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Berry phase of two atoms in the same quantized light field.

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Ann. Phys. (Leipzig) 16, No. 5 – 6, 342 – 349 (2007) / DOI 10.1002/andp.200610238
Berry phase of two atoms in the same quantized light field
Mai-Lin Liang∗ , Jin-Hua Zhang, and Bing Yuan
Physics Department, School of Science, Tianjin University, Tianjin 300072, China
Received 9 August 2006, revised 11 February 2007, accepted 13 February 2007 by U. Eckern
Published online 8 March 2007
Key words Berry phase, quantized light field, Jaynes-Cummings model.
PACS 03.65.Vf
The Berry phase of two atoms exposed to the same quantized light field has been studied. The interaction
between each atom and the light field obeys the Jaynes-Cummings model. Due to the quantization of the
light field, the interaction Hamiltonians for the two atoms interacting with the field do not commute with
each other, which induces new parts in the Berry phase. Meanwhile, there appears a phase which doesn’t
exist in the classical regime and is purely induced by the field quantization.
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1 Introduction
For any wave the phase is very important. The concept of geometric phase was first introduced by Pancharatnam [1] in his pioneering work, comparing the phases of two beams of polarized light. The discovery
of the quantum counterpart of the geometric phase by Berry in 1984 [2] strongly motivated the research on
the phases of wave functions in quantum mechanics. For a quantum system with
parame time-dependent
ters (R1 (t), R2 (t), . . .) ≡ R(t), the instantaneous eigenstate can be written as n, R(t) with n being the
quantum number. When the time-dependent parameters change slowly but periodically, the quantum state
evolves gradually and eventually returns to its initial form. The
state acquires a quantum adiabatic geomet ∂ rical phase, which is now known as the Berry phase, γ = i
n, R n, R · dR, in addition to the usual
∂R
dynamic phase. For a spin rotating in a time-dependent magnetic field, the Berry phase turns out to be the
solid angle in the parameter space. Since the work of Berry [2], there have been many studies on this quantum geometrical phase [3–5]. Theoretically, the original adiabatic and periodic evolution [2] is generalized
to, for instance, non-adiabatic evolution [6], non-adiabatic and non-cyclic evolution [7], mixed states [8,9],
open systems [10], and even with a quantized field driving [11–13]. Meanwhile, various kinds of practical
models are studied, which include the spins in a time-dependent magnetic field [2,14], the time-dependent
harmonic oscillator [15], the spin-orbit interaction [16–18], the composite systems [19–21], etc. Sjöqvist
calculated the geometric phases for a pair of entangled spins [19], elucidating how the entanglements sway
the Berry phase. Tong et al. discussed the relation between the geometric phase of the system and that
of the subsystems [20, 21]. Studies also show the importance of geometric phases in the areas of physics,
such as the Bose-Einstein condensates [22], the Jahn-Teller effect [23], the understanding of the anomaly
phenomenon in quantum field theory [24], and in many interesting topics of quantum optics [25–29]. The
geometric phases have been tested in a numerous experiments [30]. A feasible experiment to measure the
effects induced by the field quantization is proposed in [12], which may be carried out in the near future.
Besides the theoretical interest, the geometrical phase has potential applications, among which the
geometric quantum computation [31–36] is one of the most importance. The basis of geometric quantum
computation is the phase shift of quantum states [35] which is particularly attractive because of the fact
∗
Corresponding author E-mail: mailinliang@yahoo.com.cn, mailinliang@eyou.com
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Ann. Phys. (Leipzig) 16, No. 5 – 6 (2007)
343
that geometric phases depend only upon the global features and it is robust against certain errors and
