of 8
PHYSICAL REVIEW A
92
, 042104 (2015)
Mesoscopic mechanical resonators as quantum noninertial reference frames
B. N. Katz,
1
,
*
M. P. Blencowe,
1
and K. C. Schwab
2
,
3
1
Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA
2
Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
3
Kavli Nanoscience Institute, Pasadena, California 91125, USA
(Received 30 September 2014; published 8 October 2015)
An atom attached to a micrometer-scale wire that is vibrating at a frequency
100 MHz and with displacement
amplitude
1 nm experiences an acceleration magnitude
10
9
ms
2
, approaching the surface gravity of a
neutron star. As one application of such extreme noninertial forces in a mesoscopic setting, we consider a model
two-path atom interferometer with one path consisting of the 100 MHz vibrating wire atom guide. The vibrating
wire guide serves as a noninertial reference frame and induces an in principle measurable phase shift in the wave
function of an atom traversing the wire frame. We furthermore consider the effect on the two-path atom wave
interference when the vibrating wire is modeled as a quantum object, hence functioning as a quantum noninertial
reference frame. We outline a possible realization of the vibrating wire, atom interferometer using a superfluid
helium quantum interference setup.
DOI:
10.1103/PhysRevA.92.042104
PACS number(s): 03
.
65
.
Ta
,
03
.
30
.
+
p
,
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.
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.
Wk
,
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.
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.
+
j
I. INTRODUCTION
During the past decade there has been a growing effort to
demonstrate nano-to-mesoscale mechanical systems existing
in manifest quantum states [
1
5
]. One motivation is to under-
stand how classical dynamics arises from quantum dynamics
for systems with center of mass much larger than that of a
single atom, and in particular establish whether the quantum-
to-classical transition is solely a consequence of environmen-
tally induced decoherence [
6
] or perhaps ultimately due to
some as yet undiscovered, fundamental “collapse” mechanism
[
7
]. Three important milestones have been the demonstration
of a
5 GHz mechanical resonator mode in a single-phonon
Fock state [
8
], the demonstration of an entangled
10 MHz
mechanical resonator mode–microwave cavity mode state [
9
],
and the demonstration of a
4 MHz mechanical resonator in a
quadrature-squeezed state with minimum variance 0.80 times
that of the quantum ground state [
10
].
A particularly straightforward mechanical geometry is that
of a long, thin suspended beam (wire) that is supported at
both ends (i.e., doubly clamped). The wire can be driven
transversely, exciting its fundamental flexural mode resonance,
using several available actuation methods. For the example of
a crystalline silicon (Si) wire that is a few micrometers long
and a fraction of a micrometer in cross section, the mechanical
fundamental flexural frequency is

2
π
×
100 MHz [
11
].
Consider an Si atom or other atom type attached to the
surface midway along the length of such a vibrating wire.
Suppose that the midpoint displacement amplitude is
X
0
1 nm. Then assuming that the midpoint undergoes simple
harmonic motion,
X
(
t
)
=
X
0
cos (
t
+
φ
0
), we have for the
maximum acceleration experienced by the attached atom
̈
X
max
=

2
X
0
10
9
ms
2
. This is an unexpectedly large ac-
celeration, 10
8
times larger than the gravitational acceleration
g
on the surface of the Earth and closer in magnitude to the
surface gravity of a neutron star [
12
].
*
Present address: Department of Physics, Pennsylvania State
University, University Park, Pennsylvania 16802, USA.
In this paper, we analyze one possible application of these
extreme accelerations in a mesoscopic setting; there are un-
doubtedly other hitherto unexplored applications. In particular,
we shall consider a model two-path atom interferometer shown
schematically in Fig.
1
. The right path consists of a vibrating
wire, atom guide segment with micrometer dimensions similar
to those described above. The left path is fixed, without
a vibrating wire segment. Incident atom wave packets split
into left and right wave packet components. The right wave
component will accumulate a phase shift relative to the left
wave component as a consequence of the vibrating wire
functioning effectively as a noninertial reference frame for the
traversing right wave. The right wave component eventually
exits the noninertial frame and recombines with the left wave
component. As we shall see below in Sec.
II
, this results in
an in principle detectable fringe visibility for the left-right
wave component interference as the vibrating wire amplitude
is varied in the nanometer range.
Quantum wave interference due to gravitational and inertial
forces has been experimentally demonstrated for neutrons
[
13
16
], atoms [
17
], Cooper pairs [
18
], and electrons [
19
],
verifying the equivalence between these forces for quantum
matter systems [
20
]. Furthermore, a recent comprehensive
analysis takes into account also the possibility of atoms in
each path of an interferometer experiencing different inertial
forces, similar to our vibrating wire, interferometer model
[
21
]. However, in all of these experiments and analyses the
noninertial reference frames (and of course gravity) were
treated as classical systems and there was no reason to
view them otherwise. On the other hand, with mesoscopic
mechanical resonators now being prepared and measured in
manifest quantum states [
8
,
9
] as described above, it is very
natural to consider the consequences for the matter wave
interference of the vibrating wire functioning effectively as
a quantum noninertial reference frame [
22
26
]. We shall
therefore consider in Sec.
III
the effects on the fringe visibility
for the left-right wave component interference when the
vibrating wire is described quantum-mechanically; the details
of the calculations are given in the Appendix. Section
IV
outlines a possible way to realize a noninertial, vibrating
1050-2947/2015/92(4)/042104(8)
042104-1
©2015 American Physical Society
B. N. KATZ, M. P. BLENCOWE, AND K. C. SCHWAB
PHYSICAL REVIEW A
92
, 042104 (2015)
FIG. 1. Scheme of the two-path atom interferometer with nonin-
ertial, vibrating wire frame forming part of the right path.
wire-atom interferometer using a superfluid helium quantum
interference setup.
II. CLASSICAL FRAMES
Beginning first with a classical description, we approx-
imate the noninertial frame as a long, thin beam (wire)
of length
L
with hinged boundary conditions, so that for
small-amplitude transverse displacements (
X
0

