Regular Articles

Two-mode Squeezing in an
Electromechanical Resonator
Imran Mahboob, Hajime Okamoto,and Hiroshi Yamaguchi
Abstract
A mechanical resonator integrated with piezoelectric transducers enables mechanical nonlinearities tobe dynamically engineered to emulate non-degenerate parametric down-conversion. In this configuration, millions of phonons are simultaneously generated in pairs in two macroscopic vibration modes,
which results in the amplification of their motion by more than 20 dB. Mechanical two-mode squeezed
states are also created in parallel, which exhibit fluctuations 5 dB below the thermal level of their constituent modes and they harbor correlations between the modes that become perfect as their amplification
is increased. This remarkable observation of correlations between two massive phonon ensembles establishes the means to create an entangled macroscopic mechanical system at the single phonon level.
Keywords: electromechanical resonator, parametric down-conversion, squeezed-states, phonons

1. Introduction
Electromechanical systems consist of a single
degree of mechanical freedom hosting a spectrally
pure resonance that is embedded in an electricaltransduction circuit, and they have emerged as a versatile platform for a range of technological applications and to study fundamental science [1, 2]. For
instance, their small inertial mass and high quality
resonance have enabled the development of ultra-precise sensors that can detect a single electron spin andeven the mass of a proton leading to a new class of
medicinal diagnostic technology [3–5]. The electricaltransduction circuit can also enable the underlying
potential energy landscape of the mechanical resonator to be dynamically engineered, yielding a range of
nonlinear motional dynamics that can be harnessed
for both information storage and processing, whichbrings forth the concept of mechanical computation
offering both ultra-low power consumption and thecapacity for unprecedented parallel data processing[6–9].
Most tantalizingly, a high frequency mechanicalresonator operated at sufficiently low temperatures

can even condense into its quantum ground state,
where its low mass and spectral purity yield quantum
zero point fluctuations that are large enough to beobserved, enabling a macroscopic quantum system to
be studied [10–12]. In this instance, although themechanical resonator is composed of billions ofatoms, only the phonons sustained by the fundamental resonance mode are quantized, and it corresponds
to a tangible vibration of the mechanical element.
This phonon picture, namely the collective excitationof atoms in the fundamental vibration mode, enables
concepts from atomic molecular optical (AMO)
physics developed for photons, that is, a quanta ofelectromagnetic radiation, to be exploited. Indeed, itis the laser cooling techniques pioneered in AMO
physics that have been utilized most successfully tocool the macroscopic mechanical oscillator so that on
average, its fundamental mode is occupied by muchless than one phonon [13, 14].
Although the notion of a macroscopic mechanicalresonator composed of billions of atoms being considered only in terms of the number of phonons sustained by its fundamental mode might seem counterintuitive, this paradigm has successfully been exploited
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(a)

(b)
Idler
ωi

Nonlinearχ(2) crystal

Signal

ωii

(c)

ω1

ωi

Pump
ωs
ωp
ki

ks

θ

θ

ωi

(d)

ω1
ω2

kp

Fig. 1. (a) Schematic illustration of the spontaneous parametric down-conversion process used to generate non-classicallight where the signal and idler photons are entangled in a two-mode squeezed state. This process occurs at theexpense of the pump photons where the signal and idler photons conserve both momentum kp = ks + ki, where k is
the wavevector, and energy ωp = ωs + ωi. The nonlinear crystals typically used are BBO (beta-barium borate) for TypeII down-conversion (where the signal and idler photons have perpendicular polarization) and KDP (potassium dihydrogen phosphate) for type I down-conversion (where the signal and idler photons have parallel polarization). (b)Schematic depiction of a microwave cavity composed from a superconducting metal (orange) and terminated by a
SQUID with two Josephson junctions (blue) that can sustain electromagnetic standing waves, for instance ωi and ωii,
which are coupled via the SQUID. (c) Cavity electromechanical system with a microwave resonator of frequency ωiconsisting of a spiral inductor and a capacitor and a mechanical resonator of frequency ω1 which forms a compliablecapacitor. Light and sound in these two subsystems are nonlinearly coupled via the oscillating electrical energy
stored in the capacitor. (d) Phonon-cavity electromechanical system consisting of two localized mechanical vibration
modes ω1 and ω2 that are coupled via strain. In all three systems, if the respective flux penetrating the SQUID, theelectric field stored in the capacitor, or the mechanical strain are sinusoidally pumped at the sum frequency of theirelectromagnetic and/or mechanical modes, signal and idler photons and/or phonons are generated that conserve
energy but the momentum conservation constraint is relaxed due to their stationary wave nature.

