s for superconducting electrical circuits:
An architecture for quantum computation
Alexandre Blais,
1
Ren-Shou Huang,
1,2
Andreas Wallraff,
1
S. M. Girvin,
1
and R. J. Schoelkopf
1
1
Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA
2
Department of Physics, Indiana University, Bloomington, Indiana 47405, USA
(Received 7 February 2004; published 29 June 2004
)
We propose a realizable architecture using one-dimensional transmission line resonators to reach the strong-
coupling limit of cavity quantum electrodynamics in superconducting electrical circuits. The vacuum Rabi
frequency for the coupling of cavity photons to quantized excitations of an adjacent electrical circuit
(qubit)
can easily exceed the damping rates of both the cavity and qubit. This architecture is attractive both as a
macroscopic analog of atomic physics experiments and for quantum computing and control, since it provides
strong inhibition of spontaneous emission, potentially leading to greatly enhanced qubit lifetimes, allows
high-delity quantum nondemolition measurements of the state of multiple qubits, and has a natural mecha-
nism for entanglement of qubits separated by centimeter distances. In addition it would allow production of
microwave photon states of fundamental importance for quantum communication.
DOI: 10.1103/PhysRevA.69.062320
PACS number
(s): 03.67.Lx, 73.23.Hk, 74.50. r, 32.80. t
I. INTRODUCTION
Cavity quantum electrodynamics
(CQED) studies the
properties of atoms coupled to discrete photon modes in high
Q cavities. Such systems are of great interest in the study of
the fundamental quantum mechanics of open systems, the
engineering of quantum states, and measurement-induced de-
coherence
[13] and have also been proposed as possible
candidates for use in quantum information processing and
transmission
[13]. Ideas for novel CQED analogs using na-
nomechanical resonators have recently been suggested by
Schwab and collaborators
[4,5]. We present here a realistic
proposal for CQED via Cooper pair boxes coupled to a one-
dimensional
(1D) transmission line resonator, within a
simple circuit that can be fabricated on a single microelec-
tronic chip. As we discuss, 1D cavities offer a number of
practical advantages in reaching the strong-coupling limit of
CQED over previous proposals using discrete LC circuits
[6,7], large Josephson junctions [810], or 3D cavities
[1113]. Besides the potential for entangling qubits to realize
two-qubit gates addressed in those works, in the present
work we show that the CQED approach also gives strong
and controllable isolation of the qubits from the electromag-
netic environment, permits high-delity quantum nondemo-
lition
(QND) readout of multiple qubits, and can produce
states of microwave photon elds suitable for quantum com-
munication. The proposed circuits therefore provide a simple
and efcient architecture for solid-state quantum computa-
tion, in addition to opening up a new avenue for the study of
entanglement and quantum measurement physics with mac-
roscopic objects. We will frame our discussion in a way that
makes contact between the language of atomic physics and
that of electrical engineering.
We begin in Sec. II with a brief general overview of
CQED before turning to a discussion of our proposed solid-
state realization of cavity QED in Sec. III. We then discuss in
Sec. IV the case where the cavity and qubit are tuned in
resonance and in Sec. V the case of large detuning which
leads to lifetime enhancement of the qubit. In Sec. VI, a
quantum nondemolition readout protocol is presented. Real-
ization of one-qubit logical operations is discussed in Sec.
VII and two-qubit entanglement in Sec. VIII. We show in
Sec. IX how to take advantage of encoded universality and
decoherence-free subspace in this system.
II. BRIEF REVIEW OF CAVITY QED
Cavity QED studies the interaction between atoms and the
quantized electromagnetic modes inside a cavity. In the op-
tical version of CQED
[2], schematically shown in Fig. 1(a),
one drives the cavity with a laser and monitors changes in
the cavity transmission resulting from coupling to atoms fall-
ing through the cavity. One can also monitor the spontaneous
emission of the atoms into transverse modes not conned by
the cavity. It is not generally possible to directly determine
the state of the atoms after they have passed through the
cavity because the spontaneous emission lifetime is on the
scale of nanoseconds. One can, however, infer information
about the state of the atoms inside the cavity from real-time
monitoring of the cavity optical transmission.
In the microwave version of CQED
[3], one uses a very-
high-Q superconducting 3D resonator to couple photons to
transitions in Rydberg atoms. Here one does not directly
monitor the state of the photons, but is able to determine
with high efciency the state of the atoms after they have
passed through the cavity
(since the excited state lifetime is
of the order of 30 ms
). From this state-selective detection
one can infer information about the state of the photons in
the cavity.
The key parameters describing a CQED system
(see Table
I
) are the cavity resonance frequency
r
, the atomic transi-
tion frequency
, and the strength of the atom-photon cou-
pling g appearing in the Jaynes-Cummings Hamiltonian
[14]
PHYSICAL REVIEW A 69, 062320
(2004)
1050-2947/2004/69
(6)/062320(14)/$22.50
©2004 The American Physical Society
69
062320-1 H =
r
a a + 1
2 + 2
z
+ g a
+
+
a + H + H .
