Continuous quantum measurement of two coupled quantum
dots using a point contact: A quantum trajectory approach
Author
Goan, HS, Milburn, GJ, Wiseman, HM, Sun, HB
Published
2001
Journal Title
Physical Review B: Condensed Matter and Materials Physics
DOI
https://doi.org/10.1103/PhysRevB.63.125326
Copyright Statement
© 2001 American Physical Society. This is the author-manuscript version of this paper.
Reproduced in accordance with the copyright policy of the publisher. Please refer to the
journal's website for access to the definitive, published version.
Downloaded from
http://hdl.handle.net/10072/3770
Link to published version
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.63.125326
Griffith Research Online
https://research-repository.griffith.edu.au
Continuous quantum measurement of two coupled quantum dots using a point
contact: A quantum trajectory approach
Hsi-Sheng Goan
1∗
, G. J. Milburn
1
, H. M. Wiseman
2
, and He Bi Sun
1
1
Center for Quantum Computer Technology and Department of Physics, The University of Queensland, Brisbane, Qld 4072
Australia
2
School of Science, Griffith University, Nathan, Brisbane, Qld 4111 Australi a
We obtain the finite-temperature unconditional master equation of the density matrix for two
coupled quantum dots (CQD) when one dot is subjected to a measurement of its electron occupation
number using a point contact (PC). To determine how the CQD system state depends on the actual
current through the PC device, we use the so-called quantum trajectory method to derive the zero-
temperature conditional master equation. We first treat the electron tunneling through the PC
barrier as a classical stochastic point process (a quantum-jump model). Then we show explicitly
that our results can be extended to the quantum-diffusive limit when the average electron tunneling
rate is very large compared to the extra change of the tunneling rate due to the presence of the
electron in the dot closer to the PC. We find that in both quantum-jump and quantum-diffusive
cases, the conditional dynamics of the CQD system can be described by the stochastic Schr¨odinger
equations for its conditioned state vector if and only if the information carried away from the CQD
system by the PC reservoirs can be recovered by the perfect detection of the measurements.
85.30.Vw,03.65.Bz,03.67.Lx
I. INTRODUCTION
The origins and mechanisms of decoherence (dephasing) for quantum systems in condensed matter physics have
attracted much attention recently due to a number of studies in nanostructure mesosco pic sys tems
1–5
and vari-
ous proposals for quantum co mputers
6–9
. One of the issues is the connection between decoherence a nd quantum
measurement
10,11
for a quantum system. It was reported in a recent experiment
3
with a “which-path” interferometer
that Aharonov-Bohm interference is suppressed owing to the measurement of which path an electron takes through
the double-path interferometer. A biased quantum point contact (QPC) located close to a quantum dot, which is
built in one of the interferometer’s arms, acts as a measur e ment device. The change of transmission coefficient of the
QPC, which depends on the electron charge state of the quantum dot, can be detected. The decoherence rate due to
the measurement by the QPC in this experiment ha s been calculated in Refs.
12–16
.
A quantum-mechanical two-state system, coupled to a dissipative environment, provides a universal model for many
physical sy stems. The indication of quantum coherence can b e regarded as the oscillation or the interference between
the probability amplitudes of finding a particle between the two states. In this paper, we consider the problem of
an electron tunneling between two coupled quantum dots (CQDs) using a low-transparency point contact (PC) or
tunnel junction as a detector (environment) measuring the position of the electron (see Fig. 1). This problem has been
extensively studied in Refs.
16–24
. The case of measurements by a general QPC detector with arbitrary transparency has
also been investigated in Refs.
12–15,25,26
. In addition, a similar system measured by a single electron trans istor rather
than a PC has been studied in Refs.
27,21,19,22,24,28–30
. The influence of the detector (environment) on the measured
system can be determined by the r e duced density matrix obtained by tracing out the environmental degrees of the
freedom in the to tal, system plus environment, density matrix. The master equation (or rate equations) for this CQD
system have been derived and analyzed in Refs.
