Page 1
coupledsystembyapplyingpulsesofvaryinglength.InFig.3b,Rabi
oscillations are shown for the j00. to j11. transition. When the
microwave frequency is detuned from resonance, the Rabi oscil-
lations are accelerated (bottom four curves, to be compared with
the fifth curve). After a p pulse which prepares the system in the
j10. state, these oscillations are suppressed (second curve in
Fig. 3b). After a 2p pulse they are revived (first curve in Fig. 3b).
Inthecase ofFig.3c,thequbit isfirstexcitedontothej10. state by
a p pulse, and a second pulse in resonance with the red sideband
transitiondrivesthesystembetweenthej10. andj01. states.The
Rabi frequency depends linearly on the microwave amplitude, with
a smaller slope compared to the bare qubit driving. During the time
evolution of the coupled Rabi oscillations shown in Fig. 3b and c,
the qubit and the oscillator experience a time-dependent entangle-
ment, although the present data do not permit us to quantify it to a
sufficient degree of confidence.
The sideband Rabi oscillations of Fig. 3 show a short coherence
time (,3ns), whichwe attribute mostly to the oscillator relaxation.
To determine its relaxation time, we performed the following
experiment. First, we excite the oscillator with a resonant low
power microwave pulse. After a variable delay Dt, during which
the oscillator relaxes towards n ¼ 0, we start recording Rabi
oscillations on the red sideband transition (see Fig. 4a for
Dt ¼ 1ns). The decay of the oscillation amplitude as a function of
Dt corresponds to an oscillator relaxation time of ,6ns (Fig. 4b),
consistentwithaqualityfactorof100–150estimatedfromthewidth
of the upresonance. The exponential fit (continuous line in Fig. 4b)
shows an offset of ,4% due to thermal effects. To estimate the
higher bound of the sample temperature, we consider that
the visibility of the oscillations presented here (Figs 2–4) is set by
the detection efficiency and not by the state preparation. When
related to the maximum signal of the qubit Rabi oscillations of
,40%, the 4%-offset corresponds to ,10% thermal occupation of
oscillator excited states (an effective temperature of ,60mK).
Consistently, we also observe low-amplitude red sideband oscil-
lations without preliminary microwave excitation of the oscillator.
We have demonstrated coherent dynamics of a coupled super-
conducting two-level plus harmonic oscillator system, implying
that the two subsystems are entangled. Increasing the coupling
strength and the oscillator relaxation time should allow us to
quantify the entanglement, as well as to study non-classical states
of the oscillator. Our results provide strong indications that solid-
state quantum devices could in future be used as elements for the
manipulation of quantum information.
A
Received 25 May; accepted 5 July 2004; doi:10.1038/nature02831.
1. Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information (Cambridge Univ.
Press, Cambridge, 2000).
2. Nakamura, Y. et al. Coherent control of macroscopic quantum states in a single-Cooper-pair box.
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3. Vion, D. et al. Manipulating the quantum state of an electrical circuit. Science 296, 886–889 (2002).
4. Yu, Y., Han, S., Chu, X., Chu, S. & Wang, Z. Coherent temporal oscillations of macroscopic quantum
states in a Josephson junction. Science 296, 889–892 (2002).
5. Martinis, J. M., Nam, S., Aumentado, J. & Urbina, C. Rabi oscillations in a large Josephson-junction
qubit. Phys. Rev. Lett. 89, 117901 (2002).
6. Chiorescu, I., Nakamura, Y., Harmans, C. J. P. M. & Mooij, J. E. Coherent quantum dynamics of a
superconducting flux qubit. Science 299, 1869–1871 (2003).
7. Pashkin,Yu.A.etal.Quantumoscillationsintwocoupledchargequbits.Nature421,823–826(2003).
8. Berkley,A.J.etal.Entangledmacroscopicquantumstatesintwosuperconductingqubits.Science300,
1548–1550 (2003).
9. Majer, J. B., Paauw, F. G., ter Haar, A. C. J., Harmans, C. J. P. M. & Mooij, J. E. Spectroscopy on two
coupled flux qubits. Preprint at khttp://arxiv.org/abs/cond-mat/0308192l (2003).
10. Izmalkov, A. et al. Experimental evidence for entangled states formation in a system of two coupled
flux qubits. Preprint at khttp://arxiv.org/abs/cond-mat/0312332l (2003).
11. Yamamoto, T., Pashkin, Yu. A., Astafiev, O., Nakamura, Y. & Tsai, J. S. Demonstration of conditional
gate operation using superconducting charge qubits. Nature 425, 941–944 (2003).
