Download QUANTUM ENTANGLEMENT STATE OF NON

QUANTUM PHYSICS
QUANTUM ENTANGLEMENT STATE OF NON-EQUIDISTANT
THREE-LEVEL ATOM IN INTERACTION
WITH BIMODAL CAVITY FIELD
NICOLAE ENAKI, NELLU CIOBANU
Institute of Applied Physics, Academy of Sciences of Moldova,
Academiei 5, Chisinau MD-2028, Rep. of Moldova
E-mail: [email protected]
Received September 12, 2006
A new trapping effect of three-level atom in interaction with bimodal cavity
field is proposed. The problem consists in the possibility of realization of initial
separate state of atom and electromagnetic field after the flying time through the
cavity field. The quantum properties of bimodal field, which satisfy the reversible
conditions for the atom flying through the cavity, were studied.
Key words: Three-level atom, bimodal cavity field, quantum optics, quantum
informatics.
1. INTRODUCTION
Recently in quantum physics considerable attention is devoted to quantum
mixing of two subsystems which enter in resonance in the processes of interaction.
The quantum subsystems submit statistical principals of quantum mechanics, so
that separability and restoration become non-realizable with increasing the
numbers of degree of freedom. The realization of time separation of two
quantum subsystems is an attractive problem in modern physics. One of
examples is the reversible condition for two level atoms whose states are mixed
with cavity states of electromagnetic field during the flying time through the
cavity [1–4]. This problem consists in the possibility of realization of initial
separate state of atom and electromagnetic field after the flying time through the
cavity field. The exchange energy between the atom and cavity modes of two
subsystems and the special conditions for flying time and properties of cavity
electromagnetic field must be satisfied for realization of full separation of
subsystems with initial quantum properties. The reversible condition becomes
more complicated with increasing the degrees of freedom in one of the subsystems.

Paper presented at the 7th International Balkan Workshop on Applied Physics, 5–7 July
2006, Constanþa, Romania.
Rom. Journ. Phys., Vol. 52, Nos. 5– 7 , P. 545–558, Bucharest, 2007
546
Nicolae Enaki, Nellu Ciobanu
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For example, in the study of interaction of two undistinguished radiators in
resonance interaction in one cavity mode the problem of reversibility becomes
more complicated than in the case of two-level atom [5, 6].
From the physical point of view, it is interesting to realize the reversible
conditions between three-level atoms in cascade configuration in resonance with
two cavity modes. This system has more degrees of freedom in the atom
transitions description and behavior of bimodal cavity field in comparison with
the proposed effect in paper [2–12]. For experimental realization of such effects
in maser regime can be used the Rydberg atoms like Rb prepared in superposition
states relatively the cascade transitions 40 S1/ 2 ↔ 39 P3 / 2 ↔ 39 S1/ 2 [4]. The
electric dipole matrix elements in this cascade configuration are exceedingly
large. These transitions correspond to the millimeter wave length with cascade
transition frequencies νsp = 107.41587 GHz and νsp = 39.41587 GHz for which it
is necessary to realize the high-Q bimodal cavity efficiently coupled to the atomic
dipoles.
In this paper the recursion relation for amplitude of electromagnetic field in
decompositions on the Fock states was obtained. This reversible condition can
be released in the case of two undistinguished atoms flying through the cavity
simultaneously and contained with one degree of freedom more than two level
radiators.