L
) the frame
coordinates in the lowest, fundamental flexural mode are
X
(
z,t
)
=
X
0
sin (
πz/L
) cos (
t
+
φ
0
), 0

z

L
. We sup-
pose that atoms traversing the frame in the longitudinal,
z
-coordinate direction are described by localized wave packets
propagating with uniform group velocity
v
=
L/T
relative
to the vibrating frame, where
T
is the atom dwell time in
the frame. The atoms are assumed to be confined by a har-
monic potential to the frame in their transverse,
x
-coordinate
direction that is aligned with the frame-flexing
X
-coordinate
direction. The bound atom potential is then
V
(
x,t
)
=
1
2
2
[
x
X
0
sin (
πt/T
) cos (
t
+
φ
0
)]
2
, where both atom
and frame coordinates (
x
and
X
, respectively) are defined
relative to a common origin in the assumed inertial laboratory
frame. The various characteristic frequencies are assumed to
satisfy
π/T



ω
, so that the atom spends many oscilla-
tion cycles in the frame, while it is tightly bound with negligible
transverse motion relative to the frame. This assumption is
not fundamental but rather for calculational convenience. In
particular, assuming instead


ω
requires a more involved
analysis of the interference between the left and right waves,
but can still result in significant accumulated phase differences.
Micrometer-scale resonators have masses
M
10
15
kg,
while the atom mass
m
10
27
kg, i.e.,
M
m
.Itis
therefore reasonable to neglect the back action of the atom
on the classical frame. We assume that the frame is initially
excited in its fundamental transverse flexural mode and freely
oscillates with negligible change in amplitude (i.e., damping)
over the atom dwell time
T
. In terms of the transverse atom
x
and frame
X
coordinates, the Schr
̈
odinger equation for the
atom in the vibrating frame described by the right path wave
component
ψ
R
(
x,t
) is then
i

∂ψ
R
∂t
=−

2
2
m
2
ψ
R
∂x
2
+
1
2
2
[
x
X
0
sin
(
πt/T
)
cos
(
t
+
φ
0
)
]
2
ψ
R
.
(1)
This equation may be straightforwardly solved assuming
a Gaussian function form. With the atom entering the
frame initially in its transverse ground state,
ψ
R
(
x,
0)
=
(
π

)
1
/
4
exp(
mωx
2
2

), the resulting interference between the left
and right path wave components at
t
=
T
is
ψ
L
(
x,T
)
ψ
R
(
x,T
)
=
π

e
mωx
2

+
,
(2)
where the accumulated phase difference between the left and
right waves is
φ
m
2
X
2
0
T
8

m
2
X
2
0
L
8

v
.
(3)
Note that the only atom attributes that the phase difference
(
3
) depends on are its inertial mass
m
and traversal velocity
v
relative to the vibrating frame;
ω
does not appear to leading
order as a consequence of the atom assumed to be tightly
bound transversely to the frame, i.e.,
ω


. Putting in some
numbers: with
m
10
27
kg,

2
π
×
10
8
s
1
, and
X
0
10
9
m, we have
φ
0
.
5
T
(
μ
sec). Thus, significant phase
shifts result for dwell times in excess of a microsecond.
Interestingly, this estimated phase shift following from
Eq. (
3
) is of the same order of magnitude as the mea-
sured gravitational phase difference between two interfering
neutron beam paths in Ref. [
13
], despite the fact that the
vibrating wire acceleration magnitude

