apsrev4-1

# Sideband Cooling Micromechanical Motion to the Quantum Ground State

J. D. Teufel  National Institute of Standards and Technology, Boulder, CO
80305, USA  T. Donner  JILA, National Institute of Standards and Technology
and the University of Colorado, Boulder, CO 80309, USA  Dale Li  National
Institute of Standards and Technology, Boulder, CO 80305, USA  J. W. Harlow
JILA, National Institute of Standards and Technology and the University of
Colorado, Boulder, CO 80309, USA  Department of Physics, University of
Colorado, Boulder, Colorado 80309, USA  M. S. Allman  National Institute of
Standards and Technology, Boulder, CO 80305, USA  K. Cicak  National Institute
of Standards and Technology, Boulder, CO 80305, USA  A. J. Sirois  National
Institute of Standards and Technology, Boulder, CO 80305, USA  J. D. Whittaker
National Institute of Standards and Technology, Boulder, CO 80305, USA  K. W.
Lehnert  JILA, National Institute of Standards and Technology and the
University of Colorado, Boulder, CO 80309, USA  Department of Physics,
University of Colorado, Boulder, Colorado 80309, USA  R. W. Simmonds  National
Institute of Standards and Technology, Boulder, CO 80305, USA

The advent of laser cooling techniques revolutionized the study of many
atomic-scale systems. This has fueled progress towards quantum computers by
preparing trapped ions in their motional ground state Diedrich1989 , and
generating new states of matter by achieving Bose-Einstein condensation of
atomic vapors Anderson1995 . Analogous cooling techniques Braginsky1992 ;
Kippenberg2008  provide a general and flexible method for preparing
macroscopic objects in their motional ground state, bringing the powerful
technology of micromechanics into the quantum regime. Cavity opto- or electro-
mechanical systems achieve sideband cooling through the strong interaction
between light and motion Braginsky1970 ; Blair1995 ; Teufel2008 ; Thompson2008
; Groblacher2009a ; Park2009 ; Lin2009 ; Schliesser2009 ; Rocheleau2010 ;
Riviere2010 ; Li2011 . However, entering the quantum regime, less than a
single quantum of motion, has been elusive because sideband cooling has not
sufficiently overwhelmed the coupling of mechanical systems to their hot
environments. Here, we demonstrate sideband cooling of the motion of a
micromechanical oscillator to the quantum ground state. Entering the quantum
regime requires a large electromechanical interaction, which is achieved by
embedding a micromechanical membrane into a superconducting microwave resonant
circuit. In order to verify the cooling of the membrane motion into the
quantum regime, we perform a near quantum-limited measurement of the microwave
field, resolving this motion a factor of 5.1 from the Heisenberg limit
Braginsky1992 . Furthermore, our device exhibits strong-coupling allowing
coherent exchange of microwave photons and mechanical phonons Teufel2010 .
Simultaneously achieving strong coupling, ground state preparation and
efficient measurement sets the stage for rapid advances in the control and
detection of non-classical states of motion Bose1997 ; Mancini1997 , possibly
even testing quantum theory itself in the unexplored region of larger size and
mass Marshall2003 . The universal ability to connect disparate physical
systems through mechanical motion naturally leads towards future methods for
engineering the coherent transfer of quantum information with widely different
forms of quanta.

Mechanical oscillators that are both decoupled from their environment (high
quality factor $Q$) and placed in the quantum regime could allow us to explore
quantum mechanics in entirely new ways Bose1997 ; Mancini1997 ; Marshall2003 ;
Akram2010 ; Regal2011 . For an oscillator to be in the quantum regime, it must
be possible to prepare it in its ground state, to arbitrarily manipulate its
quantum state, and to detect its state near the Heisenberg limit. In order to
prepare an oscillator in its ground state, its temperature $T$ must be reduced
such that $k_{\mathrm{B}}T<\hbar\Omega_{\mathrm{m}}$, where
$\Omega_{\mathrm{m}}$ is the resonance frequency of the oscillator,
$k_{\mathrm{B}}$ is Boltzmann’s constant, and $\hbar$ is the reduced Planck’s
constant. While higher resonance frequency modes ($>1$ GHz) can meet this
cooling requirement with conventional refrigeration ($T<50$ mK), these stiff
oscillators are difficult to control and to detect within their short
mechanical lifetimes. One unique approach using passive cooling has
successfully overcome these difficulties by using a piezoelectric dilatation
oscillator coupled to a superconducting qubit OConnell2010 . Unfortunately,
this method is incompatible with the broad range of lower frequency, high Q,
flexural mechanical modes. In order to take advantage of the attractive
mechanical properties of these oscillators, an alternative active cooling
method is required, one that can reduce the oscillator’s temperature below
that of the surrounding environment.

Cavity opto- or electro-mechanical systems Kippenberg2008  naturally offer a
method for both detecting mechanical motion and cooling a mechanical mode to
its ground state Marquardt2007 ; Wilson2007 . An object whose motion alters
the resonance frequency $\omega_{\mathrm{c}}$ of an electromagnetic cavity
experiences a radiation pressure force governed by the parametric interaction
Hamiltonian: $\hat{H}_{\mathrm{int}}=\hbar G\hat{n}\hat{x}$, where
$G=d\omega_{\mathrm{c}}/dx$, $\hat{n}$ is the cavity photon number, and
$\hat{x}$ is the displacement of the mechanical oscillator. By driving the
cavity at a frequency $\omega_{\mathrm{d}}$, the oscillator’s motion produces
upper and lower sidebands at $\omega_{\mathrm{d}}\pm\Omega_{\mathrm{m}}$.
Because these sideband photons are inelastically scattered from the drive
field, they provide a way to exchange energy with the oscillator. If the drive
field is optimally detuned below the cavity resonance
$\Delta\equiv\omega_{\mathrm{d}}-\omega_{\mathrm{c}}=-\Omega_{\mathrm{m}}$,
photons will be preferentially up-converted to $\omega_{\mathrm{c}}$ because
the photon density of states is maximal there (Fig 1b). When an up-converted
photon leaves the cavity, it removes the energy of one mechanical quantum (one
phonon) from the motion. Thus, the mechanical oscillator is damped and cooled
via this radiation-pressure force. Because the mechanical motion is encoded in
scattered photons exiting the cavity, a quantum-limited measurement of this
photon field provides a near Heisenberg-limited detection of mechanical motion
Clerk2010 .

While there has been substantial progress in cooling mechanical oscillators
with radiation pressure forces, sideband cooling to the quantum mechanical
ground state has been an outstanding challenge. Cavity optomechanical systems
have realized very large sideband cooling rates Thompson2008 ; Groblacher2009a
; Park2009 ; Lin2009 ; Schliesser2009 ; Riviere2010 ; Li2011 ; however, these
rates are not sufficient to overcome the larger thermal heating rates of the
mechanical modes. Because electromechanical experiments use much lower-energy
photons Braginsky1970 ; Blair1995 ; Teufel2008 ; Rocheleau2010 , they are
naturally compatible with operation below $100$ mK, but have consequently
suffered from weak electromechanical interactions and inefficient detection of
the photon fields.

Here, we present a cavity electromechanical system where a flexural mode of a
thin aluminum membrane is parametrically coupled to a superconducting
microwave resonant circuit. Unlike previous microwave systems, this device
achieves large electromechanical coupling by concentrating nearly all the
microwave electric fields near the mechanical oscillator Teufel2010 . The
oscillator is a $100$ nm thick aluminum membrane with a diameter of 15 µm,
suspended $50$ nm above a second aluminum layer on a sapphire substrate
Cicak2010  (see Fig. 1). These two metal layers form a variable parallel-plate
capacitor that is shunted by a $12$ nH spiral inductor. This combination of
capacitor and inductor creates a microwave cavity whose resonance frequency
depends on the mechanical displacement of the membrane and is centered at
$\omega_{\mathrm{c}}=2\pi\times 7.54$ GHz. The device is operated in a
dilution refrigerator at $15$ mK, where aluminum is superconducting, and the
microwave cavity has a total energy decay rate of $\kappa\approx 2\pi\times
200$ kHz. As expected from the dimensions of the membrane,
$\Omega_{\mathrm{m}}=2\pi\times 10.56$ MHz, and we find an intrinsic damping
rate of $\Gamma_{\mathrm{m}}=2\pi\times 32$ Hz, resulting in a mechanical
quality factor
$Q_{\mathrm{m}}=\Omega_{\mathrm{m}}/\Gamma_{\mathrm{m}}=3.3\times 10^{5}$. The
oscillator mass $m=48$ pg implies that the zero-point motion is
$x_{\mathrm{zp}}=\sqrt{\hbar/(2m\Omega_{\mathrm{m}})}=4.1$ fm. With a ratio of
$\Omega_{\mathrm{m}}/\kappa>50$, our system is deep in the resolved-sideband
regime and perfectly suited for sideband cooling to the mechanical ground
state Marquardt2007 ; Wilson2007 .

To measure the mechanical displacement, we apply a microwave field, which is
detuned below the cavity resonance frequency by $\Delta=-\Omega_{\mathrm{m}}$,
through heavily attenuated coaxial lines to the feed line of our device. The
upper sideband at $\omega_{\mathrm{c}}$ is amplified with a custom-built
Josephson parametric amplifier (JPA) Castellanos-Beltran2008 ; Teufel2009
followed by a low-noise cryogenic amplifier, demodulated at room temperature,
and finally monitored with a spectrum analyzer. The thermal motion of the
membrane creates an easily resolvable peak in the microwave noise spectrum. As
described previouslyTeufel2009 , this measurement scheme constitutes a nearly
shot-noise-limited microwave interferometer with which we can measure
mechanical displacement with minimum added noise close to fundamental limits.

In order to calibrate the demodulated signal to the membrane’s motion, we
measure the thermal noise spectrum while varying the cryostat temperature
(Fig. 1c). Here a weak microwave drive ($\sim 3$ photons in the cavity) is
used in order to ensure that radiation pressure damping and cooling effects
are negligible. When $\Omega_{\mathrm{m}}\gg\kappa\gg\Gamma_{\mathrm{m}}$ and
$\Delta=-\Omega_{\mathrm{m}}$, the displacement spectral density $S_{x}$ is
related to the observed microwave noise spectral density $S$ by:
$S_{x}=2(\kappa\Omega_{\mathrm{m}}/G\kappa_{\mathrm{ex}})^{2}S/P_{\mathrm{o}}$,
where $\kappa_{\mathrm{ex}}$ is the coupling rate between the cavity and the
feed line, and $P_{\mathrm{o}}$ is the power of the microwave drive at the
output of the cavity. According to equipartition, the area under the resonance
curve of displacement spectral density $S_{x}$ must be proportional to the
effective temperature of the mechanical mode. This calibration procedure
allows us to convert the sideband in the microwave power spectral density to a
displacement spectral density and to extract the thermal occupation of the
mechanical mode. In Fig. 1c we show the number of thermal quanta in the
mechanical resonator as a function of $T$. The linear dependence of the
integrated power spectral density with temperature shows that the mechanical
mode equilibrates with the cryostat even for the lowest achievable temperature
of $15$ mK. This temperature corresponds to a thermal occupancy
$n_{\mathrm{m}}=30$, where
$n_{\mathrm{m}}=[\exp(\hbar\Omega_{\mathrm{m}}/k_{\mathrm{B}}T)-1]^{-1}$. The
calibration determines the electromechanical coupling strength $G/2\pi=49\pm
2$ MHz/nm. With the device parameters, we can investigate both the fundamental
sensitivity of our measurement as well as the effects of radiation pressure
cooling.

