Wigner Function of Thermo-Invariant Coherent State
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In this Letter, we derive the Wigner function of the thermo-invariant coherent state |z, N [3] in thermofield dynamics (TFD).[4] In TFD the original Fock
(a + a†)|ξ = ξ|ξ , (a† + a)|ξ = ξ∗|ξ , (9)
∗Supported by the Specialized Research Fund for the Doctoral Programme of Higher Education of China under Grant No
√
√
(Q + Q)|ξ = 2ξ1|ξ , (P − P )|ξ = 2ξ2|ξ , (6)
where
Q = 1/2(a + a†), P = i 1/2(a† − a),
Q = 1/2(a + a†), P = i 1/2(a† − a), (7)
so |ξ is the common eigenstate of Q + Q and P − P . Using the technique of integration within an ordered product (IWOP) of operators[6], the completeness re-
W (x, p) = Tr [ρ∆(x, p)]
1 =
∞
v
vwenku.baidu.com
x+ ρ x−
e−ipv dv,
(11)
2π −∞
2
2
where ρ is a density operator, ∆(x, p) is the usual single-mode Wigner operator, |x is the eigenvector of the coordinate operator X. In the coordinates representation ∆(x, p) takes the form
1 ∆(x, p) =
∞
v
x−
x + v e−ipvdv,
(12)
2π −∞
2
2
while in the coherent state representation the Wigner operator is
∆(x, p) → ∆(α) =
d2z π2 |α + z
α − z|e−αz∗−zα∗
= 1 : e−2(a†−α∗)(a−α): ,
In the extending Hilbert space the coherent thermo state |ξ is defined as[5]
|ξ = exp − 1 |ξ|2 + ξa† + ξ∗a† − a†a† |0, 0 , (5) 2
where ξ = ξ1 +iξ2 is a complex number. We can prove
PACS: 03. 65. −w, 03. 65. Ca
It is well known that Wigner functions are quasiprobability distributions, which describe states of quantum systems in phase space. The Wigner function[1,2] plays an important role in studying quantum statistics and quantum optics. The partial negativity of the Wigner function is indeed a good indication of the highly nonclassical character of the state. Therefore it is useful to obtain the Wigner function for any states and to measure their negative values.
CHIN.PHYS.LETT.
Vol. 25, No. 8 (2008) 2762
Wigner Function of Thermo-Invariant Coherent State ∗
XU Xue-Fen(
)1,2, ZHU Shi-Qun(
)1∗∗
1School of Physical Science and Technology, Suzhou University, Suzhou 215006 2School of Mathematics and Physics, Jiangsu Teachers University of Technology, Changzhou 213001
20050285002, and the National Natural Science Foundation of China under Grant No 10774108. ∗∗To whom correspondence should be addressed. Email: szhu@suda.edu.cn c 2008 Chinese Physical Society and IOP Publishing Ltd
(Received 12 May 2008)
By using the thermal Winger operator of thermo-field dynamics in the coherent thermal state |ξ representation and the technique of integration within an ordered product of operators, the Wigner function of the thermoinvariant coherent state |z, N is derived. The nonclassical properties of state |z, N is discussed based on the negativity of the Wigner function.
lation of the state |ξ can be proven, that is,
d2ξ
d2ξ
|ξ ξ| =
π
π
:
e−|ξ|2
+(a†
+a)ξ+(a† ee
+a)ξ∗
−a† e
a†
−aa−a† e
a−ea†ea:
=
1,
(8)
where the symbol : :denotes normal ordering. Further, from
∆T (σ, γ) =
d2ξ π3 | − ξ + γ
ξ + γ| exp(−ξσ∗ + ξ∗σ), (14)
To calculate Wigner function of any thermo states, it would be convenient to constructing the coherent thermal state and the thermo Wigner operator in the context of TFD.