2
X
0
is eight orders
of magnitude larger than
g
10 m s
2
. To understand this,
consider the gravitational phase difference expression [
20
]:
φ
=
mg
α
A/
(

v
), where
m
is the neutron mass,
g
α
is the
component of gravity in the plane of the neutron paths,
A
is the area enclosed by the two paths, and
v
is the neutron
velocity. In contrast to this expression, Eq. (
3
) does not scale
with the area enclosed by the two paths. This is because the
transverse inertial force experienced by the atom is different
in the two paths, in particular much larger in the right path
containing the vibrating wire, whereas the gravitational field is
uniform across the neutron interferometer. Comparing Eq. (
3
)
with the gravitational phase difference expression, the reason
for the similar phase shift magnitudes is that the effective
area to atom velocity ratio term
X
0
L/v
is about eight orders
of magnitude smaller than the corresponding ratio
A/v
for the
neutron interferometer. However, in contrast to the much larger
10 cm scale neutron interferometer where the acceleration
due to gravity is of course well described classically, the
micrometer scale, noninertial vibrating wire frame can also
in principle be prepared in a manifest quantum state.
A possibly more fundamental way to understand the phase
difference (
3
) follows from the original observation of de
Broglie [
27
] that the phase of a particle’s wave function can be
directly expressed in terms of the proper time along the path
042104-2
MESOSCOPIC MECHANICAL RESONATORS AS QUANTUM . . .
PHYSICAL REVIEW A
92
, 042104 (2015)
of the particle. In particular, we have [
20
,
28
]
φ
=−
mc
2

(
t
=
T
t
=
0
R
t
=
T
t
=
0
L
)
,
(4)
where
c
is the speed of light in vacuum and
τ
L
(
R
)
is the
proper time elapsed for the atom traveling along the left
(right) interferometer path. Since the frame velocity
̇
X
max
=
X
0
1ms
1
for the above considered parameters, we have
|
v
L
(
R
)
(
t
)
|
c
and hence
L
(
R
)
=
1
[
v
L
(
R
)
(
t
)
/c
]
2
dt
{
1
1
/
2[
v
L
(
R
)
(
t
)
/c
]
2
}
dt
, where
v
L
(
R
)
(
t
) is the left (right) path
atom velocity relative the laboratory frame. Substituting in the
approximation
v
R
(
t
)
≈−
X
0

sin (
πt/T
)sin(
t
+
φ
0
), valid
for the condition
π/T



ω
,Eq.(
4
) then gives the same
result as Eq. (
3
). Thus, the phase difference (
3
) can be viewed
as a consequence of the “twin paradox” [
20
,
28
], where the
right path wave packet bound to its noninertial, vibrating frame
“ages” less than the left path wave packet during the elapsed
laboratory coordinate time interval
T
.
III. QUANTUM FRAMES
Moving on now to treating the vibrating frame as a
quantum system, the Schr
̈
odinger equation for the composite
atom-frame wave function
R
(
x,X,t
)is
i

R
∂t
=−

2
2
m
2
R
∂x
2

2
2
M
2
R
∂X
2
+
1
2
M
2
X
2
R
+
1
2
2
[
x
X
sin
(
πt/T
)
]
2
R
,
(5)
where we neglect the coupling between the frame and
its dissipative environment and
M
is the effective mo-
tional mass of the frame in the fundamental flexural mode.
The atom is assumed to enter the frame initially in its
transverse ground state:
ψ
R
(
x,
0)
=
(
π

)
1
/
4
exp(
mωx
2
2

), with
R
(
x,X,
0)
=
ψ
R
(
x,
0)
R
(
X,
0) for some initial prepared
frame state
R
(
X,
0).
A potential puzzle concerns the fact that, as a quantum
object, the frame also has a phase and hence there may be an
ambiguity concerning the part of the phase that “belongs” to
the atom. This puzzle is resolved by noting that only the atoms
are detected at the interferometer output, so that the frame
state must be traced over in the interference term, which can
be expressed as [cf. the classical frame interference Eq. (
2
)]
(see Appendix)
Tr
frame
[

x
|
ˆ
U
R
(
T
)
|
ψ
R
(0)

ψ
L
(0)
|⊗
ˆ
ρ
frame
(0)
ˆ
U
L
(
T
)
|
x
]
=
π

e
mωx
2


e
i
ˆ
φ
.
(6)
Here, we allow for the possibility that the frame is initially in
a mixed state, while the unitary operators
ˆ
U
L
(
R
)
(
T
) implement
the atom-frame evolution over the time interval
T
when the
atom is either attached (
R
) or not attached (
L
) to the frame.
Formally, we write the accumulated phase difference as an
average,

e
i
ˆ
φ
, reflecting the fact that the frame is now in a
quantum state.
Equations (
5
) and (
6
) may be straightforwardly solved by
transforming (
5
) to instantaneous normal mode coordinates
such that (
5
) becomes separable, and decomposing the initial
frame state ˆ
ρ
frame
(0) in terms of a coherent state basis
|
α
,
where
α
=
M/
(2