2. SCHRÖDINGER EQUATION AND ITS SOLUTION
Let us consider a cavity with higher quality factor, into which a flux of
three-level atoms with a very low rate is pumped, so that during the flying time,
τ, only one atom in the cavity is considered. In this case, the field damping and
spontaneous emission time in other high-Q cavity modes must be larger than the
atomic flying time through the cavity. Following the conditions of experimental
realization of two level trapping effect [2] it is consider, that a lower atomic
pump rate is preferable to increase the visibility of trapping states by reducing
the impact of external fluctuations and two atom effects in the cavity [13]. In the
time moment, t, when the atom achieves the cavity the states of atomic and
cavity field are considered totally separated, so that the wave function can be
represented in the following form
ψ = ψ a ⊗ ψ EM
(1)
where ψ a and ψ EM are the wave functions of atom and cavity
electromagnetic field in the initial time moment t. The wave function of the
three-level atom is prepared in the superposition states in classical Ramsay zone
[14, 2] situated before the entrance in the cavity
3
Quantum entanglement state of non-equidistant three-level atom
547
ψ a (t ) = exp ⎡⎢ − i H 0a t ⎤⎥ ( α e + β i + γ g ) .
⎣ =
⎦
(2a)
The bimodal cavity of electromagnetic field is represented through a
general superposition of Fock states
Nu
Mu
∑ ∑
Ψ EM (t ) = exp ⎡⎢ − i H 0EM t ⎤⎥
⎣ =
⎦ n= N
d
m = Md
Sn,m n
a
m b,
(2b)
Sn, m coefficients, which must be determined from trapping conditions described
below, where Nd, Nu and Md, Mu represent the down and upper limits in the
sums of expression (2b). The free parts of Hamiltonian for atom H 0a =
= =ωa e e − =ωb g g and bimodal cavity of electromagnetic field H 0EM =
= =ωa a + a + =ωb b + b are considered in resonance one relatively other. We
emphasize that the energies of excited and ground states are measured relatively
by the positions of the intermediate state, so that Ee = =ωa , Eg = =ωb and
Ei = 0. The frequencies ωa and ωb correspond to the resonance cavity modes
described by annihilation a, b and creation a+ , b+ field operators.
In the time moment t + τ three-level atom enters into interaction with the
cavity field and the system Hamiltonian is described by the following expression
H = H 0 + Hi ,
(3)
where
H 0 = H 0a + H 0EM ,
H i = =λ a ( a + i e + a e i ) + =λ b ( b + g i + b i g
)
are the free and interactive parts of the Hamiltonian system. The coefficients
=λ a and =λ b describe the coupling energies between the modes a, b and
transitions between the levels i ↔ e and i ↔ g correspondingly. These
coefficients can be represented through the dipole active of matrix elements,
G
G
G
G
di ,e , di ,g and strength constants of electromagnetic field g(ωa ) ( g(ωa ) =
= eλ =ωa / ε 0V , eλ , the polarization of the cavity mode, and V, the volume of
the system. In the remainder of this work we assume that the polarization of the
cavity field is pre-selected, i.e., the polarization index λ is fixed to one of two
G G
G
possible directions [15]), g(ωb ) in the following way: λ a = di , e , g(ωa ) ,
G G
λ b = di , g , g(ωb ) .
(
)
(
)
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Nicolae Enaki, Nellu Ciobanu
4
Taking in to account the commutative propriety between the free and
interaction parts of Hamiltonian [ H 0 , H I ] = 0, let us represent the wave function
ψ(t + τ) =
solution of equation in the following interaction picture
(
)
(t + τ) , where the wave function
= exp − i H 0 τ ψ
=
equation
(t + τ) = Hi ψ (t + τ)
i= ∂ ψ
∂τ
ψ (t + τ)
satisfies the
(4)
We emphasize here that the interaction Hamiltonian is independent of time
and commutes with full Hamiltonian of system. In order to solve exactly this
equation let us represent the wave function ψ (t + τ) through these operators
ψ(t + τ) = [α eiϕe e + β eiϕi i + γ eiϕg g ] ⊗
Nu
Mu
∑ ∑
Sn,m n
n = N d m = Md
a
m
b
(5)
The operators i g a + , g i a + and its H.C as a time dependent of the
functions violate the solution of Schrödinger equation. Of course in interaction
picture the operators depend on the time, but these dependence are compensated
by bra-cket atomic operators, so that interaction Hamiltonian doesn't depend on
the time, t. These new operators depend only on the electromagnetic field variables
i
i
i
− Hτ
− Hτ
− Hτ
Xˆ (τ) = e = i e , Yˆ (τ) = e = i g , Zˆ (τ) = e = i i .