is accompanied by a tilde state |n in H. A similar rule holds for operator: every annihilation operator a acting on the original Hilbert space H has an image a acting on the H. By analysing the characters of TFD, we have introduced the thermo-invariant coherent state |z, N in thermal equilibrium. The physical meaning of state |z, N is: annihilating a quantum of the system and meanwhile annihilating a hole with negative energy in the reservoir will not change the energy of the whole system in thermal equilibrium. The thermo-invariant coherent state is the common eigenvector of the pair annihilation operator aa and the ‘total’ energy operator h = a†a − a†a, i.e.
space should be doubled, a fictitious Hilbert space H (tilde-conjugate field) should be introduced, i.e. the original optical field state |n in the Hilbert space H
No. 8
XU Xue-Fen et al.
2763
one can see the orthonormal property of the state |ξ ,
ξ|.ξ = πδ(ξ − ξ )δ(ξ∗ − ξ ∗) ≡ πδ(2)(ξ − ξ ). (10)
Recall that for one-dimensional case, the usual Wigner function is defined as
aa|z, N = z|z, N ,
(1)
h|z, N = N|z, N ,
(2)
where z is complex, N is an integer, [a, a†] = 1, [a, a†] = 1 and [a, a] = [a, a†] = [a†, a†] = 0.
Let us first briefly recall the expression of the thermo-invariant coherent state |z, N ,
π
√
α = (x + ip)/ 2,
(13)
where |α = exp − 1 |α|2 + αa+ |0 is the coherent 2
state[7]. Similar in form to Eq. (13) the thermo Wigner op-
erator is expressed in the ξ| representation as
where JN(x) is the ordinary N-order Bessel function. In this study, for deriving the Wigner function
of |z, N , the coherent thermo state |ξ and thermo Wigner operator are introduced, which will bring convenience for the derivation. By using the thermo Wigner operator, we derive the compact expression of the Wigner function of the state |z, N . The nonclassical properties of state |z, N is discussed based on the negativity of the Wigner function.
∞
zn
|z, N = CN
|n + N, n , (3)
n=0 n!(N + n)!
where z is a complex number, N is an integer. The
normalization constant CN is
CN = (i|z|)−NJN(2i|z|) −1/2,
(4)
(a + a†)|ξ = ξ|ξ , (a† + a)|ξ = ξ∗|ξ , (9)
∗Supported by the Specialized Research Fund for the Doctoral Programme of Higher Education of China under Grant No
√
√
(Q + Q)|ξ = 2ξ1|ξ , (P − P )|ξ = 2ξ2|ξ , (6)
where
Q = 1/2(a + a†), P = i 1/2(a† − a),
Q = 1/2(a + a†), P = i 1/2(a† − a), (7)
so |ξ is the common eigenstate of Q + Q and P − P . Using the technique of integration within an ordered product (IWOP) of operators[6], the completeness re-
W (x, p) = Tr [ρ∆(x, p)]
1 =
∞
v
vwenku.baidu.com
x+ ρ x−
e−ipv dv,
(11)
2π −∞
2
2
where ρ is a density operator, ∆(x, p) is the usual single-mode Wigner operator, |x is the eigenvector of the coordinate operator X. In the coordinates representation ∆(x, p) takes the form
1 ∆(x, p) =
∞
v
x−
x + v e−ipvdv,
(12)
2π −∞
2
2
while in the coherent state representation the Wigner operator is
∆(x, p) → ∆(α) =
d2z π2 |α + z
α − z|e−αz∗−zα∗
= 1 : e−2(a†−α∗)(a−α): ,
In the extending Hilbert space the coherent thermo state |ξ is defined as[5]
|ξ = exp − 1 |ξ|2 + ξa† + ξ∗a† − a†a† |0, 0 , (5) 2
where ξ = ξ1 +iξ2 is a complex number. We can prove
PACS: 03. 65. −w, 03. 65. Ca
It is well known that Wigner functions are quasiprobability distributions, which describe states of quantum systems in phase space. The Wigner function[1,2] plays an important role in studying quantum statistics and quantum optics. The partial negativity of the Wigner function is indeed a good indication of the highly nonclassical character of the state. Therefore it is useful to obtain the Wigner function for any states and to measure their negative values.