to demonstrate the signature feature of photonics,
namely lasing but with the localized phonons in themechanical resonator [15–18]. Consequently, theability to cool a solid-state macroscopic mechanicalresonator into its ground state opens up the possibility of generating an all-mechanical macroscopic
entangled state [10, 19], which would enable foundational aspects of quantum mechanics to be queriedsuch as the nature of the quantum to classical divide
and the absence of quantum phenomena in our everyday classical world [20, 21].
2. Two-mode squeezed states
Entangled photons in the guise of two-mode
squeezed states were among the first non-classicalstates of light to be generated in the lab and nowadays
are routinely siphoned from spontaneous parametricVol. 13 No. 8 Aug. 2015

down-conversion [22]. This typically involves a χ(2)nonlinear crystal that is exposed to a strong laser
pump beam whose dielectric polarization responds
nonlinearly to the pump’s electric field. In this process, a pair of photons—the signal and the idler—are
generated at the expense of the pump photons, whichconserve both energy and momentum, as depicted inFig. 1(a). Remarkably, the quantum fluctuations in
the amplitude and phase of the generated photon pairare correlated, as encapsulated by the Einstein,
Podolsky, and Rosen (EPR) paradox, where theirindividual fluctuations are amplified, while their relative fluctuations are reduced below the vacuum noiselevel [23–26]. The EPR paradox recast in the Bellinequality has been routinely violated via the entangled signal and idler photons whose fluctuations
remain correlated even when they are separated overdistances of kilometers [27, 28]. The non-classicalRegular Articles

light generated from spontaneous parametric downconversion has subsequently engendered a whole
host of quantum-enabled technologies including
quantum communication [29], optical quantum computing [30], quantum teleportation [31], quantum
enhanced measurements and metrology [32, 33].
This unprecedented success has led to two-mode
squeezed states being translated to other parts of theelectromagnetic spectrum, for instance microwaves
utilizing superconducting circuits [34]. Specifically, asuperconducting resonator terminated with a nonlinear element such as a superconducting quantum interference device (SQUID), as generically depicted inFig. 1(b), can enable parametric down-conversion[35, 36]. Pumping the SQUID with flux in these socalled Josephson parametric amplifiers (JPAs) yields
a nonlinear variation in the boundary conditions of
the microwave resonator which creates a frequency
modulation in the standing waves it hosts. If the pumpfrequency coincides with the sum of two modes (say
ωi and ωii as depicted in Fig. 1(b)) in the microwave
resonator, it non-degenerately amplifies their vacuum
noise, generating signal and idler photons in a process reminiscent to that described above where thesephotons are entangled [37, 38]. Indeed, JPA-likedevices have now opened the door to many of theconcepts pioneered with optical two-mode squeezed
states but in an on-chip microwave circuit that could
even be yoked into a quantum computer.
A variation on these microwave circuits has recently emerged in the form of cavity electromechanicalsystems where the cavity, namely the microwave
resonator, is composed of an inductor and capacitor
in series, and the mechanical resonator forms amechanically compliant element of the parallel plate
capacitor, thus capacitively coupling the mechanics
to the microwaves as generically depicted in Fig. 1(c)[39–41]. Electric fields in the forms of microwaves
can be pumped into this hybrid circuit via a capacitively coupled transmission line that nonlinearly
modulates the energy stored in the capacitor andhence the frequencies of the mechanics and the cavity. If the pump coincides with the sum frequency ofthe subsystems (i.e. ωi and ω1 as depicted in Fig.
1(c)), it behaves like a non-degenerate parametricamplifier and it creates photons and phonons in pairsthat are correlated in a two-mode squeezed state [42,43]. This remarkable demonstration illustrates thattwo vastly dissimilar systems, namely light andsound, can be entangled in a macroscopic context,
providing a new regime in which quantum physics
can be explored [14].