)
X
+
iP/
2
M


. Assuming Gaus-
sian solutions and taking advantage of the separation of
frequency scales,
π/T



ω
, we obtain (see Appendix
for the details of the derivation)

e
i
ˆ
φ
1
π
d
2
α

α
|
ˆ
ρ
frame
(0)
|
αe
imT/
(4
M
)
e
i
m
8
M
(

ω
)
T
.
(7)
Consider the example of a frame initially in a
displaced coherent state: ˆ
ρ
frame
(0)
=|
R
(0)

R
(0)
|=
|
α
0

α
0
|
, where
α
0
=
M/
(2

)
X
(0)
+
iP
(0)
/
2
M


,
with
X
(0)
=
X
0
cos
φ
0
and
P
(0)
=−
MX
0
sin
φ
0
. Equation
(
7
) then gives

e
i
ˆ
φ
e
|
α
0
|
2
(
e
i
m
4
M
T
1)
+
i
m
8
M
(

ω
)
T
e
i
m
4
M
|
α
0
|
2
T
+
i
m
8
M
(

ω
)
T
=
e
i
m
2
X
2
0
T
8

+
i
m
8
M
(

ω
)
T
,
(8)
where the approximation in the second line assumes that
mT /
(4
M
)

1 and
|
α
0
|
2
[
mT /
(4
M
)]
2

1. These condi-
tions put an upper limit on the magnitude of the atom dwell
time
T
such that the back action of the atom on the frame
is negligible. Equation (
8
) coincides with the classical frame
accumulated phase difference (
3
), as we would expect for a
coherent frame state.
We now give two examples of manifest quantum frame
states. As our first example, consider a frame initially in a
Fock state: ˆ
ρ
frame
(0)
=|
N

N
|
,
N
=
0
,
1
,
2
,...
. Equation (
7
)
then gives

e
i
ˆ
φ
e
i
m
4
M
NT
+
i
m
8
M
(

ω
)
T
.
(9)
Note that Eq. (
9
) can be obtained from Eq. (
8
)simplyby
replacing the frame coherent state amplitude modulus squared
|
α
0
|
2
with the frame Fock state number
N
. However, in
contrast to the leading order accumulated phase difference
for the coherent state and classical oscillating frame, the phase
difference for the Fock state frame depends explicitly on the
frame mass
M
.
As our second quantum frame example, consider a frame
initially in a quadrature squeezed vacuum state: ˆ
ρ
frame
(0)
=
|
ξ

ξ
|=
ˆ
S
(
ξ
)
|
0

0
|
ˆ
S
(
ξ
), where the squeeze operator is de-
fined as
ˆ
S
(
ξ
)
=
exp [
1
2
(
ξ
ˆ
a
2
ξ
ˆ
a
2
)] [
29
]. Equation (
7
) then
gives

e
i
ˆ
φ
e
i
m
8
M
(

ω
)
T
cosh
2
(
r
)
e
i
m
4
M
T
sinh
2
(
r
)
e
i
m
8
M
(

ω
)
T
1
i
m
4
M
T
sinh
2
(
r
)
,
(10)
where
ξ
=
re
. Note that the interference term is suppressed
for sufficiently large squeeze parameter
r
such that
sinh
2
(
r
)
e
2
r
/
4

4
M/
(
mT
)

1
.
(11)
As our final example, we consider a frame initially in
a thermal state: ˆ
ρ
frame
(0)
=
Z
1
N
=
0
e
β


(
N
+
1
/
2)
|
N

N
|
.
In this case, Eq. (
7
)gives

e
i
ˆ
φ
e
i
mωT
8
M
sinh(
β

/
2)
sinh
[
β

/
2
imT/
(8
M
)
]
.
(12)
042104-3
B. N. KATZ, M. P. BLENCOWE, AND K. C. SCHWAB
PHYSICAL REVIEW A
92
, 042104 (2015)
Note that the interference is suppressed at sufficiently large
temperatures such that
(

)
1

4
M/
(
mT
)

1
.
(13)
The interference suppression conditions (
11
) and (
13
)
can be made more transparent by expressing in terms of
the atom energy uncertainty arising from the corresponding
frame energy uncertainty. For the example of the squeezed
frame state, the frame energy uncertainty is
E
frame
=


sinh(2
r
)
/
(
2)


2
e
2
r
/
4for
r

1, while for the
thermal frame state, the frame energy uncertainty is
E
frame
β
1
for
β
1



. Substituting into the respective conditions
(
11
) and (
13
), we obtain the following common condition on
the atom dwell time for the loss of quantum interference:
T


m
M
E
frame

E
atom
,
(14)
where we have neglected numerical factors. Thus, loss of
interference occurs when the atom dwell time in the vibrating
wire frame exceeds a dephasing time scale given by the
atom’s energy uncertainty. We may speculate that Eq. (
14
)isa
general condition for dephasing, together with the requirement
that the initial frame state is such that the probabilities
P
N
=
N
|
ˆ
ρ
frame
(0)
|
N
are broadly distributed in the Fock state
number
N
, as is the case for the squeezed state with
r