dephasing [35,36]. One candidate for quantum computation is to use the neutral atoms [37,38]. The atoms
can be loaded into the optical lattices [39,40] and one can use external light fields to manipulate the atoms.
So, the study on the geometric phases of atoms under the quantized light fields [11–13] is meaningful
both theoretically and practically. On the other hand, the atom-field interaction has been a hot topic in
quantum optics [41–46]. The dynamical properties [41–43] and the entanglements [44–46] have been
widely investigated. The studies of the geometric phase make us gain new insights into such atom-field
interaction systems.
In [11], a fully quantized version of the phase for the system of an atom interacting with a quantized light
field has been given that considers the vacuum-induced effects. Under the rotating wave approximation, the
Hamiltonian is [11, 47]
H = ωsz + ωf a+ a + λ as+ + s− a+
(1.1)
where sz = (|e e| − |g g|)/2, s+ = |e g| , s− = |g e| with |g and |e being the ground and
excited states of the two level atom respectively, λ is the atom-light coupling coefficient, a+ and a are the
creation and annihilation operators for the light field. Defining sx = (s+ + s− )/2, sy = (s+ − s− )/(2i),
the operators sx , sy and sz are named the pseudo spin operators for the atom. The Hamiltonian (1.1) can
be split into two parts, H = H0 + Hint :
H0 = (ω − ∆)sz + ωf a+ a ,
Hint = ∆sz + λ as+ + s− a+
(1.2)
where ∆ = ω − ωf is the detuning between the atom and the field. One can prove that H0 and Hint
commutate with each other. So, in the interaction picture, the interaction Hamiltonian is just Hint . The state
vector obeysHint |ψ = i∂ |ψ /∂t. The stationary equation is Hint |ψ = ε |ψ, where ε is the eigenvalue
of Hint . The two eigenvales are found to be
ε± = ± 12 ∆2 + 4λ2 (n + 1)
(1.3)
where n is the photon number. The corresponding Berry phases are [11]
∆
α+ = 2πn + π 1 − ,
∆2 + 4λ2 (n + 1)
∆
α− = 2πn + π 1 + .
∆2 + 4λ2 (n + 1)
(1.4)
When n = 0, the Berry phase is nonzero, which is the vacuum-induced phase.
In this paper, we deal with the case that two atoms are exposed to the same quantized light field. The
interaction between each atom and the light field obeys the Jaynes-Cummings model [47]. The Hamiltonian
for the whole system is
(1)
(2)
+
+ (1)
H (12) = ωs(1)
+
ωs
+
ω
a
a
+
λ
as
+
a
s
f
z
z
+
−
(2)
(2)
(1) (2)
(1) (2)
+ λ as+ + a+ s− + β s+ s− + s− s+
(1.5)
where β is the atom-atom dipole coupling constant. The two terms with λ are the interaction Hamiltonians
of atom 1 with the field and atom 2 with the field respectively. The two atom-field interaction Hamiltonians
for the two atoms in (1.5) don’t commute with each other and represent a case different from the former
studies [11–13], where only one atom exists or two atoms are in different quantized light fields. It’s interesting
to see what new results there will be for the Hamiltonian (1.5). The next section is the calculation of the
exact Berry phase and the final section is the conclusion.
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c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
344
Mai-Lin Liang et al.: Berry phase of two atoms in the same quantized light field
2 The Berry phase
The Hamiltonian (1.5) for two atoms in a quantized light field can be written as the sum of the following
two parts
(12)
(2)
H0 = (ω − ∆) s(1)
+ ωf a+ a ,
z + sz
(12)
(1)
(1)
(2)
Hint = ∆ s(1)
+ λ as+ + a+ s−
(2.1)
z + sz
(2)
(2)
(1) (2)
(1) (2)
+ λ as+ + a+ s− + β s+ s− + s− s+ .
(12)
(12)
One can prove that H0 and Hint communicate with each other. So, in interaction picture, the interaction
(12)
(12)
(12)
Hamiltonian is Hint . The wave function obeys the evolution equation Hint |ψ = i∂ |ψ /∂t. As Hint
(12)
(12)
is time-independent, the stationary equation is then Hint |ψ = ε |ψ, where ε is the eigenvalue of Hint .
By some analysis, it’s found that the eigenfunction can be written as the superposition
|ψ = C1 |ee, n + C2 |eg, n + 1 + C3 |ge, n + 1 + C4 |gg, n + 2 .
(2.2)
The coefficients Cµ , µ = 1, 2, 3, 4 are to be found below. Substituting (2.2) into the eigenvalue equation
(12)
Hint |ψ = ε |ψ, we have