16,14
(here we refer to the rate equations as the first order differential
equations in time for both diagonal and off-diagonal reduced density matrix elements). This (unconditional) master
equation is obtained when the results o f all measurement records (electron current records in this case) are completely
ignored or averaged over, and describes only the ensemble average pro perty for the CQD s ystem. However, if a
measurement is made on the sy stem and the results are available, the state or density matrix is a conditional state
conditioned on the measurement results. Hence the deterministic, unconditional master equation cannot describe
the conditional dynamics of the CQ D system in a single realization of continuous measurements which reflects the
stochastic nature of an electron tunneling throug h the PC barrier. Consequently, the c onditional master equation
should be employed. In condensed matter physics usually many identical quantum systems are prepared at the
same time and a measurement is made upon the systems. For example, in nuclear or electron magnetic resonance
exp e riments, generally an ensemble of systems of nuclei and electr ons are probed to obtain the resonance signals. This
implies that the measurement result in this case is an average response of the ensemble systems. On the other hand,
1
for various proposed condensed-matter quantum computer architectures
6–9
, how to readout physical properties of a
single e lec tronic qubit, such as charge or spin at a single electron level, is demanding. This is a non-trivial problem
since it involve s an individual quantum particle measured by a practical detector in a realistic environment. It is
particularly important to take account of the decoherence introduced by the measurements on the qubit as well as
to understand how the quantum state of the q ubit, conditioned on a particular single realization of measurement,
evolves in time for the purpos e of quantum computing.
Korotkov
18,20
has obtained the Langevin rate equations for the CQD system. These rate equations describe the
random evolution of the density matrix that both conditions, and is conditioned by, the PC detector output. In
his approach, the individual electrons tunneling through the PC barrier were ignored and the tunneling current was
treated as a continuo us, diffusive variable. More precisely, he considered the change of the output current average
over some small time τ, hIi, with respect to the ave rage current I
i
, as a Gaussian white noise distribution. He then
updated hIi in the density-matrix elements using the new values of hIi after each time interval τ. However, treating
the tunneling current as a continuous, diffusive variable is valid only when the average electron tunneling rate is very
large compared to the extra change of the tunneling rate due to the presence of the electron in the dot closer to
the PC. The resulting derivation of the stochastic rate equations is semi-phenomenological, based on basic physical
reasoning to deduce the properties of the density matrix elements, r ather than microscopic.
To make contact with the measurement output, in this paper we present a quantum trajectory
31,35–42,28
measurement
analysis to the CQD system. We first use the quantum open system approach
31–34
to obtain the unconditional
Markovian master equation for the CQD system, taking into account the finite-temperature effect of the PC reservoirs.
Particularly, we assume the transparency of the PC detector is small, in the tunnel-junction limit. Subsequently, we
derive microscopically the zero-temperature conditional master equa tion by treating the electr on tunneling through
the PC as a classical stochastic point process (also called a quantum-jump model)
37,42,28
. Generally the e volution of
the system state undergoing quantum jumps (or other stochastic processes) is known as a quantum trajectory
31
. Real
measurements (for example the photon number detection) that correspond approximately to the ideal quantum-jump
(or point-process) measurement are made re gularly in experimental quantum optics. For almo st all-infinitesimal time
intervals, the measurement result is null (no photon detected). The system in this case changes infinitesimally, but
not unitarily. The nonunitary component reflects the changing probabilities for future events conditioned o n past
null events. At randomly determined times (conditionally Poisson distributed), ther e is a detection result. When this
occurs, the system undergoes a finite evolution, ca lled a quantum jump. In reality these point pr ocesses are not seen
exactly due to a finite frequency response of the circuit that averages each event over some time. Nevertheless, we
first take the zero- response time limit and consider the electron tunneling current consisting of a sequence of random
δ function pulses, i.e., a series of stochastic point processes. Then we show explicitly that our results can be extended
to the quantum-diffusive limit and reproduce the rate equations obtained by Korotkov
18,20
. We refer to the case
studied by Korotkov
18,20
as quantum diffusion, in contrast to the case of quantum jumps considered here. Hence
our quantum trajectory approach may be considered as a formal derivation
43
of the rate eq uations in Refs.
