]http://www.ioffe.ru/ferrolab/

# Spectrum evolution and chirping of laser-induced spin wave packets in thin
iron films

Ia. A. Filatov<EMAIL_ADDRESS>[ P. I. Gerevenkov  Ioffe
Institute, 26 Politekhnicheskaya st., 194021, St. Petersburg, Russian
Federation  M. Wang  A. W. Rushforth  School of Physics and Astronomy, The
University of Nottingham, NG7 2RD Nottingham, UK  A. M. Kalashnikova  N. E.
Khokhlov  Ioffe Institute, 26 Politekhnicheskaya st., 194021, St. Petersburg,
Russian Federation

###### Abstract

We present the experimental study of ultrafast optical excitation of surface
magnetostatic spin wave (MSSW) packets and their spectral properties in thin
films of pure iron. As the packets leave the excitation area and propagate in
space, their spectra evolve non-trivially. Particularly, low or high frequency
components are suppressed at the border of the excitation area depending on
the orientation of the external magnetic field with respect to the
magnetocrystolline anisotropy axes of the film. The effect is ascribed to the
ultrafast local heating of the film. Further, the time resolution of the
implemented all-optical technique allows us to extract the chirp of the MSSW
packet in the time domain via wavelet analysis. The chirp is a result of the
group velocity dispersion of the MSSW and, thus, is controlled by the film
magnetic parameters, magnetization and anisotropy, and external field
orientation. The demonstrated tunable modulation of MSSW wave packets with
femtosecond laser pulses may find application in future magnonic-photonic
hybrid devices for wave-based data processing.

Developments in neuromorphic computing and pattern recognition largely rely on
signal processing, going beyond the schemes offered by conventional
electronics. Spin waves (SWs), or coherent magnons, in magnetically ordered
structures are considered as a potential alternative to electrons in
semiconductor devices for the transmission and processing of information in a
special area of spintronics named magnonics Pirro  _et al._ (2021); Yamaguchi,
Hirohata, and Stadler (2021); Nikitov  _et al._ (2015); Mahmoud  _et al._
(2020); Barman  _et al._ (2021). The most prominent feature of SWs is the
possibility to transfer information about the magnetic moment without electric
currents, thereby minimizing ohmic losses. Furthermore, a growing number of
works refer to SWs as an effective tool for wave-based logic and calculations
Khitun, Bao, and Wang (2010); Csaba, Papp, and Porod (2017); Khivintsev  _et
al._ (2019); Mahmoud  _et al._ (2020); Barman  _et al._ (2021). Moreover, SW
transport in data processing devices is potentially available at sub-100-nm
scales. However, such a small scale for real application components demands
appropriate materials with easy-to-manufacture procedures and satisfactory
magnonic properties, such as high frequencies of the SWs, their tunability and
easy integration into semiconductor electronic circuits. These features are
native for transition-metal based micro- and nanometer sized waveguides for
SWs, making them promising candidates for nano-magnonic applicationsYamaguchi,
Hirohata, and Stadler (2021); Mahmoud  _et al._ (2020). Furthermore, the
intrinsic strong anisotropy fields in metal nano-waveguides play the role of
an additional knob to design the properties of SWs, such as the SW dispersion
and the corresponding phase evolution Sekiguchi  _et al._ (2017); Khokhlov
_et al._ (2019). The latter is crucial to make SWs effective tools for wave-
based logic and interferometric calculationsKhitun, Bao, and Wang (2010);
Csaba, Papp, and Porod (2017); Khivintsev  _et al._ (2019); Mahmoud  _et al._
(2020); Barman  _et al._ (2021), particularly, in speeding up computationally
demanding tasks such as image and speech recognitionKozhevnikov  _et al._
(2015); Albisetti  _et al._ . The described practical purposes of magnonics
require appropriate research tools to excite and inspect the wave nature of
SWs. Recently, ultrafast laser pulses were introduced as such tool because
they can excite SW with exceptional tunability Satoh  _et al._ (2012); Chernov
_et al._ (2018); Yoshimine  _et al._ (2017); Hioki, Hashimoto, and Saitoh
(2020); Khokhlov  _et al._ (2019); Au  _et al._ (2013); Khokhlov  _et al._
(2021). However, laser excitation is intrinsically broad-band and
inhomogenious, which has a significant effect on the properties of the
generated SW packets, limiting the spatial and temporal profile and leading to
large spectral width. Therefore, understanding the propagation of such wave
packets is essential for developing applications of optically generated SWs
for future magnonic-photonic hybrid devices.

In this letter, we report on the experimental observation of controllable
spectrum evolution and frequency modulation of laser-induced magnetostatic
surface waves (MSSWs) in thin films of pure iron. We show, that the spectral
features of MSSW wave packets can be tuned by choosing the orientation of an
external magnetic field with respect to the axes of the magnetocrystalline
anisotropy. As the optically excited MSSW packets are broad-band and propagate
in a dispersive medium, they obtain a chirp — a time delay between distinct
frequency components. We clearly detect the chirping at the $\mu$m spatial
scale. The findings are crucial for the design of hybrid opto-magnonic wave-
based data processing circuits.

Figure 1:  (a) Polar Kerr signal induced by propagating MSSW packet in the
time-space domain for Fe film with $d=10$ nm in the HA-configuration. (b,c)
Space-frequency maps of the packet for the HA- (b) and EA- (c) configurations.
(d, e) FFT spectra of the packet at different pump-probe distances for the HA-
(d) and EA- (e) configurations. Inset in (e) shows enlarged area of the same
spectra.