12. Leibfried, D., Blatt, R., Monroe, C. & Wineland, D. Quantum dynamics of single trapped ions. Rev.
Mod. Phys. 75, 281–324 (2003).
13. Mandel, O. et al. Controlled collisions for multi-particle entanglement of optically trapped atoms.
Nature 425, 937–940 (2003).
14. Raimond, J. M., Brune, M. & Haroche, S. Manipulating quantum entanglement with atoms and
photons in a cavity. Rev. Mod. Phys. 73, 565–582 (2001).
15. Mooij, J. E. et al. Josephson persistent-current qubit. Science 285, 1036–1039 (1999).
16. vanderWal,C.H.etal.Quantumsuperpositionofmacroscopicpersistent-currentstates.Science290,
773–777 (2000).
17. Burkard, G. et al. Asymmetry and decoherence in double-layer persistent-current qubit. Preprint at
khttp://arxiv.org/abs/cond-mat/0405273l (2004).
18. Goorden,M. C., Thorwart, M. & Grifoni, M. Entanglement spectroscopyof adrivensolid-statequbit
and its detector. Preprint at khttp://arxiv.org/abs/cond-mat/0405220l (2004).
19. Tinkham, M. Introduction to Superconductivity 2nd edn (McGraw-Hill, New York, 1996).
20. Cohen-Tannoudji, C., Dupont-Roc, J. & Grynberg, G. Atom-photon Interactions: Basic Processes and
Applications Ch. II E (Wiley & Sons, New York, 1992).
Acknowledgements WethankA.Blais,G.Burkard,D.DiVincenzo,G.Falci,M.Grifoni,S.Lloyd,
S.Miyashita,T.Orlando,R.N.Schouten,L.VandersyepenandF.K.Wilhelmfordiscussions.This
work was supported by the Dutch Foundation for Fundamental Research on Matter (FOM), the
EU Marie Curie and SQUBIT grants, and the US Army Research Office.
Competing interests statement The authors declare that they have no competing financial
interests.
Correspondence and requests for materials should be addressed to I.C. (chiorescu@pa.msu.edu)
and J.E.M. (mooij@qt.tn.tudelft.nl).
..............................................................
Strong coupling of a single photon
to a superconducting qubit using
circuit quantum electrodynamics
A. Wallraff1, D. I. Schuster1, A. Blais1, L. Frunzio1, R.- S. Huang1,2,
J. Majer1, S. Kumar1, S. M. Girvin1& R. J. Schoelkopf1
1Departments of Applied Physics and Physics, Yale University, New Haven,
Connecticut 06520, USA
2Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
.............................................................................................................................................................................
The interaction of matter and light is one of the fundamental
processes occurring in nature, and its most elementary form is
realized when a single atom interacts with a single photon.
Reaching this regime has been a major focus of research in
atomic physics and quantum optics1for several decades and
has generated the field of cavity quantum electrodynamics2,3.
Hereweperformanexperimentinwhichasuperconductingtwo-
levelsystem,playingtheroleofanartificialatom,iscoupledtoan
on-chip cavity consisting of a superconducting transmission line
resonator. We show that the strong coupling regime can be
attained in a solid-state system, and we experimentally observe
the coherent interaction of a superconducting two-level system
with a single microwave photon. The concept of circuit quantum
electrodynamics opens many new possibilities for studying the
strong interaction of light and matter. This system can also be
exploited for quantum information processing and quantum
communication and may lead to new approaches for single
photon generation and detection.
In atomic cavity quantum electrodynamics (QED), an isolated
atomwith electricdipole moment d interacts with the vacuum state
electricfieldE0ofacavity.Thequantumnatureofthefieldgivesrise
to coherent oscillations of a single excitation between the atom and
the cavity at the vacuum Rabi frequency nRabi¼ 2dE0/h, which can
be observed when nRabiexceeds the rates of relaxation and deco-
herenceofboththeatomandthefield.Thiseffecthasbeenobserved
in the time domain using Rydberg atoms in three-dimensional
microwavecavities3and spectroscopically using alkali atoms invery
small optical cavities with large vacuum fields4.
Coherent quantum effects have been recently observed in several
superconducting circuits5–10, making these systems well suited for
use as quantum bits (qubits) for quantum information processing.