The cavity electromechanical system detailed inFig. 1(c) straddles the two extremes of light andsound, and it naturally suggests the possibility of apurely mechanical analogue of a two-mode squeezed
state in a phonon-cavity electromechanical system as
depicted in Fig. 1(d). As in the microwave case, themechanical resonator can also sustain multiple
modes, say ω1 and ω2, as shown in Fig. 1(d), which if
nonlinearly coupled, could yield non-degenerateparametric amplification. In contrast to the microwave circuit that needs to be engineered with a nonlinear element, namely the SQUID, to enable themodes to couple, the modes in the mechanical resonator can dispersively couple naturally via the strain
induced from the motion of a given mode that modifies the restoring potential, that is, the spring constantand hence, the natural frequency of the other modes[44]. Consequently, if the strain in the mechanicalelement is sinusoidally pumped, it will nonlinearly
modulate the frequencies of all the localized modes,
and once this modulation coincides with a pair ofmodes, it will non-degenerately amplify their motion
and in the process generate a two-mode squeezed
state [45]. Critically, however, an entanglement will
only be generated if the constituent modes are initialized in their quantum ground state [10–12], but theability to harness this interaction, even if the modes inquestion are thermalized, would lay a pivotal marker
on the road to generating an all-mechanical macroscopic entanglement [46].
3. Electromechanical resonator
Although degenerate parametric amplification wasdemonstrated in the early nineties, it could not be
used to disentangle correlations between the signal
and idler phonons due to their having identical frequencies and being spatially localized to the samemode in the mechanical resonator [47]. Alternatively,
a non-degenerate variant of this process in the firsttwo modes of a beam resonator yielded only a modest
phonon generation rate, which was insufficient tosignificantly amplify the thermomechanical fluctuations before they were dissipated, resulting in statistically insignificant correlations [45].
To that end, the electromechanical resonator shownin Fig. 2(a) was developed, which consists of two
mechanical elements that are strongly coupled via theexaggerated overhangs between them [48]. This
results in the two lowest-order vibration modes
shown in Figs. 2(b) and 2(c) that are extracted from afinite element calculation and henceforth labeled
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(a)

(b)

(c)

(e)
240

(f)
16

240

16
Experiment

Theory∆ω/2π (Hz)

6
4

120

8

60

12

180

120

8

4

60

4

0
0
0.00 0.05 0.10 0.15 0.20 0.25
Pump amplitude (Vrms)

0

Gain

12

180

8

∆ω/2π (Hz)

10

Gain

Displacement spectral density(10−13 mHz−0.5)

(d)

2
0
244800 245600 246400 261600 262400

Frequency (Hz)

0

0
75

50

25
Λ (106)

Fig. 2. (a) Scanning electron micrograph of the electromechanical resonator. Each mechanical element is 80 µm long, 20µm wide, 800 nm thick, and is integrated with two piezoelectric transducers formed from a doped GaAs layer located
300 nm below the surface (blue) and a gold gate electrode (yellow). (b) and (c) The electromechanical resonatorsustains symmetric (red) and asymmetric (blue) vibration modes composed from both mechanical elements that arestrongly coupled via the two large overhangs between them. (d) Thermomechanical vibrations of both modes whendriven by the Langevin force (black) and demodulated in a spectrum analyzer. Activating the non-degenerate parametric down-conversion via the piezoelectric transducers on the left mechanical element from 0 to 0.24 Vrms in 0.03
Vrms increments results in the thermomechanical fluctuations of both modes being amplified. (e) The corresponding
amplification gain and spectral power bandwidth (∆ω) with pump amplitude increments of 5 mVrms. (f) This experimental response can be faithfully reproduced by numerically solving the Hamiltonian in eq. (1).

symmetric (S) and asymmetric (A), which are closely
spaced in frequency. In addition, the spatial profilesof both modes have a large overlap yielding greater
strain-mediated dispersive coupling between them.
This combination of enhanced dispersive coupling
and a smaller frequency separation, almost tending
towards a degenerate configuration, is designed toyield larger non-degenerate parametric amplification
when the strain in this system is modulated at the sum
frequency of both modes.
The piezoelectric effect is utilized in order to modulate the strain in the mechanical modes. By design,
the electromechanical system is fabricated from gallium arsenide (GaAs), which is piezoelectrically
active [6, 49]. The piezoelectric transducer is formed
from a GaAs conducting layer and a gold gate elecVol. 13 No. 8 Aug. 2015

