Download New Approach for Finding the Phase Shift Operator via the IWOP

CHINESE JOURNAL OF PHYSICS
VOL. 48, NO. 2
APRIL 2010
New Approach for Finding the Phase Shift Operator via the IWOP Technique
Gang Ren and Shuang-xi Zhang
Department of Material Science and Engineering,
University of Science and Technology of China, Hefei, Anhui 230026, China
(Received August 31, 2009)
We present a new approach for finding the explicit expression of the unitary phase operator
by the technique of integration within an ordered product (IWOP) of an operator. An
adaptive estimation scheme to measure the phase shift angle in the two-mode case is also
given.
PACS numbers: 02.30.Tb, 03.65.Ud
I. INTRODUCTION
It is well-known that the theory of the quantum phase is an important topic in
quantum optics and quantum statistics. The phases of optical fields play the decisive
role in many optical phenomena, particularly in the diffraction and interference of light.
The question of defining a quantum phase shift operator for the optical field is a great
challenge in quantum mechanics and quantum optics. Quite a few proposals are able to
define quantum phase shift operators that are consistent with quantum mechanics and
coincide with experimental measurements, of which the frequently discussed are given by
Susskind and Glogower [1], Pegg and Barnett [2], and Sperling and Voge [3]. The main
problem for displaying the quantum phase lies in the lack of a suitably defined quantum
phase shift operator which is Hermitian and is measurable experimentally. In this work,
we concentrate on the explicit expression of phase shift operators, which are based on the
technique of integration within an ordered product (IWOP) of operators [4, 5].
The quantum limitation of phase measurement is an open problem in quantum mechanics, and one which attracts the attention of physicists, though interferometry has a long
track record as a useful technique for research [6]. To date, only a few experiments have
been published concerning phase-shift measurements at low energies limited by quantum
fluctuations [7–9]. In this work we point out that the phase shift operator can be gained
by the IWOP technique.
The presentation is arranged as follows: in Sec. II, we discuss the single-mode phase
shift operator via the coherent state. In Sec. III, the two-mode phase shift operator is given
by the two-mode entangled state. In Sec. IV, we give a theoretically ideal probe state to
measure the phase shift angle, and its corresponding probability density is also discussed.
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230
c 2010 THE PHYSICAL SOCIETY
OF THE REPUBLIC OF CHINA
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GANG REN AND SHUANG-XI ZHANG
231
II. EXPLICIT FORM OF A SINGLE-MODE PHASE SHIFT OPERATOR VIA THE
COHERENT STATE
Paul [10] defined the phase operator by the diagonal state representation (called the
Glauber-Sudarshan P-representation) as
Z 2
d z iϕ
c
e |zihz| = eiϕ
z = |z|eiϕ ,
(1)
p ,
π
where |zi is the coherent state and has the form
1
|zi = exp(− |z|2 + za†1 )|0i.
2
(2)
With the help of the coherent state, we construct the single-mode phase shift operator as
Z 2
d z iθ
|ze ihz|,
(3)
U1 (θ) =
π
where θ is the shift angle, which should be measured by experiment.
Using Eq. (2), Eq. (3) can be turned into
Z 2
1
1
d z
exp(− |z|2 + zeiθ a†1 )|0ih0| exp(− |z|2 + z ∗ a1 ).
U1 (θ) =
π
2
2
(4)
By virtue of the IWOP technique, i.e.,
|0ih0| =: exp(−a†1 a1 ) : ,
(5)
where :: stands for the normal ordering for a1 and a†1 , and we use the integration formula
Z
d2 α
1
sη
exp(−h|α|2 + sα + ηα∗ ) = exp( ),
π
h
h
Re h < 0,
(6)
we can conveniently derive the single-mode phase shift operator defined in Eq. (3) as follows
Z 2
d z
U1 (θ) =
: exp(−|z|2 + zeiθ a† + z ∗ a − a† a) : .
(7)
π
Noticing the operator identity
†
eλa1 a1 =
=
∞
X
n=0
∞
X
n=0
eλn |nihn| =
:
∞
X
a†n
an
eλn √1 |0ih0| √ 1
n!
n!
n=0
1 λ † n −a† a1
(e a1 a1 ) e 1 : = : exp[(eλ − 1)a†1 a1 ] : ,
n!
(8)
NEW APPROACH FOR FINDING THE PHASE . . .
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we can have the explicit form of U1 (θ) in Eq. (3) as
†
U1 (θ) = eiθa1 a1 .
(9)
For the eigenstate of a†1 a1 is |ni , i.e., a†1 a1 |ni = n |ni , it is clear that an eigenstate
of a†1 a1 is not a suitable choice for finding the explicit form of the phase shifter operator.
However, a single-mode phase measurement which corresponds to the operational concept,
is difficult to define. In the next sections we will give an approach to this problem, which
is crucially based on entangled two-mode states.
