Dephasing due to a fluctuating fractional quantum Hall edge current
DEPHASING DUE TO A FLUCTUATING
FRACTIONAL QUANTUM HALL EDGE CURRENT
T. K. T. NGUYEN1,2, A. CRÉPIEUX1, T. JONCKHEERE1, A. V. NGUYEN2, Y. LEVINSON3,
AND T. MARTIN1
1 Centre de Physique Théorique, CNRS Luminy case 907, 13288 Marseille cedex 9, France
2 Institute of Physics and Electronics, 10 Dao Tan, Cong Vi, Ba Dinh, Hanoi, Vietnam
3 Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
The dephasing rate of an electron level in a quantum dot, placed next to a fluctuating edge
current in the fractional quantum Hall effect, is considered. Using perturbation theory, we
first show that this rate has an anomalous dependence on the bias voltage applied to the
neighboring quantum point contact, because of the Luttinger liquid physics which describes
the fractional Hall fluid. Next, we describe exactly the weak to strong backscattering crossover
using the Bethe-Ansatz solution.
1 Introduction
The presence of electrical environment influences the transport through a quantum dot: its
energy level acquires a finite linewidth if the environment has strong charge fluctuations. Several
experiments, performed with a quantum dot embedded in an Aharonov-Bohm loop, probed the
phase coherence of transport when the dot is coupled to a controlled environment, such as a
quantum point contact (QPC)1. Charge fluctuations in the QPC create a fluctuating potential
at the dot, modulate its electron level, and destroy the coherence of the transmission through
the dot 2. Theoretical studies for describing this dephasing have been developped 3,4, and were
applied to a quantum Hall geometry 5, and to a normal metal-superconductor QPC6.
In this work, we consider the case of dephasing from a QPC in the fractional quantum Hall
effect (FQHE) regime 7. QPC transmission can be described by tunneling between edge states
which represent collective excitations of the quantum Hall fluid. It is interesting because the
transport properties deviate strongly from the case of normal conductors 8,9,10: for the weak
backscattering (BS) case, the current at zero temperature may increase when the voltage bias
is lowered, while in the strong BS case, the I(V ) is highly non linear. It is thus important to
address the issue of dephasing from a Luttinger liquid.
2 Dephasing in the fractional quantum Hall regime
The system we consider is depicted in Fig. 1. The single level Hamiltonian for the dot reads
HQD = ǫ0c
†c, where c† creates an electron. This dot is coupled capacitively to a point contact
in the FQHE. The Hamiltonian which describes the edge modes in the absence of tunneling is:
H0 = (h̄vF /4π)
dx[(∂xφ1)
2 + (∂xφ2)
2], where φ1 and φ2 are the chiral Luttinger bosonic fields,
which are related to the electron density operators ρ1(2) by ∂xφ1(2)(x) = πρ1(2)(x)/
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a) b)
Figure 1: Schematic description of the setup: the quantum dot (top) is coupled capacitively to a quantum point
contact in the FQHE regime: a) case of weak backscattering, b) case of strong backscattering.
By varying the gate potential of the QPC, one can switch from a weak BS situation, where
the Hall liquid remains in one piece (Fig. 1a), to a strong BS situation where the Hall liquid
is split in two (Fig. 1b). In the former case, the entities which tunnel are edge quasiparticle
excitations. In the latter case, between the two fluids, only electrons can tunnel. We consider
the weak BS case, and then we use a duality transformation8,11 to describe the strong BS case.
The tunneling Hamiltonian between edges 1 and 2 reads Ht = Γ0e
iω0tψ+2 (0)ψ1(0) + h.c. where
we have used a Peierls substitution to include the voltage: ω0 = e
⋆V/h̄ (e⋆ = νe is the effective
charge, ν is the filling factor). The quasiparticle operator is ψi(x) = e
νφi(x)/
2πα (the spatial
cutoff is α = vF τ0, with τ0 the temporal cutoff).