The total measured displacement noise results from two sources: the membrane’s
actual mean-square motion $S_{x}^{\mathrm{th}}$ and the _apparent_ motion
$S_{x}^{\mathrm{imp}}$ due to imprecision of the measurement. Fig. 2a
demonstrates how the use of low-noise parametric amplification significantly
lowers $S_{x}^{\mathrm{imp}}$, resulting in a reduction in the white-noise
background by a factor of more than $30$. This greatly increases the signal-
to-noise ratio of the membrane’s thermal motion, reducing the required
integration time to resolve the thermal peak by a factor of $1000$. To
investigate the measurement sensitivity in the presence of dynamical
backaction, we regulate the cryostat temperature at $20$ mK and increase the
amplitude of the detuned microwave drive while observing modifications in the
displacement spectral density. We quantify the strength of the drive by the
resulting number of photons $n_{\mathrm{d}}$ in the microwave cavity. As shown
in Fig. 2b, the measurement imprecision $S_{\mathrm{x}}^{\mathrm{imp}}$ is
inversely proportional to $n_{\mathrm{d}}$. At the highest drive power
($n_{\mathrm{d}}\approx 10^{5}$), the absolute displacement sensitivity is
$5.5\times 10^{-34}$ m2/Hz.

As expected, the increased drive power also damps and cools the mechanical
oscillator Braginsky1992 ; Marquardt2007 ; Wilson2007 . The total mechanical
dissipation rate $\Gamma_{\mathrm{m}}^{\prime}$ is the sum of the intrinsic
dissipation $\Gamma_{\mathrm{m}}$ and the radiation-pressure-induced damping
resulting from scattering photons to the upper/lower sideband
$\Gamma=\Gamma_{\mathrm{+}}-\Gamma_{\mathrm{-}}$, where
$\Gamma_{\mathrm{\pm}}=4g^{2}\kappa/[\kappa^{2}+4(\Delta\pm\Omega_{\mathrm{m}})^{2}]$.
Here, $g$ is the coupling rate between the cavity and the mechanical mode,
which depends on the amplitude of the drive:
$g=Gx_{\mathrm{zp}}\sqrt{n_{\mathrm{d}}}$. Fig. 2c shows the measured values
of $\kappa$, $g$ and $\Gamma_{\mathrm{m}}^{\prime}$ as the drive increases.
The radiation-pressure damping of the mechanical oscillator becomes pronounced
above a cavity drive amplitude of approximately 75 photons, at which point
$\Gamma=\Gamma_{\mathrm{m}}$ and the mechanical linewidth has doubled.

While the absolute value of the displacement imprecision decreases with
increasing power, the visibility of the thermal mechanical peak no longer
improves once the radiation-pressure force becomes the dominant dissipation
mechanism for the membrane. By expressing the imprecision as equivalent
thermal quanta of the oscillator
$n_{\mathrm{imp}}=\Gamma_{\mathrm{m}}^{\prime}S_{x}^{\mathrm{imp}}/8x_{\mathrm{zp}}^{2}$,
we see that the visibility of the thermal noise above the imprecision no
longer improves once the drive is much greater than $n_{\mathrm{d}}\approx
100$ (Fig. 2d). This is because a linear decrease in $S_{x}^{\mathrm{imp}}$ is
balanced by a linear increase in $\Gamma_{\mathrm{m}}^{\prime}$ due to
radiation-pressure damping. The asymptotic value of $n_{\mathrm{imp}}$ is a
direct measure of the efficiency of the microwave measurement. Ideally, for a
lossless circuit, a quantum-limited microwave measurement would imply
$n_{\mathrm{imp}}=1/4$. The incorporation of the low-noise JPA improves
$n_{\mathrm{imp}}$ close to this ideal limit, reducing the asymptotic value of
$n_{\mathrm{imp}}$ from $70$ to $1.9$ quanta. This level of sensitivity is
crucial, as we will now use this measurement to resolve the residual thermal
motion of the membrane as it is cooled into the quantum regime.

Beginning from a cryostat temperature of $20$ mK and a thermal occupation of
$n_{\mathrm{m}}^{\mathrm{T}}=40$ quanta, the fundamental mechanical mode of
the membrane is cooled by the radiation-pressure forces. Figure 3a shows the
displacement spectral density of the motional sideband as $n_{\mathrm{d}}$ is
increased from 18 to 4,500 photons along with fits to a Lorentzian lineshape
(shaded area). As described above, this increased drive results in three
effects on the spectra: lower noise floor, wider resonances and smaller area.
As it is the area that corresponds to the mean-square motion of the membrane,
it directly measures the effective temperature of the mode. At a drive
intensity that corresponds to 4,000 photons in the cavity, the thermal
occupation is reduced below one quantum of mechanical motion, entering the
quantum regime.

Observing the noise spectrum over a broader frequency range reveals that there
is also a second Lorentzian peak with linewidth $\kappa$ whose area
corresponds to the finite thermal occupation $n_{\mathrm{c}}$ of the cavity.
Over a broad frequency range it is no longer valid to evaluate the cavity
parameters at a single frequency to infer the spectrum in units of $S_{x}$.
Instead, Fig. 3b shows the noise spectrum in units of sideband power
normalized by the power at the drive frequency, $S/P_{\mathrm{o}}$. These two
sources of noise originating from either the mechanical or the electrical mode
interfere with each other and result in noise squashing Rocheleau2010  and
eventually normal-mode splitting Dobrindt2008  once $2g>\kappa/\sqrt{2}$.
Using a quantum-mechanical description applied to our circuit Rocheleau2010 ;
Clerk2010 , the expected noise spectrum is

$S/\hbar\omega=\frac{1}{2}+n_{\mathrm{add}}+\frac{2\kappa_{\mathrm{ex}}\left[\kappa
n_{\mathrm{c}}(\Gamma_{\mathrm{m}}^{2}+4\delta^{2})+4\Gamma_{\mathrm{m}}n_{\mathrm{m}}^{\mathrm{T}}g^{2}\right]}{\left|4g^{2}+\left(\kappa+2j(\delta+\widetilde{\Delta})\right)\left(\Gamma_{\mathrm{m}}+2j\delta\right)\right|^{2}}\
$ (1)

where $\delta=\omega-\Omega_{\mathrm{m}}$,
$\widetilde{\Delta}=\omega_{\mathrm{d}}+\Omega_{\mathrm{m}}-\omega_{\mathrm{c}}$,
and $n_{\mathrm{add}}$ is added noise of the microwave measurement expressed
as an equivalent number of microwave photons. Fig. 3b shows the measured
spectra and corresponding fits (shaded region) to Eq. 1 as the
electromechanical system evolves first into the quantum regime
($n_{\mathrm{m}},n_{\mathrm{c}}<1$) and then into the strong-coupling regime
($2g>\kappa/2$). The results are summarized in Fig. 3c, where the thermal
occupancy of both the mechanical and electrical modes are shown as a function
of $n_{\mathrm{d}}$. For low drive power, the cavity shows no resolvable
thermal population (to within our measurement uncertainty of 0.05 quanta) as
expected for a $7.5$ GHz mode at $20$ mK. While it is unclear whether the
observed population at higher drive power is a consequence of direct heating
of the substrate, heating of the microwave attenuators preceding the circuit,
or intrinsic cavity frequency noise, we have determined that it is not the
result of frequency or amplitude noise of our microwave generator, as this
noise is reduced far below the microwave shot-noise level with narrow-band
filtering and cryogenic attenuation (see Supplementary Information). Sideband
cooling can never reduce the occupancy of the mechanical mode below that of
the cavity. Therefore, in order for the system to access the quantum regime,
the thermal population of the cavity must remain less than one quantum.
Assuming $\Omega_{\mathrm{m}}\gg\kappa$, the final occupancy of a mechanical
mode is Dobrindt2008

$n_{\mathrm{m}}=n_{\mathrm{m}}^{\mathrm{T}}\left(\frac{\Gamma_{\mathrm{m}}}{\kappa}\frac{4g^{2}+\kappa^{2}}{4g^{2}+\kappa\Gamma_{\mathrm{m}}}\right)+n_{\mathrm{c}}\left(\frac{4g^{2}}{4g^{2}+\kappa\Gamma_{\mathrm{m}}}\right).$
(2)

This equation shows that for moderate coupling
($\sqrt{\kappa\Gamma_{\mathrm{m}}}\ll g\ll\kappa$) the cooling of the
mechanical mode is linear in the number of drive photons. Beyond this regime,
the onset of normal-mode splitting abates further cooling. Here the mechanical
cooling rate becomes limited not by the coupling between the mechanical mode
and the cavity, but instead by the coupling rate $\kappa$ between the cavity
and its environment Dobrindt2008 . Thus, the final occupancy of the mechanical
mode can never be reduced to lower than
$n_{\mathrm{m}}^{\mathrm{T}}\Gamma_{\mathrm{m}}/\kappa$, and a stronger
parametric drive will only increase the rate at which the thermal excitations
Rabi oscillate between the cavity and mechanical modes. For our device we
achieve the desired hierarchy: as the coupling is increased, we first cool to
the ground state and then enter the strong-coupling regime
($n_{\mathrm{m}}^{\mathrm{T}}\Gamma_{\mathrm{m}}<\kappa<g$). Once
$n_{\mathrm{d}}$ exceeds $2\times 10^{4}$, the mechanical occupancy converges
toward the cavity population, reaching a minimum of $0.34\pm 0.05$ quanta. At
the highest power drive power ($n_{\mathrm{d}}=2\times 10^{5}$) the mechanical
mode has hybridized with the cavity, resulting in the normal-mode splitting
characteristic of the strong-coupling regime Teufel2010 . This level of
coupling is required to utilize the hybrid system for quantum information
processing, as it is only in the strong-coupling regime that a quantum state
may be manipulated faster than it decoheres from the coupling of either the
electromagnetic or mechanical modes to the environment.