Our investigations are carried out using epitaxial films of iron with
thicknesses $d$ = 10 and 20 nm. The films are deposited on 350-$\mu$m thick
GaAs (001) substrates by magnetron sputtering and are capped with a 2.5-nm-
thick chromium layer for protective purposes. The films possess pronounced
cubic and interface-induced uniaxial in-plane magnetic anisotropy similar to
other epitaxial thin films of iron and its alloys on GaAs substrates Gester
_et al._ (1996); Wastlbauer and Bland* (2005); Parkes  _et al._ (2013).

Excitation and detection of MSSWs are performed using a space-time-resolved
magneto-optical Kerr effect (STR-MOKE) pump-probe setup Filatov  _et al._
(2020). Pump and probe laser pulses with central wavelengths of 750 nm and
1050 nm, respectively, a duration of 120 fs, 70 MHz repetition rate, are
focused normally onto opposite sides of the iron film into 3.5-$\mu$m-diameter
spots using microobjectives. The fluence of the pump pulses is of 4.5 mJ/cm2,
and the probe pulses have a fluence of about 20 times lower. The temporal
resolution is achieved with an opto-mechanical delay line placed in the pump
path. The spatial scan is realized by placing the pump microobjective on a
2-axes piezo stage. The experiments are done in the Damon-Eshbach
configuration, i.e., an external static magnetic field $\mu_{0}{\bf H}=100$ mT
is applied in the film’s plane and the propagation of the MSSW is detected
perpendicularly to H with varying distance between the pump and probe spots on
the sample surface. In the sample’s plane H is aligned at different angles
with respect to the magnetocrystalline anisotropy axes of the film. The
excitation mechanism of the MSSW in the experiments is an ultrafast change of
the magnetocrystalline anisotropy caused by ultrafast heating induced by the
pump pulse, similar to the recent experiments of MSSW excitation in galfenol
(Fe0.81Ga0.19) films Khokhlov  _et al._ (2019); Filatov  _et al._ (2020). We
investigate propagating MSSWs for two directions of H: (i) H is aligned along
the sample’s hard anisotropy axis and (ii) H is at 15∘ with respect to the
easy anisotropy axis, referred further as HA- and EA-configurations,
respectively.

Figure 1(a) shows the spatial-temporal map of the polar Kerr rotation angle
for the probe measured with the STR-MOKE setup in the sample with $d=10$ nm in
the HA-configuration. The color bar spans up to 0.3 of the maximum amplitude
of the measured Kerr signal to show the propagation of the MSSW packet beyond
the distance at which its amplitude decreases by a factor of e. The shape and
slope of the phase pattern serves as evidence that the detected Kerr signal
originates from the propagating MSSW wave packet. Thus, the propagation of
laser-induced MSSW wave packets over a distance of more than 10 $\mu$m is
clearly observed. Such propagation distance is comparable and even larger than
that of typical values in other metallic materials used in real magnonic
circuits Chumak, Serga, and Hillebrands (2017).

Next, we analyze the spectral composition of the propagating MSSW wave packet
by performing fast Fourier transform (FFT) analysis of signal in the time
domain for each pump-probe distance. The result is shown in Fig. 1(b,c) for
the 10-nm sample in the HA- and EA-configurations, respectively. The color bar
is also set at 0.3 of the maximum amplitude to more clearly characterize the
decrease of the MSSW amplitude during propagation. Compression of the spectral
width and a shift of the frequency with maximal FFT amplitude are observed
during the propagation of the MSSW from the excitation spot. More explicit
confirmation of such spectral changes is possible by analyzing the individual
spectra at different pump-probe distances (Fig. 1(d,e)). The central frequency
of the propagating MSSW shifts to the lower or higher part of the spectrum
observed in the excitation area for the HA- and EA-configurations,
respectively, in agreement with previous results for galfenol films on GaAs
substrates Filatov  _et al._ (2020); Khokhlov  _et al._ (2019). The observed
spectral evolution is associated with different MSSW dispersion relations
inside and outside the pump-heated area. Indeed, effective anisotropy fields
are partially suppressed by an abrupt decrease of the anisotropy parameters
and magnetization inside the excitation area Gerevenkov  _et al._ (2021). This
leads to a shift of the MSSW dispersion inside the pump area to the higher or
lower frequency range depending on the orientation of H because of the
different manner in which the anisotropy field enters the total effective
field: i.e. the effective anisotropy field is subtracted from (added to) H in
the HA-(EA-)configuration, leading to a decrease (increase) of the total
effective magnetic field in the sample, which defines the frequency of the
MSSW Gurevich and Melkov (1996); Kalinikos  _et al._ (1990). Since the pump
has a limited diameter, it excites a limited band of wavenumbers inside the
spot Satoh  _et al._ (2012); Jäckl  _et al._ (2017). This sets an upper limit
on the spectral range of the excited MSSW frequencies, with the lower limit
being the ferromagnetic resonance frequency. MSSWs propagating outside the
pump have the same limitation on the wavenumber, but the corresponding
frequency range is shifted up (down). This, in turn, sets upper (lower) bounds
on the available frequency in the MSSW spectrum. A detailed description of the
spectrum evolution of MSSWs in the presence of strong magneto-crystalline
anisotropy is given elsewhere Khokhlov  _et al._ (2019); Filatov  _et al._
(2020).

Using FFT spectra, we determine values of the MSSW propagation length $l_{p}$
for both magnetic field configurations by fitting the dependence of the
integral of the central peak in the FFT spectra, $S$ on the pump-probe
distance, $L$ with the exponential function $S\propto exp(-L/l_{p})$. We
determine $l_{p}$ in the range of $L>3$ $\mu$m, i.e. starting from the edge of
the pump area, which represents the value of $l_{p}$ for the MSSW propagating
in the non-heated iron film. The obtained values of $l_{p}$ for all samples
are larger than 6 $\mu$m in the HA-configuration and less than 5 $\mu$m in the
EA-configuration (see Table 1). The observation that $l_{p}$ is larger in the
HA-configuration is in good agreement with previous works on MSSWs in cubic
anisotropic metal films Sekiguchi  _et al._ (2017); Khokhlov  _et al._ (2019).
The effect is connected with the reduction of the total in-plane effective
field in the HA-configuration and is discussed in detail elsewhere Sekiguchi
_et al._ (2017).

