RESEARCH
| REPORTS
desired state as the ground state. Instead of driving
only the blue sideband, we use the Hamiltonian
gineering have been proposed include super-
conducting circuits and nanomechanics (12–14).
Reservoir engineering provides access to con-
trolled dissipation, which can be used in quantum
simulations of open quantum systems (13, 23).
16. Materials and methods are available as supplementary
materials on Science Online.
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ˇ
ˇ
ˇ
ˇ
ˇ
H
¼ ℏWðK†sþ þ Ks−Þ
ð3Þ
þ
ˇ
in which the motional state operators are con-
jugated with respect to H− (Fig. 1C). This results
ˇ
ˇ
in Rabi oscillations between the states j↓〉jU; n〉
REFERENCES AND NOTES
and j↑〉jU; n þ 1〉. Because the internal states in-
volved span a two-dimensional Hilbert space, the
motional state evolution is also contracted onto
two adjacent states of the engineered basis. For
an arbitrary initial state, the internal state pop-
ulations evolve according to Eq. 2, with the
corresponding p(n) being the probability of
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8. H. Krauter et al., Phys. Rev. Lett. 107, 080503 (2011).
9. J. T. Barreiro et al., Nature 470, 486–491 (2011).
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ACKNOWLEDGMENTS
We thank J. Alonso, A. Imamoglu, and D. Wineland for comments on the
manuscript and useful discussions. We thank J. Alonso, M. Sepiol,
K. Fisher, and C. Flühmann for contributions to the experimental apparatus.
We acknowledge support from the Swiss National Science Foundation
under grant 200021_134776 and through the National Centre of
Competence in Research for Quantum Science and Technology (QSIT).
ˇ
finding the ion in the nth element of the
engineered basis before the application of H
þ
[we denote this as pU ðnÞ in the figure to avoid
confusion]. Data sets from this type of mea-
surement are shown for the coherent state and
for the squeezed state in Fig. 4 for the same
settings as used in Figs. 2 and 3. To work in the
same basis as the state engineering, we again
drive combinations of the carrier and red and blue
motional sidebands, but with the ratios of Rabi
SUPPLEMENTARY MATERIALS
Supplementary Text
Tables S1 to S3
References (24–28)
15. W. Schleich, Quantum Optics in Phase Space (Wiley-VCH,
Berlin, 2001).
9 September 2014; accepted 24 November 2014
10.1126/science.1261033
frequencies calibrated according to Wc=Wbsb
*
¼
−a =coshðrÞ and Wrsb=Wbsb ¼ e−if tanhðrÞ with x
and a corresponding to the values used for the
reservoir engineering (16).
s
ORGANIC CHEMISTRY
We fit both experimental data sets with a form
similar to Eq. 2, obtaining the probability of
being found in the ground state of 0:90 T 0:02
and 0:88 T 0:02 for the coherent and squeezed Rh-catalyzed C–C bond cleavage by
states, respectively. We take these to be lower
bounds on the fidelity with which these states
transfer hydroformylation
were prepared, because these numbers include
ˇ
errors in the analysis pulse in addition to state-
Stephen K. Murphy,1,2 Jung-Woo Park,1 Faben A. Cruz,1 Vy M. Dong1*
preparation errors (16). The H Rabi oscillations
þ
observed in our experiments involve transitions
The dehydroformylation of aldehydes to generate olefins occurs during the biosynthesis of
various sterols, including cholesterol in humans. Here, we implement a synthetic version that
that when viewed in the energy eigenstate basis,
couple Hilbert spaces that are of appreciable size.
features the transfer of a formyl group and hydride from an aldehyde substrate to a strained
olefin acceptor. A Rhodium (Xantphos)(benzoate) catalyst activates aldehyde carbon-hydrogen
(C–H) bonds with high chemoselectivity to trigger carbon-carbon (C–C) bond cleavage and
generate olefins at low loadings (0.3 to 2 mole percent) and temperatures (22° to 80°C).
This mild protocol can be applied to various natural products and was used to achieve a
three-step synthesis of (+)-yohimbenone. A study of the mechanism reveals that the benzoate
counterion acts as a proton shuttle to enable transfer hydroformylation.
ˇ
ˇ
To account for 88% of the populations in oscil-
lations between jSðxÞ; 0〉 and jSðxÞ; 1〉 for r = 1.45,
we must include energy eigenstates up to n =
26. By our choice of basis, we reduce the rele-
vant dynamics to a two-state system, greatly
simplifying the resulting evolution of the spin
populations and thus providing a high signal-
to-noise ratio. The high fidelity with which the
squeezed state is produced is a result of the
robust nature of the reservoir engineering, which
is insensitive to laser intensity and frequency
fluctuations that are common to all frequency
components of the engineered Hamiltonian. To
generate the same state produced above with
standard methods involving unitary evolution
starting from the ground state would require simul-
taneously driving both second motional side-
bands (16). We would not expect a high fidelity
because these have Rabi frequencies comparable
to our transition linewidth, which is broadened
by magnetic field fluctuations.
he cytochrome P450 enzymes have cap-
tured the imagination of chemists who
seek to emulate their reactivity. For exam-
ple, monooxygenases motivated the design
of catalysts that epoxidize olefins and oxi-
To this end, we aimed to trigger C–C bond
cleavage (7–11) by chemoselective activation of
aldehyde C–H bonds using Rh-catalysis (Fig. 1B).
Over the past 50 years, activating aldehyde C–H
bonds with Rh has been thoroughly investigated
(12); however, the resulting acyl-RhIII-hydrides
have been trapped mainly by hydroacylation (13)
or decarbonylation (14, 15). This common inter-
mediate is also implicated in hydroformylation,
which is practiced on an industrial scale using
synthesis gas (16). Thus, we needed a strategy for
diverting the acyl-RhIII-hydride toward dehydro-
formylation. To date, olefins generated by de-
hydroformylation have been observed in low
quantities during decarbonylations (15, 17, 18).
One report describes the use of stoichiometric
Ru for dehydroformylation of butyraldehyde
(19), and another uses heterogeneous Rh or Pd
catalysts for transforming steroidal aldehydes
T
dize C–H bonds (1–4). This enzyme superfamily
also includes various demethylases that break C–C
bonds (5). In particular, lanosterol demethylase
converts aldehydes to olefins by dehydroformyl-
ation during the biosynthesis of sterols in bacte-
ria, algae, fungi, plants, and animals (6) (Fig. 1A).
Inspired by this step in biosynthesis, we sought a
transition-metal catalyst for dehydroformylations
in organic synthesis.
This toolbox for generating, protecting, and
measuring quantum harmonic oscillator states
is transferrable to any physical system in which
the relevant couplings can be engineered, facil-
itating quantum computation with continuous
variables (22). Examples in which reservoir en-
1Department of Chemistry, University of California Irvine, CA
92697-2025, USA. 2Department of Chemistry, University of
Toronto, Ontario M5S 3H6, Canada.
*Corresponding author. E-mail: dongv@uci.edu
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