Together the measurements shown in Fig. 2 and 3 quantify the overall
measurement efficiency of the system. The Heisenberg limit requires that a
continuous displacement measurement is necessarily accompanied by a backaction
force Braginsky1992 ; Clerk2010 ; Schliesser2009 , such that
$\sqrt{S_{x}^{\mathrm{imp}}S_{F}}\geq\hbar$, where $S_{F}$ is the force noise
spectral density. From the thermal occupancy and damping rate of the
mechanical mode, we extract the total force spectral density
$S_{F}=4\hbar\Omega_{\mathrm{m}}m\Gamma_{\mathrm{m}}^{\prime}(n_{\mathrm{m}}+1/2)$.
This places a conservative upper bound on the quantum backaction by assuming
that it alone is responsible for the finite occupancy of the mechanical mode.
This experiment achieves the closest approach to Heisenberg-limited
displacement detection to date Clerk2010 ; Riviere2010  with a lowest
imprecision-backaction product
$\sqrt{S_{\mathrm{x}}^{\mathrm{imp}}S_{\mathrm{F}}}=4\hbar\sqrt{n_{\mathrm{imp}}(n_{\mathrm{m}}+1/2)}=(5.1\pm
0.4)\hbar$. Thus, this mechanical device simultaneously demonstrates ground-
state preparation, strong-coupling and near quantum-limited detection.

Looking forward, this technology offers a feasible route to achieve many of
the longstanding goals for quantum _mechanical_ systems. These prospects
include a direct measurement of the zero-point motion, observation of the
fundamental asymmetry between the rate of emission and absorption of phonons
Diedrich1989 , quantum nondemolition measurements Braginsky1992  and
generation of entangled states of mechanical motion Bose1997 ; Mancini1997 .
Furthermore, combining this device with a single-photon source and detector
(such as a superconducting qubit Hofheinz2009 ; OConnell2010 ) would enable
preparation of arbitrary quantum states of mechanical motion as well as
observation of a single excitation as it Rabi oscillates between a $7$ GHz
photon and a $10$ MHz phonon Akram2010 . Because the interaction between the
mechanical mode and the cavity is parametric, the coupling strength is
inherently tunable and can be turned on and off quickly. Thus, once a quantum
state is transfered into the mechanical mode, it can be stored there for a
time
$\tau_{\mathrm{th}}=1/(n_{\mathrm{m}}^{\mathrm{T}}\Gamma_{\mathrm{m}})>100$ µs
before absorbing one thermal phonon from its environment. As this timescale is
much longer than typical coherence times of superconducting qubits, mechanical
modes offer the potential for delay and storage of quantum information.
Lastly, because mechanical oscillators can couple to light of any frequency,
they could serve as a unique intermediary that transfers quantum information
between the microwave and optical domains Regal2011 .

These measurements demonstrate the power of sideband techniques to cool a
macroscopic ($\sim 10^{12}$ atoms) mechanical mode, beyond what is feasible
with conventional refrigeration techniques, into the quantum regime. These
broadly applicable methods for state preparation, manipulation and detection,
pave the way to access the quantum nature of a wide class of long-lived
mechanical oscillators. Through the strong interaction between photons and
phonons, mechanical systems can now inherit the experimental and theoretical
power of quantum optics, opening the field of quantum acoustics.

##  I Acknowledgements

We thank A. W. Sanders for taking the micrograph in Fig. 1a and thank the JILA
instrument shop for fabrication and design of the cavity filter. This work was
financially supported by NIST and the DARPA QuASAR program. T.D. acknowledges
support from the Deutsche Forschungsgemeinschft (DFG). Contribution of the
U.S. government, not subject to copyright.

##  II Author Information

Reprints and permissions information is available at www.nature.com/reprints.
The authors declare no competing financial interests. Correspondence and
requests for materials should be addressed to J.D.T (john.teufel@nist.gov).

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Figure 1: Schematic description of the experiment. a, Colorized scanning
electron micrograph showing the aluminum (grey) electromechanical circuit
fabricated on a sapphire (blue) substrate, in which a 15 µm diameter membrane
is lithographically suspended 50 nm above a lower electrode. The membrane’s
motion modulates the capacitance, and hence, the resonance frequency of the
superconducting microwave circuit. b, A coherent microwave drive inductively
coupled to the circuit acquires modulation sidebands due to the thermal motion
of the membrane. The upper sideband is amplified with a nearly quantum-limited
Josephson parametric amplifier within the cryostat. c, The microwave power in
the upper sideband provides a direct measurement of the thermal occupancy of
the mechanical mode, which may be calibrated _in situ_ by varying the
temperature of the cryostat. The mechanical mode shows thermalization with the
cryostat at all temperatures, yielding a minimum thermal occupancy of $30$
mechanical quanta without employing sideband-cooling techniques. The inset
illustrates the concept of sideband cooling. When the circuit is excited with
a detuned microwave drive such that $\Delta=-\Omega_{\mathrm{m}}$, the narrow
line shape of the electrical resonance ensures that photons are preferentially
scattered to higher energy, providing a cooling mechanism for the membrane.
Figure 2:  Displacement sensitivity in the presence of radiation-pressure
damping. a, The displacement spectral density measured with (red) and without
(blue) the Josephson parametric amplifier. As the parametric amplifier greatly
reduces the total noise of the microwave measurement, the time required to
resolve the thermal motion is reduced by a factor of $1000$. b, As the
microwave drive power is increased, the absolute displacement sensitivity,
$S_{\mathrm{x}}^{\mathrm{imp}}$ improves, reaching a minimum of $5.5\times
10^{-34}$ m2/Hz at the highest power. c, The parametric coupling $g$ between
the microwave cavity and the mechanical mode increases as
$\sqrt{n_{\mathrm{d}}}$. This coupling damps the mechanical mode from its
intrinsic linewidth of $\Gamma_{\mathrm{m}}=2\pi\times 32$ Hz until it is
increased to that of the microwave cavity $\kappa$. d, The relative
measurement imprecision, in units of mechanical quanta, depends on the product
of $S_{\mathrm{x}}^{\mathrm{imp}}$ and $\Gamma_{\mathrm{m}}^{\prime}$. Thus,
once the power is large enough that radiation-pressure damping overwhelms the
intrinsic mechanical dissipation, $n_{\mathrm{imp}}$ asymptotically approaches
a constant value ($n_{\mathrm{imp}}=1.9$), which is a direct measure of the
overall efficiency of the photon measurement. Figure 3:  Sideband cooling the
mechanical mode to the ground state. a, The displacement noise spectra and
Lorentzian fits (shaded region) for five different drive powers. With higher
power, the mechanical mode is both damped (larger linewidth) and cooled
(smaller area) by the radiation pressure forces. b, Over a broader frequency
span, the normalized sideband noise spectra clearly show both the narrow
mechanical peak and a broader cavity peak due to finite occupancy of the
mechanical and electrical modes, respectively. A small, but resolvable,
thermal population of the cavity appears as the drive power increases, setting
the limit for the final occupancy of the coupled optomechanical system. At the
highest drive power, the coupling rate between the mechanical oscillator and
the microwave cavity exceeds the intrinsic dissipation of either mode, and the
system hybridizes into optomechanical normal modes. c, Starting in thermal
equilibrium with the cryostat at $T=20$ mK, sideband cooling reduces the
thermal occupancy of the mechanical mode from $n_{\mathrm{m}}=40$ into the
quantum regime, reaching a minimum of $n_{\mathrm{m}}=0.34\pm 0.05$. These
data demonstrate that the parametric interaction between photons and phonons
can initialize the strongly coupled, electromechanical system in its quantum
ground state.

Supplementary Information for “Sideband Cooling Micromechanical Motion to the
Quantum Ground State”

##  III Noise spectrum of an optomechanical system

A mechanical degree of freedom that parametrically couples to the cavity
resonance frequency modifies the power emerging from the cavity by scattering
photons to the upper or lower mechanical sidebands. To calculate the full
noise spectrum of the optomechanical system, we follow the general method of
input-output theory S_Walls1994 . We define
$g=Gx_{\mathrm{zp}}\sqrt{n_{\mathrm{d}}}$, where $G=d\omega_{\mathrm{c}}/dx$,
$x_{\mathrm{zp}}=\sqrt{\hbar/2m\Omega_{\mathrm{m}}}$, $m$ is the mass,
$\omega_{\mathrm{c}}$ is the cavity resonance frequency, $\Omega_{\mathrm{m}}$
is the mechanical resonance frequency and $n_{\mathrm{d}}$ is the number of
photons in the cavity due to a drive at frequency $\omega_{\mathrm{d}}$.
Furthermore, we define the response functions of the mechanical and cavity
modes as $\chi_{\mathrm{c}}^{-1}=\kappa/2+j(\delta+\widetilde{\Delta})$ and
$\chi_{\mathrm{m}}^{-1}=\Gamma_{\mathrm{m}}/2+j\delta$, where
$\Gamma_{\mathrm{m}}$ is the mechanical dissipation rate, $\kappa$ is the
cavity dissipation rate, $\delta=\omega-\Omega_{\mathrm{m}}$,
$\widetilde{\Delta}=\omega_{\mathrm{d}}-\omega_{\mathrm{c}}+\Omega_{\mathrm{m}}$
and $j=\sqrt{-1}$. $\kappa$ is total cavity dissipation rate due to both the
intentional coupling to the transmission line $\kappa_{\mathrm{ex}}$ and the
intrinsic losses $\kappa_{0}$. From these parameters, we define the
optomechanical self-energy S_Marquardt2007 ; S_Clerk2010  as a function of
$\delta$:

$\displaystyle\Sigma(\delta)$
$\displaystyle=-jg^{2}\left[\chi_{\mathrm{c}}(\delta)-\chi_{\mathrm{c}}^{*}(\delta+2\Omega_{\mathrm{m}})\right]$
($\mathrm{S}$1) $\displaystyle\approx-jg^{2}\chi_{\mathrm{c}}(\delta)$
($\mathrm{S}$2)

The approximation assumes that the drive is near the optimal detuning for
cooling ($|\widetilde{\Delta}|\ll\Omega_{\mathrm{m}}$) and the system is
sufficiently in the good-cavity limit ($\Omega_{\mathrm{m}}\gg\kappa$) such
that the cavity response at $(\delta+2\Omega_{\mathrm{m}})$ may be neglected.
Now the effective mechanical response function $\widetilde{\chi}_{\mathrm{m}}$
including the optomechanical effects is:

$\displaystyle\widetilde{\chi}_{\mathrm{m}}$
$\displaystyle=\frac{\chi_{\mathrm{m}}}{1+j\chi_{\mathrm{m}}\Sigma}$
($\mathrm{S}$3)
$\displaystyle\approx\frac{\chi_{\mathrm{c}}^{-1}}{g^{2}+\chi_{\mathrm{m}}^{-1}\chi_{\mathrm{c}}^{-1}}$
($\mathrm{S}$4)

The noise at the output of the cavity is characterized by the noise operator
$\hat{b}_{\mathrm{out}}$, which is related to the cavity field operator
$\hat{a}$ by $\hat{b}_{\mathrm{out}}=\sqrt{\beta\kappa_{\mathrm{ex}}}\hat{a}$.
$\beta$ is a dimensionless factor that depends on the geometry. Our circuit
(shown schematically in Fig. S1) couples power from the cavity equally to the
output and back to the input so here $\beta=1/2$. In principle, this fraction
could be engineered by coupling asymmetrically to the input and the output, or
by using a single port cavity ($\beta=1$).

