Scheme to measure squeezing and phase properties of a harmonic oscillator

0.09 MB·English

by  G. T. Rubin-Linares

Scheme to measure squeezing and phase properties of a harmonic
oscillator
G.T. Rub´ın-Linares and H. Moya-Cessa
INAOE, Coordinaci´on de Optica, Apdo. Postal 51 y 216, 72000 Puebla, Pue., Mexico
Abstract
5 We proposea simplescheme to measure squeezing and phaseproperties of a harmonic oscillator.
0
0
We treat in particular the case of a the field, but the scheme may be easily realized in ion traps.
2

n It is based on integral transforms of measured atomic properties as atoms exit a cavity. We show
a
J that by measuring atomic polarizations it is possible, after a given integration, to measure several

6
2 properties of the field.

1
v
PACS numbers: 42.50.-p,42.65.Ky, 03.65.-w
8
5
1
1
0
5
0
/
h
p
-
t
n
a
u
q
:
v
i
X
r
a
1
The reconstruction of a quantum state is a central topic in quantum optics and related
fields [1, 2]. During the past years, several techniques have been developed to achieve such
quantum measurements, many of them having as a basic tool quantum tomography. Clas-
sical tomography is a method for building up a picture of a hidden object using various
observations from different angles. In quantum optics, tomography has been applied ex-
perimentally to reconstruct the quantum state of light from a complete set of measured
quantities, examples may be the direct sampling of the density matrix of a signal mode in
multiport optical homodyne tomography [3] and tomographic reconstruction by unbalanced
homodyning [4]. There have also been proposals to measure electromagnetic fields inside
cavities [5, 6, 7] which may be achieved through a finite set of selective measurements of
atomic states [5, 6] that make it possible to reconstruct quasiprobability distribution func-
tionssuchastheWignerfunction, thatconstituteanalternativerepresentation ofaquantum
state of the field.
Recently it was proposed a method to measure the Wigner function of a quantized field
in a cavity by using a Fresnel transformation of the atomic inversion of atoms traversing the
cavity [5]. To measure the Wigner function is a difficult task mainly because of the number
of times the field has to be prepared: to cover the complete phase space one has to displace
(in the ideal case, continuously) the initial state, i.e. the initial field state has to be prepared
a number of times. However it may be that we need information only about some features
of the field, lt us say, squeezing properties, phase properties, etc. Here we will show how
squeezing properties and phase properties may be measured by transforming data obtained
after atoms pass through the cavity and measuring them.
A. Measuring squeezing
To measure squeezing, we need to be able to measure quantities like
Xˆ = aˆ +c.c., Xˆ2 = aˆ2 + [aˆ ]2 +2 nˆ +1 (1)
†
h i h i h i h i h i h i
where aˆ and aˆ are the creation and annihilation operators for the field mode, respectively,
†
obeying [aˆ,aˆ ] = 1. Below we will show how it is possible to measure quantities like aˆk ,k =
†
h i
1,2 by utilizing an atom as a measuring device. We start by writing the Hamiltonian of
the two-level atom field resonant interaction in the rotating wave approximation and the
2
interaction picture
0 aˆ
Hˆ = λ(aˆ σˆ +σˆ aˆ) = λ (2)
† +
− aˆ 0 
†
 
where σˆ and σˆ are the raising and lowering atomic operators, respectively. The atomic
+
−
operators obey the commutation relation [σˆ ,σˆ ] = σˆ .
+ z
−
We can re-write Hamiltonian (2) with the help of Susskind-Glogower operators [8] as
0 √nˆ +1
Hˆ = λTˆ Tˆ (3)
†
√nˆ +1 0 
 
where
1 0
Tˆ = (4)
0 Vˆ 
†
 
where nˆ = aˆ aˆ and Vˆ = 1 aˆ. Note that Tˆ Tˆ = 1 but TˆTˆ = 1. This allows us to write
† √nˆ+1 † † 6
the evolution operator as
cos(λt√nˆ +1) isin(λt√nˆ +1)
Uˆ(t) = Tˆ − Tˆ (5)
†
 isin(λt√nˆ +1) cos(λt√nˆ +1) 
−
 