CHIN.PHYS.LETT.
Vol. 25, No. 8 (2008) 2762
Wigner Function of Thermo-Invariant Coherent State ∗
XU Xue-Fen(
)1,2, ZHU Shi-Qun(
)1∗∗
1School of Physical Science and Technology, Suzhou University, Suzhou 215006 2School of Mathematics and Physics, Jiangsu Teachers University of Technology, Changzhou 213001
20050285002, and the National Natural Science Foundation of China under Grant No 10774108. ∗∗To whom correspondence should be addressed. Email: szhu@suda.edu.cn c 2008 Chinese Physical Society and IOP Publishing Ltd
(Received 12 May 2008)
By using the thermal Winger operator of thermo-field dynamics in the coherent thermal state |ξ representation and the technique of integration within an ordered product of operators, the Wigner function of the thermoinvariant coherent state |z, N is derived. The nonclassical properties of state |z, N is discussed based on the negativity of the Wigner function.
lation of the state |ξ can be proven, that is,
d2ξ
d2ξ
|ξ ξ| =
π
π
:
e−|ξ|2
+(a†
+a)ξ+(a† ee
+a)ξ∗
−a† e
a†
−aa−a† e
a−ea†ea:
=
1,
(8)
where the symbol : :denotes normal ordering. Further, from
∆T (σ, γ) =
d2ξ π3 | − ξ + γ
ξ + γ| exp(−ξσ∗ + ξ∗σ), (14)
To calculate Wigner function of any thermo states, it would be convenient to constructing the coherent thermal state and the thermo Wigner operator in the context of TFD.
is accompanied by a tilde state |n in H. A similar rule holds for operator: every annihilation operator a acting on the original Hilbert space H has an image a acting on the H. By analysing the characters of TFD, we have introduced the thermo-invariant coherent state |z, N in thermal equilibrium. The physical meaning of state |z, N is: annihilating a quantum of the system and meanwhile annihilating a hole with negative energy in the reservoir will not change the energy of the whole system in thermal equilibrium. The thermo-invariant coherent state is the common eigenvector of the pair annihilation operator aa and the ‘total’ energy operator h = a†a − a†a, i.e.
space should be doubled, a fictitious Hilbert space H (tilde-conjugate field) should be introduced, i.e. the original optical field state |n in the Hilbert space H
No. 8
XU Xue-Fen et al.
2763
one can see the orthonormal property of the state |ξ ,
ξ|.ξ = πδ(ξ − ξ )δ(ξ∗ − ξ ∗) ≡ πδ(2)(ξ − ξ ). (10)
Recall that for one-dimensional case, the usual Wigner function is defined as
aa|z, N = z|z, N ,
(1)
h|z, N = N|z, N ,
(2)
where z is complex, N is an integer, [a, a†] = 1, [a, a†] = 1 and [a, a] = [a, a†] = [a†, a†] = 0.
Let us first briefly recall the expression of the thermo-invariant coherent state |z, N ,
π
√
α = (x + ip)/ 2,
(13)
where |α = exp − 1 |α|2 + αa+ |0 is the coherent 2
state[7]. Similar in form to Eq. (13) the thermo Wigner op-
erator is expressed in the ξ| representation as
where JN(x) is the ordinary N-order Bessel function. In this study, for deriving the Wigner function
of |z, N , the coherent thermo state |ξ and thermo Wigner operator are introduced, which will bring convenience for the derivation. By using the thermo Wigner operator, we derive the compact expression of the Wigner function of the state |z, N . The nonclassical properties of state |z, N is discussed based on the negativity of the Wigner function.
∞
zn
|z, N = CN
|n + N, n , (3)
n=0 n!(N + n)!
where z is a complex number, N is an integer. The
normalization constant CN is
CN = (i|z|)−NJN(2i|z|) −1/2,
(4)