1
Here H
describes the coupling of the cavity to the con-
tinuum which produces the cavity decay rate
=
r
/ Q, while
H describes the coupling of the atom to modes other than
the cavity mode which cause the excited state to decay at rate
(and possibly also produce additional dephasing effects).
An additional important parameter in the atomic case is the
transit time t
transit
of the atom through the cavity.
In the absence of damping, exact diagonalization of the
Jaynes-Cumming Hamiltonian yields the excited eigenstates
(dressed states) [15]
+ ,n = cos
n
,n + sin
n
,n + 1 ,
2
,n = sin
n
,n + cos
n
,n + 1 ,
3
and ground state
,0 with corresponding eigenenergies
E
± , n
= n + 1
r
± 2 4<i>g
2
n + 1 +
2
,
4
E
,0
= 2 .
5
In these expressions,
n
= 1
2 tan
1
2<i>g n + 1 ,
6
and r
the atom-cavity detuning.
Figure 1
(b) shows the spectrum of these dressed states for
the case of zero detuning,
= 0, between the atom and cavity.
In this situation, degeneracy of the pair of states with n + 1
quanta is lifted by 2<i>g n + 1 due to the atom-photon interac-
tion. In the manifold with a single excitation, Eqs.
(2) and (3)
reduce to the maximally entangled atom-eld states ± , 0
=
,1 ± ,0 / 2. An initial state with an excited atom and
zero photons
,0 will therefore op into a photon ,1 and
back again at the vacuum Rabi frequency g / . Since the
excitation is half atom and half photon, the decay rate of
± , 0 is
+
/ 2. The pair of states ± , 0 will be resolved in
a transmission experiment if the splitting 2<i>g is larger than
this linewidth. The value of g =
E
rms
d /
is determined by the
transition dipole moment d and the rms zero-point electric
eld of the cavity mode. Strong coupling is achieved when
g
,
[15].
FIG. 1.
(Color online) (a) Standard representation of a cavity
quantum electrodynamic system, comprising a single mode of the
electromagnetic eld in a cavity with decay rate
coupled with a
coupling strength g =
E
rms
d /
to a two-level system with spontane-
ous decay rate
and cavity transit time t
transit
.
(b) Energy spectrum
of the uncoupled
(left and right) and dressed (center) atom-photon
states in the case of zero detuning. The degeneracy of the two-
dimensional manifolds of states with n 1 quanta is lifted by
2<i>g n + 1.
(c) Energy spectrum in the dispersive regime (long-
dashed lines
). To second order in g, the level separation is indepen-
dent of n, but depends on the state of the atom.
TABLE I. Key rates and CQED parameters for optical
[2] and microwave [3] atomic systems using 3D cavities, compared against the
proposed approach using superconducting circuits, showing the possibility for attaining the strong cavity QED limit n
Rabi
1 . For the 1D
superconducting system, a full-wave L =
resonator,
r
/ 2 = 10 GHz, a relatively low Q of 10
4
, and coupling
= C
g
/ C = 0.1 are assumed.
For the 3D microwave case, the number of Rabi ops is limited by the transit time. For the 1D circuit case, the intrinsic Cooper-pair box
decay rate is unknown; a conservative value equal to the current experimental upper bound
1 / 2
s is assumed.
Parameter
Symbol
3D optical
3D microwave
1D circuit
Resonance or transition frequency
r
/ 2 ,
/ 2
350 THz
51 GHz
10 GHz
Vacuum Rabi frequency
g / , g /
r
220 MHz, 3
10
7
47 kHz, 1
10
7
100 MHz, 5
10
3
Transition dipole
d / ea
0
1
1
10
3
2
10
4
Cavity lifetime
1 / , Q
10 ns, 3
10
7
1 ms, 3
10
8
160 ns, 10
4
Atom lifetime
1 /
61 ns
30 ms
2
s
Atom transit time
t
transit
50
s
100
s
Critical atom number
N
0
= 2
/ g
2
6
10
3
3
10
6
6
10
5
Critical photon number
m
0
=
2
/ 2<i>g
2
3
10
4
3
10
8
1
10
6
Number of vacuum Rabi ops
n
Rabi
= 2<i>g /
+
10
5
10
2
BLAIS et al.
PHYSICAL REVIEW A 69, 062320
(2004)
062320-2 For large detuning, g /
1, expansion of Eq.
(4) yields
the dispersive spectrum shown in Fig. 1
(c). In this situation,
the eigenstates of the one excitation manifold take the form
[15]
,0
g/
,0 + ,1 ,
7
+ ,0
,0 + g/
,1 .
8
The corresponding decay rates are then simply given by
, 0
g/
2
+ ,
9
+ , 0
+ g/
2
.
10
More insight into the dispersive regime is gained by mak-
ing the unitary transformation
U = exp g a
+
a
11
and expanding to second order in g
(neg