18,20
. We
find in both quantum-jump and quantum-diffusive cases that the conditional dynamics of the CQD system can be
described by the s tochastic Schr¨odinger equations (SSEs)
31,35,37,40,42
for the conditioned state vector, provided that
the information carried away from the CQD system by the PC reservoirs can be recovered by the perfect detection of
the measurements.
This paper is organized as follows. In Sec. II, we sketch the derivation of the finite-temperature unconditional master
equation for the QCD system. To determine how the CQ D system s tate depends on the actual current through the
PC device, we derive in Sec. III the zero-temperature conditional mas ter equation and the SSE in the quantum-
jump model. Then in Sec. IV we extend the results to the cas e of quantum diffusion and obtain the corresponding
conditional master equation and SSE. The analytical results in terms of Bloch sphere variables for the conditional
dynamics are presented in Sec. V. Specifically, we analyz e in this sectio n the localization ra te and mixing rate
27,21,22
.
Finally, a short c onclusion is given in Sec. VI. Appendix A is devoted to the demonstr ation of the equivalence between
the conditional stochastic rate equations in Refs.
18–20
and those derived microscopically in the present paper.
II. UNCONDITIONAL MASTER EQUATION FOR THE CQD AND PC MODEL
The appropriate way to approach quantum measurement problems is to treat the measured system, the detector
(environment), and the coupling between them microscopically. Following from Refs.
16,18,20
, we describe the whole
system (see Fig. 1) by the following Hamiltonian:
H = H
CQD
+ H
P C
+ H
coup
(1)
where
2
H
CQD
= ¯h
h
ω
1
c
†
1
c
1
+ ω
2
c
†
2
c
2
+ Ω(c
†
1
c
2
+ c
†
2
c
1
)
i
, (2)
H
P C
= ¯h
X
k
ω
L
k
a
†
Lk
a
Lk
+ ω
R
k
a
†
Rk
a
Rk
+
X
k,q
T
kq
a
†
Lk
a
Rq
+ T
∗
qk
a
†
Rq
a
Lk
, (3)
H
coup
=
X
k,q
c
†
1
c
1
χ
kq
a
†
Lk
a
Rq
+ χ
∗
qk
a
†
Rq
a
Lk
. (4)
H
CQD
represents the effective tunneling Hamiltonian for the measured CQD system. For simplicity, we assume strong
inner and inter dot Coulomb repulsion, so only one electron can occupy this CQD system. We label each dot with an
index 1, 2 (see Fig. 1) and let c
i
(c
†
i
) and ¯hω
i
represent the electron annihilation (creatio n) operator and energy for
a s ingle electron state in each dot respectively. The coupling between these two dots is given by ¯hΩ. The tunneling
Hamiltonian for the PC detector is represented by H
P C
where a
Lk
, a
Rk
and ¯hω
L
k
, ¯hω
R
k
are respectively the fermion
(electron) field annihilation operators and energies for the left and right reservoir states at wave number k. One
should not be confused by the electron in the CQD with the electrons in the PC reservoirs. The tunneling matrix
element between states k and q in left and right reservoir res pectively is given by T
kq
. Eq. (4), H
coup
, describ e s the
interaction between the detector a nd the measured system, depending on which dot is occupied. When the electron
in the CQD system is close by to the PC (i.e., dot 1 is occupied), there is a change in the P C tunneling barrier. This
barrier change results in a change of the effective tunneling amplitude from T
kq
→ T
kq
+ χ
kq
. As a consequence,
the current through the PC is also modified. This changed current can be detected, and thus a measurement of the
location of the e lec tron in the CQD system is effected.