Figure S1:  Cavity coupling block diagram. Figure S2:  Detailed schematic
diagram. A microwave generator creates a tone at the drive frequency. This
signal is filtered with a resonant cavity at room temperature and split into
two arms. The first arm excites the cavity through approximately 53 dB of
cryogenic attenuation. In order to avoid saturating the low-noise amplifier
with the microwave drive tone, the second arm is used to cancel the drive
before amplification. A computer-controlled variable attenuator and phase
shifter are run in a feedback loop to maintain cancellation at the part per
million level. A second microwave generator is used to provide the pump tone
for the Josephson parametric amplifier (JPA) as well as the reference
oscillator for the mixer. This pump tone is $1.3$ MHz above
$\omega_{\mathrm{c}}$ so that the JPA is operated as a non-degenerate
parametric amplifier, which measures both quadratures of the electromagnetic
filed at the upper sideband frequency. The last stage of attenuation on all
lines occurs inside a $20$ dB directional coupler, which allows us to minimize
the microwave power dissipated on the cold stage of the cryostat. The JPA is a
reflection amplifier; a signal incident on the strongly coupled port of the
JPA is reflected and amplified. A cryogenic circulator is used to separate the
incident and reflected waves, defining the input and output ports of the JPA.
The other circulators are used to isolate the cavity from the noise emitted
from the amplifier’s input.

Following directly the theoretical analysis of previous work S_Rocheleau2010 ;
S_Clerk2010 , we consider the noise operators $\hat{\eta}_{\mathrm{m}}$ and
$\hat{\eta}_{\mathrm{c}}$ associated with the mechanical and cavity modes
respectively, which satisfy the relations
$\langle\hat{\eta}_{\mathrm{m}}^{\dagger}\hat{\eta}_{\mathrm{m}}\rangle=n_{\mathrm{m}}^{T}$
and
$\langle\hat{\eta}_{\mathrm{c}}^{\dagger}\hat{\eta}_{\mathrm{c}}\rangle=n_{\mathrm{c}}$.
Thus, the output noise is S_Rocheleau2010

$\displaystyle\hat{b}_{\mathrm{out}}=$
$\displaystyle-\sqrt{\beta\kappa_{\mathrm{ex}}}\chi_{\mathrm{c}}\sqrt{\kappa}\left(1-g^{2}\widetilde{\chi}_{\mathrm{m}}\chi_{\mathrm{c}}\right)\hat{\eta}_{\mathrm{c}}$
$\displaystyle-\sqrt{\beta\kappa_{\mathrm{ex}}}\chi_{\mathrm{c}}\sqrt{\Gamma_{\mathrm{m}}}\left(jg\widetilde{\chi}_{\mathrm{m}}\right)\hat{\eta}_{\mathrm{m}}.$

In the frequency domain, the power spectral density of the noise at the output
(in units of W/Hz) is
$S=\hbar\omega\langle\hat{b}_{\mathrm{out}}^{\dagger}\hat{b}_{\mathrm{out}}\rangle$,

$\displaystyle S$
$\displaystyle=\frac{4\hbar\omega\beta\kappa_{\mathrm{ex}}(\Gamma_{\mathrm{m}}^{2}+4\delta^{2})\kappa
n_{\mathrm{c}}}{\left|4g^{2}+\left(\kappa+2j(\delta+\widetilde{\Delta})\right)\left(\Gamma_{\mathrm{m}}+2j\delta\right)\right|^{2}}$
$\displaystyle+\frac{16\hbar\omega\beta\kappa_{\mathrm{ex}}g^{2}\Gamma_{\mathrm{m}}n_{\mathrm{m}}^{\mathrm{T}}}{\left|4g^{2}+\left(\kappa+2j(\delta+\widetilde{\Delta})\right)\left(\Gamma_{\mathrm{m}}+2j\delta\right)\right|^{2}}.$

The first term simply represents the thermal noise of a cavity with occupancy
$n_{\mathrm{c}}$ whose spectral weight is distributed over the ‘dressed’
cavity mode. The ‘dressed’ cavity mode includes the effect of optomechanically
induced transparency S_Agarwal2010; S_Weis2010; S_Teufel2010  and reduces to a
single Lorentzian lineshape in the limit of weak coupling
($g\ll\sqrt{\kappa\Gamma_{\mathrm{m}}}$). The second term is the thermal noise
of the mechanical mode with its modified mechanical susceptibility. Unlike
previous derivations S_Rocheleau2010 , we have not assumed the weak-coupling
regime. Thus, as this equation is valid in both the weak- and strong-coupling
regimes, it gives a unified description of the thermal noise spectrum even in
the presence of normal-mode splitting. Finally, the total noise at the output
of the measurement including the vacuum noise of the photon field and the
added noise of the measurement is

$\frac{S}{\hbar\omega}=\frac{1}{2}+n_{\mathrm{add}}^{\prime}+\frac{4\beta\kappa_{\mathrm{ex}}\left[\kappa
n_{\mathrm{c}}(\Gamma_{\mathrm{m}}^{2}+4\delta^{2})+4\Gamma_{\mathrm{m}}n_{\mathrm{m}}^{\mathrm{T}}g^{2}\right]}{\left|4g^{2}+\left(\kappa+2j(\delta+\widetilde{\Delta})\right)\left(\Gamma_{\mathrm{m}}+2j\delta\right)\right|^{2}},$
($\mathrm{S}$5)

where $n_{\mathrm{add}}^{\prime}$ is the total added noise of the measurement
in units of equivalent number of photons. For an ideal measurement (_i.e._ for
a quantum-limited measurement of both quadratures of the light field),
$n_{\mathrm{add}}^{\prime}=1/2$.

Before the onset of normal-mode splitting, one can directly relate the
measured microwave power spectrum $S$ to the displacement spectral density
$S_{x}$. Assuming $\widetilde{\Delta}=0$, $n_{\mathrm{c}}\ll n_{\mathrm{m}}$
and $g,\delta\ll\kappa$,

$\displaystyle\frac{S}{\hbar\omega}$
$\displaystyle=\frac{1}{2}+n_{\mathrm{add}}^{\prime}+4\beta\frac{\kappa_{\mathrm{ex}}}{\kappa}\Gamma\frac{\Gamma_{\mathrm{m}}n_{\mathrm{m}}^{\mathrm{T}}}{\left(\Gamma_{\mathrm{m}}+\Gamma\right)^{2}+4\delta^{2}}$
($\mathrm{S}$6)
$\displaystyle=\frac{1}{2}+n_{\mathrm{add}}^{\prime}+\frac{2\beta
G^{2}n_{\mathrm{d}}}{\kappa}\frac{\kappa_{\mathrm{ex}}}{\kappa}S_{x},$
($\mathrm{S}$7)

where $\Gamma=4g^{2}/\kappa$ is the optomechanical damping rate.

##  IV Microwave measurement and calibration

The detailed circuit diagram for our measurements is shown in Fig. S2. In
order to calibrate the value of $g_{0}=Gx_{\mathrm{zp}}$ for this device, we
applied a microwave drive optimally red-detuned ($\widetilde{\Delta}=0$) and
measured the thermal noise spectrum of the mechanical oscillator as a function
of cryostat temperature. Here we restricted $n_{\mathrm{d}}\approx 3$ in order
to ensure that radiation pressure effects are negligible. With the value of
$g_{0}$ now determined, we increase the drive amplitude and measure the
thermal noise spectrum at each drive power. The noise spectra are recorded and
averaged with commercial FFT spectrum analyser. Each spectrum is typically an
average of 500 traces with a measurement time of 0.5 s per trace. The cavity
response is then measured with a weak probe tone with a vector network
analyser to determine precise cavity parameters at each microwave drive power,
including the precise detuning and $\kappa$. For larger microwave drive powers
where the cavity spectrum exhibits optomechanically induced transparency
effects S_Agarwal2010; S_Weis2010; S_Teufel2010 , this spectrum also serves as
a direct measure of $g$. Finally, using additional calibration tones, each
noise spectrum is calibrated in units of absolute microwave noise quanta and
fit with Eq. 5 to determine the occupancy of both the cavity and mechanical
modes.

For our measurements, we infer that our entire measurement chain has an
effective added noise of $n_{\mathrm{add}}^{\prime}=2.1$. This value is
consistent with the independently measured value for the added noise of the
JPA ($n_{\mathrm{add}}=0.8$) and the $2.5$ dB of loss between the output of
the cavity and the JPA S_Castellanos-Beltran2008 ; S_Teufel2009 .

Figure S3:  Measured transmission of filter cavity. A tunable resonant cavity
was implemented at room temperature in order to suppress noise $\sim 10$ MHz
above the drive frequency. As shown here, this cavity reduces the noise at the
cavity frequency by more than 40 dB, ensuring that the phase or amplitude
noise of the generator is not responsible for the finite occupancy of the
cavity at large drive power.

In order to ensure that the amplitude or phase noise of the signal generator
was not responsible for the finite occupancy of the cavity at high drive
power, we designed and built a custom filter cavity S_Rocheleau2010 . As shown
in Fig. S3, when the filter cavity is tuned to precisely the frequencies of
our circuit, it provides an addition 40 dB of noise suppression at the cavity
resonance frequency. The phase and amplitude noise of our signal generator
alone are specified by the manufacturer to be less than -150 dBc at Fourier
frequencies $10$ MHz away from the drive. With the addition of filter cavity,
we lower this noise to well below the shot-noise level of our microwave drive.
Furthermore, even without the filter cavity, we could not resolve an
appreciable difference in the cavity occupation. Thus, while we do not know
the precise mechanism for this occupancy, we conclusively determine the
generator noise is not the cause.

##  V Inferring cavity parameter and number of drive photons

The measured microwave cavity parameters may be inferred from the transmitted
power spectrum. The power at the output of the cavity $P_{\mathrm{o}}$ is
related to the input power $P_{\mathrm{i}}$ by S_Teufel2010

$P_{\mathrm{o}}=P_{\mathrm{i}}\left(\frac{\kappa_{\mathrm{0}}^{2}+4\Delta^{2}}{\kappa^{2}+4\Delta^{2}}\right),$
($\mathrm{S}$8)

where $\Delta=\omega_{\mathrm{d}}-\omega_{\mathrm{c}}$ is the difference
between the frequency of the drive $\omega_{\mathrm{d}}$ and the cavity
resonance frequency $\omega_{\mathrm{c}}$. $\kappa$ is the total intensity
decay rate of the cavity (full width at half maximum) with
$\kappa=\kappa_{\mathrm{0}}+\kappa_{\mathrm{ex}}$. $\kappa_{\mathrm{0}}$ is
the coupling rate to the dissipative environment, and $\kappa_{\mathrm{ex}}$
is the coupling rate to the transmission line used to excite and monitor the
cavity.

The number of photons in the cavity due to a coherent input drive at detuning
$\Delta$ may be calculated from the stored energy $E$ in the cavity.

