# Mode competition and anomalous cooling in a multimode phonon laser

Utku Kemiktarak1,2, Mathieu Durand2, Michael Metcalfe1,2, and John Lawall2
1Joint Quantum Institute, University of Maryland, College Park, MD 20742, USA
2National Institute of Standards and Technology, 100 Bureau Drive,
Gaithersburg, MD 20899, USA

###### Abstract

We study mode competition in a multimode “phonon laser” comprised of an
optical cavity employing a highly reflective membrane as the output coupler.
Mechanical gain is provided by the intracavity radiation pressure, to which
many mechanical modes are coupled. We calculate the gain, and find that strong
oscillation in one mode suppresses the gain in other modes. For sufficiently
strong oscillation, the gain of the other modes actually switches sign and
becomes damping, a process we call “anomalous cooling.” We demonstrate that
mode competition leads to single-mode operation and find excellent agreement
with our theory, including anomalous cooling.

###### pacs:

07.10.Cm,42.55.Ah,42.50.Wk,42.79.Dj,78.67.Pt,85.85.+j,05.40.Jc,07.60.Ly

Introduction. –  While the laser was invented more than five decades ago, its
acoustic analog has only recently been realized. Following the observation of
phonon amplification in microwave-pumped rubyTucker (1961) in 1964, the
possibility of “phonon lasing” was suggested. Subsequent work in ruby studied
the emission spectrum of phonon generationGanapolskii and Makovetskii (1974)
and multimode processesGanapolskii and Makovetskii (1994). With the advent of
optical pumping, detailed studies of the coherence of phonon emission were
enabledTilstra  _et al._ (2003), culminating in a ruby “saser”Tilstra  _et
al._ (2007). Shortly thereafter, phonon lasing was realized in a harmonically
bound Mg+ ion driven by optical forcesVahala  _et al._ (2009). Subsequently,
it was recognized that a large class of optomechanical systems, in which
optically furnished gain enables self-sustained mechanical oscillation, are
properly called “phonon lasers”Khurgin  _et al._ (2012). These include
beamsKhurgin  _et al._ (2012); Bagheri  _et al._ (2011) and cantileversMetzger
_et al._ (2008) coupled to an optical cavity, microtoroidsKippenberg and
Vahala (2007); Grudinin  _et al._ (2010), and a cantilever deriving mechanical
gain from optical bandgap excitationOkamoto  _et al._ (2011). Analogous
electromechanicalBargatin and Roukes (2003); Bennett and Clerk (2006) and
purely mechanical Mahboob  _et al._ (2013) systems have also been discussed.
Various phenomena associated with lasers, such as stimulated emissionVahala
_et al._ (2009), oscillation thresholdVahala  _et al._ (2009); Khurgin  _et
al._ (2012); Grudinin  _et al._ (2010); Okamoto  _et al._ (2011); Mahboob  _et
al._ (2013), gain narrowingMahboob  _et al._ (2013), and injection
lockingKnünz  _et al._ (2010), have been demonstrated.

With few exceptions, these investigations have involved a single mechanical
mode. Multimode emission was observed in rubyGanapolskii and Makovetskii
(1974, 1994), and two-mode oscillation was observed in a photothermally
coupled optomechanical systemMetzger  _et al._ (2008). Intermodal coupling in
an electromechanical system was exploited to realize a phonon laser without an
optical pumpMahboob  _et al._ (2013). In the domain of conventional lasers, an
interesting and important feature arises when multimode operation is
considered. As shown by Lamb in 1964Lamb (1964), a saturation phenomenon
occurs in which oscillation of one mode suppresses the gain of other modes.
This has the dramatic consequence that, in the absence of inhomogeneous gain
broadening, a laser oscillates in steady state on a single mode, even when the
small-signal gain exceeds the losses for more than one modeSiegman (1986).

In view of the multimode oscillation observed in the photothermal
systemMetzger  _et al._ (2008), it is natural to ask whether a phonon laser
employing pure radiation pressure coupling would exhibit the single-mode
oscillation characteristic of a homogeneously broadened laser. Here we study
such a system, in which one cavity mirror is formed by a highly reflective
membrane supporting many mechanical modes. We find that when the mechanical
gain exceeds the losses for more than one mode, the steady-state condition is
nevertheless always that of a single oscillating mode. We calculate the gain
for the case of two modes, and find that, just as in the conventional laser, a
strongly oscillating mode tends to “steal” gain from competing modes.
Sufficient oscillation amplitude, in fact, reverses the sign of the gain of
more weakly oscillating modes, causing them to be optically damped.
Experimentally, we are able to force certain modes to oscillate, or to put the
system in a regime in which the mode that ultimately oscillates is
unpredictable and depends on thermal fluctuations. In addition, we verify our
prediction that as the oscillation strength of one mode is increased, the
quenched modes are in fact cooled.