We are neglecting a term 0 0 in the above evolution operator (in the element 22), that
| ih |
however will not affect the measurement of squeezing as we will consider the atom in the
excited state. We consider the field in an unknown state, such that the initial state of the
system is ψ(0) = e ψ (0) , the average of the operator σˆ is given by
F +
| i | i| i
σˆ = i ψ (0) cos(λt√nˆ +1)Vˆ sin(λt√nˆ +1) ψ (0) (6)
+ F † F
h i − h | | i
i
= ψ (0) Vˆ sin[λt∆ˆ (nˆ)] sin[λt∆ˆ (nˆ)] ψ (0)
F † + F
−2h | − − | i
(cid:16) (cid:17)
where
∆ˆ (nˆ) = √nˆ +2+√nˆ +1, ∆ˆ (nˆ) = √nˆ +2 √nˆ +1. (7)
+
− −
By integrating (7) by using a Fresnel integral [5, 9]:
AB πA AB2 AB2
∞dTT sin(T2/A)sin(BT) = cos +sin (8)
4 2 4 4
Z0 r (cid:18) (cid:19)
such that (with λt = T)
i
∞dTT sin(T2/A) σˆ = ψ (0) Vˆ (γˆ γˆ ) ψ (0) (9)
+ F † 1 2 F
h i −2h | − | i
Z0
3
with
A∆ˆ (nˆ) πA A∆ˆ2(nˆ) A∆ˆ2(nˆ)
γˆ(1) = + cos + +sin + (10)
1 4 2 4 4
r " # " #!
and
A∆ˆ (nˆ) πA A∆ˆ2(nˆ) A∆ˆ2(nˆ)
(1)
γˆ2 = 4− 2 cos 4− +sin 4− . (11)
r " # " #!
Now we use the approximation (nˆ +2)(nˆ +1) nˆ + 3/2 that is valid for large photon
≈
numbers (see for instance [10]). pWe then can write ∆ˆ2(nˆ) 4nˆ + 6 and ∆ˆ2(nˆ) 0. By
+ ≈ − ≈
setting A = 4π then we obtain
γˆ(1) (√nˆ +2+√nˆ +1)πcos[(4nˆ +6)π] = √2π2∆ˆ (nˆ) (12)
1 ≈ +
and
γˆ(1) √2π2∆ˆ (nˆ), (13)
2 ≈ −
so that the integral transform (9) becomes
∞dTT sin(T2/A) σˆ = i√2π2 ψ (0) Vˆ √nˆ +1 ψ (0) (14)
+ F † F
h i − h | | i
Z0
= i√2π2 ψ (0) aˆ ψ (0) .
F † F
− h | | i
To measure ψ (0) [aˆ ]2 ψ (0) it is necessary a two-photon transition. In this case,
F † F
h | | i
0 (nˆ +1)(nˆ +2)
Hˆ = λ(2)Tˆ2 [Tˆ ]2 (15)
†
 