$n_{\mathrm{d}}=\frac{E}{\hbar\omega_{\mathrm{d}}}=\frac{2P_{\mathrm{i}}}{\hbar\omega_{\mathrm{d}}}\frac{\kappa_{\mathrm{ex}}}{\kappa^{2}+4\Delta^{2}}$
($\mathrm{S}$9)

For our circuit, $\kappa_{\mathrm{ex}}=2\pi\times 133$ kHz. Thus, when the
drive is optimally detuned such that $\Delta=-\Omega_{\mathrm{m}}$, the input
power required to excite the cavity with one photon is $P_{\mathrm{i}}\approx
2\hbar\omega_{\mathrm{d}}\Omega_{\mathrm{m}}^{2}/\kappa_{\mathrm{ex}}\approx
50$ fW.

##  VI Fundamental limits of sideband cooling

Equation 2 in the main text gives an expression for the final occupancy of a
mechanical mode, assuming that the microwave drive is optimally detuned
($\Delta=-\Omega_{\mathrm{m}}$). This expression is only the lowest order
approximation in the small quantities $g/\Omega_{\mathrm{m}}$ and
$\kappa/\Omega_{\mathrm{m}}$. Up to second order, the final occupancy is
S_Dobrindt2008

$\displaystyle n_{\mathrm{m}}$
$\displaystyle=n_{\mathrm{m}}^{\mathrm{T}}\left(\frac{\Gamma_{\mathrm{m}}}{\kappa}\frac{4g^{2}+\kappa^{2}}{4g^{2}+\kappa\Gamma_{\mathrm{m}}}\right)\left[1+\frac{g^{2}}{\Omega_{\mathrm{m}}^{2}}\frac{4g^{2}+\kappa\Gamma_{\mathrm{m}}}{4g^{2}+\kappa^{2}}\right]$
$\displaystyle+n_{\mathrm{c}}\left(\frac{4g^{2}}{4g^{2}+\kappa\Gamma_{\mathrm{m}}}\right)\left[1+\frac{8g^{2}+\kappa^{2}}{8\Omega_{\mathrm{m}}^{2}}\frac{4g^{2}+\kappa\Gamma_{\mathrm{m}}}{4g^{2}}\right]$
$\displaystyle+\frac{8g^{2}+\kappa^{2}}{16\Omega_{\mathrm{m}}^{2}}.$

The last term represents the fundamental limit for sideband cooling and
demonstrates the importance of the resolved-sideband regime. For our system,
$\Omega_{\mathrm{m}}\gg\kappa,g$; and hence this last term only contributes
negligibly to the final occupancy of the mechanical mode ($<10^{-4}$ quanta).

###  VI.1 Measurement imprecision and backaction

Throughout the main text and this supplementary information, we use the
“single-sided” convention for all spectral densities in which for any quantity
$A$, the mean-square fluctuations are
$\left<A^{2}\right>=\int_{0}^{\infty}S_{A}(\omega)\frac{d\omega}{2\pi}$. This
yields the familiar classical result that an oscillator coupled to a thermal
bath of temperature $T$ will experience a random force characterized by the
force spectral density $S_{F}=4k_{B}Tm\Gamma_{\mathrm{m}}$. More generally,

$S_{F}=4\hbar\Omega_{\mathrm{m}}\left(n_{\mathrm{m}}^{\mathrm{T}}+\frac{1}{2}\right)m\Gamma_{\mathrm{m}}\,,$
($\mathrm{S}$10)

where $n_{\mathrm{m}}^{\mathrm{T}}$ is the Bose-Einstein occupancy factor
given by
$n_{\mathrm{m}}^{\mathrm{T}}=[\exp(\hbar\Omega_{\mathrm{m}}/k_{B}T)-1]^{-1}$.

Independent of any convention for defining the spectral density, the
visibility of a thermal mechanical peak of given mechanical occupancy above
the noise floor of the measurement represents a direct measure of the overall
efficiency of the detection. As shown in Fig. S4, ratio of the peak height to
the white-noise background allows us to quantify the imprecision of the
measurement in units of mechanical quanta S_Teufel2009 ,
$n_{\mathrm{imp}}\equiv
S_{x}^{\mathrm{imp}}m\Omega_{\mathrm{m}}\Gamma_{\mathrm{m}}^{\prime}/(4\hbar)$.
Inspection of Eq. S6 implies

$n_{\mathrm{imp}}=\frac{1}{4\beta}\frac{\kappa}{\kappa_{\mathrm{ex}}}\frac{4g^{2}+\kappa\Gamma_{\mathrm{m}}}{4g^{2}}\left(\frac{1}{2}+n_{\mathrm{add}}^{\prime}\right)\\\
$ ($\mathrm{S}$11)

Figure S4:  Measurement imprecision in units of mechanical quanta.

Once the drive is strong enough that $g\gg\sqrt{\kappa\Gamma_{\mathrm{m}}}$),
$n_{\mathrm{imp}}$ no longer decreases with increasing drive. It is precisely
because we are measuring with a detuned drive that also damps the mechanical
motion, that $n_{\mathrm{imp}}$ asymptotically approaches a constant value
S_Clerk2010 ; S_Schliesser2009 . For an ideal measurement
($\beta=1,\kappa=\kappa_{\mathrm{ex}}$, and $n_{\mathrm{add}}^{\prime}=1/2$),
$n_{\mathrm{imp}}\rightarrow 1/4$. Implicit in obtaining this optimal value
for $n_{\mathrm{add}}^{\prime}$ and hence $n_{\mathrm{imp}}$ is that all the
photons exiting the cavity are measured. Any losses between the cavity and the
detector can be modeled as a beam-splitter that only transmits a fraction
$\eta$ of the photons to the detector and adds a fraction $(1-\eta)$ of vacuum
noise. So the effective added noise $n_{\mathrm{add}}^{\prime}$ accounting for
these losses becomes

$n_{\mathrm{add}}^{\prime}=\frac{n_{\mathrm{add}}}{\eta}+\left(\frac{1-\eta}{\eta}\right)\frac{1}{2},$
($\mathrm{S}$12)

Thus, shot-noise limited detection of the photons ($n_{\mathrm{add}}=1/2$) is
a necessary, but not sufficient, condition for reaching the best possible
level of precision.

Quantum mechanics also requires that a continuous displacement measurement
must necessarily impart a force back on the measured object. For an optimally
detuned drive ($\widetilde{\Delta}=0$) in the resolved-sideband regime, this
backaction force spectral density $S_{F}^{\mathrm{ba}}$ approaches a constant
value as a function of increasing drive strength and asymptotically approaches
$S_{F}^{\mathrm{ba}}=2\hbar\Omega_{\mathrm{m}}m\Gamma_{\mathrm{m}}^{\prime}$.
Again, expressing the spectral density in units of mechanical quanta gives
$n_{\mathrm{ba}}\equiv
S_{F}^{\mathrm{ba}}/(4\hbar\Omega_{\mathrm{m}}m\Gamma_{\mathrm{m}}^{\prime})\rightarrow
1/2$.

Fundamentally, the Heisenberg limit does not restrict the imprecision
$S_{x}^{\mathrm{imp}}$ or the backaction $S_{F}^{ba}$ alone, but rather it
requires their product has a minimum value S_Braginsky1992 ; S_Clerk2010

$\sqrt{S_{x}^{\mathrm{imp}}S_{F}^{ba}}=4\hbar\sqrt{n_{\mathrm{imp}}n_{\mathrm{ba}}}\geq\hbar.$
($\mathrm{S}$13)

An ideal cavity optomechanical system can achieve this lower limit for a
continuous measurement with a drive applied at the cavity resonance frequency.
When considering the case where the drive is instead applied detuned below the
cavity resonance ($\widetilde{\Delta}=0$), this product never reaches this
lower limit S_Clerk2010 ; S_Schliesser2009  and is at minimum
$\sqrt{S_{x}^{\mathrm{imp}}S_{F}^{ba}}=\hbar\sqrt{2}$.

To estimate these quantities for our measurements, we can infer the total
force spectral density experienced by our oscillator as
$S_{F}^{\mathrm{total}}=4\hbar\Omega_{\mathrm{m}}m\Gamma_{\mathrm{m}}^{\prime}(n_{\mathrm{m}}+1/2)$.
As this total necessarily includes the backaction, we may make the most
conservative assumption that it was solely due to backaction that our
oscillator remained at finite occupancy. Hence, $n_{\mathrm{ba}}\leq
n_{\mathrm{m}}+1/2$. The low thermal occupancies attained in this work allow
us to place an upper bound on how large the backaction could possibly be, and
hence quantify our measurement in terms of approach to the Heisenberg limit.
Thus,
$\sqrt{S_{x}^{\mathrm{imp}}S_{F}^{ba}}=4\hbar\sqrt{n_{\mathrm{imp}}n_{\mathrm{ba}}}\leq
4\hbar\sqrt{n_{\mathrm{imp}}(n_{\mathrm{m}}+1/2)}$. At $n_{\mathrm{d}}=3\times
10^{4}$, we simultaneously achieve $n_{\mathrm{m}}=0.36$ and
$n_{\mathrm{imp}}=1.9$ ($S_{F}^{\mathrm{total}}=1.6\times 10^{-34}$ N2/Hz and
$S_{x}^{\mathrm{imp}}=1.7\times 10^{-33}$ m2/Hz) yielding an upper limit on
the measured product of backaction and imprecision of $5.1~{}\hbar$. As stated
above, the best possible backaction-imprecision product is $\hbar\sqrt{2}$
when using red-detuned excitation; thus our measurement is only a factor of
$3.6$ above this limit. It may also be noted that this factor would have been
$1.8$ except that our chosen geometry losses half of the signal back to the
input ($\beta=1/2$). In future experiments, using a single-port geometry
($\beta=1$) will improve this inefficiency.

## References

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# On the exact divisibility by $5$ of the class number of
some pure metacyclic fields

###### Abstract

Let $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is
a natural number $5^{th}$ power-free. Let
$k=\mathbb{Q}(\sqrt[5]{n},\zeta_{5})$, with $\zeta_{5}$ is a primitive
$5^{th}$ root of unit, be the normal closure of $\Gamma$, and a pure
metacyclic field of degree $20$ over $\mathbb{Q}$. When $n$ takes some
particular forms, we show that $\Gamma$ admits a trivial $5$-class group and
$5$ divides exactly the class number of $k$.