1
and thermal state with
β
1



.
While a quantum frame state with a sufficiently large energy
uncertainty can lead to a suppression of atom interference, so
too can a mixed, classical frame state, as we have just seen
for the squeezed vacuum and thermal state examples; it is
not possible to qualitatively distinguish between quantum and
classical frame states simply by measuring the atom wave
interference for the two-path interferometer model that we are
considering. Furthermore, while it is unlikely that we would be
able to realize anytime soon such substantial squeezing [
10
]as
required by (
11
) for interference suppression, mesoscale me-
chanical resonators viewed as quantum noninertial reference
frames are nevertheless of theoretical interest for the unusual
insights that are gained.
Just as for the classical frame interference, it is interesting
to determine whether the quantum frame interference follows
from a more fundamental “twin paradox” description, where
the proper time of the atom wave packet traversing the
vibrating wire frame now develops a quantum uncertainty as
a result of the frame being in a quantum state. Formally, the
averaged interference term in (
6
) might be expressed as [cf.
Eq. (
4
)]

e
i
ˆ
φ
=
T
e
i
mc
2

[
T
0
dt
(
d
ˆ
τ
R
/dt
)
T
]
,
(15)
where
T
denotes laboratory coordinate time ordering and
ˆ
τ
R
denotes a quantum proper time operator. A path integral
formulation [
30
,
31
]of(
15
) may be possible, with the class
of interfering world line paths summed over restricted by the
requirement that the formulation must reproduce the quantum
frame interference expression (
7
) in the nonrelativistic limit.
In this respect, treating mesoscopic mechanical resonators
as quantum reference frames may yield, via the equivalence
principle, possible insights concerning the nature of quantum
gravity at low energies [
22
], in particular the effect of quantum
fluctuating space-time on matter wave interference [
32
,
33
].
IV. POSSIBLE REALIZATIONS
We now outline possible methods for realizing the meso-
scopic, vibrating wire interferometer. As was discussed, to
realize a phase difference
φ
1 with an atom and an
acceleration of 10
8
g
, one needs a wire frame dwell time
of
1
μ
s. For a nanomechanical resonator with a length of
approximately 1
μ
m, this requires an atom velocity of 1 m s
1
or less. Alternatively, electrons, given their much smaller
mass, would require a dwell time approximately 1000 times
longer [see Eq. (
3
)], and hence a velocity of
<
10
3
ms
1
to
realize a substantial phase difference in a similar length. As a
consequence, atoms appear to be much more favorable for a
possible experiment.
Figure
2
shows a scheme for a possible superfluid inter-
ferometer device used to detect quantum phase differences
in superfluid helium [
34
36
]. One arm of the interferometer
is interrupted by a 100 nm diameter aperture in a thin
silicon nitride membrane, and the other arm a suspended
nanochannel mechanical resonator [
37
] or nanopipe [
38
] with
diameter-to-length dimensions 250 nm
×
1
μ
m. The quantum
phase difference at critical velocities through submicron
apertures is typically
2
π
×
10, which yields a superfluid
velocity in the suspended channel of
1ms
1
. The quantum
phase that is generated through the large acceleration of the
suspended channel will produce a mass current through the
aperture, which modifies the apparent critical velocity that
is measured with an external diaphragm. The nanochannel
diameter should be sufficiently narrow to avoid the excitation
of transverse acoustic modes at the drive frequency: the speed
of first sound in superfluid
c
He
=
240 m s
1
, yielding an
acoustic wavelength
λ
=
c
He
/f
=
2
.
4
μ
mfor
f
=
100 MHz.
FIG. 2. (Color online) Superfluid interferometer setup:
D
is the
diaphragm which is used to both impose a chemical potential through
hydrostatic pressure and measure the resulting superflow,
A
is the
100 nm diameter aperture used to monitor the quantum phase differ-
ence,
SNC
is the suspended nanochannel filled with superfluid
4
He
moving through the channel at 1 ms
1
and accelerating transversely
at 10
8
g
, driven into motion with an external drive field.
042104-4
MESOSCOPIC MECHANICAL RESONATORS AS QUANTUM . . .
PHYSICAL REVIEW A
92
, 042104 (2015)
It appears possible to integrate a circuit with these elements
on the surface of a chip [
39
]. Other possible realizations
could involve cold atomic clouds or Bose-Einstein condensates
steered on the surface of atom chips [
40
], or guided through
hollow fiber-optic atomic and optical waveguides [
41
].
V. CONCLUSION
In this paper, we have explored several consequences of
mesoscopic mechanical resonators viewed as classical and
quantum noninertial reference frames. We considered a simple
model, two-path atom interferometer set-up, where one of
the paths is furnished by a vibrating wire atom guide. For
the example of a hundred megahertz vibrating, micron-scale
wire frame, nanometer-amplitude wire displacements can
induce significant phase shifts between the interfering atom
quantum wave function components for the two paths. We also
showed that a vibrating wire with sufficiently large quantum or
classical energy uncertainty can result in the loss of quantum
interference between the two atom wave function components.
While we suggested a possible approach towards realizing
such interferometers using a superfluid helium setup, it is
unlikely that significant quantum noninertial frame signatures
in atom wave interference can be demonstrated anytime soon.
Nevertheless, viewing mesoscopic vibrating wires as quantum
reference frames is of fundamental theoretical interest. By
making the correspondence with classical frames, we raised
the possibility of a quantum proper time formulation of
the two-path interference. Such a formulation may provide
insights concerning the effects of quantum fluctuating space-
time on matter wave interference.
ACKNOWLEDGMENTS
We thank Prof. S. A. Werner for conversations that
inspired this work. We acknowledge funding provided by
the Foundational Questions Institute (FQXi), the National
Science Foundation under Grant No. DMR-1104790 (M.P.B.),
and the Institute for Quantum Information and Matter, an
NSF Physics Frontiers Center with support from the Gordon
and Betty Moore Foundation through Grant No. GBMF1250
(KCS).
APPENDIX: DERIVATION OF THE QUANTUM FRAME
ACCUMULATED PHASE DIFFERENCE
In this section, we derive the expression (7) for the aver-
aged accumulated phase difference

