Generation of any superposition of Dicke state of excitons in coupled quantum dots

5 Pages·0.07 MB·English

by  XuBo Zou

Generation of any superposition of Dicke state of
excitons in coupled quantum dots
XuBo Zou, K. Pahlke and W. Mathis
Institute TET, University of Hannover,
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Appelstr. 9A, 30167 Hannover, Germany
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n
a Abstract
J

8 We present a scheme to generate arbitrary superposition of the Dicke

states of excitons in optically driven quantum dots. This proposal is based
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v on a sequence of laser pulses, which are tuned appropriately to control
1 transitions on Dicke state. It is shown that N laser pulses are needed to
1
generate arbitrary superposition of the Dicke states of N quantum dots.
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1 PACS number(s): 03.67.-a, 03.67.Hk.
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Quantum entanglement has been intensely studied, due to its poential applications in
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/ quantumcommunicationandinformationprocessing[?]suchasquantumteleportation[2],
h
p superdense coding[3], quantum key distribution[4] and telecloning[5]. One other area
- where entangled quantum state may have a significant impact is that of the improve-
t
n
ment of frequency standard[6, 7]. Key to improvement of frequency standard beyond
a
u the short noise limit is the establishment of a entangled state of a collection of N
q
two level atoms. Initial theoretical investigation examined the use of spin squeezed
:
v
state[6, 7], which has been extensively studied[8] in recent years. It has also been
i
X shown that, in the absence of decoherence, GHZ state of N the may be used in
r high precision spectroscopy to measure the transition frequency to an uncertainty of
a
N 1[9, 10]. When the decoherence is present, the use of the maximally entangled state
−
does not provide higher resolution as compared to using independent particle and the
best sensitivity is achieved when the particles are initially prepared in highly sym-
metric but only partially entangled states[10], which is a superposition of the Dicke
states. A number of experimental protocols for generation of GHZ state of N qubits
and spin squeezed state have been proposed, including the Cavity QED[11], trapped
ion system[12] and Bose-Einstein condensates[8]. In experiment, GHZ state of three
or four qubits has been observed[13]. Recently, it has been shown that there exist
two inequivalent classes of three qubit entangled state under local operation assisted
by classical communication, namely GHZ state |GHZ >= 1 (|111 > +|000 >) and
√2
W state |GHZ >= 1 (|100 > +|010 > +|001 >)[14]. A scheme has also been pro-
√3
posed for generating W entangled state of three or four qubits[15]. In this paper, we
present a scheme to generate arbitrary superposition of the Dicke states of N atoms,
which includes GHZandWstate ofN qubits, spin squeezed state andentangled atomic
1
state proposed inRef[10]. Our scheme is based on opticaldriven quantum dots. Recent
advance in semiconductor nanostructure fabrication and measurement suggest that op-
tically generated electron-hole pairs (exciton) in semiconductor quantum dot represent
ideal candidate for achieving coherence wave function control on the nanometer and
femtosecondscalesandimplementing alargescalequantumcomputation[16]. Recently,
optically driven quantum dot has been used to prepare maximally entangled Bell and
GHZ state[17] and implement quantum teleportation[18]. In this paper, we present a
scheme to generate arbitrary superposition of the Dicke states of excitons in optically
driven quantum dots. This proposal is based on a sequence of laser pulses, which are
tuned appropriately to control transitions on Dicke state. It is shown that only one
laser pulses are needed to generate W states of N quantum dots. The schematic re-
quirement arerealizableincurrent experiment employing ultrafast opticalspectroscopy
of quantum dots.
In order to describe our scheme, we consider a system of N identical and equispaced
quantum dots containing no net charge that are radiated by long-wavelength classical
light. In the frame of the rotating wave approximation, the formation of single excitons
within the individual quantum dots and their interdot hopping are described by the
Hamiltonian[17, 18]
H = ∆ J +geiϕJ +ge iϕJ +W(J2 −J2) (1)
ω z + − z
−
where
N N 1 N
J = c h , J = c h J = (c c −h h ) (2)
+ X †p †p − X p p z 2 X †p p p †p
p=1 p=1 p=1
Here, c and c (h and h ) are the electron (hole) creation and annihilation operator
†p p †p p
in the pth quantum dot. ∆ = ǫ − ω. ǫ represent the band gap, W is the interdot
ω
interaction parameter (Forster process). ω,ϕ andgarefrequency, phaseandamplitude
of laser field, respectively. The quasispin J operators satisfy the usual commutation
relation [J ,J ] = ±J , [J ,J ] = 2J and [J2,J ] = [J2,J ] = [J2,J ] = 0. From
z + z z +
± ± − −