The Hamiltonian describing the interaction between the dot and the QPC reads Hint =
dxf(x)ρ1(x), with f(x) a Coulomb interaction kernel, which is assumed to include screening
by the nearby gates f(x) ≃ e2e−|x|/λs/
x2 + d2, where d is the distance from the dot to the
edge, λs is a screening length. The dephasing rate, expressed in terms of irreducible charge
fluctuations in the adjacent wire, is written as 3,4,5:
τ−1ϕ =
dxf(x)
dx′f(x′)〈〈ρ1(x, t)ρ1(x′, 0) + ρ1(x′, 0)ρ1(x, t)〉〉 . (1)
The equilibrium contribution to the dephasing rate corresponds to the zero order in the
tunneling amplitude Γ0:
(τ−1ϕ )
(0) =
4π2h̄2
dxf(x)
dx′f(x′)
∂2xx′G
1 (x− x
′, t) , (2)
where the bosonic Green’s function is G
i (x − x′, t1 − t2) = 〈φi(x, t
1 )φi(x
′, tη22 ) − φ2i 〉. The
coefficients η,η1,2 = ± identify the upper/lower branch of the Keldysh contour. There is no
contribution to first order in the tunneling Hamiltonian, while the non-equilibrium contribution
corresponding to the second order in Γ0 exists:
(τ−1ϕ )
(2) = − ν
4π2h̄4
2(2πα)2
dxf(x)
dx′f(x′)
η,η1,η2,ǫ
×eiǫω0(t1−t2)eνG
(0,t1−t2)eνG
(0,t1−t2)
∂2xx′G
1 (x− x
′, t)
+ ν[∂xG
1 (x, t− t1)− ∂xG
1 (x, t− t2)][∂x′G
′,−t1)− ∂x′G−ηη21 (x
′,−t2)]
. (3)
The dephasing rate depends on the geometry of the set up via the length scales d, λs, and
α. The assumption of strong screening λs ∼ α = vF τ0 is made (f(x) ≃ 2e2αδ(x)/d). Inserting
1 (x, t) = − ln
sinh[π[(x/vF −t)((η+η′)sgn(t)−(η−η′))/2+iτ0]/h̄β]
sinh[iπτ0/h̄β]
, where
β = 1/kBT , in the dephasing rate gives: (τ
(0) = 4e4τ20 ν/πh̄
3βd2 and,
(τ−1ϕ )
(2) =
e4Γ20
π2h̄4v2F d
ν2τ2ν0
Γ(2ν)
)2ν−1
ω0h̄β
ν + i
ω0h̄β
. (4)
Note that (τ−1ϕ )
(2) = (eτ0/d)
2SI(0), with SI(0) =
dt〈〈I(t)I(0)〉〉 the zero-frequency BS
noise. The non-equilibrium contribution of the dephasing rate is proportional to the zero-
frequency noise9,10,11,12 in the quantum Hall liquid. At zero temperature, the non-equilibrium
dephasing rate given by Eq. (4) leads to (τ−1ϕ )
(2) ∝ |ω0|2ν−1 and depends on the QPC bias with
the exponent 2ν − 1 < 0, in sharp contrast with the linear dependence obtained by Levinson 3.
0 0.2 0.4 0.6 0.8 1
β = 10
β = 5
β = 50
0 0.5 1 1.5 2
0 1 2 3 4 5
β = 10
β = 10
β = 100
β = 100
β = 50
β = 50
Figure 2: (Left) Dephasing rate, plotted in units of e4Γ20τ0/π
2h̄4v2F d
2, as a function of the filling factor for weak
BS (full line) and strong BS (dashed line) at QPC bias eV = 0.1. The star, diamond and circle points correspond
to the Laughlin fractions ν = 1/m, m odd integer. (Right) Dephasing rate as a function of QPC bias with ν = 1/3
for weak BS (full line) and strong BS (dashed line). The insert is the ratio of non-equilibrium contribution in
dephasing rate between the arbitrary screening and strong screening multiplied by (α/d)2 as a function of d/λs.