(6)
Taking into account the Hamiltonian (3) one can obtain the following
system of Heisenberg equation for operators (6)
dXˆ = −iλ Za
ˆ +,
a
dτ
dYˆ = −iλ Zb
ˆ ,
b
dτ
dZˆ = −λ Xa
ˆ − λ Yb
ˆ +.
a
b
dτ
(7)
In accordance with the initial conditions for these operators Xˆ (0) = e ,
X (0) = g , Z (0) = I we obtain the following solution for the system of
Eq. (7):
cos(Ωτ) − 1 +
sin(Ωτ) +
X (τ) = ⎛⎜ λ 2a
aa + 1 ⎞⎟ e − iλ a
a i +
2
Ω
Ω
⎝
⎠
cos(Ωτ) − 1 +
+ λaλb
b ag ,
Ω2
sin(Ωτ)
sin(Ωτ) +
Y (τ) = −iλ a
a e + cos(Ωτ) i − iλ b
b g ,
Ω
Ω
(8)
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Quantum entanglement state of non-equidistant three-level atom
549
cos(Ωτ) − 1
sin(Ωτ)
Z (τ) = −λ a λ b
ab e − iλ b
bi +
2
Ω
Ω
cos(Ωτ) − 1 +
+ ⎛⎜ λ 2b
b b + 1 ⎞⎟ g ,
Ω2
⎝
⎠
where Ω = λ 2a a + a + λ 2b bb + .
Taking into account these expressions, the Eq. (4) can be exactly solved
⎧⎪⎛
n + 1 [cos(Ω
2
⎫
⎨⎜⎜ Sn,m ⎧⎨λ a α 2
n +1,m +1τ) − 1] + α ⎬ −
Ω
n +1,m +1
⎩
⎭
n = N d ,m = M d ⎪
⎩⎝
(n + 1)1/ 2
sin(Ω n +1,m +1τ) +
− Sn +1,m iβλ a
Ω n +1,m +1
ψ (t + τ) =
N u , Mu
∑
+ Sn +1,m +1γλ a λ b
⎞
[(n + 1)(m + 1)]1/ 2
[cos(Ω n +1,m +1τ) − 1] ⎟⎟ e −
2
Ω n +1,m +1
⎠
1/ 2
⎡
sin(Ω n,m +1τ) −
− ⎢ Sn −1,m iαλ a n
Ω
n ,m +1
⎣
− Sn,m β cos(Ω n,m +1τ) + Sn,m +1i γλ b
(9)
⎤
(m + 1)1/ 2
sin(Ω n,m +1τ) ⎥ i +
Ω n,m +1
⎦
⎡
(nm)1/ 2
+ ⎢ Sn −1,m −1αλ a λ b
[cos(Ω n,m τ) − 1] +
Ω2n,m
⎣
⎧⎪
⎫⎪
+ Sn,m ⎨ γλ 2b m
Ω
τ
−
+
γ
[cos(
)
1]
⎬−
n
m
,
Ω2n,m
⎩⎪
⎭⎪
1/ 2
⎤ ⎫
− Sn,m −1iβλ b m sin(Ω n,m τ) ⎥ g ⎬ n
Ω n ,m
⎦ ⎭
a
m b,
where Ωn, m = λ 2a n + λ 2b m is the Rabbi frequency.
The direct derivation of exact solution (9) shows that it satisfies the
Schrödinger equation (4) in interaction picture. More than this, in the case when,
λ a = α = 0 (or λ b = γ = 0), three-level system is reduced to two-level quantum
oscillator relatively the ground and intermediate states (or the intermediate and
excited states) of atom. The exact solution obtained by us pass in the solution
proposed in papers [7, 8].