(nˆ +1)(nˆ +2) p 0
 
where λ(2) is the interaction copnstant in the two-photon case. One can find the evolu-
tion operator that will be given by an expression similar to (5), just changing √nˆ +1
→
(nˆ +1)(nˆ +2), Vˆ Vˆ2 and Vˆ [Vˆ ]2. It is then easy to calculate the average of σˆ(2),
→ † → † +
pwhich is given by
σˆ(2) = i ψ (0) cos λ(2)t (nˆ +1)(nˆ +2) [Vˆ ]2sin λ(2)t (nˆ +1)(nˆ +2) ψ (0)
h + i − h F | † | F i
= i ψ (0) [Vˆ h]2 sinp[λtδˆ (nˆ)] sin[λitδˆ (nˆ)] hψ (0)p i (16)
F † + F
−2h | − − | i
(cid:16) (cid:17)
with
ˆ
δ (nˆ) = (nˆ +4)(nˆ +3)+ (nˆ +2)(nˆ +1) 2nˆ +5, (17)
+
≈
p p
and
ˆ
δ (nˆ) = (nˆ +4)(nˆ +3) (nˆ +2)(nˆ +1) 2. (18)
− − ≈
p p
4
Again by (Fresnel) integration of the above expression
∞dTT sin(T2/A) σˆ(2) = i4π2 ψ (0) [Vˆ ]2(γˆ(2) γˆ(2)) ψ (0) (19)
h + i − h F | † 1 − 2 | F i
Z0
with
Aδˆ (nˆ) πA Aδˆ2(nˆ) Aδˆ2(nˆ)
γˆ(2) = + cos + +sin + (20)
1 4 2 4 4
r " # " #!
and
Aδˆ (nˆ) πA Aδˆ2(nˆ) Aδˆ2(nˆ)
(2)
γˆ2 = −4 2 cos −4 +sin −4 . (21)
r " # " #!
By choosing the value A = 8π we obtain
∞dTT sin(T2/8π) σˆ(2) = i8π4 ψ (0) [Vˆ ]2 (nˆ +1)(nˆ +1) ψ (0)
h + i − h F | † | F i
Z0
= i8π4 [aˆ ]2 p (22)
†
− h i
B. Measuring phase properties
Now we turn our attention to the phase properties of the field. The procedure, although
similar to the way of obtaining the quadratures of the field, will differ in the integral forms
that will be used. We compute now the average of σˆ for the one-photon transition for an
+
atom in the ground state and the arbitrary field ψ (0) , that reads
F
| i
σˆ = i ψ (0) Vˆ sin(λt√nˆ +1)Vˆ cos(λt√nˆ +1)Vˆ ψ (0) (23)
+ F † † F
h i h | | i
that for a field with large number of photons may be approximated by
σˆ = i ψ (0) Vˆ sin(λt√nˆ +1)cos(λt√nˆ)(1 0 0 ) ψ (0) (24)
+ F † F
h i h | −| ih | | i
= i ψ (0) Vˆ sin(λt√nˆ +1)cos(λt√nˆ) ψ (0)
F † F
h | | i
Using the integral [9]
sin(AT)cos(BT)
∞
dT = π/2, A > B > 0 (25)
T
Z0
we can integrate (25) as:
σˆ iπ
∞ h +id(λt) = ψ (0) Vˆ ψ (0) . (26)
F † F
λt 2 h | | i
Z0
5
To measure [Vˆ ]2 , again a two-photon process is needed. The average value of σˆ with
† +
h i h i
the atom entering in the ground state is
σˆ(2) = i ψ (0) [Vˆ ]2sin λ(2)t (nˆ +1)(nˆ +2) [Vˆ ]2cos λ(2)t (nˆ +1)(nˆ +2) Vˆ2 ψ (0)
h + i h F | † † | F i
= i ψ (0) [Vˆ ]4sin(cid:16)λ(2)tp(nˆ +3)(nˆ +4)(cid:17)cos λ(2)t(cid:16) (nˆ +p1)(nˆ +2) Vˆ2 ψ(cid:17) (0) (27)
F † F
h | | i
(cid:16) p (cid:17) (cid:16) p (cid:17)
by performing the integral
(2)
σˆ iπ
∞ h + idT = ψ (0) [Vˆ ]4Vˆ2 ψ (0) (28)
F † F
T 2 h | | i
Z0
that for large intensity field approximates
(2)
σˆ iπ
∞ h + idT = ψ (0) [Vˆ ]2 ψ (0) (29)
F † F
T 2 h | | i
Z0
Finally, the average value of σˆ may be found by finding the average value of the ob-
+
servables σˆ and σˆ , with σˆ = σˆ + iσˆ . In order to find it, we write σˆ = Tr[σˆ ρˆ] =
x y + x y x x
h i
Tr[Rˆσˆ Rˆ ρˆ] = Tr[σˆ Rˆ ρˆRˆ] = Tr[σˆ ρˆ ], i.e. the expectation value of σˆ (the atomic in-
z † z † z R z
version or the probability of finding the atom in the excited state minus the probability
of finding it in the ground state) for a rotated (in the atomic basis) density matrix with
Rˆ = exp[(σˆ σˆ )π/4]. A similar procedure may be done for the expression σˆ . Exper-
+ y
− − h i
imentally we would need to send an atom through a cavity that contains the field under
study, then (properly) rotate it after it exits the cavity and measure its energy. We would
need an experimental setup as shown in Fig. 1.
FIG. 1: Experimental setup to measure σˆ or σˆ .
x y
h i h i
[1] K. Vogel and H. Risken, Phys. Rev. A 40, 2847 (1989).
[2] U.Leonhardt, Measuring the Quantum State of Light, (Cambridge, Cambridge University
Press) 1997.
[3] A. Zucchetti, W. Vogel, M. Tasche, and D.-G. Welsch, Phys. Rev. A 54, 1678 (1996).
[4] S. Wallentowitz and W. Vogel, Phys. Rev. A 53, 4528 (1996).
[5] P. Lougovsky, E. Solano, Z.M. Zhang, H. Walther, H. Mack, and P.W. Schleich, Phys. Rev.
Lett. 91, 010401(2003).
6
[6] L.G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997).
[7] H. Moya-Cessa, S.M. Dutra, J.A. Roversi, and A. Vidiella-Barranco, J. of Mod. Optics 46,
555 (1999); H. Moya-Cessa, J.A. Roversi, S.M. Dutra, and A. Vidiella-Barranco, Phys. Rev.
A 60, 4029 (1999).
[8] L. Susskind and J. Glogower, Physics 1, 49 (1964).
[9] I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, Academic Press,
Inc., London 1980).
[10] H. Moya-Cessa, S. Cha´vez-Cerda and W. Vogel, J. Mod. Optics 66(1998).
7