Scheme for direct measurement of the Wigner characteristic function in cavity QED
XuBo Zou, K. Pahlke, and W. Mathis
Electromagnetic Theory Group at THT, Department of Electrical Engineering, University of Hannover, Hannover, Germany
Received 24 March 2003; published 29 January 2004
We propose a simple scheme for the reconstruction of the single-mode cavity field by considering the
resonant atom-cavity interaction in the presence of a strong classical field. With the aid of the strong classical
field, it is easy to realize the displacement operator for the cavity field correlated to the internal state of the
atom. It is shown that the measurement of the population of the lower internal state directly yields the Wigner
characteristic function of the cavity field.
DOI: 10.1103/PhysRevA.69.015802 PACS numbers: 42.50.Dv
In recent years there has been great interest in the prepa-
ration and measurement of quantum states 1. Cavity QED,
with Rydberg atoms crossing superconducting cavities, offer
an almost ideal system for the generation and measurement
of nonclassical states and implementation of small scale
quantum information processing 2. In the context of cavity
QED, numerous theoretical schemes 3 for generating vari-
ous nonclassical states were proposed, which led to experi-
mental realization of Schro
¨
dinger cat state 4 and Fock state
5 in a cavity mode. Thus, it is desirable to have a powerful
tool to prove that the cavity field has indeed been prepared in
the desired state. Several measurement schemes for the cav-
ity fields have been proposed by probing quantum states with
two-level atoms and subsequently measuring the atomic state
6. But only a few of the proposals have a strikingly simple
data analysis. Wilkens and Meystre proposed a scheme for
directly measuring the Wigner characteristic function of a
cavity field via the nonlinear atomic honodyne detection 7.
In Ref. 8, Kim et al. made a similar proposal based on
current experimental conditions. In Ref. 9, Lutterbach and
Davidivich presented a scheme for direct measurement of
Wigner function of cavity field, which has been experimen-
tally demonstrated in a cavity 10. This scheme is based on
the dispersive interaction of a single atom with the cavity
field. However, dispersive interaction requires that the detun-
ing between the atoms and the cavity is much bigger than the
atom-cavity coupling strength. Thus, the quantum dynamics
operates at a low speed. In Ref. 11, Bardroff et al. proposed
a simple and fast scheme for direct measurement of the
Wigner characteristic function of the motion state of a
trapped ion. This scheme can be applied to measure the
quantum state of a cavity field via realizing a displacement
operator for the cavity field correlated to the internal state of
the atom. Such displacement operation has been suggested
by Davidovich et al. in the context of quantum switches us-
ing a dispersive atom-cavity interaction, but the experimental
realization of the scheme is difficult.
In this paper, we propose an alternative scheme to directly
measure the characteristic function of the Wigner function of
single-mode cavity field. The physical system is a two-level
atom interacting with a single-mode cavity field in the pres-
ence of a strong classical field. Recently, Solano et al. stud-
ied such physical model for generating atom-field entangled
states and field superposition states 12. These authors
showed that, with the aid of the strong classical field, it is
easy to realize the displacement operator for the cavity field
correlated to the internal state of the atom. In this paper, we
show that such physical system can be used to measure the
Wigner characteristic function of single-mode cavity field,
and the phase of classical field acting as a tunable parameter
is important for the measurement of cavity field.
We consider a two-level atom interacting with a single-
mode cavity field and driven additionally by an external clas-
sical field. In the rotating-wave approximation, the Hamil-
tonian is assuming =1) 12
H =
at
2
z
+
ca v
a
†
a +g
-
e
i +i
L
t
+
+
e
-i -i
L
t
+ a
†
-
+a
+
, 1
where
z
=| e e | -| g g | ,
+
=| e g | , and
-
=| g e | ,
with | g and | e being the ground and excited states of the
two-level atom.
at
is the atomic transition frequency. a and
a
†
are the annihilation and creation operator of the single-
mode cavity field of frequency
ca v
. is the atom-cavity
interaction strength. g and are the amplitude and phase of
the classical field.
L
is the frequency of the classical field.
Here we should mention the physical system proposed by
Alsing and Carmichael 13 on the single atom cavity QED
system with a strongly driven cavity field, which was later
studied by Mabuchi and Wiseman 14. Although these au-
thors consider a strongly driven cavity rather than a strongly
driven atom, there is a canonical mapping between the two
cases, so both systems are in fact essentially identical.
In a frame, which rotates with the classical wave fre-
quency
L
, the associated Hamiltonian of the system be-
comes
H =
2
z
+ a
†
a +g
-
e
i
+
+
e
-i
+ a
†
-
+a
+
,
2
where =
0
-
L
and = -
L
.
In the following we assume that the atom, the cavity, and
the driving classical field are all resonant: = =0. In this
case, the Hamiltonian 2 can be written as
H =g
-
e
i
+
+
e
-i
+ a
†
-
+a
+
. 3
PHYSICAL REVIEW A 69, 015802 2004
1050-2947/2004/691/0158023/$22.50 ©2004 The American Physical Society 69 015802-1