We require the reversible condition for flying time, τ, which consists in the
reversible process of quantum states of atom and cavity field. In other words, we
must consider such cavity fields for which the wave function in the time
550
Nicolae Enaki, Nellu Ciobanu
6
moment, t + τ, becomes factorized ψ(t + τ) = ψ′A (t + τ) ⊗ ψCEM (t + τ) and the
states of atom and electromagnetic field are the same as in the initial time moment
t. This corresponds to the coincidence between the probabilities of atomic and
field preparation at the initial time with the same probabilities after the
flying time Ψ A (t + τ) Ψ A (t + τ) = Ψ A (t ) Ψ A (t ) , Ψ EM (t + τ) Ψ EM (t + τ) =
= Ψ EM (t ) Ψ EM (t ) .
This condition strictly determines the population on the atomic level, living
the phases of coefficients α, β, γ and Sn, m aleatory. Let us consider the situation
when the phases for atomic amplitudes are choosing accidental. In this case the
wave function can be represented as:
ψ(t + τ) = ⎡⎣α eiϕe e + β eiϕi i + γ eiϕg g ⎤⎦ ⊗
Nu
Mu
∑ ∑
Sn,m n
n= N d m = Md
a
m b.
The aleatory phases ϕe, ϕi and ϕg, are determined from reversible conditions
which will be done below. Choosing the phases of coefficients α, β, and γ in the
following way one can obtain the recursion relation (10) for coefficients Sn, m
λ
Sn, m = − α a
γ λb
λ 2a n + λ 2b m
cot
m
β
nS
−i 1
m n −1, m −1 γ λ b
(
)
λ 2a n + λ 2b m τ Sn, m −1 .
2
(10a)
The phases for coefficients α, β, and γ was chosen in the following way:
ϕe = 0, ϕi = π, ϕg = 0. In the case when, λa = α = 0 (or λb = γ = 0), this recursion
relation passes into same recursion relation proposed in papers [7, 8] for two-level
atom.
The recursion relation (10) becomes more clear, considering the situation
for which the intermediate level is not populated, β = 0. In this case for equal
number of photons in both modes m = n, the relation (10) takes the form
)
(
sin τ n(λ 2a + λ 2b ) ⎡⎣ Sn, n − ASn −1, n −1 ⎤⎦ = 0,
2
(10b)
λ
where A = − α a . Considering Sn, n − ASn −1, n −1 = 0, we observed that condition
γ λb
(10b) becomes time independently. This relation between the coefficients
corresponds to squeezed state for which the wave function is
Ψ = 1 − A2 exp( Aa + b + ) 0
a
0 b . As following from this result, the trapping
condition becomes time independently, Snn = An S0, 0 , and the EM field state in
which it is realized this effect can be regard as squeezed states of two cavity
modes
7
Quantum entanglement state of non-equidistant three-level atom
(
Ψ = 1 − A2 0
a
0
b
+ A 1 a 1 b + ... + An n
a
n
b
+ ...
551
)
+n +n
= a b 0 a 0 b . We observe that such squeezed field doesn't
n!
change the inversion of three-level during the interaction time τ. Taking in to
account that the number of photons n calculated with the wave function Ψ
where n
a
n
b
ΨnΨ =
2 ,
1 −1
A2
we can regard this process of atomic trap as a thermal-equilibrium which stopped
the notation of the quantum oscillation of inversion. Indeed introducing in such
equilibrium the effective “temperature” one can use the detailed balance
distribution of excited and ground states
⎛ Ee − Eg
N e α2
= 2 = exp ⎜ −
Ng γ
T
⎝
⎞
⎟.
⎠
In the case, when λ a = λ b , the expression for mean number of photons is
regarded in similar way as in “thermal” field for two-level system [16]
n =
2
exp ( =ω0 / T ) − 1
where =ω0 = Ee − Eg = =ωa + =ωb .
The similarity between such two different systems may be clear in the case
when intermediate state is absent. In this problem the intermediate state is nonpopulated and we have inversion relatively intermediate and excited states.
