Preparing and manipulating quantum states of mechanical resonators is a highly interdisciplinary undertaking that now receives enormous interest for its far-reaching potential in fundamental and applied science. Up to now, only nanoscale mechanical devices achieved operation close to the quantum regime. We report a new micro-optomechanical resonator that is laser cooled to a level of 30 thermal quanta. This is equivalent to the best nanomechanical devices, however, with a mass more than four orders of magnitude larger (43 ng versus 1 pg) and at more than two orders of magnitude higher environment temperature (5 K versus 30 mK). Despite the large laser-added cooling factor of 4,000 and the cryogenic environment, our cooling performance is not limited by residual absorption effects. These results pave the way for the preparation of 100-m scale objects in the quantum regime. Possible applications range from quantum-limited optomechanical sensing devices to macroscopic tests of quantum physics

Aspelmeyer, Markus

Cole, Garrett D.

Gigan, Sylvain

Gröblacher, Simon

Hertzberg, Jared B.

Schwab, K. C.

Vanner, Michael R.

Caltech Authors - Main

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Preparation and Detection of a Mechanical Resonator Near the Ground State of
Motion
T. Rocheleau,1 T. Ndukum,1 C. Macklin,1 J.B. Hertzberg,2 A.A. Clerk,3 and K.C. Schwab4, ∗
1Department of Physics, Cornell University, Ithaca, NY 14853 USA
2Department of Physics, University of Maryland, College Park, MD 20742 USA
3Department of Physics, McGill University, Montreal, QC Canada H3A 2T8
4Applied Physics, Caltech, Pasadena, CA 91125 USA
(Dated: July 19, 2009)
We have cooled the motion of a radio-frequency nanomechanical resonator by parametric coupling
to a driven microwave frequency superconducting resonator. Starting from a thermal occupation
of 480 quanta, we have observed occupation factors as low as 3.8±1.2 and expect the mechanical
resonator to be found with probability 0.21 in the quantum ground state of motion. Cooling is
limited by random excitation of the microwave resonator and heating of the dissipative mechanical
bath.
PACS numbers: 85.85.+j, 42.50.Wk, 84.40.Dc, 85.25.-j
Cold macroscopic mechanical systems are expected to
behave contrary to our usual classical understanding of
reality; the most striking and nonsensical predictions are
states where the mechanical system is located in two
places simultaneously. Various schemes have been pro-
posed to generate and detect such states[1, 2] and all
require starting from mechanical states which are close
to the lowest energy eigenstate, the mechanical ground
state.
Naively treating the motion of a mechanical resonator
quantum mechanically, one finds the elementary result
that the energy should be quantized: En = ~ωm(n+
1
2
),
where n is an integer and ωm is the resonant frequency.
In thermal equilibrium, an average occupation factor is
expected to follow the Bose-Einstein distribution: n¯Tm =
(e~ωm/kBT − 1)−1 , where T , 2pi~, and kB are the tem-
perature, Planck’s and Boltzmann’s constants respec-
tively. Cooling a resonator into the quantum regime
where n¯Tm << 1 , and measuring the very small motions
has been challenging for a number of technical reasons;
not only are very low temperatures necessary to freeze-
out the mode, but detection with sensitivity at the quan-
tum zero-point level is required: xzp =
√
~/(2mωm),
where m is the resonator mass. Furthermore, this strong
position measurement must not heat the mode with mea-
surement backaction[3].
Many strategies have been proposed[4, 5, 6, 7, 8,
9, 10, 11, 12] and applied to realize the quantum
regime with increasing success. Experiments with nano-
electromechanical structures have been able to reach
n¯m = 25 by passively cooling a nanomechanical res-
onator (NR)[3], detected with a superconducting single
electron transistor[13]. Researchers experimenting with
opto-mechanical systems have been able to utilize ultra-
sensitive optical detection and radiation pressure to both
cool and detect n¯m = 65 in a toroidal resonator[14],
n¯m = 37 in microsphere resonator[15], and n¯m = 35 in
