Stopping single photons in one-dimensional circuit quantum electrodynamics systems
Jung-Tsung Shen, M. L. Povinelli, Sunil Sandhu, and Shanhui Fan*
Ginzton Laboratory, Stanford University, Stanford, California 94305, USA
Received 9 June 2006; revised manuscript received 16 November 2006; published 12 January 2007
We propose a mechanism to stop and time reverse single photons in one-dimensional circuit quantum
electrodynamics systems. As a concrete example, we exploit the large tunability of the superconducting charge
quantum bit charge qubit to predict one-photon transport properties in multiple-qubit systems with dynami-
cally controlled transition frequencies. In particular, two qubits coupled to a waveguide give rise to a single-
photon transmission line shape that is analogous to electromagnetically induced transparency in atomic sys-
tems. Furthermore, by cascading double-qubit structures to form an array and dynamically controlling the qubit
transition frequencies, a single photon can be stopped, stored, and time reversed. With a properly designed
array, two photons can be stopped and stored in the system at the same time. Moreover, the unit cell of the
array can be designed to be of deep subwavelength scale, miniaturizing the circuit.
DOI: 10.1103/PhysRevB.75.035320 PACS numbers: 74.50.+r, 32.80.-t, 03.67.Lx, 42.50.-p
Single photon transport properties in circuit quantum
electrodynamics circuit QED systems are expected to play
a significant role in quantum information processing and
quantum computing. In contrast to cavity QED, where a
single photon is confined in a cavity with a discrete photonic
mode spectrum, in a circuit QED system, the single photon
propagates in a one-dimensional continuum. In a recent
solid-state experimental implementation of circuit QED, a
single photon on average is coupled to a superconducting
charge quantum bit charge qubit, or Cooper pair box, in a
one-dimensional coplanar waveguide geometry. Strong cou-
pling between the single photon and the qubit has been
demonstrated.
1
Theoretically, it was also shown that the one-
dimensional propagating continuum of the single photon
leads to versatile transmission and reflection profiles, includ-
ing general Fano line shapes, and allows one-photon
switching.
2,3
The proposed formalism in Refs. 2 and 3 is very
general and encompasses the experiment in Ref. 1 as a spe-
cial case.
The Cooper pair box is highly tunable, as compared to
other qubit implementations, such as hyperfine structure in
real atoms, quantum dots, or optical resonators. The transi-
tion energy of the Cooper pair box can easily be tuned by
threading the Josephson junction loop with a magnetic field
flux.
1,4–8
In this paper, we exploit the large tunability of the
qubit to predict one-photon transport properties in multiple-
qubit systems with dynamically controlled transition fre-
quencies. In particular, two qubits coupled to a waveguide
give rise to a single-photon transmission line shape that is
analogous to electromagnetically induced transparency EIT
in atomic systems. The width of the transparency peak is
strongly tunable by adjusting the transition frequencies. Fur-
thermore, by cascading double-qubit structures to form an
array, the properties of photons inside the array are now de-
termined by a photonic band structure. By dynamically tun-
ing the transition frequency of the qubits while the photon is
in the array, the band structure and the spectrum of the single
photon can be molded almost arbitrarily, leading to highly
nontrivial information processing capabilities on chip, in-
cluding stopping, storing, and time reversal of a single-
photon pulse. Moreover, with a properly designed array, two
photons can be stopped and stored in the system at the same
time. Finally, the unit cell of the array can be designed to be
of deep subwavelength scale, miniaturizing the circuit. Pos-
sible applications of single-photon manipulation schemes
demonstrated here include logic gates, memory, buffers, and
repeaters in quantum information processing and quantum
communication.
To start with, consider the simplest case of one qubit
coupled to a waveguide. In Refs. 2 and 3, it was shown that
the transfer matrix T
q
of a single photon
9
passing through a
single qubit takes the following form:
a'
b'
=
1-
iV
2
v
g
k
- + i
-
iV
2
v
g
k
- + i
+
iV
2
v
g
k
- + i
1+
iV
2
v
g
k
- + i
a
b
T
q
a
b
, 1
where is the transition frequency of the qubit,
k
= v
g
k is
the frequency of the photon, v
g
is the group velocity of the
fundamental waveguide mode, and V is the coupling constant
between the photons and the qubits. accounts for loss
mechanisms. To highlight the intrinsic behavior of the sys-
tem, we set equal to zero. Discussion of the effects of a
nonzero is provided at the end of the paper. The transfer
matrix relates the incoming and outgoing wave amplitudes
ab' and ba' on either side of the qubit. Note the form of
the transfer matrix is the same as for a waveguide side
coupled to a single-mode optical cavity.
10–12
The exact map-
ping implies that the one-photon system in discussion exhib-
its many features that have been well studied for cavity struc-
tures and classical light. The mapping is quite interesting
given that the microscopic theories that underlie these sys-
tems are very different.
The transfer matrix T
q
Eq. 1 allows the complete de-
termination of photon transport properties. The response
function of any configuration of mutiple qubits can be calcu-
lated by cascading the transfer matrices of each individual
element in the system. For example, the total transfer matrix
PHYSICAL REVIEW B 75, 035320 2007
1098-0121/2007/753/0353205 ©2007 The American Physical Society 035320-1