced as the gain coefficient, which is
determined by the amplifier bias voltage and is equal to the linewidth
compensation of the cavity mode.

Figure 4: (a)-(c) Measured isolation ratio versus $\Delta_{\rm{m}}$ and
$\Delta_{\rm{c}}$, where the amplifier bias voltage is set to be 0, 4.8 and 5
V, respectively. (d)-(f) Experimental (circles) and theoretical (solid curve)
results of the $|S_{21}|$ and $|S_{12}|$ spectra at the field detuning marked
by the gray plane in (a), (b) and (c), respectively. When $|S_{\rm{T}}|>0$,
the propagating microwave signal is amplified. On the contrary, when
$|S_{\rm{T}}|<0$, the microwave signal is attenuated. (d) A 7.6 dB isolation
ratio is realized without amplification. (e) A 8.7 dB isolation ratio is
realized with rightward 5 dB amplification and leftward 3.7 dB attenuation.
(f) A 46.2 dB isolation ratio is realized with rightward 11.5 dB amplification
and leftward 34.7 dB attenuation.

The bare cavity mode (corresponding to zero bias voltage) has an external
damping rate of $\kappa/2\pi=48$ MHz, and an intrinsic damping rate of
$\beta/2\pi=15.5$ MHz. By continuously tuning the bias voltage, the fitted
$\beta_{\rm{e}}$ and $\kappa_{\rm{e}}$ versus the bias voltage are plotted as
the green and blue circles in Fig. 2(b) and Fig. 2(c), respectively. When the
amplifier voltage is small, the intrinsic dissipation of the cavity mode is
not fully compensated ($g<\beta$, $\beta_{\rm{e}}>0$). While the amplifier
voltage reaches 2.75 V, the intrinsic damping rate of the cavity mode is
almost exactly compensated by the active circuit ($g=\beta$,
$\beta_{\rm{e}}=0$), resulting in an extremely abrupt dip in $|S_{T}|$, as
marked by an arrow in Fig. 2(a). With the increase of the amplifier bias
voltage, the gain coefficient exceeds the intrinsic damping rate, giving rise
to an effective negative intrinsic damping rate of the cavity mode ($g>\beta$,
$\beta_{\rm{e}}<0$). It is worth noting that the amplifier also affects the
external damping rate of the cavity mode, as shown in Fig. 2(c). However, its
effect on the microwave transmission is not significant, and the transmitted
signal appears as either amplification or attenuation, depending primarily on
the compensation of the intrinsic dissipation (see Supplementary Material Sec.
II for details). By introducing the active circuit, the transmission amplitude
becomes amplified around the cavity mode frequency, and the loss-compensated
cavity is a crucial element for the subsequent realization of the
nonreciprocal amplification.

With the amplification mechanism in hand, the next step is to construct the
nonreciprocity of the system. Additionally, the YIG sphere is attached to the
cavity. The magnon mode sustained by the YIG sphere is coherently and
dissipatively coupled to the cavity photon mode. The nonreciprocity originates
from the interference between the coherent and dissipative couplings, and the
phase of the interference term is related to the direction of the microwave
propagation, which leads to a direction-dependent complex coupling
$\mathbb{C}_{R(L)}$. The magnon-photon coupling induces hybridized modes with
complex frequencies

$\begin{split}\tilde{\omega}_{\pm}=&\frac{1}{2}[\omega_{\rm{c}}+\omega_{\rm{m}}-i(\beta_{\rm{e}}+\alpha+\kappa_{\rm{e}}+\gamma)\pm\\\
&\sqrt{\left[(\omega_{\rm{c}}-\omega_{\rm{m}})-i(\beta_{\rm{e}}-\alpha+\kappa_{\rm{e}}+\gamma)\right]^{2}+4\mathbb{C}^{2}}].\end{split}$
(3)

Using the input-output theory (see Supplementary Material Sec. III for
details), we obtain the transmission spectrum of the coupled system as