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Of the various superconducting qubits, the Cooper pair box11is
especially well suited for cavity QED because of its large effective
electric dipole moment d, which can be 104times larger than in an
alkali atom and ten times larger than a typical Rydberg atom12. As
suggested in our earlier theoretical study12, the simultaneous com-
bination of this large dipole moment and the large vacuum field
strength—due to the small size of the quasi one-dimensional
transmission line cavity—in our implementation is ideal for reach-
ingthestrongcoupling limitofcavityQEDinacircuit.Othersolid-
state analogues of strong coupling cavity QED have been envisaged
in superconducting13–20, semiconducting21,22, and even micro-
mechanical systems23. First steps towards realizing such a regime
have been made for semiconductors21,24,25. To our knowledge, our
experiments constitute the first experimental observation of strong
coupling cavity QED with a single artificial atom and a single
photon in a solid-state system.
The on-chip cavity is made by patterning a thin superconducting
film deposited on a silicon chip. The quasi-one-dimensional co-
planar waveguide resonator26consists of a narrowcentre conductor
oflengthlandtwonearbylateralgroundplanes,seeFig.1a.Closeto
its full-wave (l ¼ l) resonance frequency, qr¼ 2pnr¼ 1=
2p6:044GHz; where nris the bare resonance frequency, the reso-
natorcanbemodelledasaparallelcombinationofacapacitorCand
an inductor L (the internal losses are negligible). This simple
resonant circuit behaves as a harmonic oscillator described by the
hamiltonian Hr¼ "qr(a†a þ 1/2), where ka†al ¼ k ^ nl ¼ n is the
average photon number. At our operating temperature of
T , 100mK, much less than "qr/kB< 300mK, the resonator is
nearly in its ground state, with a thermal occupancy n , 0.06. The
vacuumfluctuationsoftheresonatorgiverisetoarootmeansquare
(r.m.s.) voltage Vrms¼
ffiffiffiffiffiffi
LC
p
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
"qr=2C
p
< 1mV on its centre conductor,
and an electric field between the centre conductor and the ground
plane that is a remarkable Erms< 0.2Vm21, some hundred times
larger than in the three-dimensional cavities used in atomic micro-
wave cavity QED3. The large vacuum field strength results from the
extremely small effective mode volume (,1026cubic wavelengths)
of the resonator12.
The resonator is coupled via two coupling capacitors Cin/out, one
at each end (see Fig. 1b), to the input and output transmission lines
that allow its microwave transmission to be probed (see Fig. 2a–c).
The predominant source of dissipation is the loss of photons from
the resonator through these ports at a rate k ¼ qr/Q, whereQ is the
(loaded) quality factor of the resonator. The internal (uncoupled)
loss of the resonator is negligible (Qint< 106). Thus, the average
photon lifetime in the resonator Tr¼ 1/k exceeds 100ns, even for
our initial choice of a moderate quality factor Q < 104.
The Cooper pair box (CPB) consists of a several micrometre long
and submicrometre wide superconducting island which is coupled
via two submicrometre size Josephson tunnel junctions to a much
larger superconducting reservoir, and is fabricated in the gap
betweenthecentreconductorandthegroundplaneoftheresonator,
at an antinode of the field (see Fig. 1c). The CPB is a two-state
system described by the hamiltonian13Ha¼ 2ðEeljxþEJjzÞ=2,
where Eel¼ 4ECð12ngÞ is the electrostatic energy and EJ¼
EJ;maxcosðpFbÞ is the Josephson energy. The overall energy scales
of these terms, the charging energy ECand the Josephson energy
EJ,max, can be readily engineered during the fabrication by the
choice of the total box capacitance and resistance respectively, and
then further tuned in situ by electrical means. A gate voltage Vg
applied to the input port (see Fig. 2a), induces a gate charge ng¼
VgCg*=e that controls Eel, where Cg* is the effective capacitance
betweentheinputportoftheresonatorandtheislandoftheCPB.A
flux bias Fb¼ F/F0, applied with an external coil to the loop of the
box,controlsEJ.Denotingthegroundstateoftheboxasj # landthe
first excited state as j " l (see Fig. 2d), we have a two-level system
whoseenergyseparationEa¼ "qacanbewidelyvariedasshownin
Fig. 3c. Coherence of the CPB is limited by relaxation from the
excited state at a rate g1, and by fluctuations of the level separation
giving rise to dephasing at a rate gJ, for a total decoherence rate
g ¼ g1/2 þ gJ(ref. 13).
The Cooper pair box couples to photons stored in the resonator
by an electric dipole interaction, via the coupling capacitance Cg.