∆,
 √
λ n + 1,
 √
λ n + 1,
0,
√
λ n + 1,
0,
β,
√
λ n + 2,
 
 
√
C1
λ n + 1,
0
C1
√
 
 
β,
λ n + 2 C2 
C2 
√
  = ε 
C3 
0,
λ n + 2 C3 
√
C4
C4
λ n + 2,
−∆
(2.3)
which can be solved analytically. There are totally four eigenvalues. One eigenvalue is ε1 = −β and the
coefficients are C1 = C4 = 0, C2 = −C3 = 1. The corresponding eigenfunction takes the form
1
|ψ1 = √ (|eg − |ge) |n + 1 .
2
(2.4)
The other three eigenvalues satisfy
ε3 − βε2 − aε + c = 0
(2.5)
where
a = ∆2 + 2λ2 (2n + 3) ,
c = 2λ2 ∆ + g∆2 .
(2.6)
The exact solution to (2.5) is
ϕ
β
,
+ 2R cos
3
3
ϕ
π
β
ε3 = − 2R cos
+
,
3
3
3
ϕ
β
π
ε4 = − 2R cos
−
3
3
3
ε2 =
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
(2.7)
www.ann-phys.org
Ann. Phys. (Leipzig) 16, No. 5 – 6 (2007)
where
R=
1
3
345
3a + β 2 ,
ϕ = cos−1
9βa + 2β 3 − 27(2λ2 ∆ + g∆2 )
.
2(3a + β 2 )3/2
(2.8)
Substituting the eigenvalues (2.7) into (2.3), the coefficients and so the eigenfunction (2.2) are found to be
√
√
2λ
n
+
1
n
+
2
1
2λ
ψj = √
|gg, n + 2 ,
|ee, n + |eg, n + 1 + |ge, n + 1 +
εj − ∆
∆ + εj
N
(2.9)
4λ2 (n + 1)
4λ2 (n + 2)
N =2+
+
(∆ − εj )2
(∆ + εj )2
where j = 2, 3, 4. Using (2.9), the Berry phase can be calculated. In the full quantized regime, the geometrical phase is [11–13]
+
d
ψj U (φ) U (φ) ψj dφ
(2.10)
γj = i
dφ
c
where U (φ) = exp(−iφa+ a)is a phase shift operator and φ changes slowly from 0 to2π. For the state (2.4),
the Berry phase is 2(n + 1)π or zero (mod 2π). For the state (2.9), the Berry phase is not difficult to get
γj = 2π(n + 1)
(2.11)
−16λ2 εj n∆ + 4λ2 2(∆ − εj )2 − (∆ + εj )2
.
+ 2π
2
2
2
2
2
2
(∆ + εj ) (∆ − εj ) + 4λ (n + 1) + (∆ − εj ) (∆ + εj ) + 4λ (n + 2)
Though this result is general, in the following we mainly focus our attention on the case that there is no
the dipole-dipole interaction orβ = 0 to consider the pure effects of field quantization. Classically, if there
is no the dipole-dipole interaction orβ = 0, the Berry phase of the two-atom system will be two times
that of the one-atom system. From (2.11) and (1.4), one sees that when the effect of the field quantization
is included, there does not exist such a relation. Numerical results also show the same conclusion. The
difference between the Berry phase of the two-atom system and two times that of the one-atom system is
plotted in Fig. 1 (For all the figures in this article, the vertical axis is the Berry phase and the longitudinal
axis is the photon number).
Figs. 1a and b correspond to the states with ε2 and ε4 respectively. At finite n, the difference is obvious.
In the classical limit n → ∞, the difference tends to zero, which means that now the Berry phase of the
two-atom system is two times that of the one-atom system. Such a result can be derived analytically too.
Setting n → ∞, we have
√
√
ε3 = 0 ,
ε4 = − a .
(2.12)
ε2 = a ,
The corresponding Berry phases are γ3 = 0 and
∆
,
γ2 = 2πn + 2π 1 − √
∆2 + 4λ2 n
∆
γ4 = 2πn + 2π 1 + √
.
∆2 + 4λ2 n
(2.13)
Thus, in the limit n → ∞, Berry phases in (2.13) are two times that in (1.4) (mod 2π). The Berry phase
γ3 as a function of photon number is plotted in Fig. 2 (mod 2π), which becomes zero in the classical limit.
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346
Mai-Lin Liang et al.: Berry phase of two atoms in the same quantized light field
5
10
15
20
25
30
-0.25
-0.5
-0.75
-1
-1.25
-1.5
-1.75
-2
a
4
3.5
3
2.5
2
1.5
1
Fig. 1 Differences between the Berry
phase of the two-atom system and two times
that of the one-atom system. The parameters
are ∆ = 0.5 , λ = 1.0 .
0.5
b
5
10
15
20
25
30
0.5
0.4
0.3
0.2
0.1
5
10
15
20
25
30
Fig. 2 Berry phase of the state with eigenvalue ε3 . The parameters are ∆ = 0.5 ,
λ = 1.0 .
That the phase γ3 is zero in the classical case means that there is no such a phase classically, or this phase
is generated purely by the field quantization.
Now let’s have a look at the changes of Berry phase with the dipole-dipole coupling constant. In [13], all
the Berry phases for the eigenstates go to zero in the limit β → ∞. For the present system, it is different.
In the limit β → ∞, it is found that ε2 = β and ε3 = ε4 = 0. For ε2 = β, the state (2.9) becomes
ψj → √1 [|eg, n + 1 + |ge, n + 1] = √1 [|eg + |ge] |n + 1
2
2
c 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
(2.14)
www.ann-phys.org
Ann. Phys. (Leipzig) 16, No. 5 – 6 (2007)
347
and the Berry phase is zero. For ε3 = ε4 = 0, the eigenfunction (2.9) becomes
√
√
2λ n + 1
2λ n + 2
ψj = √1
|ee, n + |eg, n + 1 + |ge, n + 1 +
|gg, n + 2 ,
−∆
∆
N
(2.15)
4λ2 (n + 1)
4λ2 (n + 2)
N =2+
+
.
∆2
∆2
The Berry phase is
γj = 2πn + 2π
∆2 + 4λ2 (n + 2)
∆2 + 2λ2 (2n + 3)
(2.16)
which is a finite value and can’t be written as 2πm(m integer) or zero. According to the definition, whether
the Berry phase of a state is zero or not is determined by the form of the eigenfunction. For the toy model
in [13], the states reduce to |ee, n, |eg, n + 1, |ge, n + 1, |gg, n + 2, and all the Berry phases for the
eigenstates are zero. Actually, for the Ising-type interaction in [13], not all the Berry phases are zero either
in the limit β → ∞ if the two atoms are both exposed to the same quantized field. Replacing the atom-atom
interaction in (1.5) by βσ1z σ2z , the equation (2.3) becomes
 