e
i
ˆ
φ
between atom wave
components traversing the two-path interferometer, where a
segment of the right path involves a noninertial, vibrating wire
quantum frame. The initial state of the atom-frame system is
1
2
{
|
ψ
L
(0)

ψ
L
(0)
|+|
ψ
R
(0)

ψ
R
(0)
|+|
ψ
L
(0)

ψ
R
(0)
|
+|
ψ
R
(0)

ψ
L
(0)
|
}
ˆ
ρ
frame
(0)
,
(A1)
where
ψ
L
(
x,
0)
=
ψ
R
(
x,
0)
=
(
π

)
1
/
4
exp(
mωx
2
2

); i.e., the
atom enters the interferometer arms initially in its transverse
ground state and the frame is initially in some (possibly mixed)
state ˆ
ρ
frame
(0). After the atom has exited the quantum frame at
time
T
, the atom-frame state is
1
2
ˆ
U
L
(
T
)
|
ψ
L
(0)

ψ
L
(0)
|⊗
ˆ
ρ
frame
(0)
ˆ
U
L
(
T
)
+
1
2
ˆ
U
R
(
T
)
|
ψ
R
(0)

ψ
R
(0)
|⊗
ˆ
ρ
frame
(0)
ˆ
U
R
(
T
)
+
1
2
ˆ
U
L
(
T
)
|
ψ
L
(0)

ψ
R
(0)
|⊗
ˆ
ρ
frame
(0)
ˆ
U
R
(
T
)
+
1
2
ˆ
U
R
(
T
)
|
ψ
R
(0)

ψ
L
(0)
|⊗
ˆ
ρ
frame
(0)
ˆ
U
L
(
T
)
,
(A2)
where the unitary operators
ˆ
U
L
(
R
)
(
T
) implement the atom-
frame evolution over the time interval
T
when the atom is
either attached to the frame, i.e., traversing the right arm
(
R
), or not attached to the frame, i.e., traversing the left
arm (
L
).
The interference between left and right path wave compo-
nents at
t
=
T
is [cf. the classical frame interference Eq. (
2
)]
Tr
frame
[

x
|
ˆ
U
R
(
T
)
|
ψ
R
(0)

ψ
L
(0)
|⊗
ˆ
ρ
frame
(0)
ˆ
U
L
(
T
)
|
x
]
=
π

e
mωx
2


e
i
ˆ
φ
.
(A3)
Implementing the trace in terms of the frame position states
|
X
and inserting the frame coherent state resolution of unity,
1
=
1
π
d
2
α
|
α

α
|
, the interference (
A3
) can be written as
1
π
2
d
2
α
d
2
α
+∞
−∞
dX

x
|
X
|
ˆ
U
R
(
T
)
|
ψ
R
(0)
|
α
×
α
|
ˆ
ρ
frame
(0)
|
α

ψ
L
(0)
|
α
|
ˆ
U
L
(
T
)
|
x
|
X
.
(A4)
Equation (
A4
) expresses the interference term for
an arbitrary initial frame state ˆ
ρ
frame
(0) in terms of
the evolution of initial atom-frame coherent states
αR
(
x,X,T
)
≡
x
|
X
|
ˆ
U
R
(
T
)
|
ψ
R
(0)
|
α
and
αL
(
x,X,T
)

x
|
X
|
ˆ
U
L
(
T
)
|
ψ
L
(0)
|
α
, where the frame coherent state wave
function is
α
(
X
)
≡
X
|
α
=
(
M
π