Fouad ELMOUHIB
Department of Mathematics and Computer Sciences,
Mohammed first University, Oujda, Morocco,
Correspondence<EMAIL_ADDRESS>
Mohamed TALBI
Regional Center of Professions of Education and Training,
<EMAIL_ADDRESS>

Abdelmalek AZIZI
Department of Mathematics and Computer Sciences,
Mohammed first University, Oujda, Morocco,
<EMAIL_ADDRESS>
##  1 Introduction

In algebraic number theory, the task to determine the class number of a number
field $k$ is until nowday a classical and difficult open problem, although we
have explicit formulas and relations of class numbers.
In the case when this task is hard to deal with it, its seems satisfactory to
determine the exact power of a prime $p$, which divides the class number of
$k$.
Let $\Gamma=\mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a
$5^{th}$ power-free natural greater than one, and
$k_{0}=\mathbb{Q}(\zeta_{5})$ be the $5^{th}$ cyclotomic field. Then
$k=\Gamma(\zeta_{5})$ is the normal closure of $\Gamma$ and a pure metacyclic
field. There is a question when the class numbers $h_{\Gamma}$ of $\Gamma$ and
$h_{k}$ of $k$ are divisible by $5$. This divisibility was studied by several
researchers until now. Subsequently to Honda’s study of the cubic case [8],
C.Parry [11] studied with difficulty this question. He presented the relation
between $h_{\Gamma}$ and $h_{k}$:

$5^{5}h_{k}=uh_{\Gamma}^{4}$

where $u$ is a divisor of $5^{6}$. He also proved that $h_{\Gamma}$ is
divisible by $5$ if and only if $h_{k}$ is divisible by $5^{2}$. The authors
of [9] gave a several results on the rank of the $5$-Sylow subgroup of the
ideal class group of $\Gamma$ and $k$, by means of tools of genus theory and
Kummer duality.
Due to an investigation of the theory developed in [9], we gave in [1] a full
classification of all normal closures $k$ whenever its $5$-class group is of
type $(5,5)$, and in order to illuminate this classification by numerical
examples obtained by PARI/GP [13], we have noticed that for large values of
some forms of $n$, the $5$-class group of $k$ is cyclic of order $5$, which
means that $h_{k}$ is exactly divisible by $5$. The done calculations allow us
to conjecture that this divisibility is verified for these forms of $n$, see
[[1], conjecture 4.1].
In this paper, we are interested in giving a proof of this conjecture. In fact
we shall prove the following main theorem:

######  Theorem 1.1.

Let $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is
a natural $5^{th}$ power-free. Let $k$ be the normal closure of $\Gamma$.
Denote by $h_{\Gamma,5}$ and $h_{k,5}$ the $5$-class number respectively of
$\Gamma$ and $k$. Let $q_{1}$ and $q_{2}$ be primes such that
$q_{1},q_{2}\,\equiv\,\pm 2\,(\mathrm{mod}\,5)$. If the natural $n$ takes one
forms as follows:

$n\,=\,\left\\{\begin{array}[]{ll}q_{1}^{e_{1}}q_{2}&\text{ with }\quad
q_{i}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\ 5q_{1}&\text{ with }\quad
q_{1}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\ 5q_{1}q_{2}&\text{ with }\quad
q_{1}\,\text{ or }\,q_{2}$ $\not\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\
\end{array}\right.$ (1)

and $q_{2}$ and $5$ are not a quintic residue modulo $q_{1}$, then
$h_{\Gamma,5}$ is trivial and $h_{k,5}=5$.

This result will be underpinned by numerical examples obtained with the
computational number theory system PARI/GP [[13]].

##  2 Norm residue symbol

Let $L/K$ an abelian extension of number fields with conductor $f$. For each
finite or infinite prime ideal $\mathcal{P}$ of $K$, we note by
$f_{\mathcal{P}}$ the largest power of $\mathcal{P}$ that divides $f$. Let
$\beta\in K^{*}$, we determine an auxiliary number $\beta_{0}$ by the two
conditions $\beta_{0}\equiv\beta\,(\mathrm{mod}\,f_{\mathcal{P}})$ and
$\beta_{0}\equiv 1\,(\mathrm{mod}\,\frac{f}{f_{\mathcal{P}}})$. Let
$\mathcal{Q}$ an ideal co-prime with $\mathcal{P}$ such that
$(\beta_{0})=\mathcal{P}^{a}\mathcal{Q}$ ($a=0$ if $\mathcal{P}$ infinite). We
note by

$\left(\frac{\beta,L}{\mathcal{P}}\right)=\left(\frac{L/K}{\mathcal{Q}}\right)$

the Artin map in $L/K$ applied to $\mathcal{Q}$.
Let $K$ be a number field containing the $m^{th}$-roots of units, where
$m\in\mathbb{N}$, then for each $\alpha,\beta\in K^{*}$ and prime ideal
$\mathcal{P}$ of $K$, we define the norm residue symbol by:

$\left(\frac{\beta,\alpha}{\mathcal{P}}\right)_{m}=\frac{\left(\frac{\beta,K(\sqrt[m]{\alpha})}{\mathcal{P}}\right)\sqrt[m]{\alpha}}{\sqrt[m]{\alpha}}$

Therefore, if the prime ideal $\mathcal{P}$ is unramified in the field
$K(\sqrt[m]{\alpha})$, then we write:

$\left(\frac{\alpha}{\mathcal{P}}\right)_{m}=\frac{\left(\frac{K(\sqrt[m]{\alpha})}{\mathcal{P}}\right)\sqrt[m]{\alpha}}{\sqrt[m]{\alpha}}$

######  Remark 2.1.

Notice that $\left(\frac{\beta,\alpha}{\mathcal{P}}\right)_{m}$ and
$\left(\frac{\alpha}{\mathcal{P}}\right)_{m}$ are two $m^{th}$-roots of unit.

Following [5], the principal properties of the norm residue symbol are given
as follows:

######  Properties 2.1.

$\left(\frac{\beta_{1}\beta_{2},\alpha}{\mathcal{P}}\right)_{m}=\left(\frac{\beta_{1},\alpha}{\mathcal{P}}\right)_{m}\left(\frac{\beta_{2},\alpha}{\mathcal{P}}\right)_{m}$;

$\left(\frac{\beta,\alpha_{1}\alpha_{2}}{\mathcal{P}}\right)_{m}=\left(\frac{\beta,\alpha_{1}}{\mathcal{P}}\right)_{m}\left(\frac{\beta,\alpha_{2}}{\mathcal{P}}\right)_{m}$;

$\left(\frac{\beta,\alpha}{\mathcal{P}}\right)_{m}=\left(\frac{\alpha,\beta}{\mathcal{P}}\right)_{m}^{-1}$;

If $\mathcal{P}$ is not divisible by the conductor $f(\sqrt[m]{\alpha})$ of
$K(\sqrt[m]{\alpha})$ and appears in $(\beta)$ with the exponent b, then:
$\left(\frac{\beta,\alpha}{\mathcal{P}}\right)_{m}=\left(\frac{\alpha}{\mathcal{P}}\right)_{m}^{-b}$
;

$\left(\frac{\beta,\alpha}{\mathcal{P}}\right)_{m}=1$ if and only if $\beta$
is norm residue of $K(\sqrt[m]{\alpha})$ modulo $f(\sqrt[m]{\alpha})$ ;

$\left(\frac{\tau\beta,\tau\alpha}{\tau\mathcal{P}}\right)_{m}=\tau\left(\frac{\beta,\alpha}{\mathcal{P}}\right)_{m}$
for each automorphism $\tau$ of $K$ ;

${\displaystyle\prod_{\mathcal{P}}\left(\frac{\beta,\alpha}{\mathcal{P}}\right)_{m}}=1$
for all finite or infinite prime ideals;

If $K^{\prime}$ is a finite extension of $K$, $\alpha\in
K^{*},\beta^{\prime}\in K^{\prime}$ then:

${\displaystyle\prod_{\mathcal{P^{\prime}}|\mathcal{P}}\left(\frac{\beta^{\prime},\alpha}{\mathcal{P^{\prime}}}\right)_{m}}=\left(\frac{\mathcal{N}_{K^{\prime}/K}(\beta^{\prime}),\alpha}{\mathcal{P}}\right)_{m}$

Let $\alpha,\beta\in K^{*}$ and the conductors $f(\sqrt[m]{\alpha})$,
$f(\sqrt[m]{\beta})$ of respectively $K(\sqrt[m]{\alpha})$,
$K(\sqrt[m]{\beta})$ are co-prime then, the classical reciprocity law:

$\left(\frac{\beta}{(\alpha)}\right)_{m}=\left(\frac{\alpha}{(\beta)}\right)_{m}$

For more basic properties of the norm residue symbol in the number fields, we
refer the reader to [5].
Notice that in the rest of the article, we will use the norm quintic residue
symbols $(m=5)$. If we deal with a principal ring of integer, we will write
the norm quintic residue symbol as follows:

$\left(\frac{\beta,\alpha}{(\pi)}\right)_{5}=\left(\frac{\beta,\alpha}{\pi}\right)_{5}$
and
$\left(\frac{\alpha}{(\pi)}\right)_{5}=\left(\frac{\alpha}{\pi}\right)_{5}$

##  3 Ambiguous ideal classes

Let $\Gamma=\mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field, where $n$ is a
natural $5^{th}$ power-free, and $k_{0}=\mathbb{Q}(\zeta_{5})$ be the
cyclotomic field generated by a primitive $5^{th}$ root of unit $\zeta_{5}$.
Then $k=\Gamma(\zeta_{5})$ is the normal closure of $\Gamma$, and a cyclic
Kummer extension of degree $5$ of $k_{0}$. By $C_{k,5}$ we denote the
$5$-ideal class group of $k$, and by $C_{k,5}^{(\sigma)}$ the subgroup of
ambiguous ideal classes under the action of
$Gal(k/k_{0})=\langle\sigma\rangle$.
In [[6], Theorem 13], Hasse specifies rank $C_{k,5}^{(\sigma)}$ as follows:

rank $C_{k,5}^{(\sigma)}$ = $d+q^{*}-(r+1+o)$

where
$\bullet$ $d=$ number of ramified primes in $k/k_{0}$.
$\bullet$ $r=$ rank of the free abelian part of the group of units $E_{k_{0}}$
of $k_{0}$, so $r=1$.
$\bullet$ $o=1$ because $k_{0}$ contains a primitive $5^{th}$ root of unit.
$\bullet$ $q^{*}$ is defined by $[N_{k/k_{0}}(k-\\{0\\})\cap
E_{k_{0}}:N_{k/k_{0}}(E_{k_{0}})]=5^{q^{*}}$. Here $N_{k/k_{0}}$ is the
relative norm from $k$ to $k_{0}$.
We obtain that rank $C_{k,5}^{(\sigma)}$ = $d+q^{*}-3$.
We note that $N_{k/k_{0}}(E_{k_{0}})=E_{k_{0}}^{5}$ and
$[E_{k_{0}}:E_{k_{0}}^{5}]=5^{2}$, so we get that $q^{*}\in\\{0,1,2\\}$.
The group $E_{k_{0}}$ of units is generated by $\zeta_{5}$ and $\zeta_{5}+1$,
then according to the definition of $q^{*}$, we see that:

$q^{*}$ = $\begin{cases}2&\text{if }\,\zeta,\zeta+1\in N_{k/k_{0}}(k^{*}),\\\
1&\text{if }\,\zeta^{i}(\zeta+1)^{j}\in N_{k/k_{0}}(k^{*})\,\text{ for some i
and j },\\\ 0&\text{if }\,\zeta^{i}(\zeta+1)^{j}\notin
N_{k/k_{0}}(k^{*})\,\text{for}\hskip 5.69054pt0\leq i,j\leq 4\text{ and}\hskip
5.69054pti+j\neq 0.\\\ \end{cases}$

The following lemma gives us some results, which allow us to determine the
value of $q^{*}$.