Figure 1: A Fabry-Perot cavity employing a highly reflective sub-wavelength
grating in a silicon nitride membrane, operated in vacuum, forms the
optomechanical system. A “probe” laser is locked to a cavity resonance, while
a “pump” laser is blue-detuned relative to the adjacent longitudinal cavity
resonance and provides mechanical gain. A third laser is used for a Michelson
interferometer to sense membrane displacements.

Experiment. –  As illustrated in Fig. 1, our optomechanical resonator is used
as the output coupler of a Fabry-Perot cavity. It is a square (side
$a=1.25\,\mathrm{mm}$) silicon nitride membrane, patterned with an array of
gratings, each $50\,\mu\mathrm{m}$ on a side. Each grating has a period
smaller than the wavelength ($1.56\,\mu\mathrm{m}$) of the light, and has
near-unity reflectivity. The optomechanical properties of the device have been
reported previouslyKemiktarak  _et al._ (2012a, b). Only one of the gratings
is used at a time, but by choosing a particular grating, one can optimize the
radiation pressure coupling to a certain set of mechanical modes. All of the
results in this paper employ a grating centered at
$(x_{0},y_{0})=(465(10)\,\mu\mathrm{m},400(10)\,\mu\mathrm{m})$ relative to a
membrane corner. The reflectivity of this particular grating yields a cavity
finesse $F\approx 1000$. The input coupler has a radius of curvature of $R=25$
mm. We use a cavity length $L_{cav}\approx R$ so that the waist $\omega_{0}$
of the optical mode is below $20\,\mu\mathrm{m}$; correspondingly, the cavity
free spectral range is $\Delta\nu\approx 6\,\mathrm{GHz}$.

We introduce two lasers into the cavity: a probe laser and a pump laser, both
with $\lambda=1.56\,\mu\mathrm{m}$, as shown in Fig. 1. The probe laser is
locked to a cavity resonance; the pump is frequency-offset from the probe
laser by $\Delta\nu+\delta\nu$, where $\delta\nu$ represents an arbitrary
detuning from the adjacent cavity mode. In order to transduce membrane
displacements, we implement a Michelson interferometer targeting the grating
used for the optical cavity. The Michelson beam employs a third laser at
$1.56\,\mu\mathrm{m}$ that is far off resonance from any cavity mode (cavity
linewidth$\,\approx 6\,\mathrm{MHz}$).

Radiation pressure-induced dynamics. –  The power circulating in the Fabry-
Perot cavity is correlated to the membrane motion, optically modifying the
dynamics. The case of a single mechanical mode has been studied
extensivelyKippenberg and Vahala (2007); the radiation pressure enables
optically modified frequency shifts, cooling, and oscillation. Here we
generalize to multiple mechanical modes. We express the membrane displacement
$z(x,y,t)$ as a sum of products of normal modes $\phi_{mn}(x,y)$ with time-
dependent factors $q_{mn}(t)$: $z(x,y,t)=\sum_{m,n}q_{mn}(t)\phi_{mn}(x,y)$.
For a uniform square membrane, the modes are given by
$\phi_{mn}(x,y)=\sin\frac{m\pi x}{a}\sin\frac{n\pi y}{a}$, and the effective
mass $m_{eff}$ is equal to one fourth of the membrane massTimoshenko (1937).
Each mode is driven by generalized force $F_{mn}(t)=\int\int
f(x,y,t)\phi_{mn}(x,y)\,dx\,dy$Timoshenko (1937), where $f(x,y,t)$ is the
radiation pressure force per unit area.