The total density operator R(t) for the entire system in the interaction picture satisfies:
˙
R
I
(t) = −
i
¯h
[H
I
(t), R
I
(0)] −
1
¯h
2
Z
t
0
dt
′
[H
I
(t), [H
I
(t
′
), R
I
(t
′
)]] . (5)
The dynamics of the entire system is determined by the time-dependent Hamiltonian
44
:
H
I
(t) =
X
k,q
T
kq
+ χ
kq
c
†
1
c
1
a
†
Lk
a
Rq
e
i(ω
L
k
−ω
R
k
)t
+ H.C., (6)
where we have treated the sum of the tunneling Hamiltonian parts in H
P C
and H
coup
as the interaction Hamiltonian
H
I
, and H.C. stands for Hermitian conjugate of the entire previous term. By tracing both sides of Eq. (5) over the
bath (rese rvoir) variables and then changing fr om the interacting picture to the Schr¨odinger picture, we obtain
31–33
the finite-temperature, Markovian master equation for the CQD system:
˙ρ(t) = −
i
¯h
[H
CQD
, ρ(t)] + D[T
+
+ X
+
n
1
]ρ(t) + D[T
∗
−
+ X
∗
−
n
1
]ρ(t), (7)
where ρ(t) = Tr
B
R(t) and Tr
B
indicates a trace over reservoir variables. In a rriving at Eq. (7), we have made the
following assumptions and approximations: (a)treating the left and right fermion reservoirs in the PC as thermal
equilibrium free electron baths, (b)weak system-bath coupling, (c)small transparency of the PC, i.e., in the tunnel-
junction limit, (d)uncorrelated and factorizable system-bath initial condition (e)relaxation time scales of the reservoirs
being much shorter than that of the system state, (f)Markovian approximation, (g)|eV |, k
B
T ≪ µ
L(R)
, and (h)energy-
independent electron tunneling amplitudes and density of states over the bandwidth of max(|eV |, k
B
T ). Here k
B
is
the Boltzmann constant, T represe nts the temperature, eV = µ
L
−µ
R
is the external bias applied across the PC, and
µ
L
and µ
R
stand for the chemical potentials in the left and right reservoirs respectively. In Eq. (7), n
1
= c
†
1
c
1
is the
occupation number operator for dot 1. The pa rameters T
±
and X
±
are given by
|T
±
|
2
= D
±
= 2πe|T
00
|
2
g
L
g
R
V
±
/¯h, (8a)
|T
±
+ X
±
|
2
= D
′
±
= 2πe|T
00
+ χ
00
|
2
g
L
g
R
V
±
/¯h, (8b)
where D
±
and D
′
±
are the average electr on tunneling rates through the PC barrier in positive and negative bias
directions at finite temperature s, without and with the presence of the electron in dot 1 respectively. Here the
effective finite-temperature external bias po tential, eV
±
is given by the following expression:
eV
±
≡
±eV
1 − exp[∓eV/(k
B
T )]
. (9)
T
00
and χ
00
are energy-independent tunneling amplitudes near the average chemical potential, and g
L
and g
R
are
the energy-independent density of states for the left and right fermion baths. Note that the average electron currents
3
through the PC barrier is proportional to the difference between the average electron tunneling rate in opposite
directions. Hence, the average currents eD = e(D
+
−D
−
) and eD
′
= e(D
′
+
−D
′
−
), following from Eq. (8) and (9), are
temper ature independent
45,46
at least for a range of low temperatures k
B
T ≪ µ
L(R)
. In addition, the current- voltage
characteristic in the linear response region |eV | ≪ µ
L(R)
is of the same form as for an Ohmic resistor, though the
nature of charge transport is quite different in both cases.
We have also introduced, in Eq. (7), an eleg ant superoperator
37,28,47–49
D, widely used in measurement theory in
quantum optics. Physically the “irreversible” part caused by the influence of the environment in the unconditiona l
master equation, is represented by the D superoperator. Generally s uperoperators transform one operator into another
operator. Mathematically, the expression D[B]ρ means that superoperator D takes its operator argument B, acting
on ρ. Its precise definition is in terms of another two superoperators J and A:
D[B]ρ = J[B]ρ −A[B]ρ, (10)
where
J[B]ρ = BρB
†
, (11)
A[B]ρ = (B
†
Bρ + ρB
†
B)/2. (12)
The form of the master equation (7), defined through the superoperator D[B]ρ(t), preserves the positivity of the
density matrix operator ρ(t). Such a Markovian master equation is ca lled a Lindblad
50
form.