######  Lemma 3.1.

Let $k_{0}=\mathbb{Q}(\zeta_{5})$ and let $k=k_{0}(\sqrt[5]{n})$, where
$n=u\lambda^{e_{\lambda}}\pi_{1}^{e_{1}}....\pi_{g}^{e_{g}}$, with $u$ is a
unit in $\mathbb{Z}[\zeta_{5}]$, $\lambda=1-\zeta_{5}$ is the unique prime
over $5$, and $\pi_{1},...,\pi_{g}$ prime elements in $\mathbb{Z}[\zeta_{5}]$.
Then:
$\bullet$ $\zeta_{5}\,\in
N_{k/k_{0}}(k-\\{0\\})\,\Longleftrightarrow\,N_{k_{0}/\mathbb{Q}}((\pi_{i}))\,\,\equiv\,1\,(\mathrm{mod}\,25)$
for all i.
$\bullet$ $\zeta_{5}^{i}(\zeta_{5}+1)^{j}\,\in
N_{k/k_{0}}(k-\\{0\\})\,\Longleftrightarrow\,$ every $\pi|x$ above has the
property that $\zeta_{5}^{i}(\zeta_{5}+1)^{j}$ is a $5^{th}$ power modulo
$(\pi)$ in $\mathbb{Z}[\zeta_{5}]$ for all $i,j$.
$\bullet$ $(\lambda)$ ramifies in $k/k_{0}$ $\Longleftrightarrow\,$
$n\not\equiv\,\pm 1,\pm 7\,(\mathrm{mod}\,\lambda^{5})$.

###### Proof.

See [[9], Lemma 5.1] ∎

The order of the subgroup $C_{k,5}^{(\sigma)}$ of ambiguous ideal classes
whenever $n$ takes forms of theorem 1.1 is given by the following proposition:

######  Proposition 3.1.

Let $\Gamma$, $k_{0}$ and $k$ as above. Let $q_{1}$, $q_{2}$ a prime numbers
such that
$q_{1},q_{2}\equiv\pm 2\,(\mathrm{mod}\,5)$. If $n$ takes one of the following
forms

$n\,=\,\left\\{\begin{array}[]{ll}q_{1}^{e_{1}}q_{2}&\text{ with }\quad
q_{i}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\ 5q_{1}&\text{ with }\quad
q_{1}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\ 5q_{1}q_{2}&\text{ with }\quad
q_{1}\,\text{ or }\,q_{2}$ $\not\equiv\,\pm 7\,(\mathrm{mod}\,25)\\\
\end{array}\right.$ (2)

Then the subgroup $C_{k,5}^{(\sigma)}$ of ambiguous ideal classes is cyclic of
order $5$.

###### Proof.

We will successively treat the three forms to calculate $d$ and $q^{*}$
defined before, in order to show that rank $C_{k,5}^{(\sigma)}=1$.
We note that if $q$ is a prime of $\mathbb{Z}$ such that $q\equiv\pm
2\,(\mathrm{mod}\,5)$, then by [[12], Theorem 2.13], $q$ is remain inert in
$k_{0}$ and $N_{k_{0}/\mathbb{Q}}(q)=q^{4}$.
Since $k$ is the normal closure of $\Gamma$, then $k=k_{0}(\sqrt[5]{n})$. We
see that if a prime $p$ of $\mathbb{Z}$ is ramified in $\Gamma$, then all
primes of $k_{0}$ above $p$ are ramified in $k$. Since in proposition 3.1 we
deal with primes $q_{i}\equiv\pm 2\,(\mathrm{mod}\,5)$, $(i=1,2)$, which are
inert in $k_{0}$, so if $q_{i}$ are ramified in $\Gamma$, then they are
ramified in $k$ too.

$\bullet$ If $n=q_{1}q_{2}$ such that $q_{i}\equiv\pm 7\,(\mathrm{mod}\,25)$,
then $n\equiv\pm 1\,(\mathrm{mod}\,25)$ and by lemma 3.1, we have that
$\lambda$ is not ramified in $k/k_{0}$. The primes $q_{i}$ are ramified in
$k$, because $disk(\Gamma/\mathbb{Q})=5^{5}n^{4}$ and $q_{i}$ divide this
discriminant. Hence we have $d=2$.
Since $q_{i}\equiv\pm 7\,(\mathrm{mod}\,25)$, then
$N_{k/\mathbb{Q}}(q)=q^{4}\equiv 1\,(\mathrm{mod}\,25)$ and by lemma 3.1 we
have $\zeta_{5}\,\in N_{k/k_{0}}(k-\\{0\\})$. By the same reasoning we have
that $\zeta_{5}+1\,\in N_{k/k_{0}}(k-\\{0\\})$, so $q^{*}=2$, whence rank
$C_{k,5}^{(\sigma)}=1$.

$\bullet$ If $n=5q_{1}$ such that $q_{1}\equiv\pm 7\,(\mathrm{mod}\,25)$, we
have $n\not\equiv\pm 1,\pm 7\,(\mathrm{mod}\,25)$, so $\lambda$ is ramified in
$k/k_{0}$. As before we have $q_{1}$ is ramified also in $k/k_{0}$, whence
$d=2$.
Since $q_{1}\equiv\pm 7\,(\mathrm{mod}\,25)$, we proceed as the previous point
to prove that $q^{*}=2$. Hence rank $C_{k,5}^{(\sigma)}=1$.

$\bullet$ If $n=5q_{1}q_{2}$ such that $q_{1}$ or $q_{2}\not\equiv\pm
7\,(\mathrm{mod}\,25)$. We see that $\lambda,q_{1}$ and $q_{2}$ are ramified
in $k/k_{0}$ so $d=3$. Since $q_{1}$ or $q_{2}\not\equiv\pm
7\,(\mathrm{mod}\,25)$, and by lemma 3.1 we have that $\zeta_{5}\,\not\in
N_{k/k_{0}}(k-\\{0\\})$, which imply that $q^{*}<2$. Precisely, according to
the proof of [[9], Theorem 5.13], we have $q^{*}=1$. Hence rank
$C_{k,5}^{(\sigma)}=1$.
We proved that for the three forms of the natural $n$, we have rank
$C_{k,5}^{(\sigma)}=1$, which means that $C_{k,5}^{(\sigma)}$ is a cyclic
subgroup of $C_{k,5}$ of order $5$. ∎

##  4 Proof of main theorem

Let $k$ be the normal closure of a pure quintic field
$\Gamma=\mathbb{Q}(\sqrt[5]{n})$, where $n$ takes the forms mentioned above,
also $k$ is cyclic Kummer extension of degree $5$ of the cyclotomic field
$k_{0}=\mathbb{Q}(\zeta_{5})$. We put $Gal(k/k_{0})=\langle\sigma\rangle$. By
$k_{5}^{(1)}$ we denote the Hilbert $5$-class field of $k$, that is the
maximal abelian unramified extension of $k$ of degree a power of $5$. By class
field theory $C_{k,5}\simeq Gal(k_{5}^{(1)}/k)$.
Next we define the genus field of $k/k_{0}$, which we denote by $k^{*}$, to be
the maximal abelian extension of $k_{0}$ contained in $k_{5}^{(1)}$. Then
using the isomorphism $C_{k,5}\simeq Gal(k_{5}^{(1)}/k)$, we see that
$Gal(k_{5}^{(1)}/k^{*})$ can be identified with a subgroup of $C_{k,5}$, which
is called the principal genus. By class field theory this subgroup is
$C_{k,5}^{1-\sigma}$. Its easy to see that $Gal(k^{*}/k)\simeq
C_{k,5}/C_{k,5}^{1-\sigma}$, and we have the following lemma:

######  Lemma 4.1.

Let $k,C_{k,5},C_{k,5}^{(\sigma)}$ and $C_{k,5}^{1-\sigma}$ as above. Then we
have:

rank $C_{k,5}^{(\sigma)}=$ rank $C_{k,5}/C_{k,5}^{1-\sigma}$ = $1$

###### Proof.

Since the $5^{th}$ cyclotomic field $k_{0}$, has a trivial class group, we can
admit the same proof of [[2], Lemma 2.3], and by proposition 3.1, we have rank
$C_{k,5}^{(\sigma)}=1$ ∎

From lemma 4.1, we deduce that rank $Gal(k^{*}/k)=$ rank
$C_{k,5}^{(\sigma)}=1$, which means that $[k^{*}:k]=5$. By Kummer theory,
there exist $x_{1}\in k$ such that $k^{*}=k(\sqrt[5]{x_{1}})$.
In [[9], section 6], we find an investigation of the $5$-class group of pure
quintic field $\Gamma$, by giving a upper bound of its rank as follows:

rank $C_{\Gamma,5}\leq min\\{t,\,t-s+rank(C_{k,5}/C_{k,5}^{1-\sigma})^{+}\\}$

, where $t$ is the rank of $C_{k,5}^{(\sigma)}$,
$s=\mathrm{rank}(C_{k,5}^{(\sigma)}.C_{k,5}^{1-\sigma})/C_{k,5}^{1-\sigma}$
and $(C_{k,5}/C_{k,5}^{1-\sigma})^{+}$ is defined as [[9], Lemma 6.1].
We note that according to [[9], Theorem 6.6], if the natural $n$ is not
divisible by any prime $p\,\equiv\,1\,(\mathrm{mod}\,5)$, we have that
$rank(C_{k,5}/C_{k,5}^{1-\sigma})^{+}=0$, which is verified in our situation,
whence the upper bound of rank $C_{\Gamma,5}$ becomes $t-s$.
The following theorem allows us to compute the value of $s$ in terms of rank
of matrix with entries in $\mathbb{F}_{5}$ the finite field of $5$ elements.

######  Theorem 4.1.

Let $k_{0}=\mathbb{Q}(\zeta_{5})$ and $k=k_{0}(\sqrt[5]{n})$, where $n$ takes
forms as above and decompose in $k_{0}$ as
$n=u\lambda^{e_{\lambda}}\pi_{1}^{e_{1}}...\pi_{g}^{e_{g}}$, with $u$ is a
unit, $\lambda=1-\zeta_{5}$ the unique prime in $k_{0}$ above $5$, $\pi_{i}$
primes of $k_{0}$, $e_{\lambda}\in\\{0,1,2,3,4\\}$ and
$e_{i}\in\\{1,2,3,4\\}$. Let $k^{*}=k(\sqrt[5]{x_{1}})$ be the genus field of
$k/k_{0}$.
Let $\alpha_{1j}\in\mathbb{F}_{5}$ such that

$\zeta_{5}^{\alpha_{1j}}$ = $\left(\frac{x_{1},n}{\pi_{j}}\right)_{5}$ for
$1\leq j\leq g$

$\zeta_{5}^{\alpha_{1(g+1)}}$ =
$\left(\frac{x_{1},\lambda}{\lambda}\right)_{5}$ if $\lambda$ is ramified in
$k/k_{0}$.