trode sandwiching an insulating GaAs layer, which is
integrated directly into the mechanical element, asshown in Fig. 2(a). Application of an electric field
across the transducer generates in-plane strain thatcan nonlinearly modulate the spring constants andhence, the frequencies of the modes supported by themechanical system [6].
The Hamiltonian for the electromechanical systemin this configuration can then be expressed as:AH=(P / 2m + m w Q / 2)+ΛQ Q cos(w t). (1)

n=S

2
n

n

2 2
n n n

S

A

p

Here, P and Q are the canonical momentum and conjugate position of the symmetric and asymmetricmodes with mass m and natural frequency ωn, where
the summation expresses their kinetic and potentialRegular Articles

energies. The last term describes the piezoelectrically
activated non-degenerate parametric down-conversion with amplitude Λ and frequency wp = wS + wA,
that is, at the sum frequency of the modes in question.
Ostensibly, this Hamiltonian is classical, which
superficially suggests the unavailability of two-mode
squeezing. However, the last term is analogous to† †0 n (ab + a b ) for non-degenerate parametricdown-conversion in cavity electromechanical systems
in the frame rotating at the pump frequency, where a,b and a†, b† are the annihilation and creation operators
for the mechanical and microwave resonators, respectively, and 0 is the intrinsic coupling rate between
them that is enhanced by n pump photons [41–43].
Consequently, Λ encapsulates the intrinsic coupling
rate between the symmetric and asymmetric modes
that is enhanced by the piezoelectric pump, which
results in phonon pairs being simultaneously generated in them [50]. Remarkably, these phonons shouldstill be correlated, even if the two modes are thermalized and far from their quantum ground state, as longas their generation rate exceeds the rate at which they
are dissipated from both modes [46].
4. Results
4.1 Non-degenerate parametric amplification
To ascertain if this latter expectation is satisfied, thethermomechanical fluctuations of both modes arespectrally measured via optical interferometry with a
3-µW helium neon laser probing the right element of
the electromechanical system at room temperatureand in a high vacuum [46]. This measurement reveals
the modes with natural frequencies wS / 2π ≈ 246 kHz
and wA / 2π ≈ 262 kHz with quality factors Qn = wn /∆wn of 1300 and 2200, respectively, from their spectral power bandwidth ∆wn, and it corresponds to bothmodes sustaining > 107 phonons. Next non-degenerate parametric down-conversion is activated, and thethermal fluctuations from both modes are monitored.
The results of this measurement shown in Fig. 2(d)indicate that as the piezoelectric pump voltage isincreased, the thermomechanical fluctuations of bothmodes are enhanced. This amplification can be referenced to the bare thermal fluctuations, and it yields
gains of more than 20 dB that are accompanied by anarrowing of the power spectral bandwidth from bothmodes, as detailed in Fig. 2(e). At the largest pump
amplitudes (>0.275 Vrms) ∆wn → 0, resulting in bothmodes undergoing non-degenerate parametric resonance [51, 52].
To confirm these experimental observations, theequations of motion for both modes are extracted
from the Hamiltonian in eq. (1) and reformulated intheir rotating frames with Qn = Xn cos(wnt) + Ynsin(wnt), where Xn and Yn are the slowly varying inphase and quadrature components. The resultant
equations are then numerically solved, as detailed ina previous study [46], and they faithfully reproduce
the experimental response as shown in Fig. 2(f), thus
verifying that the amplification can be ascribed to
non-degenerate parametric down-conversion.
The temporal dynamics of this amplification canalso be acquired by mixing the output from the optical interferometer with local oscillators locked onto
the resonances of both modes and then demodulated
in a phase sensitive detector (PSD). The PSD samples
the random displacement fluctuations of the mechanical modes driven by the thermal Langevin force at a
rate of 50 ms over a period of 300 s. This yields fourtime series for the in-phase and quadrature components of both modes, enabling their phase portraits tobe constructed as shown in Figs. 3(a) and 3(b). Thismeasurement reveals that the thermal fluctuations ofboth modes are random and uncorrelated as evidenced by their circular distribution in phase space,indicating all vibration phases are equally likely (redand blue points in Figs. 3(a) and 3(b)). Repeating thismeasurement with a pump amplitude of 0.25 Vrms
confirms that the fluctuations in both modes areenhanced via the non-degenerate parametric downconversion while retaining their random nature;
namely, this amplification is phase insensitive [53].
The observed amplification arises from phonons
generated in both modes from the same process;hence, their fluctuations should be correlated. To that
end, the cross quadratures are constructed in phasespace from the in-phase component of the symmetricmode and the quadrature component of the asymmetric mode and vice versa, as shown in Figs. 3(c) and
3(d). This unveils squeezed distributions where a particular phase orientation is amplified while the perpendicular phase is de-amplified. The narrowness of
this distribution implies the existence of correlations
between the symmetric and asymmetric modes (if nocorrelations between the modes existed, the crossquadratures would yield circular and therefore random distributions) and is the signature feature of a
two-mode squeezed state [54, 55]. However, in orderto quantitatively verify the existence of correlations
between the modes, two criteria need to be satisfied.
First, the two-mode squeezed distributions should
exhibit smaller fluctuations than the bare distributions; otherwise, the correlations would be washed
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(b)

