# Microwave cavity-free hole burning spectroscopy of Er3+:Y2SiO5 at
millikelvin temperatures

A. Mladenov  Experimentalphysik, Universität des Saarlandes, D-66123
Saarbrücken, Germany  Institute for Nuclear Research and Nuclear Energy,
Bulgarian Academy of Sciences, Sofia  N. Pankratova  Department of Physics,
Joint Quantum Institute and Center for Nanophysics and Advanced Materials,
University of Maryland, USA  D. Sholokhov  Experimentalphysik, Universität des
Saarlandes, D-66123 Saarbrücken, Germany  V. Manucharyan  Department of
Physics, Joint Quantum Institute and Center for Nanophysics and Advanced
Materials, University of Maryland, USA  R. Gross  Experimentalphysik,
Universität des Saarlandes, D-66123 Saarbrücken, Germany  Walther-Meißner-
Institut, Bavarian academy of sciences, Garching, Germany  Munich Center for
Quantum Science and Technology (MCQST), Munich, Germany  Technical University
of Munich (TUM), Munich, Germany  P. A. Bushev  Experimentalphysik,
Universität des Saarlandes, D-66123 Saarbrücken, Germany  JARA-Institute for
Quantum Information (PGI-11), Forschungszentrum Jülich, 52428 Jülich, Germany
N. Kukharchyk  Experimentalphysik, Universität des Saarlandes, D-66123
Saarbrücken, Germany  Walther-Meißner-Institut, Bavarian academy of sciences,
Garching, Germany  Munich Center for Quantum Science and Technology (MCQST),
Munich, Germany<EMAIL_ADDRESS>
###### Abstract

Efficient quantum memory is of paramount importance for long-distance quantum
communications, as well as for complex large-scale computing architectures. We
investigate the capability of Er3+:Y2SiO5 crystal to serve as a quantum memory
for the travelling microwave photons by employing techniques developed for
dense optical ensembles. In our efforts to do so, we have performed high-
resolution microwave spectroscopy of Er3+:Y2SiO5, where we identified
electronic spin as well as hyperfine transitions. Furthermore, we have
explored spectral hole burning technique and studied the spin relaxation
process at millikelvin temperatures, determined the main relaxation
mechanisms, which lay the groundwork for further studies of the topic.

##  I Introduction

Microwave regime in quantum technologies is currently best represented by the
superconducting circuits based on superconducting qubits [1]. While the
quantum processing units based on such superconducting qubits demonstrate
outstanding results [2], further enhancement of calculation capacities
addresses the necessity of effective microwave quantum memory [3]. At the same
time, the longest storage times are demonstrated with optical quantum
memories[4, 5], which, however, rely on ultra-long coherence of the spin
states of the rare earth spin ensembles.

Realization of purely microwave quantum memories has been demonstrated with
electronic spins ensembles based on donors in silicon and rare earth ions [6].
Depending on the choice of the quantization axis and magnetic field working
point, storage times in such ensembles may reach six hours [4]. Such extremely
long storage time is possible due to using the zero first-order Zeeman (ZEFOZ)
transitions, which are analogous to the clock transitions and possess strongly
reduced sensitivity to the fluctuations of the external decoherence sources.
Coupling of the microwave fields to the spin ensembles in microwave
experiments is typically performed via resonating structures [7, 8, 9, 10],
which limits the number of transitions accessible at the same time.

Ability to effectively control the electronic spins with the propagating
microwave signals [11, 12, 13, 14, 15], which can couple to the same spins in
broad frequency range, will pave the way to a large variety of microwave
quantum memory schemes including more than two electronic levels. The atomic
frequency comb (AFC) protocol is one of the most promising quantum memory
protocols established in optics[16, 17, 18], which can be realized on two
electronic levels as well as extended to a third longer living state also
allowing for frequency domain multiplexing storage protocols [19, 20].
Deterministic narrow spectral hole burning (SHB) is a necessary step in
realization of AFC. While these techniques are strongly established in optics,
they have not yet been widely studied in the microwave domain. The first
attempt in burning microwave spectral holes has been performed on spin
ensembles coupled to a resonator [21].