If $M$ is the matrix $(\alpha_{1j})$, then $s=\mathrm{rank}\,M$.

###### Proof.

See [[9], Theorem 5.10] for $t=1$ ∎

In the remainder, we compute the value of $s$ for the three forms of the
natural $n$.
By the definition of the matrix $M$ in theorem 4.1, we can see that
$s=\mathrm{rank}\,M=1$ if and only if one $\alpha_{1j}\neq 0$.

$\bullet$ If $n=q_{1}q_{2}$ such that $q_{i}\equiv\pm 7\,(\mathrm{mod}\,25)$,
$(i=1,2)$, According to [[10], Lemma 3.3], we have that
$k^{*}=k(\sqrt[5]{q_{1}})$, so we put $x_{1}=q_{1}$ and since $q_{i}$,
$(i=1,2)$, are inert in $k_{0}$, we put $\pi_{1}=q_{1}$, then we can get the
value of $\alpha_{11}$ by calculus of $\left(\frac{x_{1},n}{\\\
pi_{1}}\right)_{5}$ $=$ $\left(\frac{q_{1},\,q_{1}q_{2}}{q_{1}}\right)_{5}$.
We have
$\left(\frac{q_{1},\,q_{1}q_{2}}{q_{1}}\right)_{5}$ =
$\left(\frac{q_{1},q_{1}}{q_{1}}\right)_{5}$$\left(\frac{q_{1},q_{2}}{q_{1}}\right)_{5}$
by $(1)$, $(2)$ of properties 2.1.

and since
$\left(\frac{q_{1},q_{1}}{q_{1}}\right)_{5}$ = $1$ because $q_{1}$ is norm in
$k_{0}(\sqrt[5]{q_{1}})/k_{0}$.
$\left(\frac{q_{1},q_{2}}{q_{1}}\right)_{5}$ =
$\left(\frac{q_{2}}{q_{1}}\right)_{5}^{-1}$ by $(4)$ of properties 2.1.
we deduce that $\zeta_{5}^{\alpha_{11}}$ =
$\left(\frac{q_{2}}{q_{1}}\right)_{5}^{-1}$ $\neq 1$ because $q_{2}$ is not a
quintic residue modulo $q_{1}$, hence $\alpha_{11}\neq 0$, which imply that
$s=1$.

$\bullet$ If $n=5q_{1}$ such that $q_{1}\equiv\pm 7\,(\mathrm{mod}\,25)$, by
[[10], Lemma 3.3], we have that $k^{*}=k(\sqrt[5]{q_{1}})$, so we admit the
same reasoning as the previous point. Its sufficient to replace $q_{2}$ by
$5$. Hence we have $s=1$.

$\bullet$ If $n=5q_{1}q_{2}$ such that $q_{1}$ or $q_{2}\not\equiv\pm
7\,(\mathrm{mod}\,25)$, and without loosing generality we can assume that
$q_{2}\not\equiv\pm 7\,(\mathrm{mod}\,25)$. We have that
$k^{*}=k(\sqrt[5]{5q_{2}})$. Put $x_{1}=5q_{2}$ and $\pi_{1}=q_{1}$, then we
calculate $\left(\frac{x_{1},n}{\\\ pi_{1}}\right)_{5}$ $=$
$\left(\frac{5q_{2},\,5q_{1}q_{2}}{q_{1}}\right)_{5}$. We have
$\left(\frac{5q_{2},\,5q_{1}q_{2}}{q_{1}}\right)_{5}$ =
$\left(\frac{5,5}{q_{1}}\right)_{5}$$\left(\frac{5,q_{1}}{q_{1}}\right)_{5}$$\left(\frac{5,q_{2}}{q_{1}}\right)_{5}$$\left(\frac{q_{2},5}{q_{1}}\right)_{5}$$\left(\frac{q_{2},q_{1}}{q_{1}}\right)_{5}$$\left(\frac{q_{2},q_{2}}{q_{1}}\right)_{5}$
by $(1)$, $(2)$ of properties 2.1.

and since
$\left(\frac{5,5}{q_{1}}\right)_{5}$ = $1$ by $(2)$ and $(4)$ of properties
2.1.
$\left(\frac{5,q_{1}}{q_{1}}\right)_{5}$ = $\left(\frac{5}{q_{1}}\right)_{5}$
$\neq 1$ by $(4)$ of properties 2.1.
$\left(\frac{5,q_{2}}{q_{1}}\right)_{5}$$\left(\frac{q_{2},5}{q_{1}}\right)_{5}$
$=1$ by $(3)$ of properties 2.1.
$\left(\frac{q_{2},q_{1}}{q_{1}}\right)_{5}$ =
$\left(\frac{q_{2}}{q_{1}}\right)_{5}$ $\neq 1$ by $(4)$ of properties 2.1.
$\left(\frac{q_{2},q_{2}}{q_{1}}\right)_{5}$ = $1$ by $(2)$ and $(4)$ of
properties 2.1.
We deduce that $\zeta_{5}^{\alpha_{11}}\neq 1$, because $q_{2}$ and $5$ are
not a quintic residue modulo $q_{1}$, which imply that $s=1$.

In summary, we proved that for the three forms of the natural $n$ we have
$s=1$, and by proposition 3.1, we have $t=1$, whence the upper bound of rank
$C_{\Gamma,5}$ is $0$, which means that $\Gamma$ admits a trivial $5$-class
number.
To finish the proof, we use the results of C. Parry in [11], which states that
$5$ divides $h_{\Gamma}$ if and only if $5^{2}$ divides $h_{k}$. The fact that
$\Gamma$ admits a trivial $5$-class number, imply that $5^{2}$ does not
divides $h_{k}$, but we proved that rank $C_{k,5}^{(\sigma)}=1$, then there is
exact divisibility of $h_{k}$ by $5$.

##  5 Numerical examples

Let $\Gamma\,=\,\mathbb{Q}(\sqrt[5]{n})$ be a pure quintic field and $k$ its
normal closure. Using the system PARI/GP [13], we compute the $5$-class number
of $\Gamma$ and $k$ for each forms of the natural $n$. The following tables
illustrate out main results theorem 1.1.

Table 1: $n\,=\,q_{1}q_{2}$ with $q_{i}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)$

$q_{1}$ | $q_{1}\,(\mathrm{mod}\,25)$ | $q_{2}$ | $q_{2}\,(\mathrm{mod}\,25)$ | $n\,=\,q_{1}q_{2}$ | $h_{k,5}$ | $h_{\Gamma,5}$
---|---|---|---|---|---|---
7 | 7 | 43 | -7 | 301 | 5 | 1
7 | 7 | 193 | -7 | 1351 | 5 | 1
7 | 7 | 293 | -7 | 2051 | 5 | 1
107 | 7 | 43 | -7 | 4601 | 5 | 1
157 | 7 | 43 | -7 | 6751 | 5 | 1
457 | 7 | 43 | -7 | 19651 | 5 | 1
107 | 7 | 193 | -7 | 20651 | 5 | 1
557 | 7 | 43 | -7 | 23251 | 5 | 1
607 | 7 | 43 | -7 | 26101 | 5 | 1
157 | 7 | 193 | -7 | 30301 | 5 | 1
107 | 7 | 293 | -7 | 31351 | 5 | 1
757 | 7 | 43 | -7 | 32551 | 5 | 1
857 | 7 | 43 | -7 | 36851 | 5 | 1
907 | 7 | 43 | -7 | 39001 | 5 | 1
107 | 7 | 443 | -7 | 47401 | 5 | 1
257 | 7 | 193 | -7 | 49601 | 5 | 1
307 | 7 | 193 | -7 | 59251 | 5 | 1
157 | 7 | 443 | -7 | 69551 | 5 | 1
257 | 7 | 293 | -7 | 75301 | 5 | 1
457 | 7 | 443 | -7 | 202451 | 5 | 1

Table 2 : $n\,=\,5q_{1}$ with $q_{1}\,\equiv\,\pm 7\,(\mathrm{mod}\,25)$

$q_{1}$ | $q_{1}\,(\mathrm{mod}\,25)$ | $n\,=\,5q_{1}$ | $h_{k,5}$ | $h_{\Gamma,5}$
---|---|---|---|---
7 | 7 | 35 | 5 | 1
43 | -7 | 215 | 5 | 1
107 | 7 | 535 | 5 | 1
157 | 7 | 785 | 5 | 1
193 | -7 | 965 | 5 | 1
257 | 7 | 1285 | 5 | 1
293 | -7 | 1465 | 5 | 1
307 | 7 | 1535 | 5 | 1
443 | -7 | 2215 | 5 | 1
457 | 7 | 2285 | 5 | 1
557 | 7 | 2785 | 5 | 1
607 | 7 | 3053 | 5 | 1
643 | -7 | 3215 | 5 | 1
757 | 7 | 3785 | 5 | 1
857 | 7 | 4285 | 5 | 1
907 | 7 | 4535 | 5 | 1

Table 3: $n\,=\,5q_{1}q_{2}$ with $q_{1}$ or $q_{2}$ $\not\equiv\,\pm
7\,(\mathrm{mod}\,25)$

$q_{1}$ | $q_{1}\,(\mathrm{mod}\,25)$ | $q_{2}$ | $q_{2}\,(\mathrm{mod}\,25)$ | $n\,=\,5q_{1}q_{2}$ | $h_{k,5}$ | $h_{\Gamma,5}$
---|---|---|---|---|---|---
2 | 2 | 3 | 3 | 30 | 5 | 1
7 | 7 | 3 | 3 | 105 | 5 | 1
2 | 2 | 13 | 13 | 130 | 5 | 1
2 | 2 | 23 | -2 | 230 | 5 | 1
17 | 17 | 3 | 3 | 255 | 5 | 1
2 | 2 | 53 | 3 | 530 | 5 | 1
37 | 17 | 3 | 3 | 555 | 5 | 1
47 | -3 | 3 | 3 | 705 | 5 | 1
67 | 17 | 3 | 3 | 1005 | 5 | 1
17 | 17 | 23 | -2 | 1955 | 5 | 1
37 | 17 | 13 | 13 | 2405 | 5 | 1
47 | -3 | 13 | 13 | 3055 | 5 | 1
47 | -3 | 23 | -2 | 5405 | 5 | 1
47 | -3 | 43 | -7 | 10105 | 5 | 1
107 | 7 | 23 | -2 | 12305 | 5 | 1
67 | 17 | 53 | 3 | 17755 | 5 | 1
97 | -3 | 43 | -7 | 20855 | 5 | 1

## References

  * [1] F.Elmouhib, M.Talbi, and A.Azizi, _5-rank of ambiguous class groups of quintic Kummer extensions_ , Proc Math Sci 132, 12 (2022). https://doi.org/10.1007/s12044-022-00660-z
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