Density oscillation in highly flattened quantum elliptic rings and
  tunable strong dipole radiation
Density oscillation in highly flattened quantum elliptic rings and tunable strong
dipole radiation
S.P. Situ, Y.Z. He, and C.G. Bao∗
The State Key Laboratory of Optoelectronic Materials and Technologies,
Zhongshan University, Guangzhou, 510275, P.R. China
A narrow elliptic ring containing an electron threaded by a magnetic field B is studied. When the
ring is highly flattened, the increase of B would lead to a big energy gap between the ground and
excited states, and therefore lead to a strong emission of dipole photons. The photon frequency
can be tuned in a wide range by changing B and/or the shape of the ellipse. The particle density is
found to oscillate from a pattern of distribution to another pattern back and forth against B. This
is a new kind of Aharonov-Bohm oscillation originating from symmetry breaking and is different
from the usual oscillation of persistent current.
∗Corresponding author
It is recognized that micro-devices are important to
micro-techniques. Various kinds of micro-devices, includ-
ing the quantum rings,1 have been extensively studied
theoretically and experimentally in recent years. Quan-
tum rings are different from other devices due to their
special geometry. A distinguished phenomenon of the
ring is the Aharonov-Bohm (A-B) oscillation of the
ground state energy and persistent current2−5. It is be-
lieved that geometry would affect the properties of small
systems. Therefore, in addition to circular rings, ellip-
tic rings or other rings subjected to specific topological
transformations deserve to be studied, because new and
special properties might be found. There have been a
number of literatures devoted to elliptic quantum dots6−9
and rings10−12. It was found that the elliptic rings have
two distinguished features. (i) The avoided crossing of
the levels and the suppression of the A-B oscillation. (ii)
The appearance of localized states which are related to
bound states in infinite wires with bends.13 These feature
would become more explicit if the eccentricity is larger
and the ring is narrower.
On the other hand, as a micro-device, the optical prop-
erty is obviously essential to its application. It is guessed
that very narrow rings with a high eccentricity might
have special optical property, this is a point to be clari-
fied. This paper is dedicated to this topic. It turns out
that the optical properties of a highly flattened narrow
ring is greatly different from a circular ring due to having
a tunable energy gap, which would lead to strong dipole
transitions with wave length tunable in a very broad
range (say, from 0.1 to 0.001cm). Besides, a kind of A-B
density-oscillation originating from symmetry breaking
was found as reported as follows.
We consider an electron with an effective mass m∗ con-
fined on a one-dimensional elliptic ring with a half major
axis rax and an eccentricity ε. Let us introduce an argu-
ment θ so that a point (x, y) at the ring is related to θ as
x = rax cos θ and y = ray sin θ, where ray = rax
1− ε2 is
the half minor axis. A uniform magnetic field B confined
inside a cylinder with radius rin vertical to the plane of
the ring is applied. The associated vector potential reads
A = Br2int/2r, where t is a unit vector normal to the
position vector r. Then, the Hamiltonian reads
H = G/(1− ε2 cos2 θ)[− d
− i2α
1− ε2
(1− ε2 sin2 θ)
1− ε2 cos2 θ
1− ε2 sin2 θ
] (1)
where G = ~2/(2m∗r2ax), α = φ/φo, φ = πr
inB is the
flux, φo = hc/e is the flux quantum.
The eigen-states are expanded as Ψj =
∑kmax
k=kmin
eikθ, where k is an integer ranging
from kmin to kmax, and j = 1, 2, · · · denotes the ground
state, the second state, and so on. The coefficients C
are obtained via the diagonalization of H . In practice, B
takes positive values, kmin = −100 and kmax = 10. This
range of k assures the numerical results having at least
four effective figures. The energy of the j − th state is
Ej = 〈H〉j ≡
dθ(1 − ε2 cos2 θ)Ψ∗jHΨj (2)
where the eigen-state is normalized as
dθ(1− ε2 cos2 θ)Ψ∗jΨj (3)
In the follows the units meV, nm, and Tesla are used,
m∗ = 0.063me (for InGaAs rings), and rin is fixed at
25. When rax = 50, ε = 0 and 0.4, the evolution of
the low-lying spectra with B are given in Fig.1. When
ε = 0.4, the effect of eccentricity is still small, the spec-
trum is changed only slightly from the case ε = 0, but
the avoided crossing of levels can be seen.10,11 In par-
ticular, the A-B oscillation exists and the period of φ
remains to be φo. However, when ε becomes large, three
remarkable changes emerge as shown in Fig.2. (i) The
A-B oscillation of the ground state vanishes gradually.
(ii) The energy of the second state becomes closer and
closer to the ground state. (iii) There is an energy gap
lying between the ground state and the third state, the
http://arxiv.org/abs/0704.0867v1
0 2 4 6 8 10
E(meV)
B(Tesla)
ε =0.4rax=50(b)
ε =0rax=50(a)
FIG. 1: Low-lying spectrum (in meV) of an one-electron sys-
tem on an elliptic ring against B. rax = 50nm and ε = 0 (a)
and 0.4 (b). The period of the flux φo = hc/e is associated
with B = 2.106 Tesla.