The amplitude $u(t)$ of the electric field circulating in a high-finesse
Fabry-Perot cavity with a varying cavity length $L(t)=L_{0}+z(t)$ is governed
by the differential equation

$\dot{u}(t)+\left(\gamma-i\,(\delta\omega+\frac{4\pi
z(t)}{\lambda}\Delta\nu)\right)\,u(t)=i\Delta\nu\,\sqrt{T_{1}\,P_{in}}$ (1)

where $\Delta\nu=c\,/(2L_{0})$, $\gamma$ is the cavity field decay rate,
related to the finesse $F$ by $\gamma=\pi\Delta\nu/F$, and $T_{1}$ is the
transmission of the input coupler. The cavity is driven by laser light of
frequency $\nu_{L}$, wavelength $\lambda$, and power $P_{in}$, detuned from a
resonance frequency $\nu_{0}$ by
$\delta\omega=2\pi\delta\nu=2\pi(\nu_{L}-\nu_{0})$. The intensity distribution
is Gaussian, with spot size $\omega_{0}$, centered at $(x_{0},y_{0})$.

For a sinusoidal membrane oscillation $q_{mn}(t)\equiv z_{mn}\sin
2\pi\nu_{mn}t$, the solution to Eq. (1) contains a spectrum of sidebands
separated by $\nu_{m}$. The dimensionless quantity $\chi_{mn}\equiv
2\,\frac{\Delta\nu}{\nu_{mn}}\frac{z_{mn}}{\lambda}\phi_{mn}(x_{0},y_{0})$
appears as a natural expansion parameter, and, for $\chi_{mn}>1$, corresponds
roughly to the number of sidebands with significant amplitude. The radiation
pressure $F^{RP}=2|u(t)|^{2}/c$ associated with the circulating optical power
oscillates at $\nu_{mn}$ and all of its harmonics. For a high-$Q$ mechanical
oscillator, the dynamics are well described by

$\ddot{q}_{mn}+(\Gamma_{mn}^{intr}+\Gamma_{mn}^{RP}(\\{\chi_{rs}\\}))\dot{q}_{mn}+\omega_{mn}^{2}q_{mn}=\frac{F_{th}(t)}{m_{eff}}$
(2)

Here $\omega_{mn}=2\pi\nu_{mn}$, and $\Gamma_{mn}^{intr}$ is the intrinsic
damping of mode $mn$, related to the mechanical quality factor $Q_{mn}$ by
$\Gamma_{mn}^{intr}=\omega_{mn}/Q_{mn}$. $F_{th}(t)$ is the thermal Langevin
force, with spectral density $S_{F}(\omega)=4k_{B}Tm_{eff}\Gamma_{mn}^{intr}$.
The $\Gamma_{mn}^{RP}(\\{\chi_{rs}\\})$ are optical modifications to the
damping of mode $mn$; modifications to the $\omega_{mn}$ are also present but
not significant here. In general, $\Gamma_{mn}^{RP}(\\{\chi_{rs}\\})$ depends
on the set of amplitudes $\\{\chi_{rs}\\}$ of all of the modes.

If the amplitudes are all small ($\chi_{mn}<<1$), only the first-order
sidebands need be considered. In this case $\Gamma_{mn}^{RP}$ can be shown to
be independent of the $\\{\chi_{rs}\\}$, and the optical damping works
independently for each mode. In previous work we have optically cooled
hundreds of mechanical modes simultaneouslyKemiktarak  _et al._ (2012b).

The situation $\Gamma^{RP}_{mn}<0$ corresponds to antidamping, or optically-
furnished mechanical gain, and is obtained by blue detuning ($\delta\nu>0$).
If the optical gain exceeds the intrinsic damping,
$-\Gamma^{RP}_{mn}>\Gamma^{intr}_{mn}$, the amplitude rings up from its
thermal value, and the first-order theory loses validity. Indeed, as the
amplitude of each mode grows, it suppresses the gain of all of the other modes
as determined by the rates $\Gamma^{RP}_{mn}(\\{\chi_{rs}\\})$. This
phenomenon of intermode gain suppression has a dramatic signature: it causes
an antidamped system to oscillate on a single mode, even if the unsaturated
mechanical gain exceeds the oscillation threshold for more than one mode.