To demonstrate the equivalence between the master equation (7) and the rate equations derive d in Ref.
16
, we
evaluate the density matrix operator in the same bas is a s in Ref.
16
and obtain
˙ρ
aa
(t) = iΩ[ρ
ab
(t) − ρ
ba
(t)] , (13a)
˙ρ
ab
(t) = iερ
ab
(t) + iΩ[ρ
aa
(t) − ρ
bb
(t)] − (|X
T
|
2
/2)ρ
ab
(t) + i Im(T
∗
+
X
+
− T
∗
−
X
−
)ρ
ab
(t) (13b)
Here ¯hε = ¯h(ω
2
− ω
1
) is the energy mismatch between the two dots, ρ
ij
(t) = hi|ρ(t)|ji, and ρ
aa
(t) and ρ
bb
(t) are
the probabilities of finding the electron in dot 1 and dot 2 respectively. The rate equations for the other two density
matrix elements can be easily obtained from the relations: ρ
bb
(t) = 1 − ρ
aa
(t) and ρ
ba
(t) = ρ
∗
ab
(t). Compared to
an isolated CQ D system, the presence of the PC detector introduces two effects to the CQD system. First, the
imaginary part of (T
∗
+
X
+
− T
∗
−
X
−
) (the last term in Eq. (13b)) causes an effective temperature-independent shift in
the energy mismatch between the two dots. Here, (T
∗
+
X
+
− T
∗
−
X
−
) = T
∗
X is a temperature-independent quantity,
where T = T
+
(0), X = X
+
(0); i.e., T
+
and X
+
evaluated at zero temperature respective ly. Second, it generates a
decoherence (dephasing) rate
Γ
d
= |X
T
|
2
/2 (14)
for the off-diagonal density matrix elements, where |X
T
|
2
= |X
+
|
2
+ |X
−
|
2
. We note that the decoherence ra te
comes entirely fr om the effect of the measurement revealing where the electron in the CQDs is located. If the
PC detector do e s not disting uis h which of the dots the electro n occupies , i.e., X
±
= 0, then Γ
d
= 0. The rate
equations in Eq. (13) are exactly the same as the zero-temperature rate equations in Ref.
16
if we assume that the
tunneling amplitudes are re al, T
00
= T
∗
00
and χ
00
= χ
∗
00
. In that case, the last term in Eq. (13b) vanishes and
Γ
d
= X
2
/2 = (
√
D
′
−
√
D)
2
/2. Actually, the relative phase between the two complex tunneling amplitudes may
produce additional effects on conditional dynamics of the CQD system as well. This will be shown later when we
discuss conditional dynamics. Physically, the presence of the electron in dot 1 ra ises the effective tunneling barrier
of the PC due to electrostatic repulsion. As a consequence, the effective tunneling amplitude becomes lower, i.e.,
D
′
= |T + X|
2
< D = |T |
2
. This sets a condition on the relative phase θ between X and T : cos θ < −|X|/(2|T |).
The dynamics of the unconditional rate equations at zero temperatur e was analyzed in Ref.
16
. Here, following from
Eqs. (14), (8) and (9), we find that the temperature-dependent decoherence rate due to the PC thermal reservoirs
has the following expression:
Γ
d
(T )
Γ
d
(0)
=
e(V
+
+ V
−
)
eV
= coth
eV
2k
B
T
. (15)
As expected, Γ
d
(T ) increases with increasing temperature , although the average tunneling current through the PC is
temper ature independent
45,46
for the same range of low temperatures k
B
T ≪ µ
L(R)
. This temperature dependence
of the decoherence rate is in fact just the temperature dependence of the zero-frequency noise power spectrum of
the current fluctuatio n in a low-transparency PC or tunnel junction
51
. The CQD system weakly coupled to another
4