In this work, we explore the spectral hole burning with the propagating
microwave signals in Er:Y2SiO5 crystal in the cavity-free regime, which allows
for addressing the electronic spins in a broad frequency range. We
characterize the control efficiency in our experimental scheme, identify the
relaxation rates and processes, and extract the resulting spectral hole
profiles. We address the potential of the SHB with the propagating microwave
signals for the microwave quantum memory techniques.

##  II Experimental detail

Figure 1: (Colour online) (a) Experimental scheme: Er:Y2SiO5 crystal is placed
on top of the SC chip with meandered co-planar transmission line. The DC
magnetic field is applied along the surface of the chip. VNA is the radio-
frequency vector network analyser, which supplies the probing signal for
spectroscopy and hole recovery measurement. RFS is a radio frequency source,
which supplies the hole burning pulse. The VNA and a microwave switch are
triggered by the RFS. (b) Photo of the crystal on the meandered transmission
line. (c) Illustration of the level scheme and the relaxation processes
responsible for the relaxation. $W_{ff}$, $W_{D}$ and $\Gamma_{SD}$ stand for
the flip-flop process, direct process and spectral diffusion, respectively.
(d) Simulation of the microwave mode propagating in a coplanar transmission
line used in experiment, which is shown as an RMS value of the propagating
magnetic field component. Simulation is performed in CST Studio Suite.

We investigate the Erbium-doped Y2SiO5 (YSO) crystal, which is supplied by
Scientific Materials Inc. with 0.02$\%$ of Erbium atomic concentration. Erbium
is in its natural abundance with $\sim 23\%$ being 167Er
($I=\nicefrac{{7}}{{2}}$), and $\sim 77\%$ being 5 isotopes with zero nuclear
spin. The crystal has dimensions
$$3\text{\,}\mathrm{mm}$\times$4\text{\,}\mathrm{mm}$\times$5\text{\,}\mathrm{mm}$$
and is placed on the top of the co-planar superconducting transmission line,
see Fig. 1(a,b). The Erbium spins are inductively coupled to the transmission
line via magnetic field created by the propagating microwave signal. The
orientation of the optical crystal axes is shown in Fig. 1(a). The external
magnetic field vector and axes D2 [22] and b are located in the plane of the
SC chip with an angle of 45∘ between axis b and magnetic field vector. The
magnetic field vector is parallel/anti-parallel to the wave-vector of the
microwave field. Due to the low crystalline and point symmetry of YSO crystal,
Erbium ions occupy two crystallographic nonequivalent sites, each showing two
magnetically nonequivalent transitions. The orientation of the crystalline
axes with respect to the magnetic field vector has been selected to fully lift
the degeneracy between the magnetically nonequivalent sites.

Schematics of the measurement scheme is shown in Fig. 1(c). In the microwave
spectroscopy and in hole burning measurements, the probing signal is created
by Vector Network Analyzer (VNA) at the power of 0 dBm, which is additionally
attenuated by $-95\text{\,}\mathrm{dB}$ before reaching the sample. After the
crystal, the transmitted signal is amplified by cryogenic low noise amplifier
by $40\text{\,}\mathrm{dB}$ and by room-temperature low noise amplifiers by
$60\text{\,}\mathrm{dB}$.

Microwave hole-burning pulse is generated by a radio-frequency source (RFS) in
the power range of 5 dBm to 15 dBm with a time duration duration of
$0.5\text{\,}\mathrm{s}$ to $5\text{\,}\mathrm{s}$. The burning pulse is
attenuated on the input RF lines of the cryostat by $-55\text{\,}\mathrm{dB}$.
At the end of the burning pulse, VNA starts measuring time dependence of the
transmission at a selected frequency. In order to protect the VNA input, it is
cloaked from the high-power burning pulses by an RF switch, with both VNA and
switch being triggered by the RFS. Both, change of absorption amplitude and
phase are recorded as functions of time with intermediate frequency filter
bandwidth (IFBW) of $10\text{\,}\mathrm{Hz}$ and sampling interval of
$\sim$$50\text{\,}\mathrm{ms}$.