The vacuum voltage fluctuations Vrmson the centre conductor of
the resonator change the energy of a Cooper pair on the box island
by an amount "g ¼ dE0¼ eVrmsCg/CS. We have shown12that this
coupled system is described by the Jaynes–Cummings hamiltonian
HJC¼ Hrþ Haþ "g(a†j2þ ajþ), where jþ(j2) creates
(annihilates) an excitation in the CPB. It describes the coherent
exchange of energy between a quantized electromagnetic field and a
quantum two-level system at a rate g/2p, which is observable if g is
much larger than the decoherence rates g and k. This strong
coupling limit3g . [g, k] is achieved in our experiments. When
the detuning D ¼ qa2 qris equal to zero, the eigenstates of the
coupled system are symmetric and antisymmetric superpositions
of a single photon and an excitation in the CPB j^l ¼ ðj0;" l^
j1;# lÞ=
and the CPB are entangled in the eigenstates j ^ l, their
entangled character is not addressed in our current cavity QED
experiment which spectroscopically probes the energies E^of the
coherently coupled system.
The strong coupling between the field in the resonator and the
CPB can be used to perform a quantum nondemolition (QND)
measurement of the state of the CPB in the non-resonant (dis-
persive) limit jDj . . g: Diagonalization of the coupled quantum
system leads to the effective hamiltonian12:
?
ffiffiffi
2
p
with energies E^¼ "(qr^ g). Although the cavity
H < " qrþg2
Djz
?
a†aþ1
2" qaþg2
D
??
jz
Figure 1 Integrated circuit for cavity QED. a, The superconducting niobium coplanar
waveguide resonator is fabricated on an oxidized 10 £ 3mm2silicon chip using optical
lithography.Thewidthofthecentreconductoris10mmseparatedfromthelateralground
planes extending to the edges of the chip by a gap of width 5mm resulting in a wave
impedance of the structure of Z ¼ 50Q being optimally matched to conventional
microwave components. The length of the meandering resonator is l ¼ 24mm. It is
coupled by a capacitor at each end of the resonator (see b) to an input and output feed
line, fanning out to the edge of the chip and keeping the impedance constant. b, The
capacitivecouplingto the input and output lines and hence the coupled quality factor Q is
controlled by adjusting the length and separation of the finger capacitors formed in the
centre conductor. c, False colour electron micrograph of a Cooper pair box (blue)
fabricated onto the silicon substrate (green) into the gap between the centre conductor
(top)andthegroundplane(bottom)ofaresonator(beige)usingelectronbeamlithography
and double angle evaporation of aluminium. The Josephson tunnel junctions are formed
attheoverlapbetweenthelongthinislandparalleltothecentreconductorandthefingers
extending from the much larger reservoir coupled to the ground plane.
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The transition frequency qr^ g2/D is now conditioned by the
qubitstatejz¼ ^1.Thus,bymeasuringthetransitionfrequencyof
the resonator, the qubit state can be determined. Similarly, the level
separation in the qubit "ðqaþ2a†a g2=Dþg2=DÞ depends on the
number of photons in the resonator. The term 2a†a g2/D, linear in
n ˆ, is the alternating current (a.c.) Stark shift and g2/D is the Lamb
shift. All terms in this hamiltonian, with the exception of the Lamb
shift, are clearly identified in the results of our circuit QED
experiments.
The properties of this coupled system are determined by probing
the resonator spectroscopically12. The amplitude Tand phase f of a
microwave probe beam of power PRFtransmitted through the
resonator are measured versus probe frequency qRF. A simplified
schematic of the microwave circuit is shown in Fig. 2a. In this set-
up, the CPB acts as an effective capacitance that is dependent on its
jzeigenstate, the couplingstrength g, and detuning D. Thisvariable
capacitance changes the resonator frequency and its transmission
spectrum. The transmission T2and phase f of the resonator for a
far-detuned qubitðg2=kD , , 1Þ;that is,whenthequbitiseffectively
decoupled from the resonator, are shown in Fig. 2b and c. In this
case, thetransmission isalorentzianofwidthdnr¼ nr/Q ¼ k/2pat
nr,andthephasefdisplaysacorrespondingstepofp.Theexpected
transmission atsmallerdetuning corresponding to afrequency shift
^g2/D ¼ k are shown by dashed lines in Fig. 2b and c. Such small
shifts in the resonator frequency are sensitively measured as a phase
shift f ¼ ^tan21(2g2/kD) of the transmitted microwave at a fixed
probe frequency qRFusing beam powers PRFwhich controllably
populate the resonator with average photon numbers fromn < 103
down to the sub-photon level n , , 1: We note that both the
resonator and qubit can be controlled and measured using capaci-
tive and inductive coupling only, that is, without attaching any d.c.
connections to either system.
