Content uploaded by David Schuster
Author content
All content in this area was uploaded by David Schuster on Jun 03, 2014
Content may be subject to copyright.
coupled system by applying pulses of varying length. In Fig. 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 2
p
pulse they are revived (first curve in Fig. 3b).
In the case of Fig. 3c, the qubit is first excited onto the j10. state by
a
p
pulse, and a second pulse in resonance with the red sideband
transition drives the system between the j10. and j01. 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 (,3 ns), which we 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 1 ns). The decay of the oscillation amplitude as a function of
Dt corresponds to an oscillator relaxation time of ,6 ns (Fig. 4b),
consistent with a quality factor of 100–150 estimated from the width
of the u
p
resonance. 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 ,60 mK).
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.
Nature 398, 786–788 (1999).
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. et al. Quantum oscillations in two coupled charge qubits. Nature 421, 823–826 (2003).
8. Berkley, A. J. et al. Entangled macroscopic quantum states in two superconducting qubits. Science 300,
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. van der Wal, C. H. et al. Quantum superposition of macroscopic persistent-current states. Science 290,
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 spectroscopy of a driven solid-state qubit
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 We thank A. Blais, G. Burkard, D. DiVincenzo, G. Falci, M. Grifoni, S. Lloyd,
S. Miyashita, T. Orlando, R. N. Schouten, L. Vandersyepen and F. K. Wilhelm for discussions. 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. Wallraff
1
, D. I. Schuster
1
, A. Blais
1
, L. Frunzio
1
, R.- S. Huang
1,2
,
J. Majer
1
, S. Kumar
1
, S. M. Girvin
1
& R. J. Schoelkopf
1
1
Departments of Applied Physics and Physics, Yale University, New Haven,
Connecticut 06520, USA
2
Department 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 optics
1
for several decades and
has generated the field of cavity quantum electrodynamics
2,3
.
Here we perform an experiment in which a superconducting two-
level system, playing the role of an artificial atom, is coupled to an
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 n ew approaches for single
photon generation and detection.
In atomic cavity quantum electrodynamics (QED), an isolated
atom with electric dipole moment d interacts with the vacuum state
electric field E
0
of a cavity. The quantum nature of the field gives rise
to coherent oscillations of a single excitation between the atom and
the cavity at the vacuum Rabi frequency n
Rabi
¼ 2dE
0
/h, which can
be observed when n
Rabi
exceeds the rates of relaxation and deco-
herence of both the atom and the field. This effect has been observed
in the time domain using Rydberg atoms in three-dimensional
microwave cavities
3
and spectroscopically using alkali atoms in very
small optical cavities with large vacuum fields
4
.
Coherent quantum effects have been recently observed in several
superconducting circuits
5–10
, making these systems well suited for
use as quantum bits (qubits) for quantum information processing.
letters to nature
NATURE | VOL 431 | 9 SEPTEMBER 2004 | www.nature.com/nature162
© 2004
Nature
Publishing
Group
Of the various superconducting qubits, the Cooper pair box
11
is
especially well suited for cavity QED because of its large effective
electric dipole moment d, which can be 10
4
times larger than in an
alkali atom and ten times larger than a typical Rydberg atom
12
.As
suggested in our earlier theoretical study
12
, 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-
ing the strong coupling limit of cavity QED in a circuit. Other solid-
state analogues of strong coupling cavity QED have been envisaged
in superconducting
13–20
, semiconducting
21,22
, and even micro-
mechanical systems
23
. First steps towards realizing such a regime
have been made for semiconductors
21,24,25
. 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 resonator
26
consists of a narrow centre conductor
of length l and two nearby lateral ground planes, see Fig. 1a. Close to
its full-wave (l ¼
l
) resonance frequency, q
r
¼2pn
r
¼1=
ffiffiffiffiffiffi
LC
p
¼
2p 6:044 GHz; where n
r
is the bare resonance frequency, the reso-
nator can be modelled as a parallel combination of a capacitor C and
an inductor L (the internal losses are negligible). This simple
resonant circuit behaves as a harmonic oscillator described by the
hamiltonian H
r
¼ " q
r
(a
†
a þ 1/2), where ka
†
al ¼k
^
nl ¼ n is the
average photon number. At our operating temperature of
T , 100 mK, much less than "q
r
/k
B
< 300 mK, the resonator is
nearly in its ground state, with a thermal occupancy n , 0.06. The
vacuum fluctuations of the resonator give rise to a root mean square
(r.m.s.) voltage V
rms
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
"q
r
=2C
p
< 1
m
V on its centre conductor,
and an electric field between the centre conductor and the ground
plane that is a remarkable E
rms
< 0.2 V m
21
, some hundred times
larger than in the three-dimensional cavities used in atomic micro-
wave cavity QED
3
. The large vacuum field strength results from the
extremely small effective mode volume (,10
26
cubic wavelengths)
of the resonator
12
.