0.06

0.06

0.03

0.03
YA (V)YS (V)

(a)

0.00

−0.03

−0.06
−0.06

0.00

−0.03

−0.03

(c)

0.00
XS (V)

0.03

0.06

−0.06
−0.06

−0.03

0.00
XA (V)

0.03

0.06

−0.03

0.00
XA (V)

0.03

0.06

(d)

0.06

0.06
−

y

0.03

0.03
YS (V)YA (V)

+

x

0.00

−0.03

−0.06
−0.06

0.00

−0.03

−0.03

0.00
XS (V)

0.03

0.06

−0.06
−0.06

Fig. 3. (a) and (b) Bare thermal fluctuations of symmetric (red points) and asymmetric (blue points) modes respectively
acquired temporally on-resonance and projected in phase space via the in-phase (X) and quadrature (Y) axes.
Activating the non-degenerate parametric down-conversion with a pump amplitude of 0.25 Vrms amplifies thefluctuations of both modes while preserving their phase (black points). (c) and (d) The cross quadratures of thepumped measurement XS: YA and XA: YS respectively reconstructed in phase space (black points) reveal two-mode
squeezed states have been generated that exhibit fluctuations in a particular phase which are below the bare thermal
fluctuations from both modes (red and blue points), while fluctuations in the orthogonal phase are amplified.

out by the thermomechanical noise from both modes.
Second, the correlations should be statistically evident from the temporal data used to construct thephase portraits.
4.2 Squeezing below the thermal noise
The former condition can in fact be visibly confirmed by examining the data in Figs. 3(c) and 3(d),which indicate that the de-amplified phases encompass a narrower distribution than the bare thermal
motion from both modes. Quantitatively, new axes x+
and y− are introduced, as shown in the inset to Fig.
3(c), onto which the counts in the squeezed distributions are projected, as shown in Figs. 4(b) and 4(c)[46]. This reveals Gaussian profiles with a zero mean
displacement, as expected from random fluctuations
Vol. 13 No. 8 Aug. 2015

driven by the thermal Langevin force, whose full
width at half maximum can be used to extract theirstandard deviation σ. This analysis is repeated as afunction of non-zero pump amplitude for the crossquadratures XS: YA and XA: YS, where the standard
deviations for both the amplified and de-amplified
phases corresponding to the x+ and y− axes are determined as shown in Fig. 4(a). Naturally, as the pump
amplitude is increased, the fluctuations along the x+
axis are amplified, while concurrently they are
reduced along the y− axis where entropy is conservedin this process. The gain can then be extracted bynormalizing with the narrowest standard deviationfrom the bare thermal distribution (corresponding to
the quadrature component of the asymmetric mode).
While this slightly overestimates the amplification, itRegular Articles

(b)
200

0.018
0.016
0.014

5.715
5.08
4.445

0.012

3.81

0.01

3.175

0.008

2.54

40

0.006

1.905

0

0.004

Counts

160

Gain

σ (V)

(a)

x+

120
80

(c)

−0.04 −0.02 0.00 0.02 0.04
x+ (V)

200

1.27
160
Counts

1
y−
0.002

120
80

0.635
40

0.00

0.05

0.10

0.15

0.20

0.25
0

Pump amplitude (Vrms)−0.04 −0.02 0.00 0.02 0.04
y− (V)