In order to understand influence of flying time, τ, in recursion relation
(10a), let us consider the particular cases:
A. Reversible condition on the vacuum state of electromagnetic field
cavity, when N d = N u , M d = Mu and N u = Mu = 0. In this case we can drop
(
)
the above recursion relation S1,1 − AS0, 0 ≠ 0, considering, sin τ (λ 2a + λ 2b ) =
2
= 0, only one element is different from zero S00 = 1. From this condition we
obtain the following discrete values for the reversible time τ
τp =
2 pπ
λ 2a + λ 2b
,
(11a)
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Nicolae Enaki, Nellu Ciobanu
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B. The second explicit case corresponds to the situation, when N u = Mu = 1,
and N d = Md = 0. In this case one can choose the recursion relation between
the coefficients S22 and S11. Considering, that S2,2 − AS1,1 ≠ 0, and
(
)
sin τ 2(λ 2a + λ 2b ) = 0 we obtained
2
2 kπ .
+ λ 2b
τk =
(11b)
λ 2a
When λ a = λ b = λ the wave function of electromagnetic field is
1/ 2
2 ⎞
⎛
ψ EM (τ p ) = ⎜ 1 + α2 ⎟
γ ⎠
⎝
⎡0
⎢⎣
0
a
b
− α 1 a 1 b ⎤⎥
γ
⎦
(12)
As follows from (12) the wave function corresponds to the entanglement state of
photons belonging the modes a and b.
Taking into account that the condition (10) is quite general, we will study
the case when the matrix element Sn, m takes zero value for n > m. In this
assumption the coefficient Sn, m can be expressed through the coefficient SN d , Md .
Indeed, requiring the zero values of coefficients SN d +1, Md and SN d −1, Md −1 , from
recursion relation (10), we obtain the following condition for flying time and
down photon numbers Nd and Md
τ λ 2a N d + λ 2b M d = 2k π,
(13a)
where k = 0, 1…
It is difficult to drop the upper limits for the numbers of photons n and m. If
we consider that in the sums the variable m depends on the variable n, Mu (n),
then we can drop the domain of summation on the plane (n, m). The non-zero
point ( N u , Mu ) corresponds to the maximum value of discrete dependence of
function Mu ( n). Considering, that this point has non-zero value of the
coefficient SNu , Mu ≠ 0, one can require zero values for the element SN u +1, Mu +1 .
)
(
λ 2a ( N u + 1) + λ 2b ( Mu + 1) τ . This
2
requirement is same for all class of elements in the recursion relation (10):
SNu , Mu + 2 , SN u −1, Mu +3 , …, SN u − p, Mu + p+ 2 . All these elements in expression (10)
The condition implies zero value for tan
(
)
λ 2a ( N u + 1) + λ 2b ( Mu + 1) τ = ∞. From this requirement
2
follows that the reversible time for the factorization of the wave function
have the same cot
9
Quantum entanglement state of non-equidistant three-level atom
τ λ 2a ( N u + 1) + λ 2b ( Mu + 1) = 2 pπ ,
553
(13b)
where p is an integer number, p = 0, 1, 2, ….
Considering that three-level atoms are prepared in the superposition of two
levels, we can obtain the reversible condition similar to the traditional two-level
atom. In this case the recursion relation (10) for α = 0 becomes
S N , m = −i
λ 2a N + λ 2b m
cot
m
β 1
γ λb
(
)
λ 2a N + λ 2b m τ SN , m −1 .
2
(14)
This expression corresponds to the reversible condition of the three-level
atom relatively the ground and excited states, stimulated by photon Fock state
N . From Eq. (14) the reversible condition takes the form
τp =
( 2 p + 1) π
λ 2a N + λ 2b ( Mu + 1)
.
In the next section the numerical behavior of the moments of bimodal
cavity field will be discussed. The fluctuation and cross fluctuations of photon
numbers will be found.