##  III Microwave spectroscopy

Figure 2: (Colour online) (a) 3D plot of the microwave absorption spectrum on
magnetic field and probing frequency. The absorption lines are marked
according to the identified microwave transitions. Hole burning spectroscopy
is performed on the strongest S1b transition. Additional resonances at fixed
frequencies are due to reflections on cable connections and bonding on the
sample holder and the chip with the transmission line, see Fig. 1 for the
measurement setup details.
(b) Measured (black lines) and simulated (blue lines) absorption spectra at
selected magnetic fields of $110\text{\,}\mathrm{mT}$ and
$185\text{\,}\mathrm{mT}$. The identified transitions are highlighted with
light blue in the measured spectra. The S1b transition studied with hole
burning technique is highlighted with red color. Amplitude of the simulated
spectra is given in arbitrary units.
(c) Measured absorption profiles (circles) and Lorentzian fits (solid lines)
of the absorption line and extracted spectral hole at
$167\text{\,}\mathrm{mT}$. The inset depicts measurement sequence for the
spectral hole recovery.

Microwave spectroscopy is performed on
I${}_{15/2}(0)\ket{-\frac{1}{2}}~{}\leftrightarrow~{}$I${}_{15/2}(0)\ket{+\frac{1}{2}}$
spin transition as a function of magnetic field at a base temperature of
cryostat equal to $10\text{\,}\mathrm{mK}$, see fig. 2(a). The absorption
spectra are measured as S21 parameter of VNA. In the spectrum, three strong
absorption lines correspond to the crystallographic (S1 or S2) and magnetic
(S_a or S_b) nonequivalent sites of the erbium ions and are identified with
the available g-tensor [23] as marked in Figure 2(a,b). The fourth absorption
line, S1a, possesses the largest g-factor of $\simeq 13$ and is outside the
measured frequency range at applied magnetic fields. The hyperfine transitions
of 167Er isotope with the nuclear spin of $\nicefrac{{7}}{{2}}$ are observed
at a much smaller absorption amplitude. The most pronounced absorption line
corresponds to the S1b site, on which we investigate the hole-burning
technique in this work.

The structure of microwave spectrum is determined by the Zeeman, hyperfine and
quadrupole interactions, where the two last ones are only present for 167Er.
The full spin hamiltonian then looks the following:

$\mathcal{H}=\mu_{B}\mathbf{B}\cdot\mathbf{g_{s}}\cdot\mathbf{S}+\mathbf{S}\cdot\mathbf{A}\cdot\mathbf{I}+\mathbf{I}\cdot\mathbf{Q}\cdot\mathbf{I},$
(1)

where $\mu_{B}$ is the Bohr magneton, vectors $\mathbf{S}$ and $\mathbf{I}$
are the electron and nuclear spin numbers, respectively. For the ground state
of erbium, electron spin equals $S=\nicefrac{{1}}{{2}}$, and nuclear spin of
167Er equals $I=\nicefrac{{7}}{{2}}$. Tensors $\mathbf{g_{s}}$, $\mathbf{I}$
and $\mathbf{Q}$ represent matrix elements for electron Zeeman, hyperfine and
quadrupole interactions, respectively. To simulate the full spectrum, we take
advantage of the EasySpin simulation package [24], where we specify the
$\mathbf{g_{s}}$, $\mathbf{I}$ and $\mathbf{Q}$ tensors, taken from [23], and
fraction of erbium isotopes with $\textrm{I}=0$ and
$\textrm{I}=\nicefrac{{7}}{{2}}$ in the naturally abundant spin ensemble. When
comparing the measured g-factors to the simulated values, we find misalignment
angles, which are about $3^{\circ}$ in b-D2 plane and $0.5^{\circ}$ in D1-b
plane for site 1, and $6^{\circ}$ for site 2 in both planes. Small
misalignment in b-D2 plane can be partially attributed to a misalignment of
the crystal on the SC chip. It cannot explain, however, large discrepancy on
site 2. Similar misalignment angles have been observed by Welinski et al.
[25], where the discrepancy on the site 2 has been attributed to a rotation of
the C2 point-symmetry axis of the substituting rare earth ion with respect to
the original Yttrium ion.