Fig. 4. (a) Standard deviation (σ) of the fluctuation distribution from the two-mode squeezed states in phase space when
projected onto the rotated axes detailed in Fig. 3(c) as a function of pump amplitude. The amplification along the x+
axis is concurrently accompanied with a de-amplification along the y− axis. The red (blue) points correspond to the
XS: YA (XA: YS) cross-quadrature reconstruction, and the solid black lines denote the ideal theoretical response
extracted from the Hamiltonian in eq. (1). Also shown is the gain normalized to the narrowest bare thermal distribution
highlighted by the dashed red line. (b) and (c) The phase portrait from XA: YS with a pump amplitude of 0.2 Vrms
projected onto the rotated x+ and y− axes reveals Gaussian distributions (points), and the corresponding leastsquares Gaussian fit (line) enables their standard deviation to be extracted as quantified above.

is consistent with the spectral response detailed inFig. 2(e). However, this normalizing reference underestimates the de-amplification, yielding a conservative 5-dB suppression of the mechanical fluctuations
below the thermal level where further reduction is
inhibited by noise from the piezotransducers at large
pump voltages [46].
To confirm these observations, the thermal Langevin force, in the equations of motion extracted fromthe Hamiltonian in eq. (1), is decomposed into inphase and quadrature components [16, 45, 46]. This
leads to a new set of composite variables for thesqueezed distributions given by XS ± YA and XA ± YS
that naturally correspond to the rotated axes x+ and y−
introduced in Fig. 3(c) [37, 46]. The pump-induced
gain in the fluctuations of these composite variables
can then be extracted from the Langevin correlation
function, yielding the solid lines in Fig. 4(a). This
verifies that the noise reduction below the thermal
level arises from the non-degenerate parametricdown-conversion.


4.3 Correlation coefficient
The statistical correlations between the time series
for the in-phase and quadrature components fromboth modes can be analyzed via the absolute correlation coefficient |cov(ZiZi) /σ Ziσ Zi|, where Zi ∈ {XS, YS,
XA, YA} and the numerator describes the covariance.
Analyzing the bare thermal fluctuations (shown by
the red and blue points in Figs. 3(a) and 3(b)) enables
a correlation coefficient matrix to be constructed,
yielding 16 permutations. However, only 10 combinations are relevant due to their symmetry, as shownin Fig. 5(a). This reveals that the diagonal elements
corresponding to the auto-correlations of Zi yield acoefficient of 1, which indicates that they are perfectly correlated with themselves, as one would
expect. However, all the off-diagonal elements yielda coefficient of 0, indicating an absence of correlations; for instance |cov(XSYS) /σ XSσ YS| ≈ 0, which isunsurprising, as this maps the circular distribution in
Fig. 3(a) corresponding to the random uncorrelated
fluctuations driven by the thermal Langevin force.
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(a)

(b)

1.0

1.0

ff.

t

0.6

ela

t

0.6

0.4

rr
Co

0.4

rr
Co

ela

0.8

e
co
ion

eff
co
ion

.

0.8

0.2
0.0

0.2
0.0

Y

Xs

A

Y
s

s

Xs

YA

Xs

YA

X

A

XA

Y

XA

A

Ys

X

Ys

Y

Xs

A

(d)

(c)

1.0

t
ela

t

0.4

rr
Co

0.4

rr
Co

ela

0.6

c
ion

0.6

c
ion

0.8

oe

ff.

0.8

oe

ff.

1.0

0.2
0.0

Y

Xs

A

A

Y

s

XA

Y
s

s

YA

Xs

Xs

YA

Y

Xs
A

A

Y

XA

0.0

X

X

Y

s

0.2

Fig. 5. (a)–(d) Correlation coefficient matrix as a function of pump amplitude from 0, 0.1, 0.2, and 0.25 Vrms, respectively.
The diagonal elements (black) correspond to auto-correlations, which naturally yield a coefficient of 1. The off-diagonal elements from XS: YS and XA: YA yield a coefficient of 0, which corresponds to the uncorrelated random fluctuations corresponding to the circular distributions in phase space as shown in Figs. 3(a), 3(b) and are independent of
the pump amplitude. In contrast, the off-diagonal elements from the two-mode squeezed states XS: YA (red) and XA:
YS (blue) converge towards a coefficient of 1 as the pump amplitude is increased, implying that the two mechanicalvibration modes have become perfectly correlated via the simultaneous generation of signal and idler phonons inpairs from the non-degenerate parametric down-conversion.