a practical point of view, parameter ∆ , ϕ and g are adjustable in the experiment to
ω
give control over the system of the quantum dots. If the initial state of the system
is symmetric with respect to exchange of quantum dots, We can only consider the
symmetric state and a basis is formed by the Dicke state |J,M >, here J = N,M =
2
−J,−J + 1,···,J. |J,M > is the eigenstate of the operator J with eigenvalue M.
z
Since J2 commute with J andJ J2 is conservation constant, we may not consider the
z
J2 terms in the following. Since±the ∆ is adjustable, in Ref[17, 18], author consider
ω
resonance condition (∆ = 0) and proposed a scheme to prepare maximally entangled
ω
Bell andGHZstate. Here we choose ∆ to satisfy thecondition ∆ = −W(2J−2i−1),
ω ω
i = 0,1,···,2J −1. Thus Hamiltonian of the system can be written in the form
H = −W(J +J −i)(J +J −i−1)+g eiϕiJ +ge iϕiJ (3)
z z i + −
−
Here we have neglected a constant term. Using theDicke state|J,M >, M = −J,−J+
1,···,J the Hamiltonian(3) take the form
J
H = −W (m+J −i)(m+J −i−1)|J,m >< J,m|
X
m= J
−
2
J 1
+g eiϕi − J(J +1)−m(m+1)|J,m+1 >< J,m|
i X q
m= J
−
J
+g e iϕi J(J +1)−m(m−1)|J,m−1 >< J,m| (4)
i − X q
m= J+1
−
It is noticed that the coefficients of the Dicke state |J,−J +i > and |J,−J +i+1 >
are equal to zero in the first line of above equation. We now consider the condition
W >> Jg, which has bee assumed in Ref[17, 18]. In this case, we apply the rotating
wave approximation and discard the rapidly oscillating term in the Hamiltonian (4)
and obtain the effective interaction
H = Ω (eiϕi|J,−J +i+1 >< J,−J +i|+e iϕi|J,−J +i >< J,−J +i+1|) (5)
i i −
Here Ω = g J(J +1)−(J −i)(J −i−1).
i q
In order to generate any superposition of Dicke state of excitons
J
Ψ = C |J,m > (6)
X m
m= J
−
we consider the situation in which the quantum dots is prepared in the zero excitons
Ψ = |J,−J > (7)
initial
In the following we show, that each term of equation(1) can be generated by one laser
pulse. Fist, we derive the quantum dots with laser frequency ∆ = −W(2J −1), after
ω
an interaction time τ , the system evolves into
0
Ψ = cos(Ω τ )|J,−J > +isin(Ω τ )eϕ0|J,−J +1 > (8)
1 0 0 0 0
we chhose the amplitude (or interaction time τ ) of the laser field in such a way that
0
the following conditional are fulfilled
cos(Ω τ ) = C (9)
0 0 J
−
we obtain
Ψ = C |J,−J > +ieϕ0 1−C2 |J,−J +1 > (10)
1 J q J
− −
without loss of generality, we have assumed that C is a real number. We then tune
J
−
the laser frequency to satisfy ∆ = −W(2J − 3). After an interaction time τ , the
ω 1
system is evolved into
Ψ = C |J,−J > +ieϕ0 1−C2 (cos(Ω τ )|J,−J+1 > +isin(Ω τ )eϕ1|J,−J+2 >)
2 J q J 1 1 1 1
− − (11)
we adjust the amplitude (or interaction time τ ) of the second laser field and phase of
1
the laser field of the first laser field to satisfy
ieϕ0 1−C2 cos(Ω τ ) = C (12)
q J 1 1 J+1
− −
3
the state become
Ψ = C |J,−J > +C |J,−J +1 > +i2eiϕ0+iϕ1 1−C2 −|C |2|J,−J +2 >
2 J J+1 q J J+1
− − − − (13)
If this procedure is done for the m−th time, the quantum state of the system is
J+m 1 m 1 J+m 1
Ψm = − X− Cl|J,l > +imexp(i X− ϕl)vuu1−− X− |C1|2|J,−J +m > (14)
l= J l=0 t l= J
− −
We now consider the (m+1)−th operation by choosing ∆ = −W(2J−2m−1). After
ω
interaction τ , the quantum state becomes
m
J+m 1 m 1 J+m 1
Ψm+1 = − X− Cl|J,l > +imexp(i X− ϕl)vuu1−− X− |C1|2(cos(Ωmτm)|J,−J +m >
l= J l=0 t l= J
− −
+isin(Ω τ )eϕm|J,−J +m+1 >) (15)
m m
Wechoosethe amplitude(orinteraction timet )of the(m+1)-thlaser pulse andphase
m
of m-th laser field to satisfy
m 1 J+m 1
imexp(i X− ϕl)vuu1−− X− |C1|2cos(Ωmτm) = C J+m (16)
l=0 t l= J −
−
After the procedure is performed for N times the system’s state definitely becomes
state(6).
We present a scheme to generate arbitrary superposition of the Dicke states of excitons
in optically driven quantum dots. This proposal is based on a sequence of laser pulses,
which are tuned appropriately to control transitions on Dicke state. It is shown that
N laser pulses are needed to generate arbitrary superposition of the Dicke states of
N quantum dots. This scheme of quantum state generation requires not more than N
laser pulses, which is the smallest possible number. Thus, in respect of short laser pulse
sequences it can be considered as the optimal solution of this problem of the quantum
state generation. it is noticed that generalized W state of N qubit can be written in the
form of Dick state |J,−J +1 >, here J = N, so that it is needed only one laser pulse
2
to generate W state of N qubits. In this scheme, we require that W is much larger
than amplitude of laser pulse. This condition is exactly parallel to that in Ref[17, 18],
which can be realized in present experiment. Finally, it is worth pointing out that the
same (or similar) Hamiltonian as in Eq(1). also arise in other context, i.e. in a two
coupled Bose-Einstein condensate and in the ultrasmall Josephson Junction, which has
been used to create macroscopic quantum superposition and realize quantum logic gate
operation. Thus the present work may be viewed in a wider context.
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