Comparison of the measured microwave spectrum against the one, simulated with
corrected axes, is shown in fig. 2(b) for magnetic fields of
$110\text{\,}\mathrm{mT}$ and $185\text{\,}\mathrm{mT}$, while numeric values
for the extracted g-factors are given in table 1.

Table 1: Overview of g-factor values obtained from simulated g-tensor and from the experiment. | S1a | S1b | S2a | S2b
---|---|---|---|---
simulated [23] | 13.11 | 1.41 | 1.89 | 2.87
measured | - | 1.41 | 1.87 | 2.87

The further absorption spectrum analysis and hole-burning investigation are
performed on the strongest transition S1b with a g-factor of 1.41, which is
composed of $42\text{\,}\mathrm{\char 37\relax}$ of 168Er,
$35\text{\,}\mathrm{\char 37\relax}$ of 166Er, and $23\text{\,}\mathrm{\char
37\relax}$ 170Er. To characterize the line, we convert it to the amplitude
units and fit to the Lorentzian lineshape function, an exemplary fit is shown
in fig. 2(c). The extracted linewidth values depend on magnetic field as
$\Gamma_{\mathrm{FWHM}}=\Gamma_{0}+\overline{\delta\gamma}B$, where
$\Gamma_{\mathrm{FWHM}}$ is the full width at half maximum, $\Gamma_{0}$ is
the linewidth at zero magnetic field and is found to be
$17(4)\text{\,}\mathrm{MHz}$, and $B$ is the magnetic field. The inhomogeneity
of the gyromagnetic ratio, $\overline{\delta\gamma}$, equals to
$0.21\text{\,}\mathrm{MHz}\text{/}\mathrm{mT}$, which is
$1.1\text{\,}\mathrm{\char 37\relax}$ of the g-factor value. The
$\Gamma_{\mathrm{FWHM}}$ is also in agreement with previous publications if
taking into account the magnetic field working point [8, 26].

Temperature of a spin ensemble, often addressed as an effective temperature,
becomes an important parameter at millikelvin temperatures. Even small
excitation result into elevated effective temperatures of spin ensembles,
which can be an order of magnitude higher then the temperature measured on the
cryostat, which makes it necessary to identify real effective temperature of
the spin ensemble. The key issue with the spin temperature is that the spin
levels are strongly quantized with respect to the thermal energy level of the
surrounding thermal bath. As a result, the effective spin ensemble temperature
is defined as a probability of finding the electronic spin in a particular
energy level, and it is calculated via Boltzmann distribution for a canonical
ensemble [27, 28].

To find the temperature of the spin system in the spectroscopy measurement, we
fit the area under the absorption line to the Boltzmann distribution on
temperature, given as $\sim\exp(\nicefrac{{\Delta
E}}{{k_{B}T_{s}}})/(1+\exp(\nicefrac{{\Delta B}}{{k_{B}T_{s}}}))$, where
$\Delta E$ is the energy difference between the spin states, $k_{B}$ is the
Boltzmann constant, and $T_{s}$ = $81.9(20)\text{\,}\mathrm{mK}$ is the spin
temperature. At the same time, the base temperature on the cryostat plate is
measured by a sensor to be equal to $11\text{\,}\mathrm{mK}$. Thermal
interface from the spin ensemble to the cryostat is mediated by spin-phonon
interaction (phonon bath) and thermal boundary resistance (Kapitza
resistance). The later one lead to a significant suppression of the thermal
energy flow from spin ensemble to the cooling element and result into such a
difference in temperature.

Figure 3: The spectral hole relaxation times as a function of magnetic field
(a) and temperature (b). Dots and filled symbols represent the experimental
data, the solid lines show the fits of experimental data to the model, and red
dashed and dotted lines outline the contributions of the flip-flop and direct
processes to the total relaxation rate.