To see this, we start by calculating $\Gamma^{RP}$ for the case of two low-
order nondegenerate modes, where we use the letters $A$ and $B$ to label mode
indices $mn$. For $\omega_{0}<<a$, we findcal  the gain of mode $A$ to be

$\displaystyle\Gamma^{RP}_{A}(\chi_{A},\chi_{B})={\cal
C}\,P_{in}\left(\frac{\phi_{A}^{2}(x_{0},y_{0})}{\nu_{A}^{2}}\right)\times\mathrm{Im}\left\\{\frac{1}{\chi_{A}}\sum_{k,l=-\infty}^{\infty}\right.$
$\displaystyle\left.\frac{J_{k}(\chi_{A})J_{k-1}(\chi_{A})}{\gamma-i(\delta\omega-(k\omega_{A}+l\omega_{B}))}\frac{J_{l}^{2}(\chi_{B})}{\gamma+i(\delta\omega-((k-1)\omega_{A}+l\omega_{B}))}\right\\}$

where $\chi_{A}$ and $\chi_{B}$ describe the oscillation amplitudes, ${\cal
C}=4T_{1}\Delta\nu^{3}\,/(\pi m_{eff}\lambda c)$, and the $J_{k}$ are Bessel
functions. Clearly the gain of mode $A$ depends on the oscillation amplitude
of mode $B$. The single-mode case, which we consider initially, can be
obtained by taking $\chi_{B}\rightarrow 0$,
$\Gamma^{RP}_{A}(\chi_{A})\equiv\Gamma^{RP}_{A}(\chi_{A},0)$.

The single-mode oscillation threshold condition is given by
$\Gamma^{RP}_{A}(0)=-\Gamma^{intr}_{A}$, and the steady-state oscillation
amplitude $\chi_{A}$ is given by
$\Gamma^{RP}_{A}(\chi_{A})=-\Gamma^{intr}_{A}$. Fig. 2 shows (red curve) the
gain of mode $(m,n)=(2,1)$ normalized to its small-amplitude value,
$\Gamma^{RP}_{21}(\chi_{21})/\Gamma^{RP}_{21}(0)$. We have taken a detuning of
$\delta\omega=0.67\gamma$ and a mechanical frequency of
$\omega_{21}=0.07\,\gamma$, corresponding to values used in our experiment.
This curve shows gain saturation, as expected: $\Gamma^{RP}$ drops to half of
the small-amplitude value for an oscillation amplitude of $\chi\approx 16$.

Mode competition. –  More interesting phenomena arise when we consider how the
gain of mode $(1,2)$ is affected by the amplitude of mode $(2,1)$. Fig. 2
(blue curve) shows the gain of the (1,2) mode, in the limit of small
oscillation amplitude $\chi_{12}$, as a function of $\chi_{21}$. The curve
exhibits two key features: the gain of the weakly-oscillating mode (1,2)
diminishes more rapidly with amplitude $\chi_{21}$ than that of the stronger
mode (2,1), and, when the (2,1) mode oscillates with $\chi_{21}>18$, the gain
of the weak mode actually switches sign and provides damping. Qualitatively
similar behavior (not shown) is found for the gain of the (2,1) mode as a
function of $\chi_{12}$.

Figure 2: Gain saturation vs oscillation amplitude, calculated for detuning
$\delta\omega=0.67\gamma$. Red: Single-mode normalized antidamping
$\Gamma^{RP}_{21}(\chi_{21})/\Gamma^{RP}_{21}(0)$ of mode (2,1) vs
dimensionless amplitude $\chi_{21}$. Blue: Two-mode normalized (anti)damping
of mode (1,2) when it is oscillating weakly,
$\Gamma^{RP}_{12}(0,\chi_{21})/\Gamma^{RP}_{12}(0,0)$, for arbitrary
$\chi_{21}$. The gain of the (1,2) mode switches sign for large $\chi_{21}$.