The resonator is coupled via two coupling capacitors C
in/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 q
r
/Q, where Q is the
(loaded) quality factor of the resonator. The internal (uncoupled)
loss of the resonator is negligible (Q
int
< 10
6
). Thus, the average
photon lifetime in the resonator T
r
¼ 1/k exceeds 100 ns, even for
our initial choice of a moderate quality factor Q < 10
4
.
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
between the centre conductor and the ground plane of the resonator,
at an antinode of the field (see Fig. 1c). The CPB is a two-state
system described by the hamiltonian
13
H
a
¼2ðE
el
j
x
þE
J
j
z
Þ=2,
where E
el
¼4E
C
ð1 2 n
g
Þ is the electrostatic energy and E
J
¼
E
J;max
cosðp
F
b
Þ is the Josephson energy. The overall energy scales
of these terms, the charging energy E
C
and the Josephson energy
E
J,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 V
g
applied to the input port (see Fig. 2a), induces a gate charge n
g
¼
V
g
C
g
*=e that controls E
el
, where C
g
* is the effective capacitance
between the input port of the resonator and the island of the CPB. A
flux bias
F
b
¼
F
/
F
0
, applied with an external coil to the loop of the
box, controls E
J
. Denoting the ground state of the box as j#l and the
first excited state as j"l (see Fig. 2d), we have a two-level system
whose energy separation E
a
¼ "q
a
can be widely varied as shown in
Fig. 3c. Coherence of the CPB is limited by relaxation from the
excited state at a rate g
1
, and by fluctuations of the level separation
giving rise to dephasing at a rate g
J
, for a total decoherence rate
g ¼ g
1
/2 þ g
J
(ref. 13).
The Cooper pair box couples to photons stored in the resonator
by an electric dipole interaction, via the coupling capacitance C
g
.
The vacuum voltage fluctuations V
rms
on the centre conductor of
the resonator change the energy of a Cooper pair on the box island
by an amount "g ¼ dE
0
¼ eV
rms
C
g
/C
S
. We have shown
12
that this
coupled system is described by the Jaynes–Cummings hamiltonian
H
JC
¼ H
r
þ H
a
þ "g(a
†
j
2
þ aj
þ
), where j
þ
(j
2
) 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 limit
3
g . [g, k] is achieved in our experiments. When
the detuning
D
¼ q
a
2 q
r
is 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 ¼ðj0;" l^
j1; # lÞ=
ffiffiffi
2
p
with energies E
^
¼ " (q
r
^ g). Although the cavity
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 j
D
j.. g: Diagonalization of the coupled quantum
system leads to the effective hamiltonian
12
:
H < " q
r
þ
g
2
D
j
z
a
†
a þ
1
2
" q
a
þ
g
2
D
j
z
Figure 1 Integrated circuit for cavity QED. a, The superconducting niobium coplanar
waveguide resonator is fabricated on an oxidized 10 £ 3mm
2
silicon chip using optical
lithography. The width of the centre conductor is 10
m
m separated from the lateral ground
planes extending to the edges of the chip by a gap of width 5
m
m resulting in a wave
impedance of the structure of Z ¼ 50 Q being optimally matched to conventional
microwave components. The length of the meandering resonator is l ¼ 24 mm. 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
capacitive coupling to 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) and the ground plane (bottom) of a resonator (beige) using electron beam lithography
and double angle evaporation of aluminium. The Josephson tunnel junctions are formed
at the overlap between the long thin island parallel to the centre conductor and the fingers
extending from the much larger reservoir coupled to the ground plane.