As shown above, the spectrum of the MSSW becomes strongly asymmetric with
distance (Fig. 1(d,e)). Hence, we may expect that the wave packet of the MSSW
acquires a phase or frequency modulation, i.e., the frequency components
propagate with different velocities resulting in the temporal shift between
them. We implement continuous wavelet transform with Morlet mother wavelet
similar to opto-acoustic pump-probe experiments Dehoux and Audoin (2012);
Danworaphong  _et al._ (2015) to extract information about the frequency
evolution in the time domain. Indeed, the resulting analysis reveals the shift
of the frequency components $f$ with time $t$, increasing with $L$ (Fig.
2(c,d)).

Figure 2:  (a,b) Wave packet signals at distances of 7 and 20 $\mu$m from the
pump spot in the time domain for the 20-nm film in the HA-configuration. (c,d)
Corresponding wavelet spectrograms in the time-frequency domain. Dashed line
on (d) shows variation of the frequency peak with time. Double arrows show the
variation of the frequency $\Delta f$ and time shift $\Delta t$ schematically.

To characterise the shift, we approximate the registered frequency modulation
of the MSSW packets with a linear function $f(t)=f_{c}+\nu t$, where $f_{c}$
is a central wave packet frequency, $\nu=\frac{\partial f}{\partial t}$ is a
parameter characterising the chirp. The non-zero value of $\nu$ appears when a
wave packet propagates in a dispersive medium, and its group velocity depends
on $f$ Marcuse (1980); Akhmanov, Vysloukh, and Chirkin (1986); Agrawal (2019).
The equation for the complex packet’s envelope in this case is obtained by
expanding the dispersion relation $f(k)$, where $k$ is a wave vector, into a
Taylor series upto the derivative of the second order. Importantly, the
obtained parabolic partial-derivative equation for the complex envelope has
the same form for optical pulses Agrawal (2019); Akhmanov, Vysloukh, and
Chirkin (1986) and for magnetostatic waves Zvezdin and Popkov (1983) both. The
resulting equation for the chirp parameter of a Gaussian wave packet at
distance $L$ isAkhmanov, Vysloukh, and Chirkin (1986); Agrawal (2019):

$2\pi\nu(L)=\frac{2L}{\beta_{2}(L^{2}+L_{D}^{2})},$ (1)

where $\beta_{2}=\frac{\partial^{2}k}{\partial\omega^{2}}$, $\omega=2\pi f$,
$L_{D}=T_{0}^{2}/|\beta_{2}|$, and $T_{0}$ is an initial half duration of the
packet at level $e^{-1}$.

To compare the chirp described by Eq. (1) with experimental value of $\nu$, we
used the analytic dispersion of MSSW in the thin film limit ($kd\ll 1$) and
for small wavenumbers (the exchange contribution to the spin wave energy is
negligible) in a simplified form Arias and Mills (1999); Kalinikos  _et al._
(1990); Zakeri (2020):

$f(k)=\left(f_{FMR}^{2}+F^{2}\frac{kd}{2}\right)^{1/2},$ (2)

where $f_{\text{FMR}}$ is the ferromagnetic resonance frequency, and $F$ is a
constant. Values of $f_{\text{FMR}}$ and $F$ are determined by the
gyromagnetic ratio, saturation magnetization, and anisotropy parameters of the
film, and the mutual orientation of H and anisotropy axes. Here we obtain
$f_{\text{FMR}}$ and $F$ by fitting the experimental MSSW dispersion
reconstructed from time-space maps via two-dimensional FFT (Fig. 3(a)). The
results of the fits are presented in Table 1. Equations (1,2) give the result:

$\nu(L)=\frac{2F^{2}dL}{4L^{2}+(\pi F^{2}dT_{0}^{2})^{2}}.$ (3)

Table 1: Parameters of MSSW obtained in experiments d | HA-configuration | EA-configuration
---|---|---
(nm) |  $l_{p}$ ($\mu$m) |  $f_{FMR}$ (GHz) |  $F$ (GHz) |  $l_{p}$ ($\mu$m) |  $f_{FMR}$ (GHz) |  $F$ (GHz)
10 | $6.1\pm 0.1$ | $8.3\pm 0.1$ | $152\pm 4$ | $2.6\pm 0.1$ | $13.9\pm 0.2$ | $220\pm 10$
20 | $8.9\pm 0.3$ | $8.2\pm 0.1$ | $148\pm 5$ | $4.7\pm 0.1$ | $14.9\pm 0.3$ | $150\pm 15$

The calculations of $\nu(L)$ show the same trend with the experimental
spectrograms (Fig. 3(b)). Deviation of experiment and analytical Eq. (3) is
due to several approximations used to derive Eq. (1). One of them is $\Delta
f\ll f_{c}$, where $\Delta f$ is the spectral width of the wave packet, which
is important for Taylor series approach. The condition is not fully satisfied
in the experiments as we detect MSSW packets with $\Delta f\approx 4$ GHz at
$f_{c}\approx 9$ GHz (see Fig. 1(c-e) and Fig. 2(c,d)). Second approximation
is that, Eq. (1) is obtained for Gaussian envelope of the packet at the
entrance of the media. The experimental envelope of MSSW is visibly asymmetric
in the time domain (Fig. 2(a)). Thus, more comprehensive theoretical analysis
of the evolution of optically excited SW packets with broadband spectra is
needed, but it is out the scope of this letter. Importantly, the value of
$\nu$ depends on $d$ and $F$. The dependence on $F$ means that the resulting
chirp of MSSW wave packets is programmable with the intrinsic effective field
of the waveguide, e.g., its saturation magnetization, anisotropy parameters,
dimensions etc. Notably, the chirp of the MSSWs in iron films reported here
has been clearly seen in previous experiments with micro-stripe
antennaeSekiguchi  _et al._ (2017), but it was not discussed yet.