Measurements of the phase f versus ngare shown in Fig. 3b, and
two different cases can be identified for a Cooper pair box with
JosephsonenergyEJ,max/h . nr.Inthefirstcase,forbiasfluxessuch
that EJ(Fb)/h . nr, the qubit does not come into resonance with
theresonatorforanyvalueofgatechargeng(seeFig.3a).Asaresult,
the measured phase shift fis maximum for the smallest detuning D
atng¼ 1andgetssmallerasDincreases(seeFig.3b).Moreover,fis
periodicinngwithaperiodof2e,asexpected.Inthesecondcase,for
values of Fbresulting in EJ(Fb)/h , nr, the qubit goes through
resonance with the resonator at two values of ng. Thus, the phase
shift f is largest as the qubit approaches resonance (D ! 0) at the
points indicated by red arrows (see Fig. 3a, b). As the qubit goes
through resonance, the phase shift f changes sign when D changes
sign. This behaviour is in perfect agreement with predictions based
on the analysis of the circuit QED hamiltonian in the dispersive
regime.
InFig.3cthequbitlevelseparationna¼ Ea/hisplottedversusthe
bias parameters ngand Fb. The qubit is in resonance with the
resonator at the points [ng, Fb], indicated by the red curve in one
quadrant of the plot. The measured phase shift f is plotted versus
Figure 2 Measurement scheme, resonator and Cooper pair box. a, The resonator with
effective inductance L and capacitanceC coupled through the capacitor Cgto the Cooper
pair box with junction capacitance CJand Josephson energy EJforms the circuit QED
system which is coupled through Cin/outto the input/output ports. The value of EJis
controllablebythemagneticfluxF.TheinputmicrowaveatfrequencyqRFisaddedtothe
gate voltage Vgusing a bias-tee. After the transmitted signal at qRFis amplified using a
cryogenic high electron mobility (HEMT) amplifier and mixed with the local oscillator at
qLO, its amplitude and phase are determined. The circulator and the attenuator prevent
leakageofthermalradiationintotheresonator.Thetemperatureofindividualcomponents
is indicated. b, Measured transmission power spectrum of the resonator (blue dots), the
full linewidth dnrat half-maximum and the centre frequency nrare indicated. The solid
redlineisafittoalorentzianwithQ ¼ nr/dnr< 104.c,Measuredtransmissionphasef
(bluedots)withfit(redline).Inpanelsbandcthedashedlinesaretheorycurvesshiftedby
^dnrwith respect to the data. d, Energy level diagram of a Cooper pair box. The
electrostatic energy EC(ni2 ng)2, with charging energy EC¼ e2/2CS, is indicated for
ni¼ 0 (solid black line), 22 (dotted line) and þ2 (dashed line) excess electrons forming
Cooper pairs on the island. CSis the total capacitance of the island given by the sum of
thecapacitancesCJofthetwotunneljunctions,thecouplingcapacitanceCgtothecentre
conductor of the resonator and any stray capacitances. In the absence of Josephson
tunnelling the states with niand niþ 2 electrons on the island are degenerate at
ng¼ 1. The Josephson coupling mediated by the weak link formed by the tunnel
junctions between the superconducting island and the reservoir lifts this degeneracy and
opens up a gap proportional to the Josephson energy EJ¼ EJ,maxcos(pFb), where
EJ,max¼ hDAl/8e2RJ, with the superconducting gap of aluminium DAland the tunnel
junction resistance RJ. A ground-state band j # l and an excited-state band j " l are
formed with a gate charge and flux-bias-dependent energy level separation of Ea.
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both ngand Fbin Fig. 3d. We observe the expected periodicity
in flux bias Fbwith one flux quantum F0. The set of parameters
[ng, Fb] for which the resonance condition is met is marked by a
sudden sign change in f, which allows a determination of the
Josephsonenergy EJ,max¼ 8.0 (^0.1) GHz andthe charging energy
EC¼ 5.2 (^0.1) GHz.
These data clearly demonstrate that the properties of the qubit
can be determined in a transmission measurement of the resonator
and that full in situ control over the qubit parameters is achieved.