Repeating this analysis as a function of pump amplitude ranging from 0.1, 0.2, and 0.25 Vrms as shown in
Figs. 5(b)–(d) indicates that the auto-correlations in
the diagonal elements remain perfect. However, theoff-diagonal elements corresponding to the crossquadrature phase portraits in Figs. 3(c) and 3(d),namely, XS: YA and XA: YS, emerge and converge
towards 1, which implies that the two modes havebecome perfectly correlated from the simultaneous
generation of phonon pairs via non-degenerate paraVol. 13 No. 8 Aug. 2015

metric down-conversion.
5. Implications and outlook
Although both mechanical modes sustain more than
10 million phonons at room temperature, the simultaneous generation of phonon pairs from the nondegenerate parametric down-conversion occurs at a
rate that renders their fluctuations indistinguishable.
Thus, remarkably, these thermalized macroscopicRegular Articles

mechanical vibration modes become perfectly
entwined in a two-mode squeezed state. Ultimately, if
the modes can be operated with phonon populationsof much less than one, then these correlations will
manifest themselves in a macroscopic all-mechanicalentanglement [10, 42, 56]. Access to such a state
would be extremely tantalizing, as it would provide
an invaluable platform to investigate the absence ofquantum mechanical phenomena in our everyday
classical world [20].
From a technological point of view, these results
herald the emergence of quantum optics to phononics, and thus, concepts such as quantum cryptography
and optical quantum computing could potentially beharnessed with sound in a microchip. More immediately, the possibility of even greater squeezing
becomes available via more strongly coupled modes
in conjunction with a more efficient piezoelectric
pump [57]. This enhanced squeezing would not onlyoffer detectors that could operate below the quantum
limit, yielding unprecedented sensitivities for metrological applications [33] and fundamental science[58], but it could even be utilized to create room temperature entanglements, thus bringing quantum soundinto a more technologically accessible regime [59].
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Regular Articles

Imran Mahboob

Hiroshi Yamaguchi

Senior Research Scientist, Distinguished
Researcher, Hybrid Nanostructure Physics
Research Group, NTT Basic Research Laboratories.
He received an MPhys. in theoretical physics
from the University of Sheffield, UK, in 2001
and a Ph.D. in physics from the University of
Warwick, UK, in 2004. He joined NTT Basic
Research Laboratories in 2005 and has been
researching electromechanical systems to harness their nonlinear dynamics to develop phonon-based information technologies. He is amember of the Institute of Physics (IoP) and theAmerican Physical Society (APS).

Executive Manager of Quantum and Nano
Device Research, Group Leader of Hybrid Nanostructure Physics Research Group, Senior Distinguished Researcher, NTT Basic Research Laboratories.
He received a B.E. and a M.S. in physics and a
Ph.D. in engineering from Osaka University in
1984, 1986 and 1993, respectively. He joined
NTT Basic Research Laboratories in 1986 andstudied compound semiconductor surfaces using
electron diffraction and scanning tunnelingmicroscopy. His current interests are micro/
nanomechanical devices using semiconductor
heterostructures. He was a visiting research fellow at Imperial College, University of London,
UK, during 1995–1996 and a visiting researcher
at the Paul Drude Institute, Germany, in 2003. He
has been a guest professor at Tohoku University
since 2006 and has served on more than 40 committees of academic societies and international
conferences. He has received numerous awards
including a JSAP Fellowship (2013), Commendation for Science and Technology by MEXT
(2013), Inoue Prize for Science (2012), IoP Fellowship (2011), SSDM2009 Paper Award (2010),
MNC2008 Outstanding Paper Award (2009), andPaper Awards from JSAP (1989, 2004, 2010). He
is a member of JSAP, the Physical Society ofJapan, IoP, APS, and the Institute of Electricaland Electronics Engineers.

Hajime Okamoto
Senior Research Scientist, Hybrid Nanostructure Physics Research Group, NTT Basic
Research Laboratories.
He received a B.E., M.E., and a Ph.D. in materials science and engineering from Waseda University, Tokyo, in 1998, 2000, and 2004, respectively. He joined NTT Basic Research Laboratories in 2004 to research and develop semiconductor opto/electro-mechanical devices. He received
an Outstanding Paper Award from the Japan
Society of Applied Physics (JSAP) (2010),
SSDM2009 Paper Award (2010), and the Young
Scientists’ Prize (2014) from the Minister of
Education, Culture, Sports, Science and Technology (MEXT). He is a member of JSAP.

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NTT Technical Review