letters to nature
NATURE | VOL 431 | 9 SEPTEMBER 2004 | www.nature.com/nature 163
© 2004
Nature
Publishing
Group
The transition frequency q
r
^ g
2
/
D
is now conditioned by the
qubit state j
z
¼ ^1. Thus, by measuring the transition frequency of
the resonator, the qubit state can be determined. Similarly, the level
separation in the qubit "ðq
a
þ2a
†
ag
2
=
D
þg
2
=
D
Þ depends on the
number of photons in the resonator. The term 2a
†
ag
2
/
D
, linear in
n
ˆ
, is the alternating current (a.c.) Stark shift and g
2
/
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 spectroscopically
12
. The amplitude Tand phase f of a
microwave probe beam of power P
RF
transmitted through the
resonator are measured versus probe frequency q
RF
. 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
j
z
eigenstate, the coupling strength g, and detuning
D
. This variable
capacitance changes the resonator frequency and its transmission
spectrum. The transmission T
2
and phase f of the resonator for a
far-detuned qubit ðg
2
=k
D
,, 1Þ; that is, when the qubit is effectively
decoupled from the resonator, are shown in Fig. 2b and c. In this
case, the transmission is a lorentzian of width
d
n
r
¼ n
r
/Q ¼ k/2p at
n
r
, and the phase f displays a corresponding step of p. The expected
transmission at smaller detuning corresponding to a frequency shift
^g
2
/
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 ^tan
21
(2g
2
/k
D
) of the transmitted microwave at a fixed
probe frequency q
RF
using beam powers P
RF
which controllably
populate the resonator with average photon numbers from n < 10
3
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 n
g
are shown in Fig. 3b, and
two different cases can be identified for a Cooper pair box with
Josephson energy E
J,max
/h . n
r
. In the first case, for bias fluxes such
that E
J
(
F
b
)/h . n
r
, the qubit does not come into resonance with
the resonator for any value of gate charge n
g
(see Fig. 3a). As a result,
the measured phase shift f is maximum for the smallest detuning
D
at n
g
¼ 1 and gets smaller as
D
increases (see Fig. 3b). Moreover, f is
periodic in n
g
with a period of 2e, as expected. In the second case, for
values of
F
b
resulting in E
J
(
F
b
)/h , n
r
, the qubit goes through
resonance with the resonator at two values of n
g
. 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.
In Fig. 3c the qubit level separation n
a
¼ E
a
/h is plotted versus the
bias parameters n
g
and
F
b
. The qubit is in resonance with the
resonator at the points [n
g
,
F
b
], 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 capacitance C coupled through the capacitor C
g
to the Cooper
pair box with junction capacitance C
J
and Josephson energy E
J
forms the circuit QED
system which is coupled through C
in/out
to the input/output ports. The value of E
J
is
controllable by the magnetic flux
F
. The input microwave at frequency q
RF
is added to the
gate voltage V
g
using a bias-tee. After the transmitted signal at q
RF
is amplified using a
cryogenic high electron mobility (HEMT) amplifier and mixed with the local oscillator at
q
LO
, its amplitude and phase are determined. The circulator and the attenuator prevent
leakage of thermal radiation into the resonator. The temperature of individual components
is indicated. b, Measured transmission power spectrum of the resonator (blue dots), the
full linewidth dn
r
at half-maximum and the centre frequency n
r
are indicated. The solid
red line is a fit to a lorentzian with Q ¼ n
r
/dn
r
< 10
4
. c, Measured transmission phase f
(blue dots) with fit (red line). In panels b and c the dashed lines are theory curves shifted by
^dn
r
with respect to the data. d, Energy level diagram of a Cooper pair box. The
electrostatic energy E
C
(n
i
2 n
g
)
2
, with charging energy E
C
¼ e
2
/2C
S
, is indicated for
n
i
¼ 0 (solid black line), 22 (dotted line) and þ2 (dashed line) excess electrons forming
Cooper pairs on the island. C
S
is the total capacitance of the island given by the sum of
the capacitances C
J
of the two tunnel junctions, the coupling capacitance C
g
to the centre
conductor of the resonator and any stray capacitances. In the absence of Josephson
tunnelling the states with n
i
and n
i
þ 2 electrons on the island are degenerate at
n
g
¼ 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 E
J
¼ E
J,max
cos(p
F
b
), where
E
J,max
¼ h
D
Al
/8e
2
R
J
, with the superconducting gap of aluminium
D
Al
and the tunnel
junction resistance R
J
. 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 E
a
.