In our experiment, the mode with the lowest threshold power is the (1,2) mode,
with $\nu_{12}=192\,{\rm kHz}$, $\Gamma_{12}^{intr}=2.5(2)\,\mathrm{s}^{-1}$,
and geometrical coupling $\phi_{1,2}(x_{0},y_{0})=0.62(3)$. The (2,1) mode,
with $\nu_{21}=207\,{\rm kHz}$, $\Gamma_{21}^{intr}=4.8(3)\,\mathrm{s}^{-1}$
and $\phi_{2,1}(x_{0},y_{0})=0.83(2)$, has a slightly higher threshold power,
$P^{thresh}_{2,1}=1.05\,P^{thresh}_{1,2}$, so for incident laser power
$P^{thresh}_{1,2}<P_{in}<P^{thresh}_{2,1}$, the (1,2) mode is the only one
that will oscillate. The (2,1) mode is, however, better coupled to the
radiation pressure, and for $P_{in}>>P^{thresh}_{2,1}$, the net small-
amplitude gain,
$-(\Gamma_{21}^{RP}(\\{\chi_{rs}^{thermal}\\})+\Gamma_{21}^{intr})$, is found
to be 1.8 times larger than the corresponding gain for the (1,2) mode. For
large pump powers, then, the (2,1) mode more quickly rings up from thermal
amplitude, and as it does, the gain for the more weakly oscillating (1,2) mode
is suppressed, as indicated in Fig. 2. Thus by appropriate choice of pump
power, it is possible to deterministically force either the (1,2) or (2,1)
mode to oscillate. With different experimental parameters, we have similarly
been able to force the (1,1) and (2,2) modes to oscillate.

Figure 3: (a), (b): Time dependence of mode competition between (1,2) and
(2,1) modes, where outcome is unpredictable. Blue: (1,2) mode. Red: (2,1)
mode. Solid lines: experiment; pump is turned on at $t\approx 50$ s and turned
off at $t\approx 175$ s. Dashed lines: Simulations; pump on/off times slightly
different for clarity. (c): Set of simulations for slightly different initial
conditions; no thermal excitation. (d): Measured set of 13 trajectories such
as those shown in (a) and (b).

For pump powers exceeding the oscillation threshold of both modes (1,2) and
(2,1), but low enough that the net small-amplitude gains for the two modes are
comparable, it is not possible to predict which mode will oscillate in steady
state. We study the time dependence of the mode competition by sending the
signal from the Michelson interferometer into lockin amplifiers referenced to
$\nu_{12}$ and $\nu_{21}$. Fig. 3a and 3b (solid lines) show typical
amplitudes $z_{mn}(t)$ for such experiments. We switch on the pump at time
$t\approx 50\,\mathrm{s}$, with detuning $\delta\omega=0.67\gamma$ and power
slightly larger than $P^{thresh}_{2,1}$, and switch it back off at $t\approx
175\,\mathrm{s}$. The amplitudes of the (1,2) and (2,1) modes both initially
grow, but after several seconds, one mode grows until its gain is saturated,
while the growth of the other mode is quenched. Independent measurements
confirm that the oscillations of all other modes are likewise quenched.

From the amplitudes of the steady-state oscillation in Fig. 3a and Fig. 3b
($75\,\mathrm{s}<t<175\,\mathrm{s}$), we infer $\chi_{21}=7.4$ and
$\chi_{12}=9.2$, respectively. From the calculated dependence of gains on
$\chi_{21}$ (Fig. 2) and $\chi_{12}$, one infers
$P_{in}=1.18(2)\,P^{thresh}_{1,2}$, and one also finds the net damping
$\Gamma_{mn}^{net}=\Gamma_{mn}^{intr}+\Gamma_{mn}^{RP}(\\{\chi_{rs}\\})$ of
the modes that are quenched to be $\Gamma_{12}^{net}\approx
0.2\,\mathrm{s}^{-1}$ and $\Gamma_{21}^{net}\approx 1.1\,\mathrm{s}^{-1}$. The
fact that the net damping of the (1,2) mode is so small manifests itself in
the size of the fluctuations of the quenched (1,2) mode while the (2,1) mode
is oscillating (Fig. 3a, $75\,\mathrm{s}<t<175\,\mathrm{s}$), that are well
above the thermal level ($t<45\,\mathrm{s}$). The dashed curves in the figures
show the results of simulations based on numerical integration of Eq. (2), in
which the thermal force $F_{th}(t)$ is modeled by means of a memoryless
Gaussian stochastic process. As in the experiment, the mode that ultimately
oscillates cannot be predicted in advance. The only adjustable parameter in
the simulation is the pump laser power, taken to be
$P_{in}=1.18\,P_{1,2}^{thresh}$.

Fig. 3c shows a set of trajectories calculated by integrating Eq. (2) with
$P_{in}=1.18\,P_{1,2}^{thresh}$ for a variety of initial conditions, taking
$T=0$ for clarity. Similar curves were shown in the paper by LambLamb (1964)
in his study of multimode operation of an “optical maser.” Corresponding
curves for thirteen successive realizations of the experiment are shown in
Fig. 3d.