$S_{\rm{21(12)}}=1+\frac{\kappa_{\rm{e}}}{i(\omega-\omega_{\rm{c}})-(\kappa_{\rm{e}}+\beta_{\rm{e}})+\frac{\mathbb{C}^{2}_{R(L)}}{i(\omega-\omega_{\rm{m}})-(\alpha+\gamma)}},$
(4)

where the complex coupling strength $\mathbb{C}_{R(L)}=M+iN$ between the
cavity and magnon modes is direction-dependent as stated above, resulting in
nonreciprocal transmission spectra, i.e., $S_{21}\neq S_{12}$. It is necessary
to adjust the ratio between the coherent and dissipative coupling strengths
for an optimal nonreciprocal transmission. Experimentally, we modify the
complex coupling strength $\mathbb{C}_{R(L)}$ by fixing the height of the YIG
sphere relative to the planar device at a distance of $D=1$ mm and then
precisely controlling its position in the x-y plane.

To clearly show the existing region of the nonreciprocal transmission exists
and reveal its origin in our device, here we place the YIG sphere at three
different positions relative to the cavity, as shown in Fig. 3(a). We depict
the corresponding $|S_{21}|$ and $|S_{12}|$ mappings versus the field detuning
$\Delta_{\rm{m}}=\omega-\omega_{\rm{m}}$ and the probe frequency detuning
$\Delta_{\rm{c}}=\omega-\omega_{\rm{c}}$ in Figs. 3(c)-(e), respectively. When
the YIG sphere is at position 1, the two hybridized modes shown in Fig. 3(c)
exhibit level repulsion, a characteristic of coherent magnon-photon coupling
Huebl  _et al._ (2013); Zhang  _et al._ (2014); Tabuchi  _et al._ (2014).
Using Eq. (3) to fit the experimental results, we find that at position 1 the
real part of the complex coupling strength $M=|\rm{Re}(\mathbb{C})|$ is
significantly larger than the imaginary part $N=|\rm{Im}(\mathbb{C})|\approx
0$, as depicted by the blue dot in Fig. 3(b). Nonreciprocity cannot be
observed in this instance because the interference term between the coherent
and dissipative couplings is negligibly small. As observed in the left
($|S_{21}|$) and right panels ($|S_{12}|$) of Figure 3(c), transmission
mappings are nearly identical.

When the YIG sphere is placed at position 2, the mode hybridization as shown
in Fig. 3(d) is quite distinct from level repulsion but similar to level
attraction, indicating the coexistence of dissipative and coherent magnon-
photon couplings Wang  _et al._ (2019). As indicated by the orange dot in Fig.
3(b), when the YIG sphere is placed at position 2, the coherent and
dissipative coupling strengths are comparable. The interference between
coherent and dissipative couplings then plays a crucial role. Consequently,
the nonreciprocity is evident in Fig. 3(d), where $|S_{21}|\neq|S_{12}|$ is
observed at various detunings. As an illustration, we plot the nonreciprocal
transmission at the arrow-marked field detuning in Fig. 3(d). As depicted in
the inset of Fig. 3(b), red and blue curves correspond to $|S_{21}|$ and
$|S_{12}|$, respectively.

The coupling strength $\mathbb{C}$ tends to be zero [gray dot in Fig. 3(b)]
when the YIG sphere is placed at position 3, which is far from the cavity and
transmission line. Conceivably, nonreciprocal transmission does not occur in
this case either. The measured transmission mappings of $|S_{21}|$ and
$|S_{12}|$ are shown in Fig. 3(e). Through the measurements at the positions
above, we can find that the coexistence of coherent and dissipative couplings
is very crucial for the emergence of nonreciprocal transmission. In addition,
to obtain the maximum nonreciprocal response, the coherent and dissipative
coupling strengths should be comparable.