an optical cavity[16].
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FIG. 1: a) Shows the Nb/Al/SiN sample: the NR is 30 µm
long, 170nm wide, 140 nm thick, formed of 60nm of stoi-
chiometric, high-stress, LPCVD SiN[17] and 80nm of Al, and
located 75nm from the gate electrode connected to the SR.
The SR is fabricated from a 345nm thick Nb film and has
a characteristic impedance of 126Ω. (b) Shows the thermal
calibration of the upconverted noise power. (c) Shows the
ultra-low noise, cyrogenic measurement circuit.
The technique we employ both to cool and detect
the motion of a NR close to the ground state involves
parametrically coupling the motion to a superconduct-
ing microwave resonator (SR)[18, 19], (Fig. 1.) The
NR has a fundamental in-plane flexural resonance of
ωm = 2pi · 6.3MHz and is capacitively coupled to a
symmetric, two-port, half-wave SR which resonates at
ωsr = 2pi · 7.5GHz. The device is located in a dilution
refrigerator and pumped through carefully filtered and
cooled leads. The thermal occupation of the SR, n¯Tsr is
expected to be 0.09 at 146mK.
The NR damping rate, ΓTm, displays an unusual lin-
ear temperature dependence below 600mK, reachingQ ∼
106 at 100mK. The SR damping rate, Γsr = 2pi · 600kHz,
is essentially temperature independent below 700mK and
is a factor of 2.4 higher than expected from design due
to internal losses.
The Hamiltonian which describes the coupled res-
2onators is given by[9, 10]:
Hˆ = ~(ωsr + gxˆ− λxˆ2)(bˆ†bˆ + 1
2
) + ~ωm(aˆ
†aˆ+
1
2
)
where aˆ (aˆ†) and bˆ (bˆ†) are the NR and SR annihi-
lation (creation) operators. The first term shows the
pondermotive-like coupling of the SR′s frequency to the
mechanical motion: xˆ = xzp(aˆ
† + aˆ) and g = ∂ωsr∂x =
ωsr
2Ct
∂Cg
∂x where Cg(x) = 450 ± 50aF is the coupling ca-
pacitance and Ct = 260fF is the SR total effective ca-
pacitance. The term proportional to xˆ2 results from
the electrostatic frequency pulling of the mechanical res-
onator by the SR[20], where λ = ωsr
2Ct
∂2Cg
∂x2 , and is re-
sponsible for parametric instabilities under certain pump
configurations[21].
When pumping the SR at ωp = ωsr − ωm, har-
monic motion of the NR preferentially up-converts mi-
crowave photons to frequency ωsr, extracting one radio-
frequency NR quantum for each up-converted microwave
SR photon, a process which both damps and cools the
NR motion[9, 10, 11, 22, 23]. This cooling process is
analogous to Raman scattering and the process used
to cool an atomic ion to the quantum ground state of
motion[10, 24]. In the sideband-resolved limit, Γsr <
ωm, the rate of this up-conversion process is given by:
Γopt = 4x
2
zpg
2n¯p/Γsr, where n¯p is the occupation of the
SR from the pumping.
From detailed balance, the NR occupation factor is
expected to follow:
n¯m =
ΓTmn¯
T
m + Γoptn¯sr
ΓTm + Γopt
where n¯sr = (Γsr/(4ωm))
2+n¯Tsr[1+2(Γsr/(4ωm))
2] is the
effective occupation factor of the SR when Γopt < Γsr[25].
The first term in the expression for n¯sr is due to the
quantum fluctuations of the pump field, and the second
term due to the thermal occupation of the SR, n¯TSR. The
expressions above show that the minimum mechanical
occupation possible is the effective occupation of the SR.
The first realization of cooling in a parametrically cou-
pled, electro-mechanical microwave system was with a
kg-scale gravitational wave transducer [26], cooling from
n¯m = 10
8 to 105; cooling of an NR with an SR was re-
cently demonstrated and achieved cooling from n¯m = 700
to 120[27].
The up-converted noise power is calibrated by apply-
ing a weak pump signal, (Γopt < Γ
T
m), and measuring the
resulting integrated sideband power, Pm, versus refriger-
ator temperature, T (Fig. 1b.) For temperature above
∼ 150mK, we observe the expected behavior consistent
with Equipartition and use this curve to establish the
relationship between measured output noise power and
n¯m. For temperatures below 150mK we observe fluctu-
ations in n¯m apparently due to a non-thermal, intermit-
tent, force noise at the level of 1 · 10−18N/
√
Hz which
FIG. 2: The figure shows n¯m (•) and Γm = Γ
T
m + Γopt (•)
