desired state as the ground state. Instead of driving
only the blue sideband, we use the Hamiltonian
þ ¼ ℏ WðK
ˇ†šþþ Kˇš−Þ ð3Þ
in which the motional state operators are conjugated with respect to H
− (Fig. 1C). This results
in Rabi oscillations between the states j↓〉jU
; n þ 1〉. Because the internal states involved 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 populations evolve according to Eq. 2, with the
corresponding p(n) being the probability of
finding the ion in the nth element of the
engineered basis before the application of H
[we denote this as pUðnÞ in the figure to avoid
confusion]. Data sets from this type of measurement 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
frequencies calibrated according to Wc=Wbsb ¼
−a*=coshðr Þ and Wrsb=Wbsb ¼ e−ifs tanhðr Þ with x
and a corresponding to the values used for the
reservoir engineering ( 16).
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
states, respectively. We take these to be lower
bounds on the fidelity with which these states
were prepared, because these numbers include
errors in the analysis pulse in addition to state-preparation errors ( 16). The H
þ Rabi oscillations
observed in our experiments involve transitions
that when viewed in the energy eigenstate basis,
couple Hilbert spaces that are of appreciable size.
To account for 88% of the populations in oscillations between jS
ðxÞ; 0〉 and jS
ðxÞ; 1〉 for r = 1.45,
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-
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).
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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).
Tables S1 to S3
References ( 24–28)
9 September 2014; accepted 24 November 2014
Rh-catalyzed C–C bond cleavage by
Stephen K. Murphy,1,2 Jung-Woo Park,1 Faben A. Cruz,1 Vy M. Dong1*
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
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.
The 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 oxidize 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 dehydroformylation during the biosynthesis of sterols in bacteria, 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.
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 intermediate 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 dehydroformylation. To date, olefins generated by dehydroformylation 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
56 2 JANUARY 2015 • VOL 347 ISSUE 6217 sciencemag.org SCIENCE
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: firstname.lastname@example.org