Combining the transmission amplification caused by the dissipation-compensated
cavity and the nonreciprocity of the coupled system, we achieve the
unidirectional amplification and isolation, as shown in Fig. 4. In order to
quantify isolation, the difference between the rightward ($S_{21}$) and
leftward ($S_{12}$) transmission amplitudes is extracted and plotted as
$20{\log}_{10}|S_{21}/S_{12}|$. Here we use its absolute value as the
isolation ratio (Iso.) of the system. The experimentally observed isolation
ratios at different amplifier bias voltages ($V$=0, 4.8, 5 V) are plotted
versus $\Delta_{\rm{m}}$ and $\Delta_{\rm{c}}$ in Figs. 4(a)-(c). As the bias
voltage of the amplifier rises, the intrinsic damping rate of the hybrid
system transits from the positive to negative, which means the generation of a
net gain in the system (see Supplementary Material Sec. IV). The maximum
isolation ratio increases as more signal gain is supplied to the system. On
the premise of a high level of isolation, it is crucial to find out the
optimal operating point in order to achieve a relatively large unidirectional
signal amplification. We choose the field detuning position marked by the gray
planes in Figs. 4(a)-(c) as examples, and plot the $|S_{21}|$ and $|S_{12}|$
spectra in Figs. 4(d)-(f), respectively. The circle points are the
experimental results, while the solid curves are the theoretical results
obtained using Eq. (4), which are in good agreement. The microwave
transmission depicted in Fig. 4(d) is nonreciprocal without amplification,
because the bias voltage of the amplifier is set to zero. In Figs. 4(e) and
4(f), the unidirectional amplification is realized when bias voltage is
applied to the amplifier. At the working frequency of $\omega/2\pi=3.003$ GHz,
the rightward propagating microwave can be amplified by 11.5 dB, whereas the
leftward propagating microwave can be attenuated by 34.7 dB, giving an
isolation ratio of 46.2 dB. Fitting these spectra with Eq. (4), we obtain the
propagation direction-dependent complex coupling strengths of the system,
which are $\mathbb{C}_{R}=(5.66-4.3i)$ MHz and $\mathbb{C}_{L}=(5.6-2.74i)$
MHz.

In conclusion, we have designed a novel nonreciprocal device that performs
unidirectional amplification and reserve isolation simultaneously. The device
consists of an active circuit with an embedded amplifier, a passive cavity
made by SRR, and a YIG sphere. The unidirectional amplification of the
microwave signal is accomplished by combining the nonreciprocity of the cavity
magnonic system with the conventional active feedback of the gain medium. By
adjusting the bias voltage of the amplifier in the active feedback circuit, we
can compensate for loss of the passive cavity mode and convert it to a gain
regime. Meanwhile, the coupling strength between the cavity photon mode and
the magnon mode is controlled by the relative position between the device and
the YIG sphere. By balancing the coherent and dissipative couplings, the
interference between them can result in nonreciprocity in our system. By
optimization via amplifier bias voltage and photon-magnon coupling strength,
we are able to achieve microwave transmission amplification of 11.5 dB in one
propagation direction and attenuation of 34.7 dB in the opposite propagation
direction. Such a unidirectional amplification device may have promising
applications in quantum information technologies, as it can both amplify the
weak signal leaking from the principal quantum system and isolate the back-
scattered noise from the readout electronics. This design of integrating two
functions in just one device will provide advantages for the construction of
large-scale information networks.

###### Acknowledgements.

We thank Zi-Qi Wang for helpful discussion. This work is supported by the
National Key Research and Development Program of China (Grant No.
2022YFA1405200), the National Natural Science Foundation of China (Grants No.
92265202, No. 11934010, and No. 12174329), the Fundamental Research Funds for
the Central Universities (Grant No. 2021FZZX001-02).

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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# A Thermodynamically Universal Turing Machine

Jihai Zhu<EMAIL_ADDRESS>
###### Abstract.

Expanding upon the widely recognized notion of mathematical universality in
Turing machines, a concept of thermodynamic universality in Turing machines is
introduced. Under the physical Church-Turing thesis, the existence of a
thermodynamically universal Turing machine (TUTM) is demonstrated. A TUTM not
only has the capability to simulate the input-output behavior of any given
Turing machine but also replicate the heat production of that machine up to an
additive constant. The finding shows that the hypothesis that the physical
world is simulated by Turing machines may not be completely absurd.

## 1\. Introduction

Turing machines (TMs) possess a crucial property known as (mathematical)
universality, which means that there exists a class of TMs called universal
Turing machines (UTMs) capable of simulating any given TM with a specified
input program. However, in the physical world, every TM (or any computational
device, such as a laptop, that possesses the same computational power as a
UTM), has its own physical properties, such as heat generation and work
dissipation during operation. This leads us to inquire whether there exists a
TM that is physically universal in the sense that it can not only simulate any
other TM mathematically but also replicate certain physical properties of the
specified TM. (Note that the concept of ”physical universality” discussed in
this paper should not be confused with the concept found in papers on cellular
automata[6].)