versus n¯p. The solid black curve is a fit to the measured Γm.
The solid blue curve is the expected value of n¯m assuming
ideal values of n¯SR and n˙T = 3 · 10
4 quanta/sec.
is observed in other similar samples[21, 28] and simi-
lar to anomalous heating effects in other systems[29, 30].
Furthermore, the linear temperature dependence of ΓTm
causes the NR to decouple from the thermal environment
at the lowest measured temperatures.
The measured signal powers are consistent with our
knowledge of the attenuation and gain of our measure-
ment circuit, and estimates of the device parameters. We
find g/2pi = 84 ± 5kHz/nm, which is the largest cou-
pling strength demonstrated to date in a system of this
type. From measurements of ωm versus n¯p and pump
frequency, we determine λ/2pi = 2.1± 0.7kHz/(nm)2.
Figure 2 shows the central result of this work. With
the refrigerator stabilized at T=146mK (n¯Tm = 480) we
measure Γm and n¯m versus the SR pump occupation,
n¯p. As is clear from Fig. 2, this process dramatically
cools the motion. However, we also observe that the SR
becomes increasingly excited as n¯p is increased(Fig. 4).
Figure 3 shows the measured output noise spectra,
Sx(ω), which is composed of up-converted microwave
photons due to n¯m, SR noise due to n¯sr, and HEMT am-
plifier noise. Correlations between the NR motion and
the SR field are important in our measured noise spectra
at the lowest mechanical occupation factors. Fluctua-
tions in the SR voltage, due to n¯sr, together with the
pump, produce forces at the ωm. The resulting motion,
together with the pump, produces noise at ωsr, however
180o out of phase with the original SR fluctuations. This
correlation results in an inverted noise peak[21], simi-
lar to noise squashing[31], which adds incoherently to
the noise power driven by the thermal bath. Our anal-
ysis shows that the NR occupation factor is given by
n¯m = n¯eff + 2n¯sr, where n¯eff is the occupation mea-
sured directly from the integrated noise peak (or dip) in
3FIG. 3: shows the noise squashing effect on Sx(ω) due to the
finite occupation of the SR, in three situations: when n¯eff > 0
(top), when n¯eff ≈ 0 (middle), and when n¯eff < 0 (bottom).
The red curves show Lorentzian fits through the mechanically
up-converted side-band.
the output noise spectrum, Sx(ω). Figure 3 shows mea-
surements of Sx(ω) in three cases at low occupation fac-
tors: when n¯eff > 0, when n¯eff ≈ 0, and when n¯eff < 0
showing the squashed output noise.
Taking the effects of n¯sr into account in this way, the
lowest mechanical occupation we have observed is n¯m =
3.8±1.2, shown in Fig. 3, with the uncertainty dominated
by the uncertainty in n¯sr. At this low occupation factor
the resonator is expected to be found in the ground state
with probability P0 = 1/(n¯m + 1) = 0.21. The cooling
power of this refrigeration technique is Q˙ = ~ωm ·Γopt =
10−22W
We have lowered the refrigerator temperature to 20mK
and do not observe an decrease in the minimum n¯m. Us-
ing the detailed balance relationship and the measured
n¯m and Γsr, we can compute the bath heating rate,
n˙T = Γ
T
mn¯
T
m, versus n¯p, (Fig. 4.) It is clear that as n¯p
increases above 3 · 107, n˙T begins to increase, nullifying
the benefit of starting at low temperatures. This level
of heating is consistent with ohmic losses in the metal
film on top the NR, and the thermal conductance of a
normal-state electron gas.
Current Limitations and Future Directions: These
measurements identify three effects which work against
the cooling process: excess fluctuations of the SR (n¯SR),
heating of the NR thermal bath at high pump powers,
and the non-thermal force noise at low temperatures.
We believe that the excess SR occupation, n¯sr, is not
a result of phase or amplitude noise of our microwave
source: the pump signal is filtered with tunable, cop-
per microwave cavities (one at 300K (Q=9.5 · 103) fol-
lowed by a second at 77K (Q=2.6 · 104)) achieving bet-
FIG. 4: The upper figure shows the bath heating rate, n˙T ,
versus pump strength n¯p, and the onset of excess heating
above n¯p = 3·10
7 . The lower figure shows the measured value