letters to nature
NATURE | VOL 431 | 9 SEPTEMBER 2004 | www.nature.com/nature164
© 2004
Nature
Publishing
Group
both n
g
and
F
b
in Fig. 3d. We observe the expected periodicity
in flux bias
F
b
with one flux quantum
F
0
. The set of parameters
[n
g
,
F
b
] for which the resonance condition is met is marked by a
sudden sign change in f, which allows a determination of the
Josephson energy E
J,max
¼ 8.0 (^0.1) GHz and the charging energy
E
C
¼ 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
the Cooper pair box is most sensitive at charge degeneracy (n
g
¼ 1),
where the qubit is to first order decoupled from 1/f fluctuations in
its charge environment, which minimizes dephasing
6
. This property
is advantageous for quantum control of the qubit at n
g
¼ 1, a point
where traditional electrometry, using a single electron transistor
(SET) for example
27
, is unable to distinguish the qubit states. We
note that this dispersive QND measurement of the qubit state
12
is
the complement of the atomic microwave cavity QED measurement
in which the state of the cavity is inferred non-destructively from the
phase shift in the state of a beam of atoms sent through the cavity
3,28
.
Making use of the full control over the qubit hamiltonian, we
then tune the flux bias
F
b
so that the qubit is at n
g
¼ 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
Fig. 4b. Owing to the coupling, the first excited states become a
doublet j ^ l. Similarly to ref. 4, we probe the energy splitting of this
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 T
2
is first measured for large detuning
D
with a probe
beam populating the resonator with a maximum of n < 1at
resonance; see Fig. 4a. From the lorentzian line the photon decay
rate of the resonator is determined as k/2p ¼ 0.8 MHz. The probe
beam power is subsequently reduced by 5 dB and the transmission
spectrum T
2
is measured in resonance (
D
¼ 0); see Fig. 4b. We
clearly observe two well-resolved spectral lines separated by the
vacuum Rabi frequency n
Rabi
< 11.6 MHz. 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
the given value k/2p ¼ 0.8 MHz and the single adjustable parameter
g/2p ¼ 0.7 MHz.
The transmission spectrum shown in Fig. 4b is highly sensitive to
the photon number in the cavity. The measured transmission
spectrum is consistent with the expected thermal photon number
of n & 0:06 (T , 100 mK); 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 n
a
¼ q
a
/2p ¼ E
a
/h between ground j#l and excited state j"l of qubit for
two values of flux bias
F
b
¼ 0.8 (orange line) and
F
b
¼ 0.35 (green line). The resonator
frequency n
r
¼ q
r
/2p is shown by a blue line. Resonance occurs at n
a
¼ n
r
symmetrically around degeneracy n
g
¼ ^1; also see red arrows. The detuning
D
/2p ¼ d ¼ n
a
2 n
r
is indicated. b, Measured phase shift f of the transmitted
microwave for values of
F
b
in a. Green curve is offset by 225 deg for visibility.
c, Calculated qubit level separation n
a
versus bias parameters n
g
and
F
b
. The resonator
frequency n
r
is 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 n
g
and
F
b
. Light colours
indicate positive f (d . 0), dark colours negative f (d , 0). The red line is a fit of the
data to the resonance condition n
a
¼ n
r
.Inc and d, the line cuts presented in a and b are
indicated by the orange and the green line, respectively. The microwave probe power P
RF
used to acquire the data is adjusted such that the maximum intra-resonator photon
number n at n
r
is about ten for g
2
=k
D
,, 1: The calibration of the photon number has
been performed in situ by measuring the a.c.-Stark shift of the qubit levels.
letters to nature
NATURE | VOL 431 | 9 SEPTEMBER 2004 | www.nature.com/nature 165
© 2004
Nature
Publishing
Group
mission and give rise to additional peaks in the spectrum owing to
transitions between higher excited doublets
30
. The transmission
spectrum calculated for a thermal photon number of n ¼ 0.5 (see
green curve in Fig. 4b) is clearly incompatible with our experimental
data, indicating that the coupled system has in fact cooled to near its
ground state, and that we measure the coupling of a single qubit to a
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 n
g
¼ 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 n
g
¼ 1 (as shown in Fig. 4b) to predominantly photon
states for n
g
.. 1orn
g
,, 1: The observed peak positions agree well
with calculations considering the qubit with level separation n
a
,a
single photon in the resonator with frequency n
r
and a coupling
strength of g/2p; see solid lines in Fig. 4c. For a different value of flux
bias
F
b
such that E
a
/h , n
r
at n
g
¼ 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 for quantum optical experiments with circuits. 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.
1. Walls, D. & Milburn, G. Quantum Optics (Springer, Berlin, 1994).
2. Mabuchi, H. & Doherty, A. Cavity quantum electrodynamics: Coherence in context. Science 298,
1372–1377 (2002).