When discussing physical universality, it may be better to restrict the scope
of physical properties. This paper specifically focuses on the thermodynamic
properties of TMs, building upon the research that has been conducted since
the 1960s regarding the relationship between thermodynamics and computation.
Early studies on the thermodynamics of Turing machines primarily concentrated
on logically reversible TMs[1, 2]. However, more recently, the work by
Kolchinski and Wolpert[3] has broadened the scope to encompass general TMs,
which serves as the main foundation for this paper. In Kolchinski and
Wolpert’s paper, a TM in the physical world is identified as a discrete-state
physical system that is coupled to a heat bath at temperature $T$ and that
evolves under the influence of a driving protocol, and the inputs and outputs
of the TM are identified as the initial and final states of the physical
system, so that the computation performed by the TM is simulated by the
dynamics over the status of the physical system. Following Kolchinski and
Wolpert’s notation, a physical process that adheres to the physical laws and
exhibits dynamics corresponding to the input-output map of a TM is referred to
as a realization of that TM.

With the above identification, techniques from stochastic thermodynamics[7]
can be applied to the analysis of the thermodynamics properties of
realizations of TMs. In this paper, we particularly focus on the amount of
heat, denoted as $Q(x)$, generated when a TM $M$ is executed on each input $x$
and gives the output $M(x)$ after a finite number of steps. We refer to $Q(x)$
as the heat function of $M$, and the pair $(M,Q)$ as a thermodynamic Turing
machine(TTM).

The pair $(M,Q)$ must obey some condition in order that there exists a
realization of $M$ with $Q(x)$ as its heat function, or that the TTM $(M,Q)$
is realizable. The complete condition for the realizability of a TTM has been
derived in[3], as is recapitulated in Section 3.

The main result of this paper is that, under the assumption of physical
Church-Turing thesis(PCTT), there exists a thermodynamically universal Turing
machine(TUTM) which is a (limiting) realizable TTM $(U,Q_{U})$ and that, given
any realizable TTM $(M,Q)$, it can not only simulate the input-output map $M$,
but also replicate the heat function $Q$ (up to an additive constant
determined by $M$ and $Q$), with a specified input program.

The paper is structured as follows: Section 2 provides a review of the basic
concepts of TMs and algorithmic information theory (AIT), which will be
utilized later in the paper. Section 3 and Section 4 recapitulate the results
derived in [3] regarding the realizability conditions of a TTM and the
dominating realizations of TMs under PCTT. Finally, in Section 5, the
existence of the TUTM under PCTT is demonstrated.

## 2\. Background of TMs and AIT

### A. Turing machines

A TM is an abstract model of computation consisting 3 variables:

  1. (1)

a tape variable which is a semi-infinite string $s\in\\{0,1,b\\}^{\infty}$,
where 0 and 1 are the binary symbols and $b$ is the blank symbol;

  2. (2)

a pointer variable $v\in\mathbb{N}$, which is interpreted as specifying a
“position” on the tape;

  3. (3)

a head variable $q$ belongs to a finite set $Q$ of states, which includes a
specially designated start state and a specially designated halt state.

The TM starts with its head in the start state, the pointer set to position 0,
and its tape containing some finite string of non-blank symbols, followed by
blank symbols. The joint state of the tape, pointer, and head evolves over
time according to a discrete-time update function, which must be bounded in a
finite region. If during that evolution the head ever enters its halt state,
the computation considered completed and the contents of the tape at that time
serves as the output. However, for certain inputs, a TM may never reach the
halt state, thereby failing to complete the computation.

The input-output map of a TM $M$ can be associated with a (partial) function
$M:\mathbb{N}\rightarrow\mathbb{N}$ when a one-to-one correspondence between
$\mathbb{N}$ and the set of finite strings $\\{0,1\\}^{*}$ is established.

A significant characteristic of TMs is their (mathematical) universality. This
means that there exists a TM
$U:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N}$ which, given any TM $M$,
there is an index $\ulcorner M\lrcorner\in\mathbb{N}$ such that for every
input $x$, $U(\ulcorner M\lrcorner,x)=M(x)$. Essentially, TM $U$ can simulate
the computation carried out by any TM and is referred to as a UTM.