Figure 3:  (a) Dispersion of MSSW wave packet in 20-nm iron film in HA-
configuration, reconstructed from experimental time-space maps (pseudo-color)
and calculated analytically with Eq. (2) (line). (b) Chirp parameter $\nu$
versus distance for 10 and 20 nm films in HA-configuration. Symbols show
experimental data, lines – analytical calculations with Eq. (3) with
$T_{0}=50$ ps.

In conclusion, we show experimentally the propagation of optically excited
MSSWs in thin iron films with propagation distances of order tens of microns,
which may find the application in magnonic waveguides and logic gates.
Additional advantages of iron as a metal for magnonic applications include the
relatively easy manufacturing procedure of the thin films and the strong in-
plane magneto-crystalline anisotropy. We use the latter here to tune the
character of the MSSW wave packet spectrum evolution in the presence of laser-
induced thermal gradients and its chirping with distance. Generated wave
packets are spectrally broad and hardly to describe analytically. Nonetheless,
the analysis of optically generated ultra broadband SW packets is crucial for
wave-based data processing in future hybrid magnonic-photonic devices.
Furthermore, the optical pump-probe technique gives a powerful toolbox for the
detailed investigation of the SWs packet including: the extraction of its
spectral evolution in space, the dispersion law, and the chirping in the time
domain. The first two options are available with Brillouin light scattering
(BLS) techniques as well Demokritov, Hillebrands, and Slavin (2001); Kalyabin
_et al._ (2019), while the chirping investigation is accessible in BLS with
introduction phase resolution with corresponding complication of the
measurement scheme Fohr  _et al._ (2009), or, alternatively, with the scanning
antennae technique Annenkov, Gerus, and Lock (2018); Annenkov and Gerus
(2012). The ultrafast laser pulses provide all above possibilities out-of-box
with excellent spatial and temporal resolution.

###### Acknowledgements.

The work of Ia.A.F. and P.I.G. was funded by RFBR (grant 20-32-70149), A.M.K.
acknowledges RSF (grant 20-12-00309), N.E.Kh. acknowledges the ”BASIS”
Foundation (grant 19-1-3-42-1).

## Author declarations

### Conflict of Interest

The authors have no conflicts to disclose.

## Data availability

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

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# A shortcut to adiabaticity in quantum field theory

Nicolás F. Del Grosso1 Fernando C. Lombardo1 Francisco D. Mazzitelli2 Paula I.
Villar1 1 Departamento de Física Juan José Giambiagi, FCEyN UBA and IFIBA
CONICET-UBA, Facultad de Ciencias Exactas y Naturales, Ciudad Universitaria,
Pabellón I, 1428 Buenos Aires, Argentina  2 Centro Atómico Bariloche and
Instituto Balseiro, Comisión Nacional de Energía Atómica, R8402AGP Bariloche,
Argentina

(today)

###### Abstract

Shortcuts to adiabaticity constitute a powerful alternative that speed up
time-evolution while mimicking adiabatic dynamics. They are also relevant to
clarify fundamental questions such as a precise quantification of the third
principle of thermodynamics and quantum speed limits. In this letter we
describe, for the first time, how to implement shortcuts to adiabaticity in
quantum field theory, for the particular case of a massless scalar field
inside a cavity with a moving wall, in $1+1$ dimensions. The approach is based
on the known solution to the problem that exploits the conformal symmetry, and
the shortcuts take place whenever there is no dynamical Casimir effect. We
obtain a fundamental limit for the efficiency of an Otto cycle with the
quantum field as a working system, that depends on the maximum velocity that
the mirror can attain. We describe possible experimental realizations of the
shortcuts using superconducting circuits.

In recent times, as new technologies allow us to manipulate smaller and
smaller systems, such as trapped ions, nano resonators and electronic circuits
ion ; circuitqed ; nanoresonator , a natural question has emerged about
whether it is possible to use them to produce machines and what their
properties would be. The novelty comes from the fact that these small systems
can exhibit quantum properties that could potentially be exploited to get an
advantage over classical machines or present new obstacles to the operation.
These questions constitute the back bone of a new area of physics that has
come to be called quantum thermodynamics. In some cases of discrete stroke
quantum machines, such as a quantum harmonic oscillator or a quantum field
undergoing an Otto cycle, it has been shown that the efficiency of the
resulting machine is maximum for an adiabatic (i.e. infinitely slow) driving
Otto_nos . The problem is that under these conditions the power of the machine
vanishes and so it becomes necessary to understand how to operate these
machines in a finite time. However this leads to a friction work on each
stroke $\langle w_{\text{fric}}\rangle=\langle w\rangle-\langle
w_{\text{ad}}\rangle$, defined as the difference between the actual work and
the adiabatic work, that is always non-negative friction . This has the effect
of reducing the efficiency of a heat engine. For example, for a quantum Otto
cycle comprised of four stokes (A, cooling at constant volume; B, adiabatic
expansion; C, heating at constant volume; D, adiabatic compression) the
efficiency in finite time is given by

$\eta=\frac{W}{Q}=\frac{W_{\text{ad}}-\langle
w_{\text{fric}}\rangle_{AB}-\langle
w_{\text{fric}}\rangle_{CD}}{Q_{\text{ad}}-\langle
w_{\text{fric}}\rangle_{AB}}\leq\frac{W_{\text{ad}}}{Q_{\text{ad}}}=\eta_{\text{ad}},$
(1)

where $W_{\text{ad}}=\langle w_{\text{ad}}\rangle_{AB}+\langle
w_{\text{ad}}\rangle_{CD}$, and is always lower or equal to the adiabatic
efficiency. Therefore, it is paramount to understand if it is possible to
implement an adiabatic evolution in finite time.