We note that in the dispersive regime this new read-out scheme for
theCooperpairboxismostsensitiveatchargedegeneracy(ng¼ 1),
where the qubit is to first order decoupled from 1/f fluctuations in
itschargeenvironment,whichminimizesdephasing6.Thisproperty
is advantageous for quantum control of the qubit at ng¼ 1, a point
where traditional electrometry, using a single electron transistor
(SET) for example27, is unable to distinguish the qubit states. We
note that this dispersive QND measurement of the qubit state12is
thecomplementoftheatomicmicrowavecavityQEDmeasurement
inwhichthestateofthecavityisinferrednon-destructivelyfromthe
phaseshiftinthestateofabeamofatomssentthroughthecavity3,28.
Making use of the full control over the qubit hamiltonian, we
then tune the flux bias Fbso that the qubit is at ng¼ 1 and in
resonance with the resonator. Initially, the resonator and the qubit
are cooled into their combined ground state j0, # l; see inset in
NATURE|VOL 431|9 SEPTEMBER 2004|www.nature.com/nature
Fig. 4b. Owing to the coupling, the first excited states become a
doubletj ^ l.Similarlytoref.4,weprobetheenergysplittingofthis
doublet spectroscopically using a weak probe beam so that n , , 1:
The intra-resonator photon number, n, is calibrated by measuring
the a.c.-Stark shift of the qubit in the dispersive case. The resonator
transmission T2is first measured for large detuning D with a probe
beam populating the resonator with a maximum of n < 1 at
resonance; see Fig. 4a. From the lorentzian line the photon decay
rate of the resonator is determined as k/2p ¼ 0.8MHz. The probe
beam power is subsequently reduced by 5dB and the transmission
spectrum T2is measured in resonance (D ¼ 0); see Fig. 4b. We
clearly observe two well-resolved spectral lines separated by the
vacuum Rabi frequency nRabi< 11.6MHz. The individual lines
have a width determined by the average of the photon decay rate k
and the qubit decoherence rate g. The data are in excellent agree-
ment with the transmission spectrum numerically calculated using
thegivenvaluek/2p ¼ 0.8MHzandthesingleadjustableparameter
g/2p ¼ 0.7MHz.
The transmission spectrum shown in Fig. 4b is highly sensitiveto
the photon number in the cavity. The measured transmission
spectrum is consistent with the expected thermal photon number
of n & 0:06 (T , 100mK); see red curve in Fig. 4b. Owing to the
anharmonicity of the coupled atom-cavity system in the resonant
case, an increased thermal photon number would reduce trans-
Figure 3 Strong coupling circuit QED in the dispersive regime. a, Calculated level
separation na¼ qa/2p ¼ Ea/h between ground j # l and excited state j " l of qubit for
twovaluesoffluxbiasFb¼ 0.8(orangeline)andFb¼ 0.35(greenline).Theresonator
frequency nr¼ qr/2p is shown by a blue line. Resonance occurs at na¼ nr
symmetrically around degeneracy ng¼ ^1; also see red arrows. The detuning
D/2p ¼ d ¼ na2 nris indicated. b, Measured phase shift f of the transmitted
microwave for values of Fbin a. Green curve is offset by 225 deg for visibility.
c, Calculated qubit level separation naversus bias parameters ngand Fb. The resonator
frequency nris indicated by the blue plane. At the intersection, also indicated by the red
curve in the lower right-hand quadrant, resonance between the qubit and the resonator
occurs (d ¼ 0). For qubit states below the resonator plane the detuning is d , 0, above
d . 0. d, Density plot of measured phase shift f versus ngand Fb. Light colours
indicate positive f (d . 0), dark colours negative f (d , 0). The red line is a fit of the
datatotheresonanceconditionna¼ nr.Incandd,thelinecutspresentedinaandbare
indicated by the orange and the green line, respectively. The microwaveprobe power PRF
used to acquire the data is adjusted such that the maximum intra-resonator photon
number n at nris about ten for g2=kD , , 1: The calibration of the photon number has
been performed in situ by measuring the a.c.-Stark shift of the qubit levels.
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mission and give rise to additional peaks in the spectrum owing to
transitions between higher excited doublets30. The transmission
spectrum calculated for a thermal photon number of n ¼ 0.5 (see
greencurveinFig.4b)isclearlyincompatiblewithourexperimental
data,indicatingthatthecoupledsystemhasinfactcooledtonearits
groundstate,andthatwemeasurethecouplingofasinglequbittoa
single photon. The nonlinearity of the cavity QED system is also
observed at higher probe beam powers, as transitions are driven
between states higher up the dressed state ladders (not shown).