##  IV Dynamics of spectral holes

Spectral hole burning is performed in a saturation recovery experiment, when a
microwave burning pulse of a fixed frequency is sent to the crystal for a
fixed time. The population of the ground state is then transferred to the
excited state, as indicted with the red arrow in fig. 1(c). As a result, the
absorption amplitude is reduced. The recovery of the saturation amplitude over
time is detected with a week probing signal of VNA and is governed by the
dynamics of relaxation processes.

### Relaxation processes.

Below $1\text{\,}\mathrm{K}$, spin relaxation is limited to two main
processes: flip-flop, $R_{\textrm{FF}}$, and direct process,
$R_{\textrm{d}}=\frac{1}{\tau_{\textrm{d}}}$. These processes are
schematically shown in fig. 1(c). Third process depicted in fig. 1 is spectral
diffusion, $\Gamma_{SD}$, which contributes into broadening of the spectral
hole.

The spin-spin relaxation rate given by flip-flop process is defined as [29]

$\displaystyle
R_{\textrm{FF}}=w_{\textrm{FF}}\tanh[2](\frac{g\mu_{\textrm{B}}B}{2k_{\textrm{B}}T_{\textrm{s}}}),$
(2)

where $w_{\textrm{FF}}$ is the flip-flop coefficient, which, following the
approach from Car et al. [30], we estimate to be equal
$50\text{\,}\mathrm{mHz}$. Constants $\mu_{\textrm{B}}$ and $k_{\textrm{B}}$
are Bohr magneton and Boltzmann constant, respectively, and $T_{\textrm{s}}$
is the effective temperature of the spin system.

Rate of the direct process over the magnetic field and temperature in absence
of the phonon bottleneck is given by [31]

$\displaystyle
R_{\textrm{d}}=\frac{1}{\tau_{\textrm{d}}}=w_{\textrm{d}}g^{5}\textrm{B}^{5}\coth(\frac{g\mu_{\textrm{B}}B}{2k_{\textrm{B}}T_{\textrm{s}}}),$
(3)

where the spin-phonon relaxation rate coefficient $w_{\textrm{d}}$ is
estimated to be equal to $23\text{\,}\mathrm{Hz}\mathrm{T}^{-5}$ [31, 32, 7].

Thus, the final equation including flip-flop and direct process relaxation
rates reads as $R=R_{\textrm{FF}}+R_{\textrm{d}}$, which is then fit
independently to the magnetic field, fig. 3(a), and temperature, fig. 3(b),
dependent data. Derived coefficients are listed in table 2 and show very good
agreement with analytically derived values.

### Effective temperature.

The effective temperature of the spin ensemble $T_{s}$, which is derived from
relaxation processes, is in a good agreement with the spin temperature derived
from the microwave spectrum, $\sim$70\text{\,}\mathrm{mK}$$, see table 2 for
detail. It also means that below magnetic fields of
$B_{pol}\simeq$2\text{\,}\frac{\mathrm{T}}{\mathrm{K}}$\cdot
T_{s}\simeq$140\text{\,}\mathrm{mT}$$ the spin ensemble is not polarized to
the ground state. This results into stronger flip-flop rate as well as into
smaller hole amplitudes leading to smaller signal-to-noise ratio in the data.
Below the magnetic field of $70\text{\,}\mathrm{mT}$, splitting of the spin
states is smaller than the thermal energy level,
${k_{B}T_{s}}\geq{g\mu_{B}B}$, and saturation recovery is not measurable.