Figure 4: Effective temperature of (1,2) mode while (2,1) mode oscillates,
illustrating anomalous cooling for large pump powers. Black: Temperature
inferred without pump, $T_{0}=278(21)\,\mathrm{K}$. Blue circles: Effective
temperature with pump. Red line: Analytic prediction
$T_{eff}=293~{}\mathrm{K}\times\Gamma_{12}^{intr}/(\Gamma_{12}^{intr}+\Gamma_{12}^{RP})$.
Inset: Histograms of amplitude of (1,2) mode. Black: No pump. Green: Weak
pump, $P_{in}=1.3\,P_{2,1}^{thresh}$. Violet: Strong pump,
$P_{in}=6.4\,P_{2,1}^{thresh}$.

Anomalous cooling. –  While the amplitudes of the fluctuations in the quenched
mode are above the thermal level in Fig. 3a and Fig. 3b, the calculated
antidamping shown in Fig. 2 shows that we expect cooling of the quenched mode
when the amplitude of the oscillating mode is large enough. To study this
matter, we set the (2,1) mode into oscillation with a sequence of ten pump
powers from $P_{in}=1.3\,P_{2,1}^{thresh}$ to $P_{in}=6.4\,P_{2,1}^{thresh}$.
For each power, we measured the amplitude $z_{mn}(t)$ of both the (1,2) and
(2,1) modes for 450 seconds, then extinguished the pump, allowed the
transients to die away, and measured the thermal amplitudes for another 450
seconds. The inset to Fig. 4 shows histograms of the amplitudes $z_{12}(t)$
for the cases of no pump, low ($P_{in}=1.3\,P_{2,1}^{thresh}$) and high
($P_{in}=6.4\,P_{2,1}^{thresh}$) pump powers. In thermal equilibrium, the
amplitude $z_{mn}$ is distributed according to a Boltzmann distribution

$p(z_{mn})=\frac{m_{eff}\omega_{mn}^{2}}{k_{B}T}z_{mn}\,e^{-\frac{m_{eff}\omega_{mn}^{2}z_{mn}^{2}}{2k_{B}T}}$
(3)

Each of the curves in the inset is fit to Eq. (3). The statistics of the set
of ten such thermal (pump off) measurements yields a mean of
$T_{0}=278\,\mathrm{K}$ with a standard deviation of $\approx 12\,\mathrm{K}$.
The largest uncertainty in the inferred temperature arises from
$\phi_{12}(x_{0},y_{0})$, used to infer $z_{12}(t)$, and contributes an
uncertainty of 6%; adding the statistical contribution in quadrature, we
assign an uncertainty of 7.5% to the temperature measurements. At
$P_{in}=1.3\,P_{2,1}^{thresh}$, the statistics of the fluctuations in the
(1,2) mode, while the (2,1) mode is oscillating, correspond to an effective
temperature of $1040(77)\,\mathrm{K}$. At $P_{in}=6.4\,P_{2,1}^{thresh}$, the
effective temperature is $180(14)\,\mathrm{K}$, illustrating the anomalous
cooling predicted in Fig. 2. Fig. 4 shows the effective temperature inferred
from fits to (3) for all ten pump powers. Also shown is the analytic
prediction
$T_{eff}=293~{}\mathrm{K}\times\Gamma_{12}^{intr}/(\Gamma_{12}^{intr}+\Gamma_{12}^{RP})$Kippenberg
and Vahala (2007).

Conclusion. –  We have studied the problem of mode competition in a multimode
phonon laser both theoretically and experimentally. By using a highly
reflective membrane as the end mirror of an optical cavity, in which many
mechanical modes are coupled to the intracavity radiation pressure, we
demonstrate that oscillation of one mode tends to “steal” gain from more
weakly oscillating modes, culminating in single-mode steady-state operation.
Remarkably, strong oscillation of one mode even causes optical damping of the
other modes. In addition to more fully illuminating the analogy between phonon
lasers and their optical counterparts, the insights gained here can be used to
force a particular mode to oscillate when multiple modes are capable of
oscillation, which may be useful as applications of phonon lasers appear.

###### Acknowledgements.

We acknowledge useful discussions with Jake Taylor and National Science
Foundation support through the Physics Frontier Center at the Joint Quantum
Institute. Research performed in part at the NIST Center for Nanoscale Science
and Technology.

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