of n¯sr versus n¯p; the structure is suspected to be related to
temporal dynamics of the transition between superconducting
and normal states of the metal films and resulting microwave
side-band generation[32].
ter than L(+6.3MHz) < −195dbc/Hz, and contributing
less than 0.04 photons into the SR at our highest value
of n¯p. Without these cavities the SR would be excited to
n¯sr = 35. We also believe that this excess SR occupation
is not due to ohmic heating of and resulting thermal ra-
diation from the cyrogenic attenuator network since n¯sr
increases only weakly over a wide span of n¯p. Tests of Nb
SR devices at 1.2K before the surface micromaching of
the NR do not show excess dissipation and suggest that
the excess losses are related to our fabrication process.
Increasing Γopt by engineering larger coupling
strength, g, and/or decreasing Γsr should be very ben-
eficial since it will lead to higher cooling rates at lower
pump powers, minimizing the effect of excess bath heat-
ing, n˙T . By increasing Γopt a factor of 10, maintaining
the same n˙T , we expect n¯m ≈ 0.5, with P0 = .67. This
approach will be limited when Γopt becomes comparable
to Γsr which limits the rate of cooling[25, 33].
The deep quantum limit, n¯m ≪ 1, will be accessi-
ble when it is possible to utilize lower refrigerator tem-
peratures and lower mechanical damping rates at these
temperatures. Understanding and eliminating the ex-
cess bath heating and the non-thermal force noise will
be required. Furthermore, superconducting metals on
the NR also appear to be required due to the expected
mechanical force noise from transport and electron mo-
mentum scattering in diffusive conductors[34, 35]. We
estimate that this heating mechanism will limit n¯m > 3
at n¯p = 3 · 108 assuming our current device parameters
and a resistance of 100Ω through the NR.
4These measurements show that detection with sensi-
tivity to resolve motions approaching the ground state
is possible with existing HEMT-based amplifiers. Elimi-
nating internal SR losses, unbalancing the SR couplings,
and implementing improved microwave amplifiers[36, 37]
would significantly reduce the measurement time.
Nonetheless, the production and detection of a NR
with n¯m = 3.8 is sufficient to enable future experiments.
Due to Uncertainy Principle fluctuations of the mechan-
ical motion and resulting spontaneous emission, the rate
of microwave photon up-conversion is expected to differ
from the rate of down-conversion. This difference can be
used as a fundamental thermometry technique[9, 10, 24],
and would be the first quantitative measurement of the
zero-point motion of a mechanical structure.
This level of cooling is essential to realize entangled
states between superconducting quantum bits and the
motion of a nanomechanical device[1, 38, 39]. Similar to
procedures in atomic physics, such an experiment would
involve preparing the cold state of the mechanical device
and after the refrigeration is complete, the cooling can
be turned off. The state of the cold beam could then
be manipulated before thermallization of the motion. In
our realization, we expect cooling from n¯m = 500 to 4
quanta in ∼ 200µs, and one thermal quantum to enter
the resonator in τ = (n˙T )
−1 = 2µs which exceeds super-
conducting qubit manipulation times and is comparable
to qubit measurement and relaxation times.
We acknowledge helpful conversations with M. As-
pelmeyer, R. Ilic, M. Skvarla, M. Metzler, M. Shaw and
assistance from M. Savva, S. Rosenthal, and M. Corbett.
This work has been supported by the Fundamental Ques-
tions Institute fqxi.org (RFP2-08-27), and the US NSF
(DMR-0804567). Device fabrication was performed at
the Cornell Nanoscale Facility, a member of the NNIN
(NSF Grant ECS-0335765).
∗ schwab@caltech.edu; www.kschwabresearch.com
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Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity

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Demonstration of an ultracold micro-optomechanical oscillator in a cryogenic cavity

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