### B. Algorithimic information theory

AIT studies complexity measures on strings. Because most objects with interest
can be described in terms of strings, or as the limit of a sequence of
strings, AIT can be applied to lots of areas.

The algorithmic complexity of a string $x$ is defined as the length of the
shortest program that, when fed into a UTM $U$, produces $x$ as output:

$C_{U}(x)\equiv\min_{p:U(p)=x}l(p)$ (1)

Because any two UTMs $U$ and $U^{\prime}$ can simulate each other with
programs of finite length, $|C_{U}(x)-C_{U^{\prime}}(x)|\leq
c_{U,U^{\prime}}$. As the algorithmic complexity of a string approaches
infinity as its length increase, an additive constant may be ignored and we
simply write $C(x)$ for $C_{U}(x)$.

Similarly, The conditional algorithmic complexity of $x\in\\{0,1\\}^{*}$ given
$y\in\\{0,1\\}^{*}$ is defined as the length of the shortest program that,
when paired with $y$ and then fed into a UTM $U$, produces $x$ as output:

$C_{U}(x|y)\equiv\min_{p:U(y,p)=x}l(p)$ (2)

For the same reason, we may omit the subscript $U$ and write $C(x|y)$ for
$C_{U}(x|y)$.

While the algorithmic complexity of a string well characterizes the intuition
complexity of a string, it lacks some good properties we want a complexity
concept to observe, such as the semi-additivity[4]:

$C(x,y)\nleq C(x)+C(y)$ (3)

To derive a concept of complexity which has better mathematical properties, we
have to first turn to the concept of prefix sets:A set of strings
$A\subset\\{0,1\\}^{*}$ is called prefix if there is no two strings in $A$
such that one is the prefix of the other, formally:

$\forall x,y\in A\forall z\in\\{0,1\\}^{*}(xz\neq y)$ (4)

Every prefix set satisfies the Kraft’s inequality:

$\sum_{x\in A}2^{-l(x)}\leq 1$ (5)

Now it is able to define the concept of prefix Turing machines: a prefix TM is
a TM whose domain is a prefix set. There also exists prefix UTMs, so we can
define the prefix algorithmic complexity of $x\in\\{0,1\\}^{*}$ as:

$K_{U}(x)\equiv\min_{p:U(p)=x}l(p)$ (6)

where $U$ is a prefix UTM.

The prefix conditional algorithmic complexity $K_{U}(x|y)$ can be similarly
defined and we may write $K(x)$, $K(x|y)$ for $K_{U}(x)$, $K_{U}(x|y)$.

## 3\. A Theorem on the Realizability of a TTM

We consider a physical system with a countable state space $X$, which is
connected to a heat bath at temperature $T$ and a work reservoir. The system
evolves under the influence of a driving protocol, and we are interested in
its dynamics over some fixed interval $t\in[0,1]$. In this paper, only
deterministic dynamics is considered, so every dynamics on $X$ can be
associated to a (partial) function $f:X\to X$, and we define $Q(x)$ as the
heat transferred to the heat bath during $[0,1]$. A pair $(f,Q)$ of a
(partial) function $f:X\to X$ together with a (partial) function
$Q:X\to\mathbb{R}$ is called realizable if there exists a physical system with
$f$ as its dynamics and $Q$ as its heat function.

To simplify matters, we assume $k_{B}T=1$ henceforth, allowing us to equate
heat transfer into (or out of) the heat bath with the entropy flow into (or
out of) the heat bath.

As is the common practice in stochastic thermodynamics, we denote $p_{X}$ as
the initial distribution of the system over its state space, and $p_{f(X)}$ as
the final distribution.

The following technical result links the logical properties of a (partial)
function f with the heat function of any realization of that $f$ . This result
will be central to our analysis, as it will allows us to establish
thermodynamic constraints on processes that realize TMs.

######  Theorem 1.

Given a countable set $X$, let $f:X\to X$ and $Q:X\to\mathbb{R}$ be two
(partial) functions with the same domain of definition. The following are
equivalent:

  1. (1)

For all $p_{X}$,

$\langle Q\rangle_{p_{X}}+S(p_{f(X)})-S(p_{X})\geq 0$ (7)

  2. (2)

For all $y\in\operatorname{im}f$,

$\sum_{x:f(x)=y}2^{-Q(x)}\leq 1$ (8)

  3. (3)

(f,Q) is realizable.