 
√
√
C1
∆ + β,
λ n + 1, λ n + 1,
0
C1
√
 
 √
 
λ
n
+
1,
−β,
0,
λ
C
C
n
+
2
 2

  2
(2.17)
√
 √
  = ε  .
C3 
λ n + 1,
0,
−β,
λ n + 2 C3 
√
√
0,
λ n + 2, λ n + 2, −∆ + β
C4
C4
One of the eigenvalues is ε1 = −β and the eigenfunction is the same as (2.4). The corresponding Berry
phase is zero. The other three eigenvalues satisfy
ε3 − βε2 − β 2 + ∆2 + 2λ2 (2n + 3) ε + β 3 − β∆2 + 4nβλ2 + 2λ2 ∆ = 0 .
(2.18)
The corresponding eigenfunctions can be written as
√
√
2λ n + 2
2λ n + 1
ψj = √1
|ee, n + |eg, n + 1 + |ge, n + 1 +
|gg, n + 2 ,
∆ + εj − β
N εj − β − ∆
4λ2 (n + 1)
4λ2 (n + 2)
N =2+
+
.
(∆ + β − εj )2
(∆ + εj − β)2
(2.19)
2
In the limit β → ∞, Eq. (2.18) reduces to (ε + β) (ε − β) = 0, solutions of which are ε2 = ε3 = β and
ε4 = −β. For ε4 = −β, the state (2.19) tends to (2.14) and the Berry phase is 2π(n + 1) or zero. But for
ε2 = ε3 = β, the state (2.19) becomes
√
√
2λ
n
+
1
n
+
2
1
2λ
ψ j = √
|ee, n + |eg, n + 1 + |ge, n + 1 +
|gg, n + 2 ,
−∆
∆
N
(2.20)
4λ2 (n + 2)
4λ2 (n + 1)
+
.
N =2+
∆2
∆2
The Berry phase is the same as (2.16) and is nonzero.
One may notice that, in the strong coupling limit β → ∞, the parametersλ, ∆disappear from the energy
eigenvalues, but still appear in the eigenfunctions. This phenomenon may be understood as follows. In
the total energy, different forms of energy are summed up. If one part is large enough, other parts will
be negligible. But for the eigenstates, different interactions stand for different actions on the states. One
interaction can’t replace all other ones, even if this interaction is very strong.
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348
Mai-Lin Liang et al.: Berry phase of two atoms in the same quantized light field
3 Conclusions
New results for the Berry phase of two atoms interacting with the same quantized light field are presented.
The atom-field interaction Hamiltonians for the two atoms don’t commute with each other, which induces
new phenomena for the geometric phases: (1) The geometric phase for the two-atom system is not two times
that of the one-atom system even if there is no atom-atom interaction. But, in the large quantum limit, the
geometric phase for the two-atom system is two times that of the one-atom system, which is just the classical
case. (2) When the atom-atom coupling is infinite, the geometric phase for the eigenstate |ψ2 is not zero,
which is different from the results in [13], where the geometric phase for each eigenstate is zero. In [13],
the atom-field interaction Hamiltonians for the two atoms commute with each other. The above calculations
(2.17) to (2.20) show that, even if for the toy model in [13], not all the geometric phases for the eigenstates
are zero when the two atoms are in the same quantized light field. (3) For the state with the eigenvalue ε3
in (2.7), the corresponding geometric phase γ3 vanishes in the classical limit. Or, the phase γ3 is purely
induced by quantum effects. So, non-commutation between the two atom-field interaction Hamiltonians in
(1.5) has important effects on the geometric phase.
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