III. EXPLICIT FORM OF TWO-MODE PHASE SHIFT OPERATOR VIA ENTANGLED STATE
In order to avoid the difficulty of starting from a single-mode phase description,
we now use a two-mode entangled state to find the explicit form of the phase shifter.
We first briefly review some properties of the two-mode entangled state. The two-mode
Hilbert space can be spanned by eigenstates of the quadratures Qi = √12 (al + a†l ) and
Pl = √12i (al − a†l ), l = 1, 2, where al and a†l are the corresponding destruction and creation
operators. We define the two-mode operators
Qr ≡ Q1 + Q2 ,
Pc = P1 − P2 .
(10)
From the commutation relation of Qr and Pc , i.e., [Qr , Pc ] = 0, we can derive the
common eigenstates of Qr and Pc as
1
|ξi = |ξ|eiϕ i = exp(− |ξ|2 + ξa†1 + ξ ∗ a†2 − a†1 a†2 )|00i,
2
(11)
in which ξ = ξ1 + iξ2 is an arbitrary complex number. Qc and Pr satisfy the following
relation
√
√
(Q1 + Q2 )|ξi = 2ξ1 |ξi,
(P1 − P2 )|ξi = 2ξ2 |ξi.
(12)
Using the IWOP technique, we can prove that
Z 2
d ξ
|ξihξ|
π
Z 2
d ξ
=
: exp[−|ξ|2 + ξ(a†1 + a2 ) + ξ ∗ (a1 + a†2 ) − (a†1 + a2 )(a1 + a†2 )] : = 1,
π
(13)
and
hξ 0 |ξi = πδ(2) (ξ 0 − ξ).
(14)
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GANG REN AND SHUANG-XI ZHANG
233
With the help of the two-mode entangled state, we construct a two-mode phase shift
operator as
Z 2
d ξ iθ
|ξe ihξ|.
(15)
U2 (θ) =
π
Noticing that |00i h00| =: exp(−a†1 a1 − a†2 a2 ) :, we can perform the integration as
Z 2
d ξ
: exp[−|ξ|2 +ξ(eiθ a†1 +a2 )+ξ ∗ (e−iθ a†2 +a1 )−a†1 a†2 −a1 a2 −a†1 a1 −a†2 a2 ] : . (16)
U2 (θ) =
π
From the integration formula in Eq. (6), we get
U2 (θ) =: exp[(eiθ − 1)a†1 a1 − (eiθ − 1)a†2 a2 ] : .
Using Eq. (8), we have
h
i
U2 (θ) = exp iθ(a†1 a1 − a†2 a2 ) ,
(17)
(18)
which can be considered
asan explicit formof the two-mode
phase shift operator. Since the
†
†
†
†
eigenstate of a1 a1 − a2 a2 is |mni , i.e., a1 a1 − a2 a2 |mni = (m − n) |mni , it is clear
that an eigenstate of a†1 a1 − a†2 a2 is not a suitable choice for finding the explicit form of
the phase shifter operator.
IV. THEORETICAL IDEAL TWO-MODE PROBE STATE
We construct a theoretical probe state which can be guided by the properties of the
state |ξi. Formally we can create this entangled two-mode state with the help of a 50 : 50
beam splitter described by the unitary transformation
π
UBS ≡ exp[− (a†1 a2 − a†2 a1 )]
4
(19)
on the two-mode entangled state |ξi.
Let us imagine that we send the two quadrature states |x = X0 i1 and |p = P0 i2 onto
this beam splitter. They fulfill the eigenvalue equations x1 |xi1 = x|xi1 and p2 |pi2 = p|pi2 .
If we make use of the relations
1
1
†
UBS x1 UBS
= √ (x1 + x2 ) = √ Qr ,
2
2
1
1
†
UBS p2 UBS = √ (p1 − p2 ) = √ Pc ,
2
2
(20)
we see that the state
|ξi = UBS |x = X0 i1 |p = P0 i2
(21)
NEW APPROACH FOR FINDING THE PHASE . . .
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describes the output of the beam splitter.
It would allow us to determine an unknown phase shift θ by just preparing the state
|ξi and performing a single measurement of the observable X and P in Eq. (20). Based
on these probe states, we will then discuss an adaptive measurement scheme for the phase
shift θ.
The probability density describing the phase shift angle measurement is given by
ρ(ξ) = |hξeiθ |ξi|2 = π 2 [δ2 (ξeiθ − ξ)]2 .
(22)
This is the fundamental quantity which allows us to retrieve the unknown phase shift θ .
In summary, we constructed single-mode and two-mode phase shift operators via the
coherent state and the two-mode entangled state accordingly. A theoretical ideal two-mode
probe state to measure the phase shift angle is also presented. The technique of integration
within an ordered product of operators is a consequence for this derivation. We hope this
method can be extended to multi-modes and will be helpful for making an experiment on
measuring the phase shift angle.
Acknowledgements
The authors would like to thank the referees for their constructive suggestions and
comments in this passage.
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