Although in most cases the finite time operation causes the emergence of
coherences in the state of the system that result in an efficiency loss, in
many cases it is possible to implement protocols, named shortcuts to
adiabaticity (STA), that evolve the initial state into the final state that
would have been obtained with an adiabatic evolution, but in a finite time
berry ; STA1 ; STA2 ; STA3 . These protocols typically require a full control
of the quantum system and end up being extremely challenging from an
experimental stand point.

In previous works, STA have been considered from a theoretical and/or an
experimental point of view for different physical systems: trapped ions
Palmero , cold atoms Torrontegui , ultracold Fermi gases Dowdall , Bose-
Einstein condensates in atom chips Amri , etc. In this letter we explore the
possibility of applying STA in quantum field theory. In particular, we will
consider a scalar quantum field in a one-dimensional cavity with a moving
wall, whose state undergoes a unitary evolution. We will show that given a
wall trajectory $L_{\rm ref}(t)$ we can find a shortcut to adiabaticity given
by an effective trajectory $L_{\text{eff}}(t)$ that, when implemented in
finite time, results in the same state as if the original had been performed
adiabatically. As we will see, this protocol has the advantage that it can be
easily implemented experimentally using superconducting circuits, since it
does not require additional exotic potentials. Moreover, the effective
trajectory can be computed from the original quite simply, paving the way for
more efficient quantum field thermal machines.

There are several ways of introducing adiabatic shortcuts in quantum systems.
A natural approach is to impose the instantaneous eigenstates of a time
dependent Hamiltonian to be solutions of a time-dependent Schrödinger equation
of a different Hamiltonian, which is the original one plus a “counter
diabatic” Hamiltonian STA2 . This procedure leads in general to an effective
non-local Hamiltonian which is difficult to implement experimentally. For some
systems (such as a harmonic oscillator with a time-dependent frequency
$\omega(t)$), an additional unitary transformation simplifies the effective
Hamiltonian, without modifying the initial and final states STA5 . Also for
these systems, an alternative approach calzetta  is based on the following
observation: the adiabatic WKB solution for the position operator $\hat{q}(t)$
can be written in terms of annihilation and creation operators as

$\hat{q}(t)=\hat{a}\frac{e^{-i\int^{t}\omega_{\rm
ref}(t^{\prime})dt^{\prime}}}{\sqrt{2\omega_{\rm
ref}(t)}}+\hat{a}^{\dagger}\frac{e^{i\int^{t}\omega_{\rm
ref}(t^{\prime})dt^{\prime}}}{\sqrt{2\omega_{\rm ref}(t)}}$ (2)

(we stress that along this paper we will be working with natural units where
$c=\hbar=k_{B}=1$). This is an approximate solution for the oscillator with a
reference frequency $\omega_{\rm ref}(t)$, valid if it is slowly varying, but
an exact solution of a system with an effective frequency

$\omega_{\rm eff}^{2}(t)=\omega_{\rm
ref}^{2}+\frac{1}{2}\left(\frac{\ddot{\omega}_{\rm ref}}{\omega_{\rm
ref}}-\frac{3}{2}\left(\frac{\dot{\omega}_{\rm ref}}{\omega_{\rm
ref}}\right)^{2}\right)\,,$ (3)

which turns out to be the external evolution that leads to a STA.

In the context of quantum field theory under the influence of time dependent
backgrounds, free fields can be described as a set of interacting harmonic
oscillators with time-dependent frequencies. Assuming that the time dependence
takes place in a finite time period, the departure from adiabaticity is
measured by the Bogoliubov transformation that connects the IN and OUT Fock
spaces. When non-trivial, this transformation indicates the presence of
particle creation, which is related to the friction work mentioned before. In
the particular case of quantum fields in flat spacetime with time dependent
boundary conditions and/or moving boundaries this effect is named motion
induced radiation or dynamical Casimir effect (DCE) DCE .

A naive generalization of the WKB approach described above, applied to each
harmonic oscillator, does not work in general, because the effective frequency
$\omega_{\rm eff}(t)$ turns out to be “mode-dependent”, that is, it produces a
STA for a given mode, but not to the full field. However, for a conformal
field in $1+1$ dimensions, it is possible to follow a different approach to
the STA, based on conformal transformations. We will consider a massless
scalar field confined to a cavity of variable size $L(t)$. We assume the left
mirror located at $x=0$ and the right mirror at $x=L(t)$, where we impose
Dirichlet boundary conditions. The idea is to perform a coordinate conformal
transformation such that, in the new coordinates, both mirrors are at rest.
The transformation is given by Moore

$\displaystyle\bar{t}+\bar{x}=R(t+x)$ $\displaystyle\bar{t}-\bar{x}=R(t-x)\,,$
(4)

where the function $R$ is fixed in such a way that $x=0$ corresponds to
$\bar{x}=0$ and $x=L(t)$ to $\bar{x}=1$. These conditions imply that $R$ must
satisfy the so called Moore equation Moore

$R(t+L(t))-R(t-L(t))=2\,.$ (5)

Due to the conformal invariance of the classical action, the Klein-Gordon
equation for the massless scalar field is the usual wave equation in the new
coordinates. The field modes can be written as

$\displaystyle f_{n}(x,t)$ $\displaystyle=$ $\displaystyle\frac{1}{\sqrt{4\pi
n}}\big{(}\exp\left(-in\pi R(t+x)\right)$ (6) $\displaystyle-$
$\displaystyle\exp\left(-in\pi R(t-x)\right)\big{)}.$

Being functions of $t\pm x$, these modes are solutions of the wave equation.
Moreover, the Moore equation implies that the modes satisfy Dirichlet boundary
conditions on the mirrors.