Apart from extracting temperature of the spin ensemble, we directly measure
the temperature of the cryostat itself which reaches down to
$11\text{\,}\mathrm{mK}$. Temperature of the cryostat remains unchanged during
the sweep of the magnetic field and is controllably changed during the
temperature-dependent measurement. For fitting the relaxation processes to
experimental data, we introduce the correction of the actual temperature of
the spin ensemble with respect to that of the cryostat [32, 33]:

$T_{s}=T_{min}(1+(\nicefrac{{T}}{{T_{min}}})^{2})^{\nicefrac{{1}}{{2}}},$ (4)

where $T$ is the temperature of the cryostat and $T_{min}$ is minimal
temperature attainable by the spin system. At the limit of $T<T_{min}$,
$T_{s}=T_{min}$, while for $T>T_{min}$, $T_{s}=T$. Relaxation rates extracted
thus from the temperature dependent data are in good agreement with the
magnetic field dependence. Largest declination of the data points from the
fit-curves are observed for the lowest and higher fields, see fig. 3(b).
However, such a straggle of the data points is similar to that observed for
the multiple measurements in the magnetic field dependence, fig. 3(a).

Table 2: Coefficients of relaxation processes and effective temperature values | $w_{D}$ | $w_{ff}$ | $T_{s}$ | R-Sq.
---|---|---|---|---
| $\mathrm{Hz}$ | $\mathrm{mHz}$ | $\mathrm{mK}$ | %
Fit of magnetic field dependence | $23(1)$ | $50(1)$ | $69(8)$ | 89
Fit of temperature dependence | $22(2)$ | $30(1)$ | $73(21)$ | 85
Analytically derived | $23$ |  $51$ |  |
Spectroscopy |  |  | $81.9$ | 50

### Rabi frequencies and acting microwave power.

For efficient control of the a spin ensemble over the transmission line, we
need to know the effective acting power of the magnetic field component of the
microwave signal and correct Rabi frequency. To identify the Rabi frequency,
we vary the length of the burning pulse and measure the maximally achieved
amplitude of the spectral hole. From the observed oscillations, we derive the
Rabi frequency to be equal to $\simeq$3.9\text{\,}\mathrm{Hz}$$. Taking into
account the size of the propagating microwave mode, see fig. 1(d), we derive
the RMS amplitude of the magnetic component of the microwave field
$B_{AC}\simeq$0.35\text{\,}\mathrm{nT}$$, or total acting microwave power
$P_{act-R}\simeq$45\text{\,}\mathrm{fW}$~{}(-105\,\textrm{dBm})$.

From the spectroscopy measurement, we also derive additional losses on the
sample and SC to be in the order of $-40\text{\,}\mathrm{dB}$. Simulating the
propagation of the microwave mode as shown in fig. 1(d), we estimate the mean
loss of amplitude in the main acting volume of the microwave mode to be in the
order of $-20\text{\,}\mathrm{dB}$. Applying the total loss from the
spectroscopy measurement, comprising of $\sim$-115\text{\,}\mathrm{dB}$$, see
fig. 1, to the microwave power of $15\,\textrm{dBm}$ generated by the RFS, we
obtain expected acting power of $P_{act-a}=-100\,\textrm{dBm}$ in the
microwave mode. We thus see that the acting powers from Rabi frequency,
$P_{act-R}$, and from analytical calculations, $P_{act-a}$, are in good
agreement. The further hole burning experiment is performed with
$5\,\textrm{dBm}$ output power of RFS, with Rabi frequency of
$\simeq$1\text{\,}\mathrm{Hz}$$, at which we have not observed any pronounced
oscillations when varying the pulse duration.

##  V Spectral hole profiles

By keeping the burning frequency fixed and changing the probing frequency, we
identify the profiles of the spectral holes. Such a three-dimensional spectrum
of a spectral hole is shown in fig. 4(a), where the pump frequency was kept
fixed at $3.651\text{\,}\mathrm{GHz}$ and probing frequency was scanned around
the pump with total span of $20\text{\,}\mathrm{MHz}$. When increasing the
burning time, the amplitude of the spectral hole is increased, while the width
of the spectral hole and relaxation time are independent of the burning time,
which has been confirmed at several magnitudes of magnetic field. Therefore,
the burning pulse length of $3\text{\,}\mathrm{s}$ has been selected for the
hole burning experiment as an optimal one. Independence of the spectral hole
width of the burning pulse length is attributed to the rate of spectral
diffusion, which is faster ($\sim$1\text{\,}\mathrm{kHz}$$) than possible
shortest burning pulse ($0.1\text{\,}\mathrm{s}$-$1\text{\,}\mathrm{s}$).