3. Raimond, J., Brune, M. & Haroche, S. Manipulating quantum entanglement with atoms and photons
in a cavity. Rev. Mod. Phys. 73, 565–582 (2001).
4. Thompson, R. J., Rempe, G. & Kimble, H. J. Observation of normal-mode splitting for an atom in an
optical cavity. Phys. Rev. Lett. 68, 1132–1135 (1992).
5. Nakamura, Y., Pashkin, Y. A. & Tsai, J. S. Coherent control of macroscopic quantum states in a single-
Cooper-pair box. Nature 398, 786–788 (1999).
6. Vion, D. et al. Manipulating the quantum state of an electrical circuit. Science 296, 886–889 (2002).
7. Martinis, J. M., Nam, S., Aumentado, J. & Urbina, C. Rabi oscillations in a large Josephson-junction
qubit. Phys. Rev. Lett. 89, 117901 (2002).
8. Chiorescu, I., Nakmura, Y., Harmans, C. J. P. M. & Mooij, J. E. Coherent quantum dynamics of a
superconducting flux qubit. Science 299, 1869–1871 (2003).
9. Yamamoto, T., Pashkin, Y. A., Astafiev, O., Nakamura, Y. & Tsai, J. S. Demonstration of conditional
gate operation using superconducting charge qubits. Nature 425, 941–944 (2003).
10. Berkley, A. J. et al. Entangled macroscopic quantum states in two superconducting qubits. Science 300,
1548–1550 (2003).
11. Bouchiat, V., Vion, D., Joyez, P., Esteve, D. & Devoret, M. H. Quantum coherence with a single Cooper
pair. Phys. Scr. T76, 165–170 (1998).
12. Blais, A., Huang, R.-S., Wallraff, A., Girvin, S. & Schoelkopf, R. Cavity quantum electrodynamics for
superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320
(2004).
13. Makhlin, Y., Scho
¨
n, G. & Shnirman, A. Quantum-state engineering with Josephson-junction devices.
Rev. Mod. Phys. 73, 357–400 (2001).
14. Buisson, O. & Hekking, F. in Macroscopic Quantum Coherence and Quantum Computing (eds Averin,
D. V., Ruggiero, B. & Silvestrini, P.) (Kluwer, New York, 2001).
15. Marquardt, F. & Bruder, C. Superposition of two mesoscopically distinct quantum states: Coupling a
Cooper-pair box to a large superconducting island. Phys. Rev. B 63, 054514 (2001).
16. Al-Saidi, W. A. & Stroud, D. Eigenstates of a small Josephson junction coupled to a resonant cavity.
Phys. Rev. B 65, 014512 (2001).
17. Plastina, F. & Falci, G. Communicating Josephson qubits. Phys. Rev. B 67, 224514 (2003).
18. Blais, A., Maassen van den Brink, A. & Zagoskin, A. Tunable coupling of superconducting qubits.
Phys. Rev. Lett. 90, 127901 (2003).
19. Yang, C.-P., Chu, S.-I. & Han, S. Possible realization of entanglement, logical gates, and quantum-
information transfer with superconducting-quantum-interference-device qubits in cavity QED. Phys.
Rev. A 67, 042311 (2003).
20. You, J. Q. & Nori, F. Quantum information processing with superconducting qubits in a microwave
field. Phys. Rev. B 68, 064509 (2003).
21. Kiraz, A. et al. Cavity-quantum electrodynamics using a single InAs quantum dot in a microdisk
structure. Appl. Phys. Lett. 78, 3932–3934 (2001).
22. Childress, L., Sørensen, A. S. & Lukin, M. D. Mesoscopic cavity quantum electrodynamics with
quantum dots. Phys. Rev. A 69, 042302 (2004).
23. Irish, E. K. & Schwab, K. Quantum measurement of a coupled nanomechanical resonator–Cooper-
pair box system. Phys. Rev. B 68, 155311 (2003).
24. Weisbuch, C., Nishioka, M., Ishikawa, A. & Arakawa, Y. Observation of the coupled exciton-photon
mode splitting in a semiconductor quantum microcavity. Phys. Rev. Lett. 69, 3314–3317 (1992).
25. Vuckovic, J., Fattal, D., Santori, C., Solomon, G. S. & Yamamoto, Y. Enhanced single-photon emission
from a quantum dot in a micropost microcavity. Appl. Phys. Lett. 82, 3596 (2003).
26. Day, P. K., LeDuc, H. G., Mazin, B. A., Vayonakis, A. & Zmuidzinas, J. A broadband superconducting
detector suitable for use in large arrays. Nature 425, 817–821 (2003).
