# Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics

**Abstract**

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 for several decades and has generated the field of cavity quantum electrodynamics. 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 new approaches for single photon generation and detection.

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.

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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.

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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.

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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

.

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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.

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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.

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2. Mabuchi, H. & Doherty, A. Cavity quantum electrodynamics: Coherence in context. Science 298,

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in a cavity. Rev. Mod. Phys. 73, 565–582 (2001).

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optical cavity. Phys. Rev. Lett. 68, 1132–1135 (1992).

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gate operation using superconducting charge qubits. Nature 425, 941–944 (2003).

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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).

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superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320

(2004).

13. Makhlin, Y., Scho

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n, G. & Shnirman, A. Quantum-state engineering with Josephson-junction devices.

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D. V., Ruggiero, B. & Silvestrini, P.) (Kluwer, New York, 2001).

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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).

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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).

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quantum dots. Phys. Rev. A 69, 042302 (2004).

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pair box system. Phys. Rev. B 68, 155311 (2003).

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from a quantum dot in a micropost microcavity. Appl. Phys. Lett. 82, 3596 (2003).

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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.

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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 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.

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- CitationsCitations1802
- ReferencesReferences43

- "The condition J c (ω dip ) > J free (ω dip ) results in the celebrated Purcell enhancement [12] of the decay rate of an excited dipole. Moreover, any asymmetry of J c (ω) with respect to ω dip results in a cavity contribution to the Lamb shift of ω dip [3, 11, 13] . When J c (ω) is nonconstant in a frequency window of order J c (ω)/, the cavity vacuum can be considered as a non-Markovian bath, resulting in reversible processes such as the vacuum Rabi oscillations [11]. "

[Show abstract] [Hide abstract]**ABSTRACT:**Virtually all interactions that are relevant for atomic and condensed matter physics are mediated by the quantum fluctuations of the electromagnetic field vacuum. Consequently, controlling the latter can be used to engineer the strength and the range of inter-particle interactions. Recent experiments have used this premise to demonstrate novel quantum phases or entangling gates by embedding electric dipoles in photonic cavities or waveguides which modify the electromagnetic fluctuations. Here, we demonstrate theoretically that the enhanced fluctuations in the anti-squeezed quadrature of a squeezed vacuum state allows for engineering interactions between electric dipoles without the need for any photonic structure. Thus, the strength and range of the resulting dipole-dipole coupling can be engineered by dynamically changing the spatial profile of the squeezed vacuum in a travelling-wave geometry.- "This Hamiltonian has been demonstrated with numerous effective two-level systems coupling to cavity fields, eg. atoms [11], quantum dots [26] and in particular superconducting qubits in circuit QED [16, 17]. In the regime where the detuning, ∆ = Ω q − ω r , is much larger than the coupling strength, g, we arrive at the dispersive Hamiltonian "

[Show abstract] [Hide abstract]**ABSTRACT:**We propose a minimal scheme for a quantum teleportation protocol using only two qubits and one cavity field, with the cavity field used both for encoding discrete information and for readout. An implementation of the scheme with superconducting qubits and resonators is presented. Our compact scheme restricts the accessible information to the signal emitted from the resonator and, hence, it relies on continuous probing rather than an instantaneous projective Bell state measurement. We show that the past quantum state formalism [S. Gammelmark et. al., Phys. Rev. Lett. 111, 160401] can be successfully applied to estimate what would have been the most likely Bell measurement outcome conditioned on our continuous signal. This inferred outcome determines the local qubit operation on the target qubit which yields the optimal teleportation fidelity.- "Working at high frequencies also allows one to measure effects related to quantum capacitance [4,5] , or investigate electronic transitions resonant with the cavity. By avoiding transport, cQED readout techniques could yield to QND [6,7] (quantum non-demolition) measurements of charge or spin states in quantum dot devices. Another significant potential of cQED architectures is the scalability, which, combined with recently demonstrated spin–photon coupling [8], could allow us to tackle fundamental problems such as the coupling and entanglement of distant spins [9][10][11][12][13][14][15][16]. "

[Show abstract] [Hide abstract]**ABSTRACT:**Cavity quantum electrodynamics allows one to study the interaction between light and matter at the most elementary level. The methods developed in this field have taught us how to probe and manipulate individual quantum systems like atoms and superconducting quantum bits with an exquisite accuracy. There is now a strong effort to extend further these methods to other quantum systems, and in particular hybrid quantum dot circuits. This could turn out to be instrumental for a noninvasive study of quantum dot circuits and a realization of scalable spin quantum bit architectures. It could also provide an interesting platform for quantum simulation of simple fermion–boson condensed matter systems. In this short review, we discuss the experimental state of the art for hybrid circuit quantum electrodynamics with quantum dots, and we present a simple theoretical modeling of experiments.

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