For a static cavity of size $L_{0}$, the general solution to the Moore
equation reads $R(t)=t/L_{0}+r(t)$, where $r(t)$ is an arbitrary function of
period $2L_{0}$. The usual modes for the static cavity are obtained setting
$r(t)=0$. Assuming that the right mirror is at rest in the IN and OUT regions,
i.e. for $t\to\pm\infty$, we can define the IN Moore function as the solution
of Eq.(5) such that $R_{\rm IN}(t)\to t/L_{0}$ as $t\to-\infty$, and $R_{\rm
OUT}$ as the solution that satisfies $R_{\rm OUT}(t)\to t/L_{1}$ as
$t\to+\infty$, where $L_{0}$ and $L_{1}$ are the initial and final sizes of
the cavity, respectively. These functions determine the IN and OUT modes
through Eq.(6). In general we will have that, as $t\to+\infty$, $R_{\rm
IN}(t)=R_{\rm OUT}(t)+r(t)$, with a non-vanishing periodic function $r(t)$.
The IN and OUT field modes will be connected by a non-trivial Bogoliubov
transformation, which describes physically the creation of particles induced
by the motion of the right mirror, or DCE, and is encoded in the non-vanishing
function $r(t)$.

To evaluate the time evolution of the mean value of the energy-momentum tensor
in a thermal state, one can follow the traditional approach based on point-
splitting regularization. The result is Fulling ; Birrel

$\displaystyle\langle T_{tt}\rangle$ $\displaystyle=\langle
T_{xx}\rangle=G(t-x)+G(t+x)$ $\displaystyle\langle T_{tx}\rangle$
$\displaystyle=\langle T_{xt}\rangle=-G(t-x)+G(t+x),$ (7)

with

$\displaystyle G$ $\displaystyle=$
$\displaystyle-\frac{1}{24\pi}[(R^{\prime\prime\prime}/R^{\prime})-\dfrac{3}{2}(R^{\prime\prime}/R^{\prime})^{2}]$
(8) $\displaystyle+$
$\displaystyle\dfrac{(R^{\prime})^{2}}{2}[-\dfrac{\pi}{24}+F(TL_{0})]\,,$

where $F(TL_{0})$ is related to the initial thermal energy of the cavity

$F(TL_{0})=\sum_{n\geq 1}\frac{n\pi}{(\exp(\frac{n\pi}{L_{0}T}-1)}\,.$ (9)

These equations are a slight generalization of the zero temperature results in
Fulling , see also Ref.Alves . For the static case, the terms proportional to
$(R^{\prime})^{2}$ in the different components of the stress-tensor reproduce
the Casimir energy density and force in $1+1$ dimensions.

The total energy inside the cavity is given by

$E(t)=\int_{0}^{L(t)}\langle T_{tt}(x,t)\rangle\,dx\,,$ (10)

while for an adiabatic evolution the energy is obtained using the
approximation $R^{\prime}(t)\simeq 1/L(t)$ and neglecting derivatives of
$L(t)$:

$E_{\rm ad}(L,T)=-\frac{\pi}{24L}+\frac{F(TL_{0})}{L}\,.$ (11)

The above results suggest a simple way to introduce STA in our conformal field
theory, using an “inverse engineering” approach: any Moore function $R$ that
satisfies at the same time the two conditions $R\to t/L_{0}$ as $t\to-\infty$
and $R\to t/L_{1}$ as $t\to+\infty$ will produce an evolution in which the
Bogoliubov transformation between the IN and OUT bases is the identity
Castagnino . Therefore, there will be no particle creation, and the occupation
numbers will be the same in the initial and final states. The associated
trajectory is defined implicitly by the Moore Eq. (5). If $R$ is defined as a
piecewise function, it should be smooth enough in order to avoid divergences
in the mean value of the stress tensor.

In order to make contact with previous approaches for STA in other contexts,
it is useful to discuss the adiabatic (or WKB) solutions for the field modes.
For an adiabatic trajectory $L(t)$, the Moore function is approximately given
by Moore

$R_{\rm WKB}[L]=\int^{t}\frac{dt^{\prime}}{L(t^{\prime})}\,.$ (12)

Let us consider a (non-adiabatic) reference trajectory $L_{\rm ref}(t)$ that
starts at $L_{0}$ and ends at $L_{1}$. We would like to find an effective
trajectory, $L_{\rm eff}(t)$, such that the time evolution of the modes is the
WKB - like solution evaluated on $L_{\rm ref}(t)$. If we insert $R_{\rm
WKB}[L_{\rm ref}]$ in the field modes Eq.(6), the modes will satisfy exactly
the wave equation, but will not satisfy the Dirichlet boundary conditions on
$x=L_{\rm ref}(t)$. Instead, the modes will vanish on an effective trajectory
that satisfies

$\int_{0}^{t+L_{\rm eff}(t)}\frac{1}{L_{\rm
ref}(t^{\prime})}dt^{\prime}-\int_{0}^{t-L_{\rm eff}(t)}\frac{1}{L_{\rm
ref}(t^{\prime})}dt^{\prime}=2.$ (13)

We stress that this equation defines the effective trajectory $L_{\rm
eff}(t)$. Therefore, implementing the effective trajectory, the field modes
will be described by the WKB modes of the reference trajectory $L_{\rm ref}$.
Moreover, the initial and final lengths of the cavity will be the same for
both trajectories, and the population of the modes will also be the same at
the initial and final times: $L_{\rm eff}(t)$ is the adiabatic shortcut
associated with reference trajectory $L_{\rm ref}$, just as $\omega_{\rm eff}$
is the STA for the quantum harmonic oscillator with frequency $\omega_{\rm
ref}$.