27. Lehnert, K. et al. Measurement of the excited-state lifetime of a microelectronic circuit. Phys. Rev. Lett.
90, 027002 (2003).
Figure 4 Vacuum Rabi mode splitting. a, Measured transmission T
2
(blue line) versus
microwave probe frequency n
RF
for large detuning ðg
2
=
D
k ,, 1Þ and fit to lorentzian
(dashed red line). The peak transmission amplitude is normalized to unity. The inset shows
the dispersive dressed states level diagram. b, Measured transmission spectrum for the
resonant case
D
¼ 0atn
g
¼ 1 (blue line) showing the vacuum Rabi mode splitting
compared to numerically calculated transmission spectra (red and green lines) for thermal
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 0.8 MHz. The inset shows the resonant dressed
states level diagram. c, Resonator transmission amplitude T plotted versus probe
frequency n
RF
and gate charge n
g
for
D
¼ 0atn
g
¼ 1. Blue colour corresponds to small
T, red colour to large T. Dashed lines are uncoupled qubit level separation n
a
and
resonator resonance frequency n
r
. Solid lines are level separations found from exact
diagonalization of H
JC
. Spectrum shown in b corresponds to line cut along red arrows.
d,Asinc, but for E
J
/h , n
r
. The dominant character of the corresponding eigenstates is
indicated.
letters to nature
NATURE | VOL 431 | 9 SEPTEMBER 2004 | www.nature.com/nature166
© 2004
Nature
Publishing
Group
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 Edamatsu
1,2
, Goro Oohata
1,3
, Ryosuke Shimizu
2
& Tadashi Itoh
4,2
1
Research Institute of Electrical Communication, Tohoku University, Sendai
980-8577, Japan
2
CREST, Japan Science and Technology Agency (JST), Japan
3
ERATO Semiconductor Spintronics Project, JST, Japan
4
Graduate 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 photons
1,2
is parametric down-conversion. Shor t-wavelength entangled
photons are desirable for generating further entanglement
between three or four photons, but it is difficult to use parametric
down-conversion to generate suitably energetic entangled pho-
ton pairs. One method that is expected to be applicable for
the generation of such photons
3
is resonant hyper-parametric
scattering (RHPS): a pair of entangled photons is generated in a
semiconductor via an electronically resonant third-order non-
linear optical process. Semiconductor-based sources of entangled
photons would also be advantageous for practical quantum
technolo gies, but attem pts to generate entangled photons in
semiconductors have not yet been successful
4,5
. Here we report
experimental evidence for the generation of ultraviolet entangled
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 entang led photons by
current injection, analogous to current-driven single photon
sources
6,7
.
The material we used in this study was copper chloride (CuCl)
single crystal. Because CuCl has a large bandgap (,3.4 eV), 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 (,200 meV) and biexciton (,30 meV). 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
(refs 8, 9 and ref. 10 and references therein). Figure 1a schematically
shows the RHPS process in resonance to the biexciton state. The
two pump (parent) photons (frequency q
i
) resonantly create the
biexciton, and are converted into the two scattered (daughter)
photons (q
s
, q
s
0
). The biexciton state (G
1
) created in this process
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,
instead of the oversimplified picture in Fig. 1a, we must consider the
exciton-polariton picture; the RHPS obeys the phase-matching
condition that takes into account the polariton dispersion relation
8
.
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
whose photon energies, as well as the sum of momenta, is conserved
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 lately
12
.In
addition, a large parametric gain via the biexcitonic resonance in
CuCl was reported recently
13
. Similar stimulated parametric scatter-
ing of polaritons has also been observed in semiconductor micro-
cavities, even at high temperatures
14
.
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. The RHPS is very efficient (a few orders of magnitude higher
than that of typical parametric down-conversion): We got of the
order of 10
10
photons s
21
sr
21
by 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-
interval analyser recorded the time interval (t) between the detected
Figure 1 Schematic diagram of the resonant hyper-parametric scattering (RHPS) via
biexciton. a , Two pump (parent) photons of frequency q
i
are converted to the two
scattered (daughter) photons (q
s
, q
s
0
). b, The polariton dispersion drawn in two
dimensions of momentum space. The biexciton decays into two polaritons that satisfy the
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.
letters to nature
NATURE | VOL 431 | 9 SEPTEMBER 2004 | www.nature.com/nature 167
© 2004
Nature
Publishing
Group