3. NUMERICAL RESULTS
Let us discuss the quantum properties of bimodal cavity field in the
reversible states (13b). For simplicity we will study the behavior of quantum
fluctuations and cross fluctuations of photon number as function of level
2
2
2
2
2
2
populations α , β and γ , where α + β + γ = 1. The fluctuations of
photon numbers in the first and second modes can be determined by the
expression
2
2
σ2n = : nˆ 2 : − nˆ , σ2m = : mˆ 2 : − mˆ ,
where n̂ = a + a and mˆ = b + b. Analogically the cross fluctuation is definite in the
following way
ˆ ˆ − nˆ mˆ .
σ2n, m = nm
The numerical results for the small and large number of coefficients Sn, m
are plotted in Fig. 1. The dependences of cross-fluctuations (see Fig. 1a) and
photon numbers fluctuations, σ2n (see Fig. 1b), σ2m (see Fig. 1c) as function of
2
2
parameters α and γ are plotted. This numerical simulation is obtained for
twenty-five non-zero values of photon numbers of coefficients, Sn, m.
554
Nicolae Enaki, Nellu Ciobanu
Fig. 1a
Fig. 1b
10
11
Quantum entanglement state of non-equidistant three-level atom
Fig. 1c
Fig. 1d
555
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Nicolae Enaki, Nellu Ciobanu
12
Fig. 1 – (a) Cross fluctuations, σ2n, m , as function of the populations α
2
and γ .
2
(b), (c) Fluctuations of the photon numbers σ2n and σ2m . In the pictures (d) and (e)
the black region represent sub-Poisson statistic
( σ2n , σ2m < 1)
and white region
represents super-Poisson statistic ( σ2n , σ2m > 1) . For these dependences were used
the following parameters: Nd = 0, Nu = 4, p = 1.
Considering the situation when the interaction constants are equal,
λa = λb = λ, we represented the negative and positive values of these normal
ordered quantum fluctuations by means of the white and black regions in
Figs. 1d, e. These regions correspond to geodesic cross-sections of fluctuations
σ2n and σ2m plotted in Figs. 1b, c. More evidently grey geological zones in
Figs. 1d, e represent zero values of these normal ordered fluctuations. The
negative and positive values correspond to sub-Poissonian and super-Poissonian
photon-statistic regions respectively. The growth from sub-Poissonian into
super-Poissonian photon-statistic regions as function of non-zero coefficients
Sn, m can be compared in both cases (see Figs. 1d, e). It is observed that the
fluctuations of the mean number of photons are growing with increase the
2
probability of population α on the excited state. This growth of the fluctuation
reaches the maximum value after which it decreases down to a certain value.
13
Quantum entanglement state of non-equidistant three-level atom
557
From our numerical results it follows that a new qualitative behavior of
fluctuations and cross fluctuations for more large numbers of photon states, Sn, m,
was not observed with increasing the upper limits Nu and Mu in the summation.
4. CONCLUSIONS
This effect gives us the possibility to use the prepared atomic inversion of
three-level atom flying through the cavity in resonance with two modes of
quantum electromagnetic field in quantum computing. Choosing the flying time
according with Eq. (11), one can factorize the wave function of two subsystems:
atom and cavity vacuum. In the case of non-vacuum state of electromagnetic
field the decomposed coefficients on the states Fock Snm must satisfy the
reversible conditions described by recursion relations (10) and (13). The problem
of realization of such states can be similar to the problem of preparation of one
photon optical trapping effect [2]. Taking in to account the good arrangement in
cascade configuration of energetic levels 40 S1/ 2 ↔ 39 P3 / 2 ↔ 39 S1/ 2 of Rydberg
atom used in two-photon maser emission [4] we consider that optical trapping
effects can be experimentally realized. The progress in experiments with trapped
atoms improves the possibilities of experimental realization of time reversibility
in interaction of three-level systems with quantum bimodal cavity field.
It is well known that quantum phase’s effect is used in quantum algorithms
of processing information [17, 18]. The phase’s separability between the three
level atom and bimodal cavity field after interaction can be one of the candidates
systems for logical information processing.
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