In the left of Fig. 2, we plot the dependence of the non-equilibrium contribution of the
dephasing rate on the filling factor ν for weak BS and strong BS for several temperatures
(β = 5, 10, 50) at fixed QPC bias. ν is considered as a continuous variable, while it has physical
meaning only at Laughlin fractions 7. For the strong BS case, the dephasing rate increases
when the ν increases. For weak BS and 1/β ≪ eV , the dephasing rate has a local maximum at
ν < 1/2, the position of which depends on temperature: when the temperature increases, it gets
closer to ν = 1/2. The rate at ν = 1 is smaller than that at ν = 1/3. This result demonstrates
that for two different filling factors, we can have comparable dephasing rates. For weak BS and
1/β > eV , the dephasing rate increases when the filling factor increases. In the right of Fig. 2,
the dependence of the dephasing rate on the QPC bias voltage is plotted for ν = 1/3 and several
temperatures. In the case of strong BS, the dephasing rate increases when the bias increases.
For 1/β ≪ eV , the dephasing rate saturates, whereas for 1/β > eV , the dephasing rate increases
when eV increases, but it increases from a finite value (not shown), which is proportional to the
temperature. Things are quite different at weak BS. At high temperatures, the dephasing rate
decreases when we increase eV : this behavior is symptomatic of current and noise characteristic
in a Luttinger liquid. In the low temperature case 1/β ≪ eV , for small eV , the lower the
temperature, the bigger the dephasing rate and the faster it decreases when we increase eV .
3 General formula for the decoherence rate
The charge fluctuations are directly related to the current fluctuations along the edges which are
identical to the fluctuations of the tunneling current. The tunneling current fluctuations were
computed non pertubatively using Bethe-Ansatz techniques13. We can therefore invoke current
conservation at the point contact to derive a general formula for the decoherence rate, which
describes the crossover from weak to strong BS: (τ−1ϕ )
(2) = (e3τ20 /d
2)(V Gdiff−I)ν/(1−ν) where
Gdiff = ∂V I is the differential conductance and I is the current
14. This expression allows us
to describe the crossover in the dephasing rate from the weak to the strong BS regime.
Remarkably, it is possible to go beyond the strong screening limit, and one can compute
Eq. (3) for an arbitrary Coulomb kernel f(x). The result can be displayed in terms of the ratio
between the arbitrary screening dephasing rate and the strong screening dephasing rate:
(τ−1ϕ )
(τ−1ϕ )
(eα)2
dxf(x)
. (5)
If the Coulomb interaction kernel f(x) is chosen as suggested before, the dephasing rate at
arbitrary λs has an analytical expression: F = (πd/2α)
2[E0(d/λs)+N0(d/λs)], where E0(d/λs)
and N0(d/λs) are the Weber and the Neumann functions of zero order. F is plotted in the
insert of Fig. 2: F is infinite in the absence of screening. However, the presence of metallic gates
always imposes a finite screening length. F decreases with d/λs and approaches 1 when λs is
close to the spatial cutoff α. The dephasing rate increases when the screening decreases.
4 Conclusion
We have established a general formula for the dephasing rate of a quantum dot located in the
proximity of a fluctuating fractional edge current. For strong screening, we have shown that the
dephasing rate is given by the tunneling current noise, for both weak and strong BS. For weaker
screening, the spatial dependence of the density-density correlation function has to be taken into
account, but we have shown explicitly that the long range nature of the Coulomb interaction
can be included as a trivial multiplicative factor. The fact that the dephasing rate decreases
with increasing voltage can be reconciled with the fact that the charge noise is directly related
to the BS current noise in the FQHE. There it is known, and seen experimentally, that when
the bias voltage dominates over the temperature, both the tunneling current and noise bear a
power law dependence ∼ V 2ν−1 with a negative exponent. The fact that at low temperatures,
the dephasing rate for filling factors can be lower than that of the integer quantum Hall effect
comes as a surprise and is a consequence of chiral Luttinger liquid theory.
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	Introduction
	Dephasing in the fractional quantum Hall regime
	General formula for the decoherence rate
	Conclusion