)
1
/
4
exp
[
1
4
(
X
X
zp
)
2
+
α
(
X
X
zp
)
α
Re(
α
)
]
,
(A5)
with
X
zp
=

2
M
the frame zero point uncertainty. The wave
function
αR
describing the attached atom-frame composite
system is a solution to the Schr
̈
odinger equation (
5
), repro-
duced here:
i

αR
∂t
=−

2
2
m
2
αR
∂x
2
+
1
2
2
[
x
X
sin
(
πt/T
)
]
2
αR

2
2
M
2
αR
∂X
2
+
1
2
M
2
X
2
αR
.
(A6)
On the other hand, the wave function
αL
, describing the
atom-frame composite system with the atom not attached to
042104-5
B. N. KATZ, M. P. BLENCOWE, AND K. C. SCHWAB
PHYSICAL REVIEW A
92
, 042104 (2015)
the frame, is a solution to the following Schr
̈
odinger equation:
i

αL
∂t
=−

2
2
m
2
αL
∂x
2
+
1
2
2
x
2
αL

2
2
M
2
αL
∂X
2
+
1
2
M
2
X
2
αL
,
(A7)
which simply describes two decoupled harmonic oscillators.
It is convenient to work in terms of dimensionless coordinates
̃
x
=
x/x
zp
, with
x
zp
=

2
,
̃
X
=
X
2
Mω/

, and
τ
=
ωt
.
The Schr
̈
odinger equations (
A6
) and (
A7
) then become
respectively
i
αR
∂τ
=
{
2
̃
x
2
2
̃
X
2
+
1
4
(
̃
x
̃
X
)
V
(
τ
)
(
̃
x
̃
X
)}
αR
(A8)
and
i
αL
∂τ
=
{
2
̃
x
2
+
1
4
̃
x
2
2
̃
X
2
+
1
4
(

ω
)
2
̃
X
2
}
αL
,
(A9)
where the coupled potential energy term in the attached
atom-frame Schr
̈
odinger equation (
A8
) has been put in matrix
form to indicate more clearly the possibility to diagonalize the
matrix (i.e., transform to decoupled coordinates):
V
(
τ
)
=
(
1
m
M
sin
(
πτ
ωT
)
m
M
sin
(
πτ
ωT
)(

ω
)
2
+
m
M
sin
2
(
πτ
ωT
)
)
.
(A10)
In particular, noting that
V
(
τ
) is symmetric and hence has real
eigenvalues
λ
±
(
τ
) with associated orthonormal eigenvectors
v
±
(
τ
):
V
(
τ
)
v
±
(
τ
)
=
λ
±
(
τ
)
v
±
(
τ
)
,
(A11)
we have that
S
T
(
τ
)
V
(
τ
)
S
(
τ
)
=
(
λ
+
(
τ
)0
0
λ
(
τ
)
)
,
(A12)
where the transformation matrix is
S
(
τ
)
=
(v
+
(
τ
)
,
v
(
τ
)
)
.
Introducing coordinates
ξ
±
,
(
̃
x
̃
X
)
=
S
(
τ
)
(
ξ
+
ξ
)
,
(A13)
Schr
̈
odinger equation (
A8
) becomes
i
αR
∂τ
=
{
2
∂ξ
2
+
+
1
4
λ
+
(
τ
)
ξ
2
+
2
∂ξ
2
+
1
4
λ
(
τ
)
ξ
2
}
αR
.
(A14)
From Eq. (
A5
), the initial, attached atom-frame composite
coherent state is
αR
(
ξ
+
,
0
)
=
αR
(
̃
x,
̃
X,
0)
=
(
π

)
1
/
4
exp
(
̃
x
2
4
)
×
(
M
π

)
1
/
4
exp
(

4
ω
̃
X
2
+
α

ω
̃
X
α
Re(
α
)
)
,
(A15)
where the first equality follows from the fact that
S
(
τ
)issimply
the identity matrix at
τ
=
0. Given the form of Eq. (
A14
) and
the fact that the initial state (
A15
) is a product state, we have
for the final state
αR
(
̃
x,
̃
X,ωT
)
=
αR
(
ξ
+
,ωT
)
=
ψ
R
(
ξ
+
,ωT
)
αR
(
ξ
,ωT
)
,
(A16)
where the first equality follows from the fact that
S
(
τ
)is
simply the identity matrix at
τ
=
ωT
, and where
ψ
R
and
αR
are solutions to the decoupled Schr
̈
odinger equations:
i
∂ψ
R
∂τ
=
{
2
∂ξ
2
+
+
1
4
λ
+
(
τ
)
ξ
2
+
}
ψ
R
(A17)
and
i
αR
∂τ
=
{
2
∂ξ
2
+
1
4
λ
(
τ
)
ξ
2
}
αR
,
(A18)
with the respective initial conditions following from Eq. (
A15
):
ψ
R
(
ξ
+
,
0)
=
ψ
R
(
̃
x,
0)
=
(
π