We also observe the anti-crossing between the single photon
resonator state and the first excited qubit state by tuning the qubit
into and out of resonance with a gate charge near ng¼ 1 and
measuring the transmission spectrum (see Fig. 4c). The vacuum
Rabi peaks evolve from a state with equal weight in the photon and
qubit at ng¼ 1 (as shown in Fig. 4b) to predominantly photon
statesforng. . 1orng, , 1:Theobservedpeakpositions agreewell
with calculations considering the qubit with level separation na, a
single photon in the resonator with frequency nrand a coupling
strengthofg/2p;seesolidlinesinFig.4c.Foradifferentvalueofflux
bias Fbsuch that Ea/h , nrat ng¼ 1, two anti-crossings are
observed (see Fig. 4d) again in agreement with theory.
The observation of the vacuum Rabi mode splitting and the
corresponding avoided crossings demonstrates that the strong
coupling limit of cavity QED has been achieved, and that coherent
superpositions of a single qubit and a single photon can be
generated on a superconducting chip. This opens up many new
possibilities forquantumoptical experiments withcircuits.Possible
applications include using the cavity as a quantum bus to couple
widely separated qubits in a quantum computer, or as a quantum
memory to store quantum information, or even as a generator
and detector of single microwave photons for quantum
communication.
A
Received 11 June; accepted 12 July 2004; doi:10.1038/nature02851.
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2. Mabuchi, H. & Doherty, A. Cavity quantum electrodynamics: Coherence in context. Science 298,
1372–1377 (2002).
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Figure 4 Vacuum Rabi mode splitting. a, Measured transmission T2(blue line) versus
microwave probe frequency nRFfor large detuning ðg2=Dk , , 1Þ and fit to lorentzian
(dashedredline).Thepeaktransmissionamplitudeisnormalizedtounity.Theinsetshows
the dispersive dressed states level diagram. b, Measured transmission spectrum for the
resonant case D ¼ 0 at ng¼ 1 (blue line) showing the vacuum Rabi mode splitting
comparedtonumericallycalculatedtransmissionspectra(redandgreenlines)forthermal
photon numbers of n ¼ 0.06 and 0.5, respectively. The dashed red line is the calculated
transmission for g ¼ 0 and k/2p ¼ 0.8MHz. The inset shows the resonant dressed
states level diagram. c, Resonator transmission amplitude T plotted versus probe
frequencynRFandgatechargengforD ¼ 0atng¼ 1.Bluecolourcorrespondstosmall
T, red colour to large T. Dashed lines are uncoupled qubit level separation naand
resonator resonance frequency nr. Solid lines are level separations found from exact
diagonalization of HJC. Spectrum shown in b corresponds to line cut along red arrows.
d,Asinc,butforEJ/h , nr.Thedominantcharacterofthecorrespondingeigenstatesis
indicated.
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28. Nogues, G. et al. Seeing a single photon without destroying it. Nature 400, 239–242 (1999).
29. Schuster, D. I. et al. AC-Stark shift and dephasing of a superconducting quibit strongly coupled to a
cavity field. Preprint at http://www.arXiv.org/cond-mat/0408367 (2004).
30. Rau, I., Johansson, G. & Shnirman, A. Cavity QED in superconducting circuits: susceptibility at
elevated temperatures. Preprint at http://www.arXiv.org/cond-mat/0403257 (2004).
Acknowledgements We thank J. Teufel, B. Turek and J. Wyatt for their contributions to the
project and are grateful to P. Day, D. DeMille, M. Devoret, S. Weinreb and J. Zmuidzinas for
numerous conversations. This work was supported in part by the National Security Agency and
Advanced Research and Development Activity under the Army Research Office, the NSF, the
David and Lucile Packard Foundation, the W. M. Keck Foundation, and the Natural Science and
Engineering Research Council of Canada.
Competing interests statement The authors declare that they have no competing financial
interests.
Correspondence and requests for materials should be addressed to A. W.
(andreas.wallraff@yale.edu).
..............................................................
Generation of ultraviolet entangled
photons in a semiconductor
Keiichi Edamatsu1,2, Goro Oohata1,3, Ryosuke Shimizu2& Tadashi Itoh4,2
1Research Institute of Electrical Communication, Tohoku University, Sendai
980-8577, Japan
2CREST, Japan Science and Technology Agency (JST), Japan
3ERATO Semiconductor Spintronics Project, JST, Japan
4Graduate School of Engineering Science, Osaka University, Toyonaka 560-8531,
Japan
.............................................................................................................................................................................