Proof.  See[3]. $\Box$

## 4\. The Dominating Realization of a TM

The physical Church-Turing thesis(PCTT) states that any function that can be
computed by a physical system can be computed by a TM[5]. In the subsequent
sections of this paper, we adopt the assumption that PCTT holds. This
assumption requires that the heat function $Q(x)$ of every realizable TTM must
be computable.

[3] demonstrated that, for every TM $M$, under PCTT, there exists a (limiting)
realizable TTM $(M,Q_{dom})$, where $Q_{dom}(x)\equiv K(x|M(x))$ is smaller
than any other realizable heat function $Q$, up to an additive constant
determined by $M$ and $Q$. Formally:

######  Theorem 2.

The heat function $Q_{dom}(x)\equiv K(x|M(x))$ is physically realizable and is
better than any other realization of $M$ with heat function $Q$, in the sense
that

$Q_{dom}(x)\leq Q(x)+K(Q)+K(M)+O(1)$ (9)

Proof.  See[3]. $\Box$

It should be noted that $K(x|M(x))$ is normally uncomputable, but rather upper
semi-computable, which means that there is a decreasing sequence $Q_{0}\geq
Q_{1}\geq\cdots$ whose limit $\lim_{n\to\infty}Q_{n}=Q_{dom}$. Thus we can
consider the dominating heat function as the limit of a sequence of realizable
heat functions. Generally, we say a TTM $(M,Q)$ is limiting realizable if
there exists decreasing realizable heat functions of $M$: $Q_{0}\geq
Q_{1}\geq\cdots$, whose limit $\lim_{n\to\infty}Q_{n}=Q$.

## 5\. The Existence of TUTM

We now demonstrate the main result of this paper, that there exists a UTUM
$(U,Q_{U})$, which can simulate both the input-output map and the heat
function of any realizable TTM $(M,Q)$.

Before the formal statement and proof of the result, it is better to grasp
some intuition of it. As a UTM has an input parameter $\ulcorner M\lrcorner$
to simulate the input-output map of any given TM $M$, a UTUM should have two
input parameters, $\ulcorner M\lrcorner$ and $\ulcorner Q\lrcorner$, so as to
simulate both the input-output map and the heat function of any realizable TTM
$(M,Q)$.

######  Theorem 3.

There exists a TUTM $U$ with heat function $Q_{U}$ that is limiting realizable
and that for any other realizable TTM $(M,Q)$:

$U(\ulcorner M\lrcorner,\ulcorner Q\lrcorner,x)=M(x)$ (10)

$|Q_{U}(\ulcorner M\lrcorner,\ulcorner Q\lrcorner,x)-Q(x)|\leq K(M)+K(Q)+O(1)$
(11)

Proof.  The existence of a UTM which has an invalid second parameter as in
Eq.10 is obvious. We only have to define a limiting realizable heat function
$Q_{U}$ for $U$ to satisfy Eq.11.

Define

$Q_{U}(\ulcorner M\lrcorner,\ulcorner Q\lrcorner,x)\equiv Q(x)+K(M)+K(Q)$ (12)

$Q_{U}$ is upper semi-computable, so we only have to prove that $Q_{U}$
satisfies Eq.8. As the definition of the prefix algorthmic complexity include
a prefix set, it satisfies the Kraft’s inequality:

$\sum_{x}2^{-K(x)}\leq 1$ (13)

So the following inequality holds, thus completing the proof (the last
inequality comes from the realizability of $Q$):

$\sum_{e,i,x:M_{e}(x)=y}2^{-Q_{U}(e,i,x)}\leq\sum_{e,i,x}2^{-Q(x)}2^{-K(e)}2^{-K(i)}\leq\sum_{x}2^{-Q(x)}\leq
1$ (14)

$\Box$

## 6\. Discussion

We have demonstrated the existence of TUTM in the last section, which tells us
that, a physically universal TM may in some sense exist and with that machine,
we may simluate the physical world with only numeric input. It shows that the
hypothesis that the physical world is simulated by Turing machines may not be
completely absurd.

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