)
1
/
4
exp
(
̃
x
2
4
)
(A19)
and
αR
(
ξ
,
0)
=
αR
(
̃
X,
0)
=
(
M
π

)
1
/
4
exp
(

4
ω
̃
X
2
+
α

ω
̃
X
α
Re(
α
)
)
.
(A20)
The eigenvalue solutions to Eq. (
A11
)are
λ
±
(
τ
)
=
1
2
[
1
+
(

ω
)
2
+
m
M
sin
2
(
πτ
ωT
)
]
±
1
2
[
1
+
(

ω
)
2
+
m
M
sin
2
(
πτ
ωT
)
]
2
4
(

ω
)
2
,
(A21)
which under the conditions
m
M
and


ω
can be
approximated as
λ
+
(
τ
)
1
+
m
M
sin
2
(
πτ
ωT
)
(A22)
and
λ
(
τ
)
(

ω
)
2
[
1
m
M
sin
2
(
πτ
ωT
)]
.
(A23)
Solving the decoupled Schr
̈
odinger equations (
A17
), (
A18
)
with approximate eigenvalues (
A22
), (
A23
), and initial condi-
tions (
A19
), (
A20
), it is convenient to assume that the solutions
are Gaussian:
ψ
R
(
ξ
+
)
=
(
π

)
1
/
4
exp[
a
+
(
τ
)
ξ
2
+
+
b
+
(
τ
)
ξ
+
+
c
+
(
τ
)]
(A24)
and
αR
(
ξ
)
=
(
M
π

)
1
/
4
exp[
a
(
τ
)
ξ
2
+
b
(
τ
)
ξ
+
c
(
τ
)]
.
(A25)
042104-6
MESOSCOPIC MECHANICAL RESONATORS AS QUANTUM . . .
PHYSICAL REVIEW A
92
, 042104 (2015)
Substituting Eqs. (
A24
) and (
A25
) into their respective
Schr
̈
odinger equations (
A17
) and (
A18
), we obtain the
following equations for the time-dependent coefficients:
da
±
4
ia
2
±
+
i
4
λ
±
(
τ
)
=
0
,
db
±
4
ib
±
a
±
=
0
,
(A26)
dc
±
2
ia
±
ib
2
±
=
0
.
The approximate solutions to these coefficients at
τ
=
ωT
are
a
+
(
ωT
)
≈−
1
4
,b
+
(
ωT
)
0
,
(A27)
c
+
(
ωT
)
≈−
i
ωT
2
(
1
+
m
4
M
)
,
and
a
(
ωT
)
≈−

4
ω
,
b
(
ωT
)
α

ω
exp
[
iT
(
1
m
4
M
)]
,
c
(
ωT
)
≈−
i
T
2
(
1
m
4
M
)
,
+
α
2
2
(1
e
2
iT
[
1
m/
(4
M
)
]
)
α
Re(
α
)
.
(A28)
Substituting the solutions (
A27
) and (
A28
) into Eq. (
A16
), we
obtain for the evolution of the attached atom-frame coherent
states
αR
(
x,X,T
)
(
π

)
1
/
4
exp
(
mωx
2
2

)
e
i
ωT
2
(
1
+
m
4
M
)
×
X
|
αe
iT
[
1
m/
(4
M
)
]
e
i
T
2
(
1
m
4
M
)
.
(A29)
An analogous but considerably more straightforward analysis
gives the evolution of the atom-frame coherent states when the
atom traverses the left path (i.e., is not attached to the frame):
αL
(
x,X,T
)
=
(
π

)
1
/
4
exp
(
mωx
2
2

)
e
i
ωT
2
×
X
|
αe
iT
e
i
T
2
.
(A30)
Substituting the solutions (
A29
) and (
A30
) into the interfer-
ence expression (
A4
), performing the trace over the frame
subsystem (i.e., integration with respect to the
X
coordinate),
we obtain

e
i
ˆ
φ
1
π
2
d
2
α
d
2
α

α
e
iT
|
αe
iT
[
1
m/
(4
M
)
]
×
α
|
ˆ
ρ
frame
(0)
|
α
e
i
m
8
M
(

ω
)
T
.
(A31)
From the equality

α
e
iT
|
αe
iT
[1
m/
(4
M
)]
=

α
|
αe
imT/
(4
M
)
,the
α
integral can be carried out and
we finally obtain Eq. (
7
) for the averaged accumulated phase
difference between the left and right path atom waves:

e
i
ˆ
φ
1
π
d
2
α

α
|
ˆ
ρ
frame
(0)
|
αe
imT/
(4
M
)
e
i
m
8
M
(

ω
)
T
.
(A32)
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