Entanglement is one of the key features of quantum information
and communications technology. The method that has been used
most frequently to generate highly entangled pairs of photons1,2
is parametric down-conversion. Short-wavelength entangled
photons are desirable for generating further entanglement
betweenthreeorfourphotons,butitisdifficulttouseparametric
down-conversion to generate suitably energetic entangled pho-
ton pairs. One method that is expected to be applicable for
the generation of such photons3is resonant hyper-parametric
scattering (RHPS): a pair of entangled photons is generated in a
semiconductor via an electronically resonant third-order non-
linearopticalprocess.Semiconductor-basedsourcesofentangled
photons would also be advantageous for practical quantum
technologies, but attempts to generate entangled photons in
semiconductors have not yet been successful4,5. Here we report
experimentalevidencefor thegenerationofultravioletentangled
photon pairs by means of biexciton resonant RHPS in a single
crystal of the semiconductor CuCl. We anticipate that our results
will open the way to the generation of entangled photons by
current injection, analogous to current-driven single photon
sources6,7.
The material we used in this study was copper chloride (CuCl)
single crystal. Because CuCl has a large bandgap (,3.4eV), it is
suitable for generating photon pairs in the short wavelength region
near ultraviolet. Furthermore, the material has large binding ener-
gies for the exciton (,200meV) and biexciton (,30meV). These
characteristics have made CuCl one of the most thoroughly inves-
tigated materials on the physics of excitons and biexcitons (ref. 8
and references therein). In particular, the ‘giant oscillator strength’
in the two-photon excitation of the biexciton results in a large
increase in RHPS efficiency, which is advantageous for our experi-
ment. In fact the RHPS in CuCl has been observed since the 1970s
(refs8, 9 and ref.10 and referencestherein). Figure1a schematically
shows the RHPS process in resonance to the biexciton state. The
two pump (parent) photons (frequency qi) resonantly create the
biexciton, and are converted into the two scattered (daughter)
photons (qs, qs
has zero angular momentum (J ¼ 0), so we expected the polariza-
tions of the daughter photons to be entangled so that their total
angular momentum is also zero. With this expectation in mind, we
note that polarization correlation between two classical pump
beams has been known since the early 1980s (ref. 11). In practice,
insteadoftheoversimplifiedpictureinFig.1a,wemustconsiderthe
exciton-polariton picture; the RHPS obeys the phase-matching
condition that takesinto account the polariton dispersion relation8.
The RHPS in this case is also called two-photon resonant polariton
scattering or spontaneous hyper-Raman scattering. In this process,
shown in Fig. 1b, the biexciton is created from a pair of parent
photons (polaritons, more accurately). The sum of the parent
photons’ energies matches the biexciton energy. The biexciton
progressively coherently decays into two polaritons, the sum of
whosephotonenergies,aswellasthesumofmomenta,isconserved
as that of the biexciton. Although the RHPS in CuCl has been
known for decades, the possibility of generating entangled photons
by this process was theoretically pointed out only lately12. In
addition, a large parametric gain via the biexcitonic resonance in
CuClwasreportedrecently13.Similarstimulatedparametricscatter-
ing of polaritons has also been observed in semiconductor micro-
cavities, even at high temperatures14.
In the present experiment, we used a vapour-phase-grown thin
single crystal of CuCl. Figure 2 presents the schematic drawing of
our experimental set-up and Fig. 3 shows the spectrum of light
emitted from the sample. The large peak at the downward arrow in
Fig. 3 is the Rayleigh scattered light of the pump beam that was
tuned to the two-photon excitation resonance of the biexciton. The
two peaks indicated by LEP and HEP (lower and higher energy
polaritons) on either side of the pump beam originate from the
RHPS.TheRHPSisveryefficient(afewordersofmagnitudehigher
than that of typical parametric down-conversion): We got of the
order of 1010photons s21sr21by using pump light of ,2mW. A
pair of photons, one from LEP and the other from HEP, is emitted
into different directions according to the phase-matching con-
dition, so we placed two optical fibres at appropriate positions
and led each photon within the pair into two independent mono-
chromators followed by two photomultipliers (PMTs). A time-
intervalanalyserrecordedthetimeinterval(t)betweenthedetected
0). The biexciton state (G1) created in this process
Figure 1 Schematic diagram of the resonant hyper-parametric scattering (RHPS) via
biexciton. a, Two pump (parent) photons of frequency qiare converted to the two
scattered (daughter) photons (qs, qs
dimensionsof momentumspace. The biexciton decays into twopolaritons that satisfythe
phase-matching condition so that both energy and momentum are conserved. The red
curve on the polariton-dispersion surface indicates the states on which the phase-
matching condition can be satisfied.
0). b, The polariton dispersion drawn in two
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