0 4 8 12 16 20
B (Tesla)
E(meV)
rax= 50,   ε = 0.8
FIG. 2: Similar to Fig.1 but ε = 0.8. The lowest eight
levels are included, where a great energy gap lies between the
ground and the third states.
gap width increases nearly linearly with B. The exis-
tence of the gap is a remarkable feature which has not
yet been found before from the rings with a finite width.
This feature is crucial to the optical properties as shown
later. Fig.3 demonstrates further how the gap varies
with ε, rax, and B , where B is from 0 to 30 (or φ from 0
to 14.24φo). One can see that, when ε is large and rax is
small, the increase of B would lead to a very large gap.
0 5 10 15 20 25 30
0 5 10 15 20 25 30
(b)  ε = 0.8
ε =0.8
B (Tesla)
E3-E1 
(meV)
(a)  r
ε =0.6
ε=0.4
FIG. 3: Evolution of the energy gap E3 − E1 when rax and
ε are given.
The A-B oscillation of the ground state energy is given
in Fig.4. The change of ε does not affect the period
(2.106 Tesla). However, when ε is large, the amplitude of
the oscillation would be rapidly suppressed. Thus, for a
highly flattened elliptic ring, the A-B oscillation appears
only when B is small.
0 2 4 6 8 10
B (Tesla)
ε =0,   0.4,   0.8rax=50
FIG. 4: The A-B oscillation of the ground state energy. The
solid, dash-dot-dot, and dot lines are for ε = 0, 0.4, and 0.8,
respectively.
The persistent current of the j − th state reads14
Jj = G/~[Ψ
1− ε2
(1− ε2 sin2 θ)
)Ψj + c.c.] (4)
The A-B oscillation of Jj is plotted in Fig.5. When ε is
small (≤ 0.4), just as in Fig.4, the effect of ε is small as
shown in 5a. When ε is large there are three noticeable
points: (i) The oscillation of the ground state current
would become weaker and weaker when B increases. (ii)
The current of the second state has a similar amplitude
as the ground state, but in opposite phase. (iii) The
third (and higher) state has a much stronger oscillation
of current.
0 2 4 6 8 10
B (Tesla)
(b)   rax=50,      ε =0.8
ε=0,  0.4,  0.8(a)  r
FIG. 5: The A-B oscillation of the persistent current J . (a)
is for the ground state with ε = 0 (solid line), 0.4 (dash-dot-
dot), and 0.8 (dot). (b) is for the first (ground), second and
third states (marked by 1,2, and 3 by the curves) with ε fixed
at 0.8. The ordinate is 106 times J/c in nm−1.
For elliptic rings, the angular momentum L is not con-
served. However, it is useful to define (L)j = 〈−i ∂∂θ 〉j
(refer to eq.(2)). This quantity would tend to an integer
if ε → 0. It was found that (i) When ε is small (≤ 0.4),
(L)1 of the ground state decreases step by step with B,
each step by one, just as the case of circular rings. How-
ever, when ε is large, (L)1 decreases continuously and
nearly linearly. (ii) When ε is small, |(L)i− (L)1| is close
(not close) to 1 if 2 ≤ i ≤ 3 (otherwise). Since L would
be changed by ±1 under a dipole transition, the ground
state would therefore essentially jump to the second and
third states. Accordingly, the dipole photon has essen-
tially two energies, namely, E2 −E1 and E3 −E1 . How-
ever, this is not exactly true when ε is large.
There is a relation between the dipole photon energies
and the persistent current.15 For ε = 0, the ground state
with L = k1 would have the current J1 = G(k1 + α)/π~,
while the ground state energy E(k1) = G(k1 + α)
2. Ac-
cordingly the second and third states would have L =
k1 ± 1, therefore we have
|E3 − E2| = |E(k1 + 1)− E(k1 − 1)| = 2hJ1 (5)
This relation implies that the current can be accurately
measured simply by measuring the energy difference of
the photons emitted in dipole transitions. For elliptic
rings, this relation holds approximately when ε is small
(≤ 0.4), as shown in Fig.6a. However, the deviation is
quite large when ε is large as shown in 6c.
0 2 4 6 8 10
(c)   ε = 0.7
B (Tesla)
(b)   ε = 0.5
(a)     ε = 0.3
FIG. 6: E3 − E2 and the persistent current of the ground
state. The solid line denotes (E3 − E2)/(2hc)10
6, the dash-
dot-dot line denotes |J |/c·106 . They overlap nearly if ε < 0.3.
The probability of dipole transition from Ψj to Ψj′
reads
(ω/c)3|〈x∓ iy〉j′,j |2 (6)
where ~ω = Ej′ − Ej is the photon energy,
〈x∓ iy〉j′,j = rax
dθ(1 − ε2 cos2 θ)Ψ∗j′
[cos θ ∓ i
1− ε2 sin θ]Ψj (7)
The probability of the transition of the ground state to
the j′ − th state is shown in Fig.7. When ε is small
(≤ 0.4) and B is not very large (≤ 10), the allowed final
states essentially Ψ2 and Ψ3, and the oscillation of the
probability is similar to the case of circular rings with the
same period as shown in 7a and 7b. In particular, P±3,1 is
considerably larger than P±2,1 due to having a larger pho-
ton energy, thus the third state is particularly important
to the optical properties. When ε is large (Fig.7c), the
oscillation disappears gradually with B,while the prob-
ability increases very rapidly due to the factor (ω/c)3.