It is simple to see that, if $t+L_{\rm eff}(t)<0$, then $L_{\rm eff}(t)=L_{\rm
ref}(0)=L_{0},\ \text{for }t<-L_{\rm ref}(0)$. Analogously, if $t-L_{\rm
eff}(t)>\tau$ then $L_{\rm eff}(t)=L_{\rm ref}(\tau)=L_{1},\ \text{for
}t>\tau+L_{\rm ref}(\tau).$ This tells us that this effective trajectory is
static before $t=-L_{0}$ and after $t=\tau+L_{1}$. Moreover, it coincides with
the reference trajectory $L_{\rm ref}$ before and after. A relevant question
is whether this is in fact a real trajectory meaning that its speed is always
below $c=1$. We can answer this question taking the time derivative of the
defining equation and solving for the speed of the wall

$\frac{d}{dt}L_{\rm eff}(t)=\frac{L(t+L_{\rm eff}(t))-L(t-L_{\rm
eff}(t))}{L(t-L_{\rm eff}(t))+L(t+L_{\rm eff}(t))}\leq 1,$ (14)

from which we see that its speed is indeed bounded by the speed of light.

A simple smooth trajectory that interpolates between $L_{0}$ and $L_{1}$ is
given by

$L_{\rm ref}(t)=\begin{cases}\begin{array}[]{c}L_{0}\,,\\\
L_{0}\,(1-\epsilon\delta(t))\,,\\\
L_{1}\,,\end{array}&\begin{array}[]{c}t<0\\\ 0<t<\tau\\\
0<t\,,\end{array}\end{cases}.$ (15)

where $\epsilon$ is a positive constant less than 1, the final distance is
given by $L_{1}=L_{0}(1-\epsilon)$ and the evolution function is

$\displaystyle\delta(t)=10(t/\tau)^{3}-15(t/\tau)^{4}+6(t/\tau)^{5}.$ (16)

In Fig.1 we show the effective trajectory $L_{\rm eff}$ associated to this
$L_{\rm ref}$. The energy density inside the cavity evolves from the static
Casimir thermal energy corresponding to a cavity of length $L_{0}$ to that
corresponding to $L_{1}$. The evolution is non-adiabatic at intermediate
times, as depicted in Fig.2 for the particular case $T=0$: indeed, for an
adiabatic evolution the energy density would be constant inside the cavity at
each time, and equal to the static Casimir energy density corresponding to the
instantaneous size of the cavity. Note also that, for an arbitrary trajectory
(not a shortcut), the final energy density would also contain the contribution
of the created particles. In Fig. 3, we plot the adiabaticity parameter
$Q^{*}(t)=E(t)/E_{\text{ad}}(t)$ for the cavity as a function of time, during
the shortcut, at different temperatures for the reference trajectory as well
as for the corresponding STA. This parameter effectively measures the distance
between the adiabatic energy for an infinitely slow motion and the actual
energy in the cavity for the STA found, being of course equal to 1 when these
two coincide. We can see that although both of them depart from the adiabatic
result at intermediate times, the STA returns to 1 at the end of the motion
confirming the implementation of an adiabatic shortcut and its deviation from
adiabaticity at intermediate times is relatively small.

Figure 1:  A reference trajectory (blue solid line) given by Eq. (15) with
initial length $L_{0}=1$, $\epsilon=0.3$ and $\tau=1$. The corresponding
effective trajectory (orange dashed line) calculated from Eq. (13) gives the
shortcut to adiabaticity.

To summarize, solving the equation for $L_{\rm eff}(t)$ we get a trajectory
that, when implemented, generates an evolution of the state of the quantum
field that in the end coincides with that of an exact adiabatic evolution
along $L_{\rm ref}(t)$. That is, we have found a shortcut to adiabaticity for
a scalar quantum field in a cavity with a moving wall. The results can be
generalized to other two dimensional conformal fields confined in a cavity of
variable length, as a massless Dirac field satisfying bag boundary conditions
on the boundaries MPC87 .

In order to discuss some fundamental limits for the power and efficiency of a
quantum Otto cycle, we will consider the following limiting trajectory $L_{\rm
ref}(t)=L_{0}\theta(-t)+L_{1}\theta(t)$, which corresponds to a very small
displacement time, $\tau\ll L_{0},L_{1}$.

In this case we can replace in Eq. (13) to find that the effective trajectory
is given by

$L_{\rm eff}(t)=\begin{cases}\begin{array}[]{c}L_{0}\\\
\frac{2L_{0}L_{1}-t(L_{0}-L_{1})}{L_{0}+L_{1}}\\\
L_{1}\end{array}&\begin{array}[]{c}t<-L_{0}\\\ -L_{0}<t<L_{1}\\\
L_{1}<t\end{array}\end{cases}.$ (17)

This means in the limit case where we want to implement an instantaneous
length change without spurious photon generation, we need at least
$L_{0}+L_{1}$ units of real time to implement it and it consists of a linear
motion. This is consistent with previous results for this particular case
Fulling ; Castagnino .