Figure 4: (a) Time-dependent profile of the microwave spectral hole a function
of time and frequency detuning from the central burning frequency, and (b)
relaxation time at different detuning frequencies as compared to the
absorption line profile, which are measured at $185\text{\,}\mathrm{mT}$. (c)
Spectral hole width as a function of relaxation time at
$167\text{\,}\mathrm{mT}$ and $185\text{\,}\mathrm{mT}$.

The profiles of spectral holes have been probed for a set of magnetic filed
values. By fitting the profiles to the Lorentzian shape, we extract amplitude
and width of the created spectral holes as functions of time, fig. 4, and of
magnetic field. In the range of magnetic fields below
$170\text{\,}\mathrm{mT}$, area and amplitude of the holes are nearly
constant. At increase of magnetic field above $170\text{\,}\mathrm{mT}$, the
width of spectral hole increases while amplitude decreases. The relative
amplitude of spectral holes is at the level of $\sim$7.5\text{\,}\mathrm{\char
37\relax}$$ of the spectral line amplitude, and similarly the relative width
slightly varies around $\sim$65\text{\,}\mathrm{\char 37\relax}$$ of the
spectral line width.

Measuring the relaxation of the spectral hole over its profile, we observed it
to be faster when measured outside of width at half maximum for magnetic field
of $185\text{\,}\mathrm{mT}$, see fig. 4(b). Inside of the spectral hole width
region, the relaxation time can be considered constant. Such behavior of
spectral holes appears then as narrowing of the spectral hole with time as
shown in fig. 4(c). Interestingly, the spectral hole burned at
$167\text{\,}\mathrm{mT}$, which profile is demonstrated in fig. 2(c), has
shown nearly no change of the width over time. Moreover, there is a narrowing
of the spectral hole after $20\text{\,}\mathrm{s}$ of relaxation, after which
the width returns to it’s initial value of $\simeq$37\text{\,}\mathrm{MHz}$$.
The wider spectral hole observed at $185\text{\,}\mathrm{mT}$ converges to a
width value of $\simeq$40\text{\,}\mathrm{MHz}$$, which is similar to that at
$167\text{\,}\mathrm{mT}$ resulting approximately $60\text{\,}\mathrm{\char
37\relax}$ of the absorption line width. At other magnetic fields, width of
spectral holes is also $\sim 60\%$, revealing no change over the time. We
suggest that the large initial broadening on spectral hole at
$185\text{\,}\mathrm{mT}$ and it’s subsequent narrowing is related to
interference of one of the on-chip parasitic resonance at a close frequency
($\sim$3.75\text{\,}\mathrm{GHz}$$) to the transition in-study.

## Conclusions

In conclusion, we have for the first time demonstrated a cavity-free microwave
spectral hole burning on Erbium-doped Y2SiO5 crystal. We were able to achieve
efficiency of spectral hole of $60\text{\,}\mathrm{\char 37\relax}$ in width
and $7.5\text{\,}\mathrm{\char 37\relax}$ in amplitude in the range of
magnetic fields up to $300\text{\,}\mathrm{mT}$. From the spectral hole
recovery over time, main relaxation processes are identified: flip-flop
process and direct process. The effective temperature of the spin ensemble
from both spectral hole burning and spectroscopy converged to
$\simeq$70\text{\,}\mathrm{mK}$$. The achieved in this work results on
spectral hole width and depth are limited by the duration of the hole-burning
pulse and the acting microwave power. We have demonstrated the hole burning
over a large span of magnetic fields and frequencies, accessible in absence of
resonator. Further enhancement of the control of the spin systems in cavity-
free regime require exploiting the zero first-order Zeeman transitions and
increase of the acting microwave power in burning pulse. Results presented in
this publication pave the way towards the realization of cavity-free microwave
quantum memory based on rare-earth spin ensembles.

## Acknowledgement

### Funding information

This work is supported by DFG through the grants INST 256/415-1, BU 2510/2-1
and EXC 2111.

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