Since E3 − E1 is nearly proportional to B as shown in
Fig.3, the probability is nearly proportional to B3. This
leads to a very strong emission (absorption). Further-
more, in Fig.7c the black solid curve is much higher than
the dash-dot-dot curve, it implies that the final states
can be higher than Ψ3, this leads to an even larger prob-
ability.
0 2 4 6 8 10
B(Tesla)
c) ε =0.8
b)  ε =0.4
a)   ε =0
FIG. 7: Evolution of the probability of dipole transition of
the ground state. The green line is for Ψ1 to Ψ2 transition,
red line for Ψ1 to Ψ3, dash-dot-dot line is for the sum of the
above two, solid line in black is for the total probability.
For circular rings, the particle densities ρ of all the
eigen-states are uniform under arbitraryB. However, for
elliptic rings, ρ is no more uniform as shown in Fig.8. For
the ground state (8a), when φ =0, the non-uniformity is
slight and ρ is a little smaller at the two ends of the major
axis (θ = 0, π). When φ increases, the density at the
two ends of the minor axis (θ = π/2, 3π/2) increases as
well. When φ = 4φo the non-uniformity is very strong as
shown by the curve 9, where ρ ≈ 0 when θ ≈ 0 or π. The
second state has a parity opposite to the ground state,
0 20 40 60 80 100 120
0 20 40 60 80 100 120
0 20 40 60 80 100 120
Arc length (nm)
                      fourth state 
    third state
8ε =0.8,   r
ground state
FIG. 8: Particle densities ρ as functions of the arc length
(the according change of θ is 0 to π). The fluxes are given as
φ = (i− 1)φo/2, where i is an integer from 1 to 9 marked by
the curves. The first group of curves (in violet) have φ/φo =
integer, the second group (in green) have φ/φo = half-integer.
When φ increases, the curve of ρ jumps from the first group
to the second group and jumps back, and repeatedly.
but their densities are similar. For the third state (8b), ρ
is peaked not at the ends of the major and minor axes but
in between. In particular, when B increases, ρ oscillates
from one pattern (say, in violet line) to another pattern
(in green line), and repeatedly. The density oscillation
would become stronger in higher states (8c). The period
of oscillation remains to be φo, thus it is a new type of
A-B oscillation without analogue in circular rings (where
ρ remains uniform). Incidentally, the density oscillation
does not need to be driven by a strong field, instead, a
small change of φ from 0 to φo is sufficient.
Let us evaluate Ej roughly by using (L)j to replace
the operator −i ∂
in eq.(2) Then,
Ej ≈ G
dθ{[(L)j + α
1− ε2
1− ε2 sin2 θ
αε2 sin 2θ
2(1− ε2 sin2 θ)
]2}Ψ∗jΨj (8)
There are two terms at the right each is a square of a
pair of brackets (for circular rings the second term does
not exist). It is reminded that, while α = φ/φo is given
positive, (L)j is negative. Thus there is a cancellation
inside the first term. Therefore, when ε and α are large,
the second term would be more important. It is recalled
that both Ψ1 and Ψ2 are mainly distributed around θ =
π/2 and 3π/2 (refer to Fig.8a), where the second term
is zero due to the factor sin 2θ. Accordingly the energies
of Ψ1 and Ψ2 are lower. On the contrary, both Ψ3 and
Ψ4 are distributed close to the peaks of the second term
(refer to Fig.8b and 8c), this leads to a higher energy.
This effect would be greatly amplified by αε2 , this leads
to the large energy gap shown in Fig.3.
In summary, the optical property of highly flattened
elliptic narrow rings was found to be greatly different
from circular rings. For the latter, both the energy of
the dipole photon and the probability of transition are
low, and they are oscillating in small domains. On the
contrary, for the former, both the energy and the prob-
ability are not limited, the energy (probability) is nearly
proportional to B (B3), they are tunable by changing
ε, rax and/or B. It implies that a strong source of light
with frequency adjustable in a wide domain can be de-
signed by using highly flattened, narrow, and small rings.
Furthermore, a new type of A-B oscillation, namely, the
density oscillation, originating from symmetry breaking,
was found. This is a noticeable point because the density
oscillation might be popular for the systems with broken
symmetry (e.g., with C3 symmetry).
Acknowledgment: The support under the grants
10574163 and 90306016 by NSFC is appreciated.
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14, Eq.(4) originates from a 2-dimensional system via
the following steps. (i) the components of the current
along X- and Y-axis are firstly obtained from the conser-
vation of mass as well known. (ii) Then, the component
along the tangent of ellipse jθ can be obtained. (iii) jθ
is integrated along the normal of the ellipse under the
assumption that the wave function is restricted in a very
narrow region along the normal, then it leads to eq.(4).
15, Y.Z. He, C.G. Bao (submitted to PRB)