We have previously mentioned that finite time driving is necessary to improve
the power delivered but usually results in additional friction energy on the
working medium that diminishes the efficiency of the engine. In the case of a
scalar quantum field with a moving boundary we have found a STA, which
maximizes the work delivered, that turns out to be $\langle
w_{\text{ad}}\rangle$, and the efficiency, $\eta=\eta_{\text{ad}}$. Moreover,
the power produced is bounded by the minimum time that it takes to implement
the STA (twice for compression and expansion and under the assumption that the
thermalization times are much shorter than the compression and expansion ones)

$P=\frac{W_{\text{ad}}}{2(L_{0}+L_{1}+\tau)}\leq\frac{W_{\text{ad}}}{2(L_{0}+L_{1})}.$
(18)

In Fig. 4 we compare the power given by an engine under a quantum Otto cycle
whose expansion and compression strokes given either by the reference or the
effective STA trajectory. We can see that for slow motions (ie, large $\tau$)
they converge to the same value, since $\tau\ll L_{0}$ and $W\approx
W_{\text{ad}}$, but for extremely fast motion the power of the reference
trajectory decreases rapidly and becomes negative while the STA, which always
has a superior efficiency, also provides a higher power. Perturbative
calculations to second order in $\epsilon$ also show that the power associated
with the reference trajectory decays as $1/\tau^{4}$ for short cycle times
Otto_nos  in concordance with the figure presented here.

Even though we have given an explicit simple way to compute an STA that
maximizes both the efficiency and the power delivered, it may be very
difficult to implement in practice. For example for an Otto cycle, the
efficiency is given by

$\eta_{\text{otto}}=1-\frac{L_{1}}{L_{0}}\leq\eta_{\text{carnot}}=1-\frac{T_{1}}{T_{0}},$
(19)

and is bounded by Carnot’s efficiency. Then given a cavity of length $L_{0}$
and thermal baths $T_{1}$ and $T_{0}$, if we want to achieve the best possible
efficiency the final length is fixed to $L_{1}/L_{0}=T_{1}/T_{0}$.
Implementing then the best possible STA forces us to move the mirror at the
speed $v=(L_{0}-L_{1})/(L_{0}+L_{1})$, which for $T_{0}\gg T_{1}$ gives
$L_{0}\gg L_{1}$, and the speed $v\approx 1$ approximates the speed of light,
leading to an experimental impossibility.

Figure 2:  Energy density for an adiabatic shortcut corresponding to the
polynomial trajectory with lengths $L_{0}=1$, $L_{1}=0.7$ and $\tau=1$ from
$t=-L_{0}$ to $t=L_{1}+\tau$ and the field initially in the vacuum state. The
energy density is a negative constant before and after the compression and it
is smaller at the end.

Alternatively, we can set the problem with the constraint that the speed of
the mirror should be at most a constant $v$. Then, to get the most possible
power we would take an STA with constant speed
$v=(L_{0}-L_{1})/(L_{0}+L_{1})$. This would set the final length to
$L_{1}=L_{0}(1-v)/(1+v)$. We can now clearly see that the efficiency

$\eta_{\text{otto}}=1-\frac{(1-v)}{(1+v)}$ (20)

is directly bounded by the maximum speed the mirror can reach.

Figure 3: Adiabaticity parameter as a function of time for the reference
(solid line) and effective trajectory (dashed line) at three different
temperatures. An inset plot has been added to better visualize the curves for
$T=0$ and $T=5$. The parameters used for the reference trajectory were
$L_{0}=1$, $\epsilon=0.3$ and $\tau=1$. Figure 4:  Power for an Otto cycle as
a function of the timescale, $\tau$, implementing the reference (solid line)
or effective STA trajectory (dashed line) for the expansion and compression
strokes. The parameters used for the trajectory were $L_{0}=1$ and
$\epsilon=0.3$, while the thermal baths used for the cycle had temperatures
$T_{0}=1$ and $T_{1}=5$.

We now address some related problems: Is it possible to perform a translation
of the cavity avoiding the excitation of the modes? For a non-rigid cavity
this can be trivially achieved by a two step STA, moving first the right
mirror and then the left mirror, both following STA. This translation without
excitation may be relevant for discussions on relativistic quantum information
where it has been shown the DCE can degrade the entanglement between observers
in relative motion RQI . More generally, the STA protocol presented here could
be useful for other conformal field theories satisfying time dependent
boundary conditions, providing a route for modifying these conditions without
altering the state of the field in a bounded region.

Is it possible to have a STA for a single accelerated mirror? In other words,
what happens with the population of the field modes outside the cavity? In
this case there are no STA: assuming that the mirror is initially static,
accelerates during some time, and finally becomes at rest, the total energy
outside the cavity is strictly positive Fulling , and given by the energy of
the created particles Walker . Unlike what happens inside the cavity, the DCE
cannot be reversed.

Finally, we describe a possible experimental realization of the shortcut.
Superconducting circuits have proved to be useful to simulate a one
dimensional cavity with a moving mirror Nori . Indeed, the DCE has been
observed experimentally using a superconducting waveguide ended by a SQUID,
and the time dependent external condition is implemented by varying the
magnetic flux $\Phi(t)$ on the SQUID Delsing . In order to perform a STA, one
should consider a closed waveguide ended by a SQUID, and apply an effective
time dependent magnetic flux $\Phi_{\rm eff}(t)$. When the field is initially
in the vacuum state, no photons should be detected after applying the magnetic
field on the SQUID. Indeed, we have recently proposed to implement an Otto
cycle in this system Otto_nos , and showed that, for expansion and compression
of small amplitude, it is possible to avoid the DCE and maximize the
efficiency of the cycle. The results of this paper show that the DCE can be
avoided for trajectories of arbitrary amplitude.

This research was supported by Agencia Nacional de Promoción Científica y
Tecnológica (ANPCyT), Consejo Nacional de Investigaciones Científicas y
Técnicas (CON- ICET), Universidad de Buenos Aires (UBA) and Univer- sidad
Nacional de Cuyo (UNCuyo). P.I.V. acknowledges ICTP-Trieste Associate Program.

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